Mathematics

mathematics

EDITORIAL BOARD Editor in Chief Barry Max Brandenberger, Jr. Curriculum and Instruction Consultant, Cedar Park, Texas Associate Editor Lucia Romberg McKay, Ph.D. Mathematics Education Consultant, Port Aransas, Texas Consultants L. Ray Carry, Ph.D. Pamela D. Garner EDITORIAL AND PRODUCTION STAFF Cindy Clendenon, Project Editor Linda Hubbard, Editorial Director Betz Des Chenes, Managing Editor Alja Collar, Kathleen Edgar, Monica Hubbard, Gloria Lam, Kristin May, Mark Mikula, Kate Millson, Nicole Watkins, Contributing Editors Michelle DiMercurio, Senior Art Director Rita Wimberley, Buyer Bill Atkins, Phil Koth, Proofreaders Lynne Maday, Indexer Barbara J. Yarrow, Manager, Imaging and Multimedia Content Robyn V. Young, Project Manager, Imaging and Multimedia Content Pam A. Reed, Imaging Coordinator Randy Bassett, Imaging Supervisor Leitha Etheridge-Sims, Image Cataloger Maria L. Franklin, Permissions Manager Lori Hines, Permissions Assistant Consulting School Douglas Middle School, Box Elder, South Dakota Teacher: Kelly Lane Macmillan Reference USA Elly Dickason, Publisher Hélène G. Potter, Editor in Chief

ii

mathematics VOLUME

4

Sc-Ze

Barry Max Brandenberger, Jr., Editor in Chief

Copyright © 2002 by Macmillan Reference USA, an imprint of the Gale Group All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the Publisher. Macmillan Reference USA 300 Park Avenue South New York, NY 10010

Macmillan Reference USA 27500 Drake Rd. Farmington Hills, MI 48331-3535

Library of Congress Cataloging-in-Publication Data Mathematics / Barry Max Brandenberger, Jr., editor in chief. p. cm. Includes bibliographical references and index. ISBN 0-02-865561-3 (set : hardcover : alk. paper) - ISBN 0-02-865562-1 (v. 1 : alk. paper) ISBN 0-02-865563-X (v. 2 : alk. paper) - ISBN 0-02-865564-8 (v. 3 : alk. paper) ISBN 0-02-865565-6 (v. 4 : alk. paper) 1. Mathematics-Juvenile literature. [1. Mathematics.] I. Brandenberger, Barry Max, 1971QA40.5 .M38 2002 510-dc21 00-045593 Printed in the United States of America 1 2 3 4 5 6 7 8 9 10

Table of Contents VOLUME 1: Preface List of Contributors

A Abacus Absolute Zero Accountant Accuracy and Precision Agnesi, Maria Gaëtana Agriculture Air Traffic Controller Algebra Algebra Tiles Algorithms for Arithmetic Alternative Fuel and Energy Analog and Digital Angles, Measurement of Angles of Elevation and Depression Apollonius of Perga Archaeologist Archimedes Architect Architecture Artists Astronaut Astronomer Astronomy, Measurements in Athletics, Technology in

B Babbage, Charles Banneker, Benjamin Bases Bernoulli Family Boole, George Bouncing Ball, Measurement of a

Brain, Human Bush, Vannevar

C Calculators Calculus Calendar, Numbers in the Carpenter Carroll, Lewis Cartographer Census Central Tendency, Measures of Chaos Cierva Codorniu, Juan de la Circles, Measurement of City Planner City Planning Comets, Predicting Communication Methods Compact Disc, DVD, and MP3 Technology Computer-Aided Design Computer Analyst Computer Animation Computer Graphic Artist Computer Information Systems Computer Programmer Computer Simulations Computers and the Binary System Computers, Evolution of Electronic Computers, Future of Computers, Personal Congruency, Equality, and Similarity Conic Sections Conservationist Consistency Consumer Data Cooking, Measurement of Coordinate System, Polar v

Table of Contents

Coordinate System, Three-Dimensional Cosmos Cryptology Cycling, Measurements of

Flight, Measurements of

PHOTO AND ILLUSTRATION CREDITS GLOSSARY TOPIC OUTLINE VOLUME ONE INDEX

Fractions

Form and Value Fractals Fraction Operations Functions and Equations

G Galileo Galilei Games

VOLUME 2:

Gaming Gardner, Martin

D

Genome, Human

Dance, Folk Data Analyst Data Collxn and Interp Dating Techniques Decimals Descartes and his Coordinate System Dimensional Relationships Dimensions Distance, Measuring Division by Zero Dürer, Albrecht

Geography

E Earthquakes, Measuring Economic Indicators Einstein, Albert Electronics Repair Technician Encryption End of the World, Predictions of Endangered Species, Measuring Escher, M. C. Estimation Euclid and his Contributions Euler, Leonhard Exponential Growth and Decay

F Factorial Factors Fermat, Pierre de Fermat’s Last Theorem Fibonacci, Leonardo Pisano Field Properties Financial Planner vi

Geometry Software, Dynamic Geometry, Spherical Geometry, Tools of Germain, Sophie Global Positioning System Golden Section Grades, Highway Graphs Graphs and Effects of Parameter Changes

H Heating and Air Conditioning Hollerith, Herman Hopper, Grace Human Body Human Genome Project Hypatia

I IMAX Technology Induction Inequalities Infinity Insurance agent Integers Interest Interior Decorator Internet Internet Data, Reliability of Inverses

Table of Contents

K Knuth, Donald Kovalevsky, Sofya

Morgan, Julia Mount Everest, Measurement of Mount Rushmore, Measurement of Music Recording Technician

L Landscape Architect Leonardo da Vinci Light Light Speed Limit Lines, Parallel and Perpendicular Lines, Skew Locus Logarithms Lotteries, State Lovelace, Ada Byron Photo and Illustration Credits Glossary Topic Outline Volume Two Index

VOLUME 3:

M Mandelbrot, Benoit B. Mapping, Mathematical Maps and Mapmaking Marketer Mass Media, Mathematics and the Mathematical Devices, Early Mathematical Devices, Mechanical Mathematics, Definition of Mathematics, Impossible Mathematics, New Trends in Mathematics Teacher Mathematics, Very Old Matrices Measurement, English System of Measurement, Metric System of Measurements, Irregular Mile, Nautical and Statute Millennium Bug Minimum Surface Area Mitchell, Maria Möbius, August Ferdinand

N Nature Navigation Negative Discoveries Nets Newton, Sir Isaac Number Line Number Sets Number System, Real Numbers: Abundant, Deficient, Perfect, and Amicable Numbers and Writing Numbers, Complex Numbers, Forbidden and Superstitious Numbers, Irrational Numbers, Massive Numbers, Rational Numbers, Real Numbers, Tyranny of Numbers, Whole Nutritionist

O Ozone Hole

P Pascal, Blaise Patterns Percent Permutations and Combinations Pharmacist Photocopier Photographer Pi Poles, Magnetic and Geographic Polls and Polling Polyhedrons Population Mathematics Population of Pets Postulates, Theorems, and Proofs Powers and Exponents Predictions Primes, Puzzles of vii

Table of Contents

Probability and the Law of Large Numbers Probability, Experimental Probability, Theoretical Problem Solving, Multiple Approaches to Proof Puzzles, Number Pythagoras

Q Quadratic Formula and Equations Quilting

R Radical Sign Radio Disc Jockey Randomness Rate of Change, Instantaneous Ratio, Rate, and Proportion Restaurant Manager Robinson, Julia Bowman Roebling, Emily Warren Roller Coaster Designer Rounding Photo and Illustration Credits Glossary Topic Outline Volume Three Index

VOLUME 4:

S Scale Drawings and Models Scientific Method, Measurements and the Scientific Notation Sequences and Series Significant Figures or Digits Slide Rule Slope Solar System Geometry, History of Solar System Geometry, Modern Understandings of Solid Waste, Measuring Somerville, Mary Fairfax Sound viii

Space, Commercialization of Space Exploration Space, Growing Old in Spaceflight, Mathematics of Sports Data Standardized Tests Statistical Analysis Step Functions Stock Market Stone Mason Sun Superconductivity Surveyor Symbols Symmetry

T Telescope Television Ratings Temperature, Measurement of Tessellations Tessellations, Making Time, Measurement of Topology Toxic Chemicals, Measuring Transformations Triangles Trigonometry Turing, Alan

U Undersea Exploration Universe, Geometry of

V Variation, Direct and Inverse Vectors Virtual Reality Vision, Measurement of Volume of Cone and Cylinder

W Weather Forecasting Models Weather, Measuring Violent Web Designer

Table of Contents

Z Zero

Topic Outline Cumulative Index

Photo and Illustration Credits Glossary

ix

Scale Drawings and Models Scale drawings are based on the geometric principle of similarity. Two figures are similar if they have the same shape even though they may have different sizes. Any figure is similar to itself, so, in this specific case, the similar figures do actually have the same size as well as the same shape.

S

Scale Drawing and Models in Geometry In traditional Euclidean geometry, two polygons are similar if their corresponding angles are equal and their corresponding sides are in proportion. For example, the two right triangles shown below are similar, because the length of each side in the large triangle is twice the length of the corresponding side in the small triangle.

10 5

3

polygon a geometric figure bounded by line segments

8

4

6

In transformational geometry, two figures are said to be similar if one of them can be “mapped” onto the other one by a transformation that expands or contracts all dimensions by the same factor. Such a transformation is usually called a dilation, a size change, or a size transformation. For example, in the preceding illustration, the small triangle can be mapped onto the large triangle by a dilation with scale factor 2. The ratio of the length of a side in the large triangle to the corresponding side in the small triangle is 2-to-1, often written as 2:1. Another example is that a square with sides of length 2 centimeters (cm) is similar to a square with sides of length 10 cm, since the 2 cm square can be mapped onto the 10 cm square by a dilation with magnitude (scale factor) 5. This simply means that each dimension of the 2 cm square is multiplied by 5 to produce the 10 cm square. We call 5 the scale factor or magnitude of the dilation and we say that the ratio of the dimensions of the 1

Scale Drawings and Models

larger square to those of the smaller square is 5:1 (or “5-to-1”). The smaller square has been “scaled up” by a factor of 5.

✶This process of scaling up or down to produce similar figures with different sizes has numerous applications in many fields, such as in copying machines where one enlarges or reduces the original.

A figure can also be “scaled down”.✶ Starting with a square with sides of length 10 cm, we can produce a square whose sides have length 2 cm by 1 scaling the larger square by a factor of 5. Note that all squares are similar because any given square can be mapped onto any other square by a dilation of magnitude b/a where b/a is the ratio of the sides of the second square to that of the first. This is not true for polygons in general. For example, any two triangles whose corresponding angles are unequal cannot be similar.

Maps One familiar type of scale drawing is a map. When someone plans a trip, it is convenient to have a map of the general region in which that trip will take place. Obviously, the map will not be the actual size, but, if it is to be helpful, it should look similar to the region, but on a much smaller scale. Any map that is designed to help people plan out a journey will have a “legend” printed on it showing, for example, how many miles or kilometers are represented by each inch or centimeter. It would be terribly confusing to the traveler to have the scale vary on different parts of the same map, so the scale is uniform for the entire map, which returns to the mathematical concept of similarity. The ratio of each distance in the real world to its corresponding distance on the map will be the same for all corresponding distances between the two. It is no coincidence that mathematicians have adopted the word “mapping” to refer to the correspondence between the points in one set and the points of another set. The dilation referred to in the earlier paragraph is often called a “similarity mapping.”

Models Almost everyone is familiar with toys that are scale models of cars, airplanes, boats, and space shuttles, as well as many other familiar objects. Here again, similarity is applied in the manufacturing of these scale models. A popular series of scale model cars has the notation “1:18” on the box. This is a ratio that tells how the model compares in size to the real car. In this case, all 1 the dimensions of the model car are 18 the size of the corresponding di1 1 1 mensions of the real car. So the model is 18 as long, 18 as wide, and 18 as high as the real car. If the real car is 180 inches long, then the model car 1 will be 18 as long or 10 inches in length. Because all the dimensions of the model car are scaled from the real car by the same factor, the model looks realistic. This same similarity principle is used in reverse by the people who design automobiles. They usually start with a sketch, drawn more or less to scale, of the car they want to design; translate this sketch into a true scale drawing, often with the help of computer-assisted design software; and build a scale model to the specifications in the drawing. Then, if everything is approved by the executives of the automobile company, the scale drawings and models are given to automotive engineers who design and build a full-size prototype of the vehicle. If the prototype also receives approval from senior executives, then the vehicle is put into full-scale production. 2

Scale Drawings and Models

Other Uses Scale drawings and models are also very important for architects, builders, carpenters, plumbers, and electricians. Architects work in a similar manner as automotive designers. They start with sketches of ideas showing what they want their buildings to look like, then translate the sketches into accurate scale drawings, and finally build scale models of the buildings for clients to evaluate before any construction on the actual structure is begun. If the client approves the design, then detailed scale drawings, called blueprints, are prepared for the whole building so that the builders will know where to put walls, doors, restrooms, stairwells, elevators, electrical wiring, and all other features of the building. On each blueprint is a legend similar to that found on a map that tells how many feet are represented by each inch on the blueprint. In addition, a blueprint tells the scale of the drawing to the actual building. All of the people involved in the construction of the building have access to these blueprints so that every feature of the building ends up being the right size and in the right place. Artists such as painters and sculptors often use scale drawings or scale models as preliminary guides for their work. An artist commissioned to paint a large mural often makes preliminary small sketches of the drawings she plans to put on the wall. These are essentially scale drawings of the final work. Similarly, a sculptor usually makes a small scale model, perhaps of clay or plaster, before beginning the actual sculpture, which might be in marble, granite, or brass. Mount Rushmore, one of the great national monuments in the United States, is a gigantic sculpture featuring the heads of Presidents George Washington, Thomas Jefferson, Abraham Lincoln, and Theodore Roosevelt carved into the natural granite of the Black Hills of

Architect Frank Lloyd Wright (1867–1959) views a model of the Guggenheim Museum in 1945. The actual facility was completed in New York City in 1959. Such models help designers view a project in three dimensions, whereas blueprints show only two.

3

Scientific Method, Measurements and the

South Dakota. The sculptor of this imposing work, Gutzon Borglum, first created plaster scale models of the heads in his studio before ever going to the mountain. His models had a simple 1:12 scaling ratio; one inch on the model would represent one foot on the mountain. He made very careful measurements on the models, multiplied these measurements by 12, and used these latter values to guide his work in sculpting the mountain. When Borglum’s work was finished, each of the faces of the four presidents was more than 60 feet high from the chin to the top of the head. From arts and crafts to large-scale manufacturing processes, the use of similarity in scale drawings and models is fundamental to creating products that are both useful and pleasing to the eye. S E E A L S O Computer-Aided Design; Mount Rushmore, Measurement of; Photocopier; Ratio, Rate and Proportion; Transformations. Stephen Robinson Bibliography Chakerian, G. D., Calvin D. Crabill, and Sherman K. Stein. Geometry: A Guided Inquiry. Pleasantville, NY: Sunburst Communications, Inc. 1987. Yaglom, I. M. Geometric Transformations. Washington, D.C.: Mathematical Association of America, 1975. Internet Resources

“Creating Giants in the Black Hills.” Travelsd.com. . Scale Drawings. New York State Regents Exam Prep Center. .

Scientific Method, Measurements and the hypothesis a proposition that is assumed to be true for the purpose of proving other propositions

A common misconception about the scientific method is that it involves the use of exact measurements to prove hypotheses. What is the real nature of science and measurement, and how are these two processes related?

The Nature of Scientific Inquiry According to one definition, the scientific method is a mode of research in which a hypothesis is tested using a controlled, carefully documented, and replicable (reproducible) experiment. In fact, scientific inquiry is conducted in many different ways. Even so, the preceding definition is a reasonable summary or approximation of much scientific work. Science, at heart, is an effort to understand and make sense of the world outside and inside ourselves. Carl Sagan, in The Demon-Haunted World: Science as a Candle in the Dark, noted that the scientific method has been a powerful tool in this effort. He concluded that the knowledge generated by this method is the most important factor in extending and bettering our lives. The steps of the scientific method follow. 1. Recognize a problem. 2. Formulate a hypothesis using existing knowledge. 3. Test the hypothesis by gathering data.

4

Scientific Method, Measurements and the

4. Revise the hypothesis (if necessary). 5. Test the new hypothesis (if necessary). 6. Draw a conclusion. Scientists often use controlled experiments. Theoretically, controlled experiments eliminate alternative explanations by varying only one factor at a time. Does this guarantee that all alternative explanations are eliminated? To answer this question, consider the role measurement plays in the scientific method.

The Nature of Measurement Measurement addresses a fundamental question that arises in many scientific studies (and everyday activities): “How big or small is it?” Answering this question can involve either direct or indirect measurements. For example, determining the size of a television screen can be done directly by using a tape measure. Many other measurements, including most of those done by scientists, must be done indirectly. For example, if a teacher wants to know how much algebra her students learned during the semester, she cannot examine their brains directly for this knowledge. So she gives them a test, which indicates what they have stored in their brains. Likewise, astronomers cannot directly measure the distance to a faraway star, its temperature or chemical composition, or whether it has orbiting planets. They must devise ways of measuring such attributes indirectly. Even when measurement is done directly, it is essentially an estimation rather than an exact process. One reason for this is the difference between discrete quantities and continuous quantities. To determine how much there is of a discrete quantity, one simply has to count the number of items in a collection. To determine how much there is of a continuous quantity, however, one must measure it. Although measurement is a more complicated process than counting, it involves the following fundamentally simple idea: Divide the continuous quantity into equal-size parts (units) and count the number of units. In effect, the measurement process transforms a continuous quantity into one that is countable (a discrete quantity). For example, measuring the diagonal of a television involves, in effect, subdividing it into segments 1-inch in length and counting the number of these units.

discrete quantities amounts composed of separate and distinct parts continuous quantities amounts composed of continuous and undistinguishable parts

Unlike counting a discrete quantity, measurement can involve a part of a unit. A unit of measurement can be subdivided into smaller units (such as tenths of an inch), which can be further subdivided (such as hundredths of an inch), and so on, forever. In other words, if one measures the diagonal of a 34-inch television, it is extremely unlikely to be exactly 34 inches in length. It could be, for instance, 33.9, 34.1, 33.999, or 34.0001 inches in length. Absolutely precise measurement is impossible because measurement devices cannot exactly replicate a standard unit. For example, the inch demarcations on a yardstick are not exactly 1 inch each, even when the stick was first manufactured. Nonexact units further result from repeated expansion and contraction of the stick because of repeated heating and cooling. 5

Scientific Method, Measurements and the

Accurately measuring small amounts of chemical solutions requires the proper measuring devices and attention to detail. Replicate samples, as shown here, are a mainstay of scientific measurement.

Even if measuring devices were perfectly accurate, it would not be humanly possible to read them with perfect precision. In brief, even direct measurement involves some degree or margin of error.

The Relation Between Measurement and the Scientific Method Testing a hypothesis requires collecting data. There are basically two types of data: categorical (name) data, such as gender, and numerical (number) data. The latter can involve a discrete quantity (such as the number of men who had a particular voting preference) or a continuous one (such as the amount of math anxiety). Continuous data must be measured on a scale. Because measurements are always imprecise to some degree, scientists’ conclusions may be incorrect. So scientists work hard to develop increasingly accurate measurement devices. The possibility of measurement error is also a reason why scientific experiments need to be replicated. Indeed, scientific conclusions based on any type of data may be imperfect for a number of reasons. One is that scientists’ instruments or tests do not always measure what the scientists intend. Another problem is that scientists frequently cannot collect data on every possible case, and so only a sample of cases are measured, and a sample is not always representative of all the cases. For example, a biologist may find a result that is true for her sample of men, but the conclusions may not apply to women or even all men. Measurement is an integral aspect of the scientific method. However, using the scientific method does not guarantee exact results and definitive proof of a hypothesis. Rather, it is a sound approach to constructing an increasingly accurate and thorough understanding of our world. S E E A L S O Accuracy and Precision; Data Collection and Interpretation; Problem Solving, Multiple Approaches to. Arthur J. Baroody 6

Scientific Notation

Bibliography Bendick, Jeanne. Science Experiences Measuring. Danbury, CT: Franklin Watts, 1971. Nelson, D., and Robert Reys. Measurement in School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 1976. Sagan, Carl. The Demon-Haunted World: Science as a Candle in the Dark. New York: Ballantine Books, 1997. Steen, Lynn Arthur, ed. On the Shoulder of Giants: New Approaches to Numeracy. Washington, D.C.: National Academy Press, 1990.

Scientific Notation Scientific notation is a method of writing very large and very small numbers. Ordinary numbers are useful for everyday measurement, such as daily temperatures and automobile speeds, but for large measurements like astronomical distances, scientific notation provides a way to express these numbers in a short and concise way. The basis of scientific notation is the power of ten. Since many large and small numbers have a few integers with many zeros, the power of ten can be used to shorten the length of the written number. A number written in scientific notation has two parts. The first part is a number between 1 and 10, and the second part is a power of ten. Mathematically, writing a number in scientific notation is the expression of the number in the form n  10 x where n is a number greater than 1 but less than 10 and x is an exponent of 10. An example is 15,653 written as 1.5653  104. This type of notation is also used for very small numbers such as 0.0000000072, which, in scientific notation, is rewritten as 7.2  109. Some examples of measurements where scientific notation becomes useful follow. • The wavelength for violet light is 40-millionths of a centimeter  4  105cm. • Some black holes are measured by the amount of solar masses they could contain. One black hole was measured as 10,000,000 or 1.0  107 solar masses. • The weight of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is 0.000,000,000,000,000,000,000,000,006,645 kilograms (6.645  1027 kilograms). • A computer hard disk could hold 4 gigabytes (about 4,000,000,000 bytes) of information. That is 4.0 x 109 bytes. • Computer calculation speeds are often measured in nanoseconds. A nanosecond is 0.000000001 seconds, or 1.0  109 seconds.

Converting Numbers into Scientific Notation Complete the following steps to convert large and small numbers into scientific notation. First, identify the significant digits and move the decimal place to the right or left so that only one integer is on the left side of the decimal. Rewrite the number with the new decimal place and include only the identified significant digits. Then, following the number, write a mul-

integer a positive whole number, its negative counterpart, or zero

solar masses dimensionless units in which mass, radius, luminosity, and other physical properties of stars can be expressed in terms of the Sun’s characteristics

POWERS, EXPONENTS, AND SCIENTIFIC NOTATION The process of calculating scientific notation comes from the rules of exponents. • Any number raised to the power of zero is one. • Any number raised to the power of one is equal to itself. • Any number raised to the nth power is equal to the product of the number used as a factor n times.

7

Sequences and Series

tiplication sign and the number 10. Raise the 10 to the exponent that represents the number of places you moved the decimal point. If the number is large and you moved the decimal point to the left, the exponent is positive. Conversely, if the number is small and you moved the decimal point to the right, the exponent is negative. If a chemist is trying to discuss the number of electrons expected in a sample of atoms, the number may be 1,100,000,000,000,000,000,000, 000,000. Using three significant figures the number is written 1.10  1027. Along the same lines the weight of a particular chemical may be 0.0000000000000000000721 grams. In scientific notation the example would be written 7.21  1020. S E E A L S O Measurement, Metric System of; Numbers, Massive; Powers and Exponents. Brook E. Hall Bibliography Devlin, Keith J. Mathematics, the Science of Patterns: The Search for Order in Life, Mind, and the Universe. New York: Scientific American Library, 1997. Morrison, Philip, and Phylis Morrison. Powers of Ten: A Book About the Relative Size of Things in the Universe and the Effect of Adding Another Zero. New York: Scientific American Library, 1994. Pickover, Clifford A. Wonders of Numbers: Adventures in Math, Mind, and Meaning. Oxford, U.K.: Oxford University Press, 2001.

Sequences and Series A sequence is an ordered listing of numbers such as {1, 3, 5, 7, 9}. In mathematical terms, a sequence is a function whose domain is the natural number set {1, 2, 3, 4, 5, . . .}, or some subset of the natural numbers, and whose range is the set of real numbers or some subset thereof. Let f denote the sequence {1, 3, 5, 7, 9}. Then f (1)  1, f (2)  3, f (3)  5, f (4)  7, and f (5)  9. Here the domain of f is {1, 2, 3, 4, 5} and the range is {1, 3, 5, 7, 9}. The terms of a sequence are often designated by subscripted variables. So for the sequence {1, 3, 5, 7, 9}, one could write a1  1, a2  3, a3  5, a4  7, and a5  9. A shorthand designation for a sequence written in this way is {an}. It is sometimes possible to get an explicit formula for the general term of a sequence. For example, the general nth term of the sequence {1, 3, 5, 7, 9} may be written an  2n  1, where n takes on values from the set {1, 2, 3, 4, 5}.

✶A famous infinite sequence is the so-called Fibonacci sequence {1, 1, 2, 3, 5, 8, 13, 21, . . .} in which the first two terms are each 1 and every term after the second is the sum of the two terms immediately preceding it.

8

The sequence {1, 3, 5, 7, 9} is finite, since it contains only five terms, but in mathematics, much work is devoted to the study of infinite sequences.✶ For instance, the sequence of “all” odd natural numbers is written {1, 3, 5, 7, 9, . . .}, where the three dots indicate that there is an infinite number of terms following 9. Note that the general term of this sequence is also an  2n  1, but now n can take on any natural number value, however large. Other examples of infinite sequences include the even natural numbers, all multiples of 3, the digits of pi (), and the set of all prime numbers.

Shapes, Efficient

The amount of money in a bank account paying compound interest at regular intervals is a sequence. If $100 is deposited at an annual interest rate of 5 percent compounded annually, then the amounts of money in the account at the end of each year form a sequence whose general term can be computed by the formula an  100(1.05)n, where n is the number of years since the money was deposited.

Series A series is just the sum of the terms of a sequence. Thus {1, 3, 5, 7, 9} is a sequence, but 1  3  5  7  9 is a series. So long as the series is finite the sum may be found by adding all the terms, so 1  3  5  7  9  25. If the series is infinite, then it is not possible to add all the terms by the ordinary addition algorithm, since one could never complete the task. Nevertheless, there are infinite series that have finite sums. An exam1 1 1 1 ple of one such series is: 1  2  4  8  16  . . . , where, again, the three dots indicate that this series continues according to this pattern without ever ending. Now, obviously someone cannot sit down and complete the term by term addition of this series, even in a lifetime, but mathematicians reason in the following way. The sum of the first term is 1; the sum of the first two terms is 1.5; the sum of the first three terms is 1.75; the sum of the first four terms is 1.875; the sum of the first five terms is 1.9375; and so on. These are called the first five partial sums. Mathematically, it is possible to argue that no matter how many partial sums one takes, the nth partial sum will always be slightly less than 2 (in the above example), for any value of n. On the other hand, each new partial sum will be closer to 2 than the one before it. Therefore, one would say that the limiting value of the sequence of partial sums is 2 and the sum is defined 1 1 1 1 as the infinite series 1  2  4  8  16 . . . . This infinite series is said to converge to 2. A case where the partial sums grow without bound is described as a series that diverges or that has no finite sum. S E E A L S O Fibonacci, Leonardo Pisano. Stephen Robinson

algorithm a rule or procedure used to solve a mathematical problem

partial sum with respect to infinite series, the sum of its first n terms

Bibliography Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.

Shapes, Efficient 1 Eighteen ounces of Salubrious Cereal is packaged in a box 28 (inches)  11 716  11. The box’s volume is about 180 cubic inches; its total surface area, A, is about 249 square inches. Could the manufacturer keep the same volume but reduce A by changing the dimensions of the package? If so, the company could save money.

Consider 3  6  10 and 4  5  9 boxes. Their volume is 180 cubic inches, but A  216 and 202 square inches, respectively. Clearly, for a fixed volume, surface area can vary, suggesting that a certain shape might produce a minimum surface area. The problem can be rephrased this way: Let x, y, and z be the sides of a box such that xyz  V for a fixed V. Find the minimum value of the function A(x, y, z)  2xy  2xz  2yz. This looks complicated. An appropriate 9

Shapes, Efficient

strategy would be to first solve a simpler two-dimensional problem in order to develop methods for attacking the three-dimensional problem. Consider the following: Let the area of a rectangular region be fixed at 100 square inches. Find the minimum perimeter of the region. Letting x and y be the sides of the rectangle means that xy  100, and you are asked to minimize the perimeter, P(x, y)  2x  2y. Begin by solving the first equation for y so that the second can be expressed as a function of one vari2(100) able, namely P(x)  2x  x. The problem can now be solved in several ways. First, graph the function, obtaining the graph below. P

x 10

Using the techniques available on the graphing calculator, you could find that the minimum function value occurs at x  10, making y  10, and P  40. Consider yet another simplification to our “minimum area” problem— let the base of the box be a square, thereby eliminating one variable. If the base is x by x and the height is y, then a fixed V  x 2y and a variable 4V V A(x, y)  2x 2  4xy. Since y  x2 , A(x)  2x2  x. Using calculus, 4V 3 A (x)  4x  x2  0, giving x  y  V . The optimal solution is a cube. Calculus makes it clear that the minimum surface area occurs when the cereal box is a cube, which means the area equals 6V 2/3. In the original prob3 lem, a cube with edges equal to 180   5.65 gives a surface area of approximately 192 square inches, a savings of 23% from the original box. It should be noted that nonrectangular shapes may reduce the surface area further. For example, a cylinder with a radius of 2.7 inches and a volume of 180 cubic inches would have a total surface area of 179 square inches. A sphere, while not a feasible shape for a cereal box, would have a surface area of 154 square inches if the volume is 180 cubic inches. The larger question then becomes: Of all three dimensional shapes with a fixed volume, which has the least surface area?

History of the Efficient Shape Problem Efficient shapes problems emerged when someone asked the question: Of all plane figures with the same perimeter, which has the greatest area? Aristotle (384 B.C.E.–322 B.C.E.) may have known this problem, having noted that a geometer knows why round wounds heal the slowest. Zenodorus (c. 180 B.C.E.) wrote a treatise on isometric figures and included the following propositions, not all of which were proved completely.

10

Shapes, Efficient

1. Of all regular polygons with equal perimeter, the one with the most sides has the greatest area. 2. A circle has a greater area than any regular polygon of the same perimeter. 3. A sphere has a greater volume than solid figures with the same surface area. Since Zenodorus draws on the work of Archimedes (287 B.C.E.–212 it is possible that Archimedes contributed to the solution of the problem. Zenodorus used both direct and indirect proofs in combination with inequalities involving sides and angles in order to establish inequalities among areas of triangles. Pappus (c. 320 C.E.) extended Zenodorus’s work, adding the proposition that of all circular segments with the same circumference, the semicircle holds the greatest area. In a memorable passage, Pappus wrote that through instinct, living creatures may seek optimal solutions. In particular he was thinking of honey bees, noting that they chose a hexagon for the shape of honeycombs—a wise choice since hexagons will hold more honey for the same amount of material than squares or equilateral triangles. B.C.E.),

After Pappus, problems such as these did not attract successful attention until the advent of calculus in the late 1600s. While calculus created tools for the solution of many optimization problems, calculus was not able to prove the theorem that states that of all plane figures with a fixed perimeter, the circle has the greatest area. Using methods drawn from synthetic geometry, Jacob Steiner (1796–1863) advanced the proof considerably. Schwarz, in 1884, finally proved the theorem. Research on efficient shapes is still flourishing. One important aspect of these problems is that they come in pairs. Paired with the problem of minimizing the perimeter of a rectangle of fixed area is the dual problem of maximizing the area of a rectangle with fixed perimeter. For example, the makers of a cereal box may want to have a large area for advertising. Similarly, paired with the problem of minimizing the surface area of a shape of fixed volume is the problem of maximizing the volume of a shape of fixed surface area. Standard calculus problems involve finding the dimensions of a cylinder that minimizes surface area for a fixed volume or its pair. The cylinder whose height equals its diameter has the least surface area for a fixed volume. S E E A L S O Minimum Surface Area; Dimensional Relationships. Don Barry Bibliography Beckenbach, Edwin and Richard Bellman. An Introduction to Inequalities. New York: L. W. Singer, 1961. Chang, Gengzhe and Thomas W. Sederberg. Over and Over Again. Washington, D.C.: Mathematical Association of America, 1997. Heath, Sir Thomas. A History of Greek Mathematics. New York: Dover Publications, 1981. Kazarinoff, Nicholas D. Geometric Inequalities. New York: L. W. Singer, 1961. Knorr, Wilbur Richard. The Ancient Tradition of Geometric Problems. New York: Dover Publications, 1993.

11

Significant Figures or Digits

Significant Figures or Digits Imagine that a nurse takes a child’s temperature. However, the mercury thermometer being used is marked off in intervals of one degree. There is a 98° F mark and a 99° F mark, but nothing in between. The nurse announces that the child’s temperature is 99° F. How does someone interpret this information? The answer becomes clear when one has an understanding of significant digits.

The Importance of Precision One possibility, of course, is that the mercury did lie exactly on the 99° F mark. What are the other possibilities? If the mercury were slightly above or below the 99° F mark (so that his actual temperature was, say, 99.1° F or 98.8° F), the nurse would probably still have recorded it as 99° F. However, if his actual temperature was 98.1° F, the nurse should have recorded it as 98° F. In fact, the temperature would have been recorded as 99° F only if the mercury lay within half a degree of 99° F—in other words, if the actual temperature was between 98.5° F and 99.5° F. (This interval includes its left endpoint, 98.5° F, but not its right endpoint, because 99.5° F would be rounded up to 100° F. However, any number just below 99.5° F, such as 99.49999° F, would be rounded down to 99° F.) One can conclude from this analysis that, in the measurement of 99° F, only the first 9 is guaranteed to be accurate—the temperature is definitely 90-something degrees. The second 9 represents a rounded value. Now suppose that, once again, the child’s temperature is being taken, but this time, a more standard thermometer is being used, one that is marked off in intervals of one tenth of a degree. Once again, the nurse announces that his temperature is 99° F. This announcement provides more information than did the previous measurement. This time, if his temperature were actually 98.6° F, it would have been recorded as such, not as 99° F. Following the reasoning of the previous paragraph, one can conclude that his actual temperature lies at most halfway toward the next marker on the thermometer—that is, within half of a tenth of a degree of 99° F. So the possible range of values this time is 98.95° F to 99.05° F—a much narrower range than in the previous case. When marking down a patient’s temperature in a medical chart, it may not be necessary to make it clear whether a measurement is precise to one half of a degree or one twentieth of a degree. But in the world of scientific experiments, such precision can be vital. It is clear that simply recording a measurement of 99° F does not, by itself, give any information as to the level of precision of this measurement. Thus, a method of recording data that will incorporate this information is needed. This method is the use of significant digits (also called significant figures). In the second of the two scenarios described earlier, the nurse is able to round off the measurement to the nearest tenth of a degree, not merely to the nearest degree. This fact can be communicated by including the tenths digit in the measurement—that is, by recording the measurement as 99.0° F rather than 99° F. The notation 99.0° F indicates that the first two digits are accurate and the third is a rounded value, whereas the notation 99° F indicates that the first digit is accurate and the second is a rounded value. 12

Significant Figures or Digits

The number 99.0° F is said to have three significant digits and the number 99° F to have two.

What Is Significant? When one looks at a measurement, how can she tell how many significant digits it has? Any number between 1 and 9 is a significant digit. When it comes to zeros, the situation is a little more complicated. To see why, suppose that a certain length is recorded to be 90.30 meters long. This measurement has four significant digits—the zero on the end (after the three) indicates that the nine, first zero, and three are accurate and the last zero is an estimate. Now suppose that she needs to convert this measurement to kilometers. The measurement now reads 0.09030 kilometers. She now has five digits in the measurement, but she has not suddenly gained precision simply by changing units. Any zero that appears to the left of the first nonzero digit (the 9, in this case) cannot be considered significant. Now suppose that she wants to convert that same measurement to millimeters, so that it reads 90,300 millimeters. Once again, she has not gained any precision, so the rightmost zero is not significant. However, the rule to be deduced from this situation is less clear. How should she consider zeros that lie to the right of the last nonzero digit? The zero after the three in 90.30 is significant because it gives information about precision—it is not a “necessary” zero, in the sense that mathematically 90.30 could be written 90.3. However, the two zeros after the three in 90300 are mathematically necessary as place-holders; 90300 is certainly not the same number as 9030 or 903. So you cannot conclude that they are there to convey information about precision. They might be—in the example, the first one is and the second one is not—but it is not guaranteed. The following rules summarize the previous discussion. 1. Any nonzero digit is significant. 2. Any zero that lies between non-zero digits is significant. 3. Any zero that lies to the left of the first non-zero digit is not significant. 4. If the number contains a decimal point, then any zero to the right of the last non-zero digit is significant. 5. If the number does not contain a decimal point, then any zero to the right of the last non-zero digit is not significant. Here are some examples: 0.030700 — five significant digits (all but the first two zeros); 400.00 — five significant digits; 400 — one significant digit (the four); and 5030 — three significant digits. Consider the last example. One has seen earlier that the zero in the ones column cannot be considered to be significant, because it may be that the measurement was rounded to the nearest multiple of ten. If, however, someone is measuring with an instrument that is marked in intervals of one unit, then recording the measurement as 5030 does not convey the full level of 13

Sine

precision of the measurement. One cannot record the measurement as 5030.0, because that indicates a jump from three to five significant digits, which is too many. In such a circumstance, avoid the difficulty by recording the measurement using scientific notation, i.e. writing 5030 as 5.030  103. Now the final zero lies to the right of the decimal point, so by rule four, it is a significant digit.

Calculations Involving Significant Figures Often, when doing a scientific experiment, it is necessary not only to record data but to do computations with the information. The main principle involved is that one can never gain significant digits when computing with data; a result can only be as precise as the information used to get it. In the first example, two pieces of data, 10.30 and 705, are to be added. Mathematically, the sum is 715.30. However, the number 705 does not have any significant digits to the right of the decimal point. Thus the three and the zero in the sum cannot be assumed to have any degree of accuracy, and so the sum is simply written 715. If the first piece of data were 10.65 instead of 10.30, then the actual sum would be 715.65. In this case, the sum would be rounded up and recorded as 716. The rule for numbers to be added (or subtracted) with significant digits is that the sum/difference cannot contain a digit that lies in or to the right of any column containing an insignificant digit. In a second example, two pieces of data, 10.30 and 705, are to be multiplied. Mathematically, the product is 7261.5. Which of these digits can be considered significant? Here the rule is that when multiplying or dividing, the product/quotient can have only as many significant digits as the piece of data containing the fewest significant digits. So in this case, the product may only have three significant digits and would be recorded as 7260. In the final example, the diameter of a circle is measured to be 10.30 centimeters. To compute the circumference of this circle, this quantity must be multiplied by . Since  is a known quantity, not a measurement, it is considered to have infinitely many significant digits. Thus the result can have as many significant digits as 10.30—that is, four. (To do this computation, take an approximation of  to as many decimal places as necessary to guarantee that the first four digits of the product will be accurate.) A similar principle applies if someone wishes to do any strictly mathematical computation with data: for instance, if one wants to compute the radius of the circle whose diameter she has measured, multiply by 0.5. The answer, however, may still have four significant digits, because 0.5 is not a measurement and hence is considered to have infinitely many significant digits. S E E A L S O Accuracy and Precision; Estimation; Rounding. Naomi Klarreich Bibliography Gardner, Robert. Science Projects About Methods of Measuring. Berkeley Heights, NJ: Enslow Publishers, Inc., 2000.

Sine 14

See Trigonometry.

Slide Rule

Slide Rule Pocket calculators only came into common use in the 1970s. Digital computers first appeared in the 1940s, but were not in widespread use by the general public until the 1980s. Before pocket calculators, there were mechanical desktop calculators, but these could only add and subtract and at best do the most basic of multiplications. A tool commonly used by engineers and scientists who dealt with math in their work was the slide rule. A slide rule is actually a simple form of what is called an analog computer, a device that does computation by measuring and operating on a continuously varying quantity, as opposed to the discreet units used by digital computers. While many analog computers work by measuring and adding voltages, the slide rule is based on adding distances.

Rule A

1

3 3

2 Rule B

4

5

6

7 7

1

2

3

4

5

6

7

4

A simple slide rule looks like two rulers that are attached to each other so that they can slide along one edge. In fact any two number lines that can be moved past each other make up a slide rule. To add two numbers on a slide rule, look at the above illustration, which shows how to add 3 and 4. Start by finding 3 on rule A. Then slide rule B so that the zero point of rule B lines up with 3 on rule A. Find 4 on rule B and it will be lined up with the sum on rule A, which is 7. Reverse these steps to subtract two numbers. The slide rule was invented by William Oughtred (1574–1660), an English mathematician. Oughtred designed the common slide rule, which has a movable center rule between fixed upper and lower rules. Many different scales are printed on it, and a sliding cursor helps the user better view the alignment between scales. He also designed less common circular slide rules that worked on the same principles. The previous example has already illustrated how to add and subtract using a slide rule. One can also multiply and divide with a slide rule using logarithms. In fact, slide rules were invented specifically for multiplying and dividing large numbers using logarithms shortly after John Napier invented logarithms. To multiply 3,750 and 225 using a slide rule, first find log10 3,750 and log10 225. Use the slide rule to add these logarithms. Then find the antilog of the sum, which is the product of 3,750 and 225.

logarithm the power to which a certain number called the base is to be raised to produce a particular number

Division using a slide rule is the same as multiplication, except one subtracts the logarithms. Of course, one must have logarithm tables handy when multiplying or dividing with a slide rule, to look up the logarithms. One can compute powers on a slide rule by taking the log of a log. Since Ax is the same as x times log A, one can do this multiplication by adding log x and log (log A). Special log-log scales on a slide rule make it possible to calculate powers. Roots, such as 2  can be considered fractional powers, so the slide rule with a log-log scale could be used to compute roots as

15

Slope

sine if a unit circle is drawn with its center at the origin and a line segment is drawn from the origin at angle theta so that the line segment intersects the circle at (x, y), then y is the sine of theta cosine if a unit circle is drawn with its center at the origin and a line segment is drawn from the origin at angle theta so that the line segment intersects the circle at (x, y), then x is the cosine of theta tangent if a unit circle is drawn with its center at the origin at angle theta so that the line segment intersects the circle at (x, y ), then the y ratio  is the tangent of x theta

well. Additional scales were provided to look up trigonometric functions such as sine, cosine, and tangent. The accuracy of a slide rule is limited by its size, the quality to which its scales are marked, and the ability of the user to read the scales. Typical engineering slide rules were accurate to within 0.1 percent. This degree of accuracy was considered sufficient for many engineering and scientific applications. However, the use of slide rules declined rapidly after the electronic calculator became inexpensive and widely used. This was due, in large part, to the fact that it is much easier and convenient to perform applications, such as multiplication, on a calculator than it is to look up logarithms. S E E A L S O Analog and Digital; Bases; Logarithms; Mathematical Devices, Early. Harry J. Kuhman Bibliography Hopp, Peter M. Slide Rules: Their History, Models, and Makers. Mendham, NJ: Astragal Press, 1999. Sommers, Hobart H., Harry Drell, and T. W. Wallschleger. The Slide Rule and How to Use It. New York: Follett, 1964.

Slope Slope is a quantity that measures the steepness of a line. To figure out how to define slope, think about what it means for one line to be steeper than another. Intuitively, one would say the steeper line “climbs faster.” To make this mathematically precise, consider the following. If two points on the steeper line that are horizontally one unit apart are compared with two points on the less-steep line that are horizontally one unit apart, the pair on the steeper line will be farther apart vertically. Thus the slope of a line is defined to be the ratio of the vertical distance to the horizontal distance between any two points on that line. Sometimes this ratio is called “rise to run,” meaning the vertical change relative to the horizontal change. Several interesting things can be noticed about the slope of a line. First of all, the definition does not tell us which two points on the line to choose. Fortunately, this does not matter because any pair of points on the same line will yield the same slope. In fact, this is really the defining characteristic of a line: When a line is “straight,” this means that its steepness never varies, unlike a parabola or a circle, which “bend” and therefore climb more steeply in some places than in others. The slope of a line indicates whether the line slants upwards from left to right, slants downwards from left to right, or is flat. If a line slants upwards, movement from one point to a second point is such that one unit to the right of the first will yield an increase (that is, a positive difference) in the y-coordinates of the points. If the line slants downward, the points’ ycoordinates will decrease (a negative difference) as one unit is moved to the right. If the line is horizontal, the y-coordinate will not change. Therefore, an upward-slanting line has positive slope, a downward-slanting line has negative slope, and a horizontal line has a slope of zero.

16

Solar System Geometry, History of

y

Undefined slope

Larger positive slope

Negative slope Still larger positive slope

Small positive slope

Slope = 0 x

A slope also involves division, which raises the possibility of dividing by zero. This would occur whenever the slope of a vertical line is computed (and only then). Therefore, vertical lines are considered to have an undefined slope. Alternatively, some people consider the slope of a vertical line to be infinite, which makes sense intuitively because a vertical line climbs as steeply as possible. S E E A L S O Graphs, Effects of Parameter Changes; Infinity; Lines, Parallel and Perpendicular; Lines, Skew. Naomi Klarreich Bibliography Charles, Randall I., et al. Focus on Algebra. Menlo Park, CA: Addison-Wesley, 1996.

Soap Bubbles

See Minimum Surface Area.

Solar System Geometry, History of Humanity’s understanding of the geometry of the solar system has developed over thousands of years. This journey towards a more accurate model of our planetary system has been marked throughout by controversy and misconception. The science of direct observation would ultimately play the most important role in bringing order to our understanding of the universe.

Ancient Conceptions and Misconceptions Archaeologists have found artifacts demonstrating that ancient Babylonians and Egyptians made numerous observations of celestial events such as full moons, new moons, and eclipses, as well as the paths of the Sun and Moon relative to the stars. The ancient Greeks drew upon the work of the ancient Babylonians and Egyptians, and they used that knowledge to learn even more about the heavens. As far back as the sixth century B.C.E., Greek astronomers and mathematicians attempted to determine the structure of the universe. Anaximander (c. 611 B.C.E.–546 B.C.E.) proposed that the distance of the Moon from Earth is 19 times the radius of Earth. Pythagoras (c. 572 B.C.E.–497 B.C.E.),

celestial relating to the stars, planets, and other heavenly bodies eclipse occurrence when an object passes in front of another and blocks the view of the second object; most often used to refer to the phenomenon that occurs when the Moon passes in front of the Sun or when the Moon passes through Earth’s shadow radius the line segment originating at the center of a circle or sphere and terminating on the circle or sphere; also, the measure of that line segment

17

Solar System Geometry, History of

HOW FAR IS THE MOON FROM EARTH? In contrast to Anaximander’s smaller estimate, Claudius Ptolemy would later prove that the distance of the Moon from Earth is actually approximately 64 times the radius of Earth.

Pythagorean Theorem a mathematical statement relating the sides of right triangles; the square of the hypotenuse is equal to the sum of the squares of the other two sides circumference the distance around a circle stade an ancient Greek measurement of length, one stade is approximately 559 feet (about 170 meters) congruent exactly the same everywhere longitude one of the imaginary great circles beginning at the poles and extending around Earth; the geographic position east or west of the prime meridian circumnavigation the act of sailing completely around a globe

trigonometry the branch of mathematics that studies triangles and trigonometric functions square root with respect to real or complex numbers s, the number t for which t2  s

18

whose name is given to the Pythagorean Theorem, was perhaps the first Greek to believe that the Earth is spherically shaped. The ancient Greeks largely believed that Earth was stationary, and that all of the planets and stars rotated around it. Aristarchus (c. 310 B.C.E.–230 B.C.E.) was one of the few who dared question this hypothesis, and is known as the “Copernicus of antiquity.” His bold statement was recorded by the famous mathematician Archimedes (c. 287 B.C.E.–212 B.C.E.) in the book The Sand Reckoner. Aristarchus is quoted as saying that the Sun and stars are motionless, Earth orbits around the Sun, and Earth rotates on its axis. Neither Archimedes nor any of the other mathematicians of the time believed Aristarchus. Consequently, Aristarchus’s remarkable insight about the movements of the solar system would go unexplored for the next 1,800 years. Eratosthenes (c. 276 B.C.E.–194 B.C.E.), a Greek mathematician who lived during the same time as Archimedes, was another key contributor to the early understanding of the universe. Eratosthenes was the librarian at the ancient learning center in Alexandria in Egypt. Among his many mathematical and scientific contributions, Eratosthenes used an ingenious system of measurement to produce the most accurate estimate of Earth’s circumference that had been computed up to that time. At Syene, which is located approximately 5,000 stades from Alexandria, Eratosthenes observed that a vertical stick had no shadow at noon on the summer solstice (the longest day of the year). In contrast, a vertical stick at Alexandria produced a shadow a little more than 7° from vertical (approxi1 mately 50th of a circle) on the same day. Using the property that two parallel lines (rays of the Sun) cut by a transversal (the line passing through the center of Earth and Alexandria) form congruent alternate interior angles, Eratosthenes deduced that the distance between Syene and Alexandria must 1 be 50 th of the circumference of Earth. As a result, he computed the circumference to be approximately 250,000 stades. Thus, Eratosthenes’s estimate equates to about 25,000 miles, which is remarkably close to Earth’s actual circumference of 24,888 miles. Apparently, Eratosthenes’s interest in this calculation stemmed from the desire to develop a convenient measure of longitude to use in the circumnavigation of the world. The most far-reaching contribution during the Greek period came from Claudius Ptolemy (c. 85 C.E.–165 C.E.). His most important astronomical work, Syntaxis, (better known as Almagest) became the premier textbook on astronomy until the work of Kepler and Galileo over 1,500 years later. Mathematically, Almagest explains both plane and spherical trigonometry, and includes tables of trigonometric values and square roots. Ptolemy’s text firmly placed Earth at the center of the universe without any possibility of its movement. His work also reflected the Greek belief that celestial bodies generally possess divine qualities that have spiritual significance for humans. Ptolemy synthesized previous explanations for planetary motion given by the Greek mathematicians Apollonius, Eudoxus, and Hipparchus into a grand system capable of accounting for almost any orbital phenomenon. This scheme included the use of circular orbits (cycles), orbits whose center lies on the path of a larger orbit (“epicycles”), and eccentric orbits, where Earth is displaced slightly from the center of the orbit. Despite its explanatory power, however, the sheer complexity of Ptolemy’s model of

Solar System Geometry, History of

the universe ultimately drove Nicolas Copernicus (1473–1543) to a completely different conclusion about Earth’s place in the solar system.

The Dawn of Direct Observation By the first half of the sixteenth century, Ptolemy’s geocentric model of the universe had taken on an almost mystical quality. Not only were scientists unquestioning in their devotion to the model and its implications, but the leadership of the Catholic Church in Europe regarded Ptolemy’s view as doctrine nearly equal in status to the Bible. It is ironic, therefore, that the catalyst for the scientific revolution would come from Copernicus, a Catholic trained as a mathematician and astronomer. Copernicus’s primary goal in his astronomical research was to simplify and improve the techniques used to calculate the positions of celestial objects. To accomplish this, he replaced Ptolemy’s geocentric model and its complicated system of compounded circles with a radically different heliocentric view that placed the Sun at the center of the solar system. He retained Ptolemy’s concept of uniform motion of the planets in circular orbits, but now the orbits were concentric rather than overlapping circles. Copernicus’s ideas were very controversial given the faith in Ptolemy’s model. Consequently, despite the fact that his model was completed in 1530, his theory was not published until after his death in 1543. The book containing Copernicus’s theory, De Revolutionibus Orbium Caelestium (“On the Revolutions of the Heavenly Spheres”), was so mathematically technical that only the most competent astronomer could understand it. Yet its effects would be felt throughout the scientific community. Copernicus’s new theory was published at a critical juncture, at the intersection of the ancient and modern worlds. It would set the stage for the new science of direct observation known as empiricism, which would follow less than a century later. Johannes Kepler (1571–1630) was a product of the struggle between ancient and modern beliefs. A teacher of mathematics, poetry and writing, he made remarkable breakthroughs in cosmology as a result of his work with the astronomer Tycho Brahe (1546–1601). Brahe had hired Kepler as an assistant after reading his book Mysterium Cosmographicum. In it, Kepler posits a model of the solar system in which the orbits of the six known planets around the Sun are shown as inspheres and circumspheres of the five regular polyhedra: the tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron. Kepler’s fascination with these geometric solids reflected his devotion to ancient Greek tradition. Plato had given a prominent role to the regular polyhedra, also known as Platonic solids, when he mystically associated them with fire, air, water, earth and the universe, respectively. When Brahe died suddenly and Kepler inherited his fastidious observations of the planets, Kepler set about to verify his own model of the solar system using the data. This began an eight-year odyssey that, far from confirming his model, completely changed the way Kepler looked at the universe. To his credit, unlike the vast majority of mathematicians and scientists of his time, Kepler recognized that if the model and the data did not agree, then the data should be assumed to be correct. Little by little, Kepler’s romantic notions about

geocentric Earthcentered

heliocentric Suncentered

empiricism the view that the experience of the senses is the single source of knowledge cosmology the study of the origin and evolution of the universe

inspheres spheres that touch all the “inside” faces of a regular polyhedron; also called “enspheres” circumspheres spheres that touch all the “outside” faces of a regular polyhedron

19

Solar System Geometry, History of

This illustration of a model shows the orbits of the planets as suggested by Johannes Kepler.

focus one of the two points that define an ellipse; in a planetary orbit, the Sun is at one focus and nothing is at the other focus radius vector a line segment with both magnitude and direction that begins at the center of a circle or sphere and runs to a point on the circle or sphere semi-major axis onehalf of the long axis of an ellipse; also equal to the average distance of a planet or any satellite from the object it is orbiting

20

the structure of the solar system gave way to new, more scientific ideas, resulting in Kepler’s Three Laws of Planetary Motion: First Law

The planets move about the Sun in elliptical orbits with the Sun at one focus.

Second Law

The radius vector joining a planet to the Sun sweeps over equal areas in equal intervals of time.

Third Law

The square of the time of one complete revolution of a planet about its orbit is proportional to the cube of the orbit’s semi-major axis.

The first two laws were published in Astronomia Nova in 1609, with the third law following ten years later in Harmonice Mundi. Kepler’s calculations were so new and different that he had to invent his mathematical tools as he went along. For example, the primary method used to confirm the sec-

Solar System Geometry, History of

ond law involved summing up triangle-shaped slices to compute the elliptical area bounded by the orbit of Mars. Kepler gives one account in Astronomia Nova of computing and summing 180 triangular slices, and repeating this process forty times to ensure accuracy! This work with infinitesimals, which was adapted from Archimedes’s method of computing circular areas, directly influenced Isaac Newton’s development of the calculus about a half-century later. Interestingly, Kepler’s work with the second law led him to the first law, which finally convinced him that planetary orbits were not circles at all (as Copernicus had assumed). However, as Kepler’s account in the Astronomia Nova makes clear, once he had deduced the equation governing Mars’s orbit, due to a computation error he did not quite realize that the equation was that of an ellipse. This is not as strange as it might first appear, considering that Kepler discovered his laws prior to Descartes’s development of analytical geometry, which would occur about a decade later. Thus, with the elliptical equation in hand, he suddenly decided that he had come to a dead end and chose to pursue the hypothesis that he felt all along must be true—that the orbit of Mars was elliptical! In short order Kepler realized that he had already confirmed his own hypothesis. In the publication of his second law, Kepler coined the term “focus” in referring to an elliptical orbit, though the importance of the foci for defining an ellipse was well understood in Greek times. No discussion of either the geometry of the solar system or the science of direct observation is complete without noting the key contributions of Galileo Galilei (1564–1642). Like Kepler, Galileo’s science was closely tied to mathematics. In his investigations of the velocity of falling objects, the parabolic flight paths of projectiles, and the structures of the Moon and the planets Galileo introduced new ways of mathematically describing the world around us. As he wrote in his book Saggiatore, “The Book [of Nature] is . . . written in the language of mathematics, and its characters are triangles, circles, and geometric figures without which it is . . . impossible to understand a single word of it; without these one wanders about in a dark labyrinth.” Using the latest technology (which included telescopes), Galileo’s observations of Jupiter and its moons as well as the phases of Venus left no doubt about the validity of Copernicus’s heliocentric theory. His close inspection of the surface of the Moon also convinced him that the ancient Greek assumption about the Moon and the planets being perfect spheres was simply not tenable. Galileo paid dearly for his findings, incurring the wrath of the Catholic Church and a skeptical scientific community. But as a result of his struggle, mathematics was no longer viewed as either the servant of philosophy or theology as it had been for millennia. Instead, mathematics came to be regarded as a means of validating theoretical models, ushering in a revolutionary era of scientific discovery.

elliptical a closed geometric curve where the sum of the distances of a point on the curve to two fixed points (foci) is constant infinitesimals functions with values arbitrarily close to zero calculus a method of dealing mathematically with variables that may be changing continuously with respect to each other ellipse one of the conic sections, it is defined as the locus of all points such that the sum of the distances from two points called the foci is constant analytical geometry describes the study of geometric properties by using algebraic operations

velocity distance traveled per unit of time in a specific direction parabolic an open geometric curve where the distance of a point on the curve to a fixed point (focus) and a fixed line (directrix) is the same

tenable defensible, reasonable

Through the work of Copernicus, Kepler and Galileo, the mistaken beliefs, pseudoscience, and romanticism of the ancients were gradually replaced by modern observational science, whose goal was and continues to be the congruence of theory with available data. Though the scientific view of the solar system continues to evolve to this day, it does so resting firmly on the foundation of the breakthroughs of the sixteenth and seventeenth centuries. 21

Solar System Geometry, Modern Understandings of

Galilei, Galileo; Solar System Geometry, Modern Understandings of. Paul G. Shotsberger

SEE ALSO

Bibliography Boyer, Carl B. A History of Mathematics, 2nd ed., rev. Uta C. Merzbach. New York: John Wiley and Sons, 1991. Callinger, Ronald. A Contextual History of Mathematics. Upper Saddle River, NJ: Prentice-Hall, 1999. Cooke, Roger. The History of Mathematics: A Brief Course. New York: John Wiley and Sons, 1997. Drake, Stillman. Discoveries and Opinions of Galileo. Garden City, NY: Doubleday, 1957. Eves, Howard. An Introduction to the History of Mathematics. New York: Saunders College Publishing, 1990. Fields, Judith. Kepler’s Geometrical Cosmology. Chicago: University of Chicago Press, 1988. Heath, Thomas L. Greek Astronomy. New York: Dover Publishing, 1991. Koestler, Arthur. The Watershed. New York: Anchor Books, 1960. Taub, Liba C. Ptolemy’s Universe. Chicago: Open Court, 1993.

Solar System Geometry, Modern Understandings of

geocentric Earthcentered

heliocentric Suncentered velocity distance traveled per unit of time in a specific direction elliptical a closed geometric curve where the sum of the distances of a point on the curve to two fixed points (foci) is constant

calculus a method of dealing mathematically with variables that may be changing continuously with respect to each other

22

Aristarchus (c. 310 B.C.E.–230 B.C.E.), an ancient Greek mathematician and astronomer, made the first claim that the planets of the solar system orbit the Sun rather than Earth. However, it was Nicolas Copernicus (1473–1543) who would spur modern investigations that would ultimately overthrow the ancient view of a geocentric universe. Johannes Kepler (1571–1630) and Galileo Galilei (1564–1642) initially carried out these investigations. Through the work of Copernicus, Kepler, and Galileo, the geocentric view of circular orbits with constant velocities was gradually replaced by a heliocentric perspective in which planets travel in elliptical orbits of changing velocities. Kepler and Galileo worked during the beginning of what has come to be known as the “Century of Genius,” a remarkable time of mathematical and scientific discovery lasting from the early 1600s through the early 1700s. Isaac Newton (1642–1727), perhaps the greatest mathematician of the modern period, lived his entire life during the Century of Genius, building on the foundation laid by Kepler and Galileo.

Advances in Celestial Mechanics Isaac Newton was born the year that Galileo died, so there is a sense that the investigations begun by Kepler and Galileo continued in an unbroken chain through the life of Newton. Newton was a mathematician and scientist who was profoundly influenced by the work of his predecessors. Mathematically, he was able to synthesize the work of Kepler and others into perhaps the most useful mathematical tool ever devised, the calculus. Scientifically, among Newton’s many accomplishments, his work in celestial mechanics can be considered a generalization and explanation of Kepler’s three laws of planetary motion. As early as 1596, Kepler had made refer-

Solar System Geometry, Modern Understandings of

ence to a force (he used the word “souls”) that seemed to emanate from the Sun, resulting in the planets’ orbits. Despite many years of work, Kepler was unable to specifically identify this force. Newton not only named the force, but developed a way of mathematically describing its magnitude in a two-body system: Gm1m2 F   r2 where m1 and m2 are the masses of the two bodies (for instance, the Sun and Earth), r is the distance between the bodies, and G is the gravitational constant. This formula is known now as the “Universal Law of Gravity.” As a result of his work on the two-body problem, Newton concluded that such a system may be bound or unbound or in “steady-state,” referring to the situation in which the potential energy of the system is greater than, less than, or equal to (respectively) the kinetic energy of the system. If bound, then one body will orbit the central mass on an elliptical path, as Kepler had stated in his second law of motion; however, if unbound, the relative orbit will be hyperbolic; and if steady-state, the relative orbit will be parabolic. The two-body problem was applicable not only to planets of the solar system, but to the paths of asteroids and comets as well. Of course, the reality of the solar system is that forces beyond just the two bodies in question influence gravitation and orbital paths. Newton noted in his influential book Principia that he believed the solution to a three-body problem would exceed the capacity of the human mind. From the 1700s onward, mathematical contributions to our understanding of the solar system came from individuals who, unlike before, were not necessarily astronomers. Lagrange, Euler and Laplace—all considered eminent mathematicians—provided new tools for more accurately describing the complexities of the universe. Joseph-Louis Lagrange (1736–1813) studied perturbations of planetary orbits. He lent his name to what has come to be called “Lagrangian points,” which are equally spaced locations in front and behind a planet in its orbit, containing bodies (such as asteroids) that obey the principles of the three-body problem. Lagrange and Leonhard Euler (1707–1783) shared the 1772 prize from the Académie des Sciences of Paris for their work on the three-body problem, though it appears that Euler was the first to work on the general problem for bodies outside the solar system. In his masterwork Mécanique Céleste, Pierre-Simon Laplace (1749–1827) showed that the eccentricities and inclinations of planetary orbits to each other always remain small, constant, and selfcorrecting. He also addressed the question of the stability of the solar system and sowed the seeds for the discovery of Neptune.

kinetic energy the energy an object has as a consequence of its motion hyperbolic an open geometric curve where the difference of the distances of a point on the curve to two fixed points (foci) is constant parabolic an open geometric curve where the distance of a point on the curve to a fixed point (focus) and a fixed line (directrix) is the same

perturbations small displacements in an orbit

The Discoveries of Neptune and Pluto Despite the seismic nature of the mathematical and scientific discoveries through the seventeenth and eighteenth centuries, mathematicians and astronomers of the time could scarcely have imagined the discoveries that would be made in the following two centuries. Perhaps the greatest surprise of the period following the time of Newton, Lagrange, Euler, and Laplace was the extent to which mathematics would be used not only to confirm astronomical theories but also to predict as yet undetected phenomena. 23

Solar System Geometry, Modern Understandings of

Until 1781, there were only six known planets in the solar system. That year, an amateur astronomer discovered the planet that would later be named Uranus. By about 1840 enough irregularities had been noted in the orbit of Uranus, including those contained in Laplace’s book, that astronomers were actively seeking the cause. One popular theory was that Newton’s law of gravitation began to break down over large distances. The other primary theory assumed the opposite—it used Newton’s law to predict the existence of an eighth, previously undetected planet that was affecting the orbit of Uranus. In fact, observations of the planet that would come to be known as Neptune had been made by various astronomers as far back as Galileo, but it was assumed to be another star. It was John Adams (1819–1892) of England and Urbain Le Verrier (1811–1877) of France, working simultaneously using Newton’s formulas and the calculus, who accomplished what seemed impossible at the time: making accurate predictions of the position of an unknown planet based solely on the gravitational effects on a known planet. Remarkably, the astronomers were even able to determine the new planet’s mass without direct observation. In 1846, an astronomer in Berlin very close to Le Verrier’s predicted position observed Neptune. This use of mathematics as a prediction, rather than just a confirmation tool, established it as an essential method for astronomical investigation.

ecliptic the plane of the Earth’s orbit around the Sun

Interestingly, even Neptune’s orbit did not behave as expected, and the existence of Neptune did not completely account for the irregularities in the orbits of either Uranus or Jupiter. As a result, in 1905 Percival Lowell (1855–1916) hypothesized the existence of a ninth planet. Lowell would die before finding the planet, but in 1930 the discovery of Pluto was made using the telescope at Lowell’s observatory in Flagstaff, Arizona. The planet’s orbit was found to be highly eccentric and inclined at a much greater angle to the ecliptic than the other planets. In addition, its orbit actually intersects the interior of the orbit of Neptune, allowing it to approach Uranus more closely than Neptune. Though Pluto’s orbit obeys Kepler’s and Newton’s laws, astronomers consider the orbit to be unpredictable over a long period of time (2  107 years). This unpredictability, or sensitivity to initial conditions, is what mathematicians and astronomers refer to as chaos. In a chaotic region, nearby orbits diverge exponentially with time. Though a recent innovation, chaos theory confirms the work of the mathematician Henri Poincaré (1854–1912) on perturbation series, which he did at the beginning of the last century. Though Poincaré’s work called into question Laplace’s conclusion about the stability of the solar system, it is now believed that our planetary scheme is an example of bounded chaos, where motion is chaotic but the orbits are relatively stable within a billion year timeframe. The realization that, as a result of chaos, even few-body systems can in fact be very complex in their behavior is arguably the greatest discovery about the geometry of the solar system in the final two decades of the twentieth century.

Relativity and the New Geometries A major discovery of the first two decades of the twentieth century was Albert Einstein’s (1879–1955) theory of relativity. In 1859, Le Verrier had found a small discrepancy between the predicted orbit of Mercury and its 24

Solar System Geometry, Modern Understandings of

actual path. Le Verrier believed the cause of the difference was some undetected planet or ring of material that was too close to the Sun to be observed. However, if this was not the case, Newton’s theory of gravitation appeared to be inadequate to explain the discrepancy. In 1905, Poincaré posited an initial form of the special theory of relativity, and argued that Newton’s law was not valid and proposed gravitational waves that moved with the velocity of light. Only weeks later, Einstein published his first paper on special relativity, where he suggested that the velocity of light is the same whether the light is emitted by a body at rest or by a body in uniform motion. Then, in 1907, Einstein began to wonder how Newtonian gravitation would have to be modified to fit in with special relativity. This ultimately resulted in his general theory of relativity, whose litmus test was the discrepancy in the orbit of Mercury. Einstein applied his theory of gravitation and discovered that the difference was not due to unseen planets or a ring of material, but could be accounted for by his theory of relativity. Einstein found Euclidean geometry, the geometry typically taught in high school, to be too limited for describing all gravitational systems that might be encountered in space. As a result, he ascribed a prominent role to the non-Euclidean geometries that had recently been developed during the nineteenth century. In the early 1800s, Janos Bolyai (1802–1860) and Nicolai Lobachevsky (1793–1856) had boldly published their revolutionary views of one kind of non-Euclidean geometry. Essentially, they independently confirmed that a geometry was possible where there existed not just one but an infinite number of parallel lines to a given line through a point not on the line. In the classical mechanics of Newton, a key assumption was that rays of light travel in parallel lines that are everywhere equidistant from each other. In contrast, Bolyai and Lobachevsky claimed that parallel lines converged and diverged, though still not intersecting. This kind of geometry is known as Lobachevskian or Hyperbolic, whereas Euclidean or plane geometry is referred to as Parabolic. There is also another kind of non-Euclidean geometry, known as Elliptic, which resulted from the later work of Poincaré.

litmus test a test that uses a single indicator to prompt a decision Euclidean geometry the geometry of points, lines, angles, polygons, and curves confined to a plane non-Euclidean geometry a branch of geometry defined by posing an alternate to Euclid’s fifth postulate equidistant at the same distance

The great mathematician Karl Freidrich Gauss (1777–1855) had early on advanced the idea that space was curved, even suggesting a method for measuring the curvature. However, the connection between Gauss’s work with the curvature of space and the new geometries was not immediately apparent. In fact, the immediate effect of the non-Euclidean geometries was to cause a crisis in the mathematical community: Was there any longer a relationship between mathematics and reality? Euclidean geometry had been the basis not only for all of mathematics up to and including the calculus, but also for all scientific observation and measurement, and even the philosophies of Emmanuel Kant (1724–1804) and his followers. Einstein immediately saw how the various geometries co-existed in space. The genius of his insight was the realization that, whereas light might travel in a straight line in some restricted situation (say, on Earth) when the vast expanse of the universe is considered, light rays are actually traveling in non-Euclidean paths due to gravity from masses such as the Sun. In fact, a modern theoretical model of the universe based on Einstein’s general theory of relativity depends on a more complex non-Euclidean geometry than either Hyperbolic or Elliptic. The assumption is that the universe is not Eu-

Karl Freidrich Gauss made a number of contributions to the field astronomy, including his theories of perturbations between planets, which led to the discovery of Neptune.

25

Solid Waste, Measuring

clidean, and it is the concentration of matter in space (which is not uniform) that determines the extent of the deviation from the Euclidean model. The modern period of observational science has revolutionized the way we view our universe. The geometry of our solar system is infinitely more complex than was considered possible during the time of Kepler, Galileo or even Newton. With new tools such as the Hubble telescope available to twenty-first century astronomers, it is difficult to imagine the discoveries that await us. S E E A L S O Astronomer; Chaos; Solar System Geometry, History of. Paul G. Shotsberger Bibliography Drake, Stillman. Discoveries and Opinions of Galileo. Garden City, NY: Doubleday, 1957. Eves, Howard. An Introduction to the History of Mathematics. New York: Saunders College Publishing, 1990. Henle, Michael. Modern Geometries: Non-Euclidean, Projective, and Discrete, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2001. Koestler, Arthur. The Watershed. New York: Anchor Books, 1960. Smart, James R. Modern Geometries, 4th ed. Pacific Grove, CA: Brooks/Cole, 1994. Weissman, Paul R., Lucy-Ann McFadden, and Torrence V. Johnson, eds. Encyclopedia of the Solar System. New York: Academic Press, 1999.

Solid Waste, Measuring We usually know where things come from. The food we buy comes from farms, practically everything else we own comes from factories, and the raw materials to make them come from mines and forests. But were does it all go once we finish using it or it breaks? We put it on the curb and—thanks to trash collectors—it seems to disappear. Yet in reality, nothing disappears: We just change useful things into trash, just as we change natural resources into things we can use. This article examines how mathematics lets us calculate the amounts of trash we must either divert to beneficial uses or dispose in landfills or incinerators.

Industry Nature

materials

Home goods

waste

Types and Fates of Solid Waste In general, there are two kinds of trash, or solid waste. Industrial solid waste is generated when factories convert raw materials into products. Municipal solid waste is the trash we throw away after we buy and use those products. This article discusses only municipal solid waste, or MSW. The U.S. Environmental Protection Agency estimates that about 223,230,000 tons of MSW were generated in the United States in 2000. Almost all the waste met one of two fates: 26

Solid Waste, Measuring

• Disposal at a landfill or incinerator; or • Recycling or reuse (called diversion). A small amount of the generated waste was dumped illegally, but these amounts are difficult to quantify, and are not addressed in this article. Waste Disposal. The most common waste disposal method in the United States is landfilling. A landfill essentially is a big hole in the ground where we bury our garbage. Engineers design a landfill based on the amount and types of solid waste it needs to receive over its projected “lifetime.” Once it is full and cannot hold any more trash, a new landfill must be dug. Knowing how much waste is generated by a community helps waste managers and community planners determine how big to make a new landfill, as well as how many trash trucks are needed to collect and transport the waste. Calculations for a community start with individual households. In 1998 the average person in the United States produced 4.46 pounds of waste per day. From this, the amount of waste produced by a household of four people over a year is calculated as follows. 4.46 lbs / person / day  365 days / year  4 people / household  6,511.6 lbs / household / year To calculate how much waste must be managed by a community of 160,000 people (that is, 40,000 households of four people each), it is more convenient to convert pounds to tons, starting with the number derived from above. First, 6,511.6 lbs / household / year  1 ton / 2,000 lbs  3.3 tons / household / year So, 3.3 tons / household / year  40,000 households  132,000 tons / year Assuming that engineers designed a landfill to be 60 feet from bottom to top, and the community has equipment to compact the waste into the landfill at a rate of 2 cubic yards per ton, how many acres of land will be needed each year to accommodate the community’s waste? The next equation shows the calculation. 132,000 tons / year  2 yd3 / ton  9 ft3 / yd3  1 / 60 ft  1 acre / 43,560 ft2  0.91 acres / year This is equivalent to filling a football field with waste each year. And, if the landfill charges the community a disposal fee of $30 for every ton, that generates $3,960,000 every year (132,000 tons / year  $30 / ton). This number illustrates how expensive landfill disposal can be, not only from disposal fees, but also in terms of land use constraints and the costs of protecting the environment. Reuse and Recycling. As the U.S. population continues to increase, Americans keep throwing away more trash each year. To decrease the amount of municipal solid waste (MSW) that is disposed in landfills, consumers, businesses, and industries can divert certain materials from disposal via reuse and recycling. Even better, they can reduce the amount of trash generated in the first place. The more materials that are reduced, reused, and recycled, the less quickly landfills will fill up. 27

Solid Waste, Measuring

The equations below show how to calculate the percentage of MSW produced (generated) that is diverted from disposal via reuse and recycling. Reduction is difficult to quantify, and is not considered in the following equations. Generation  Disposal  ( Recycling  Reuse )  Disposal  Diversion So, % Diversion  (Diversion / Generation)  100% Suppose the community planner contacted the disposal and recycling facilities in her area and found how many tons of MSW were disposed, recycled, and reused in recent years. (Most states require landfills to weigh all waste they receive, and to account for its origin.) Substituting these numbers into the series of equations above, she can derive the percentage of waste generated in 2002 that was being kept from landfills by recycling and reuse. Her results are shown in the table, and the graph is shown below. 250,000 Disposal

Tons of waste

200,000

150,000 Recycling

100,000

50,000 Reuse 0 1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

Year

As the table shows, by 2002 our sample community was able to keep nearly one-third of its MSW out of its landfill. This number is fairly representative of most communities in the United States that have strong MSW recycling and reuse programs. “Pay-As-You-Throw.” Most U.S. communities have added recycling to their waste management programs to help divert waste from disposal. But community leaders know that not everything can be reused or recycled. A better alternative is to keep things from becoming waste in the first place. And yet there is a catch: consumers like to buy things! Although most people are genuinely concerned about the environment, it will take more than concern to prompt citizens into reducing or rethinking what they buy, and therefore what they will eventually throw away. SOLID WASTE DIVERTED FROM LANDFILL DISPOSAL

Disposal Recycle Reuse Total Generation % Diversion

28

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

185,545 14,118 2,017 201,680 8%

191,065 19,107 2,123 212,295 10%

196,652 24,581 2,235 223,468 12%

199,945 32,932 2,352 235,229 15%

200,564 44,570 2,476 247,610 19%

203,301 54,735 2,606 260,642 22%

205,770 65,846 2,744 274,360 25%

207,936 77,976 2,888 288,800 28%

215,840 85,120 3,040 304,000 29%

224,000 92,800 3,200 320,000 30%

Solid Waste, Measuring

One motivation for reducing waste is money: namely, making consumers pay for what they throw away. Some communities have implemented such programs, called “Pay-As-You-Throw,” or PAYT. Without PAYT, a household either pays a company every month or year to take their trash, or the household pays for its trash collection in its taxes. In other words, the household pays the same amount no matter whether the family throws away one bag or twenty bags. But what if the household must pay for every bag or can it sets out for the trash collector? Suppose $300 of its taxes has gone toward trash pickup. Now, what if instead the household pays $1.00 for every bag set out? If the family sets out four bags a week, that would be 4 bags  $1.00 / bag  52 wks / year  $208 per year. Although this yields a savings of $92 ($300  $208), it seems like the cost is greater because the household sees the $4.00 disappear every week and will not see the tax savings until annual tax season. But what if the household could further reduce its garbage from four bags to three bags a week by being more careful about what is thrown away and by recycling more? In this case, the household would pay only $3.00 per week (3 bags  $1.00 per bag) instead of $4.00. Compared to the $4.00 per week fee, householders would see their dollar savings every time they put out the trash. This way of looking at monetary benefits is called a market incentive. If appealing to our concern for the environment is not enough to motivate some of us to reduce waste generation, then most of us will do it if we can save money (and hence if it costs us money to not do it). Applying PAYT in a Sample Community. Essentially, PAYT is as much psychology as it is mathematics, but still there is math involved. The amount charged must be affordable yet enough to cover the cost of disposing the waste generated. How does a community decide a fair charge? First the community must determine how much waste there will be and how much it will cost to dispose. To find out, start with the amount of waste generated under the old system; that is, before PAYT measures were implemented. Using 2002 as a base year, and using numbers from the previous table, we know this is 320,000 tons. Divide this amount by the number of people in the community to find how much waste is generated by each resident. For this calculation, the 2002 population must first be estimated. Suppose that census records showed a 1990 population of 154,000 and a 2000 population of 159,000. This is an increase of 5,000 residents over 10 years, or 500 people per year. Because the most recent census year was 2000, and the base year is 2002, there are 2 years’ worth of people to add. Hence, 159,000  (500  2)  160,000. The amount of waste generated per resident can now be determined.

Recycling household materials such as glass containers can save residents money, especially if a community has a “Pay-As-You-Throw” program. Disposing of less trash also benefits the environment, reduces the consumption of natural resources, and creates recycling-related jobs.

320,000 tons 160,000  2 tons per resident It is important to note that this amount includes the trash that business like offices and restaurants generate. This is why the rate of trash generation is so much greater than the previously stated rate of 4.46 pounds, which represents only what people throw away from their homes. 29

Solid Waste, Measuring

It usually takes a year or two to start a PAYT program, and often the community will grow during that time. How many people will be in the community when the program starts, say in 2004? Use the previous peryear increase in population to make the adjustment for 2 more years into the future. That is, 160,000  (500  2 years)  161,000. Now, how much waste would those people generate in 2004, the first year of PAYT, if there were no PAYT prior to that year? The answer is 2 tons / person  161,000 persons  322,000 tons. But once the community has a PAYT program in place, the amount of waste generated should start to decrease. Usually a community starting a PAYT program will look to other communities of about the same size that already have such programs, and then assume they will have a similar decrease. Suppose our sample community discovered that waste generation in nearby Payville decreased from 320,000 tons to 176,000 tons after they implemented their PAYT program. The decrease for Payville is 176,000 tons

320,000 tons  100 percent  55 percent. Now our sample community can apply this rate to the waste it estimates will be generated annually with PAYT, and then determine the monthly total. The annual waste expected with PAYT is 322,000 tons  0.55  177,100 tons. Hence, 177,100 tons / year  1 year / 12 months  14,758 tons / month. Next, estimate how much the PAYT program will cost. Some costs, like building rent, probably will not change much. Others generally will increase over time. For instance, the salary of city employees will probably be greater in 2004 than in 2002. These increases can be estimated the same way population increases were projected: namely, by taking an average of historical increases over a period of time and projecting the annual increase to future years. Conversely, certain costs will decrease as waste decreases. The city may not need as many trucks or as many people to pick up trash as before. Assume that these scenarios have already been considered when estimating costs in the list below. Building mortgage or rent per month Salaries of secretary and file clerk per month Salaries of truck drivers, waste collectors per month

$100,000 $5,000 $250,000

Truck gasoline & maintenance per month

$55,000

Cost of garbage bags to your city per month

$20,000

Landfill fees per ton per month ($30 / ton  14,758 tons / month) Total Costs

$442,740 ________ 872,740

Now the dollar charge per bag can be determined. First you need to know how much an average bag weighs. A bag full of cat litter will weigh more than a bag of the same size filled with tissue. A simple way to find an average weight per bag is to go to a landfill, select several bags at random, weigh them, add the weight of each, and then divide by the number of bags. Suppose five bags weighed 10, 20, 5, 25, and 40 lbs. Hence, (10  20  5  25  40) / 5  20 lbs per bag. Converting to tons, 20 lbs / bag 2,000 lbs / ton  0.01 ton / bag. 30

Somerville, Mary Fairfax

Next, the number of bags that will result from the 14,758 tons that are expected is calculated by 14,758 tons / month  1 bag / 0.01 ton  1,475,800 bags / month. Finally, knowing that total costs per month are $872,740, how much is that per bag? $872,740 1,475,800 bags  $0.59 per bag This price will cover the estimated costs. Now the community leaders must use psychology again and ask whether it is too high, in which case citizens may refuse to accept PAYT. But if the price is too low, the idea of market incentives applies, and citizens won’t be enticed to save money by reducing the trash they throw out (that is, because the money potentially saved would be minimal). So the community leaders must adjust the price up or down to yield the best result. This example gives insight into the mathematics needed to create a waste disposal system that can pay for itself, is good for the environment, and is fair. People who throw away less trash should be paying less than people who throw away more trash. Richard B. Worth and Minerva Mercado-Feliciano Bibliography U.S. Environmental Protection Agency. Characterization of Municipal Solid Waste in the United States: 1998 Update. Prairie Village, KS: Franklin Associates, a service of McLaren/Hart, 1999. ———. Pay-As-You-Throw: A Fact Sheet for Environmental and Civic Groups. September, 1996. ———. Pay-As-You-Throw—Lessons Learned About Unit Pricing. April, 1994. ———. Pay-As-You-Throw: Throw Away Less and Save Fact Sheet. September, 1996. ———. Pay-As-You-Throw Workbook. September, 1996. Internet Resources

Your Guide to Waste Reduction, Reuse, and Recycling in Monroe County, Indiana. Monroe County Solid Waste Management District. .

algebra the branch of mathematics that deals with variables or unknowns representing the arithmetic numbers

Somerville, Mary Fairfax Scottish-born English Mathematics Writer 1780–1872 Mary Fairfax Somerville was born December 26, 1780, the fifth of seven children, to Margaret Charters and Lieutenant William George Fairfax. The family lived in Scotland during an era in which it was customary to provide daughters with an education emphasizing domestic skills and social graces. When Somerville was nine years old, her father expressed disappointment with her reading, writing, and arithmetic skills, and the following year, Somerville was sent to an expensive and exclusive boarding school. She was very unhappy and left there, her only full-time formal schooling, at the end of the year. When she was 13, Somerville encountered algebra by chance when looking at a ladies’ fashion magazine. At this time, such things as riddles, puzzles, and simple mathematical problems were common features in such publications. As a result of her curiosity, Somerville persuaded her younger brother’s tutor to buy her copies of Bonny Castle’s Algebra and Euclid’s Elements.

Mary Fairfax Somerville was called the “Queen of Nineteenth-Century Science” by the London Morning Post.

31

Somerville, Mary Fairfax

Somerville’s father was unhappy with the books she was reading, fearing negative effects on her domestic skills and social graces, and forbade her to read such materials. Despite her father’s restrictions, Somerville secretly continued to read about mathematics by candlelight during the night while others in her family slept. In 1804, at the age of 24, she married her cousin, Captain Samuel Greig of the Russian Navy, and they eventually had two sons, Woronzow (1805–1865) and William George (1806–1814). After the death of Greig in 1807, Somerville began, for the first time, to openly study mathematics and investigate physical astronomy. One of her most helpful mentors during this time was William Wallace (1768–1843), a Scottish mathematics master at a military college. It was upon his advice that Somerville obtained a small library of French books to provide her with a sound background in mathematics. These works were important in forming her mathematical style and account, in part, for her mastery of French mathematics. Another supporter of her mathematical studies was her cousin, Dr. William Somerville (1777–1860), who was a surgeon in the British Navy. They were married in 1812 and had four children. With the encouragement of her husband, Somerville explored her interest in science and mathematics that steadily deepened and expanded into four books that would become her most important works. In 1831, she published The Mechanism of Heavens, which was a “translation” of the first four books of the French mathematician, Laplace’s Mécanique Céleste. Her work was noted as being far superior to previous attempts (mere translations of only Laplace’s first book) at bringing such work to English readers. In 1834 Somerville’s second book, On the Connexion of the Physical Sciences, an account of the connections among the physical sciences, was met with even greater success and distinctions. She and Caroline Herschel were elected to the Royal Astronomical Society in 1835, becoming the first women so honored. In 1838, William Somerville’s failing health caused the family to move to the warmer climate of Italy, where William died in 1860. Mary spent the remaining years of her life in Italy and in 1848, at age 68, published her third and most successful book, Physical Geography, which was widely used in schools and universities for many years. Her last scientific book, On Molecular and Microscopic Science, a summary of the most recent discoveries in chemistry and physics, was published in 1869 when she was 89. quaternion a form of complex number consisting of a real scalar and an imaginary vector component with three dimensions

During the last years of her life, Mary Somerville was involved in many projects, including writing a book on quaternions and reviewing a volume, On the Theory of Differences. Her reading, studying, and writing about mathematics continued until the very day of her death. Somerville died on November 29, 1872, at the age of 92. Gay A. Ragan Bibliography Cooney, Miriam P., ed. Celebrating Women in Mathematics and Science. Reston, VA: National Council of Teachers of Mathematics, 1996. Grinstein, Louise S., and Paul J. Campbell. Women of Mathematics. New York: Greenwood Press, 1987.

32

Sound

Osen, Lynn M. Women in Mathematics. Cambridge, MA: MIT Press, 1974. Perl, Teri. Math Equals: Biographies of Women Mathematicians and Related Activities. Menlo Park, CA: Addison-Wesley Publishing Company, Inc., 1978. ———. Women and Numbers: Lives of Women Mathematicians plus Discovery Activities. San Carlos, CA: Wide World Publishing/Tetra, 1993. Internet Resources

The MacTutor History of Mathematics archive. University of St Andrews. . “Mary Fairfax Somerville.” Biographies of Women Mathematicians. .

Sound Sound is produced by the vibration of some sort of material. In a piano, a hammer strikes a steel string, causing the string to vibrate. A guitar string is plucked and vibrates. A violin string vibrates when it is bowed. In a saxophone, the reed vibrates. In an organ pipe, the column of air vibrates. When a person speaks or sings, two strips of muscular tissue in the throat vibrate. All of the vibrating objects produce sound waves.

Characteristics of Waves All waves, including sound waves, share certain characteristics: they travel through material at a certain speed, they have frequency, and they have wavelength. The frequency of a wave is the number of waves that pass a point in one second. The wavelength of a wave is the distance between any two corresponding points on the wave. For all waves, there is a simple mathematical relationship between these three quantities called the wave equation. If the frequency is denoted by the symbol f, and the wavelength is denoted by the symbol , and the symbol v denotes the velocity, then the wave equation is v  f . In a given medium, a sound wave with a shorter wavelength will have a higher frequency. Waves also have amplitude. Amplitude is the “height” of the wave, or how “big” the wave is. The amplitude of a sound wave determines how loud is the sound. Longitudinal Waves. Sound waves are longitudinal waves. That means that the part of the medium vibrating moves back and forth instead of up and down or from side to side. Regions where the particles of the medium are pushed together are called compressions. Places where the particles are pulled apart are called rarefactions. A sound wave consists of a series of compressions and rarefactions moving through the medium. As with all types of waves, the medium is left in the same place after the wave passes. A person listening to a radio across the room receives sound waves that are moving through the air, but the air is not moving across the room. Speed of Sound Waves. Sound travels through solids, liquids, and gases at different speeds. The speed depends on the springiness of the medium. Steel, for example, is much springier than air, so sound travels through steel about fifteen times faster than it travels through air. 33

Sound

At 0° C, sound travels through dry air at about 331 meters per second. The speed of sound increases with temperature and humidity. The speed of sound in air is related to many important thermodynamic properties of air. Since the speed of sound in air measures how fast a wave of pressure will move through air, anything that depends on air pressure will be expected to behave differently near the speed of sound. This characteristic caused designers of high-speed aircraft many problems. Before the design problems were overcome, several test pilots lost their lives while trying to “break the sound barrier.” The speed of sound changes with temperature. The speed increases by 0.6 meters per second (m/s) for each Celsius degree rise in temperature (T ). This information can be used to construct a formula for the speed (v) of sound at any temperature: v  (331  0.60T )m/s At average room temperature of 20° C, the speed of sound is close to 343 meters per second.

logarithmic scale a scale in which the distances that numbers are positioned, from a reference point, are proportional to their logarithms

Frequency and Pitch. Sounds can have different frequencies. The frequency of the sound is the number of times the object vibrates per second. Frequency is measured in vibrations per second, or hertz (Hz). One Hz is one vibration per second. We perceive different frequencies of sound as different pitches; as a consequence, the higher the frequency, the higher the pitch. The normal human ear can detect sounds with frequencies between 20 Hz and 20,000 Hz. As humans age, the upper limit usually drops. Dogs can hear much higher frequencies, up to 50,000 Hz. Bats can detect frequencies up to 100,000 Hz.

Intensity and Perception of Sound Sound waves, like all waves, transport energy from one place to another. The rate at which energy is delivered is called power and is measured in watts. Sound intensity is measured in watts per square meter (W/m 2). The human ear can detect sound intensity levels as low as 1012 W/m2 and as high as 1.0 W/m2. This is an incredible range. Because of the wide range of sound intensity that humans can hear, humans perceive loudness instead of intensity. A sound with ten times the intensity in watts per square meter is perceived as only about twice as loud.

The sound at a heavy metal concert is about a billion times the intensity of a whisper, but is perceived as only about 800 times as loud because humans perceive sound on a logarithmic scale.

34

Since humans do not directly perceive sound intensity, a logarithmic scale for loudness was developed. The unit of loudness is the decibel, named after Alexander Graham Bell. The threshold of hearing—0 dB—represents a sound intensity of 1012 W/m2. Each tenfold increase in intensity corresponds to 10 dB on the loudness scale. Thus 10 dB is ten times the sound intensity of 0 dB. A sound of 20 dB is ten times the intensity of a 10 dB sound and one hundred times the intensity of a 0 dB sound. The list below shows the loudness of some common sounds. Source

Loudness (dB)

Jet engine

140

Threshold of pain

120

Rock concert

115

Space, Commercialization of

Source Subway train

Loudness (dB) 100

Factory

90

Busy street

70

Normal speech

60

Library

40

Whisper

20

Quiet breathing

10

Threshold of hearing

0

Even short exposure to sounds above 120 dB will cause permanent damage to hearing. Longer exposure to sounds just below 120 dB will also cause permanent damage. S E E A L S O Logarithms; Powers and Exponents. Elliot Richmond Bibliography Epstein, Lewis Carroll. Thinking Physics. San Francisco: Insight Press, 1990. Giancoli, Douglas C. Physics, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1991. Haber-Schaim, Uri, John A. Dodge, and James A. Walter. PSSC Physics, 7th ed. Dubuque, IA: Kendall/Hunt, 1990. Hewitt, Paul G. Conceptual Physics. Menlo Park, CA: Addison-Wesley, 1992.

Space, Commercialization of Since the space program began in the early 1960s, there has been talk of the possible commercial exploitation of space. One of the earliest satellites launched into space was the giant metallic Mylar balloon called “Echo.” Radio signals were reflected from the satellite in an early test of satellite communications. By the early part of the twenty-first century, the space around Earth was filled with commercial communications satellites. Television, radio, telephone, data transfer, and many other forms of information pass through these satellite systems. Excluding the use of two cans and a piece of string, it would be hard to find any form of information transfer that does not involve satellites. In the early days of space exploration, many people thought that the apparent weightlessness in orbiting satellites would allow for types of manufacturing that would not be possible in a normal gravity environment. However, improved processing techniques in Earth-based facilities have proven to be more economical than space-based facilities. Manufacturing remains an underdeveloped form of space commercialization. Nonetheless, commercial exploitation of space remains a real possibility. In recent years, several small companies have been formed. Former astronauts or former National Aeronautics and Space Administration (NASA) scientists or engineers lead many of these small companies. These people believe that the commercial exploitation of space has economic potential. Whatever their goals as a company, they are all convinced that a profit can be made in space. 35

Space, Commercialization of

Space Stations In 1975, participants in an Ames Research Center (a part of NASA) summer workshop created the design for what they considered to be a commercially viable space station. The space station was to be a toroidal structure (like a wheel) 2 kilometers (km) in diameter, and it was supposed to be built in orbit at the same distance as the Moon. Lagrange points two positions in which the motion of a body of negligible mass is stable under the gravitational influence of two much larger bodies (where one larger body is moving about the other) centrifugal the outwardly directed force a spinning object exerts on its restraint; also the perceived force felt by persons in a rotating frame of reference microgravity the apparent weightless conditions of objects in free fall payloads the passengers, crew, instruments, or equipment carried by an aircraft, spacecraft, or rocket

The group selected either the L4 or L5 lunar Lagrange points as the planned station’s orbital position. These are points in space 60° ahead of or behind the Moon in orbit. These points were chosen because they are gravitationally stable and objects placed at one of these two points tend to stay in position. The station would be large enough to hold 10,000 colonists, who would live and work in a habitat that formed the rim of the wheel. At one revolution per minute, centrifugal acceleration at the rim would equal 9.8 m/s2, which would simulate Earth’s gravity. Engineers thought that the colonists could breathe oxygen extracted from moon rocks. The habitat tube would include 63 hectares of farmland, enough to grow food to feed the colonists. The team calculated that the colony would cost $200 billion, which would be equivalent to $500 billion in 2001. The engineers predicted this enormous cost could be recovered within 30 years in savings achieved by having the colonists assemble solar-powered satellites in orbit. This was an ambitious, even grandiose plan. Current space station plans are much more modest by comparison. The International Space Station holds seven crew members and generates negligible income. It will never generate enough income to cover its $60-billion construction cost. NASA is still trying to form alliances with companies interested in using the space station’s microgravity environment for commercial purposes such as renting research modules on the space station to pharmaceutical, biotechnology, or electronics companies. Thus far, private companies have shown little interest. In contrast to other forms of space commercialization, the satellite communications business is booming and shows no signs of slowing down. The strong demand for cellular telephone service and satellite television broadcasts keeps that part of the space industry commercially successful. In all, space-related businesses accounted for over $121 billion per year in 1998. The International Space Station is expected to grow to a mass of over 500,000 kilograms by 2004. As of 2001, all items are shipped from Earth, with space-based operations limited to assembly. This is obviously a major constraint in mission planning. Demand for space-based construction is projected to cost $20 billion per year in 2005, rising to $500 billion per year by 2020 as human space exploration progresses. This cost is primarily due to the high cost of launching payloads from Earth’s surface.

Low-Cost Launch Vehicles

Successful commercial exploitation of space may still be speculative, with the one exception of satellite communications.

36

One major obstacle to commercially successful space manufacturing is the enormous cost of launching a vehicle into Earth orbit. Launch costs using either expendable rockets or the Space Shuttle are currently between $10,000 and $20,000 per kilogram. This high cost led NASA to invest in a prototype launch vehicle called the “X-33.” It was hoped that this prototype would lead to a lightweight, fully reusable space plane. Lockheed Mar-

Space, Commercialization of

tin was building the prototype and originally planned to follow it with a commercial vehicle called the “VentureStar.” However, NASA withdrew funding for the project, leaving it 75 percent complete. Instead NASA created a new program, the Space Launch Initiative, to continue research and development of reusable launch vehicles.

Space-Based Manufacturing While the communications and weather satellite business is profitable, the space industry may need to diversify. For many years, NASA has tried to interest private companies in space-based manufacturing, with the idea that medicines involving large protein molecules, certain types of ultra-pure semiconductor materials, and other products could be manufactured with better quality in an orbital station with near-zero gravity. However, many experts now believe that space-based manufacturing will never be profitable, even if cheaper launch vehicles become available. In-orbit assembly of solar-powered satellites—the main purpose of the giant space colony conceived in 1975—seems to offer more hope of profitability, especially with the anticipated growth in global energy consumption. A solar collector in geostationary orbit would be in full sunlight all the time, except when Earth is near the equinox points in its orbit. It would also be above the atmosphere, so it would receive about 8 times as much light as a solar collector on the ground. Engineers have calculated that a dish 10 kilometers in diameter could generate about 5 billion watts of power, 5 times the output of a conventional power plant. The power could be transmitted by microwave beams from the satellites to antenna arrays on Earth that would convert the microwave energy directly into electrical energy with high efficiency. By using a large antenna array, the microwaves can be kept at a safe intensity level. The array could be mounted several meters off of the ground. The space underneath would receive about 50 percent of normal sunlight, so it could be used to grow shade-tolerant crops.

geostationary orbit an Earth orbit made by an artificial satellite that has a period equal to the Earth’s period of rotation on its axis (about 24 hours)

While promising, solar power is a long way from reality. In 1997, NASA completed a study that reexamined the costs of solar-powered satellites. NASA’s conclusion was that solar power could be made profitable only if launch costs drop below $400 per kilogram, a fraction of the current $10,000 to $20,000 per kilogram launch costs. This seems like an impossible cost reduction, but new technologies are always expensive before their costs rapidly decrease with the innovations brought forth by healthy competition.

Space Tourism Until solar power or some other form of space manufacturing becomes more practical, the most likely source of income might be tourism. Former Apollo astronaut Edwin “Buzz” Aldrin, Jr., the second man to walk on the Moon, has formed a private foundation, the ShareSpace Foundation, to promote space tourism. “People have come up to me and asked, ‘When do we get a chance to go?’” Aldrin says. A 1997 survey of 1,500 Americans showed that 42 percent were interested in flying on a space cruise. Space tourism was encouraged by a 1998 NASA study that concluded it could grow into a $10billion-a-year industry in a few decades. The First Space Tourist. Space tourism became a reality in 2001, at least for one wealthy individual. Dennis Tito, an investment banker from Cali37

Space, Commercialization of

fornia, originally paid an estimated $20 million to the Russian Space Agency, RKK Energia, for the privilege of being transported to the Russian Space Station Mir. However, before that event could take place, the aging Mir was allowed to burn up in the atmosphere. Tito subsequently made arrangements with the Russian Space Agency to be transported to the International Space Station (ISS) as a passenger on a Russian Soyuz flight. The other agencies operating the International Space Station originally objected strongly, but when it became apparent that Russia was going to fly Tito to the Space Station in spite of their objections, they reluctantly agreed to allow him to board. On April 30, 2001, Tito officially became the world’s first space tourist when he boarded the International Space Station Alpha. Tito was required to sign liability releases and to agree to pay for anything he damaged while on board. NASA argued that the presence of Tito on the space station, which was not designed to receive paying guests, would hamper scientific work. The schedule of activities on board the ISS was curtailed during Tito’s visit to allow for possible disruption. In spite of these difficulties, NASA and the other agencies operating the space station anticipate further requests from paying tourists.

Mining in Space Many space entrepreneurs are also considering the possibility of mining minerals and valuable ores from Earth’s Moon and the asteroid belt. There are strong hints from radar data that there may be ice caps at the lunar poles in deep craters that never receive sunlight. This discovery has raised the possibility of a partially self-sustaining Moon colony. However, the most promising commercial possibilities for mining in space may come from nearEarth asteroids (NEAs). These are asteroids that intersect Earth’s orbit. Many of these asteroids are easier to reach than the Moon. Certain kinds of asteroids are rich in iron, nickel, cobalt and platinum-group metals. A two-kilometer wide asteroid of the correct kind holds more ore than has been mined since the beginning of civilization. It would be difficult, impractical, expensive, and dangerous to transport this ore to Earth’s surface. Thus, asteroid ores would be processed in space, and the metals used to build satellites, spaceships, hotels, and solar power satellites. Surprisingly, the most valuable resource to be mined from the asteroids might turn out to be water. This water could be supplied to the space hotels and other satellites, or solar energy could be used to break down the water into hydrogen and oxygen that could then be used as rocket fuel. Since all of the materials are already in orbit, rockets built in space and launched using fuel from water would cost much less than rockets launched from Earth. SpaceDev is a publicly-held commercial space exploration and development company that already has plans in the works for asteroid prospecting. The company has plans to send a spacecraft known as the “Near Earth Asteroid Prospector” (NEAP) to the asteroid 4660 Nereus by 2002. NEAP will cost about $50 million, but this is inexpensive compared to other spacecraft. If this mission is successful, it will be the first commercial deep-space mission. NEAP will carry several scientific instruments, and SpaceDev hopes to make a profit by selling the data collected by these instruments to other entrepreneurs, governments, and university researchers. SpaceDev also 38

Space, Commercialization of

hopes to carry other scientific instruments. The company currently offers space for one ejectable payload for $9 million, and three custom nonejectable payloads for $10 million each. NASA classified NEAP as a Mission of Opportunity, which means that research groups can compete for NASA funds to place scientific instruments onboard the spacecraft.

The Space Enterprise Zone If any of the companies interested in the commercial exploitation of space are going to make a profit, they must avoid having to compete with NASA. Several companies and interest groups, such as the Space Frontier Foundation, have proposed that NASA turn over all low-orbit space endeavors to private companies. This would include the Space Shuttle and the International Space Station. NASA, they feel, should concentrate on deep space exploration. Rick Tumlinson, the president of the Space Frontier Foundation, says “NASA astronauts shouldn’t be driving the space trucks, they should be going to Mars!” NASA is also studying how to transfer technology to private industry. This has been a goal of NASA from its very beginning. In 1998 Congress passed the Commercial Space Act, which requires that NASA draft a plan for the privatization of the space shuttle. The law also established a regulatory framework for licensing the next generation of reusable launch vehicles. To encourage even more privatization of space, several space entrepreneurs have proposed that Congress create a space enterprise zone similar to the enterprise zones in many inner cities. Companies beginning new spacerelated businesses, such as building reusable launch vehicles, would not be taxed for several years. In spite of the obvious problems, most people in the space industry remain optimistic. As a group, they are convinced that the commercial exploitation of space will eventually become profitable, and that commercial operations will be established in Earth orbit, on the Moon, in the asteroid belt and on other planets. For years, science fiction writers and established scientists have looked far ahead and predicted such extraordinary ideas as transforming Mars into a habitable planet. There is even a word invented to describe this process, “terraforming.” The number of space entrepreneurs may remain small, but they share a vision that commercial exploits like terraforming Mars are possible and even necessary if we are to survive as a species. S E E A L S O Space Exploration; Spaceflight, History of; Spaceflight, Mathematics of. Elliot Richmond Bibliography Chaisson, Eric, and Steve McMillan. Astronomy Today, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1993. Criswell, D., and P. Harris. “An Alternative Solar Energy Source.” Earth Space Review, 2 (1993): 11–15. Giancoli, Douglas C. Physics, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1991. Strangway, D. “Moon and Asteroid Mines Will Supply Raw Material for Space Exploitation.” Canadian Mining Journal, 100 (1979): 44–52.

39

Space Exploration

Internet Resources

Buzz Aldrin. . SpaceDev. “The World’s First Commercial Space Exploration and Development Company.” . Space Frontier Foundation Online. .

Space Exploration On July 20, 1969, the people of Earth looked up and saw the Moon in a way they had never seen it before. It was the same Moon that humans had been observing in the sky since the dawn of their existence, but on that July evening, for the first time in history, two members of their own species walked on its surface. At that time it seemed that Neil Armstrong’s “giant leap for mankind” would mark the beginning of a bold new era in the exploration of other worlds by humans. In 1969, some people believed that human scientific colonies would be established on the lunar surface by the 1980s, and that a manned mission to Mars would surely be completed before the turn of the twenty-first century. By the end of 1972, a dozen humans would explore the surface of the Moon. However, 28 years later, as revelers around the world greeted the new millenium, the number of lunar visitors remained at twelve, and a manned mission to Mars seemed farther away than it did in the summer of 1969. How is it that the United States could arguably produce the greatest engineering feat in human history in less than a decade and then fail to follow through with what seemed to be the next logical steps? The answers are complex, but have mostly to do with the tremendous costs of human missions and the American public’s willingness to pay for them.

The Space Race Fuels Space Exploration Gross Domestic Product a measure in the change in the market value of goods, services, and structures produced in the economy

The Apollo program, which sent humans to the Moon, was hugely expensive. At its peak in 1965, 0.8 percent of the Gross Domestic Product (GDP) of the United States was spent on the program. In 2000, the budget for all of the National Aeronautics and Space Administration (NASA) was about 0.25 percent of GDP, with little public support for increased spending on space programs. The huge budgets for the Apollo program were made palatable to Americans in the 1960s because of ongoing competition between the United States and the Soviet Union. The United States had been humiliated by the early successes of the Soviet Union’s space program, including the launch of Sputnik. The Soviet launch of Sputnik, the first man-made satellite, in 1957, and the early failures of American rockets put the United States well behind its greatest rival in what is now commonly known as the “space race.” In 1961, when Soviet cosmonaut Yuri Gagarin became the first human to ride a spacecraft into orbit around the Earth, the leaders of the United States felt the need to respond in a very dramatic way. In less than a year, then-President John F. Kennedy set a national goal of sending a man to the Moon and returning him safely to Earth by the end of the decade. The American public, feeling increasingly anxious about perceived Soviet superiority in science and engineering, gave their support to gigantic

40

Space Exploration

spending increases for NASA to fund the race to the Moon. Neil Armstrong’s first footprint in the lunar dust on July 20, 1969, achieved the first half of Kennedy’s goal. Four days later, splashdown in the Pacific Ocean of the Apollo 11 capsule carrying Armstrong, fellow Moon-walker Edwin “Buzz” Aldrin, and command pilot Michael Collins completed the achievement. Americans were jubilant over this success but growing increasingly weary of spending such huge sums of money on the space program when there were pressing needs at home. By 1972, with the United States mired in an unpopular war in Vietnam, the public’s willingness to fund any more forays, whether to the Moon or southeast Asia, had been exhausted. On December 14, 1972, astronaut Eugene Cernan stepped onto the ladder of the lunar module Challenger, leaving the twentieth century’s final human footprint on the lunar surface.

Robotic Exploration of Space Some scientists contend that the same amount of scientific data could have been obtained from the Moon by robotic missions costing a tenth as much as the Apollo missions. As the twenty-first century dawns, the enormous expense and complexity of human space flight to other worlds has scientists relying almost solely on robotic space missions to explore the solar system and beyond. Many scientists believe that the robotic exploration of space will lead to more discoveries than human missions since many robotic missions may be launched for the cost of a single human mission. NASA’s current estimate of the cost of a single human mission to Mars is about $55 billion, more than the cost of the entire Apollo program. In the early-twentyfirst century, human space travel is limited to flights of NASA’s fleet of space shuttles, which carry satellites, space telescopes, and other scientific instruments to be placed in Earth’s orbit. The shuttles also ferry astronaut-scientists to and from the Earth-orbiting International Space Station. Numerous robotic missions are either in progress or are being planned for launch through the year 2010. In October of 1997, the Cassini mission to Saturn was launched. Cassini flew by Jupiter in December of 2000 and will reach Saturn in July of 2004. Cassini is the best instrumented spacecraft ever sent to another planet. It carries the European built Huygens probe, which will be launched from Cassini to land on Saturn’s giant moon, Titan. The probe carries an entire scientific laboratory, which will make measurements of the atmospheric structure and composition, wind and weather, energy flux, and surface characteristics of Titan. It will transmit its findings to the Cassini orbiter, which will relay them back to Earth. The orbiter will continue its four-year tour of Saturn and its moons, measuring the system’s magnetosphere and its interaction with the moons, rings, and the solar wind; the internal structure and atmosphere of Saturn; the rings; the atmosphere and surface of Titan; and the surface and internal structure of the other moons of Saturn. In April, 2001, NASA launched the Mars 2001 Odyssey with a scheduled arrival in October of 2001. Odyssey carries instruments to study the mineralogy and structure of the Martian surface, the abundance of hydrogen, and the levels of radiation that may pose a danger to future human explorers of Mars. NASA’s Genesis mission launched on August 8, 2001. This

When Neil Armstrong became the first person to walk on the Moon, his first words on that historic occasion were: “That’s one small step for a man, one giant leap for mankind.”

magnetosphere an asymmetric region surrounding the Earth in which charged particles are trapped, their behavior being dominated by Earth’s magnetic field solar wind a stream of particles and radiation constantly pouring out of the Sun at high velocities; partially responsible for the formation of the tails of comets

41

Space Exploration

coma the cloud of gas that first surrounds the nucleus of a comet as it begins to warm up

spacecraft was sent to study the solar wind, collecting data for two years. This information will help scientists understand more about the composition of the Sun. In July of 2002, Contour is scheduled to launch with the purpose of flying by and taking high-resolution pictures of the nucleus and coma of at least two comets as they make their periodic visits to the inner solar system. In November of 2002, the Japanese Institute of Space and Aeronautical Science is scheduled to launch the MUSES-C spacecraft, which will bring back surface samples from an asteroid. The European Space Agency will launch Rosetta in January of 2003 for an eight-year journey to rendezvous with the comet 46 P/Wirtanen. Rosetta will spend two years studying the comet’s nucleus and environment and will make observations from as close as one kilometer. Six international robotic missions to Mars, one to Mercury, one to Jupiter’s moons Europa and Io, and one to the comet P/ Tempel 1 are planned for the period between 2003–2004. Scheduled for an early 2007 launch will be the Solar Probe, which will fly through the Sun’s corona sending back information about its structure and dynamics. The Solar Probe is the last launch currently scheduled for the decade 2001–2010, although this could change as scientists propose new missions and governments grant funding for these missions.

The Future of Space Exploration

The Andromeda galaxy (M31) and its two dwarf elliptical satellite galaxies. Its trillions of stars gathered to form a large elliptical shape.

nuclear fission a reaction in which an atomic nucleus splits into fragments

42

Beyond the first decade of the twenty-first century, plans for the exploration of space remain in the “dream” stage. Scientists at the world’s major space facilities, universities, and other scientific laboratories are working to find breakthroughs that will make space travel faster, less expensive, and more hospitable to humans. One technology that is being met with some enthusiasm in the space science community is the “solar sail,” which works somewhat like the wind sail on a sailboat except that it gets its propulsion from continuous sunlight. A spacecraft propelled by a solar sail would need no rockets, engines, or fuel, making it significantly lighter than rocket-propelled crafts. It could be steered in space in the same manner as a sailboat is steered in water, by tilting the sail at the correct angle. The sail itself would be a thin aluminized sheet. Powered by continuous sunlight, the solar sailcraft could cruise to the outermost reaches of the solar system and even into interstellar space. Estimates are that this technology will not be fully developed until sometime near the end of 2010. In January of 2001, scientists at Israel’s Ben Gurion University reported that they had shown that the rare nuclear material americium-242m (Am242m) could sustain a nuclear fission reaction as a thin metallic film less than a thousandth of a millimeter thick. If true, this could enable Am-242m to act as a nuclear fuel propellant for a relatively lightweight spacecraft that could reach Mars in as little as two weeks instead of the six to eight months required for current spacecraft. However, like the solar sail, this method of propulsion is also far from the implementation stage. Producing enough of it to power spacecraft will be difficult and expensive, and no designs for the type of reactor necessary for this type of nuclear fission have even been proposed. Other scientists are looking into the possibility that antimatter could

Space Exploration

be harnessed and used in jet propulsion, but the research is still in the very early stages. It may be that one or more of these technologies will propel spacecraft around the solar system in the decades ahead, but none of them hold the promise of reaching the holy grail of interstellar travel at near light speed. Moving at the Speed of Light. The difficulty with interstellar space travel is the almost incomprehensible vastness of space. If, at some future time, scientists and engineers could build a spacecraft that could travel at light speed, it would still require more than four years to reach Proxima Centauri, the nearest star to our solar system. Consequently, to achieve the kind of travel among the stars made popular in science fiction works, such as Star Trek or Star Wars, an entirely different level of technology than any we have yet seen is required. The Pioneer 10 and Voyager 1 spacecraft launched in the 1970s have traveled more than 6.5 billion miles and are on their way out of our solar system, but at the speed they are traveling it would take them tens of thousands of years to reach Proxima Centauri. Incremental increases in speed, while helpful within the solar system, will not enable interstellar travel. That will require a major breakthrough in science and technology. Einstein’s relativity theory excludes bodies traveling at or beyond the speed of light. As a result, scientists are looking for ways to circumvent this barrier by distorting the fabric of spacetime itself to create wormholes, which are shortcuts in space-time, or by using warp drives, which are moving segments of space-time. These would require the expenditure of enormous amounts of energy. To study the feasibility of such seemingly far-fetched proposals, NASA, in 1996, established the Breakthrough Propulsion Physics Program. It may seem to the people of Earth, at the beginning of the twenty-first century, that these dreams of creating reality from science fiction are next to impossible—but we should remember that sending humans to the Moon was nothing more than science fiction at the turn of the twentieth century. S E E A L S O Light Speed; Space, Growing Old in; Spaceflight, History of; Spaceflight, Mathematics of. Stephen Robinson

relativity the assertion that measurements of certain physical quantities such as mass, length, and time depend on the relative motion of the object and observer

Bibliography Cowen, Roger. “Travelin’ Light.” Science News 156 (1999): 120. Friedman, James. Star Sailing: Solar Cells and Interstellar Travel. New York: John Wiley & Sons, 1988. Harrison, Albert A. Spacefaring: The Human Dimension. Berkeley: University of California Press, 2001. Wagner, Richard, and Howard Cook. Designs on Space: Blueprints for 21st Century Space Exploration. New York: Simon and Shuster, 2000. lnternet Resources

Millis, Mark. “The Big Mystery: Interstellar Travel.” MSNBC.COM. . Exotic Element Could Fuel Nuclear Fusion for Space Flight. . Solar System Exploration: Missions: Saturn: Cassini. . Solar System Exploration: News: Launch and Events Calendar. .

43

Space, Growing Old in

Space, Growing Old in Radiation and vacuum, large variations in temperature, and lack of oxygen make it impossible for humans to survive unprotected in space. Therefore, mini-ecosystems are needed to sustain life. But would there be any benefit to living and growing old in mini-ecosystems in space? Would the elderly live longer or suffer less from ailments such as arthritis? g a common measure of acceleration; for example 1 g is the acceleration due to gravity at the Earth’s surface, roughly 32 feet per second per second microgravity the apparent weightless condition of objects in free fall

On Earth, the direction of gravity (G) is perpendicular to the surface, and the intensity is 1 g. The intensity of gravity on the Earth’s Moon is 0.17 g and on Mars it is 0.3 g. Even 200 miles away from Earth, the forward motion of a spacecraft counterbalances Earth’s gravity, resulting in continuous free-fall around the planet with a resultant acceleration of about 106 g or 1 micro g. This is not weightlessness but microgravity. Astronauts orbit the Earth in about 90-minute cycles and go through 16 day/night cycles every 24 hours. With the clock seemingly ticking faster and the reduced influence of g, what are the possibilities of growing old in space? Data collected so far indicate that the clock of living organisms ignores the 90-minute cycle and “freeruns” at slightly more than 24 hours, relying on the internal genetic clock. But microgravity also has more obvious consequences. All life on Earth has evolved in 1 g and, therefore, has developed systems to sense and use g. Only by going into space can one fully understand how g has affected life on Earth. Biological organisms respond to changes in g direction and intensity. Plants grow and align with the direction of g. Humans change the direction g enters their body by changing position with respect to the surface of the Earth. For example, when standing up, g intensity (Gz) is greatest; when lying down, g pulling across the chest (Gx) is least intense.

The Effects of Microgravity on the Human Body Microgravity is not a threat to life. Astronauts endure physical and psychological stresses for short periods of time and survive to complete a mission. The longer they live in space, however, the more difficult they find it to recover and re-adapt to Earth. After less than five months on the Mir space station, David Wolf lost 23 lbs, 40 percent of his muscle mass, and 12 percent of his bone mass. It took one year to recover this bone mass. In space, adult humans lose 2 percent bone mass per month, compared to about 1 percent per year on Earth. When astronauts return to Earth, long rehabilitation is needed: the weight of the body cannot be supported because of loss of calcium from bone; there is degradation of ligaments and cartilage in joints; and astronauts experience decreased lower limb muscle mass and strength. It is even hard to sit up without fainting because in microgravity blood volume is reduced, the heart grows smaller, and unused blood vessels in the legs are now no longer able to resist gravitational pull by pumping blood up towards the head. Astronauts find that even standing and walking can be a chore. They may need to first walk with their feet apart for balance. Because the vestibular system in the inner ear that senses g and acceleration received no such input in space, leg movement coordination and proper balance are disturbed. 44

Space, Growing Old in

Shannon Lucid, shown here with Yuri Usachev, managed to walk off the space shuttle Atlantis, after 188 days in space. Lucid logged almost 400 hours on the stationary bicycle and treadmill aboard the space station Mir.

Exercise alone may not prevent adaptation to microgravity nor help maintain Earth-health-status to reduce this long rehabilitation. One way to help astronauts prepare for the increased gravitational pull back on Earth may be to provide “artificial gravity” while on the spacecraft, using an on-board centrifuge for short exposures to g. Because the effects of microgravity are similar to the effects of prolonged exposure to Gx (the gravitational pull across the chest while lying in bed), tests on Earth have used healthy sleeping volunteers to mimic the effects of spaceflight microgravity, and the results have been used to research the best treatments. On Earth, the symptoms astronauts suffer during spaceflight are associated with growing old. In astronauts the symptoms are fully reversible after their return to Earth. Their age has not changed, nor does it affect their life span back on Earth. In typical, earth-bound aging, however, these same symptoms are believed to be inevitable and irreversible. Because the symptoms are also seen after prolonged bed rest, it is believed that even healthy inactive people develop aging symptoms much earlier than had they been more active and used g to greater advantage. If a human were to live in space forever, the body would adapt completely to microgravity. Some functions needed on Earth but not necessary in space would degrade or even disappear. Changes would progress until the body reached a new steady state, what would then be “normal” for space. At that point, the body may never be able to re-adapt to Earth. On the other hand, consider someone on Earth with the same symptoms, pinned to a wheelchair by g—perhaps because of paralysis after a spinal cord injury. Such a person may experience tremendous freedom in being able to move about in microgravity using only the upper body as do the astronauts.

Would We Live Longer in Space? These appropriate changes to the space environment tell us nothing about whether humans would live less or longer in space. To find out, scientists 45

Spaceflight, History of

are using specimens with much shorter life spans. Experiments with fruit flies (Drosophila melanogaster) exposed for short periods of their adult life to space and then studied back on Earth have shown no change in their life span. Many genes related to aging in organisms like fruit flies and the roundworm (Caenorhabditis elegans) are similar to the aging genes of humans. Because fruit flies and roundworms live 4-8 weeks, it is easier to study them in space both from birth and over several generations. Markers of biological aging like telomeres studied extensively in aging research on Earth will also be studied in space. Telomeres are fragments of non-coding deoxyribonucleic acid (DNA; that is, DNA that does not give rise to proteins) found on the ends of each chromosome. When a cell divides, telomeres shorten until they eventually become so short that the cell can no longer divide and dies. The role of g and microgravity on such mechanisms may help unravel the mysteries of growing old both in space and on Earth. So what is growing old? On Earth we use it to describe the debilitating changes that occur as one’s age increases, eventually leading to death. Clearly, this does not apply to what happens in space, suggesting that perhaps growing old on Earth should be redefined. Space has taught scientists a great deal about growing old on Earth. The degenerative changes seen in astronauts and in humans aging on Earth indicate how important it is to be active and to use g to its fullest on Earth. In so doing, humans may live healthier, if not longer, lives. S E E A L S O Space Exploration; Spaceflight, History of. Joan Vernikos Bibliography Long, Michael E. “Surviving in Space.” National Geographic January (2001): 6–29. Miquel, Jaime, and Ken A. Souza. “Gravity Effects on Reproduction, Development and Aging.” Advances in Space Biology and Medicine 1 (1991): 71–97. Nicogossian, Arnauld E., Carolyn Leach Huntoon, and Sam L. Pool. Space Physiology and Medicine. Philadelphia, PA: Lea and Febiger, 1994. Vernikos, Joan. “Human Physiology in Space.” Bioessays 18 (1996): 1029–1037. Warshofsky, Fred. Stealing Time: The New Science of Aging. New York: TV Books, 1999.

Spaceflight, History of In the early-thirteenth century, the Chinese invented the rocket by packing gunpowder into a tube and setting fire to the powder. Half a millenium later, the British Army colonel, William Congreve, developed a rocket that could carry a 20-pound warhead nearly three miles. In 1926, the American physicist Robert Goddard built and flew the first liquid-fueled rocket, becoming the father of rocket science in the United States. In Russia, a mathematics teacher named Konstantin Tsiolkovsky derived and published the mathematical theory and equations governing rocket propulsion. Tsiolkovsky’s work was largely ignored until the Germans began to employ it to build rocket-based weapons in the 1920s under the leadership of 46

Spaceflight, History of

mathematician and physicist Hermann Oberth. In October of 1942, the Germans launched the first rocket to penetrate the lower reaches of space. It reached a speed of 3,500 miles per hour, giving it a range of about 190 miles. The technology and design of the German rocket were essentially the same as those pioneered by Goddard. The successors of this first rocket would be the infamous V-2 ballistic missiles, which the Nazis launched against England in 1944 and 1945. This assault came too late to turn the course of the war in Germany’s favor, but devastating enough to get the full attention of the U.S. government. Many of the German V-2 rocket scientists, led by Werhner von Braun, surrendered to the Americans and were brought to the United States, where they became the core of the American rocket science program in the late1940s, 1950s, and 1960s. Von Braun and his team took the basic technology of the V-2 missiles and scaled it up to build larger and larger rockets fueled by liquid hydrogen and liquid oxygen.

The Weapons Race Leads to a Space Race In the beginning, the focus of the American rocket program was on national defense. After World War II, the United States and the Soviet Union emerged as the world’s two most formidable powers. Each country was determined to keep its arsenal of weapons at least one step ahead of the other’s. Chief among those arsenals would be intercontinental ballistic missiles, huge rockets that could carry nuclear warheads to the cities and defense installations of the enemy. To reach a target on the other side of the world, these rockets had to make suborbital flights, soaring briefly into space and back down again. In the 1950s, as rocket engines became more and more powerful, both nations realized that it would soon be possible to send objects into orbit around Earth and eventually to the Moon and other planets. Thus, the Cold War missile race gave birth to the space race and the possibility of eventual space flight by humans. Werhner von Braun and his team of rocket scientists were sent to the Redstone Army Arsenal near Huntsville, Alabama and given the task of designing a super V-2 type rocket, which would be called the Redstone, named for its home base. The first Redstone rocket was launched from Cape Canaveral, Florida on August 20, 1953. Three years later, on September 20, 1956, the Jupiter C Missile RS-27, a modified Redstone, became the first missile to achieve deep space penetration, soaring to an altitude of 680 miles above Earth’s surface and traveling more than 3,300 miles. Meanwhile, in the Soviet Union, rocket scientists and engineers, led by Sergei Korolyev, were working to develop and build their own intercontinental ballistic missile system. In August of 1957, the Soviets launched the first successful Intercontinental Ballistic Missile (ICBM), called the R-7. However, the shot heard round the world would be fired two months later, on October 4, 1957, when the Soviets launched a modified R-7 rocket carrying the world’s first man-made satellite, called Sputnik. Sputnik was a small, 23-inch aluminum sphere carrying two radio transmitters, but its impact on the world, and especially on the United States, was enormous. The “space race” had begun in earnest—there would be no turning back. One month later, the Soviets launched Sputnik 2, which carried Earth’s first space traveler, a dog named Laika. The United States government and 47

Spaceflight, History of

its people were stunned by the Soviet successes and the first attempt by the United States to launch a satellite was pushed ahead of schedule to December 6, 1957, just a month after Laika’s journey into space. The Vanguard launch rocket, which was supposed to propel the satellite into orbit, exploded shortly after lift-off. It was one of a series of U.S. failures, and dealt a serious blow to Americans’ confidence in the country’s science and engineering programs, which were previously thought to be the best in the world. Success finally occurred on January 31, 1958 with the launch of the first U.S. satellite, Explorer 1, which rode into space atop another Jupiter C rocket.

The Race to the Moon Both the Soviet Union and the United States wanted to be the first to send a satellite to the Moon. In 1958, both countries attempted several launches targeting the Moon, but none of the spacecraft managed to reach the 25,000 miles per hour speed necessary to break free of Earth’s gravity. On January 2, 1959, the Soviet spacecraft Luna 1 became the first artificial object to escape Earth’s orbit, although it did not reach the Moon as planned. However, eight months later, on September 14, 1959, Luna 2 became the first man-made object to strike the lunar surface, giving the Soviets yet another first in the space race. A month later, Luna 3 flew around the Moon and radioed back the first pictures of the far side of the Moon, which is not visible from Earth.

cosmonaut the term used by the Soviet Union and now used by the Russian Federation to refer to persons trained to go into space; synonymous with astronaut

The United States did not resume its attempts to send an object to the Moon until 1962, but by then, the space race had taken on a decidedly human face. On April 12, 1961, an R-7 rocket boosted a spacecraft named Vostok that carried 27-year-old Soviet cosmonaut Yuri Gagarin into Earth orbit and into history as the first human in space. Gagarin’s 108-minute flight and safe return to Earth placed the Soviet Union clearly in the lead in the space race. Less than a month later, on May 5, 1961, a Redstone booster rocket sent the U.S. Mercury space capsule, Freedom 7, which carried American astronaut Alan Shepard, on a 15-minute suborbital flight. The United States was in space, but just barely. An American would not orbit Earth until February of 1962, when astronaut John Glenn’s Mercury capsule, Friendship 7, would be lifted into orbit by the first of a new generation of American rockets known as Atlas. Just weeks after Shepard’s suborbital flight in 1961, U.S. President John F. Kennedy announced that a major goal for the United States was to send a man to the Moon and return him safely to Earth before the end of the decade. The fulfillment of that goal was to be the result of one of the greatest scientific and engineering efforts in the history of humanity. The project cost a staggering $25 billion, but Congress and the American people had become alarmed by the Soviets’ early lead in the space race and were more than ready to fund Kennedy’s bold vision. The Mercury and Vostok programs became the first steps in the race to the Moon, showing that humans could survive in space and be safely brought back to Earth, but the Mercury and Vostok spacecraft were not designed to take humans to the Moon. By 1964, the Soviet Union was ready to take the next step with a spacecraft named Voskhod. In October of 1964,

48

Spaceflight, History of

Voskhod 1 carried three cosmonauts, the first multi-person space crew, into orbit. They stayed in orbit for a day and then returned safely to Earth. In March of 1965, the Soviets launched Voskhod 2 with another three-man crew. One of the three, Alexei Leonov, became the first human to “walk” in space. Within weeks of Leonov’s walk, the United States launched the first manned flight of its second-generation Gemini spacecraft. The Gemini was built to carry two humans in considerably more comfort than the tiny Mercury capsule. Although Gemini was not the craft that would ultimately take men to the Moon, it was the craft that would allow American astronauts to practice many of the maneuvers they would need to use on lunar missions. Ten Gemini missions were flown in 1965 and 1966. On these missions, astronauts would walk in space, steer their spacecraft into a different orbit, rendezvous with another Gemini craft, dock with an unmanned spacecraft, reach a record altitude of 850 miles above Earth’s surface, and set a new endurance record of 14 days in space. When the Gemini program came to an end in November of 1966, the United States had taken the lead in the race to the Moon. The Soviets had abandoned their Voskhod program after just two missions to concentrate their efforts on reaching the Moon before the United States. Both countries developed a third-generation spacecraft designed to fly to the Moon and back. The Soviet version was called Soyuz, while the American version was named Apollo. Both could accommodate a three-person crew. In November of 1967, Werhner von Braun’s giant Saturn V rocket was ready for its first test flight. The three-stage behemoth stood 363 feet high, including the Apollo command module perched on top. Its first stage engines delivered 7.5 million pounds of thrust, making it the most powerful rocket ever to fly. On its first test flight, the mighty Saturn launched an unmanned Apollo spacecraft to an altitude of 11,000 miles above Earth’s surface. On December 21, 1968, the Saturn V boosted astronauts Frank Borman, Jim Lovell, and Bill Anders inside their Apollo 8 spacecraft into Earth orbit. After two hours in orbit, the Saturn’s third stage engines fired one more time, increasing Apollo 8’s velocity to 25,000 miles per hour. For the first time in history, humans had escaped the pull of Earth’s gravity and were on their way to the Moon. On December 24, Apollo 8 entered lunar orbit, where it would stay for the next twenty hours mapping the lunar surface and sending back television pictures to Earth. Apollo 8 was not a landing mission, so on December 25, the astronauts fired their booster rockets and headed back to Earth, splashing down in the Pacific Ocean two days later. The last year of the decade was about to begin and the stage was set to fulfill President Kennedy’s goal. Two more preparatory missions, Apollo 9 and Apollo 10, would be used to do a full testing of the Apollo command module and the lunar module, which would take the astronauts to the surface of the Moon. The astronauts chosen to ride Apollo 11 to the Moon were Neil Armstrong, Edwin “Buzz” Aldrin, and Michael Collins. Liftoff occurred on July 16, 1969, with Apollo 11 reaching lunar orbit on July 20. Armstrong and Aldrin squeezed into the small spider-like lunar module named Eagle, closed the hatch, separated from the command module, and began their descent 49

Spaceflight, History of

to the lunar surface. Back on Earth, the world watched in anticipation as Neil Armstrong guided his fragile craft toward the gray dust of the Moon’s Sea of Tranquility. Armstrong’s words came back across 239,000 miles of space: “Houston. Tranquility Base here. The Eagle has landed.” At 10:56 P.M., eastern standard time, July 20, 1969, Armstrong became the first human to step on the surface of another world. “That’s one small step for a man,” he proclaimed, “and one giant leap for mankind.” Aldrin followed Armstrong onto the lunar surface, where they planted an American flag and then began collecting soil and rock samples to bring back to Earth. The Apollo 11 astronauts returned safely to Earth on July 24, fulfilling President Kennedy’s goal with five months to spare. Five more teams of astronauts would walk and carry out experiments on the Moon through 1972. A sixth team, the crew of Apollo 13, was forced to abort their mission when an oxygen tank exploded inside the service module of the spacecraft. On December 14, 1972, after a stunning three-day exploration of the lunar region known as Taurus-Littrow, astronauts Eugene Cernan and Harrison Schmitt fired the ascent rockets of the Lunar module Challenger to rejoin crewmate Ron Evans in the Apollo 17 command module for the journey back to Earth. Apollo 17 was the final manned mission to the Moon of the twentieth century. The American people and their representatives in Congress had exhausted their patience with paying the huge sums necessary to send humans into space. The Soviets never did send humans to the Moon. After the initial Soyuz flights, their Moon program was plagued by repeated failures in technology, and once the United States had landed men on the Moon, the Soviet government called off any additional efforts to achieve a lunar landing. Although the Apollo program did not lead to an immediate establishment of scientific bases on the Moon or human missions to Mars as was once envisioned, human spaceflight did not end. The three decades following the final journey of Apollo 17 have seen the development of the space shuttle program in the United States, as well as active space programs in Russia, Europe, and Japan. Scientific work is currently under way aboard the International Space Station, with space shuttle flights ferrying astronauts, scientists, and materials to and from the station. Deep spaceflights by humans to the outer planets and the stars await significant breakthroughs in rocket propulsion. S E E A L S O Astronaut; Space, Commercialization of; Space Exploration; Space, Growing Old in; Spaceflight, Mathematics of. Stephen Robinson Bibliography Hale, Francis J. Introduction to Space Flight. Upper Saddle River, NJ: Prentice Hall, 1998. Heppenheimer, T. A. Countdown: A History of Spaceflight. New York: John Wiley and Sons, 1999. Watson, Russell and John Barry. “Rockets.” In Newsweek Special Issue: The Power of Invention. Winter (1997–1998): 64–66. lnternet Resources

NASA Goddard Space Flight Center Homepage. .

50

Spaceflight, Mathematics of

Spaceflight, Mathematics of The mathematics of spaceflight involves the combination of knowledge from two different areas of physics and mathematics: rocket science and celestial mechanics. Rocket science is the study of how to design and build rockets and how to efficiently launch a vehicle into space. Celestial mechanics is the study of the mathematics of orbits and trajectories in space. Celestial mechanics uses the laws of motion and the law of gravity as first stated by Isaac Newton.

A Short History of Rockets People have wondered about the possibility of space flight since ancient times. Centuries ago the Chinese used rockets for ceremonial and military purposes. In the latter half of the twentieth century, we have been able to develop rockets powerful enough to launch vehicles into space with sufficient energy to achieve or exceed orbital velocities. The rockets mentioned in the American national anthem, the “Star Spangled Banner,” were British rockets fired at the American fortress in Baltimore. However, these so-called “Congreves,” named for their inventor, Sir William Congreve, were not the first war rockets. Some six centuries before, the Chinese connected a tube of primitive gunpowder to an arrow and created an effective rocket weapon to defend against the Mongols.

orbital velocities the speed and direction necessary for a body to circle a celestial body, such as the Earth, in a stable manner

The Physics Behind Rockets and Their Flight The early Chinese “fire arrows,” the British Congreves, and even our familiar Fourth of July skyrockets, are propelled by exactly the same principles of physics that propelled the Delta, Atlas, Titan, and Saturn rockets, as well as the modern space shuttle. In each case, some kind of fuel is burned with an oxidant in the combustion chamber, creating hot gases accompanied by high pressure. In the combustion chamber, these gases try to expand, pushing equally in all directions. At the bottom of the combustion chamber, there is an exhaust nozzle that flares out like a bell. The expanding gases push on all sides of the combustion chamber and the nozzle, except at the bottom. This results in a net force on the rocket, accelerating it in a direction opposite to the nozzle. This idea is summarized in Newton’s Third Law of Motion: “To every action, there is an equal and opposite reaction.” The gases “thrown” out of the bottom of the nozzle constitute the action. The reaction is the rocket’s acceleration in the opposite direction. This system of propulsion does not depend on the presence of an atmosphere—it works just as well in empty space.✶ According to Newton’s First Law of Motion, “A body at rest will remain at rest unless acted on by an unbalanced force.” The unbalanced force is provided by the rocket motor, so the rocket motor and attached spacecraft are accelerated. Because Newton’s laws were postulated in the seventeenth century, some people have argued, “All the basic scientific problems of spaceflight were solved 300 years ago. Everything since has been engineering.” However, many difficult problems of mathematics, chemistry, and engineering had to be solved before humans could apply Newton’s basic principles to achieve controlled spaceflight.

oxidant a chemical reagent that combines with oxygen combustion chemical reaction combining fuel with oxygen accompanied by the release of light and heat net force the final, or resultant, influence on a body that causes it to accelerate

✶One of the early astronauts answered a question from mission control asking who was flying with the words, “Isaac Newton is flying this thing.”

51

Spaceflight, Mathematics of

Using Rockets in Spaceflight

oxidizer the chemical that combines with oxygen or is made into an oxide

Robert Goddard was an early American rocket pioneer, who initially encountered resistance to his ideas. Some people, not understanding Newton’s laws very well, insisted rockets would not operate in the vacuum of space because there would be no air for them to push against. They were thinking in terms of a propeller-driven airplane. But a rocket needs no air to push against; action-reaction forces propel it. Nor does a rocket need air to burn the fuel. An oxidizer is mixed with the fuel in the engine. In 1926 in Massachusetts, Goddard successfully launched the world’s first liquid-fuel rocket. By modern standards it was very rudimentary: It flew 184 feet in two and a half seconds. But the mighty Atlas, Titan, and Saturn liquid-fueled rockets are all direct descendents of Goddard’s first rocket— even the Space Shuttle uses liquid fuel rockets as its main engine. At the same time that Goddard was doing his pioneering work on rockets in the United States, two other pioneers were beginning their work in other countries. In the early Soviet Union, Konstantin Tsiolkovski was conducting experiments very similar to Goddard’s. In Germany, Hermann Oberth was studying the same ideas. Military leaders in Nazi Germany recognized the possibilities of using long-range rockets as weapons. This led to the German V-2 missiles that could be launched from Germany to a height of 100 kilometers across the English Channel. These missiles could reach speeds greater than 5,600 kilometers per hour. After World War II, the United States and the Soviet Union used captured German V-2 rockets, as well as the knowledge of German rocket scientists, to start their own missile programs. In October 1957, the Soviets propelled a satellite named Sputnik into space. Four years later, the first human orbited the Earth.

Now rockets are used to launch vehicles into space, but in World War II the Germans developed them for use as longrange weapons.

✶In total, six Apollo expeditions landed on the Moon and returned to Earth between 1969 and 1972.

trajectory the path followed by a projectile; in chaotic systems, the trajectory is ordered and unpredictable

The first U.S. satellite, Explorer I, was launched into orbit in January of 1958. On May 5, 1961, Alan Shepard became the first American to fly into space with a fifteen-minute flight of the Freedom 7 capsule. Four years later, on February 20, 1962, John Glenn became the first American to orbit Earth in the historic flight of the Friendship 7 capsule. In 1961, President John F. Kennedy had set a national goal of sending and landing a man on the Moon and then returning him safely to Earth within the decade. In July 1969, Astronaut Neil Armstrong took “a giant step for mankind” as he stepped onto the surface of the Moon. It was an event few Americans living at the time will ever forget.✶

Navigating in Space Getting to the Moon, Mars, or Venus is not simply a matter of aiming a rocket in the desired direction and hitting a launch button. The process of launching a spacecraft on a trajectory that will take it to another planet has been compared to trying to shoot at a flying duck from a moving speedboat. Launching a spacecraft into orbit is relatively simpler. The spacecraft is launched toward the east so that it can take advantage of Earth’s rotation, and gain some extra speed. For the same reason, satellites are usually launched from sites as close to the equator as possible. On the Apollo missions to the Moon, the spacecraft was boosted into an Earth orbit where it coasted until it was in the proper position to start

52

Spaceflight, Mathematics of

the second leg of the trip. At the appointed time, the Apollo spacecraft was boosted out of its parking orbit and into a trajectory or path to the Moon. This meant firing the spacecraft’s rocket engines when it was at a point in orbit on the opposite side of Earth from the Moon. Orbital paths are ellipses, parabolas, or some other conic section. In the case of a spacecraft launched from Earth into orbit, the spacecraft follows an elliptical orbit with the center of Earth at one focus. The point in the orbit closest to Earth is known as the “perigee” and the point on the elliptical orbit farthest from Earth is called the “apogee.” The Apollo’s engines were fired when it was at perigee. For Apollo to reach the Moon, the point of maximum altitude above Earth, the apogee, had to intersect the orbit of the Moon and the arrival had to be timed so that the Moon would arrive at the same position when the spacecraft appeared. When the Moon’s gravitational pull on the spacecraft was greater than Earth’s pull, it made sense to shift the points of reference. Thus, the position of the Apollo spacecraft began to be considered in terms of its where it was relative to the Moon. At that point, the spacecraft was more than half way to the Moon. Its orbit was now calculated in relation to the Moon’s orbit. As the spacecraft approached the Moon, it swung around the Moon in an elliptical orbit, with the Moon at one focus. Apollo then fired its engines again to slow down and enter orbit.

ellipse one of the conic sections, it is defined as the locus of all points such that the sum of the distances from two points called the foci is constant parabola a conic section; the locus of all points such that the distance from a fixed point called the focus is equal to the perpendicular distance from a line conic of or relating to a cone, that surface generated by a straight line, passing through a fixed point, and moving along the intersection with a fixed curve

Orbital Velocity

WHAT EXACTLY IS “FREE-FALL”?

The velocities needed for specific space missions were calculated long before space flight was possible. To put an object into orbit around the Earth, for instance, a velocity of at least 7.9 kilometers/second (km/s) (about 18,000 miles per hour), depending on the precise orbit desired, must be achieved. This is called “orbital velocity.”

A satellite in orbit around Earth is falling towards Earth; it is accelerating downward. Since there is nothing to stop its fall, it is said to be in “free-fall.” Astronauts on board the satellite are also falling downward, and this produces a sensation of weightlessness. However, it would be incorrect to say there is no gravity or that the astronauts are weightless, since the satellite is only a few hundred kilometers above Earth’s surface. NASA likes to use the term “microgravity,” which refers to the fact that the objects in the spacecraft actually attract each other. Astronauts feel weightless not because they have “escaped gravity” but because they are freely falling toward the surface of Earth.

In effect, an object in Earth orbit is falling around the Earth. A satellite or missile with a speed less than orbital velocity moves in an elliptical orbit with the center of Earth at one focus. However, the orbit intersects Earth’s surface. With a slightly increased speed, the satellite still falls back toward Earth. In simply moves sideways fast enough that it does not actually hit the Earth. It keeps missing. To leave Earth orbit for distant space missions, a velocity greater than 11.2 km/s (25,000 miles per hour) is required. This is called “escape velocity.” In Earth orbit, the downward force of Earth’s gravity maintains the satellite’s acceleration towards Earth, but Earth keeps curving away. At still higher speeds, the orbit becomes more elliptical and the apogee point is at a greater distance. If the launch speed reaches escape velocity, the apogee is infinitely far away and the ellipse has become a parabola in Earth’s frame of reference. At even greater speed, the orbit is a hyperbola in Earth’s frame of reference. S E E A L S O Solar System Geometry, Modern Understandings of; Space, Commercialization of; Space Exploration; Space, Growing Old in; Spaceflight, History of. Elliot Richmond

53

Sports Data

elliptical orbit a planet, comet, or satellite follows a curved path known as an ellipse when it is in the gravitational field of the Sun or another object; the Sun or other object is at one focus of the ellipse hyperbola a conic section; the locus of all points such that the absolute value of the difference in distance from two points called foci is a constant

Bibliography Chaisson, Eric, and Steve McMillan. Astronomy Today, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1993. Giancoli, Douglas C. Physics, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1991.

Sports Data Who was the greatest home run hitter: Babe Ruth or Roger Maris? How does Mark McGwire compare? What professional football team won the most games in one season? Sports fans argue such questions and use data collected about players and teams to try to answer them. Every sport from baseball to golf, from tennis to rodeo competitions keep statistics about individual and team performances. These statistics are used by coaches to help plan strategies, by team owners to match salaries with performance, and by fans who simply enjoy knowing about their favorite sports and players. Most statistics are either about players or teams. There are many different statistical measures used, but the most common ones are maxima and minima, rankings, averages, ratios, and percentages.

Maximum or Minimum Perhaps the simplest of all statistics answers the question, “Who won the most competitions?” or “Who made the fewest errors?” In all professional sports, data are kept on the number of games won and lost by teams and the number of errors made by players. A listing of the number of wins quickly shows which team won the most games. Mathematically, one is seeking the largest number (or the smallest number) in a list. These numbers are called the maximum (or the minimum) values of the list.

Rankings Another common way to compare players or teams is to rank them. The number of games won each season often ranks teams. Some sports rank individual players as well. Tennis, for example, ranks players (called their seed) based on the number of tournaments they have completed, the strength of their opponents, and their performance in each. Wins against higher-ranked opponents affect a player’s seed more than wins against lower-ranked opponents.

Average Averaging is one of the most well known ways to create a statistic from data, and one of the most well known of all sports statistics is a baseball player’s batting average. To average a series of numbers, add them up and divide by the number of numbers; a batting average is no different. Each time a player comes to the plate, he either gets 1 hit or he does not hit. To compute the batting average, add up all the numbers (that is, the total number of hits in a season) and divide by the total number of times he was at bat. Averages are common in other sports as well. In football, for example, one can compute a player’s average punt returns (total number of yards / number of attempts), his average yards rushing (total number of yards / 54

Sports Data

In addition to mathematics being involved in technical aspects of snowboarding, it is also relevant to the score a participant receives in competition.

number of attempts), and his average yards receiving (total number of yards / number of times received). In basketball, players are compared by their scoring average (total number of points scored divided by number of games played), their rebound average (total number of rebounds / number of games played), and their stealing average (total number of balls stolen / number of games played). In other sports, the scores are based on averages. In snowboarding, for example, each competitor is given a score from five different judges who look at different characteristics of a run. The average of these numbers determines a snowboarder’s score on a scale of 1.0–10.0. Averages sometimes need to be adjusted. One may want to compare the performance of two pitchers, for example, who played a different number of innings. To compute the pitcher’s earned run average, divide the total number of earned runs by the total number of innings pitched and then multiply by 9. This gives a number that represents the number of runs that would have been earned if the player had completed nine innings of play.

Ratios and Percentages Often interesting information cannot be obtained from a simple average. A baseball coach, for example, might want his next batter to get a walk and he must decide which of his pinch hitters is most likely to get a walk. In this case the most useful statistic is a percentage called the base on balls percentage. It is computed by dividing the number of times a player is at bat by the number of times he was walked. This decimal is then written as a percent. Similarly, the stolen base percentage is computed by dividing the number of stolen bases by the number of attempts at a stolen base. Percentages are a commonly used statistic in other sports as well. In basketball, for example, a player’s field-goal percentage is the ratio of the number of field goals attempted to the number of field goals made; a player’s free-throw percentage and three-point field-goal percentage is calculated 55

Sports Data

similarly. In football, a quarterback’s efficiency is measured by the percentage of passes that are completed (number of passes completed / number of passes attempted).

Weighted Average The batting average compares the number of hits to the number of times a player was at bat. This allows a comparison of players who have been at bat a different number of times. In this average, however, both singles and home runs are counted the same—as hits. A player who hits mainly singles could have a higher batting average than a player who mostly belts home runs. In order to average numbers where some of the numbers are more important than others, a statistician will often prefer to use a weighted average. A baseball player’s slugging average is a weighted average of all of the player’s hits. To compute a weighted average, each number is assigned a weight. For the slugging average, a home run is assigned a weight of 4, a triple 3, a double 2, a single 1, and an out 0 points. The slugging average is computed as follows. Multiply 4 (number of home runs). Multiply 3 (number of triples). Multiply 2 (number of doubles). Multiply 1 (number of singles). Multiply 0 (number of outs). Add these numbers and divide by the number of at-bats. For example, a player who has 80 at-bats and got 20 hits would have a batting average of 20 / 80  .250 whether those hits were home runs or singles. Suppose those 20 hits included 4 home runs, 3 triples, 8 doubles, and 5 singles. To compute the player’s slugging average, first compute 4(4)  3(3)  8(2)  1(5)  46 and then divide by the number of at-bats to get SA  46 / 80  .575. Another player who had 15 singles and 5 doubles in 80 times at bat would have the same batting average but a slugging average of only [2(5)  1(15)] / 80  .3125.

Threshold Statistics

A player’s batting average is found by dividing the total number of hits by the total times at bat. A pitcher’s earned run average is the ratio of earned runs allowed to innings pitched, scaled to the level of a game.

56

In many situations there is no need for any computation. Often there is an interest in knowing whether a player has reached some mark of excellence. These are called threshold statistics. For example, pitchers who have pitched a perfect game (no one gets on base) or bowlers who have scored 300 (a perfect score) have reached a threshold of excellence. Other examples of threshold statistics are pitchers who have pitched a no-hitter, pitchers who have attained 3,000 strikeouts in their careers, and batters who have hit “for the cycle” (a single, double, triple, and home run in the same game).

Algebraic Formula Many sports enthusiasts and coaches have come up with other, sometimes quite complex measures of individual and team performance. These are often expressed as algebraic formulas that combine other more elementary statistics. A simple example from hockey, for example, is a player’s /

Sports Data

TOTAL TURNOVERS GAINED IN THE BIG TEN 2001 Gained Team

Games Played Fumbles

Indiana Ohio State Wisconsin Michigan Northwestern Penn State Illinois Minnesota Purdue Michigan State Iowa

11 12 13 12 12 12 11 12 12 11 12

31 10 6 12 13 7 8 13 9 6 8

Lost

Interceptions Total Fumbles Interceptions Total Margin 4 18 20 14 12 17 12 6 10 12 9

35 28 26 26 25 24 20 19 19 18 17

13 6 11 10 6 11 12 9 7 9 9

15 12 9 5 7 9 10 8 12 16 11

28 18 20 15 13 20 22 17 19 25 20

7 10 6 11 12 4 –2 2 0 –7 –3

score. This is computed from the formula PM  (T)  (T) where T stands for the number of minutes the player was on the ice when his team scored a goal and T stands for the number of minutes he was on the ice when a goal was scored against his team. The difference in these numbers is the / score or PM. This gives a measure of the efficiency of different players in both offensive and defensive play. A similar computation is used in football to determine a team’s total turnovers (called the margin in the table.) It is computed by subtracting the number of times a team lost the ball through fumbles or interceptions from the number of the times the team gained the ball through fumbles or interceptions. A more complex algebraic computation is a baseball player’s adjusted production statistic. It is computed from the formula APRO  OBP / LOBP  SA / LSA  1. That is, a player’s On-Base Percentage is compared to the League’s On-Base Percentage and his Slugging Average is compared to the League’s Slugging Average. These two ratios are then added and 1 is subtracted.

Estimation The distance of a home run may be estimated using math. At Wrigley Field in Chicago, there is a person responsible for estimating the distance of each home run. When one occurs, about 10 seconds are needed to determine the distance and relay the information to the scoreboard operator for display. The dimensions of Wrigley are 355 feet down the left-field line, 353 feet down the right-field line, and 400 feet to dead center. The power alleys are both marked at 368 feet. When a home run is hit and lands inside the park, these dimensions are used to estimate the distance. If the ball lands in the stands, 3 feet are added for each row of bleacher seats. For example, if a ball is hit into the left side of the field between the foul pole and left center, four rows into the bleachers, then the total distance is estimated as 362 feet (the midpoint of 355 and 368) plus 12 feet (4 rows multiplied by 3 feet per row), which equals 374 feet. Of course, it gets even harder to estimate the distance when a ball leaves the park! S E E A L S O Athletics, Technology in. Alan Lipp and Denise Prendergast Bibliography Gutman, Dan, and Tim McCarver. The Way Baseball Works. New York: Simon & Schuster, 1996.

57

Standardized Tests

1998 Sports Almanac. New York: Little, Brown and Company, 1998. Internet Resources

About Snowboarding. Salt Lake 2002: Official Site of the 2002 Olympic Winter Games. . “Statistical Formulas Page.” Baseball Almanac. 2001 .

Standardized Tests Standardized tests are administered in order to measure the aptitude or achievement of the people tested. A distribution of scores for all test takers allows individual test takers to see where their scores rank among others. Well-known examples of standardized tests include “IQ” (Intelligence Quota) tests, the PSAT (Preliminary Scholastic Achievement Test) and SAT (Scholastic Achievement Test) tests taken by high school students, the GRE (Graduate Requirements Examination) test taken by college students applying to graduate school, and the various admission tests required for business, law, and medical schools.

The “Normal” Curve

mean the arithmetic average of a set of data

standard deviation a measure of the average amount by which individual items of data might be expected to vary from the arithmetic mean of all data

The mathematics behind the distribution of scores on standardized tests comes from the fields of probability theory and mathematical statistics. A cornerstone of this mathematical theory is the “Central Limit Theorem,” which states that for large samples of observations (or scores in the case of standardized tests), the distribution of the observations will follow the bellshaped normal probability curve illustrated below. This means that most of the observations will cluster symmetrically around the mean or average value of all the observations, with fewer observations farther away from the mean value. One measure of the spread or dispersion of the observations is called the standard deviation. According to statistical theory illustrated above, about 68 percent of all observations will lie within plus or minus one standard deviation of the mean; 95 percent will lie within plus or minus two standard deviations of the mean (see graph below); and 99.7 percent will lie within plus or minus three standard deviations of the mean. Standardized test scores are examples of observations that have this property. 95% 68%

–2

–1

+1

Mean Median Mode

58

+2

Standardized Tests

Consider, for example, a standardized test for which the mean score is 500 and the standard deviation is 100. This means that about 68 percent of all test takers will have scores that fall between 400 and 600; 95 percent will have scores between 300 and 700; and virtually all of the scores will fall between 200 and 800. In fact, many standardized tests, including the PSAT and SAT, have just such a scale on which 200 and 800 are the minimum and maximum scores, respectively, that will be given.

Scaled Scores The “standardized” in standardized tests means that similar scores must represent the same level of performance from year to year. Statisticians and test creators work together to ensure that, for example, if a student scores 650 on one version of the SAT as a junior and 700 on a different version as a senior, that this truly represents a gain in achievement rather than one version of the test being more difficult than the other. By “embedding” some questions that are identical in all versions of a test and analyzing the performance of each group on those common questions, test creators can ensure a level of standardization. If one group scores significantly lower on the common questions, this is interpreted to mean that the lower scoring group is not as strong as the higher scoring group. If group A scores higher than group B on questions identical to both their tests but then scores the same or lower than group B on the complete test, it would be assumed that the test given to group A was more difficult than that given to group B. Statisticians can develop a mathematical formula that will correct for such a variance in the difficulty of tests. Such a formula would be applied to the “raw” scores of the test takers in order to obtain “scaled” scores for both groups. These scaled scores could then be compared. A scaled score of 580 on version A means the same thing as a scaled score of 580 on version B, even though the raw scores may be different. In this sense the scores are said to have been “standardized.”

Statistical Scores A second meaning of “standardized” is more subtle, more mathematically involved, and not well understood by the general public. This meaning has to do with the bell-shaped normal probability curve mentioned at the beginning of this article. Theoretically, there are an infinite number of normal curves—one for each different set of observations that might be made. Mathematicians would say that there is an entire “family” of normal curves, and, the members of the normal curve family share similarities as well as differences. All bell-shaped curves are high in the middle and slope down to long “tails” to the right and left. Although different types of observations will have different mean values, those mean values will always occur at the middle of the distributions. They may also have different standard deviations as discussed earlier, but the percentage of values lying between plus or minus one of those standard deviations will still be about 68 percent, the percentage of values lying between plus or minus two standard deviations will still be about 95 percent, and so on. 59

Statistical Analysis

In order to make the analysis of normal distributions simpler, statisticians have agreed upon one particular normal curve that will represent all the rest. This special normal curve has a mean of 0 and a standard deviation of 1 and is called the “standard normal curve.” A “standardized” test result, therefore, is one based on the use of a standard normal curve as its reference. The advantage of having the standard normal curve represent all the other normal curves is that statisticians can then construct a single table of probabilities that can be applied to all normal distributions. This can be done by “mapping” those distributions onto the standard normal curve and making use of its probability table. The term “mapping” in mathematics refers to the transformation of one set of values to a different set of values. To illustrate, consider the test with a mean of 500 and a standard deviation of 100. The mean of this set of scores lies 500 units to the right of the standard normal distribution’s mean of 0. So to “map” the mean of the test scores onto the standard normal mean, 500 is subtracted from all the test scores. Now there is a new distribution with the correct mean but the wrong standard deviation. To correct this, all of the scores in the new distribution are divided by 100   1, which is the standard deviation of the standard normal 100, since  100 distribution. The two distributions are now identical. In mathematical terms the test scores have been “mapped” onto the standard normal values. This mapping is composed of two transformations: a translation of 500 to the left and a scale change of 1/100. This composition can be represented (x  500) , where x is any test score. by  100 Building on this example, suppose one wants to know the percentage of (650  500)   1.5. Then test takers who scored 650 or above. First, compute  100 go to a standard normal table, look up a standard score of 1.5, and see that about 6.88 percent of standard normal scores are at 1.5 or above. This means that about 6.88 percent of the test scores are 650 or higher. This procedure may be used with any normally distributed data set for which the mean and standard deviation are known. S E E A L S O Central Tendency, Measures of; Mapping, Mathematical; Statistical Analysis; Transformations. Stephen Robinson Bibliography Angoff, William. “Calibrating College Board Scores.” In Statistics: A Guide to the Unknown, ed. Judith Tanur, Frederick Mosteller, William H. Kruskal, Richard F. Link, Richard S. Pieters, and Gerald R. Rising. San Francisco: Holden-Day, Inc., 1978. Blakeslee, David W., and William Chin. Introductory Statistics and Probability. Boston: Houghton Mifflin Company, 1975. Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.

Statistical Analysis You may have heard the saying “You can prove anything with statistics,” which implies that statistical analysis cannot to be trusted, that the conclu60

Statistical Analysis

sions that can be drawn from it are so vague and ambiguous that they are meaningless. Yet the opposite is also true. Statistical analysis can be reliable and the results of statistical analysis can be trusted if the proper conditions are established.

What Is Statistical Analysis? Statistical analysis uses inductive reasoning and the mathematical principles of probability to assess the reliability of a particular experimental test. Mathematical techniques have been devised to allow measurement of the reliability (or fallibility) of the estimate to be determined from the data (the sample, or “N”) without reference to the original population. This is important because researchers typically do not have access to information about the whole population, and a sample—a subset of the population— is used. Statistical analysis uses a sample drawn from a larger population to make inferences about the larger population. A population is a well-defined group of individuals or observations of any size having a unique quality or characteristic. Examples of populations include first-grade teachers in Texas, jewelers in New York, nurses at a hospital, high school principals, Democrats, and people who go to dentists. Corn plants in a particular field and automobiles produced by a plant on Monday are also populations. A sample is the group of individuals or items selected from a particular population. A random sample is taken in such a way that every individual in the population has an equal opportunity to be chosen. A random sample is also known as an unbiased sample.

inductive reasoning drawing general conclusions based on specific instances or observations; for example, a theory might be based on the outcomes of several experiments

inferences the act or process of deriving a conclusion from given facts or premises

Most mail surveys, mall surveys, political telephone polls, and other similar data gathering techniques generally do not meet the proper conditions for a random, unbiased sample, so their results cannot to be trusted. These are “self-selected” samples because the subjects choose whether to participate in the survey and the subjects may be picked based on the ease of their availability (for example, whoever answers the phone and agrees to the interview).

Selecting a Random Sampling The most important criterion for trustworthy statistical analysis is correctly choosing a random sample. For example, suppose you have a bucket full of 10,000 marbles and you want to know how many of the marbles are red. You could count all of the marbles, but that would take a long time. So, you stir the marbles thoroughly and, without looking, pull out 100 marbles. Now you count the red marbles in your random sample. There are 10. Thus you could conclude that approximately 10 percent of the original marbles are red. This is a trustworthy conclusion, but it is not likely to be exactly right. You could improve your accuracy by counting a larger sample; say 1,000 marbles. Of course if you counted all the marbles, you would know the exact percentage, but the point is to pick a sample that is large enough (for example, 100 or 1,000) that gives you an answer accurate enough for your purposes. Suppose the 100 marbles you pulled out of the bucket were all red. Would this be proof that all 10,000 marbles in the bucket were red? In science, statistical analysis is used to test a hypothesis. In the example we are testing, the hypothesis would be “all the marbles in the bucket are red.”

hypothesis a proposition that is assumed to be true for the purpose of proving other propositions

61

Statistical Analysis

Statistical inference makes it possible for us to state, given a sample size (100) and a population size (10,000), how often false hypotheses will be accepted and how often true hypotheses are rejected. Statistical analysis cannot conclusively tell us whether a hypothesis is true; only the examination of the entire population can do that. So “statistical proof” is a statement of just how often we will get “the right answer.”

Using Basic Statistical Concepts Statistics is applicable to all fields of human endeavor, from economics to education, politics to psychology. Procedures worked out for one field are generally applicable to the other fields. Some statistical procedures are used more often in some fields than in others. Example 1. Suppose the Wearemout Pants Company wants to know the average height of adult American men, an important piece of information for a clothing manufacturer producing pants. The population is all men over the age of 25 who live in the United States. It is logistically impossible to measure the height of every man who lives in the United States, so a random sample of around 1,000 men is chosen. If the sample is correctly chosen, all ethnic groups, geographic regions, and socioeconomic classes will be adequately represented. The individual heights of these 1,000 men are then measured. An average height is calculated by dividing the sum of these individual heights by the total number of subjects (N  1,000). By doing so, imagine that we calculate an average height is 1.95 meters (m) for this sample of adult males in the United States. If a representative sample was selected, then this figure can be generalized to the larger population. The random sample of 1,000 men probably included some very short men and some very tall men. The difference between the shortest and the tallest is known as the “range” of the data. Range is one measure of the “dispersion” of a group of observations. A better measure of dispersion is the “standard deviation.” The standard deviation is the square root of the sum of the squares of the differences divided by one less than the number of observations.

S

MEAN, MEDIAN, AND MODE The average in the clothing example is known as the “arithmetic mean,” which is one of the measures of central tendency. The other two measures of central tendency are the “median” and the “mode.” The median is the number that falls in the mid-point of an ordered data set, while the mode is the most frequently occurring value.

62



n

 (x i1

i

 x)

 n1

In this equation, xi is an observed value and x is the arithmetic mean. In our example, if a smaller height interval is used (1.10 m, 1.11 m, 1.12 m, 1.13 m, and so on) and the number of men in each height interval plotted as a function of height a smooth curve can be drawn which would have a characteristic shape, known as a “bell” curve or “normal frequency distribution.” A normal frequency distribution can be stated mathematically as (x  ) 1 f (x)   e 2   2 2

2

The value of sigma ( ) is a measure of how “wide” the distribution is. Not all samples will have a normal distribution, but many do, and these distributions are of special interest. The following figure shows three normal probability distributions. Because there is no skew, the mean, median, and mode are the same. The mean

Statistical Analysis

of curve (a) is less than the mean of curve (b), which in turn is less than the mean of (c). Yet the standard deviation, or spread, of (c) is least, whereas that of (a) is greatest. This is just one illustration of how the parameters of distributions can vary. f(x) (c) (b) (a)

x

Example 2. One of the most common uses of statistical analysis is in determining whether a certain treatment is efficacious. For example, medical researchers may want to know if a particular medicine is effective at treating the type of pain resulting from extraction of third molars (known as “wisdom” teeth). Two random samples of approximately equal size would be selected. One group would receive the pain medication while the other group received a “placebo,” a pill that looked identical but contained only inactive ingredients. The study would need to be a “double-blind” experiment, which is designed so that neither the recipients nor the persons dispensing the pills knew which was which. Researchers would know who had received the active medicines only after all the results were collected. Example 3. Suppose a student, as part of a science fair project, wishes to determine if a particular chemical compound (Chemical X) can accelerate the growth of tomato plants. In this sort of experiment design, the hypothesis is usually stated as a null hypothesis: “Chemical X has no effect on the growth rate of tomato plants.” In this case, the student would reject the null hypothesis if she found a significant difference. It may seem odd, but that is the way most of the statistical tests are set up. In this case, the independent variable is the presence of the chemical and the dependent variable is the height of the plant. The next step is experiment design. The student decides to use height as the single measure of plant growth. She purchases 100 individual tomato plants of the same variety and randomly assigns them to 2 groups of 50 each. Thus the population is all tomato plants of this particular type and the sample is the 100 plants she has purchased. They are planted in identical containers, using the same kind of potting soil and placed so they will receive the same amount of light and air at the same temperature. In other words, the experimenter tries to “control” all of the variables, except the single variable of interest. One group will be watered with water containing a small amount of the chemical while the other will receive plain water. To make the experiment double-blind, she has another student prepare the watering cans each day, so that she will not know until after the experiment is complete which group was receiving the treatment. After 6 weeks, she plans to measure the height of the plants.

null hypothesis the theory that there is no validity to the specific claim that two variations of the same thing can be distinguished by a specific procedure independent variable in the equation y  f(x), the input variable is x (or the independent variable) dependent variable in the equation y  f(x), if the function f assigns a single value of y to each value of x, then y is the output variable (or the dependent variable)

63

Statistical Analysis

The next step is data collection. The student measures the height of each plant and records the results in data tables. She determines that the control group (which received plain water) had an average (arithmetic mean) height of 1.3 m (meters), while the treatment group had an average height of 1.4 m. Now the student must somehow determine if this small difference was significant or if the amount of variation measured would be expected under no treatment conditions. In other words, what is the probability that 2 groups of 50 tomato plants each, grown under identical conditions would show a height difference of 0.1 m after 6 weeks of growth? If this probability is less than or equal to a certain predetermined value, then the null hypothesis is rejected. Two commonly used values of probability are 0.05 or 0.01. However, these are completely arbitrary choices determined mostly by the widespread use of previously calculated tables for each value. Modern computer analysis techniques allow the selection of any value of probability. The simplest test of significance is to determine how “wide” the distribution of heights is for each group. If there is a wide variance ( ) in heights (say,  25), then small differences in mean are not likely to be significant. On the other hand, if the dispersion is narrow (for example, if all the plants in each group were close to the same height, so that  0.1) then the difference would probably be significant.

chi-square test a generalization of a test for significant differences between a binomial population and a multinomial population nominal scales a method for sorting objects into categories according to some distinguishing characteristic, then attaching a label to each category

There are several different tests the student could use. Selecting the right test is often a tricky problem. In this case, the student can reject several tests outright. For example, the chi-square test is suitable for nominal scales (yes or no answers are one example of nominal scales), so it does not work here. The F-test measures variability or dispersion within a single sample. It too is not suitable for comparing two samples. Other statistical tests can also be rejected as inappropriate for various reasons. In this case, since the student is interested in comparing means, the best choice is a t test. The t-test compares two means using this formula: M1  M2  ( 1  2) t   SM1M2 In this case, the null hypothesis assumes that 1  2  0 (no difference in the sample groups), so that we can say: M1  M2 t   SM1M2 The quantity SM1M2 is known as the standard error of the mean difference. 2  SM22 The stanWhen the sample sizes are the same, SM1M2  S M1 . dard error of the mean difference is the square root of the sums of the squares of the standard errors of the means for each group. The standard S error of the mean for each group is easily calculated from Sm  . N is  N the sample size, 50, and the student can calculate the standard deviation by the formula for standard deviation given above. The final experimental step is to determine sensitivity. Generally speaking, the larger the sample, the more sensitive the experiment is. The choice of 50 tomato plants for each group implies a high degree of sensitivity.

64

Step Functions

Students, teachers, psychologists, economists, politicians, educational researchers, medical researchers, biologists, coaches, doctors and many others use statistics and statistical analysis every day to help them make decisions. To make trustworthy and valid decisions based on statistical information, it is necessary to: be sure the sample is representative of the population; understand the assumptions of the procedure and use the correct procedure; use the best measurements available; keep clear what is being looked for; and to avoid statements of causal relations if they are not justified. S E E A L S O Central Tendency, Measures of; Data Collection and Interpretation; Graphs; Mass Media, Mathematics and the. Elliot Richmond

causal relation a response to an input that does not depend on values of the input at later times

Bibliography: Huff, Darrell. How to Lie With Statistics. New York: W. W. Norton & Company, 1954. Kirk, Roger E. Experimental Design, Procedures for the Behavioral Sciences. Monterrey, CA: Brooks/Cole Publishing Company, 1982. Paulos, John Allen. Innumeracy: Mathematical Illiteracy and Its Consequences. New York: Hill & Wang, 1988. Tufte, Edward R. The Visual Display of Quantitative Information. Cheshire, CT: Graphics Press, 1983.

Step Functions In mathematics, functions describe relationships between two or more quantities. A step function is a special type of relationship in which one quantity increases in steps in relation to another quantity. For example, postage cost increases as the weight of a letter or package increases. In the year 2001 a letter weighing between 0 and 1 ounce required a 34-cent stamp. When the weight of the letter increased above 1 ounce and up to 2 ounces, the postage amount increased to 55 cents, a step increase. A graph of a step function f gives a visual picture to the term “step function.” A step function exhibits a graph with steps similar to a ladder. The domain of a step function f is divided or partitioned into a number of intervals. In each interval, a step function f(x) is constant. So within an interval, the value of the step function does not change. In different intervals, however, a step function f can take different constant values. One common type of step function is the greatest-integer function. The domain of the greatest-integer function f is the real number set that is divided into intervals of the form . . .[2, 1), [1, 0), [0, 1), [1, 2), [2, 3),. . . The intervals of the greatest-integer function are of the form [k, k  1), where k is an integer. It is constant on every interval and equal to k. f(x)  0 on [0, 1), or 0x1 f(x)  1 on [1, 2), or 1x2 f(x)  2 on [2, 3), or 2x3 For instance, in the interval [2, 3), or 2x3, the value of the function is 2. By definition of the function, on each interval, the function equals the greatest integer less than or equal to all the numbers in the interval. Zero, 65

Stock Market

1, and 2 are all integers that are less than or equal to the numbers in the interval [2, 3), but the greatest integer is 2. Therefore, in general, when the interval is of the form [k, k  1), where k is an integer, the function value of greatest-integer function is k. So in the interval [5, 6), the function value is 5. The graph of the greatest integer function is similar to the graph shown below.

y 3 2 1 –3

–2

–1

1

2

3

x

–1 –2 –3

There are many examples where step functions apply to real-world situations. The price of items that are sold by weight can be presented as a cost per ounce (or pound) graphed against the weight. The average selling price of a corporation’s stock can also be presented as a step function with a time period for the domain. Frederick Landwehr Bibliography Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.

Stock Market The term “stock market” refers to a large number of markets worldwide in which shares of stock are bought and sold. In 1999 there were 142 such markets in major cities around the world, where hundreds of millions of shares of stock changed hands daily. Some stock trading takes place in traditional markets, where representatives of buyers and sellers meet on a trading floor to bid on and sell stocks. The stocks traded in these markets are sold to the highest bidder by auction. Increasingly, however, stocks are traded in markets that have no physical trading floor, but are bought or sold electronically. Electronic stock trading involves networks of computers and/or telephones used to link brokers. Stock price is determined by negotiation between representatives of buyers and sellers, rather than by auction. Hence, these markets are called “negotiated markets.” All the stock exchanges around the world, including auction markets at centralized locations and electronic negotiated markets, are collectively called the stock market. 66

Stock Market

Shares of stock are the basic elements bought and sold in the stock market. In order to raise working capital, a company may offer for sale to the public “shares” or “portions of ownership” in the company. These shares are represented by a certificate stating that the individual owns a given number of shares. Revenue from this initial public offering of stock goes to the issuing corporation. Investors buy stock hoping that its value will increase. If it does, the shareholder may sell the stock at a profit. If the stock price declines, selling the stock results in a loss for the stockholder. The issuing company does not receive any income when stocks are traded after the initial public offering.

The Price of Stocks A stock is worth whatever a buyer will pay for it. Because stock market prices reflect the varying opinions and perceptions of millions of traders each day, it is sometimes difficult to pinpoint why a certain stock rises or falls in price. A wide range of factors affects stock prices. These may include the company’s actual or perceived performance, overall economic climate and market conditions, philosophy and image projected by the firm’s corporate management, political action around the world, the death of a powerful world leader, a change in interest rates, and many other influences. Investors bid on a stock if they believe it is a good value for the current price. If there are more buyers than sellers, the price of the stock rises. When investors collectively stop buying a given stock or begin selling the stock from their portfolios, the stock price falls.

Understanding Stock Tables Perceptions of a stock’s value are further shaped by information that is available in the financial press, at financial web sites, and from brokers either online or in person. Most major newspapers print this information in stock tables. Some stock charts are printed daily; others contain weekly or quarterly summaries of trading activity. Because each stock table reports data from only one stock exchange, many sources print more than one table. Although the format varies, most stock tables contain, at a minimum, certain important figures. 52 Week High Low 45 1/4 21 66 3/4 20 3/8

Stock Goodrch Goodyr

Div 1.10 1.20

Yld 25 20.1

PE 18 16

Vol 1953 3992

High 28 1/4 24 1/2

Low Close 27 1/16 27 1/2 23 24 1/8

Change –1 3/4 +11/16

The high and low prices for the stock during the last 52 weeks, shown in the first two columns of the table above, are indicators of the volatility of the stock. Volatility is a measure of risk. The more a stock’s price varies over a short time, the more potential there is to make or lose money trading that stock. Although the prices of all stocks go up and down with market fluctuations, some stocks tend to rise more quickly and fall more sharply than the overall market. Other stocks generally move up and down more slowly than the market as a whole. 67

Stock Market

✶Stock exchanges are increasingly moving to decimal notation.

In the United States, stock prices are traditionally reported in eighths, sixteenths, or even thirty-seconds of a dollar.✶ This practice dates from the time when early settlers cut European coins into pieces to use as currency. 1 Goodrich’s 52-week high of 454 in the above example means the highest trading price for this stock in the last year was $45.25. The third column in the table holds an abbreviated name for the stock’s issuing corporation. Some financial papers also print the ticker symbol, a very short code name for the company. Brokers often use ticker symbols when ordering or selling stock. Ticker symbols are abbreviated names; for example, “GM” is used for General Motors or “INTC” for Intel, depending on which stock market exchange is being consulted.

The New York Stock Exchange trading floor became jammed with traders as stocks plunged in the biggest one-day sellout in history on October 19, 1987.

68

The total cash dividend per share paid by the company in the last year is found in the fourth column of the table. The company’s board of directors determines the size and frequency of dividends. Traditionally, only older and more mature companies pay dividends, whereas smaller, new companies tend to reinvest profits to finance continued growth. Some dividends are paid in the form of additional stock given to shareholders rather than in cash. Stock tables do not report the value of these non-cash dividends. The dividend-to-price ratio, or yield, is shown in the table’s fifth column. This is the annual cash dividend per share divided by the current price of the stock and is expressed as a percentage. In this example, Goodrich’s closing stock price of $27.50 divided into the annual dividend of $1.10 shows

Stock Market

a current yield of 4 percent. Yield can be used to compare income from a particular stock with income from other investments. It does not reflect the gain or loss realized if the stock is sold. The price-to-earnings ratio (P/E ratio) is the price of the stock divided by the company’s earnings per share for the last twelve months. A P/E ratio of 18 means that the closing price of a given stock was eighteen times the amount per share that the firm earned in the preceding twelve months. Volume in a stock table is the total number of shares traded in the reported period. Daily volumes are usually reported in hundreds. This chart shows a trading volume of 195,300 shares of Goodrich stock. Unusually high volumes can indicate either that investors are enthusiastic about a stock and likely to bid it up, or that wary stockholders are selling their holdings. The next three columns show the high, low, and closing price for a particular trading period, such as a day or a week. The final column reports the net change in price since the last reporting period. In this daily chart, the 3 closing price for Goodrich stock was 14 ($1.75) lower than the previous day’s 11 close, whereas Goodyear closed 16 of a dollar ($0.6875) higher.

Stock Exchanges The New York Stock Exchange, sometimes called the “Big Board,” is the largest and oldest auction stock market in the United States. Organized in 1792, it is a traditional market, where representatives of buyers and sellers meet face to face. More than 3,000 of the largest and most prestigious companies in the United States had stocks listed for sale on the NYSE in early 2000, and it is not unusual for a billion shares of stock to change hands daily, making it by far the busiest traditional market in the world.

UNPRECEDENTED CLOSURES OF FINANCIAL STOCK MARKETS On Tuesday, September 11, 2001, terrorist assaults demolished the World Trade Center in New York City and damaged the Pentagon in Washington, D.C. These attacks on the structure of the U.S. financial and governmental systems led to the closing of the U.S. stock markets, including the New York Stock Exchange (NYSE), the National Association of Securities Dealers Automated Quotations (NASDAQ), and the American Stock Exchange (AMEX). These financial stock exchanges reopened the following Monday after remaining shut down for the longest period of time since 1929.

Requirements for a corporation to list its stock on the NYSE are stringent. They include having at least 1.1 million publicly traded shares of stock, with an initial offering value of at least 40 million dollars. The American Stock Exchange (AMEX) in New York and several regional stock exchanges are traditional markets with less stringent listing requirements. The NASDAQ is an advanced telecommunications and computer-based stock market that has no centralized location and no single trading floor. However, more shares trade daily on the NASDAQ than on any other U.S. exchange. The NASDAQ is a negotiated market, rather than an auction market. Nearly 5,000 companies had stocks listed on the NASDAQ in 1999. Companies listing on the NASDAQ range from relatively small firms to large corporations such as Microsoft, Intel, and Dell. Stocks not listed on any of the exchanges are traded over the counter (OTC). They are bought and sold online or through brokers. These are generally stocks from smaller or newer companies than those listed on the exchanges. Stocks in about 11,000 companies are traded in the OTC market.

Stock Market Indexes A large number of indexes, or summaries of stock market activity, track and report stock market fluctuations. The most widely known and frequently quoted index is the Dow Jones Industrial Average (DJIA). The DJIA is an adjusted average of the change in price of thirty major industrial stocks listed on the NYSE. 69

Stone Mason

Another widely used stock market index is the Standard and Poor’s 500 (S&P 500). It measures performance of stocks in 400 industrial, 20 transportation, 40 utility, and 40 financial corporations listed on the NYSE, AMEX, and NASDAQ. Because of this, many analysts feel that the S&P 500 gives a much broader picture of the stock market than does the Dow Jones. Indexes for the NASDAQ and AMEX track the performance of stocks listed on those exchanges. Lesser known indexes monitor specific sectors of the market (transportation or utilities, for example). The Wall Street Journal publishes twenty-nine different stock market indexes each day. S E E A L S O Economic Indicators. John R. Wood and Alice Romberg Wood Bibliography Case, Samuel. The First Book of Investing. Rocklin, CA: Prima Publishing, 1999. Dowd, Merle E. Wall Street Made Simple. New York: Doubleday, 1992. Morris, Kenneth M., and Virginia B. Morris. Guide to Understanding Money & Investing. New York: Lightbulb Press, 1999.

Stone Mason algebra the branch of mathematics that deals with variables or unknowns representing the arithmetic numbers geometry the branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, planes, and solids

Stone masons rely on mathematics and specially designed tools to calculate and measure exact dimensions.

70

Masonry is the building material used for the construction of brick, concrete, and rock structures. A stone mason constructs, erects, installs, and repairs structures using brick, concrete block insulation, and other masonry units. Walls, arches, floors, and chimneys are some of the structures that a stone mason builds or repairs. Part of the responsibility of a stone mason is to cut and trim masonry materials to certain specifications. A stone mason’s work tools include a framing square for setting project outlines, levels for setting forms, a line level for making layouts and setting slope, and a tape measure. Masons need a working knowledge of ratios for mixing concrete and mortar. A foundation in algebra and geometry is helpful for laying out a

Sun

building site, and for building the frames that will contain the object being built. An ability to calculate slope, volume, and area is important for a stone mason. A basic knowledge of trade math is required for most stone mason training programs and apprenticeships. Marilyn Schwader Bibliography Masonry Essentials. Minnetonka, MN: Cowles Creative Publishing, Inc., 1997.

Sun The Sun is located in the “suburbs” of the Milky Way galaxy, around 30,000 light-years from the center and within one of its spiral arms. It revolves around the galaxy’s center at an average speed of 155 miles per second, taking 225 million years to complete each circuit. Although the Sun is just one star among an estimated 400 billion stars in the Milky Way galaxy, it is the closest star to Earth. More importantly, it is the only star that provides Earth with enough light, heat, and energy to sustain life. Also, the strong gravitational pull of the Sun holds the Earth in orbit within its solar system. Many people forget that the Sun is a star because it looks so big and different when compared to other stars and because the Sun appears in the sky during the day, whereas other stars only appear at night. Because the Sun is so close to the Earth, its luminosity (brightness) overwhelms the brightness of other stars, drowning out their daytime light.✶

Size and Distance

✶Because of its relative nearness, the Sun appears about 10 billion times brighter than the next brightest star.

Archimedes (287 B.C.E.–212 B.C.E.) placed the Sun at the center of the solar system. Observations of Galileo (1564–1642) supported this heliocentric theory, nullifying the ancient belief that the Earth was the center of the solar system. Before the Sun’s distance was known, Aristarchus (310 B.C.E.–230 B.C.E.) knew that the Moon shines by reflected sunlight. Aristarchus decided that if he measured the angle between the Moon and the Sun when the Moon is half-illuminated, he could then compute the ratio of their distances from the Earth. Aristarchus estimated that this angle was 87, resulting in the ratio of their distances at sin 3. Before the invention of trigonometry, Aristarchus used a similar method 1 1 to calculate the inequality 18  sin 3  20 , reasoning that the Sun was 18 to 20 times farther away from the Earth than the Moon. As calculations were refined, the angle between Moon and Sun was shown to be 89°50 . Astronomers learned that the Sun is actually 400 times farther away from the Earth than the Moon. Scientists now know that the Earth-Sun distance is 93 million miles. This distance was discovered when radar signals were bounced off Venus’s surface to determine the Earth-to-Venus distance. At a speed of 500 mph, a journey from the Earth to the Sun would take 21 years. Ancient civilizations thought that the Sun and Moon were the same size. Yet the Sun’s diameter is really 864,338 miles across, which is more than 71

Sun

400 times the Moon’s diameter and about 109 times the Earth’s diameter. The Sun has a volume 1.3 million times the Earth’s volume. Thus, a million Earths could be packed within the Sun. Although enormous compared to Earth, the Sun is an average-sized star. German mathematician Johannes Kepler (1531–1630) devised his laws of planetary motion while studying the motion of Mars around the Sun. The Sun’s mass can be calculated from his third law with the equation T 2 ()r3 , where T is the period of Earth’s revolution (3.15  107 seconds),  (GMs) r is the radius of Earth’s revolution (1.51011m), and G is the planetary constant (6.67  1011 Newton’s m2/kg). To find Ms (Sun’s mass) insert these values into the equation and solve Ms  ()(1.5  1011m)3/(6.67  1011Nm2/kg)(3.15  107s)2  2.0  1030 kilograms. Therefore, the Sun’s mass is about 300,000 times the Earth’s mass. Yet with respect to other stars, the Sun’s mass is just average.

Time and Temperature The Sun’s rotation is similar to the Earth’s rotation (one rotation every day), but because the Sun is gaseous not all parts rotate at the same speed. Galileo first noticed that the Sun rotates when he observed sunspots moving across the disk. He found that a particular spot took 27 days to make a complete circuit. Later observations found that the Sun at its equator rotates in slightly over 24.5 days. At locations two-thirds of the distance above and below the equator, the rotation take nearly 31 days. In order to support its large mass, the Sun’s interior must possess extremely large pressures and temperatures. The force of gravity at the core’s surface is about 250 million times as great as Earth’s surface gravity. No solids or liquids exist under these conditions, so the Sun’s body primarily consists of the gases hydrogen (73 percent) and helium (25 percent).

photosphere the very bright portion of the Sun visible to the unaided eye; the portion around the Sun that marks the boundary between the dense interior gases and the more diffuse cooler gases in the outer realm of the Sun chromosphere the transparent layer of gas that resides above the photosphere in the atmosphere of the Sun corona the upper, very rarefied atmosphere of the Sun that becomes visible around the darkened Sun during a total solar eclipse

72

Within the Sun’s core, nuclear fusion reactions release huge amounts of energy. About 5 billion kilograms of hydrogen convert to helium, releasing energy each second. The core temperature is about 15 million kelvin, with a density of 160 grams per cubic centimeter. Based on mathematical calculations, the solar core is approximately the size of Jupiter, or approximately 75,000 to 100,000 miles in diameter. The amount of hydrogen within the Sun’s core should sustain fusion for another 7 billion years. For now, the Sun is bathing the Earth with just the right amount of heat. High-energy gamma rays (the particles created by fusion reactions) travel outward from the core and ultimately through the outer layers of the photosphere, chromosphere, and corona, losing most of their energy in the process. Along the way to the Sun’s surface, the temperature of the gamma rays has dropped from 15 million to 6,000 kelvin. Yet even at these temperatures, the Sun is in the middle range of stellar surface temperatures.

Measuring the Sun Information available about the Sun has increased with revolutionary scientific discoveries. Early telescopic observations allowed scientific study to begin, showing that the Sun is a dynamic, changing body. Later develop-

Superconductivity

ments within spectroscopy, and the discovery of elementary particles and nuclear fusion, allowed scientists to further understand its composition and the processes that fuel it.✶ Recent developments of artificial satellites and other spacecraft now allow scientists to continuously study the Sun. Among the advances that have significantly influenced solar physics are the spectroheliograph, which measures the spectrum of individual solar features; the coronagraph, which permits study of the solar corona without an eclipse; and the magnetograph, which measures magnetic-field strength over the solar surface. Space instruments have revolutionized solar study and continue to add to increased, but still incomplete, knowledge about the Sun. S E E A L S O Archimedes; Galileo Galilei; Solar System Geometry, History of; Solar System Geometry, Modern Understandings of. William Arthur Atkins (with Philip Edward Koth)

✶Astronomers now believe that the Sun is about 4.6 billion years old and will shine for another 7 billion years.

Bibliography Nicolson, Iain. The Sun. New York: Rand McNally and Company, 1982. Noyes, Robert W. The Sun: Our Star. Cambridge, MA: Harvard University Press, 1982. Washburn, Mark. In the Light of the Sun. New York: Harcourt Brace Jovanovich, 1981. Internet Resources

“The Sun.” NASA’s Goddard Space Flight Center. . “Today from Space: Sun and Solar System.” Science at NASA. .

Superconductivity In the age of technology, with smaller and smaller electronic components being used in a growing number of applications, one pertinent application of mathematics and physics is the study of superconductivity. All elements and compounds possess intrinsic physical properties including a melting and boiling point, malleability, and conductivity. Conductivity is the measure of a substance’s ability to allow an electrical current to pass from one end of a sample to the other. The measurement of resistivity (the inverse of conductivity) is called resistance and is measured in the unit ohms ().

intrinsic of itself; the essential nature of a thing; originating within the thing malleability the ability or capability of being shaped or formed

Ohm’s Law An important and useful formula in science is called Ohm’s Law, E  IR. E is voltage in volts, I is current in amperes, and R is resistance in ohms. To visualize this, imagine a wire conducting electrons along its length, like a river flowing on the surface of the wire. The resistance acts like rocks in the river, slowing the flow. Current is equivalent to the amount of water flowing (or the number of electrons per unit time), and voltage is equivalent to the slope of the river. As the water hits the rocks, it splashes up and away. This is equivalent to resistance generating heat in a circuit. All normal materials have some sort of resistance, thus all circuits generate heat to a greater or lesser degree. 73

Superconductivity

An interesting thing happens as the temperature of the wire changes. As the temperature elevates, the resistance increases; as the temperature lowers, the resistance decreases, but only to a point, then it goes back up again. In 1911 Kamerlingh Onnes discovered that mercury cooled to 4 kelvin (4 K) (that is 269.15° C , about 453° F) suddenly loses all resistance. He called this phenomenon superconductivity. Superconductivity is the ability of a substance to conduct electricity without resistance. If applied to Ohm’s Law, a voltage (E ) is applied, the current (I ) should continue on its own if the voltage is then removed and the resistance is zero. This makes sense in terms of Ohm’s Law, as E (0)  I (X)  R (0). When tested, it was found that this does indeed take place, with the current value (X) dropping over time as a function of the voltage applied, with this current being referred to as a Josephson current. superconduction the flow of electric current without resistance in certain metals and alloys while at temperatures near absolute zero

✶The zero point of the kelvin scale is the temperature at which all molecular motion theoretically stops, sometimes called absolute zero.

At the time Onnes discovered superconduction, it was believed that superconductivity was simply an intrinsic property of a given material.✶ However, Onnes soon learned that he could turn superconduction on and off with the application of a large current, or strong magnetic field. Other than a lack of resistance, it was believed for many years that superconducting materials possessed the same properties as their normal counterparts. Then in 1933, it was discovered that superconducting materials are highly diamagnetic (that is, highly repelled by and exerting a great influence on magnetic fields), even if their normal counterparts were not. This led one of the discoverers, W. Meissner, to make scientific predictions regarding the electromagnetic properties of superconductors and have his name assigned to the effect, the Meissner effect. Meissner’s predictions were confirmed in 1939, paving the way for further discoveries. In 1950 it was demonstrated for the first time that the movement of electrons in a superconductor must take atomic vibrational effects into account. Finally in 1957 a fundamental theory presented by physicists J. Bardeen, Leon Cooper, and J. R. Schrieffer, called BCS theory, allowed predictions of possible superconducting materials, and the behavior of these materials.

BCS Theory BCS theory explains superconductivity in a manner similar to the river of electrons example. When the material becomes superconducting, the electrons become grouped into pairs called Cooper pairs. These pairs dance around the rocks (resistance), like two people holding hands around a pole. This symmetry of movement allows the electrons to move without resistance. The first superconductors were experimental rarities, for research only. During the first 75 years of research, the temperature at which materials could be made to superconduct did not rise very much. Before 1986 the highest temperature superconductor worked at a temperature of 23 K. Karl Muller and Johannes Bednorz found a material that had a transition temperature (the temperature where a material becomes superconducting) of nearly 30 K, in 1986. Their research and discovery allowed even higher temperature superconductors to be made of ceramics containing various ratios of, usually, barium or strontium, copper, and oxygen. These ceramics allowed superconductivity to be done at liquid nitrogen temperatures (77 K)— 74

Symbols

a much more obtainable temperature than 4.2 K for liquid helium. However, ceramics are difficult to produce, break easily, and do not readily lend themselves to mass production. The newest generation of superconductors are nearing 148° C (125 K), which is the high temperature record as of 1998. Superconductivity already touches the world, with its use in MRI (magnetic resonance imaging) magnets, chemical analytical tools such as NMR (nuclear magnetic resonance) spectroscopy, and unlimited electrical and electronic uses. If a high temperature superconductor could be mass produced cheaply, it would revolutionize the electronics industry. For example, one battery could be made to last years. In the future, people may look back at this basic research and compare it to the first discovery of fire. S E E A L S O Absolute Zero; Temperature, Measurement of. Brook E. Hall Bibliography Mendelssohn, K. The Quest for Absolute Zero: The Meaning of Low Temperature Physics, 2nd ed. Boca Raton, FL: CRC Press, 1977. Shachtman, T. Absolute Zero and the Conquest of Cold. New York: Houghton Mifflin Co., 1999.

Surveyor

See Cartographer.

Symbols Symbols are part of the language of mathematics. The use of symbols gives mathematics the power of generality: The ability to focus not only on a specific object, but also on a collection of objects. For instance, variables— symbols with the ability to vary in value used in place of numbers—make it possible to write general equations. Consider the equation 3  4  7. This equation is specific. Suppose we are interested in all pairs of numbers whose sum is equal to 7. Using variables, x and y, to denote the two numbers, we would write the general equation x  y  7. The variables, x and y, stand for many different pairs of numbers; x can be 5 and y can be 2 or x can be 9 and y can be 2. The only condition is that the sum of x and y is 7. Symbols also include special notation for mathematical operations, like  for addition and  for taking the square root. Some common symbols and their uses are explained in the accompanying table.

Basic Mathematical Symbols The Set. The foundation of mathematics rests on the concept of a set, which is a collection of objects that share a well-defined characteristic. The elements of a set are written in braces {}. An element x that belongs to a set A is written as x  A. An element y that does not belong to A is written as y ∉ A. {0} is a set that contains zero as its only element. This set has one element. {} is the empty set and does not contain any element. The empty set is also called a null set and sometimes written with the symbol ∅. Operations. Numbers are the building blocks of mathematics. They can be combined by four operations: addition and its inverse operation, 75

Symbols

MATHEMATICAL SYMBOLS General Arithmetic and Algebra Symbol

Meaning

Differential Calculus and Integral Calculus Symbol

Meaning

Trigonometry and Geometry Symbol

n

Meaning

Constants Symbol

Meaning

subtraction, and multiplication and its inverse operation, division. The additive inverse of a number b is b and the sum of a number and its additive inverse is always 0, that is, b  (b)  0. The additive inverse of 4 is 4. 1 Similarly, the multiplicative inverse of a nonzero number b is b1, or b, and the product of a number and its multiplicative inverse is always 1, that is, 1 1 bb1  b(b)  1. The multiplicative inverse of 7 is 7. Size. Numbers can be compared according to their magnitude, or size. Given any two numbers, say x and y, there are three possibilities: x is equal to y (x  y), or x is less than y (x  y), or x is greater than y (x  y). Two numbers, x and y, can also be compared by taking their ratio, x:y, or x/y. A number percent, say 5 percent denotes the ratio of 5 to 100 or 5/100. In some problems, it is not important to know if a given number, say x, is positive or negative. The only important thing is its magnitude and it is represented by its absolute value, |x|. For instance, |5|  5 and |5|  5. Exponents. If a number is multiplied by itself two or more times, it can be written more compactly using an exponent that appears as a superscript to the right of the number. For example, if 4 is multiplied by itself three times, then it can be written as 4  4  4, or 43, where 3 is the exponent. An exponent is also called a power. In general, xn is read as “x raised to the 1 1 power n.” Raising a number to powers that are fractional, like 2 or 3, is n called a radical, or root of the number. b  denotes the nth root of the number b; where n is called the index and b is called the radicand. For n  3  is 2, the index is omitted and b is the square root of b. For n  3, b called the cube root of b. 76

Symmetry

Functions. A fundamental idea in mathematics is that of a function that describes a relationship between two quantities. In symbols, it is written as y  f (x) and read as “y is the function of x,” which means that the value of y depends on x. A function is sometimes called a map, or mapping. Logarithm, y  logax, is a particular type of function. By definition, a logarithm is an exponent. If 10x  y, then log10 y  x. Algebra. In algebra, numbers are represented as both variables and constants. A variable is a number whose value can change. Variables are mostly denoted by lowercase, terminating letters of the alphabet like x, y, or z. A constant is a fixed number, denoted mostly by first few lower case letters a, b, and c, or k. Algebraic expressions called polynomials contain one or more 5 variable and/or constant terms. For example, 3x, 2x  2, and x2  3x  5 2 are all polynomials: 3x is a monomial (one term), 2x  2 is a binomial 2 (two terms), and x  3x  2 is a trinomial (three terms). A constant number that multiplies a variable in an equation, or polynomial, is called a coefficient. For example, 2 is a coefficient of 2x in the equation 2x  3  0. In ax2, a is the coefficient of x2. Equations. Solving different types of equations appears in many situations in mathematics. The simplest equation is linear, in which the highest power of the variable, x, is 1, for example, 3x  2  1. If the highest power of the variable is 2, then the equation is a quadratic equation. For example, x2  4 and 2x2  3x  1  0 are quadratic equations. Calculus. Calculus is a branch of mathematics that partly deals with rate y of changes. For a variable x, x denotes a small change in x. Further, x measures the change in variable y with respect to change in x. When x bey comes increasingly smaller and approaches 0, the rate of change x equals dy the instantaneous rate of change and is denoted by dx and is called the derivative of y with respect to x. Also, calculus deals with finding the area of a region under the curve, or graph, of a function f(x). If the region is bounded by lines x  a and x  b, then the area is denoted by the integral of the function f. In symbols, it is written as ab f(x)dx. S E E A L S O Algebra; Calculus; Functions and Equations; Number System, Real. Rafiq Ladhani Bibliography Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995. Dugopolski, Mark. Elementary Algebra, 2nd ed. Boston: Addison-Wesley Publishing Company, 1996. Dunham, William. The Mathematical Universe. New York: John Wiley & Sons Inc., 1994. Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 6th ed. New York: HarperCollins Publishers, 1990.

Symmetry Symmetry is a visual characteristic in the shape and design of an object. Take a starfish for an example of an object with symmetry. If the arms of the starfish are all exactly the same length and width, then the starfish could be folded in half and the two sides would exactly match each other. The 77

Symmetry

line on which the starfish was folded is called a line of symmetry. Any object that can be folded in half and each side matches the other is said to have line symmetry. Line Symmetry

Line of Symmetry

When the object is folded on the line of symmetry, the two sides match each other exactly

When an object has line symmetry, one half of the object is a reflection of the other half. Just imagine that the line of symmetry is a mirror. The actual object and its reflection in the “mirror” would exactly match each other. A human face, if vertically bisected into two halves (left and right), would reveal its bilateral symmetry: that is, the left half ideally matches the right half. It is possible for an object to have more than one line of symmetry. Imagine a square. A vertical line of symmetry could be drawn down the middle of the square and the left and right sides would be symmetrical. Also, a horizontal line could be drawn across the middle of the square and the top and bottom would be symmetrical. As shown in the figure below, a star or a starfish has five lines of symmetry.

Five lines of symmetry

Alternately, the ability of an object to match its original figure with less than a full turn about its center is called rotational or point symmetry. For example, if a starfish embedded in the sand on a beach is picked up, rotated less than 360, and then set back down exactly into its original imprint in the sand, the starfish would be said to have rotational symmetry.

78

Symmetry

Rotational Symmetry

Original

Rotated 72˚

Rotated 144˚

Many natural and manmade items have reflection and/or rotation symmetry. Some examples are pottery, weaving, and quilting designs; architectural features such as windows, doors, roofs, hinges, tile floors, brick walls, railings, fences, bridge supports, or arches; many kinds of flowers and leaves; honeycomb; the cross section of an apple or a grapefruit; snowflakes; an open umbrella; letters of the alphabet; kaleidoscope designs; a pinwheel, windmill, or ferris wheel; some national flags (but not the U.S. flag); a ladder; a baseball diamond; or a stop sign.

The design of this starfish demonstrates both line symmetry and rotational symmetry.

Mathematicians use symmetries to reduce the complexity of a mathematical problem. In other words, one can apply what is known about just one of multiple symmetric pieces to other symmetric pieces and thereby learn more about the whole object. S E E A L S O Transformations. Sonia Woodbury Bibliography Schattschneider, D. Visions of symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher. New York: W. H. Freeman and Company, 1990. Internet Resources

Search for Math: Symmetry. The Math Forum at Swarthmore College. .

79

Telescope There is much confusion and debate concerning the origin of the telescope. Many notable individuals appear to have simultaneously and independently discovered how to make a telescope during the last months of 1608 and the early part of 1609. Regardless of its origins, the invention of the telescope has led to great progress in the field of astronomy.

T

The Origin of the Telescope Contrary to popular belief, Galileo Galilei (1564–1642) did not invent the telescope, and he was probably not even the first person to use this instrument in astronomy. Instead, the latter honor may be attributed to Thomas Harriot (1560–1621). Harriot developed a map of the Moon several months before Galileo began observations. Nevertheless, Galileo distinguished himself in the field through his patience, dedication, insight, and skill. The actual inventor of the telescope may never be known with certainty. Its invention may have been by a fortuitous occurrence when some spectacle maker happened to look through two lenses at the same time. Several accounts report that Hans Lipperhey of Middelburg in the Netherlands had two lenses set up in his spectacle shop to allow visitors to look through them and see the steeple of a distant church. However, this story cannot be verified. It is known that the first telescopes were shown in the Netherlands. Records show that in October 1608, the national government of the Netherlands examined the patent application of Lipperhey and a separate application by Jacob Metius of Alkmaar. Their devices consisted of a convex and concave lens mounted in a tube. The combination of the two lenses magnified objects by 3 or 4 times. However, the government of the Netherlands considered the devices too easy to copy to justify awarding a patent. The government did vote a small award to Metius and employed Lipperhey to devise binocular versions of his telescope. Another citizen of Middelburg, Zacharias Janssen, had also made a telescope at about the same time but was out of town when Lipperhey and Matius made their applications.

convex curved outward, bulging concave hollowed out or curved inward

News of the invention of the telescope spread rapidly throughout Europe. Within a few months, simple telescopes, called “spyglasses,” could be purchased at spectacle-maker’s shops in Paris. By early 1609, four or five 81

Telescope

telescopes had made it to Italy. By August of 1609, Thomas Harriot had observed and mapped the Moon with a six-power telescope.

What Galileo Discovered Despite Harriot’s honor as the first telescopic astronomer, it was Galileo who made the telescope famous. At the time, Galileo was Professor of Mathematics at the University of Padua in Italy. Somehow, he learned of the new instrument that had been invented in Holland, although there is no evidence that he actually saw one of the telescopes known to be in Italy. Over the next several months in 1609 and 1610, Galileo made several progressively more powerful and optically superior telescopes using lenses he ground himself. Galileo used these instruments for a systematic study of the night sky. He saw mountains and craters on the Moon, discovered four satellites of Jupiter, viewed sunspots, observed and recorded the phases of Venus, and found that the Milky Way galaxy consisted of clouds of individual stars. Galileo summarized his discoveries in the book Sidereus Nuncius (The Starry Messenger) published in March of 1610. Others working at around the same time claimed to have made similar discoveries—others certainly observed sunspots—but Galileo gathered all of his observations together and wrote about them first. Consequently, he is generally credited with their discovery. Ptolemaic theory the theory that asserted Earth was a spherical object at the center of the universe surrounded by other spheres carrying the various celestial objects

The observation of Venus’s phases was especially important to Galileo. According to Ptolemaic theory, Venus would show crescent and “new” phases, but it would not go through a complete cycle of phases. The Ptolemaic model never placed Venus on the opposite side of the Sun as seen from Earth, so Venus would never appear “full.” Yet Galileo clearly observed a nearly full Venus. He also observed that the four satellites of Jupiter orbited the planet, conclusively demonstrating that there was at least one instance of an orbital center other than Earth, a clear contradiction to the Ptolemaic model.

Adapting the Telescope Galileo apparently had no real knowledge of how the telescope worked but he immediately recognized its military value, as well as its entertainment value. He set about building a version that is commonly known as a “Galilean telescope.” It had a convex lens as a primary objective (the lens in front) and a concave lens as the eyepiece. The focal point of the objective lens was behind the eyepiece, and the eyepiece served primarily to form the upright image desired for terrestrial observation. Johannes Kepler was arguably the first person to give a concise theory of how light passed through the telescope and formed an image. Kepler also discussed the various ways in which the lenses could be combined in different optical systems, improving on Galileo’s design. Kepler’s design used convex lenses for both the primary objective and the eyepiece. However, in spite of his theoretical understanding, there is no evidence that Kepler ever actually tried to put together a telescope. Telescopes built following Kepler’s design were not practical for military applications or everyday use because they inverted and reversed the images and showed people upside-down. However, their greater magnification, 82

Telescope

brighter image, and wider angle of view made them best for astronomical observations where the inverted image made no difference. The telescope rapidly came into common astronomical use during the 20 years after it was invented. Unfortunately, it soon became evident that the refracting telescope had a great disadvantage. The main problem with early telescopes was the low quality of their glass and the poor manner in which the lenses were ground. However, even the best lenses had two inherent defects. One defect resulted because the objective lens did not bend all wavelengths equally, and this resulted in the red part of the light-beam being brought to a focus at a greater distance from the objective. An image of a star viewed through an astronomical telescope from this period seemed to be surrounded by colored fringes. This defect is known as “chromatic aberration.” The other problem resulted when the surface of the lens was ground to a spherical shape. Rays passing through the edge of the lens were brought to a focus at a different distance than rays passing near the center of the lens. This defect is called “spherical aberration.” A lens can be ground to a different shape (all modern optical instruments use “aspheric” lenses) but the lens grinders of Galileo’s time did not possess the technology to do this. One remedy for both chromatic and spherical aberrations was to make telescopes with extremely long focal length lenses (so that the lenses did not have much curvature), but this required telescopes several meters long. These telescopes were cumbersome and difficult to use. Another solution for chromatic aberration, unknown at the time, was to use an objective lens made from two different kinds of glass glued together. This method greatly reduces chromatic aberration.

Development of the Reflecting Telescope During the 1680s, Cambridge University was often closed for fear of the Plague. When this occurred, physicist Isaac Newton would retreat to his country home in Lincolnshire. During one of these intervals, Newton began trying to unravel the problem of chromatic aberration. Newton allowed a beam of sunlight to pass through a glass prism and observed that the beam was split into a rainbow of colors. On the basis of this and other experiments, he decided (incorrectly, it turns out) that the refractor could never be cured of chromatic aberration. Newton consequently developed a new type of telescope, “the reflector,” in which there is no objective lens. The light from the object under observation is collected by a curved mirror, which reflects all wavelengths equally. Newton likely did not originate the idea of a reflecting telescope. Earlier, in 1663, Scottish mathematician James Gregory had suggested the possibility of a reflector. However, Newton was apparently the first person to actually build a working reflector. Newton created a reflecting telescope with a 2.5-cm (centimeter) metal mirror and presented it to the Royal Society in 1671. But using a reflecting mirror instead of a reflecting lens created another problem. The light beam is reflected back up the tube but it cannot be observed without blocking the light entering the tube. Gregory had suggested inserting a curved secondary mirror that would reflect the light back through a hole in the center of the 83

Telescope

primary mirror. However, the technology needed to grind the complex mirror surfaces required for Gregory’s design did not exist at the time. Newton solved this problem by introducing a second flat mirror in the middle of the tube mounted at a 45-degree angle so that the light beam is reflected out the side of the tube. The eyepiece was then mounted to the side of the tube. This design is still known as a “Newtonian reflector.” Since Newton’s time, several other reflecting telescope designs have been developed. Newtonian reflectors were not free of problems. Metal mirrors were hard to grind. The mirror surface tarnished quickly and had to be polished every few months. These problems kept the Newtonian reflector from being widely accepted until after 1774, when new designs, polishing techniques, the use of silvered glass, and other innovations were developed by William Herschel. Herschel discovered the planet Uranus in 1781 using a telescope he had made. He continued to build reflecting telescopes over the next several years, culminating in an enormous 122-cm instrument completed in 1789. Herschel’s 122-cm telescope remained the largest in the world until 1845, when the Irish astronomer, William Parsons, the third Earl of Rosse, completed an instrument known as the Leviathan, which had a mirror diameter of 180 cm. Lord Rosse used this instrument to observe “spiral nebulae,” which are now known to be other galaxies. Throughout the eighteenth and nineteenth centuries, telescopes of ever-increasing size were built.

Modern Telescopes The twentieth century saw continued improvement in telescope size and design. Larger telescopes are preferred for two reasons. Larger instruments gather more light. Astronomical distances are so great that most objects are not visible to the unaided eye. The Andromeda galaxy (M31) is generally considered the most distant object that can be seen with the naked eye, and it is the closest galaxy to Earth outside of the Milky Way. To see very far out into space requires large telescope objectives. This is another reason for the general preference of astronomers for reflecting telescopes. It is easier to build large mirrors than it is to build large lenses. A second reason for the general trend toward large instruments is resolving power. The ability of a telescope to separate two closely spaced stars (or see fine detail in general) is known as resolving power. If R is the resolving power (in arc seconds),  is the wavelength (in micrometers) and d  is the diameter of the objective (in meters) then: R  0.25 . As d gets d larger, R gets smaller (smaller is better). Percival Lowell peers through a telescope at his namesake observatory in Flagstaff, Arizona. The historic observatory would be the site of a number of significant findings, among them Clyde Tombaugh’s discovery of the planet Pluto in 1930.

84

During most of the twentieth century, astronomical images were recorded on photographic film. Later in the century, most observatories and research astronomers switched to solid state devices called CCDs (chargecoupled devices). These devices are much more sensitive to low light levels, and they also have the advantage of creating an electronic image that can be fed directly into a computer where it can be immediately processed. Earth’s atmosphere continues to challenge the progress of astronomy. Infrared and ultraviolet wavelengths do not pass through the atmosphere,

Television Ratings

so astronomy in those parts of the spectrum is generally done by balloonbased telescopes or by satellites. A bigger problem with the atmosphere is its inherent instability. Even on the clearest of nights, images jiggle and quiver due to small variations in the atmosphere. One way to get around this problem is to get above the atmosphere. The Hubble Space Telescope (named after Edwin Hubble, who discovered the galactic redshift) is a satellite based optical and infrared observatory. The spectacular images from “Hubble” have pushed back the frontiers of astronomical knowledge, while raising many new questions. Ground-based large telescope design also continues to evolve. Very large mirrors suffer from many problems. To make them stiff, they must be very thick, which makes them very heavy. Modern telescopes use thin mirrors with many different supports that can be independently controlled by a computer. As the mirror is moved, the supports continually adjust to compensate for gravity, keeping the shape of the mirror within precise tolerances. As of 2001, the largest telescope in the world is the twin mirror Keck Telescope. The Keck consists of two matched mirrors housed in separate buildings. Each telescope housing stands eight stories tall and each mirror weighs 300 tons. The mirrors are not made of one solid piece of glass. Instead, each mirror combines thirty-six 1.8-m (meter) hexagonal cells combined to form a collecting area equivalent to one 10-m mirror. Since the twin mirrors are separated by a wide distance, the Keck telescope has a much greater resolving power than either mirror alone would have. The other advantage of the Keck telescope is its position on the top of an extinct volcano, Mauna Kea, in Hawaii. The atmosphere is very stable and the mountaintop, at 4 km (kilometers), is so high that the telescopes are above most of Earth’s atmosphere. The European Southern Observatory in Chile is constructing a similar telescope that will combine light from four 8.2-m mirrors working as a single instrument. These and similar instruments around the world promise to reveal even more about our universe. S E E A L S O Astronomer. Elliot Richmond

The Hubble Space Telescope was launched in 1990 as part of NASA’s Great Observatories program. It is capable of recording images in the visible, near-ultraviolet, and near-infrared.

redshift motion-induced change in the frequency of light emitted by a source moving away from the observer

Bibliography Chaisson, Eric, and Steve McMillan. Astronomy Today, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1993. Giancoli, Douglas C. Physics, 3rd ed., Englewood Cliffs, NJ: Prentice Hall, 1991 Pannekoek, Anton. A History of Astronomy. New York: Dover Publications, 1961. Sagan, Carl. Cosmos. New York: Random House, 1980.

Television Ratings Suppose your favorite television program has just been cancelled because of bad ratings. In the world of television ratings, the number of people watching a show is the important factor, not the intrinsic, qualitative value of the show. So who is rating television shows, and how is it done? Nielsen Media Research was founded in 1923 to measure the number of people listening to radio programs. Early on, the number of people lis85

Television Ratings

tening to the radio was low enough to actually count. However, with the advent of television, cable, satellite, and digital television, the number of viewers has greatly increased. In fact, counting the number of viewers of a television show is impossible, so the Nielsen company has devised a way to estimate the number of viewers.

How Ratings Are Determined Nielsen Media Research measures a sample of television viewers to determine when and what they are watching. Nielsen uses a sample of more than 5 thousand households, which is over 13 thousand people. With close to 100 million households in the United States, one might wonder whether the sample of 5,000 households is large enough to provide a cross-section of the whole market. The company illustrates the sampling procedure with this example: . . .You don’t need to eat an entire pot of vegetable soup to know what kind of soup it is. A cup of soup is more than adequate to represent what is in the pot. If, however, you don’t stir the soup to make sure that all of the various ingredients have a good chance of ending up in the cup, you might just pour a cup of vegetable broth. Stirring the soup is a way to make sure that the sample you draw represents all the different parts of what is in the pot. A truly random list of households will statistically provide a variety of demographic groups. Using demographic information from the U.S. Census Bureau, Nielsen Media Research is able to compare characteristics of the entire population with the characteristics of the sample population. Their findings indicate that the two samples compare favorably. In addition, Nielsen Media Research occasionally completes special tests in which thousands of randomly selected telephone numbers are called, and the people are asked if their televisions are on and who is watching. This information provides a check on television usage and viewing. The Nielsen company provides information on the amount of television set usage, the programs viewed, the commercials watched, and the people watching. To measure television set usage in sample homes, the Nielsen company installs meters to televisions which automatically log when the sets are on and to which programs the sets are tuned. To measure which programs are being watched, the Nielsen company collects network and local programming information in order to know what programs are being shown on any given television set in a sample home. Each commercial aired is marked with an invisible “fingerprint” that enables trackers to determine when and where commercials are aired. Combining this information with the programming information allows trackers to note which commercials aired during which television programs. The Nielsen company provides each sample home with what is known as a “people meter” in addition to the television meter. The people meter is a box with a button for each member of the household and one for visitors. When a viewer begins watching television, he or she presses a button that logs in the television usage. When the program is completed, the viewer again presses the button to register the end the viewing time. In addition to measuring national network programs, Nielson Media Research also measures local viewership. The Nielsen company uses pro86

Temperature, Measurement of

gram diaries to measure the local audiences. These diaries are records of who in the sample group watches what programs and what channels during a one-week period. Diaries are collected in 210 television markets. In addition, in the 48 largest television markets, there is a sample of homes with set meters that record when the television is on and the channel to which it is tuned. Homes with set meters installed on television sets do not have people meters installed.

Using Ratings This information regarding what television shows are being watched and by whom is used by two industries: marketing companies and television production companies. Marketing companies use the information to determine the demographic characterization of the viewers. The information generates questions such as: Are men or women watching the show? Is the audience mainly older adults or teenagers? Does this show appeal to minorities? Marketing companies want this information in order to produce commercials that will appeal to the group that is watching a given program. The television production companies cannot pay more for a program than they earn from selling advertising during the program. The larger the audience of a particular program, the more the commercial time during that program is worth. The basic formula for commercial time is a negotiated rate per thousand viewers, multiplied by the Nielsen audience estimate. Nielsen ratings are also used by public broadcasting stations, enabling them to learn who their audience is and, therefore, to make more informed programming decisions.

The television show “The Crocodile Hunter” relies on the outlandish adventures of its host to attract viewers and boost ratings.

All that is left to understand, is who are the Nielsen families and how are they selected? The sample population includes homes in all fifty states— in cities, towns, suburbs, and rural areas. Some households include renters; others own their home. These households may or may not include children. Various ethnic and income groups are represented. Overall, the sample population is very similar to the true population of the United States. The selection of the sample families is random; therefore it is not possible to increase one’s chances. In fact, including all those who ask to be included would skew the sample, making it nonrepresentative of the population at large. Only by chance can a household become a member of the Nielsen family. S E E A L S O Randomness; Statistical Analysis. Susan Strain Mockert Internet Resources Who We Are and What We Do. Nielsen Media Research. .

Temperature, Measurement of From the Eskimo in Alaska to the Nigerian living near the equator, people of various cultures are exposed to variations in temperature. Our early experiences help us to develop the concept of temperature as a measure of how hot or cold something is. Warmth may be associated with a season of the year or an object such as a stove or a fireplace. The measurement of

87

Temperature, Measurement of

temperature is important to everyday life, providing information about our health and regulating our outdoor activities with accurate weather reports.

Defining and Measuring Temperature Temperature is the number assigned to an object to indicate its warmth. The concept of temperature came about because people wanted to quantify and measure differences in warmth. When an object with a higher temperature comes in contact with a cooler object, a transfer of heat occurs until the two objects are the same temperature. When the heat transfer is complete, it can be said that that the two objects are in thermal equilibrium. Temperature can hence be defined as the quantity of warmth that is the same for two or more objects that are in thermal equilibrium. The temperature 0° C, 273.16 K (kelvin), is the point at which ice, water, and water vapor are all present and in thermal equilibrium. This is known as the triple point of water. Absolute zero temperature occurs when the motion of atoms and molecules practically stops, which occurs at 273° C, or 0 K. Accurate temperature readings are necessary in maintaining meteorological records. Meteorology is the science that deals with the study of weather, which is the condition of the atmosphere at a specific time and place. A meteorologist is a professional who studies and forecasts the weather. The accuracy of weather forecasts is dependent upon collecting data that includes temperature. Air temperature is measured by using a thermometer.

Evolution of the Thermometer Thermometers are instruments that are used to measure degrees of heat. Ferdinand II, Grand Duke of Tuscany, used the first sealed alcohol-in-glass thermometer in 1641. Robert Hook used a thermometer containing red dye in alcohol in 1664. Hook’s thermometer used water as the fixed freezing point and became the standard thermometer that was used by Gresham College and the Royal Society until 1709.

In this modern replica of a Galilean thermometer (also known as a slow thermometer), differently sized weights float within a liquid. Because a cool liquid is denser than a warm one, it supports more weight and more floats will rise to the top of the cylinder. The lowest of the floating weights indicates the room temperature.

88

In 1702, Ole Roemer of Copenhagen developed a thermometer that used a scale with two fixed points: snow and boiling water. This thermometer became a calibrated instrument that contained a visible substance, either mercury or alcohol, that traveled along a narrow passageway in a tube that used two-fixed points on a scale. The passageway through the tube is called a bore. At the bottom of the bore is a bulb that contains the liquid. The thermometer operates on the principle that these fluids expand (swell) when heated and contract (shrink) when cooled. When the liquid is warmed, it expands and moves up the bore, but when the liquid cools, it contracts and moves down the bore in the opposite direction. When a thermometer is placed in contact with an object and reaches thermal equilibrium, we have a quantitative measure for the temperature of the object. For instance, when a thermometer is placed under the arm of an infant, heat is transferred until thermal equilibrium is reached. Thus, when you observe how much the mercury or alcohol has expanded in the bore, you can find the baby’s temperature by reading the scale on the thermometer. Other thermometers besides alcohol and mercury thermometers are metal thermometers, digital thermometers, and strip thermometers. Rather than containing liquid, the metal thermometer uses a strip of metal made

Temperature, Measurement of

up of two different heat-sensitive metals that have been welded together. A thermograph is a metal thermometer that records the temperature continuously all day. However, today’s meteorologists use electronic computers rather than thermographs to obtain permanent records. Digital thermometers are highly sensitive and give precise readings on electronic meters. The strip thermometer is a celluloid tape made with heat-sensitive liquid crystal chemicals that react by changing the color of the tape according to the temperature. Strip thermometers are useful in taking the temperature of infants because one only need put the strip on the baby’s forehead for a few seconds to obtain an accurate temperature reading.

Thermometer Scales The scale on the thermometer tells us how high the liquid is in the thermometer and gives the temperature of whatever is around the bulb. Thus, the thermometer can be used to measure heat in any object including solids, liquids, and gases. The scale on a thermometer is divided equally with divisions called degrees. The most commonly used scales are Fahrenheit (F) and Celsius (C). The Fahrenheit scale (English system of measurement) is used in the United States while the Celsius scale (metric system of measurement) is used by most other countries. Gabriel Fahrenheit (1685–1736), a German physicist, developed the Fahrenheit scale, which was first used in 1724. In 1745, Carolus Linnaeus of Sweden described a scale of a hundred steps or centigrade, which became known as the Centigrade scale. In 1948, the term “centigrade” was changed to Celsius. The Celsius scale was named after the Swedish scientist, Anders Celsius (1701–1744). The Celsius scale was used to develop the Kelvin scale. William Thomson (1824–1907), also known as Lord Kelvin, was a British physicist and proposed the Kelvin scale in 1848. The scale is based on the concept that a gas will lose 1/273.16 of its volume for every one-degree drop in Celsius temperature. Thus, the volume would become zero at 273.16 degrees Celsius—a temperature known as absolute zero. Thus, the Kelvin scale is used primarily to measure gases.

WATER AND TEMPERATURE SCALES Water is the medium that is used as an international standard for determining temperature scales. On the Fahrenheit scale, water freezes at 32 degrees and boils at 212 degrees. Water freezes at 0 degrees and boils at 100 degrees Celsius.

Mathematics is used to convert temperatures from one scale to another. To convert from Celsius to Fahrenheit, multiply by 1.8 and add 32 degrees (° F  1.8° C  32). To convert from Fahrenheit to Celsius, subtract 32 (°F  32) ). To convert from Celsius to Kelvin, degrees and divide by 1.8 (°C   1.8 simply add 273 degrees to the Celsius temperature (K  °C  273). S E E A L S O Absolute Zero; Measurement, Tools of; Meterology, Measurements in. Jacqueline Leonard Bibliography Jones, Edwin R., and Richard L. Childers. Contemporary College Physics, Second Edition. Addison-Wesley Publishing Co., Inc., 1993. National Council of Teachers of Mathematics. Measurement in School Mathematics: 1976 Yearbook. Reston, VA: NCTM, 1976. Victor, Edward, and Richard D. Kellough. Science for the Elementary and Middle School, Ninth Edition. Upper Saddle River, NJ: Prentice-Hall, 2000. Yaros, Ronald A. Weatherschool: Teacher Resource Guide. St. Louis: Yaros Communications, Inc, 1991.

89

Tessellations

Internet Resources

Lynds, Beverly T. “About Temperatures.” .

Tessellations For centuries, mathematicians and artists alike have been fascinated by tessellations, also called tilings. Tessellations of a plane can be found in the regular patterns of tiles in a bathroom floor, or flagstones in a patio. They are also widely used in the design of fabric or wallpaper. Tessellations of three-dimensional space play an important role in chemistry, where they govern the shapes of crystals.

plane generally considered an undefinable term, a plane is a flat surface extending in all directions without end, and that has no thickness

A tessellation of a plane (or of space) is any subdivision of the plane (or space) into regions or “cells” that border each other exactly, with no gaps in between. The cells are usually assumed to come in a limited number of shapes; in many tessellations, all the cells are identical. Tessellations allow an artist to translate a small motif into a pattern that covers an arbitrarily large area. For mathematicians, tessellations provide one of the simplest examples of symmetry.

translate to move from one place to another without rotation

In everyday language, the word “symmetric” normally refers to an object with dihedral or mirror symmetry. That is, a mirror can be placed exactly in the middle of the object and the reflection of the mirrored half is the same as the half not mirrored. Such is the case with the leftmost tessellation in the figure. If an imaginary mirror is placed along the axis shown, then every seed-shaped cell, such as the one shown in color, has an identical mirror image on the other side of the axis. Likewise, every diamondshaped cell has an identical diamond-shaped mirror image.

Examples of symmetry in tessellations show that the transformation moves each cell to an identical cell.

MIRROR

mirror axis

90

Mirror symmetry is not the only kind of symmetry present in tessellations. Other kinds include translational symmetry, in which the entire pattern can be shifted; rotational symmetry, in which the pattern can be rotated about a central point; and glide symmetry, in which the pattern can first be reflected and then shifted (translated) along the axis of reflection. Examples of these three kinds of symmetry are shown in the other three blocks of the figure. In each case, the tessellation is called symmetric under a transformation only if that transformation moves every cell to an exactly matching cell.

TRANSLATION

direction of translation

ROTATION

indicates center of rotation

GLIDE

glide axis

Tessellations

In the rightmost block of the figure, the tessellation has glide symmetry but does not have mirror symmetry because the mirror images of the shaded cells overlap other cells in the tessellation. The collection of all the transformations that leave a tessellation unchanged is called its symmetry group. This is the tool that mathematicians traditionally use to classify different types of tilings. The classification of patterns can be further refined according to whether the symmetry group contains translations in one dimension only (a frieze group), in two dimensions (a wallpaper group), or three dimensions (a crystallographic group). Within these categories, different groups can be distinguished by the number and kind of rotations, reflections, and glides that they contain. In total, there are seven different frieze groups, seventeen wallpaper groups, and 230 crystallographic groups.

Exploring Tessellations To an artist, the design of a successful pattern involves more than mathematics. Nevertheless, the use of symmetry groups can open the artist’s eyes to patterns that would have been hard to discover otherwise. A stunning variety of patterns with different kinds of symmetries can be found in the decorations of tiles at the Alhambra in Spain, built in the thirteenth and fourteenth centuries. In modern times, the greatest explorer of tessellation art was M. C. Escher. This Dutch artist, who lived from 1898 to 1972, enlivened his woodcuts by turning the cells of the tessellations into whimsical human and animal figures. Playful as they appear, such images were based on a deep study of the seventeen (two-dimensional) wallpaper groups. One of the most fundamental constraints on classical wallpaper patterns is that their rotational symmetries can only include half-turns, one-third turns, quarter-turns, or one-sixth turns. This constraint is related to the fact that regular triangles, squares, and hexagons fit together to cover the plane, whereas regular pentagons do not. However, one of the most exciting developments of recent years has been the discovery of simple tessellations that do exhibit five-fold rotational symmetry. The most famous are the Penrose tilings, discovered by English mathematician Roger Penrose in 1974, which use only two shapes, called a “kite” and a “dart.” Three-dimensional versions of Penrose tilings✶ have been found in certain metallic alloys. These “non-periodic tilings,” or “quasicrystals,” are not traditional wallpaper or crystal patterns because they have no translational symmetries. That is, if the pattern is shifted in any direction and any distance, discrepancies between the original and the shifted patterns appear.

✶A Penrose tiling was used to design a mural in the science building of Denison University in Granville, Ohio.

Nevertheless, Penrose tilings have a great deal of long-range structure to them, just as ordinary crystals do. For example, in any Penrose tiling the ratio of the number of kites to the number of darts equals the “golden ratio,” 1.618. . .. Mathematicians are still looking for new ways to explain such patterns, as well as ways to construct new non-periodic tilings. S E E A L S O Escher, M. C.; Tessellations, Making; Transformations. Dana Mackenzie 91

Tessellations, Making

Bibliography Beyer, Jinny. Designing Tessellations. Chicago: Contemporary Books, 1999. Gardner, Martin. Penrose Tiles to Trapdoor Ciphers. New York: W. H. Freeman, 1989. Grunbaum, Branko, and G. C. Shephard. Tilings and Patterns. New York: W. H. Freeman, 1987.

Tessellations, Making

rhombus a parallelogram whose sides are equal

The Dutch artist Maurits Cornelius Escher was, more than anyone else, responsible for bringing the art of tessellations to the public eye. True, tessellations have been used for decorative purposes for centuries, but the cells were usually very bland: squares, equilateral triangles, regular hexagons, or rhombuses. The visual interest of such a tessellation lies in the pattern as a whole, not in the individual cells. But Escher discovered that the cells themselves could become an interesting part of the pattern. His tessellations feature cells in the shape of identifiable figures: a bird, a lizard, or a rider on horseback. To create an Escher-style tessellation, start with one of the polygons mentioned above. (This is not essential, but it is easiest.) Next, choose two sides and replace them with any curved or polygonal path. The only rule is that anything that is added to the polygon on one side has to be taken away from the other. If a protruding knob has been added to one side, the other side must have an identically shaped dimple. This procedure can be repeated with other pairs of sides. Usually, the result will vaguely resemble something, perhaps a lizard. The outline can then be revised so that the result becomes more recognizable. But every time one side is modified, it is imperative that its corresponding side is modified in precisely the same way. This makes it nearly impossible to draw a convincing lizard, but by adding a few embellishments—eyes, scales and paws—a reasonable caricature can be produced. Escher, thanks to his years of experience, was a master at this last step.

translation to move from one place to another without rotation glide reflection a rigid motion of the plane that consists of a reflection followed by a translation parallel to the mirror axis

The figure illustrates the process of making tessellations. Starting with a hexagon, the top and bottom sides are replaced with identical curves that are related by a translation. The upper curve suggests a head, and the bottom curve suggests two heels. Next the top left and top right sides are replaced with two identical curves that are related by a glide reflection. (Note: A translation will only work if the two sides are parallel.) One curve bulges outward and the other inward, in accordance with the rule that anything removed from one side must be given back to the other. Then the bottom left and bottom right sides are replaced with S-shaped curves that represent the feet and legs. Again, these two curves are identical and related by a glide reflection. Finally, the figure is decorated so that it looks a little bit like a ghost wearing boots. (The complete tessellation is shown here, but obviously Escher did it better.)

92

Tessellations, Making

translate s

The figure shows how to transform a hexagon into a pattern that will tile the plane. The completed tessellation is decorated to suggest ghosts with boots.

glide glide

Decorate!

Another way of creating interesting tessellations has been called the kaleidoscope method. In this method, one begins with a triangular “primary cell,” which may have angles of 60°, 60°, 60°; 30°, 60°, 90°; or 45°, 45°, 90°. After it is decorated in any way, the triangle should be reflected through one side after another until a hexagon or a square is completed. Finally, this figure can be used as a tile to cover the plane. The kaleidoscope method can be modified by using rotations about the midpoint of a side or about a vertex, instead of reflections, to generate the additional copies of the decorated primary cell. Different patterns of rotation or reflection can yield a dazzling variety of different tessellations, all arising from the same primary cell. S E E A L S O Escher, M. C.; Tessellations. Dana Mackenzie Bibliography Beyer, Jinny. Designing Tessellations. Chicago: Contemporary Books, 1999.

rotation a rigid motion of the plane that fixes one point (the center of rotation) and moves every other point around a circle centered at that point reflection a rigid motion of the plane that fixes one line (the mirror axis) and moves every other point to its mirror image on the opposite side of the line

93

Time, Measurement of

Time, Measurement of The history of time measurement is the story of the search for more consistent and accurate ways to measure time. Early human groups recorded the phases of the Moon some 30,000 years ago, but the first minutes were counted accurately only 400 years ago. The atomic clocks that allowed mankind to track the approach of the third millennium (in the year 2001) by a billionth of a second are less than 50 years old. Thus, the practice and accuracy of time measurement has progressed greatly through humanity’s history. The study and science of time measurement is called horology. Time is measured with instruments such as a clock or calendar. These instruments can be anything that exhibits two basic components: (1) a regular, constant, or repetitive action to mark off equal increments of time, and (2) a means of keeping track of the increments of time and of displaying the result. Imagine your daily life—getting ready in the morning, driving your car, working at your job, going to school, buying groceries, and other events that make up your day. Now imagine all the people in your neighborhood, in your city, and in your country doing these same things. The social interaction that our existence involves would be practically impossible without a means to measure time. Time can even be considered a common language between people, and one that allows everybody to proceed in an orderly fashion in our complicated and fast-paced world. Because of this, the measurement of time is extremely important to our lives.

The Origins of Modern Time Measurement The oldest clock was most likely Earth as it related to the Sun, Moon, and stars. As Earth rotates, the side facing the Sun changes, leading to its apparent movement from one side of Earth, rising across the sky, reaching a peak, falling across the rest of the sky, and eventually disappearing below Earth on the opposite side to where it earlier appeared. Then, after a period of darkness, the Sun reappears at its beginning point and makes its journey again. This cyclical phenomenon of a period of brightness followed by a period of darkness led to the intervals of time now known as a day. Little is known about the details of timekeeping in prehistoric eras, but wherever records and artifacts are discovered, it is found that these early people were preoccupied with measuring and recording the passage of time. A second cyclical observation was likely the repeated occurrence of these days, followed by the realization that it took about 30 of these days (actually, a fraction over 27 days) for the Moon to cycle through a complete set of its shape changes: namely, its main phases of new, first quarter, full, and last quarter. European ice-age hunters over 20,000 years ago scratched lines and gouged holes in sticks and bones, possibly counting the days between phases of the Moon. This timespan was eventually given the name of “month.” Similarly, the sum of the changing seasons that occur as Earth orbits around the Sun gave rise to the term “year.” For primitive peoples it was satisfactory to divide the day into such things as early morning, mid-day, late afternoon, and night. However, as societies became more complex, the need developed to more precisely divide the day. 94

Time, Measurement of

The modern convention is to divide it into 24 hours, an hour into 60 minutes, and a minute into 60 seconds. The division into 60 originated from the ancient Babylonians (1900 B.C.E.–1650 B.C.E.), who attributed mystical significance to multiples of 12, and especially to the multiple of 12 times 5, which equals 60. The Babylonians divided the portion that was lit by the Sun into 12 parts, and the dark interval into 12 more, yielding 24 divisions now called hours. Ancient Arabic navigators measured the height of the Sun and stars in the sky by holding their hand outstretched in front of their faces, marking off the number of spans. An outstretched hand subtends an angle of about 15 degrees at eye level. With 360 degrees in a full circle, 360° divided by 15° equals 24 units, or 24 hours. Babylonian mathematicians also divided a complete circle into 360 divisions, and each of these divisions into 60 parts. Babylonian astronomers also chose the number 60 to subdivide each of the 24 divisions of a day to create minutes, and each of these minutes were divided into 60 smaller parts called seconds.

MARKING CELESTIAL TIME Some archeologists believe the complex patterns of lines on the desert floor in Nazca, Peru were used to mark celestial time. Medicine Wheels—stone circles found in North America dating from over 1,000 years ago— also may have been used to follow heavenly bodies as they moved across the sky during the year.

subtend to extend past and mark off a chord or arc

Early Time-Measuring Instruments The first instrument to measure time could have been something as simple as a stick in the sand, a pine tree, or a mountain peak. The steady shortening of its shadow would lead to the noon point when the Sun is at its highest position in the sky, and would then be followed by shadows that lengthen as darkness approaches. This stick eventually evolved into an obelisk, or shadow clock, which dates as far back as 3500 B.C.E. The Egyptians were able to divide their day into parts comparable to hours with these slender, four-sided monuments (which look similar to the Washington Monument). The moving shadows formed a type of sundial, enabling citizens to divide the day into two parts. Sundials evolved from flat horizontal or vertical plates to more elaborate forms. One version from around the third century B.C.E.) was the hemispherical dial, or hemicycle, a half-bowl-shaped depression cut into a block of stone, carrying a central vertical pointer and marked with sets of hour lines for different seasons. Apparently 5,000 to 6,000 years ago, civilizations in the Middle East and North Africa initiated clock-making techniques to organize time more efficiently. Ancient methods of measuring hours in the absence of sunlight included fire clocks, such as the notched candle and the Chinese practice of burning a knotted rope. All fire clocks were of a measured size to approximate the passage of time, noting the length of time required for fire to travel from one knot to the next. Devices almost as old as the sundial include the hourglass, in which the flow of sand is used to measure time intervals, and the water clock (or “clepsydra”), in which the water flow indicates the passage of time.

The Mechanization of Clocks By about 2000 B.C.E., humans had begun to measure time mechanically. Eventually, a weight falling under the force of gravity was substituted for the flow of water in time devices, a precursor to the mechanical clock. The first recorded examples of such mechanical clocks are found in the four95

Time, Measurement of

This sundial in a Bolivian town in 1986 illustrates the endurance and utility of ancient time-measuring devices.

teenth century. A device, called an “escapement,” slowed down the speed of the falling weight so that a cogwheel would move at the rate of one tooth per second. The scientific study of time began in the sixteenth century with Italian astronomer Galileo Galilei’s work on pendulums. He was the first to confirm the constant period of the swing of a pendulum, and later adapted the pendulum to control a clock. The use of the pendulum clock became popular in the 1600s when Dutch astronomer Christiaan Huygens applied the pendulum and balance wheel to regulate the movement of clocks. By virtue of the natural period of oscillation from the pendulum, clocks became accurate enough to record minutes as well as hours. Although British physicist Sir Isaac Newton continued the scientific study of time in the seventeenth century, a comprehensive explanation of time did not exist until the early twentieth century, when Albert Einstein proposed his theories of relativity. Einstein defined time as the fourth dimension of a four-dimensional world consisting of space (length, height, depth) and time. Quartz-crystal clocks were invented in the 1930s, improving timekeeping performance far beyond that of pendulums. When a quartz crystal is placed in a suitable electronic circuit, the interaction between mechanical stress and electric field causes the crystal to vibrate and generate a constant frequency that can be used to operate an electronic clock display. But the timekeeping performance of quartz clocks has been substantially surpassed by atomic clocks. An atomic clock measures the frequency of electromagnetic radiation emitted by an atom or molecule. Because the atom or molecule can only emit or absorb a specific amount of energy, the radiation emitted or absorbed has a regular frequency. This allowed the National Institute of Standards and Technology to establish the second as the amount of time radiation would take to go through 9,192,631,770 cycles at the frequency emitted by cesium atoms making the transition from one state 96

Topology

to another. Cesium clocks are so accurate that they will be off by only one second after running for 300 million years.

Time Zones In the 1840s, the Greenwich time standard was established with the center of the first time zone set at the Royal Greenwich Observatory in England, located on the 0-degree longitude meridian. In total, twenty-four time zones—each fifteen degrees wide—were established equidistant from each other east and west of Greenwich’s prime meridian. Today, when the time is 12:00 noon in Greenwich, it is 11:00 A.M. inside the next adjoining time zone to the west, and 1:00 P.M. inside the next adjoining time zone to the east. The term “A.M.” means ante meridiem (“before noon”), while “P.M.” means post meridiem (“after noon”). But military time is measured differently. The military clock begins its day with midnight, known as either or 0000 hours (“zero hundred hours”) or 2400 hours (“twenty-four hundred hours”). An early-morning hour such as 1:00 A.M. is known in military time as 0100 hours (pronounced “oh one hundred hours”). In military time, 12:00 noon is 1200 (“twelve-hundred hours”). An afternoon or evening hour is derived by adding 12; hence, 1:00 p.m. is known as 1300 (“thirteen-hundred hours”), and 10:00 p.m. is known as 2200 (“twenty-two hundred hours”). S E E A L S O Calendar, Numbers in the. William Arthur Atkins (with Philip Edward Koth) Bibliography Gibbs, Sharon L. Greek and Roman Sundials. New Haven, CT: Yale University Press, 1976. Landes, David S. Revolution in Time: Clocks and the Making of the Modern World. Cambridge, MT: The Belknap Press of Harvard University Press, 1983. Tannenbaum, Beulah, and Myra Stillman. Understanding Time: The Science of Clocks and Calendars. New York: Whittlesey House, McGraw-Hill Book Company, Inc., 1958.

ATOMIC TIME? Variations and vagaries in the Earth’s rotation eventually made astronomical measurements of time inadequate for scientific and military needs that required highly accurate timekeeping. Today’s standard of time is based on atomic clocks that operate on the frequency of internal vibrations of atoms within molecules. These frequencies are independent of the Earth’s rotation, and are consistent from day to day within one part in 1,000 billion.

longitude one of the imaginary great circles beginning at the poles and extending around Earth; the geographic position east or west of the prime meridian meridian a great circle passing through Earth’s poles and a particular location equidistant at the same distance

lnternet Resources

ABC News Internet Ventures. “What is Time: Is It Man-Made Concept, Or Real Thing?” .

Topology Topology is sometimes called “rubber-sheet geometry” because if a shape is drawn on a rubber sheet (like a piece of a balloon), then all of the shapes you can make by stretching or twisting—but never tearing—the sheet are considered to be topologically the same. Topological properties are based on elastic motions rather than rigid motions like rotations or inversions. Mathematicians are interested in the qualities of figures that remain unchanged even when they are stretched and distorted. The qualities that are unchanged by such transformations are said to be topologically invariant because they do not vary, or change, when stretched. 97

Topology

A MUSICAL COMPARISON A soprano and a baritone can sing the same song even though one sings high notes and the other low notes. Shifting a song up or down the musical scale changes some qualities of the music but not the pattern of notes that creates the song. Similarly, topology deals with variations that occur without changing the underlying “melody” of the shape.

As an example, the figure shows a triangle, a square, a rough outline of the United States, and a ring. The first three shapes are topologically equivalent; we can stretch and pull the boundary of the square until it becomes a circle or the U.S. shape. But no matter how much we pull or stretch this basic outline we cannot make it look like a ring. Since a triangle is topologically the same as a square, and a sphere is the same as a cone, the idea of angle, length, perimeter, area, and volume play no role in topology. What remains the same is the number of boundaries that a shape has. A triangle has an inside and an outside separated by a closed boundary line. Every possible distortion of a triangle will also have an inside and an outside separated by a boundary. The ring, on the other hand, has two boundaries forming an inside region separated from two disconnected outside regions. No matter how you transform a ring it will always have two boundaries, one inside region, and two outside regions. It can be quite challenging and surprising to discover whether two shapes are topologically the same. For example, a soda bottle is the same as a soup bowl, which is the same as a dinner plate. A coffee cup and a donut are topologically the same. But a coffee cup is topologically different from a soup bowl because the hole in the cup’s handle does not occur in the bowl. Because topology treats shapes so differently from the way we are accustomed to thinking about them, some of the most interesting objects studied in topology may seem very strange. One of the most well known objects is called the Möbius Strip, named for the German mathematician August Ferdinand Möbius who first studied it. This curious object is a twodimensional surface that has only one side. A Möbius Strip can be easily constructed by taking the two ends of a long, rectangular strip of paper, giving one end a half twist, and gluing the two ends together. The Klein Bottle, which theoretically results from sewing together two Möbius Strips along their single edge, is a bottle with only one side. In our threedimensional world, it is impossible to construct because a true Klein Bottle can exist only in four dimensions. The concepts of sidedness, boundaries, and invariants have been generalized by topologists to higher dimensions. Although difficult to visualize, topologists will talk about surfaces in four, five, and even higher dimensions. While much of the study of topology is theoretical, it has deep connections to relativity theory and modern physics which also imagine our universe as having more than three dimensions. S E E A L S O Dimensions; Mathematics, Impossible; Möbius, August Ferdinand. Alan Lipp Bibliography Chinn, W.G. and Steenrod, N.E. First Concepts of Topology. New York: Random House, 1966.

98

Toxic Pollutants, Measuring

Internet Resources

“Beginnings of Topology.” In Math Forum. August 98. .

Toxic Pollutants, Measuring The amount of pollution in our environment continues to pose a challenge for industry, business, and decisionmakers. Some of the major chemicals of concern are those that harm the environment on a large scale, such as chlorofluorocarbons, which destroy the atmospheric ozone layer. Other chemicals that are harmful to human health are lead in water, cyanide (a deadly poison) leaching from landfills into water supplies, ammonia from car exhaust, and a long list of carbon-based chemicals that are byproducts of industrial processes. Pollutants may start out as a solid, liquid, or gas, but eventually are counted as individual molecules within another substance. For instance, if a contaminant contains many toxic chemicals, each one will be measured separately. Some chemicals are more toxic to living organisms than others, so they are measured independently to discover their presence in the environment. A toxic substance is usually measured in “parts per million” or “ppm.” This measure states that the number of units of the particular chemical under survey occurs in comparison to one million units of the surrounding natural chemicals. When amounts of toxic chemicals are measured over large areas over a specific period of time, the amounts of pollutants can be measured in pounds.

An environmental technician monitors the underground water level at an abandoned gas station. These sites may contain underground tanks of hazardous petroleum products.

99

Toxic Pollutants, Measuring

Categories of Toxic Pollutants Identifying dangerous wastes can be a complicating factor in measuring certain pollutants. There are four major categories of toxic pollutants that cover a wide range of materials. The first major grouping of pollutants is based on ignitability, or its ability to catch on fire or explode. These are wastes and pollutants that may be a fire hazard if they collect in landfills or other storage sites. Liquids and solids are not usually ignitable, but the vapors or fumes they make can be very dangerous. Each type of ignitable pollutant has a flash point. The flash point is the temperature at which the flammable vapors will interact with a spark or flame to create an explosion. Each pollutant has its own flash point which has been determined in a laboratory. Another measure of toxic chemicals is their corrosivity, which is a chemical process in which metals and minerals are converted into undesirable byproducts. One of the ways in which corrosiveness is measured is by the pH, which is an indicator of the acidity or alkalinity of solution. Mathematically, pH is the negative logarithm of the hydrogen ion (a single proton with no electron) concentration in water. The pH scale ranges from 1 to 14. A pH of between 1 and 3 is highly acidic while values between 11 and 14 indicate a highly alkaline solution. A pH of 7 is considered neutral. The third general category of pollutants is based on the reactivity of the substance. Reactive chemicals are identified as pollutants because they may explode or make toxic fumes that pose a threat to human health and the environment. Soils are often acidic or basic, so the potential reactivity of chemicals in soils is of great concern. Among other properties, reactive pollutants: • readily undergo violent chemical change; • react violently or form potentially explosive mixtures with water; and • generate toxic fumes when exposed to mildly acidic or basic conditions. The last of the four general categories of toxic pollutants is based on the toxicity of substances. Toxicity is the ability of a substance to produce adverse health effects by exposure to the substance.

Measuring Toxic Pollutants An example of how some pollutants are measured, including ignitable and toxic substances, is the Toxicity Characteristic Leaching Procedure (TCLP). This procedure and others, sometimes called Extraction Procedures (EP), are based on how a leaching pollutant may decompose and distribute to the water supply or soil from a landfill containing city solid waste. The major concern about the leaching process is the possibility that toxic chemicals may reach the groundwater and migrate to other sources of drinking water. There are numerous chemical processes used to obtain a sample for analysis. These are many stepped (phased) processes and all have very clear guidelines on how they are to be performed. Once the TCLP extract has been obtained it is immediately tested for pH. All individual chemicals are analyzed separately. The initial analysis starts with the following formula: 100

Transformations

where

(V1)(C1)  (V2)(C2) Final Analyte Concentration   V1  V2

V1  volume of the first phase (L) C1  concentration of the analyte of concern in the first phase (mg/L) V2  volume of the second phase (L) and C2  concentration of the analyte of concern in the second phase (mg/L). To assure quality control over of initial measurements there are many additional mathematical and chemical measurement techniques that are performed after the first analysis. In one type of quality control one in every 20 samples is remeasured. This particular sample is called spike. It is measured against a sample without the potential toxin in it (the control). The spikes are calculated by the following formula: Percent R (percent recovery)  100 (Xs  Xu)/K where Xs  measured value for the spiked sample Xu  measured value for the unspiked sample and K  known of the spike in the sample (this comes from the first or known measurement of the sample). SEE ALSO

Logarithms. Brook E. Hall

Bibliography Washington State Department of Ecology. Washington State Toxics Release Inventory: Summary Report 1997. Washington State, 1999. Washington State Department of Ecology. Reducing Toxics in Washington: A Report to the Legislature. Washington State, 1999. Washington State Department of Ecology. Biological Testing Methods 80-12 for the Designation of Dangerous Waste. Washington State, 1999.

Transformations A transformation is a mathematical function that repositions points in a one-dimensional, two-dimensional, three-dimensional, or any other ndimensional space. In this article, only transformations in the familiar twodimensional rectangular coordinate plane will be discussed. Transformations map one set of points onto another set of points, generally with the purpose of changing the position, size, and/or shape of the figure made up by the first set of points. The first set of points, from the domain of the transformation, is called the set of pre-images, whereas the second set of points, from the range of the transformation, is called the set of images. Therefore, a transformation maps each pre-image point to its image point.

domain the set of all values of a variable used in a function range the set of all values of a variable in a function mapped to the values in the domain of the independent variable

101

Transformations

Reflections Reflections are transformations that involve “flipping” points over a given line; hence, this type of transformation is sometimes called a “flip.” When a figure is reflected in a line, the points on the figure are mapped onto the points on the other side of the line which form the figure’s mirror image. For example, in the first figure below, the triangle ABC, the pre-image figure, is reflected in the x-axis to produce the image triangle A’B’C’. Note that if triangle ABC is traversed from A to B to C and back to A, the direction of this movement is counterclockwise. If triangle A’B’C’ is traversed from A’ to B’ to C’ and back to A’, the direction is clockwise. This “reversal of orientation” is similar to the way images in a mirror are reversed, and is a fundamental property of all reflections.

y C

A

B x

A'

B'

C'

In the case of the reflection in the x-axis, as seen in the first figure, the first coordinate of each image point is the same as the first coordinate of its pre-image point, but the second coordinate of any image point is the opposite, or negative, of the second coordinate of its pre-image point. Mathematically, we say that for a reflection in the x-axis, pre-image points of the form (x, y) are mapped to image points of the form (x, y), or, more compactly, r(x, y)  (x, y), where r represents the reflection. Such a formula is sometimes called an image formula, since it shows how a transformation acts on a pre-image point to produce its image point. A reflection in the y-axis leaves all second coordinates the same but replaces each first coordinate with its opposite; therefore, an image formula for a reflection in the y-axis may be written r(x, y)  (x, y). The image formula r(x, y)  (y, x) represents a reflection in the line y  x. When this reflection is done, the coordinates of each pre-image point are reversed to give the image.

Rotations A second type of transformation is the rotation, also known as a “turn.” A rotation, as its name suggests, takes pre-image points and rotates them by some specified angle measure about some fixed point. In the figure below, the pre-image triangle CDE has been rotated 90° about the origin of the coordinate system to produce the image triangle C’D’E’.

102

Transformations

E'

D'

y

E

C'

C

D x

The image formula for a 90° rotation is R(x, y)  (y, x). It can be shown that, in general, the image formula for a rotation of angle measure t about the origin has image formula R(x, y)  [xcos(t)  ysin(t)], [xsin(t)  ycos(t)]. If t is positive, the direction of the rotation is counterclockwise; if t is negative, then the rotation is clockwise.

Translations Another type of transformation is the translation or “slide.” Translations take pre-image points and move them a given number of units horizontally and/or a given number of units vertically. A translation image formula has the form T(x, y)  (x  a, y  b), where a is the number of units moved horizontally and b is the number of units moved vertically. If a is positive, the horizontal shift is to the right. If a is negative, then it is to the left. Similarly, if b is positive, the vertical shift is upward; but if b is negative, the vertical shift is downward. In the figure below, the pre-image triangle CDE has been translated 4 units to the left and 2 units upward to give the image triangle C’D’E’. The image formula would be T(x, y)  (x  (4), y  2)  (x  4, y  2).

E'

C'

y

D'

E

C

D

x

Reflections, rotations, and translations change only the location of a figure. They have no effect on the size of the figure or on the distance between points in the figure. For this reason they are called “isometries” from the Greek words meaning “same measure.” An isometry is also known as a “distance-preserving transformation.” The next transformation discussed—the dilation—is not an isometry; that is, it does not preserve distance.

103

Transformations

In transformations known as dilations, the resulting image may be larger or smaller than its original pre-image.

(a)

(b)

y

C'

y

E' C

E

D' C

E

E'

C'

D k=3

(c)

(d)

y

E

C E D

D

x k = –3

E'

D'

x

k = –1/2 E'

D'

x

k = 1/2

y

C

D

D'

x

C'

C'

Dilations A dilation is also known as a “stretch” or “shrink” depending on whether the image figure is larger or smaller than the pre-image figure. The image formula for a dilation is d(x, y)  (kx, ky), where k is a real number, called the magnitude or scale factor, and where the center of the dilation is the origin. If k  1, the image of the dilation is an enlargement of the preimage on the same side of the center as the pre-image. Part (a) of the boxed figure illustrates this for k  3. If 0  k  1, the image of the dilation is a reduction in size of the preimage on the same side of the center as the pre-image. Part (b) of the fig1 ure illustrates this for k  2. If k  1, the image of the dilation is an enlargement of the pre-image on the opposite side of the center from the pre-image. Part (c) of the figure illustrates this for k  3. If 1  k  0, the image of the dilation is a reduction in size of the pre-image on the opposite side of the center from the pre-image. Part (d) 1 of the figure illustrates this for k  2. Notice that the effect of a negative value for k is equivalent to that of a dilation with a magnitude equal to the absolute value of k followed by a rotation of 180° about the center of the dilation.

104

Triangles

All of the transformations discussed above can easily be extended into 3-dimensional space. For example, a dilation in the 3-dimensional xyz-coordinate system would have an image formula of the form d(x, y, z)  (kx, ky, kz). When transformations—reflections, rotations, translations, and dilations—are expressed as matrices (as is taught in linear algebra), they can then be combined to create the movement of figures in computer animation programs. S E E A L S O Mapping, Mathematical; Tessellations. Stephen Robinson

matrix a rectangular array of data in rows and columns

Bibliography Serra, Michael. Discovering Geometry. Emeryville, CA: Key Curriculum Press, 1997. Narins, Brigham, ed. World of Mathematics, Detroit: Gale Group, 2001. Usiskin, Zalman, Arthur Coxford, Daniel Hirschhorn, Virginia Highstone, Hester Lewellen, Nicholas Oppong, Richard DiBianca, and Merilee Maeir. Geometry. Glenview, IL: Scott Foresman and Company, 1997.

Triangles A triangle is a closed three-sided, three-angled figure, and is the simplest example of what mathematicians call polygons (figures having many sides). Triangles are among the most important objects studied in mathematics owing to the rich mathematical theory built up around them in Euclidean geometry and trigonometry, and also to their applicability in such areas as astronomy, architecture, engineering, physics, navigation, and surveying. In Euclidean geometry, much attention is given to the properties of triangles. Many theorems are stated and proved about the conditions necessary for two triangles to be similar and/or congruent. Similar triangles have the same shape but not necessarily the same size, whereas congruent triangles have both the same shape and size. One of the most famous and useful theorems in mathematics, the Pythagorean Theorem, is about triangles. Specifically, the Pythagorean Theorem is about right triangles, which are triangles having a 90° or “right” angle. The theorem states that if the length of the sides forming the right angle are given by a and b, and the side opposite the right angle, called the hypotenuse, is given by c, then c 2  a2  b2. It is almost impossible to overstate the importance of this theorem to mathematics and, specifically, to trigonometry.

Triangles and Trigonometry Trigonometry literally means “triangle measurement” and is the study of the properties of triangles and their ramifications in both pure and applied mathematics. The two most important functions in trigonometry are the sine and cosine functions, each of which may be defined in terms of the sides of right triangles, as shown below.

Euclidean geometry the geometry of points, lines, angles, polygons, and curves confined to a plane trigonometry the branch of mathematics that studies triangles and trigonometric functions hypotenuse the long side of a right triangle; the side opposite the right angle sine if a unit circle is drawn with its center at the origin and a line segment is drawn from the origin at angle theta so that the line segment intersects the circle at (x, y), then y is the sine of theta cosine if a unit circle is drawn with its center at the origin and a line segment is drawn from the origin at angle theta so that the line segment intersects the circle at (x, y), then x is the cosine of theta

105

Triangles

B

sin (A) =

a c

cos (A) =

b c

c a

C

A b

These functions are so important in calculating side and angle measures of triangles that they are built into scientific calculators and computers. Although the sine and cosine functions are defined in terms of right triangles, their use may be extended to any triangle by two theorems of trigonometry called the Law of Sines and the Law of Cosines (see page 108). These laws allow the calculation of side lengths and angle measures of triangles when other side and angle measurements are known.

Ancient and Modern Applications triangulation the process of determining the distance to an object by measuring the length of the base and two angles of a triangle parallax the apparent motion of a nearby object when viewed against the background of more distant objects due to a change in the observer’s position

algorithm a rule or procedure used to solve a mathematical problem

Perhaps the most ancient use of triangles was in astronomy. Astronomers developed a method called triangulation for determining distances to far away objects. Using this method, the distance to an object can be calculated by observing the object from two different positions a known distance apart, then measuring the angle created by the apparent “shift” or parallax of the object against its background caused by the movement of the observer between the two known positions. The Law of Sines may then be used to calculate the distance to the object. The Greek mathematician and astronomer, Aristarchus (310 B.C.E.–250 B.C.E.) is said to have used this method to determine the distance from the Earth to the Moon. Eratosthenes (c. 276 B.C.E.–195 B.C.E.) used triangulation to calculate the circumference of Earth. Modern Global Positioning System (GPS) devices, which allow earthbound travelers to know their longitude and latitude at any time, receive signals from orbiting satellites and utilize a sophisticated triangulation algorithm to compute the position of the GPS device to within a few meters of accuracy. S E E A L S O Astronomy, Measurements in; Global Positioning System; Polyhedrons; Trigonometry. Stephen Robinson Bibliography Foerster, Paul A. Algebra and Trigonometry. Menlo Park, CA: Addison-Wesley Publishing Company, 1999. Serra, Michael. Discovering Geometry. Emeryville, CA: Key Curriculum Press, 1997. Narins, Brigham, ed. World of Mathematics, Detroit: Gale Group, 2001.

106

Trigonometry

Trigonometry The word “trigonometry” comes from two Greek words meaning “triangle measure.” Trigonometry concerns the relationships among the sides and angles of triangles. It also concerns the properties and applications of these relationships, which extend far beyond triangles to real-world problems. Evidence of a knowledge of elementary trigonometry dates to the ancient Egyptians and Babylonians. Led by Ptolemy, the Greeks added to this field of knowledge during the first millennium B.C.E.; simultaneously, similar work was produced in India. Around 1000 C.E., Muslim astronomers made great advances in trigonometry. Inspired by advances in astronomy, Europeans contributed to the development of this important mathematical area from the twelfth century until the days of Leonhard Euler in the eighteenth century.

Trigonometric Ratios To understand the six trigonometric functions, consider right triangle ABC with right angle C. Although triangles with identical angle measures may have sides of different lengths, they are similar. Thus, the ratios of the corresponding sides are equal. Because there are three sides, there are six possible ratios.

B

c

A

a

C

b

Working from angle A, label the sides as follows: side c represents the hypotenuse; leg a represents the side opposite angle A; and leg b is adjacent to angle A. The definitions of the six trigonometric functions of angle A are listed below.

a opposite = c hypotenuse

Cotangent

cot A =

b adjacent = a opposite

cos A =

b adjacent = c hypotenuse

Secant

sec A =

c hypotenuse = b adjacent

tan A =

a opposite = b adjacent

Cosecant

csc A =

c hypotenuse = a opposite

Sine

sin A =

Cosine

Tangent

For any angle congruent to angle A, the numerical value of any of these ratios will be equal to the value of that ratio for angle A. Consequently, for any given angle, these ratios have specific values that are listed in tables or can be found on calculators.

hypotenuse the long side of a right triangle; the side opposite the right angle

congruent exactly the same everywhere, superposable

107

Trigonometry

Basic Uses of Trigonometry. The definitions of the six functions and the Pythagorean Theorem provide a powerful means of finding unknown sides and angles. For any right triangle, if the measures of one side and either another side or angle are known, the measures of the other sides and angles can be determined. For example, suppose the measure of angle A is 36° and side c measures 12 centimeters (and angle C measures 90°). To determine the measure of angle B, subtract 36 from 90 because the two non-right angles must sum to a 90°. To determine sides a and b, solve the equations sin 36°  12 and cos b 36°  12 , keeping in mind that sin 36° and cos 36° have number values. The results are a  12sin 36°  7.1 cm and b  12cos 36°  9.7 cm. Two theorems that are based on right-triangle trigonometry—the Law of Sines and the Law of Cosines—allow us to solve for the unknown parts of any triangle, given sufficient information. The two laws, which can be expressed in various forms, follow.

Law of Sines:

b c a   sin B sin C sin A

Law of Cosines: a  b  c  2bc cos A 2

2

2

Expanded Uses of Trigonometry The study of trigonometry goes far beyond just the study of triangles. First, the definitions of the six trigonometric functions must be expanded. To accomplish this, establish a rectangular coordinate system with P at the origin. Construct a circle of any radius, using point P as the center. The positive horizontal axis represents 0°. As one moves counter-clockwise along the circle, a positive angle A is generated.

acute sharp, pointed; in geometry, an angle whose measure is less than 90 degrees

Consider a point on the circle with coordinates (u, v). (The reason for using the coordinates (u, v) instead of (x, y) is to avoid confusion later on when constructing graphs such as y  sin x.) By projecting this point onto the horizontal axis as shown below, a direct analogy to the original trigonometric functions can be made. The length of the adjacent side equals the u-value, the length of the opposite side equals the v-value, and the length of the hypotenuse equals the radius of the circle. Thus the six trigonometric functions are expanded because they are no longer restricted to acute angles.

(u,v) r A P

108

Trigonometry

v a opposite = = r c hypotenuse

cot A =

u b adjacent = = v a opposite

cos A =

b adjacent u = = c hypotenuse r

sec A =

c hypotenuse r = = b adjacent u

tan A =

a opposite v = = b adjacent u

csc A =

r hypotenuse = c = v a opposite

sin A =

For any circle, similar triangles are created for equal central angles. Consequently, one can choose whatever radius is most convenient. To simplify calculations, a circle of radius 1 is often chosen. Notice how four of the functions, especially the sine and cosine functions, become much simpler if the radius is 1. These expanded definitions, which relate an angle to points on a circle, allow for the use of trigonometric functions for any angle, regardless of size. So far the angles discussed have been measured in degrees. This, however, limits the applicability of trigonometry. Trigonometry is far less restricted if angles are measured in units called radians. Using Radian Measure. Because all circles are similar, for a given central angle in any circle, the ratio of an intercepted arc to the radius is constant. Consequently, this ratio can be used instead of the degree measure to indicate the size of an angle.

radian an angle measure approximately equal to 57.3 degrees, it is the angle that subtends an arc of a circle equal to one radius

Consider for example a semicircle with radius 4 centimeters. The arc length, which is half of the circumference, is exactly 4 centimeters. In radians, therefore, the angle is the ratio 4 centimeters to 4 centimeters, or simply . (There are no units when radian measure is used.) This central angle also measures 180°. Recognizing that 180° is equivalent to (when measured in radians), there is now an easy way of converting to and from degrees and radians. This can also be used to determine that an angle of 1 radian, an angle which intercepts an arc that is precisely equal to the radius of the circle, is approximately 57.3°. Now the domain for the six trigonometric functions may be expanded beyond angles to the entire set of real numbers. To do this, define the trigonometric function of a number to be equivalent to the same function of an angle measuring that number of radians. For example, an expression such as sin 2 is equivalent to taking the sine of an angle measuring 2 radians. With this freedom, the trigonometric functions provide an excellent tool for studying many real-world phenomena that are periodic in nature. The figure below shows the graphs of the sine, cosine, and tangent functions, respectively. Except for values for which the tangent is undefined, the domain for these functions is the set of real numbers. The domain for the parts of the graphs that are shown is 2 x 2 . Each tick mark on the x-axis represents 2 units, and each tick mark on the y-axis represents one unit.

109

Trigonometry

y = cos x

y = sin x

h–scale =

h–scale = 2 Cosine curve

Sine curve

y = tan x

h–scale = Tangent curve

To understand the graphs, think back to a circle with radius 1. Because v the radius is 1, the sine function, which is defined as r, simply traces the vertical value of a point as it moves along the circumference of the circle. It starts at 0, moves up as high as 1 when the angle is 2 (90°), retreats to 0, goes down to 1, returns to 0, and begins over again. The graph of the co sine function is identical except for being 2 (90°) out of phase. It records the horizontal value of a point as it moves along the unit circle.

asymptote the line that a curve approaches but never reaches

The tangent is trickier because it concerns the ratio of the vertical value to the horizontal value. Whenever the vertical component is 0, which happens at points along the horizontal axis, the tangent is 0. Whenever the horizontal component is 0, which happens at points on the vertical axis, the tangent is not defined—or infinite. Thus, the tangent has a vertical asymptote every units.

Temperature

A Practical Example. By moving the sine, cosine, and tangent graphs left or right and up or down and by stretching them horizontally and vertically, these trigonometric functions serve as excellent models for many things. For example, consider the function, in which x represents the month of the year and y  54  22 cos(6 (x  7)), in which x represents the average monthly temperature measured in Fahrenheit.

v–scale = 10 h–scale = 1

Month Graph of y = 54 + 22 cos

110





(x – 7)

Turing, Alan

The “parent” function is the cosine, which intercepts the vertical axis at its maximum value. In our model, we find the maximum value shifted 7 units to the right, indicating that the maximum temperature occurs in the seventh month, July. The normal period of the cosine function is 2 units, but our trans formed function is only going 6 as fast, telling us that it takes 12 units, in this case months, to complete a cycle. The amplitude of the parent graph is 1; this means that its highest and lowest points are both 1 unit away from its horizontal axis, which is the mean functional (vertical) value. In our example, the amplitude is 22, indicating that its highest point is 22 units (degrees) above its average and its lowest is 22 degrees below its average. Thus, there is a 44-degree difference between the average temperature in July and the average temperature in January, which is half a cycle away from July. Finally, the horizontal axis of the parent function is the x-axis; in other words, the average height is 0. In this example, the horizontal average has been shifted up 54 units. This indicates that the average spring temperature—in April to be specific—is 54 degrees. So too is the average temperature in October. Combining this with the amplitude, it is found that the average July temperature is 76 degrees, and the average January temperature is 32 degrees. Trigonometric equations often arise from these mathematical models. If, in the previous example, one wants to know when the average temperature is 65 degrees, 65 is substituted for y, and the equation is solved for x. Any of several techniques, including the use of a graph, can work. Similarly, if one wishes to know the average temperature in June, 6 is substituted for x, and the equation is solved for y. S E E A L S O Angles, Measurement of. Bob Horton Bibliography Boyes, G. R. “Trigonometry for Non-Trigonometry Students.” Mathematics Teacher 87, no. 5 (1994): 372–375. Klein, Raymond J., and Ilene Hamilton. “Using Technology to Introduce Radian Measure.” Mathematics Teacher 90, no. 2 (1997): 168–172. Peterson, Blake E., Patrick Averbeck, and Lynanna Baker. “Sine Curves and Spaghetti.” Mathematics Teacher 91, no. 7 (1998): 564–567. Swetz, Frank J., ed. From Five Fingers to Infinity: A Journey through the History of Mathematics. Chicago: Open Court Publishing Company, 1994.

Turing, Alan British Mathematician and Cryptanalyst 1912–1954 In the October 1950 issue of Mind, the brilliant thinker Alan Turing wrote, “We may hope that machines will eventually compete with men in all purely intellectual fields.” Turing believed that machines could mimic the processes of the human brain but acknowledged that people would have difficulty accepting such a machine—a problem that still plagues artificial intelligence today. 111

Turing, Alan

Turing also proposed a test to measure whether a machine could be considered “intelligent.” The widely acclaimed “Turing test” involved a connecting a human by teletype (later a computer keyboard) to either another human or a machine. The first human would then ask questions that are translated through these mechanical links. If the respondent on the other end was indeed a machine, but the human asking questions could not tell whether the responses came from a human or machine, then the machine should be regarded as intelligent. While attending graduate school in mathematics, Turing wrote On Computable Numbers (1936), in which he described hypothetical devices (later dubbed “Turing machines”) that presaged today’s computers. A Turing machine could perform logical operations and systematically read, write, or erase symbols written on paper tape.

In a footnote to his 1936 paper on computable numbers, Alan Turing proposed a machine that could solve mathematical problems—a model for what later would become the modern computer.

Upon completing his doctorate at Princeton University in 1938, Turing was invited to the Institute of Advanced Studies (also at Princeton) to become assistant to John von Neumann, the brilliant mathematician, synthesizer, and promoter of the stored program concept. Turing declined and instead returned to Cambridge, England. Turing began to work for the wartime cryptanalytic headquarters at Bletchley Park, halfway between Oxford and Cambridge, after the British declared war with Germany on September 3, 1939. At Bletchley Park, Turing and his colleagues utilized electric and mechanical forms of decryption to break the code of the Enigma cipher machines, on which almost all German communications were enciphered (computed arithmetically). Turing helped construct decoders (called Bombes) that could more rapidly test key codes until correct combinations were found, cutting the time it took to decipher German codes from weeks to hours. As a result, many Allied convoys were saved, and it has been estimated that World War II might have lasted 2 more years if not for the work at Bletchley Park. Turing’s life was not without tumult or a sad conclusion. In 1952, Turing was tried and convicted of “gross indecency” for being homosexual. He was sentenced to probation and hormone treatments. In June 1954, he died from cyanide poisoning. Although he possessed the cyanide in connection with chemical experiments he was performing, it is widely believed he committed suicide. Turing’s life has inspired writers, artists, and sculptors. Breaking the Code, a play based on his life, opened in London in 1986. Although the dialogue and scenes are mostly invented, the play ably conveys the remarkable life of this extraordinary man. S E E A L S O Computers, Evolution of Electronic; Cryptology. Marilyn K. Simon Bibliography Hinsley, F. H., and Alan Stripp. Codebreakers. New York: Oxford University Press, 1993. Hodges, Andrew. Alan Turing: The Enigma. New York: Simon & Schuster, 1983. Turing, Alan. “Computing Machinery and Intelligence.” Mind 59, no. 236 (1950):433–460.

112

Turing, Alan

Williams, Michael R. A History of Computing Technology. Englewood Cliffs, NJ: Prentice Hall, 1985. Internet Resources

The Alan Turing Home Page. Ed Andrew Hodges. .

113

Undersea Exploration Over the last two centuries there has been an explosion in our knowledge of the global underwater environment. This understanding of the world’s oceans and seas has been accomplished by a variety of methods, including the use of both manned and unmanned (i.e., robotic) vehicles. When people use any of these modern methods to explore the undersea realm, they must, in one way or another, use a mathematical description of the undersea environment in order to be successful. For example, scuba divers must be aware of such quantities as the depth below the surface and the total time they can remain submerged. Each of these quantities is expressed as a number: for example, depth in feet and total dive time in minutes. The use and application of mathematics is therefore an essential part of undersea exploration.

U

Oceanography is the branch of earth science concerned with the study of all aspects of the world’s oceans, such as the direction and strength of currents, and variations in temperatures and depths. Oceanography also encompasses the study of the marine life inhabiting the oceans. As noted earlier, there are multiple ways to explore the oceans and oceanic life. When people and/or machines are sent to directly explore the ocean environment, several different properties of the oceans must be taken into account in order for the dive to be successful and safe. These properties include—but are not limited to—ocean depth and pressure, temperature, and illumination (visibility).

Ocean Properties “Pressure” is a measurement related to force, and is especially useful when dealing with gases or liquids. When an object is immersed in a swimming pool, in a lake, or in the ocean, it experiences a force pushing inward over its entire surface. For instance, a balloon that is inflated, sealed, and then submerged will be pushed inward by water pressure. The pressure on the balloon is the force exerted by the water on every square inch of the balloon’s surface. Pressure is therefore measured as the force-per-unit-area, such as pounds-per-square-inch. Due to Earth’s atmosphere, all objects at sea level experience a pressure of approximately 14.7 pounds-per-squareinch (abbreviated 14.7 lbs/in2) over their entire surface.

IS AN OCEAN DIFFERENT FROM A SEA? The words “ocean” and “sea” are often used interchangeably to refer to any of the large bodies of salt water that together cover a majority of Earth’s surface. The word “undersea” refers to everything that lies under the surface of the world’s seas or oceans, down to the seafloor.

115

Undersea Exploration

atmosphere a unit of pressure equal to 14.7 lbs/in2, which is the air pressure at mean sea level

Perhaps the greatest limitation to underwater exploration is water pressure, which increases with depth (i.e., the deeper one goes, the greater the pressures encountered). Moreover, the pressures within the ocean depths can be immense. A balloon lowered deeper and deeper into the ocean will continuously shrink in size as the water pressure increases. Just under the ocean’s surface the water pressure is 14.7 lbs/in2, equal to one atmosphere. For every 10 meters (about 33 feet) of additional ocean depth, the pressure increases by one atmosphere. “Visibility” refers to how well one can see through the water. As light from the Sun passes through seawater, it is both absorbed and scattered so that visibility due to sunlight decreases rapidly with increasing depth. Below about 100 meters (approximately 330 feet), most areas of the ocean appear dark. “Temperatures” at the ocean surface vary greatly across the globe. Tropical waters can be quite comfortable to swim in, while polar waters are much colder. However, below a depth of about 250 meters most seawater across the world hovers around the freezing point, from 3 C to –1 C.

The Development of Diving Equipment In the past, people who wished to dive down and explore the oceans were limited by the extremes of pressure, visibility and temperature found there. Nevertheless, for thousands of years people have dove beneath the waves to exploit the sea’s resources for things like pearls and sponges. However, divers without the benefit of mechanical devices are limited to the amount of air stored in their lungs, and so the duration of their dives is quite restricted. In such cases, diving depths of around 12 meters (less than 40 feet) and durations of less than two minutes are normal. Various devices have been invented to extend the depth and duration of divers. All these devices provide an additional oxygen supply. Over 2,300 years ago Alexander the Great (356 B.C.E–323 B.C.E) was supposedly lowered beneath the waves in a device that allowed him an undersea view. Alexander may have used an early form of the diving bell, which is like a barrel with one end open and the other sealed. With the closed end at the top, air becomes trapped inside the barrel, and upon being lowered into the water the occupant is able to breathe. About 2,000 years after Alexander’s undersea excursion, Englishman Edmund Halley (1656–1742) constructed a modified diving bell, similar to Alexander’s apparatus, but with additional air supplied to the occupant through hoses. Modern diving stations, like diving bells, are containers with a bottom opening to the sea. Water is prevented from entering the station because, like a diving bell, the air pressure inside the station is equal to the water pressure outside. Researchers can live in a diving station for long periods of time, allowing them immediate access to the ocean environment at depths of up to 100 meters (328 feet).

116

Undersea Exploration

The Emergence of Diving Suits To allow greater freedom of movement when submerged, various underwater suits and breathing devices have been invented. In the early 1800s, diving suits were constructed, and modern versions are still in use. These suits possess a hard helmet with a transparent front for visibility; an airtight garment that keeps water out but still allows movement; and connecting hoses linked to a surface machine that feeds a continuous flow of fresh air to the suit. Over the years, additional improvements have been added to diving suits, such as telephone lines running between the surface and diver for communication. From an exploratory point-of-view, diving suits are limited because of their bulkiness, the cumbersome connecting lines that run to the surface, and the restriction of the diver to the seafloor. Skin diving (or “free diving”) overcomes the mobility problem by allowing the diver to freely swim underwater. As the name implies, most skin divers are not completely covered by a suit, but rather use minimal equipment such as a facemask, flippers, and often a snorkel for breathing. A snorkel has tubes that allow the diver to breathe surface air while underwater. This type of diving (called snorkeling) must be performed close to the surface. As compared to snorkeling, scuba diving considerably increases the diving depth of snorkeling by employing an oxygen supply contained in tanks (scuba is derived from “self contained underwater breathing apparatus”). Varying versions of scuba gear have been invented. Probably the most popular is the aqualung, introduced by the undersea explorer Jacques Cousteau (1910–1997). When the diver begins to inhale, compressed air is automatically fed into her or his mouthpiece by valves designed to assure a constant airflow at a pressure that matches the outside water pressure. The compressed air is contained in cylinders that are carried on the diver’s back. Using conventional mixtures of oxygen and compressed air, divers can safely submerge to around 75 meters (almost 250 feet). With special breathing mixtures, such as oxygen with helium, scuba dives of greater than 150 meters (about 500 feet) have been safely accomplished. To illuminate their surroundings, scuba divers sometimes carry battery-powered lights. Station, suit, and scuba divers utilize compressed gases for breathing. As divers go deeper, the pressure on their lungs increases, requiring that they breathe higher-pressure air. This process forces gases such as nitrogen into the cells of the diver’s body. Divers who do not follow the proper procedure upon returning to the surface risk experiencing a painful, possibly fatal condition known as “the bends.” The bends are the result of the pressurized gases inside the diver’s body being released quickly, like carbon dioxide being released from a shaken soda can. A diver can avoid the bends by ascending to the surface in a controlled and timed manner, so that the gas buildup inside his or her cells is slowly dissipated. This procedure is called decompression. A competent diver uses gauges to be aware at all times of the depth of the dive, and ascends to the surface in a timed manner. It is recommended that divers ascend no faster than 30 feet per minute. For example, if a diver is at a depth of 90 feet, it should take 3 minutes to

USING PROPORTIONS WHEN DIVING Scuba divers know that the deeper the dive, the greater the pressure—a direct proportion. Yet the deeper the dive, the less time they can safely stay underwater to avoid “the bends” —an inverse proportion.

117

Undersea Exploration

Alvin, a manned submersible vehicle, explores the submerged hull of the Titanic.

ascend. The use of numbers and measurements are therefore of utmost importance in all types of diving.

Underseas Exploration Using Submersibles Although scuba divers can explore some shallow parts of the ocean, especially environments such as coral reefs, submersibles allow exploration of every part of the ocean, regardless of depth. Submersibles are airtight, rigid diving machines built for underwater activities, including exploration. They may be classified as being either manned or remotely operated. Manned submersibles require an onboard crew. Unlike scuba or suit divers, the people inside submersibles usually breathe air at sea-level pressure. Therefore, people in submersibles are not concerned with decompressing at the end of a dive. In 1960, the Trieste submersible made a record dive of 10,914 meters (35,800 feet) into an area of the Pacific Ocean known as Challenger Deep. Mathematics was essential for both the construction of Trieste and for its record-breaking dive. For instance, the maximum pressure anticipated for its dives (over 80 tons per-square-inch) had to be correctly calculated, and then the vessel walls had to be built accordingly, or the craft would be destroyed. Today there are many new and sophisticated manned submersibles possessing their own propulsion system, life support system, external lighting, mechanical arms for sampling, and various recording devices, such as video and still cameras. Remotely operated submersibles do not need a human crew onboard and they are guided instead by a person at a different location (hence its name). These vehicles are equipped with video cameras from which the remote operator views the underwater scene. Remote submersibles can byand-large duplicate the operations of a manned submersible but without endangering human life in the case of an accident. Philip Edward Koth (with William Arthur Atkins) 118

Universe, Geometry of

Bibliography Carey, Helen H., and Judith E. Greenberg. Under the Sea. Milwaukee, WI: Raintree Publishers, 1990. Davies, Eryl. Ocean Frontiers. New York: Viking Press, 1980. Oleksy, Walter G. Treasures of the Deep: Adventures of Undersea Exploration. New York: J. Messner, 1984. Stephen, R. J. Undersea Machines. London, U.K.: F. Watts, 1986.

Universe, Geometry of For centuries, mathematicians and physicists believed that the universe was accurately described by the axioms of Euclidean geometry, which now form a standard part of high school mathematics. Some of the distinctive properties of Euclidean geometry follow: • Straight lines go on forever. • Parallel lines are unique. That is, given a line and a point not on that line, there is one and only one parallel line that passes through the given point. • Parallel lines are equidistant. That is, two points moving along parallel lines at the same speed will maintain a constant distance from each other. • The angles of a triangle add to 180°. • The circumference of a circle is proportional to its radius, r, (C  2r) and the area, A, of a circle is proportional to the square of the radius c (A  r 2), where pi () is defined to be approximately 3.14 (2r ). The last four properties are characteristic of a “flat” space that has no intrinsic curvature. (See figure below.)

Flat Geometry

Other types of geometry, called non-Euclidean geometries, violate some or all of the properties of Euclidean geometry. For example, on the surface of a sphere (a positively curved space), the closest thing to a straight path is a great circle, or the path you would follow if you walked straight forward without turning right or left. But great circles do not go on forever: They loop around once and then close up. If “lines” are interpreted to mean “great circles,” the other properties of Euclidean geometry are also false in spherical geometry. Parallel great circles do not exist at all. The positive curvature causes great circles that start out in the same direction to converge. For example, the meridians on a globe converge at the north and south poles. The angles of a spherical triangle add up to more than 180°, and the circumference and area of spherical circles are smaller than 2r and r 2, respectively. (See figure on p. 120.) 119

Universe, Geometry of

Closed Geometry

In the early years of the nineteenth century, three mathematicians— Karl Friedrich Gauss, Janos Bolyai, and Nikolai Lobachevski—independently discovered non-Euclidean geometries with negative curvature. In these geometries, lines that start out in the same direction tend to diverge from each other. The angles of a triangle add to less than 180°. The circumference of a circle grows exponentially faster than the radius. Although negatively curved spaces are difficult to visualize and depict because we are accustomed to looking at the world through “Euclidean eyes,” you can get some idea by looking at the figure directly below.

Open Geometry

Curved Space and General Relativity

✶An oft-repeated aphorism is: “Matter tells space how to curve, and space tells matter how to move.”

For a long time curved geometries remained in the realm of pure mathematics. However, in 1915 Albert Einstein proposed the theory of general relativity, which stated that our universe is actually curved, at least in some places. Any large amount of matter tends to impart a positive curvature to the space around it. The force that we call gravity is a result of the curvature of space.✶ In general relativity, rays of light play the role of lines. Because space is positively curved near the Sun, Einstein predicted that rays of light from a distant star that just grazed the surface of the Sun would bend towards the Sun. (See figure below.) To observers on Earth, such stars would appear displaced from where they really were. Most of the time, of course, the light from the Sun completely obscures any light that comes from distant stars, so we cannot see this effect. But Einstein’s prediction was dramatically confirmed by observations made during the total solar eclipse of 1919. Such “gravitational lensing” effects are now commonplace in astronomy. Observed position of star

Actual position of star

Sun

120

Earth

Universe, Geometry of

Einstein, like many later scientists, was also very interested in the overall geometry of the universe. The stars are, after all, aberrations in a universe that is mostly empty space. If you distributed all the matter in the universe evenly, would the curvature be positive, negative, or zero? Interestingly, on this issue Einstein was not a revolutionary. He assumed, in his first model of the universe, that the overall geometry would be flat. It was a Russian physicist, Alexander Friedmann,✶ who first pointed out in 1922 that Einstein’s equations could be solved just as well by positively or negatively curved universes.

✶Alexander Alexandrovitch Friedmann (1888–1925) was a founder of dynamic meteorology.

Friedmann also showed that the curvature was not just a mathematical curiosity: The fate of the entire universe depended on it. This dependence occurs because the curvature of the universe is closely related to the density of matter. Namely, if there is less than the critical density of matter in the universe, then the universe is negatively curved, but if there is more than the critical density of matter, then it is positively curved. If the universe is negatively curved, it keeps expanding forever because the gravitational attraction caused by the small amount of matter is not enough to rein in the universe’s expansion. But if the universe is positively curved, the universe ends with a “Big Crunch” because the gravitational attraction causes the expansion of space to slow down, stop, and reverse. If the amount of matter is “just right,” then the curvature is neither positive nor negative, and space is Euclidean. In a flat universe, the growth of space slows down but never quite stops.

Big Bang Theory Both of Friedmann’s models of the universe assume that it began with a bang. Although Einstein was skeptical of these theories at first, the Big Bang theory received powerful support in 1929, when Edwin Hubble showed that distant galaxies are receding from Earth at a rate proportional to their distance. For the first time, it was possible to determine the age of the universe. (Hubble’s original estimate was 2 billion years; the currently accepted range is 10 to 15 billion years.) Another piece of evidence for the Big Bang theory came in 1964, in the form of the cosmic microwave background, a sort of afterglow of the Big Bang. The discovery of this radiation, which comes from all directions of the sky, confirmed that the early universe (about 300,000 years after the Big Bang) was very hot, and also very uniform. Every part of the cosmic fireball was at almost exactly the same temperature.

big bang the singular event thought by most cosmologists to represent the beginning of our universe; at the moment of the big bang, all matter, energy, space, and time were concentrated into a single point

After 1964, the Big Bang theory became a kind of scientific orthodoxy. But few people realized that it was really several theories, not one—the question of the curvature and ultimate fate of the universe was still wide open. Over the 1960s and 1970s, a few physicists began to have misgivings about the Friedmann models. One centered on the assumption of homogeneity, which Friedmann (and Einstein) had made as a mathematical convenience. Though the microwave data had confirmed this nicely, there still was no good explanation for it. And, in fact, a careful look at the night sky seems to show the distribution of mass in the universe is not very uniform. Our immediate neighborhood is dominated by the Milky Way. The Milky Way is itself part of the Local Group of galaxies, which in turn lies on the 121

Universe, Geometry of

light-year the distance light travels within a vacuum in one year

fringes of the Virgo Cluster, whose center is 60 million light-years away. Astronomers have found evidence for even larger structures, such as a “Great Wall” of galaxies 300 million light-years across. How could a universe that started out uniform and featureless develop such big clumps? A second problem was the “flatness problem.” In cosmology, the branch of physics that deals with the beginnings (and endings) of the universe, the density of matter in the universe is denoted by Omega (abbreviated O or ). By convention, a flat (Euclidean) universe has a density of 1, and this is called the critical density. In more ordinary language, the critical density translates to about one hydrogen atom per cubic foot, or one poppy seed spread out over the volume of the Earth. It is an incredibly low density. The trouble is that Omega does not stay constant unless it equals 1 or 0. If the universe started out with a density a little less than 1, then Omega would rapidly decrease until it was barely greater than 0. If it started out with a density of 1, then Omega would stay at 1. It is very difficult to create a universe where, after 10 billion years, Omega is still partway between 0 and 1. And yet that is what the observational data suggested: The most careful estimates to date put the density of ordinary matter, plus the density of dark matter that does not emit enough light to be seen through telescopes, at about 0.30. This seemed to require a very “fine-tuned” universe. Physicists view with suspicion any theory that suggests our universe looks the way it does because of a very specific, or lucky, choice of constants. They would rather deduce the story of our universe from general principles.

Revising the Big Bang Theory In 1980, Alan Guth proposed a revised version of the Big Bang that has become the dominant theory, because it answers both the uniformity and flatness paradoxes. And unlike Friedmann’s models, it makes a clear prediction about the geometry of the universe: The curvature is 0 (or very close to it), and the geometry is Euclidean. Guth’s theory, called inflation, actually came out of particle physics, not cosmology. At an unimaginably small time after the Big Bang—roughly a trillionth of a trillionth of a nanosecond—the fundamental forces that bind atoms together went through a “phase transition” that made them repel rather than attract. For just an instant, a billionth of a trillionth of a nanosecond, the universe flew apart at a phenomenal, faster-than-light rate. Then, just as suddenly, the inflationary epoch was over. The universe reached another phase transition, the direction of the nuclear forces reversed, and space resumed growing at the normal pace predicted by the ordinary Big Bang models. Inflation solves the uniformity problem because everything we can see today came from an incredibly tiny, and therefore homogeneous, region of the pre-inflationary universe. And it solves the flatness problem because, like a balloon being stretched taut, the universe lost any curvature that it originally had. The prediction that space is flat has become one of the most important tests for inflation. Until recently, the inflationary model has not been consistent with astronomical observations. Its supporters had to argue that the observed clumps of galaxies, up to hundreds of millions of light-years in size, can be 122

Universe, Geometry of

reconciled with a very homogeneous early universe. They also had to believe that something is out there to make up the difference between the observed Omega of 0.3 and the predicted Omega of 1. A series of new measurements in 1999 and 2000, however, provided some real victories for inflation. First, they showed that the microwave background does have some slight variations, and the size of these variations is consistent with a flat universe. More surprisingly, the data suggest that most of the “missing matter” is not matter but energy, in an utterly mysterious form called dark energy or vacuum energy. In essence, this theory claims that there is something about the vacuum itself that pushes space apart. The extra energy causes space to become flat, but it also makes space expand faster and faster (rather than slower and slower, as in Friedmann’s model). Until these measurements, the theory of vacuum energy was not accepted. It had originally been proposed by Einstein, who later abandoned it and called it his greatest mistake. Yet years later, cosmologists said that the vacuum energy was the most accurately known constant in physics, to 120 decimal places, and composes 70 percent of the matter-energy in our universe. There is no way to tell at this time whether vacuum energy is the final word or whether it will be discarded again. It is clear, though, that the geometry of space will continue to play a crucial role in understanding the most distant past and the most distant future of our universe. S E E A L S O Cosmos; Einstein, Albert; Euclid and His Contributions; Geometry, Spherical; Solar System Geometry, History of; Solar System Geometry, Modern Understandings of. Dana Mackenzie Bibliography Lightman, Alan, and Roberta Brawer. Origins: The Lives and Worlds of Modern Cosmologists. Cambridge, MA: Harvard University Press, 1990. Steele, Diana. “Unveiling the Flat Universe.” Astronomy (August) 2000: 46–50. Weeks, Jeffrey. The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds. New York: Marcel Dekker, 1985.

123

Variation, Direct and Inverse A variable is something that varies among components of a set or population, such as the height of high school students. Two types of relationships between variables are direct and inverse variation. In general, direct variation suggests that two variables change in the same direction. As one variable increases, the other also increases, and as one decreases, the other also decreases. In contrast, inverse variation suggests that variables change in opposite directions. As one increases, the other decreases and vice versa.

V

Direct Variation Consider the case of someone who is paid an hourly wage. The amount of pay varies with the number of hours worked. If a person makes $12 per hour and works two hours, the pay is $24; for three hours worked, the pay is $36, and so on. If the number of hours worked doubles, say from 5 to 10, the pay doubles, in this case from $60 to $120. Also note that if the person works 0 hours, the pay is $0. This is an important component of direct variation: When one variable is 0, the other must be 0 as well. So, if two variables vary directly and one variable is multiplied by a constant, then the other variable is also multiplied by the same constant. If one variable doubles, the other doubles; if one triples, the other triples; if one is cut in half, so is the other. Algebraically, the relationship between two variables that vary directly can be expressed as y  kx, where the variables are x and y, and k represents what is called the constant of proporx y tionality. (Note that this relationship can also be expressed as x  k or y 1 1  k, with k also representing a constant.) In the preceding example, the equation is y  12x, with x representing the number of hours worked, y representing the pay, and 12 representing the hourly rate, the constant of proportionality. Graphically, the relationship between two variables that vary directly is represented by a ray that begins at the point (0, 0) and extends into the first quadrant. In other words, the relationship is linear, considering only positive values. See part (a) of the figure on the next page. The slope of the ray depends on the value of k, the constant of proportionality. The bigger k is, the steeper the graph, and vice versa.

125

Variation, Direct and Inverse

(b)

(a) y = 12x

y

y=

y

v–scale = 20

100 x

v–scale = 5 h–scale = 1

h–scale = 5 x

x

Direct variation

Indirect variation

Inverse Variation When two variables vary inversely, one increases as the other decreases. As one variable is multiplied by a given factor, the other variable is divided by that factor, which is, of course, equivalent to being multiplied by the reciprocal (the multiplicative inverse) of the factor. For example, if one variable doubles, the other is divided by two (multiplied by one-half); if one triples, the other is divided by three (multiplied by one-third); if one is multiplied by two-thirds, the other is divided by two-thirds (multiplied by three-halves). Consider a situation in which 100 miles are traveled. If traveling at an average rate of 5 miles per hour (mph), the trip takes 20 hours. If the average rate is doubled to 10 mph, then the trip time is halved to 10 hours. If the rate is doubled again, to 20 mph, the trip time is again halved, this time to 5 hours. If the average rate of speed is 60 mph, this is triple 20 mph. Therefore, if it takes 5 hours at 20 mph, 5 is divided by 3 to find the travel 5 2 time at 60 mph. The travel time at 60 mph equals 3, or 13 hours. Algebraically, if x represents the rate (in miles per hour) and y represents the time it takes (in hours), this relationship can be expressed as xy  100 100 100 or y  x or x  y. In general, variables that vary inversely can be k k expressed in the following forms: xy  k, y  x, or x  y. hyperbola a conic section; the locus of all points such that the absolute value of the difference in distance from two points called foci is a constant asymptotic describes a line that a curve approaches but never reaches

The graph of the relationship between quantities that vary inversely is one branch of a hyperbola. See part (b) of the figure. The graph is asymptotic to both the positive x- and y-axes. In other words, if one of the quantities is 0, the other quantity must be infinite. For the example given here, if the average rate is 0 mph, it would take forever to go 100 miles; similarly, if the travel time is 0, the average rate must be infinite. Many pairs of variables vary either directly or inversely. If variables vary directly, their quotient is constant; if variables vary inversely, their product is constant. In direct variation, as one variable increases, so too does the other; in inverse variation, as one variable increases, the other decreases. S E E A L S O Inverses; Ratio, Rate, and Proportion. Bob Horton Bibliography Foster, Alan G., et al. Merrill Algebra 1: Applications and Connections. New York: Glencoe/ McGraw Hill, 1995. ———. Merrill Algebra 2 with Trigonometry Applications and Connections. New York: Glencoe/McGraw Hill, 1995. Larson, Roland E., Timothy D. Kanold, and Lee Stiff. Algebra I: An Integrated Approach. Evanston, IL: D.C. Heath and Company, 1998.

126

Vectors

Vectors A vector in the Cartesian plane is an ordered pair (a, b) of real numbers. This is the mathematician’s concise definition of a two-dimensional vector. Physicists and engineers like to develop this concept a bit more for the purpose of applying vectors in their disciplines. Thus, they like to think of the mathematician’s ordered pairs as representing displacements, velocities, accelerations, forces, and the like. Since such things have magnitude and direction, they like to imagine vectors as arrows in the plane whose magnitudes are their lengths and whose directions are the directions that the arrows are pointing. The two small dark arrows shown in part (a) of the drawing below form our point of departure in the study of vectors in the plane. These are called the unit basis vectors, or unit vectors. From them all other vectors arise. To the mathematician, they are simply the ordered pairs (1, 0) and (0, 1). To the physicist, they really are the arrows extending from the origin to the points (1, 0) and (0, 1). The horizontal one is commonly named i and the vertical one is commonly named j. To say that all other vectors in the plane arise from these two means that all vectors are linear combinations of i and j. For instance, the mathematician’s vector (2, 3) is 2(1, 0)  3(0, 1), while the physicist’s arrow with horizontal displacement of 2 units and vertical displacement 3 units is more compactly written as 2i  3j and is represented as in part (b) below. (a)

(b)

y

2i + 3j

y

(c)

(d) (2, 3)

(0, 1) j i 0

(1, 0)

x

0

(1, 0)

(2, 3)

w

v

(0, 1) j i

v+w x

0

w

v

(6, 2)

(6, 2) v+w

0

In the vocabulary of physics, the end of the arrow with the arrowhead is called the “head” of the vector, while the end without the arrowhead is called the “tail.” The tail does not necessarily have to be at the origin as in the picture. When it is, the vector is said to be in standard position, but any arrow with horizontal displacement 2 and vertical displacement 3 is regarded by physicists as 2i 3j, as shown in part (b) of the drawing above. The 2 and 3 are called horizontal and vertical components (respectively) of the vector 2i 3j. By definition, if v  (a, b)  ai  bj and w  (c, d)  ci dj, then v  w  (a  c, b  d)  (a  c)i  (b  d)j. This definition of addition for vectors leads to a very convenient geometrical interpretation. In part (c) of the drawing above, v  2i  3j and w  4i  1j. Then v  w  6i  2j. 127

Vectors

Physicists call their convention for adding geometric vectors “head-totail” addition. Notice in the part (d) of the drawing above that v is in standard position and extends to the point (2, 3). If the tail of w is placed at the head of v, then the head of w ends up at (6, 2). Now if we draw the arrow from the origin to (6, 2), we have an appropriate representation of v  w in standard position. The vector v  w is usually called the resultant vector of this addition. So, in general, the resultant of the addition of two vectors will be represented geometrically as an arrow extending from the tail of the first vector in the sum to the head of the second vector. This convention is often called the parallelogram law because the resultant vector always forms the diagonal of a parallelogram in which the two addend vectors lie along adjacent sides.

Applications of Vectors As an example from physics, consider an object being acted upon by two forces: a 30 pound (lb) force acting horizontally and a 40 lb force acting vertically. The physicist wants to know the magnitude and direction of the resultant force. The schematic diagram below represents this situation.

40 lb.

50 lb.

53.13˚ 0

30 lb.

By the parallelogram law, the resultant of the two forces is the diagonal of the rectangle. The Pythagorean Theorem may be used to calculate that this diagonal has a length of 50, representing a 50-lb. resultant force. The inverse cosine of 30/50 is 53.13°, giving the angle to the horizontal at which the resultant force acts on the object. Consider another example in which vectors represent velocities. An airplane is attempting to fly due east at 600 mph (miles per hour), but a 50mph wind is blowing from the northeast at a 45° angle to the intended due east flight path. If the pilot does not take corrective action, in what direction and with what velocity will the plane fly? The drawing below is a vector representation of the situation. 3.58˚

600mph 566.75mph 45˚ 50mph

Law of Cosines for a triangle with angles A, B, C and sides a, b, c, a2  b2  c2  2bc cos A Law of Sines if a triangle has sides a, b, and c and opposite angles A, B, and C, then sin A/a  sin B/b  sin C/c

128

The vector representing the plane’s intended velocity points due east and is labeled 600 mph, while the vector representing the wind velocity points from the northeast at a 45° angle to the line heading due east. Note the head-to-tail positioning of these two vectors. Now the parallelogram law gives the actual velocity vector for the plane, the resultant vector, as the diagonal of the parallelogram with two sides formed by these two vectors. In this case, the Pythagorean Theorem may not be used, because the triangle formed by the three vectors is not a right triangle. Fortunately, some advanced trigonometry using the so-called Law of Cosines and Law of Sines

Virtual Reality

can be used to determine that the plane’s actual velocity relative to the ground is 566.75 mph at an angle of 3.58° south of the line representing due east. In navigation, angles are typically measured clockwise from due north, so a navigator might report that this plane was traveling at 566.75 mph on a heading, or bearing, of 93.58°. As another example of how mathematicians and physicists use vectors, consider a point moving in the xy-coordinate plane so that it traces out some curve as the path of its motion. As the point moves along this curve, the x and y coordinates are changing as functions of time. Suppose that x  f(t) and y  g(t). Now the mathematician will say that the position at any time t is ( f(t), g(t)) and that the position vector for the point is R(t)  ( f(t), g(t))  f(t)i  g(t)j. The physicist will say that the position vector R(t) is an arrow starting at the origin and ending with the head of the arrow at the point ( f(t), g(t)). (See the figure below.) It is now possible to define the velocity and acceleration vectors for this motion in terms of ideas from calculus, which are beyond the scope of this article. (x, y) = (f(t), g(t)) R(t)

The mathematician’s definition of a vector may be extended to three or more dimensions as needed for applications in higher dimensional space. For example, in three dimensions, a vector is defined as an ordered triple of real numbers. So the vector R(t)  ( f(t), g(t), h(t))  f(t)i  g(t)j  h(t)k could be a position vector that traces out a curve in three-dimensional space. S E E A L S O Flight, Measurements of; Numbers, Complex. Stephen Robinson Bibliography Dolciani, Mary P., Edwin F. Beckenbach, Alfred J. Donnelly, Ray C. Jurgensen, and William Wooten. Modern Introductory Analysis. Boston: Houghton Mifflin Company, 1984. Foerster, Paul A. Algebra and Trigonometry. Menlo Park, CA: Addison-Wesley Publishing Company, 1999. Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.

Virtual Reality “Virtual Reality,” or VR (also known as “artificial reality” (AR) or “cyberspace”), is the creation of an interactive computer-generated spatial environment. This simulated environment can represent what one might encounter in everyday experience, represent pure fantasy, or be a combination of both. Early “first-generation” computer interfaces handled only simple onedimensional (1D) streams of text. The second generation, developed in the late 1970s and early 1980s for two-dimensional (2D) environments, started to use a computer screen’s windows, icons, menus, and pointers (WIMP), including sliders, clicking to select, dragging to move, and so on. The third generation of interfaces is characterized by three-dimensional (3D) models and expressiveness. 129

Virtual Reality

GENERATIONS AND DIMENSIONS OF COMPUTER–HUMAN USER INTERFACES

realtime occuring immediately, allowing interaction without significant delay position-tracking sensing the location and/or orientation of an object multimodal input/output (I/O) multimedia control and display that uses various senses and interaction styles stereographics presenting slightly different views to left and right eyes, so that graphic scenes acquire depth spatial sound audio channels endowed with directional and positional attributes (like azimuth, elevation, and range) and room effects (like echoes and reverberation) bits binary digits: the smallest units of information that a digital computer can represent (high/low, on/off, true/false)

130

Generation/ Dimensions

Mode

Input

Output

First/1D

textual

keyboard line editor

teletype monaural sound

Second/2D

planar

screen editor mouse joystick trackball touchpad light pen

image-based visuals stereo panning

Third/3D

aural

speech understanding head-tracking

speech synthesis MIDI spatial sound

haptic: tactile and kinesthetic

3D joystick, spaceball DataGlove mouse, bird, bat, wand gesture recognition handwriting recognition

tactile displays Braille devices force-feedback displays motion platforms

olfactory

gas detectors

smell emitters

gustatory

??

?

visual

head- and eye-tracking

3D graphic-based visuals stereoscopic systems; head-worn displays holograms vibrating mirrors

VR uses various computer techniques, including realtime 3D computer graphics and animation, position-tracking, and multimodal input/output (I/O), especially stereographics and spatial sound. Fully developed VR will use all the senses: vision (seeing), audition (hearing), haptics (feeling, including pressure, position, temperature, texture, vibration), and eventually olfaction (smell) and gustation (taste). Humans have the capacity to absorb a great deal of data, but traditional two-dimensional computer interfaces rarely generate more than a few hundred bits per second worth of data. Traditional computer interfaces force a user to interact graphically, in the plane of the screen. Virtual reality, however, opens up this interaction between user and computer by creating an 3D-environment in which the user can manipulate volumetric objects and navigate through spaces.

Virtual Reality and Immersive Hypermedia If simple, linear (1D) text, such as a story, is augmented with extra media— such as sound, graphics, images, animation, video, and so on—it becomes multimedia. If this same story is extended with nonlinear attributes—such as annotations, cross-references, footnotes, marginalia, bibliographic citations, and hyperlinks—it becomes hypertext. The combination of multimedia and hypertext is called “hypermedia.” VR is interactive hypermedia because it gives users a sense of immersion or presence in a virtual space, including a flexible perspective or point of view, with the ability to move about.

Virtual Reality

data (presented textually)

non-sequential access: (interactive) links & cross-references (footnotes, bibliographic citations, marginalia...)

hypertext r rtext

voice & sound, graphics & images, animation & video, ...

multimedia

(interactive) hypermedia = (via spatial data organization) {virtual reality, artificial reality, cyberspace}

Multimedia x Hyper text = Hypermedia

“Classic” VR uses a head-worn display (HWD), also known as a headmounted display (HMD), that presents a synthetic environment via stereophonic audio and miniature displays in front of the eyes. Such a helmet, or “brain bucket,” often uses a position tracker to determine the orientation of the wearer and adjust the displays accordingly. Users may also wear “DataGloves,” sets of finger and hand sensors that can be used to virtually grasp, manipulate, and move virtual objects. Full-body VR-suits are manufactured and sold, but they are less common because they are still expensive and cumbersome.

Mixed Reality Reality and virtuality are not opposites, but rather two ends of a spectrum. What is usually thought of as “reality” is actually filled with sources of virtual information (such as telephones, televisions, and computers), and virtuality has many artifacts of the real world, such as gravity. Along the reality-virtuality spectrum are techniques called “mixed reality,” also variously known as or associated with augmented, enhanced, hybrid, mediated, or virtualized reality/virtuality. In a mixed reality system, sampled data (from the “real world”) and synthesized data (generated by computer) are combined. An example of a mixed reality is a computer graphic scene that includes images captured by a camera. Head-mounted displays (HMDs) are important for mixed reality systems. A typical mixed reality system adds simulation to reality by either overlaying computer graphics on a transparent display (“optical see-through,” which uses half-silvered mirrors) or mixing computer graphics with a video signal captured by a camera mounted in front of the face, presenting the result via an opaque display (“video see-through”). Similarly, computergenerated spatial sound can be added to the user’s environment to provide navigation cues. Mixed reality systems encourage “synesthesia,” or crosssensory experiences. For example, infrared heat sensor data could be shifted into the visible spectrum.

131

Virtual Reality

camera phone ...

sampled (with optional synethesia)

tele-existence telepresence human operator

real environment

robot (hardware) remote control

effectors

sensors avatar puppet

artificial reality virtual virtuality simulation (software)

virtual space

synthesized synthesized sound CG ...

Telepresence and Virtual Reality

Liquid Presence A rich multimedia environment, like that enabled by (but unfortunately not always provided by) VR, carries the danger of overwhelming a user with too much information. Control mechanisms are needed to limit the media streams and help the user focus attention. For example, “radio buttons,” like those used in a car to select a station, automatically cancel any previous selection (since only one station is listened to at once). On an audio mixing console, controls are associated with every channel, so they may be selectively disabled with “mute” and exclusively heard with “solo” (in the spirit of “anything not mandatory is forbidden”). Predicate calculus provides a mathematical notation for describing logical relations. For these mixing console functions, the relation can be written: active(sourcex)   mute(sourcex) ^ (y solo(sourcey) ⇒ solo(sourcex) ), where  means “not,” ^ means “and,”  means “there exists,” and ⇒ means “implies.” This expression means that a particular source channel x is active unless it has been explicitly excluded (turned off) or another channel y has been included (focused upon) when the original source x is ignored. Symmetrically, the opposite of an information or media source is a sink. In an articulated sound system, equivalents of mute and solo are deafen and attend: active(sinkx)   deafen(sinkx) ^ (y attend(sinky) ⇒ attend(sinkx) ). In general, a user might want to listen to multiple audio channels simultaneously. For example, one might want to have a private conversation with a small number of people while simultaneously monitoring an ongoing conference as well as a nursery intercom, installing pairs of virtual ears in each interesting space. Such an omnipresence capability is equally useful 132

Virtual Reality

for other sensory modalities. For instance, a guard might generally have access to many video channels from security cameras but may sometimes want to focus on some subset of them or disable some of them (for privacy). Virtual environments allow users to control the quality and quantity of their presence. For example, some interfaces allow a user to cut/paste a representative icon between windows symbolizing rooms, using the pasteboard as a science-fiction teleporter. Coupling such capability with a metaphorical replicator, a user can copy/paste their icon, allowing them to clone their avatar and distribute their presence. A general predicate calculus expression modeling multimedia attributes and such liquid presence is: active(x)   exclude(x) ^ (y include(y) ⇒ include(x) ).

avatar representation of user in virtual space (after the Hindu idea of an incarnation of a deity in human form)

Related Technology and Approaches Important core technologies for VR include computer graphics and animation, multimedia, hypermedia, spatial sound, and force-feedback displays. Trackers (including magnetic, inertial, ultrasonic, and optical), along with global positioning systems (GPSs), are needed to sense users’ positions and, in the case of mixed reality systems, align the displays with the real world. VR systems are often integrated with other advanced interface technology, like speech understanding and synthesis. Virtual Reality is also encouraged by Internet technology, including Java (and Java3D), XML (eXtensible Markup Language), MPEG-4 (for low bit rate videoconferencing and videophony), and QTVR (QuickTime for Virtual Reality, enabling panoramic image-based rendering), as well as the VRML (Virtual Reality Modeling Language) and its successor X3D. Expressive figures are increasingly important in virtual environments, and virtual humans, also known as “vactors” (virtual actors), integrate natural body motion (sometimes using “mocap,” or motion capture, along with kinematics and physics), facial expressions, skin, muscles, hair, and clothes. As such technology matures, its realtime performance will approach in realism the prerendered special effects of Hollywood movies. A related idea is “A-life” (for “artificial life”), the programming of organic or natural-seeming processes, including genetic algorithms, cellular automata, and artificial intelligence.

Applications Applications of virtual reality and mixed reality are unlimited, but are currently focused on allowing users to explore and design spaces before they are built; “infoviz,” or information visualization (of financial or scientific data, for example); simulations (of vehicles, factories, etc.); entertainment and games (like massively multiplayer online role playing games [MMORPG], modern equivalents of the classic Dungeons and Dragons games); conferencing (chatspaces and collaborative systems); and, especially for mixed reality, medicine.

In the Future Future goals of virtual reality include a whole-body user interface paradigm and ultimately a system that allows people to enjoy completely virtual worlds without any restrictions, like the Holodeck on the television show Star Trek: 133

Virtual Reality

The helmet, visor, and glove in this virtual reality (VR) simulation give the user the sensation of driving a car.

chromakey photographing an object shot against a known color, which can be replaced with an arbitrary background (like the weather maps on television newscasts)

Next Generation. Current VR systems are more like 3D extensions to 2D interfaces, in which the world has become a mouse pad and the user has become a mouse. For example, the Vivid Mandala system (www.vivid.com) uses “mirror VR,” in which users watch a chromakey reflection of themselves placed in computer graphics and photographic context, gesturing through fantasy scenarios. On the horizon are full-immersion photorealistic and sonorealistic interfaces in shared virtual environments via high-bandwidth wireless Internet connections, perhaps using visual display technology that writes images directly to the retina from one’s eyewear, or head-mounted projective systems that reflect images back to each wearer via retroreflective surfaces in a room. Virtual reality will be extended by mixed reality and complemented by pervasive or ubiquitous computing (also known as “ubicomp”)—that exploits an environment saturated with networked computers and transparent interfaces, including information furniture and appliances sensitive to human intentions—and wearable computers. S E E A L S O Computer Animation; Computers, Future. Michael Cohen Bibliography Barfield, Woodrow, and Thomas A. Furness III, eds. Virtual Environments and Advanced Interface Design. New York: Oxford University Press, 1995. Begault, Durand R. 3-D Sound for Virtual Reality and Multimedia. Academic Press, 1994. Durlach, Nathaniel I., and Anne S. Mavor, eds. Virtual Reality—Science and Technological Challenges. Washington, D.C.: National Research Council, National Academy Press, 1995. Krueger, Myron W. Artificial Reality II. Reading, MA: Addison-Wesley, 1991. Kunii, Tosiyasu L., and Annie Luciani, eds. Cyberworlds. Tokyo: Springer-Verlag, 1998.

134

Vision, Measurement of

McAllister, David F. Stereo Computer Graphics and Other True 3D Technologies. Princeton, NJ: Princeton University Press, 1993. Nelson, Theodor H. Computer Lib/Dream Machines, 1974. Redmond, WA: Tempus Books of Microsoft Press, 1998. Thalman, Nadia Magnenat and Daniel Thalman, eds. Artificial Life and Virtual Reality. New York: John Wiley & Sons, 1994. Wolfram, Stephen. A New Kind of Science. Champaign, IL: Wolfram Science, 2001. Internet Resources

IEEE-VR: IEEE Virtual Reality Conference. . Sponsored by the IEEE Computer Society. . Java3D: 3D framework for Java. . MPEG Home Page. . Presence: Teleoperators and Virtual Environments. . QuickTime for Virtual Reality. . SIGGRAPH: Special Interest Group on Computer Graphics. . Web 3D Consortium. . XML: eXtensible Markup Language. .

Vision, Measurement of Many people wear glasses or contact lenses. In most cases, this is to correct some inherent defect in vision. The three most common vision defects are farsightedness, nearsightedness, and astigmatism.

Detecting Vision Defects The most common test for detecting vision defects is the visual acuity test. This is the well-known eye chart with a big letter E at the top. The test is conducted by having the patient stand or sit a certain distance from the chart. The person being tested covers one eye at a time and reads the smallest line of text she or he can see clearly. Each line of text is identified by a pair of numbers such as 20/15, 20/40, and so on. The first number is the distance (in feet) from the eye chart. The second number is the distance at which a person with “normal” vision can read that line. So a visual acuity of 20/40 means that a person with normal vision could read that line from 40 feet away. Normal vision refers to average vision for the general population and is considered reasonably good vision. Many people have vision that is significantly better than normal. A visual acuity of 20/15 (common in young people) means that the person can read a line at 20 feet that a person with normal vision could only read at 15 feet. Metric units are used in countries that have adopted the metric system. In those countries, normal visual acuity would be reported as 6/6 (that is, in units of meters where 1 meter equals about 3.28 feet). Minimum levels of visual acuity (for example, required for driving without corrective lenses) would be 6/12. Astigmatism can be easily detected by having the patient examine a chart such as the one shown on the next page. Astigmatism will cause some of the lines to appear darker or lighter. Some lines may appear to be blurry or double. 135

Vision, Measurement of

Presbyopia is a common condition among older adults. This is the progressive inability to focus on nearby objects. This condition is evident when a person tries to read something by holding it at arm’s length. The condition occurs when the lens in the eye becomes less flexible with age.

Corrective Lenses If a person has visual acuity that is less than adequate, corrective lenses may be appropriate. Eyeglasses and contact lenses are available to correct nearsightedness, farsightedness, and astigmatism. The human eye is much like a camera. It has a lens and a diaphragm (called the iris) for controlling the amount of light that enters the eye. Instead of film or a Charge Coupled Device (CCD) chip, the eye has a layer of light-sensitive nerve tissue at the back called the retina. The eye does not have a shutter. However, the human nervous system acts much like a shutter by processing individual images received on the retina at the rate of about 30 images per second. The retina is highly curved (instead of flat, like the film in a camera) but our nervous system is trained from infancy to correctly interpret images from this curved surface. So straight lines are seen as straight lines, vertical lines appear vertical, and the horizon looks like a horizontal line.

focal length the distance from the focal point (the principle point of focus) to the surface of a lens or concave mirror

The bending of light rays necessary to get the image correctly focused on the curved retina is done mostly by the cornea at the front of the eye. Modern surgical techniques have been developed for altering the shape of the cornea to correct many vision defects without the need for corrective lenses. The lens of the eye provides the small adjustments necessary for focusing at different distances. To focus on distant objects, the ciliary muscles around the lens relax and the lens gets thinner. This lengthens the focal length of the lens so that distant objects are correctly focused. To focus on objects that are near, the ciliary muscles contract, forcing the lens to get thicker and shortening its focal length. The closest distance at which objects can be sharply focused is called the near point. The far point is the maximum distance at which objects can be seen clearly. An eye is considered normal with a near point of 25 cm (centimeters) and a far point at infinity (able to focus parallel rays). However, these distances change with age. Children can focus on objects as close as 10 cm, while older adults may be able to focus on objects no nearer than 50 cm. Calling any eye “normal” is somewhat misleading, because 67 percent of adults in the United States use glasses or other corrective lenses. Farsightedness, or hyperopia, refers to a vision defect that causes the eye to be unable to focus on nearby objects because the eye is either too short or the cornea is the wrong shape. For persons with hyperopia, the near point is greater than 25 cm. A similar vision defect is known as pres-

136

Vision, Measurement of

byopia. Hyperopia is due to a misshapen eyeball or cornea, whereas presbyopia is due to the gradual stiffening of the crystalline lens. Presbyopia also prevents people from being able to focus on nearby objects, but it affects only the near point. Converging lenses (magnifying lenses) will correct for both hyperopia and presbyopia by making light rays appear to come from objects that are farther away. Nearsightedness, or myopia, refers to the vision defect that causes an eye to be able to only focus on nearby objects. The far point will be less than infinity. It is caused by an eye that is too long or by a cornea that is incorrectly shaped. In both cases, the images of distant objects fall in front of the retina and appear blurry. Diverging lenses will correct this problem, because they cause parallel rays to diverge as if they came from an object that is closer.

Nearsighted eye/corrected with diverging lens

Farsighted eye/corrected with converging lens

Astigmatism is caused by a lens or cornea that is somewhat distorted so that point objects appear as short lines. The image may be spread out horizontally, vertically, or obliquely. Astigmatism is corrected by using a lens that is also “out of round.” The lens is somewhat cylindrical instead of spherical. Frequently, astigmatism occurs along with hyperopia or myopia, so the lens must be ground to compensate for both problems. Contact lenses work in a way very similar to eyeglasses. However, they have one advantage over eyeglasses. Since the contact lens is much closer to the eye, less correction is needed. This gives the user of contact lenses a wider field of view. Calculating the power of corrective lenses can be done in some cases using the familiar lens formula. If the focal length of a lens is f, the distance from the lens to the object is do, and the distance from the lens to the image is di, then: 1 1 1      f di do As an example, suppose a farsighted person has a near point of 100 cm. What power of glasses will this person need to be able to read a newspaper at 25 cm? To solve this we must recognize that the image will be virtual (as in a magnifying glass) so di, will have a negative value. The glasses must provide an image that appears to be 100 cm away when the object is 25 cm away. 1 1 1 1 1 1            f di do 100 25 33 137

Volume of Cone and Cylinder

diopter a measure of the power of a lens or a prism, equal to the reciprocal of its focal length in meters

The power, P, of a lens in diopters (D) is P  1 1   3.0D. S E E A L S O Light. P  f   0.3333

1  f

when f is in meters, so Elliot Richmond

Bibliography Epstein, Lewis Carroll. Thinking Physics. San Francisco: Insight Press, 1990. Giancoli, Douglas C. Physics, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1991. Hewitt, Paul. Conceptual Physics. Menlo Park, CA: Addison-Wesley, 1992.

Volume of Cone and Cylinder Picture a rectangle divided into two right triangles by a diagonal. How is the area of the right triangle formed by the diagonal related to the area of the rectangle? The area of any rectangle is the product of its width and length. For example, if a rectangle is 3 inches wide and 5 inches long, its area is 15 square inches (length times width). The figure below shows a rectangle “split” along a diagonal, demonstrating that the rectangle can be thought of as two equal right triangles joined together. The areas of rectangles and right triangles are proportional to one another: a rectangle has twice the area of the right triangle formed by its diagonal.

In a similar way, the volumes of a cone and a cylinder that have identical bases and heights are proportional. If a cone and a cylinder have bases (shown in color) with equal areas, and both have identical heights, then the volume of the cone is one-third the volume of the cylinder.

h

h

Imagine turning the cone in the figure upside down, with its point downward. If the cone were hollow with its top open, it could be filled with a liquid just like an ice cream cone. One would have to fill and pour the contents of the cone into the cylinder three times in order to fill up the cylinder. The figure above also illustrates the terms height and radius for a cone and a cylinder. The base of the cone is a circle of radius r. The height of 138

Volume of Cone and Cylinder

the cone is the length h of the straight line from the cone’s tip to the center of its circular base. Both ends of a cylinder are circles, each of radius r. The height of the cylinder is the length h between the centers of the two ends. The volume relationship between these cones and cylinders with equal bases and heights can be expressed mathematically. The volume of an object is the amount of space enclosed within it. For example, the volume of a cube is the area of one side times its height. The figure below shows a cube. The area of its base is indicated in color. Multiplying this (colored) area by the height L of the cube gives its volume. And since each dimension (length, width and height) of a cube is identical, its volume is L  L  L, or L3, where L is the length of each side.

L

L L

The same procedure can be applied to finding the volume of a cylinder. That is, the area of the base of the cylinder times the height of the cylinder gives its volume. The bases of the cylinder and cone shown previously are circles. The area of a circle is r 2, where r is the radius of the circle. Therefore, the volume Vcyl is given by the equation: Vcyl  r 2h (area of its circular base times its height) where r is the radius of the cylinder and h is its height. The volume of the cone (Vcone) is one-third that of a cylin1 der that has the same base and height: Vcone3r 2h. The cones and cylinders shown previously are right circular cones and right circular cylinders, which means that the central axis of each is perpendicular to the base. There are other types of cylinders and cones, and the proportions and equations that have been developed above also apply to these other types of cylinders and cones. Philip Edward Koth (with William Arthur Atkins) Bibliography Abbott, P. Geometry. New York: David Mckay Co., Inc., 1982. Internet Resources

The Method of Archimedes. American Mathematical Society. .

139

Weather Forecasting Models The weather has an astonishing impact on our lives, ranging from the frustration of being caught in a sudden downpour to the trillions of dollars spent in weather-sensitive businesses. Consequently, a great deal of time, effort, money, and technology is used to predict the weather. In the attempt to improve weather prediction, meteorologists rely on increasingly sophisticated computers and computer models. When researchers created the first computer models of Earth’s atmosphere in the mid-twentieth century, they worked with computers that were extremely limited compared to the supercomputers that exist today. As a result, the first weather models were oversimplified, although they still provided valuable insights. As computer technology advanced, more of the factors that influenced the atmosphere, such as physical variables, could be taken into account, and the complexity and accuracy of weather forecast models increased.

W meteorologists people who study the atmosphere in order to understand weather and climate

Today’s computer-enhanced weather forecasts can be successful only if the data observations are accurate and complete. Ground-based weather stations are equipped with many instruments, including barometers (which measure atmospheric pressure, or the weight of the air); wind vanes (which measure wind direction), anemometers (which measure wind velocity, or speed); hygrometers (which measure the moisture, or humidity, in the air); thermometers; and rain gauges. Data obtained through these instruments are combined with data from aircraft, ships, satellites, and radar networks. These data are then put into complex mathematical equations that model the physical laws governing atmospheric conditions. Solving these equations with the aid of computers yields a description—a forecast—of the atmosphere derived from its current state (that is, the initial values). This can then be interpreted in terms of weather—rain, temperature, sunshine, and wind.

Early Attempts to Forecast Weather Several decades prior to the advent of the modern computer, Vilhelm Bjerknes, a Norwegian physicist-turned-meteorologist, advocated a mathematical approach to weather forecasting. This approach was based on his belief that it was possible to bring together the full range of observation and theory to predict weather changes. British scientist Lewis Fry Richardson was the first person to attempt to work out Bjerknes’s program. During and

Early computer models relied heavily on weather station data and measurements obtained from weather balloons. A gondola carries instruments that measure wind speed, temperature, humidity, and other atmospheric conditions.

141

Weather Forecasting Models

algorithm a rule or procedure used to solve a mathematical problem

shortly after World War I, he went on to devise an algorithmic scheme of weather prediction. Unfortunately, Richardson’s method required 6 weeks to calculate a 6-hour advance in the weather. Moreover, the results from his method were inaccurate. Consequently, Richardson’s work, which was widely noticed, convinced contemporary meteorologists that a computational approach to weather prediction was completely impractical. Yet it laid the foundation for numerical weather prediction. Following World War II, a Hungarian-American mathematician named John von Neumann began making plans to build a powerful and versatile electromechanical computer devoted to the advancement of the mathematical sciences. Von Neumann’s objective was to demonstrate, through a particular scientific problem, the revolutionary potential of the computer. He chose weather prediction for that problem, and in 1946 established the Meteorology Project at the Institute for Advanced Study in Princeton, New Jersey, to work on its resolution.

fluid dynamics the science of fluids in motion

chaos theory the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems

The Chaos Connection. By the 1960s the work of American atmospheric sciences researcher Edward Lorenz showed that no matter how good the data or forecasting methods, accurate day-by-day forecasts of the weather may not be possible for future periods beyond about two weeks. Because Earth’s atmosphere follows the complex, nonlinear rules of fluid dynamics, the smallest error in the determination of initial conditions (such as wind, temperature, and pressure) will amplify because of the nonlinear dynamics of atmospheric processes, so that a forecast at some point becomes nearly useless. This dependence of prediction on the precision of input values is a basic tenet of mathematical chaos theory. The fundamental ideas of chaos theory are captured in what is known as the “butterfly effect,” which, roughly stated, holds that a butterfly flapping its wings can set in motion a storm on the other side of the world. Yet with the next flap, nothing of meteorological significance may happen. Hence, chaos places a limit on the accuracy of computer-enhanced weather forecasts as the forecast period and geographic scale increases.

Relationship between Forecasting and Computers The desire to make better weather predictions coincided with the evolution of today’s modern electronic computers. In fact, computers and meteorology have helped shape one another. Moreover, weather forecasting has helped drive the development of better multimedia software, both for forecasting purposes and for communicating forecasts to specific audiences. Forecasting the weather by computer is called numerical weather prediction. To make a skillful forecast, weather forecasters must understand the strengths and limitations of computer models and recognize that models can still be mistaken. Forecasters study the output from models over time and compare the forecast output to the weather that actually occurs. This is how they determine the biases, strengths, and weaknesses of the models. They also may modify what they learn from computer models based on their own experience forecasting the weather through physical and dynamical processes. Hence, good forecasters must not only have knowledge of meteorological variables, but they must also be able to use their intuition.

142

Weather Forecasting Models

Increasingly powerful computers have enabled more sophisticated models of numerical weather prediction. Here a geophysicist studies weather patterns using maps and visualization software.

Improvements in Forecasts. Continued improvements in computer technology allows more of the dynamical and physical factors influencing the atmosphere to be added to models, thereby improving weather forecasts, including those for specific conditions. For example, some forecasting models help predict tropical weather features such as hurricanes, typhoons, and monsoons. Other models help forecast smaller-scale features such as thunderstorms and lightning. As these forecasting models improve, forecasters should be able to issue more accurate and timely storm warnings and advisories. Improvements in Communication. Even if weather data were extraordinarily precise, weather forecasts would have little meaning if consumers of the data did not receive them in an effective and timely manner. Local and national weather reports demonstrate the strong relationship between weather forecasting and computer technology. Today’s state-of-the-art visualization techniques for weather forecasts, including three-dimensional data, time-series, and virtual scenery, have made the weather segment a highly watchable part of television newscasts. A three-dimensional representation of weather patterns is typically superimposed on local and national geographic maps. With a click of a computer mouse, cloud formations and fronts appear, and predictions about temperature, precipitation, and other weather-related data march across the screen. The twentieth century saw the inventions of radio broadcasting, television, and the Internet. The twenty-first century undoubtedly will bring other innovations in technology and media that will play an equally significant role

143

Weather, Violent

in the dissemination of weather information. S E E A L S O Chaos; Computer Simulations; Predictions; Temperature, Measurement of; Weather, Violent. Marilyn K. Simon Bibliography Hodgson, Michael. Weather Forecasting. Guildford, CT: The Globe Pequot Press, 1999. Lockhart, Gary. The Weather Companion. New York: John Wiley & Sons, 1988. Lorenz, Edward. The Essence of Chaos. Seattle: University of Washington Press, 1996. Internet Resources

American Meteorological Society. . “Forecasting by Computers.” User Guide to ECMWF. ECMWF—European Centre for Medium-Range Weather Forecasts. . Ostro, Steve. “Weather Forecasting in the Next Century.” .

Weather, Violent

meteorologist a person who studies the atmosphere in order to understand weather and climate

Violent weather can occur anywhere. Depending on where an outbreak occurs, there will be different types of storms. However, not all storms are violent. There are, for example, thunderstorms that are not severe, and tornadoes that do not cause extensive damage. Common measurement scales do not provide an understanding of the strength of a weather disturbance. Consequently, meteorologists must use different scales to measure the strength of weather systems, and to make predictions about when violent weather may strike.

Thunderstorms There are three ingredients necessary for a thunderstorm to form: 1. moisture must be present in the lower levels of the atmosphere; 2. cold air must be present in the upper atmosphere; and 3. there must be a catalyst to push the warm air into the cold air. This catalyst is usually in the form of a front, which is the interface between air masses at different temperatures. As the warm air rises it cools and some of the water vapor turns into clouds. Eventually the air mass will reach an area where it is the same temperature as the area surrounding it, and thunderstorms can occur. Thunderstorms can be measured as strong or severe. Some thunderstorms, though, may be neither. Severe thunderstorms are classified as having winds greater than or equal to 58 mph (miles per hour), and/or hail greater than or equal to three-quarters of an inch. If a severe thunderstorm is strong enough, it can become a supercell thunderstorm. A supercell thunderstorm is the type that is most likely to spawn a tornado. In a supercell, as the air moves upward it begins to rotate, which does not happen in other types of thunderstorms. When the whole cell rotates, it is called a mesocyclone.

144

Weather, Violent

Tracking Thunderstorms. Many meteorologists use Doppler radar to track thunderstorms. Christian Doppler suggested the idea behind Doppler radar over 150 years ago. It differs from traditional weather radar because it can detect not only where a storm is located, but also how fast the air inside the storm is moving. Doppler radar transmits pulses of waves that are reflected by water droplets in the air. Some of the energy reflected by the droplets then returns to the radar unit and shows how the storm is moving and what it looks like. Doppler radar also shows where moisture is located in a storm because where there is a higher moisture content, there is greater reflectivity since there is more for the waves to reflect off of. The ability to determine the structure of storms is particularly useful in the detection of severe weather. For example, on radar, areas of a storm containing hail will display denser than areas with just rain. Doppler radar is therefore an invaluable tool in storm detection and prediction. Meteorologists can monitor a storm across large areas and make sure the public stays informed about its progress.

Tornadoes Tornadoes are formed in thunderstorms when winds meet at varying angles. If low-level winds flow in a southerly direction and upper-level wind flows towards the west, the air masses can interact violently. The two paths will begin to swirl against each other, forming a vortex. As this vortex increases it will appear as a funnel to someone on the ground. Once this funnel touches the ground, it becomes a tornado. One of the recent scales created to measure the severity of a tornado is the Fujita scale. Developed in 1971, the Fujita scale is used to measure the intensity of a tornado by examining the damage caused after it has passed over man-made structures. Tetsuya Theodore Fujita (1920–1998) took into account the wind speed of a tornado and the damage it caused, and from that he was able to assign a number to represent the intensity of the tornado. On the Fujita scale, an F2 tornado is classified as a significant tornado with winds between 113 and 157 mph. An F2 tornado can cause considerable damage. Roofs may be torn off frame houses, mobile homes may be demolished, and large trees can be uprooted. In an F5 tornado winds can reach between 261 and 318 mph. A tornado of this strength can lift frame houses from their foundation and carry them considerable distances, send automobile-sized objects through the air, and cause structural damage to steel-reinforced structures.

HOW REALISTIC IS HOLLYWOOD? It is possible to measure a tornado from the inside, just as the characters were trying to do in the 1996 movie Twister. The idea for the device that they used, Dorothy, actually came from a similar instrument, TOTO (TOtable Tornado Observatory), invented by Howie Bluestein. TOTO was retired after 1987 and is now on display in Silver Spring, Maryland, at a facility of the National Oceanic and Atmospheric Administration.

Not long after the inception of the Fujita scale, it was combined with the work of Allen Pearson to also consider the components of path length and width. Adding these two measurements together provides more accurate tornado readings. The Fujita scale is not perfect. Wind speed does not always accurately correlate with the amount of damage caused by a tornado. For example, a tornado with a great intensity that touches down in a rural area may rank lower on the Fujita scale than a less intense tornado that touches down in an urban area. This is because a tornado is a rural area could cause less damage

145

Weather, Violent

Scientists use current data measurements and historical information to predict the likelihood of tornadoes.

since there is less to be damaged. An accurate F-scale rating often depends on the experience of the person who assigns the rating. Though the Fujita scale does have flaws, it is an improvement over the rating system that had previously been in place. The scale is simple enough to be put into daily practice without substantial added expense, and a rating can be assigned in a reasonable amount of time. Predicting Tornadoes. While tornadoes can occur during any time of year in the United States, they most frequently occur during the months of March through May. Organizations such as the National Weather Service keep detailed records of when and where tornadoes occur. From these records, they can make predictions about when the most favorable times are for tornadoes to occur in different areas of the country. Different areas have different times when tornado development is most likely. For example, by keeping records, it has been shown that the mid-section of the country is much more likely to have tornadoes than the west. The amount of tornadoes seen in the mid-section has earned it the nickname “tornado alley.”

Hurricanes Hurricanes form when there is a pre-existing weather disturbance, warm tropical oceans, and light winds. If these conditions persist, they can combine and produce violent wind, waves, and torrential rain. A hurricane is a type of tropical cyclone. These cyclones are classified by their top wind speeds as a tropical depression, tropical storm, or a hurricane. In a hurricane, walls of wind and rain form a circle of spinning wind and water. In the very center of the storm there is a calm called the “eye.” At the edges of the eye, the winds are the strongest. Hurricanes cover huge areas of water and are the source of considerable damage when they come onshore.

146

Weather, Violent

Like tornadoes, hurricanes are classified according to their wind speed. They are measured on the Saffir-Simpson hurricane scale. The number rating assigned from the scale is based on a hurricane’s intensity. A category one hurricane has winds between 74 and 95 mph. This type of hurricane will not have a very high storm surge, and will cause minimal damage to building structures. In a category five hurricane, top sustained winds will be greater than 155 mph. The storm surge in a hurricane like this can be greater than 18 feet above normal, flooding will occur, and evacuation will be necessary in low-lying areas. The number assigned to a hurricane is not always a reliable indicator of how much damage a hurricane can cause. Lower-category storms can cause more damage than high-category storms depending on where they make landfall. In most hurricanes, loss of life occurs due to flooding, and lesser hurricanes, and even tropical storms, can often bring more extensive rainfall than a higher-category hurricane.

storm surge the front of a hurricane, which bulges because of strong winds; can be the most damaging part of a hurricane

Predicting Hurricanes. For years meteorologists have been tracking and keeping records on where and when hurricanes occur. The official start of hurricane season in the Atlantic Basin is June 1. The peak time for hurricane formation is from mid-August to late October. Hurricanes can form outside of the season, but it is uncommon. Scientists have been able to designate a season for hurricanes from careful observation of storms over many years. Not only can scientists predict the favorable time of year for hurricanes to form, but they can also make predictions on the path a hurricane may take depending on where it forms at certain times in the season. To make these predictions, scientists must use statistical analysis to find out probabilities on where and when a hurricane can form.

Floods Floods happen when the ground becomes so saturated with moisture that it cannot hold any more. Often, floods occur when too much rain falls in a short period of time, or when snow melts quickly. Initially the ground can absorb the moisture, but eventually there is a point where the soil cannot absorb anymore. At this point, water begins to collect above ground, and flooding occurs. A flood is measured by how much the water is above the flood stage. Flood stages are different everywhere. Measurements of a flood can be done in inches, feet, or larger units depending on the severity. We have also begun to quantify a flood by the economic damage it causes. Predicting Floods. The prediction of floods can be statistically described by frequency curves. Flood frequency is expressed as the probability that a flood of a given magnitude will be equaled or exceeded in any one year. The 100-year flood, for example, is the peak discharge that is expected to be equaled or exceeded on the average of once in 100 years. In other words, there is a 1 percent chance that a flood of peak magnitude will occur in a given year. Similarly, a 50-year flood has a 2 percent chance of occurring in any given year, and a 25-year flood has a 4 percent chance. This recurrence interval represents a long-term average, and these types of floods

147

Web Designer

could occur with a greater or lesser frequency. Many floods that people consider to be record-breaking are actually much lower in magnitude. Violent weather is unavoidable. With reliable measurements and predictions, though, humans can increase their preparedness to minimize the damage and destruction violent weather can cause. S E E A L S O Predictions; Weather Forcasting Models. Brook Ellen Hall Internet Resources Hurricane Basics. National Hurricane Center. . Questions and Answers about Thunderstorms. National Severe Storms Laboratory. . The Fujita Scale of Tornado Intensity. The Tornado Project. . The Saffir-Simpson Hurricane Scale. National Hurricane Center. . Tracking Severe Weather with Doppler Radar. Forecast Systems Laboratory. . Tracking Tornadoes. The Weather Channel. .

Web Designer World Wide Web the part of the Internet allowing users to examine graphic “web” pages

geometry the branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, planes, and solids Hypertext Markup Language the computer markup language used to create documents for the World Wide Web

algebra the branch of mathematics that deals with variables or unknowns representing the arithmetic numbers

A web designer puts information on the World Wide Web in the form of web pages for use by individuals and businesses. Web designers use the principles that a graphic artist would use to create their artwork. The difference, however, is that a web designer’s tool is a computer, and their “paints” are computer software. Web designers use mathematics in a variety of ways. The drawing applications used by web designers require a thorough knowledge of geometry. Additional applications require knowledge of spatial measurements and the ability to fit text and pictures in a page layout. Web designers who use source codes such as Hypertext Markup Language (HTML) need to know basic math for calculating the widths and heights of objects and pages. Accounting and bookkeeping are important areas of math that web designers should study if they plan to establish their own businesses. Because web designers must be able to plan, design, program, and maintain web sites, a knowledge of computers and how they work is very important. Most computer science courses require algebra and geometry as prerequisites. S E E A L S O Computer Graphic Artist; Internet. Marilyn Schwader Bibliography McGuire-Lytle, Erin. Careers in Graphic Arts and Computer Graphics. New York: The Rosen Publishing Group, Inc., 1999.

148

Zero The idea of nothingness and emptiness has inspired and puzzled mathematicians, physicists, and even philosophers. What does empty space mean? If the space is empty, does it have any physical meaning or purpose?

Z

From the mathematical point of view, the concept of zero has eluded humans for a very long time. In his book, The Nothing That Is, author Robert Kaplan writes, “Zero’s path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travellers who first brought it to the West.” But our own familiarity with zero makes it difficult to imagine a time when the concept of zero did not exist. When the last pancake is devoured and the plate is empty, there are zero pancakes left. This simple example illustrates the connection between counting and zero. Counting is a universal human activity. Many ancient cultures, such as the Sumerians, Indians, Chinese, Egyptians, Romans, and Greeks, developed different symbols and rules for counting. But the concept of zero did not appear in number systems for a long time; and even then, the Roman number system had no symbol for zero. Sometime between the sixth and third centuries B.C.E., zero made its appearance in the Sumerian number system as a slanted double wedge. To appreciate the significance of zero in counting, compare the decimal and Roman number system. In the decimal system, all numbers are composed of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. After counting to nine, the digits are repeated in different sequences so that any number can be written with just ten digits. Also, the position of the number indicates the value of the number. For example, in 407, 4 stands for four hundreds, 0 stands for no tens, and 7 stands for seven. The Roman number system consists of the following few basic symbols: I for 1, V for 5, and X for 10. Here are some examples of numbers written with Roman numerals. IV  4

XV  15

VIII  8

XX  20

XIII  13

XXX  30

Without a symbol for zero, it becomes very awkward to write large numbers. For 50, instead of writing five Xs, the Roman system has a new symbol, L. 149

Zero

Performing a simple addition, such as 33  22, in both number systems further shows the efficiency of the decimal system. In the decimal number system, the two numbers are aligned right on top of each other and the corresponding digits are added. 33  22 55 In the Roman number system, the same problem is expressed as XXXIII  XXII, and the answer is expressed as LV. Placing the two Roman numbers on top of each other does not give the digits LV, and therefore when adding, it is easier to find the sum with the decimal system.

Properties of Zero All real numbers, except 0, are either positive (x  0) or negative (x  0). But 0 is neither positive nor negative. Zero has many unique and curious properties, listed below. Additive Identity: Adding 0 to any number x equals x. That is, x  0  x. Zero is called the additive identity. Multiplication property: Multiplying any number b by 0 gives 0. That is, b  0  0. Therefore, the square of 0 is equal to zero (02  0). Exponent property: Any number other than zero raised to the power 0 equals 1. That is, b0  1. Division property: A number cannot be divided by 0. Consider the problem 12/0  x. This means that 0  x must be equal to 12. No value of x will make 0  x  12. Therefore, division by 0 is undefined.

Undefined Division Because division by 0 is undefined, many functions in which the denominator becomes 0 are not defined at certain points in their domain sets. For 1 1  is not defined at x instance, f(x)  x is not defined at x  0; g(x)   (x  1) 1    1; and h(x)  (x2 1) is not defined at either x  1 or x  1. 1

Even though the function f(x)  x is not defined at 0, it is possible to see the behavior of the function around 0. Points can be chosen close to 0; for instance, x equal to 0.001, 0.0001, and 0.00001. The function values at these points are f(0.001)  1/0.001  1,000; f(0.0001)  10,000; and f(0.00001)  100,000. As x becomes smaller and approaches 0, the function values become 1 larger. In fact, the function f(x)  x grows without bound; that is, the function values has no upper ceiling, or limit, at x  0. In mathematics, this be1 havior is described by saying that as x approaches 0, the function f(x)  x approaches infinity.

Approaching Zero 1

1

1

1

1

1

1

Consider a sequence of numbers 1, 2, 3, 4, 5, 6, 7, . . . , n, which in decimal notation is expressed as 1, 0.5, 0.33, 0.25, 0.2, 0.16, 0.14, and so on.

150

Zero 1

Each number in the sequence {n} is called a term. As n becomes larger, 1 becomes increasingly smaller. When n  10,000, n is 0.0001.

1  n

1

The sequence n approaches 0, but its terms never equals 0. However, the terms of the sequence can be as close to 0 as wanted. For instance, it is possible for the terms of the sequence to get close enough to 0 so that the 1  or 106. If one difference between the two is less than a billionth,  1,000,000 1   1,000,000, then the sequence terms will be smaller than takes n   106 6 10 . S E E A L S O Division by Zero; Limit. Rafiq Ladhani Bibliography Dugopolski, Mark. Elementary Algebra, 3rd ed. Boston: McGraw-Hill, 2000. Dunham, William. The Mathematical Universe. John Wiley & Sons Inc., 1994. Kaplan, Robert. The Nothing That Is. New York: Oxford University Press, 1999. Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.

151

Photo and Illustration Credits The illustrations and tables featured in Mathematics were created by GGS Information Services. The photographs appearing in the text were reproduced by permission of the following sources:

Volume 4 AP/Wide World Photos: 3, 34, 52, 55, 87, 118, 134, 146; Will/Deni McIntyre/ Photo Researchers, Inc.: 6; New York Public Library Picture Collection: 20; Corbis-Bettmann: 25; © Martha Tabor/Workins Images Photographs: 29, 143; Archive Photos: 31, 45, 85; Robert J. Huffman. Field Mark Publications: 36; U.S. National Aeronautics and Space Administration (NASA): 41; Photograph by Tony Ward. Photo Researchers, Inc.: 42; David Schofield/ Trenton Thunder: 56; Associated Press/AP/Wide World Photos: 68; Martha Tabor/Working Images Photographs: 70; Photograph by Robert J. Huffman. Field Mark Publications: 79; Lowell Observatory: 84; © Roger Ressmeyer/ Corbis: 88; Dana Mackenzie: 90, 93; © John Slater/Corbis: 96; Sara Matthews/ SGM Photography: 99; Photo Researchers, Inc.: 112; © Bettmann/Corbis: 141.

153

Glossary abscissa: the x-coordinate of a point in a Cartesian coordinate plane absolute: standing alone, without reference to arbitrary standards of measurement absolute dating: determining the date of an artifact by measuring some physical parameter independent of context absolute value: the non-negative value of a number regardless of sign absolute zero: the coldest possible temperature on any temperature scale; 273° Celsius abstract: having only intrinsic form abstract algebra: the branch of algebra dealing with groups, rings, fields, Galois sets, and number theory acceleration: the rate of change of an object’s velocity accelerometer: a device that measures acceleration acute: sharp, pointed; in geometry, an angle whose measure is less than 90 degrees additive inverse: any two numbers that add to equal 1 advection: a local change in a property of a system aerial photography: photographs of the ground taken from an airplane or balloon; used in mapping and surveying aerodynamics: the study of what makes things fly; the engineering discipline specializing in aircraft design aesthetic: having to do with beauty or artistry aesthetic value: the value associated with beauty or attractiveness; distinct from monetary value algebra: the branch of mathematics that deals with variables or unknowns representing the arithmetic numbers algorithm: a rule or procedure used to solve a mathematical problem algorithmic: pertaining to an algorithm ambiguity: the quality of doubtfulness or uncertainty analog encoding: encoding information using continuous values of some physical quantity 155

Glossary

analogy: comparing two things similar in some respects and inferring they are also similar in other respects analytical geometry: describes the study of geometric properties by using algebraic operations anergy: spent energy transferred to the environment angle of elevation: the angle formed by a line of sight above the horizontal angle of rotation: the angle measured from an initial position a rotating object has moved through anti-aliasing: introducing shades of gray or other intermediate shades around an image to make the edge appear to be smoother applications: collections of general-purpose software such as word processors and database programs used on modern personal computers arc: a continuous portion of a circle; the portion of a circle between two line segments originating at the center of the circle areagraph: a fine-scale rectangular grid used for determining the area of irregular plots artifact: something made by a human and left in an archaeological context artificial intelligence: the field of research attempting the duplication of the human thought process with digital computers or similar devices; also includes expert systems research ASCII: an acronym that stands for American Standard Code for Information Interchange; assigns a unique 8-bit binary number to every letter of the alphabet, the digits, and most keyboard symbols assets: real, tangible property held by a business corporation including collectible debts to the corporation asteroid: a small object or “minor planet” orbiting the Sun, usually in the space between Mars and Jupiter astigmatism: a defect of a lens, such as within an eye, that prevents focusing on sharply defined objects astrolabe: a device used to measure the angle between an astronomical object and the horizon astronomical unit (AU): the average distance of Earth from the Sun; the semi-major axis of Earth’s orbit asymptote: the line that a curve approaches but never reaches asymptotic: pertaining to an asymptote atmosphere (unit): a unit of pressure equal to 14.7 lbs/in2, which is the air pressure at mean sea level atomic weight: the relative mass of an atom based on a scale in which a specific carbon atom (carbon-12) is assigned a mass value of 12 autogiro: a rotating wing aircraft with a powered propellor to provide thrust and an unpowered rotor for lift; also spelled “autogyro” 156

Glossary

avatar: representation of user in virtual space (after the Hindu idea of an incarnation of a deity in human form) average rate of change: how one variable changes as the other variable increases by a single unit axiom: a statement regarded as self-evident; accepted without proof axiomatic system: a system of logic based on certain axioms and definitions that are accepted as true without proof axis: an imaginary line about which an object rotates axon: fiber of a nerve cell that carries action potentials (electrochemical impulses) azimuth: the angle, measured along the horizon, between north and the position of an object or direction of movement azimuthal projections: a projection of a curved surface onto a flat plane bandwidth: a range within a band of wavelengths or frequencies base-10: a number system in which each place represents a power of 10 larger than the place to its right base-2: a binary number system in which each place represents a power of 2 larger than the place to its right base-20: a number system in which each place represents a power of 20 larger than the place to the right base-60: a number system used by ancient Mesopotamian cultures for some calculations in which each place represents a power of 60 larger than the place to its right baseline: the distance between two points used in parallax measurements or other triangulation techniques Bernoulli’s Equation: a first order, nonlinear differential equation with many applications in fluid dynamics biased sampling: obtaining a nonrandom sample; choosing a sample to represent a particular viewpoint instead of the whole population bidirectional frame: in compressed video, a frame between two other frames; the information is based on what changed from the previous frame as well as what will change in the next frame bifurcation value: the numerical value near which small changes in the initial value of a variable can cause a function to take on widely different values or even completely different behaviors after several iterations Big Bang: the singular event thought by most cosmologists to represent the beginning of our universe; at the moment of the big bang, all matter, energy, space, and time were concentrated into a single point binary: existing in only two states, such as “off” or “on,” “one” or “zero” 157

Glossary

binary arithmetic: the arithmetic of binary numbers; base two arithmetic; internal arithmetic of electronic digital logic binary number: a base-2 number; a number that uses only the binary digits 1 and 0 binary signal: a form of signal with only two states, such as two different values of voltage, or “on” and “off” states binary system: a system of two stars that orbit their common center of mass; any system of two things binomial: an expression with two terms binomial coefficients: coefficients in the expansion of (x  yn, where n is a positive integer binomial distribution: the distribution of a binomial random variable binomial theorem: a theorem giving the procedure by which a binomial expression may be raised to any power without using successive multiplications bioengineering: the study of biological systems such as the human body using principles of engineering biomechanics: the study of biological systems using engineering principles bioturbation: disturbance of the strata in an archaeological site by biological factors such as rodent burrows, root action, or human activity bit: a single binary digit, 1 or 0 bitmap: representing a graphic image in the memory of a computer by storing information about the color and shade of each individual picture element (or pixel) Boolean algebra: a logic system developed by George Boole that deals with the theorems of undefined symbols and axioms concerning those symbols Boolean operators: the set of operators used to perform operations on sets; includes the logical operators AND, OR, NOT byte: a group of eight binary digits; represents a single character of text cadaver: a corpse intended for medical research or training caisson: a large cylinder or box that allows workers to perform construction tasks below the water surface, may be open at the top or sealed and pressurized calculus: a method of dealing mathematically with variables that may be changing continuously with respect to each other calibrate: act of systematically adjusting, checking, or standardizing the graduation of a measuring instrument carrying capacity: in an ecosystem, the number of individuals of a species that can remain in a stable, sustainable relationship with the available resources 158

Glossary

Cartesian coordinate system: a way of measuring the positions of points in a plane using two perpendicular lines as axes Cartesian plane: a mathematical plane defined by the x and y axes or the ordinate and abscissa in a Cartesian coordinate system cartographers: persons who make maps catenary curve: the curve approximated by a free-hanging chain supported at each end; the curve generated by a point on a parabola rolling along a line causal relations: responses to input that do not depend on values of the input at later times celestial: relating to the stars, planets, and other heavenly bodies celestial body: any natural object in space, defined as above Earth’s atmosphere; the Moon, the Sun, the planets, asteroids, stars, galaxies, nebulae central processor: the part of a computer that performs computations and controls and coordinates other parts of the computer centrifugal: the outwardly directed force a spinning object exerts on its restraint; also the perceived force felt by persons in a rotating frame of reference cesium: a chemical element, symbol Cs, atomic number 55 Chandrasekhar limit: the 1.4 solar mass limit imposed on a white dwarf by quantum mechanics; a white dwarf with greater than 1.4 solar masses will collapse to a neutron star chaos theory: the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems chaotic attractor: a set of points such that all nearby trajectories converge to it chert: material consisting of amorphous or cryptocrystalline silicon dioxide; fine-grained chert is indistinguishable from flint chi-square test: a generalization of a test for significant differences between a binomial population and a multinomial population chlorofluorocarbons: compounds similar to hydrocarbons in which one or more of the hydrogen atoms has been replaced by a chlorine or fluorine atom chord: a straight line connecting the end points of an arc of a circle chromakey: photographing an object shot against a known color, which can be replaced with an arbitrary background (like the weather maps on television newscasts) chromosphere: the transparent layer of gas that resides above the photosphere in the atmosphere of the Sun chronometer: an extremely precise timepiece 159

Glossary

ciphered: coded; encrypyted circumference: the distance around a circle circumnavigation: the act of sailing completely around the globe circumscribed: bounded, as by a circle circumspheres: spheres that touch all the “outside” faces of a regular polyhedron client: an individual, business, or agency for whom services are provided by another individual, business, or industry; a patron or customer clones: computers assembled of generic components designed to use a standard operation system codomain: for a given function f, the set of all possible values of the function; the range is a subset of the codomain cold dark matter: hypothetical form of matter proposed to explain the 90 percent of mass in most galaxies that cannot be detected because it does not emit or reflect radiation coma: the cloud of gas that first surrounds the nucleus of a comet as it begins to warm up combinations: a group of elements from a set in which order is not important combustion: chemical reaction combining fuel with oxygen accompanied by the release of light and heat comet: a lump of frozen gas and dust that approaches the Sun in a highly elliptical orbit forming a coma and one or two tails command: a particular instruction given to a computer, usually as part of a list of instructions comprising a program commodities: anything having economic value, such as agricultural products or valuable metals compendium: a summary of a larger work or collection of works compiler: a computer program that translates symbolic instructions into machine code complex plane: the mathematical abstraction on which complex numbers can be graphed; the x-axis is the real component and the y-axis is the imaginary component composite number: an integer that is not prime compression: reducing the size of a computer file by replacing long strings of identical bits with short instructions about the number of bits; the information is restored before the file is used compression algorithm: the procedure used, such as comparing one frame in a movie to the next, to compress and reduce the size of electronic files concave: hollowed out or curved inward 160

Glossary

concentric: sets of circles or other geometric objects sharing the same center conductive: having the ability to conduct or transmit confidence interval: a range of values having a predetermined probability that the value of some measurement of a population lies within it congruent: exactly the same everywhere; having exactly the same size and shape conic: of or relating to a cone, that surface generated by a straight line, passing through a fixed point, and moving along the intersection with a fixed curve conic sections: the curves generated by an imaginary plane slicing through an imaginary cone continuous quantities: amounts composed of continuous and undistinguishable parts converge: come together; to approach the same numerical value convex: curved outward, bulging coordinate geometry: the concept and use of a coordinate system with respect to the study of geometry coordinate plane: an imaginary two-dimensional plane defined as the plane containing the x- and y-axes; all points on the plane have coordinates that can be expressed as x, y coordinates: the set of n numbers that uniquely identifies the location of a point in n-dimensional space corona: the upper, very rarefied atmosphere of the Sun that becomes visible around the darkened Sun during a total solar eclipse corpus: Latin for “body”; used to describe a collection of artifacts correlate: to establish a mutual or reciprocal relation between two things or sets of things correlation: the process of establishing a mutual or reciprocal relation between two things or sets of things cosine: if a unit circle is drawn with its center at the origin and a line segment is drawn from the origin at angle theta so that the line segment intersects the circle at (x, y), then x is the cosine of theta cosmological distance: the distance a galaxy would have to have in order for its red shift to be due to Hubble expansion of the universe cosmology: the study of the origin and evolution of the universe cosmonaut: the term used by the Soviet Union and now used by the Russian Federation to refer to persons trained to go into space; synonomous with astronaut cotton gin: a machine that separates the seeds, hulls, and other undesired material from cotton 161

Glossary

cowcatcher: a plow-shaped device attached to the front of a train to quickly remove obstacles on railroad tracks cryptography: the science of encrypting information for secure transmission cubit: an ancient unit of length equal to the distance from the elbow to the tip of the middle finger; usually about 18 inches culling: removing inferior plants or animals while keeping the best; also known as “thinning” curved space: the notion suggested by Albert Einstein to explain the properties of space near a massive object, space acts as if it were curved in four dimensions deduction: a conclusion arrived at through reasoning, especially a conclusion about some particular instance derived from general principles deductive reasoning: a type of reasoning in which a conclusion necessarily follows from a set of axioms; reasoning from the general to the particular degree: 1/360 of a circle or complete rotation degree of significance: a determination, usually in advance, of the importance of measured differences in statistical variables demographics: statistical data about people—including age, income, and gender—that are often used in marketing dendrite: branched and short fiber of a neuron that carries information to the neuron dependent variable: in the equation y  f(x), if the function f assigns a single value of y to each value of x, then y is the output variable (or the dependent variable) depreciate: to lessen in value deregulation: the process of removing legal restrictions on the behavior of individuals or corporations derivative: the derivative of a function is the limit of the ratio of the change in the function; the change is produced by a small variation in the variable as the change in the variable is allowed to approach zero; an inverse operation to calculating an integral determinant: a square matrix with a single numerical value determined by a unique set of mathematical operations performed on the entries determinate algebra: the study and analysis of equations that have one or a few well-defined solutions deterministic: mathematical or other problems that have a single, well-defined solution diameter: the chord formed by an arc of one-half of a circle differential: a mathematical quantity representing a small change in one variable as used in a differential equation 162

Glossary

differential calculus: the branch of mathematics primarily dealing with the solution of differential equations to find lengths, areas, and volumes of functions differential equation: an equation that expresses the relationship between two variables that change in respect to each other, expressed in terms of the rate of change digit: one of the symbols used in a number system to represent the multiplier of each place digital: describes information technology that uses discrete values of a physical quantity to transmit information digital encoding: encoding information by using discrete values of some physical quantity digital logic: rules of logic as applied to systems that can exist in only discrete states (usually two) dihedral: a geometric figure formed by two half-planes that are bounded by the same straight line Diophantine equation: polynomial equations of several variables, with integer coefficients, whose solutions are to be integers diopter: a measure of the power of a lens or a prism, equal to the reciprocal of its focal length in meters directed distance: the distance from the pole to a point in the polar coordinate plane discrete: composed of distinct elements discrete quantities: amounts composed of separate and distinct parts distributive property: property such that the result of an operation on the various parts collected into a whole is the same as the operation performed separately on the parts before collection into the whole diverge: to go in different directions from the same starting point dividend: the number to be divided; the numerator in a fraction divisor: the number by which a dividend is divided; the denominator of a fraction DNA fingerprinting: the process of isolating and amplifying segments of DNA in order to uniquely identify the source of the DNA domain: the set of all values of a variable used in a function double star: a binary star; two stars orbiting a common center of gravity duodecimal: a numbering system based on 12 dynamometer: a device that measures mechanical or electrical power eccentric: having a center of motion different from the geometric center of a circle eclipse: occurrence when an object passes in front of another and blocks the view of the second object; most often used to refer to the phenomenon 163

Glossary

that occurs when the Moon passes in front of the Sun or when the Moon passes through Earth’s shadow ecliptic: the plane of the Earth’s orbit around the Sun eigenvalue: if there exists a vector space such that a linear transformation onto itself produces a new vector equal to a scalar times the original vector, then that scalar is called an eigenfunction eigenvector: if there exists a vector space such that a linear transformation onto itself produces a new vector equal to a scalar times the original vector, then that vector is called an eigenvector Einstein’s General Theory of Relativity: Albert Einstein’s generalization of relativity to include systems accelerated with respect to one another; a theory of gravity electromagnetic radiation: the form of energy, including light, that transfers information through space elements: the members of a set ellipse: one of the conic sections, it is defined as the locus of all points such that the sum of the distances from two points called the foci is constant elliptical: a closed geometric curve where the sum of the distances of a point on the curve to two fixed points (foci) is constant elliptical orbit: a planet, comet, or satellite follows a curved path known as an ellipse when it is in the gravitational field of the Sun or another object; the Sun or other object is at one focus of the ellipse empirical law: a mathematical summary of experimental results empiricism: the view that the experience of the senses is the single source of knowledge encoding tree: a collection of dots with edges connecting them that have no looping paths endangered species: a species with a population too small to be viable epicenter: the point on Earth’s surface directly above the site of an earthquake epicycle: the curved path followed by planets in Ptolemey’s model of the solar system; planets moved along a circle called the epicycle, whose center moved along a circular orbit around the sun epicylic: having the property of moving along an epicycle equatorial bulge: the increase in diameter or circumference of an object when measured around its equator usually due to rotation, all planets and the sun have equatorial bulges equidistant: at the same distance equilateral: having the property that all sides are equal; a square is an equilateral rectangle equilateral triangle: a triangle whose sides and angles are equal 164

Glossary

equilibrium: a state of balance between opposing forces equinox points: two points on the celestial sphere at which the ecliptic intersects the celestial equator escape speed: the minimum speed an object must attain so that it will not fall back to the surface of a planet Euclidean geometry: the geometry of points, lines, angles, polygons, and curves confined to a plane exergy: the measure of the ability of a system to produce work; maximum potential work output of a system exosphere: the outermost layer of the atmosphere extending from the ionosphere upward exponent: the symbol written above and to the right of an expression indicating the power to which the expression is to be raised exponential: an expression in which the variable appears as an exponent exponential power series: the series by which e to the x power may be approximated; ex  1  x  x2/2!  x3/3!  . . . exponents: symbols written above and to the right of expressions indicating the power to which an expression is to be raised or the number of times the expression is to be multiplied by itself externality: a factor that is not part of a system but still affects it extrapolate: to extend beyond the observations; to infer values of a variable outside the range of the observations farsightedness: describes the inability to see close objects clearly fiber-optic: a long, thin strand of glass fiber; internal reflections in the fiber assure that light entering one end is transmitted to the other end with only small losses in intensity; used widely in transmitting digital information fibrillation: a potentially fatal malfunction of heart muscle where the muscle rapidly and ineffectually twitches instead of pulsing regularly fidelity: in information theory a measure of how close the information received is to the information sent finite: having definite and definable limits; countable fire: the reaction of a neuron when excited by the reception of a neurotransmitter fission: the splitting of the nucleus of a heavy atom, which releases kinetic energy that is carried away by the fission fragments and two or three neutrons fixed term: for a definite length of time determined in advance fixed-wing aircraft: an aircraft that obtains lift from the flow of air over a nonmovable wing floating-point operations: arithmetic operations on a number with a decimal point 165

Glossary

fluctuate: to vary irregularly flue: a pipe designed to remove exhaust gases from a fireplace, stove, or burner fluid dynamics: the science of fluids in motion focal length: the distance from the focal point (the principle point of focus) to the surface of a lens or concave mirror focus: one of the two points that define an ellipse; in a planetary orbit, the Sun is at one focus and nothing is at the other focus formula analysis: a method of analysis of the Boolean formulas used in computer programming Fourier series: an infinite series consisting of cosine and sine functions of integral multiples of the variable each multiplied by a constant; if the series is finite, the expression is known as a Fourier polynomial fractal: a type of geometric figure possessing the properties of self-similarity (any part resembles a larger or smaller part at any scale) and a measure that increases without bound as the unit of measure approaches zero fractal forgery: creating a natural landscape by using fractals to simulate trees, mountains, clouds, or other features fractal geometry: the study of the geometric figures produced by infinite iterations futures exchange: a type of exchange where contracts are negotiated to deliver commodites at some fixed price at some time in the future g: a common measure of acceleration; for example 1 g is the acceleration due to gravity at the Earth’s surface, roughly 32 feet per second per second game theory: a discipline that combines elements of mathematics, logic, social and behavioral sciences, and philosophy gametes: mature male or female sexual reproductive cells gaming: playing games or relating to the theory of game playing gamma ray: a high-energy photon general relativity: generalization of Albert Einstein’s theory of relativity to include accelerated frames of reference; presents gravity as a curvature of four-dimensional space-time generalized inverse: an extension of the concept of the inverse of a matrix to include matrices that are not square generalizing: making a broad statement that includes many different special cases genus: the taxonomic classification one step more general than species; the first name in the binomial nomenclature of all species geoboard: a square board with pegs and holes for pegs used to create geometric figures 166

Glossary

geocentric: Earth-centered geodetic: of or relating to geodesy, which is the branch of applied mathematics dealing with the size and shape of the earth, including the precise location of points on its surface geometer: a person who uses the principles of geometry to aid in making measurements geometric: relating to the principles of geometry, a branch of mathematics related to the properties and relationships of points, lines, angles, surfaces, planes, and solids geometric sequence: a sequence of numbers in which each number in the sequence is larger than the previous by some constant ratio geometric series: a series in which each number is larger than the previous by some constant ratio; the sum of a geometric sequence geometric solid: one of the solids whose faces are regular polygons geometry: the branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, planes, and solids geostationary orbit: an Earth orbit made by an artificial satellite that has a period equal to the Earth’s period of rotation on its axis (about 24 hours) geysers: springs that occasionally spew streams of steam and hot water glide reflection: a rigid motion of the plane that consists of a reflection followed by a translation parallel to the mirror axis grade: the amount of increase in elevation per horizontal distance, usually expressed as a percent; the slope of a road gradient: a unit used for measuring angles, in which the circle is divided into 400 equal units, called gradients graphical user interface: a device designed to display information graphically on a screen; a modern computer interface system Greenwich Mean Time: the time at Greenwich, England; used as the basis for universal time throughout the world Gross Domestric Product: a measure in the change in the market value of goods, services, and structures produced in the economy group theory: study of the properties of groups, the mathematical systems consisting of elements of a set and operations that can be performed on that set such that the results of the operations are always members of the same set gyroscope: a device typically consisting of a spinning wheel or disk, whose spin-axis turns between two low-friction supports; it maintains its angular orientation with respect to inertial conditions when not subjected to external forces Hagia Sophia: Instanbul’s most famous landmark, built by the emperor Justinian I in 537 C.E. and converted to a mosque in 1453 C.E. 167

Glossary

Hamming codes: a method of error correction in digital information headwind: a wind blowing in the opposite direction as that of the course of a vehicle Heisenberg Uncertainty Principle: the principle in physics that asserts it is impossible to know simultaneously and with complete accuracy the values of certain pairs of physical quantities such as position and momentum heliocentric: Sun-centered hemoglobin: the oxygen-bearing, iron-containing conjugated protein in vertebrate red blood cells heuristics: a procedure that serves to guide investigation but that has not been proven hominid: a member of family Hominidae; Homo sapiens are the only surviving species Huffman encoding: a method of efficiently encoding digital information hydrocarbon: a compound of carbon and hydrogen hydrodynamics: the study of the behavior of moving fluids hydrograph: a tabular or graphical display of stream flow or water runoff hydroscope: a device designed to allow a person to see below the surface of water hydrostatics: the study of the properties of fluids not in motion hyperbola: a conic section; the locus of all points such that the absolute value of the difference in distance from two points called foci is a constant hyperbolic: an open geometric curve where the difference of the distances of a point on the curve to two fixed points (foci) is constant Hypertext Markup Language: the computer markup language used to create documents on the World Wide Web hypertext: the text that contains hyperlinks, that is, links to other places in the same document or other documents or multimedia files hypotenuse: the long side of a right triangle; the side opposite the right angle hypothesis: a proposition that is assumed to be true for the purpose of proving other propositions ice age: one of the broad spans of time when great sheets of ice covered the Northern parts of North America and Europe; the most recent ice age was about 16,000 years ago identity: a mathematical statement much stronger than equality, which asserts that two expressions are the same for all values of the variables implode: violently collapse; fall in inclination: a slant or angle formed by a line or plane with the horizontal axis or plane 168

Glossary

inclined: sloping, slanting, or leaning incomplete interpretation: a statistical flaw independent variable: in the equation y  f(x), the input variable is x (or the independent variable) indeterminate algebra: study and analysis of solution strategies for equations that do not have fixed or unique solutions indeterminate equation: an equation in which more than one variable is unknown index (number): a number that allows tracking of a quantity in economics by comparing it to a standard, the consumer price index is the best known example inductive reasoning: drawing general conclusions based on specific instances or observations; for example, a theory might be based on the outcomes of several experiments Industrial Revolution: beginning in Great Britain around 1730, a period in the eighteenth and nineteenth centuries when nations in Europe, Asia, and the Americas moved from agrarian-based to industry-based economies inertia: tendency of a body that is at rest to remain at rest, or the tendency of a body that is in motion to remain in motion inferences: the act or process of deriving a conclusion from given facts or premises inferential statistics: analysis and interpretation of data in order to make predictions infinite: having no limit; boundless, unlimited, endless; uncountable infinitesimals: functions with values arbitrarily close to zero infinity: the quality of unboundedness; a quantity beyond measure; an unbounded quantity information database: an array of information related to a specific subject or group of subjects and arranged so that any individual bit of information can be easily found and recovered information theory: the science that deals with how to separate information from noise in a signal or how to trace the flow of information through a complex system infrastructure: the foundation or permanent installations necessary for a structure or system to operate initial conditions: the values of variables at the beginning of an experiment or of a set at the beginning of a simulation; chaos theory reveals that small changes in initial conditions can produce widely divergent results input: information provided to a computer or other computation system inspheres: spheres that touch all the “inside” faces of a regular polyhedron; also called “enspheres” 169

Glossary

integer: a positive whole number, its negative counterpart, or zero integral: a mathematical operation similar to summation; the area between the curve of a function, the x-axis, and two bounds such as x  a and x  b; an inverse operation to finding the derivative integral calculus: the branch of mathematics dealing with the rate of change of functions with respect to their variables integral number: integer; that is, a positive whole number, its negative counterpart, or zero integral solutions: solutions to an equation or set of equations that are all integers integrated circuit: a circuit with the transistors, resistors, and other circuit elements etched into the surface of a single chip of silicon integration: solving a differential equation; determining the area under a curve between two boundaries intensity: the brightness of radiation or energy contained in a wave intergalactic: between galaxies; the space between the galaxies interplanetary: between planets; the space between the planets interpolation: filling in; estimating unknown values of a function between known values intersection: a set containing all of the elements that are members of two other sets interstellar: between stars; the space between stars intraframe: the compression applied to still images, interframe compression compares one image to the next and only stores the elements that have changed intrinsic: of itself; the essential nature of a thing; originating within the thing inverse: opposite; the mathematical function that expresses the independent variable of another function in terms of the dependent variable inverse operations: operations that undo each other, such as addition and subtraction inverse square law: a given physical quality varies with the distance from the source inversely as the square of the distance inverse tangent: the value of the argument of the tangent function that produces a given value of the function; the angle that produces a particular value of the tangent invert: to turn upside down or to turn inside out; in mathematics, to rewrite as the inverse function inverted: upside down; turned over ionized: an atom that has lost one or more of its electrons and has become a charged particle 170

Glossary

ionosphere: a layer in Earth’s atmosphere above 80 kilometers characterized by the existence of ions and free electrons irrational number: a real number that cannot be written as a fraction of the form a/b, where a and b are both integers and b is not zero; when expressed in decimal form, an irrational number is infinite and nonrepeating isometry: equality of measure isosceles triangle: a triangle with two sides and two angles equal isotope: one of several species of an atom that has the same number of protons and the same chemical properties, but different numbers of neutrons iteration: repetition; a repeated mathematical operation in which the output of one cycle becomes the input for the next cycle iterative: relating to a computational procedure to produce a desired result by replication of a series of operations iterator: the mathematical operation producing the result used in iteration kinetic energy: the energy an object has as a consequence of its motion kinetic theory of gases: the idea that all gases are composed of widely separated particles (atoms and molecules) that exert only small forces on each other and that are in constant motion knot: nautical mile per hour Lagrange points: two positions in which the motion of a body of negligible mass is stable under the gravitational influence of two much larger bodies (where one larger body is moving) latitude: the number of degrees on Earth’s surface north or south of the equator; the equator is latitude zero law: a principle of science that is highly reliable, has great predictive power, and represents the mathematical summary of experimental results law of cosines: for a triangle with angles A, B, C and sides a, b, c, a2  b2  c2  2bc cos A law of sines: if a triangle has sides a, b, and c and opposite angles A, B, and C, then sin A/a  sin B/ b  sin C/c laws of probability: set of principles that govern the use of probability in determining the truth or falsehood of a hypothesis light-year: the distance light travels within a vaccuum in one year limit: a mathematical concept in which numerical values get closer and closer to a given value linear algebra: the study of vector spaces and linear transformations linear equation: an equation in which all variables are raised to the first power linear function: a function whose graph on the x-y plane is a straight line or line segment 171

Glossary

litmus test: a test that uses a single indicator to prompt a decision locus (pl: loci): in geometry, the set of all points, lines, or surfaces that satisfies a particular requirement logarithm: the power to which a certain number called the base is to be raised to produce a particular number logarithmic coordinates: the x and y coordinates of a point on a cartesian plane using logarithmic scales on the x- and y-axes. logarithmic scale: a scale in which the distances that numbers are positioned, from a reference point, are proportional to their logarithms logic circuits: circuits used to perform logical operations and containing one or more logic elements: devices that maintain a state based on previous input to determine current and future output logistic difference equation: the equation x(n1)  r  xn(1xn) is used to study variability in animal populations longitude: one of the imaginary great circles beginning at the poles and extending around Earth; the geographic position east or west of the prime meridian machine code: the set of instructions used to direct the internal operation of a computer or other information-processing system machine language: electronic code the computer can utilize magnetic trap: a magnetic field configured in such a way that an ion or other charged particle can be held in place for an extended period of time magnetosphere: an asymmetric region surrounding the Earth in which charged particles are trapped, their behavior being dominated by Earth’s magnetic field magnitude: size; the measure or extent of a mathematical or physical quantity mainframes: large computers used by businesses and government agencies to process massive amounts of data; generally faster and more powerful than desktops but usually requiring specialized software malfunctioning: not functioning correctly; performing badly malleability: the ability or capability of being shaped or formed margin of error: the difference between the estimated maximum and minimum values a given measurement could have mathematical probability: the mathematical computation of probabilities of outcomes based on rules of logic matrix: a rectangular array of data in rows and columns mean: the arithmetic average of a set of data median: the middle of a set of data when values are sorted from smallest to largest (or largest to smallest) 172

Glossary

megabyte: term used to refer to one million bytes of memory storage, where each byte consists of eight bits; the actual value is 1,048,576 (220) memory: a device in a computer designed to temporarily or permanently store information in the form of binomial states of certain circuit elements meridian: a great circle passing through Earth’s poles and a particular location metallurgy: the study of the properties of metals; the chemistry of metals and alloys meteorologist: a person who studies the atmosphere in order to understand weather and climate methanol: an alcohol consisting of a single carbon bonded to three hydrogen atoms and an O–H group microcomputers: an older term used to designate small computers designed to sit on a desktop and to be used by one person; replaced by the term personal computer microgravity: the apparent weightless condition of objects in free fall microkelvin: one-millionth of a kelvin minicomputers: a computer midway in size between a desktop computer and a main frame computer; most modern desktops are much more powerful than the older minicomputers and they have been phased out minimum viable population: the smallest number of individuals of a species in a particular area that can survive and maintain genetic diversity mission specialist: an individual trained by NASA to perform a specific task or set of tasks onboard a spacecraft, whose duties do not include piloting the spacecraft mnemonic: a device or process that aids one’s memory mode: a kind of average or measure of central tendency equal to the number that occurs most often in a set of data monomial: an expression with one term Morse code: a binary code designed to allow text information to be transmitted by telegraph consisting of “dots” and “dashes” mouse: a handheld pointing device used to manipulate an indicator on a screen moving average: a method of averaging recent trends in relation to long term averages, it uses recent data (for example, the last 10 days) to calculate an average that changes but still smooths out daily variations multimodal input/output (I/O): multimedia control and display that uses various senses and interaction styles multiprocessing: a computer that has two or more central processers which have common access to main storage nanometers: billionths of a meter 173

Glossary

nearsightedness: describes the inability to see distant objects clearly negative exponential: an exponential function of the form y  ex net force: the final, or resultant, influence on a body that causes it to accelerate neuron: a nerve cell neurotransmitters: the substance released by a neuron that diffuses across the synapse neutron: an elementary particle with approximately the same mass as a proton and neutral charge Newtonian: a person who, like Isaac Newton, thinks the universe can be understood in terms of numbers and mathematical operations nominal scales: a method for sorting objects into categories according to some distinguishing characteristic, then attaching a label to each category non-Euclidean geometry: a branch of geometry defined by posing an alternate to Euclid’s fifth postulate nonlinear transformation: a transformation of a function that changes the shape of a curve or geometric figure nonlinear transformations: transformations of functions that change the shape of a curve or geometric figure nuclear fission: a reaction in which an atomic nucleus splits into fragments nuclear fusion: mechanism of energy formation in a star; lighter nuclei are combined into heavier nuclei, releasing energy in the process nucleotides: the basic chemical unit in a molecule of nucleic acid nucleus: the dense, positive core of an atom that contains protons and neutrons null hypothesis: the theory that there is no validity to the specific claim that two variations of the same thing can be distinguished by a specific procedure number theory: the study of the properties of the natural numbers, including prime numbers, the number theorem, and Fermat’s Last Theorem numerical differentiation: approximating the mathematical process of differentiation using a digital computer nutrient: a food substance or mineral required for the completion of the life cycle of an organism oblate spheroid: a spheroid that bulges at the equator; the surface created by rotating an ellipse 360 degrees around its minor axis omnidirectional: a device that transmits or receives energy in all directions Öort cloud: a cloud of millions of comets and other material forming a spherical shell around the solar system far beyond the orbit of Neptune 174

Glossary

orbital period: the period required for a planet or any other orbiting object to complete one complete orbit orbital velocity: the speed and direction necessary for a body to circle a celestial body, such as Earth, in a stable manner ordinate: the y-coordinate of a point on a Cartesian plane organic: having to do with life, growing naturally, or dealing with the chemical compounds found in or produced by living organisms oscillating: moving back and forth outliers: extreme values in a data set output: information received from a computer or other computation system based on the information it has received overdubs: adding voice tracks to an existing film or tape oxidant: a chemical reagent that combines with oxygen oxidizer: the chemical that combines with oxygen or is made into an oxide pace: an ancient measure of length equal to normal stride length parabola: a conic section; the locus of all points such that the distance from a fixed point called the focus is equal to the perpendicular distance from a line parabolic: an open geometric curve where the distance of a point on the curve to a fixed point (focus) and a fixed line (directrix) is the same paradigm: an example, pattern, or way of thinking parallax: the apparent motion of a nearby object when viewed against the background of more distant objects due to a change in the observer’s position parallel operations: separating the parts of a problem and working on different parts at the same time parallel processing: using at least two different computers or working at least two different central processing units in the same computer at the same time or “in parallel” to solve problems or to perform calculation parallelogram: a quadrilateral with opposite sides equal and opposite angles equal parameter: an independent variable, such as time, that can be used to rewrite an expression as two separate functions parity bits: extra bits inserted into digital signals that can be used to determine if the signal was accurately received partial sum: with respect to infinite series, the sum of its first n terms for some n pattern recognition: a process used by some artificial-intelligence systems to identify a variety of patterns, including visual patterns, information patterns buried in a noisy signal, and word patterns imbedded in text 175

Glossary

payload specialist: an individual selected by NASA, another government agency, another government, or a private business, and trained by NASA to operate a specific piece of equipment onboard a spacecraft payloads: the passengers, crew, instruments, or equipment carried by an aircraft, spacecraft, or rocket perceptual noise shaping: a process of improving signal-to-noise ratio by looking for the patterns made by the signal, such as speech perimeter: the distance around an area; in fractal geometry, some figures have a finite area but infinite perimeter peripheral vision: outer area of the visual field permutation: any arrangement, or ordering, of items in a set perpendicular: forming a right angle with a line or plane perspective: the point of view; a drawing constructed in such a way that an appearance of three dimensionality is achieved perturbations: small displacements in an orbit phonograph: a device used to recover the information recorded in analog form as waves or wiggles in a spiral grove on a flat disc of vinyl, rubber, or some other substance photosphere: the very bright portion of the Sun visible to the unaided eye; the portion around the Sun that marks the boundary between the dense interior gases and the more diffuse photosynthesis: the chemical process used by plants and some other organisms to harvest light energy by converting carbon dioxide and water to carbohydrates and oxygen pixel: a single picture element on a video screen; one of the individual dots making up a picture on a video screen or digital image place value: in a number system, the power of the base assigned to each place; in base-10, the ones place, the tens place, the hundreds place, and so on plane: generally considered an undefinable term, a plane is a flat surface extending in all directions without end, and that has no thickness plane geometry: the study of geometric figures, points, lines, and angles and their relationships when confined to a single plane planetary: having to do with one of the planets planisphere: a projection of the celestial sphere onto a plane with adjustable circles to demonstrate celestial phenomena plates: the crustal segments on Earth’s surface, which are constantly moving and rotating with respect to each other plumb-bob: a heavy, conical-shaped weight, supported point-down on its axis by a strong cord, used to determine verticality in construction or surveying 176

Glossary

pneumatic drill: a drill operated by compressed air pneumatic tire: air-filled tire, usually rubber or synthetic polar axis: the axis from which angles are measured in a polar coordinate system pole: the origin of a polar coordinate system poll: a survey designed to gather information about a subject pollen analysis: microscopic examination of pollen grains to determine the genus and species of the plant producing the pollen; also known as palynology polyconic projections: a type of map projection of a globe onto a plane that produces a distorted image but preserves correct distances along each meridian polygon: a geometric figure bounded by line segments polyhedron: a solid formed with all plane faces polynomial: an expression with more than one term polynomial function: a functional expression written in terms of a polynomial position tracking: sensing the location and/or orientation of an object power: the number of times a number is to be multiplied by itself in an expression precalculus: the set of subjects and mathematical skills generally necessary to understand calculus predicted frame: in compressed video, the next frame in a sequence of images; the information is based on what changed from the previous frame prime: relating to, or being, a prime number (that is, a number that has no factors other than itself and 1) Prime Meridian: the meridian that passes through Greenwich, England prime number: a number that has no factors other than itself and 1 privatization: the process of converting a service traditionally offered by a government or public agency into a service provided by a private corporation or other private entity proactive: taking action based on prediction of future situations probability: the likelihood an event will occur when compared to other possible outcomes probability density function: a function used to estimate the likelihood of spotting an organism while walking a transect probability theory: the branch of mathematics that deals with quantities having random distributions processor: an electronic device used to process a signal or to process a flow of information 177

Glossary

profit margin: the difference between the total cost of a good or service and the actual selling cost of that good or service, usually expressed as a percentage program: a set of instructions given to a computer that allows it to perform tasks; software programming language processor: a program designed to recognize and process other programs proliferation: growing rapidly proportion: the mathematical relation between one part and another part, or between a part and the whole; the equality of two ratios proportionately: divided or distributed according to a proportion; proportional protractor: a device used for measuring angles, usually consisting of a half circle marked in degrees pseudorandom numbers: numbers generated by a process that does not guarantee randomness; numbers produced by a computer using some highly complex function that simulates true randomness Ptolemaic theory: the theory that asserted Earth was a spherical object at the center of the universe surrounded by other spheres carrying the various celestial objects Pythagorean Theorem: a mathematical statement relating the sides of right triangles; the square of the hypotenuse is equal to the sums of the squares of the other two sides Pythagorean triples: any set of three numbers obeying the Pythogorean relation such that the square of one is equal to the sum of the squares of the other two quadrant: one-fourth of a circle; also a device used to measure angles above the horizon quadratic: involving at least one term raised to the second power quadratic equation: an equation in which the variable is raised to the second power in at least one term when the equation is written in its simplest form quadratic form: the form of a function written so that the independent variable is raised to the second power quantitative: of, relating to, or expressible in terms of quantity quantum: a small packet of energy (matter and energy are equivalent) quantum mechanics: the study of the interactions of matter with radiation on an atomic or smaller scale, whereby the granularity of energy and radiation becomes apparent quantum theory: the study of the interactions of matter with radiation on an atomic or smaller scale, whereby the granularity of energy and radiation becomes apparent 178

Glossary

quaternion: a form of complex number consisting of a real scalar and an imaginary vector component with three dimensions quipus: knotted cords used by the Incas and other Andean cultures to encode numeric and other information radian: an angle measure approximately equal to 57.3 degrees, it is the angle that subtends an arc of a circle equal to one radius radicand: the quantity under the radical sign; the argument of the square root function radius: the line segment originating at the center of a circle or sphere and terminating on the circle or sphere; also the measure of that line segment radius vector: a line segment with both magnitude and direction that begins at the center of a circle or sphere and runs to a point on the circle or sphere random: without order random walks: a mathematical process in a plane of moving a random distance in a random direction then turning through a random angle and repeating the process indefinitely range: the set of all values of a variable in a function mapped to the values in the domain of the independent variable; also called range set rate (interest): the portion of the principal, usually expressed as a percentage, paid on a loan or investment during each time interval ratio of similitude: the ratio of the corresponding sides of similar figures rational number: a number that can be written in the form a/b, where a and b are intergers and b is not equal to zero rations: the portion of feed that is given to a particular animal ray: half line; line segment that originates at a point and extends without bound real number: a number that has no imaginary part; a set composed of all the rational and irrational numbers real number set: the combined set of all rational and irrational numbers, the set of numbers representing all points on the number line realtime: occuring immediately, allowing interaction without significant delay reapportionment: the process of redistributing the seats of the U. S. House of Representatives, based on each state’s proportion of the national population recalibration: process of resetting a measuring instrument so as to provide more accurate measurements reciprocal: one of a pair of numbers that multiply to equal 1; a number’s reciprocal is 1 divided by the number 179

Glossary

red shift: motion-induced change in the frequency of light emitted by a source moving away from the observer reflected: light or soundwaves returned from a surface reflection: a rigid motion of the plane that fixes one line (the mirror axis) and moves every other point to its mirror image on the opposite side of the line reflexive: directed back or turning back on itself refraction: the change in direction of a wave as it passes from one medium to another refrigerants: fluid circulating in a refrigerator that is successively compressed, cooled, allowed to expand, and warmed in the refrigeration cycle regular hexagon: a hexagon whose sides are all equal and whose angles are all equal relative: defined in terms of or in relation to other quantities relative dating: determining the date of an archaeological artifact based on its position in the archaeological context relative to other artifacts relativity: the assertion that measurements of certain physical quantities such as mass, length, and time depend on the relative motion of the object and observer remediate: to provide a remedy; to heal or to correct a wrong or a deficiency retrograde: apparent motion of a planet from east to west, the reverse of normal motion; for the outer planets, due to the more rapid motion of Earth as it overtakes an outer planet revenue: the income produced by a source such as an investment or some other activity; the income produced by taxes and other sources and collected by a governmental unit rhomboid: a parallelogram whose sides are equal right angle: the angle formed by perpendicular lines; it measures 90 degrees RNA: ribonucleic acid robot arm: a sophisticated device that is standard equipment on space shuttles and on the International Space Station; used to deploy and retrieve satellites or perform other functions Roche limit: an imaginary surface around a star in a binary system; outside the Roche limit, the gravitational attraction of the companion will pull matter away from a star root: a number that when multiplied by itself a certain number of times forms a product equal to a specified number rotary-wing design: an aircraft design that uses a rotating wing to produce lift; helicopter or autogiro (also spelled autogyro) 180

Glossary

rotation: a rigid motion of the plane that fixes one point (the center of rotation) and moves every other point around a circle centered at that point rotational: having to do with rotation round: also to round off, the systematic process of reducing the number of decimal places for a given number rounding: process of giving an approximate number sample: a randomly selected subset of a larger population used to represent the larger population in statistical analysis sampling: selecting a subset of a group or population in such a way that valid conclusions can be made about the whole set or population scale (map): the numerical ratio between the dimensions of an object and the dimensions of the two or three dimensional representation of that object scale drawing: a drawing in which all of the dimensions are reduced by some constant factor so that the proportions are preserved scaling: the process of reducing or increasing a drawing or some physical process so that proper proportions are retained between the parts schematic diagram: a diagram that uses symbols for elements and arranges these elements in a logical pattern rather than a practical physical arrangement schematic diagrams: wiring diagrams that use symbols for circuit elements and arranges these elements in a logical pattern rather than a practical physical arrangement search engine: software designed to search the Internet for occurences of a word, phrase, or picture, usually provided at no cost to the user as an advertising vehicle secant: the ratio of the side adjacent to an acute angle in a right triangle to the side opposite; given a unit circle, the ratio of the x coordinate to the y coordinate of any point on the circle seismic: subjected to, or caused by an earthquake or earth tremor self-similarity: the term used to describe fractals where a part of the geometric figure resembles a larger or smaller part at any scale chosen semantic: the study of how words acquire meaning and how those meanings change over time semi-major axis: one-half of the long axis of an ellipse; also equal to the average distance of a planet or any satellite from the object it is orbiting semiconductor: one of the elements with characteristics intermediate between the metals and nonmetals set: a collection of objects defined by a rule such that it is possible to determine exactly which objects are members of the set set dancing: a form of dance in which dancers are guided through a series of moves by a caller 181

Glossary

set theory: the branch of mathematics that deals with the well-defined collections of objects known as sets sextant: a device for measuring altitudes of celestial objects signal processor: a device designed to convert information from one form to another so that it can be sent or received significant difference: to distinguish greatly between two parameters significant digits: the digits reported in a measure that accurately reflect the precision of the measurement silicon: element number 14, it belongs in the category of elements known as metalloids or semiconductors similar: in mathematics, having sides or parts in constant proportion; two items that resemble each other but are not identical sine: if a unit circle is drawn with its center at the origin and a line segment is drawn from the origin at angle theta so that the line segment intersects the circle at (x, y), then y is the sine of theta skepticism: a tendency towards doubt skew: to cause lack of symmetry in the shape of a frequency distribution slope: the ratio of the vertical change to the corresponding horizontal change software: the set of instructions given to a computer that allows it to perform tasks solar masses: dimensionless units in which mass, radius, luminosity, and other physical properties of stars can be expressed in terms of the Sun’s characteristics solar wind: a stream of particles and radiation constantly pouring out of the Sun at high velocities; partially responsible for the formation of the tails of comets solid geometry: the geometry of solid figures, spheres, and polyhedrons; the geometry of points, lines, surfaces, and solids in three-dimensional space spatial sound: audio channels endowed with directional and positional attributes (like azimuth, elevation, and range) and room effects (like echoes and reverberation) spectra: the ranges of frequencies of light emitted or absorbed by objects spectrum: the range of frequencies of light emitted or absorbed by an object sphere: the locus of points in three-dimensional space that are all equidistant from a single point called the center spin: to rotate on an axis or turn around square: a quadrilateral with four equal sides and four right angles square root: with respect to real or complex numbers s, the number t for which t2  s 182

Glossary

stade: an ancient Greek measurement of length, one stade is approximately 559 feet (about 170 meters) standard deviation: a measure of the average amount by which individual items of data might be expected to vary from the arithmetic mean of all data static: without movement; stationary statistical analysis: a set of methods for analyzing numerical data statistics: the branch of mathematics that analyzes and interprets sets of numerical data stellar: having to do with stars sterographics: presenting slightly different views to left and right eyes, so that graphic scenes acquire depth stochastic: random, or relating to a variable at each moment Stonehenge: a large circle of standing stones on the Salisbury plain in England, thought by some to be an astronomical or calendrical marker storm surge: the front of a hurricane, which bulges because of strong winds; can be the most damaging part of a hurricane stratopause: the boundary in the atmosphere between the stratosphere and the mesosphere usually around 55 kilometers in altitude stratosphere: the layer of Earth’s atmosphere from 15 kilometers to about 50 kilometers, usually unaffected by weather and lacking clouds or moisture sublimate: change of phase from a solid to a gas sublunary: “below the moon”; term used by Aristotle and others to describe things that were nearer to Earth than the Moon and so not necessarily heavenly in origin or composition subtend: to extend past and mark off a chord or arc sunspot activity: one of the powerful magnetic storms on the surface of the Sun, which causes it to appear to have dark spots; sunspot activity varies on an 11-year cycle superconduction: the flow of electric current without resistance in certain metals and alloys while at temperatures near absolute zero superposition: the placing of one thing on top of another suspension bridge: a bridge held up by a system of cables or cables and rods in tension; usually having two or more tall towers with heavy cables anchored at the ends and strung between the towers and lighter vertical cables extending downward to support the roadway symmetric: to have balanced proportions; in bilateral symmetry, opposite sides are mirror images of each other symmetry: a correspondence or equivalence between or among constituents of a system synapse: the narrow gap between the terminal of one neuron and the dendrites of the next 183

Glossary

tactile: relating to the sense of touch tailwind: a wind blowing in the same direction of that of the course of a vehicle tangent: a line that intersects a curve at one and only one point in a local region tectonic plates: large segments of Earth’s crust that move in relation to one another telecommuting: working from home or another offsite location tenable: defensible, reasonable terrestrial refraction: the apparent raising or lowering of a distant object on Earth’s surface due to variations in atmospheric temperature tessellation: a mosaic of tiles or other objects composed of identical repeated elements with no gaps tesseract: a four-dimensional cube, formed by connecting all of the vertices of two three-dimensional cubes separated by the length of one side in four-dimensional space theodolite: a surveying instrument designed to measure both horizontal and vertical angles theorem: a statement in mathematics that can be demonstrated to be true given that certain assumptions and definitions (called axioms) are accepted as true threatened species: a species whose population is viable but diminishing or has limited habitat time dilation: the principle of general relativity which predicts that to an outside observer, clocks would appear to run more slowly in a powerful gravitational field topology: the study of those properties of geometric figures that do not change under such nonlinear transformations as stretching or bending topspin: spin placed on a baseball, tennis ball, bowling ball, or other object so that the axis of rotation is horizontal and perpendicular to the line of flight and the top of the object is rotating in the same direction as the motion of the object trajectory: the path followed by a projectile; in chaotic systems, the trajectory is ordered and unpredictable transcendental: a real number that cannot be the root of a polynomial with rational coefficients transect: to divide by cutting transversly transfinite: surpassing the finite transformation: changing one mathematical expression into another by translation, mapping, or rotation according to some mathematical rule 184

Glossary

transistor: an electronic device consisting of two different kinds of semiconductor material, which can be used as a switch or amplifier transit: a surveyor’s instrument with a rotating telescope that is used to measure angles and elevations transitive: having the mathematical property that if the first expression in a series is equal to the second and the second is equal to the third, then the first is equal to the third translate: to move from one place to another without rotation translation: a rigid motion of the plane that moves each point in the same direction and by the same distance tree: a collection of dots with edges connecting them that have no looping paths triangulation: the process of determining the distance to an object by measuring the length of the base and two angles of a triangle trigonometric ratio: a ratio formed from the lengths of the sides of right triangles trigonometry: the branch of mathematics that studies triangles and trigonometric functions tropopause: the boundry in Earth’s atmosphere between the troposphere and the stratosphere at an altitude of 14 to 15 kilometers troposphere: the lowest layer of Earth’s atmosphere extending from the surface up to about 15 kilometers; the layer where most weather phenomena occur ultra-violet radiation: electromagnetic radiation with wavelength shorter than visible light, in the range of 1 nanometer to about 400 nanometer unbiased sample: a random sample selected from a larger population in such a way that each member of the larger population has an equal chance of being in the sample underspin: spin placed on a baseball, tennis ball, bowling ball, or other object so that the axis of rotation is horizontal and perpendicular to the line of flight and the top of the object is rotating in the opposite direction from the motion of the object Unicode: a newer system than ASCII for assigning binary numbers to keyboard symbols that includes most other alphabets; uses 16-bit symbol sets union: a set containing all of the members of two other sets upper bound: the maximum value of a function vaccuum: theoretically, a space in which there is no matter variable: a symbol, such as letters, that may assume any one of a set of values known as the domain variable star: a star whose brightness noticeably varies over time 185

Glossary

vector: a quantity which has both magnitude and direction velocity: distance traveled per unit of time in a specific direction verify: confirm; establish the truth of a statement or proposition vernal equinox: the moment when the Sun crosses the celestial equator marking the first day of spring; occurs around March 22 for the northern hemisphere and September 21 for the southern hemisphere vertex: a point of a graph; a node; the point on a triangle or polygon where two sides come together; the point at which a conic section intersects its axis of symmetry viable: capable of living, growing, and developing wavelengths: the distance in a periodic wave between two points of corresponding phase in consecutive cycles whole numbers: the positive integers and zero World Wide Web: the part of the Internet allowing users to examine graphic “web” pages yield (interest): the actual amount of interest earned, which may be different than the rate zenith: the point on the celestial sphere vertically above a given position zenith angle: from an observer’s viewpoint, the angle between the line of sight to a celestial body (such as the Sun) and the line from the observer to the zenith point zero pair: one positive integer and one negative integer ziggurat: a tower built in ancient Babylonia with a pyramidal shape and stepped sides

186

Topic Outline APPLICATIONS Agriculture Architecture Athletics, Technology in City Planning Computer-Aided Design Computer Animation Cryptology Cycling, Measurements of Economic Indicators Flight, Measurements of Gaming Grades, Highway Heating and Air Conditioning Interest Maps and Mapmaking Mass Media, Mathematics and the Morgan, Julia Navigation Population Mathematics Roebling, Emily Warren Solid Waste, Measuring Space, Comercialization of Space, Growing Old in Stock Market Tessellations, Making CAREERS Accountant Agriculture Archaeologist Architect Artist Astronaut Astronomer Carpenter Cartographer Ceramicist City Planner Computer Analyst Computer Graphic Artist Computer Programmer Conservationist Data Analyst

Electronics Repair Technician Financial Planner Insurance Agent Interior Decorator Landscape Architect Marketer Mathematics Teacher Music Recording Technician Nutritionist Pharmacist Photographer Radio Disc Jockey Restaurant Manager Roller Coaster Designer Stone Mason Web Designer DATA ANALYSIS Census Central Tendency, Measures of Consumer Data Cryptology Data Collection and Interpretation Economic Indicators Endangered Species, Measuring Gaming Internet Data, Reliability of Lotteries, State Numbers, Tyranny of Polls and Polling Population Mathematics Population of Pets Predictions Sports Data Standardized Tests Statistical Analysis Stock Market Television Ratings Weather Forecasting Models FUNCTIONS & OPERATIONS Absolute Value Algorithms for Arithmetic Division by Zero 187

Topic Outline

Estimation Exponential Growth and Decay Factorial Factors Fraction Operations Fractions Functions and Equations Inequalities Matrices Powers and Exponents Quadratic Formula and Equations Radical Sign Rounding Step Functions GRAPHICAL REPRESENTATIONS Conic Sections Coordinate System, Polar Coordinate System, Three-Dimensional Descartes and his Coordinate System Graphs and Effects of Parameter Changes Lines, Parallel and Perpendicular Lines, Skew Maps and Mapmaking Slope IDEAS AND CONCEPTS Agnesi, Maria Gaëtana Consistency Induction Mathematics, Definition of Mathematics, Impossible Mathematics, New Trends in Negative Discoveries Postulates, Theorems, and Proofs Problem Solving, Multiple Approaches to Proof Quadratic Formula and Equations Rate of Change, Instantaneous MEASUREMENT Accuracy and Precision Angles of Elevation and Depression Angles, Measurement of Astronomy, Measurements in Athletics, Technology in Bouncing Ball, Measurement of a Calendar, Numbers in the Circles, Measurement of Cooking, Measurements of Cycling, Measurements of Dance, Folk Dating Techniques Distance, Measuring 188

Earthquakes, Measuring End of the World, Predictions of Endangered Species, Measuring Flight, Measurements of Golden Section Grades, Highway Light Speed Measurement, English System of Measurement, Metric System of Measurements, Irregular Mile, Nautical and Statute Mount Everest, Measurement of Mount Rushmore, Measurement of Navigation Quilting Scientific Method, Measurements and the Solid Waste, Measuring Temperature, Measurement of Time, Measurement of Toxic Chemicals, Measuring Variation, Direct and Inverse Vision, Measurement of Weather, Measuring Violent NUMBER ANALYSIS Congruency, Equality, and Similarity Decimals Factors Fermat, Pierre de Fermat’s Last Theorem Fibonacci, Leonardo Pisano Form and Value Games Gardner, Martin Germain, Sophie Hollerith, Herman Infinity Inverses Limit Logarithms Mapping, Mathematical Number Line Numbers and Writing Numbers, Tyranny of Patterns Percent Permutations and Combinations Pi Powers and Exponents Primes, Puzzles of Probability and the Law of Large Numbers Probability, Experimental Probability, Theoretical Puzzles, Number Randomness

Topic Outline

Ratio, Rate, and Proportion Rounding Scientific Notation Sequences and Series Significant Figures or Digits Step Functions Symbols Zero NUMBER SETS Bases Field Properties Fractions Integers Number Sets Number System, Real Numbers: Abundant, Deficient, Perfect, and Amicable Numbers, Complex Numbers, Forbidden and Superstitious Numbers, Irrational Numbers, Massive Numbers, Rational Numbers, Real Numbers, Whole SCIENCE APPLICATIONS Absolute Zero Alternative Fuel and Energy Astronaut Astronomer Astronomy, Measurements in Banneker, Benjamin Brain, Human Chaos Comets, Predicting Cosmos Dating Techniques Earthquakes, Measuring Einstein, Albert Endangered Species, Measuring Galileo, Galilei Genome, Human Human Body Leonardo da Vinci Light Light Speed Mitchell, Maria Nature Ozone Hole Poles, Magnetic and Geographic Solar System Geometry, History of Solar System Geometry, Modern Understandings of Sound

Space Exploration Space, Growing Old in Spaceflight, History of Spaceflight, Mathematics of Sun Superconductivity Telescope Temperature, Measurement of Toxic Chemicals, Measuring Undersea Exploration Universe, Geometry of Vision, Measurement of SPATIAL MATHEMATICS Algebra Tiles Apollonius of Perga Archimedes Circles, Measurement of Congruency, Equality, and Similarity Dimensional Relationships Dimensions Escher, M. C. Euclid and his Contributions Fractals Geography Geometry Software, Dynamic Geometry, Spherical Geometry, Tools of Knuth, Donald Locus Mandelbrot, Benoit B. Minimum Surface Area Möbius, August Ferdinand Nets Polyhedrons Pythagoras Scale Drawings and Models Shapes, Efficient Solar System Geometry, History of Solar System Geometry, Modern Understandings of Symmetry Tessellations Tessellations, Making Topology Transformations Triangles Trigonometry Universe, Geometry of Vectors Volume of Cone and Cylinder SYSTEMS Algebra Bernoulli Family 189

Topic Outline

Boole, George Calculus Carroll, Lewis Dürer, Albrecht Euler, Leonhard Fermat, Pierre de Hypatia Kovalevsky, Sofya Mathematics, Very Old Newton, Sir Isaac Pascal, Blaise Robinson, Julia Bowman Somerville, Mary Fairfax Trigonometry TECHNOLOGY Abacus Analog and Digital Babbage, Charles Boole, George Bush, Vannevar Calculators Cierva Codorniu, Juan de la Communication Methods Compact Disc, DVD, and MP3 Technology

190

Computer-Aided Design Computer Animation Computer Information Systems Computer Simulations Computers and the Binary System Computers, Evolution of Electronic Computers, Future of Computers, Personal Galileo, Galilei Geometry Software, Dynamic Global Positioning System Heating and Air Conditioning Hopper, Grace IMAX Technology Internet Internet Data, Reliability of Knuth, Donald Lovelace, Ada Byron Mathematical Devices, Early Mathematical Devices, Mechanical Millennium Bug Photocopier Slide Rule Turing, Alan Virtual Reality

Cumulative Index Page numbers in boldface indicate article titles. Those in italics indicate illustrations and photographs; those with an italic “f ” after the page numbers indicate figures, and those with an italic “t” after the page numbers indicate tables. The boldface number preceding the colon indicates the volume number.

A Abacuses, 1:1, 1–3, 71, 3:12–13 See also Calculators Abscissa, defined, 3:10 and Glossary Absolute, defined, 1:9 and Glossary Absolute-altitude, 2:60 Absolute dating, 2:10, 11–13 Absolute error. See Accuracy and precision Absolute temperature scales, 1:5 Absolute value, 1:3, 3:91 Absolute zero, 1:3–7, 4:74 Abstract algebra, defined, 1:158, 2:1 and Glossary Abstraction, 3:79 Abundant numbers, 3:83–84 Acceleration, 1:72–73 defined, 3:186 and Glossary Accelerometer, defined, 3:187 and Glossary Accountants, 1:7–8 Accuracy and precision, 1:8–10 Adding machines, 1:71 Addition algebra tiles, 1:17 algorithms, 1:21 complex numbers, 3:88–89 efficiency of Base-10 system, 4:150 inverses, 2:148–149 matrices, 3:32–33 precision, 1:8–9

significant digits, 4:14 slide rules, 4:15 Additive identity, 3:88–89, 4:150 Additive inverse, 4:76 Adjusted production statistics, computing, 4:57 Aecer (saxon term), 3:35 Aerial photography, 2:24, 3:7 Aerodynamics, 1:90 athletics, 1:53–55 cycling, 1:164 defined, 1:53 and Glossary flight, 1:90–91 Aesara of Lucania, 3:160 Agnesi, Maria Gaëtana, 1:10–11 Agriculture, 1:11–14 AIDS epidemic and population projections, 3:137–138 Air conditioning and heating, 2:117–119 Air pressure drag, 1:164 Air traffic controllers, 1:14, 14–15 Airspeed, 2:60–61 Aleph null, 2:136–137 Alexander the Great, 4:116 Algebra, 1:15–17 defined, 1:10, 2:18, 4:31, and Glossary indeterminate, 2:47 mathematical symbols in, 4:77 See also Abstract algebra; Boolean algebra Algebra tiles, 1:17–20, 17f, 18f, 19f, 20f Algorithms, 1:20–23, 159 defined, 1:20, 2:143, 3:74, 4:9 and Glossary Allen, Paul, 1:136 Allometry, 2:124–125 Almagest (Ptolemy), 4:18 Altair 8800 computer, 1:136 Alternative fuel and energy, 1:23–25

Altitude, 2:60 Aluminum bicycle frames, 1:163 Ambiguity, 2:1 American Standard Code for Information Interchange. See ASCII American Stock Exchange (AMEX), 4:69 Americium-252m (Am-242m), 4:42–43 Amicable numbers, 3:85 Amplitude, 2:157, 4:33 Amundsen, Roald, 3:127–128 Analog and digital information, 1:25–28, 102 Analog computers, 4:15 Analytical Engine, 1:57, 3:18 Analytical geometry, defined, 2:18, 4:21 and Glossary Analytical Institutions (Instituzioni analitiche ad uso della gioventù italiana) (Agnesi), 1:10 Ancient mathematical devices, 3:11–15 Ancient mathematics, 3:27–32 Anderson, Michael P., 1:45 Andromeda galaxy, 4:42 Anergy, defined, 2:117 and Glossary Angle of rotation, 2:3 Angles elevation and depression, 1:31–33, 32f, 2:23 measurement, 1:28–31 See also Trigonometry Animal population. See Population of pets Antarctica ozone levels, 3:109 Anthrobot, 2:156 Anti-aliasing, defined, 1:115 and Glossary Apollo spacecraft, 4:40–41, 49–50, 52–53 Apollonius of Perga, 1:33, 33–35 Apple I and II computers, 1:136

191

Cumulative Index

Applewhite, Marshall, 2:36 Application programs, 1:129 Approximating primes, 3:147 Approximation. See Irregular measurements Archaeologists, 1:35 Archaeology, 2:9 Archimedean screw, 1:35 Archimedean solids, 3:135 Archimedes, 1:35–36, 93 Archimedes’ Principle, 1:36 Architects, 1:36–37, 4:3 Architecture, 1:37–39 Arcs, 1:91 defined, 1:30, 3:37 and Glossary Area, 1:73–74, 74f, 2:165, 3:41–42 Area-minimizing surfaces, 3:49–50 Areagraph, defined, 1:95 and Glossary Aristarchus, 4:18, 71 Aristotle, 2:134 Arithmometer, 1:71, 3:17 Armstrong, John, 3:56 Armstrong, Neil, 4:49–50 ARPANET, 2:141 Ars Conjectandi (Bernoulli), 1:61 Artifact, defined, 1:35, 2:9 and Glossary Artificial gravity, 4:45 Artificial intelligence, 1:132 defined, 1:112 and Glossary Artificial low temperatures, 1:6–7 Artists, 1:39–40, 4:3 ASCII, 1:126 defined, 1:103 and Glossary Assets, 1:7 Associative property, 2:57 Asteroid ores, 4:38 Asteroids, defined, 1:47 and Glossary Astigmatism, 4:135, 137 Astrolabe, 3:67 defined, 2:126 and Glossary Astronaut Candidate Program, 1:44 Astronaut training, 1:43 Astronauts, 1:40–46 Astronomers, 1:46–47 Astronomical units, defined, 1:99 and Glossary Astronomy cosmos, 1:153–158 Greek, 3:161, 4:17–18 measurements in, 1:47–52 Asymptotes, defined, 2:47, 4:110 and Glossary Atanasoff-Berry Computer (ABC), 1:129

192

Athletics, technology in, 1:52–55 See also Sports data Atmosphere, defined, 4:116 and Glossary Atomic clocks, 4:96–97 Atomic weight, defined, 3:40 and Glossary Attitude of aircraft, 2:61–62 Auroras, 1:156 Autogiro, 1:90, 90–91 defined, 1:89 and Glossary Automobile design and scaling, 4:2 Avatars, defined, 4:133 and Glossary Average, 4:54–55 See also Central tendency, measures of; Weighted average “Average” height, 2:124 Average rate of change, 3:178–179, 179f, 180f See also Functions Axiomatic systems, defined, 2:43–44, 3:160 and Glossary Axioms, defined, 3:140 and Glossary Axis, 1:148 defined, 2:3 and Glossary Axon, 1:67–68 Azimuth, defined, 1:38 and Glossary Azimuthal projections, 3:6–7

B Babbage, Charles, 1:57–58, 2:175, 3:18–19 Babylonian mathematics algebra, 1:15 angle measurement, 1:29–30 astronomical measurements, 1:47 calendar, 1:77, 2:34 number system, 3:28–29 pi, 3:125 place value concept, 3:82 timekeeping, 4:95 Baculum, 1:28–29 Baking soda and baking powder, 1:147 Ball speed and ball spin, 1:53–54 Ballots for the blind, 3:132 Bandwidth, defined, 1:135, 2:144 and Glossary Banneker, Benjamin, 1:58, 58–59 Bar codes. See Universal Product Codes Bar graphs, 2:110 See also Pie graphs Base-2 number system, 1:26, 60, 2:170 See also Binary numbers

Base-10 number system, 1:20, 26, 59–60, 3:12, 28, 4:149–150 defined, 2:15, 3:12 and Glossary Base-20 number system, defined, 3:28 and Glossary Base-60 number system, 3:28–29 defined, 1:59, 2:71, 3:29 and Glossary Baselines, 1:47 Bases. See Base-2 number system; Base-10 number system; Base-20 number system; Base-60 number system BASIC language, 1:136 Batting average, 4:56 BCS theory, 4:74–75 Bell, Jocelyn, 1:155 Bell curves, 4:58f, 59–60 Bernoulli, Daniel, 1:60–62 Bernoulli, Jakob, 1:60–62, 3:148 Bernoulli, Johann, 1:60–62, 61 Bernoulli’s Equation, 1:61 Best-fit lines, 2:111 Bias in news call-in polls, 2:6 Biased sampling, defined, 2:6 and Glossary Bicycles, 1:162–163 Bidirectional frames, 1:108 Bifurcation values, 1:88 Big Bang theory, 4:121–123 defined, 1:6, 4:121 and Glossary Big Ten football statistics, 4:57f Binary, defined, 2:120 and Glossary Binary arithmetic, 3:20 Binary numbers, 1:25, 102–103, 126 See also Base-2 number system Binary signals, 1:126 Binary star systems, 1:154 Binomial form. See Form and value Binomial theorem, defined, 1:16, 3:75 and Glossary Bioengineering, defined, 3:26 and Glossary Biomechanics, defined, 1:53 and Glossary Bitmap images, 1:111 Bits and bytes, 1:26–27, 103 Bits, defined, 4:130 and Glossary Bytes, defined, 3:96 and Glossary Bjerknes, Vilhelm, 4:141 Black holes, 1:156–157, 2:161 Blackjack, 2:79–80 Block pattern of city planning. See Grid pattern of city planning Blocks and function modeling, 1:19–20 Blueprints, 4:3

Cumulative Index

Blurring in computer animation, 1:115 Body Mass Index (BMI), 2:125 Bolyai, János, 3:71, 4:25 Boole, George, 1:62–64, 63 Boolean algebra, 1:63, 64f See also Boolean operators Boolean operators, 1:118–119 Borglum, John Gutzon de la Mothe, 3:57–58, 60 Bouncing balls, measurement of, 1:64–67 Brache, Tycho, 1:49, 4:19 Brain, comparison with computers, 1:67–69 Braun, Werhner von, 4:47 Bridge (card game), 2:77 Briggs, Henry, 2:171 British Imperial Yard, 3:35 British thermal units (BTUs), 2:118 Brooklyn Bridge, 3:184–185 Bush, Vannevar, 1:69, 69–70 Butterfly Effect, 1:87, 4:142 Bytes and bits, 1:26–27, 103, 3:96, 4:130 Bytes, defined, 3:96 and Glossary Bits, defined, 4:130 and Glossary

C Cable modems, 2:144 See also Modems CAD (Computer-aided design), 1:109–112 Cadavers, 2:26 Cadence (pedaling), 1:163 Caesar, Julius, 1:77–78, 158 Caissons, defined, 3:185 and Glossary Calculators, 1:71–72, 72 Calculus, 1:72–76 defined, 2:1, 3:17 and Glossary Bernoulli family and, 1:61 defined, 2:1, 3:17, 4:21 and Glossary geography and, 2:90–91 mathematical symbols in, 4:77 See also Differential calculus; Functions; Limit (calculus) Calendars, 1:76–79, 2:33–34 Calibration, defined, 1:12, 3:7 and Glossary Calibration of radiocarbon dates, 2:13–14 California Raisins, 3:8 Cantor, Georg, 2:135–136 Capture-recapture method, 3:145 Carbon-12 and carbon-14, 2:13

Carbon fiber bicycle frames, 1:163 Carpenter, Loren, 2:69 Carpenters, 1:79 Carroll, Lewis, 1:79–80, 80 Carrying capacity, 1:87 Cartesian coordinate system, 2:17–19, 18f See also Vectors Cartesian philosophy, 2:17–18 Cartesian plane, 2:165 defined, 2:67 and Glossary Cartographers, 1:80–81 defined, 3:5 and Glossary See also Maps and mapmaking CASH 3 lottery, 2:173 Cash dividend per share, 4:68 Casino games. See Gaming Cassini Mission to Saturn, 4:41 Catenary curves, defined, 3:185 and Glossary Catenoid, 3:49 Causal relation, defined, 4:65 and Glossary CDs (Compact discs), 1:106–109, 107 Celestial, defined, 4:17 and Glossary Celestial bodies, 2:24 Celestial mechanics, 4:51 Celestial navigation, 3:67 Celsius temperature scale, 1:4, 4:89 Census. See U.S. Census Central Limit Theorem, 4:58 Central tendency, measures of, 1:83–85 Centrifugal force, 1:91 Centrigrade temperature scale, 4:89 Ceramicists, 1:85–86 Cerf, Vinton, 2:142 Cesium, defined, 3:40 and Glossary Chain drives and gears, 1:163 Challenger explosion, 1:43 Chandler Circle, 3:128 Chandrasekhar limit, 1:154–155 Change of perspective in problem solving, 3:154 Chaos and chaos theory, 1:86–89 dance and, 2:3 defined, 2:1, 3:20, 4:142 and Glossary role in predictions, 3:144 solar system geometry, 4:24 weather forecasting and, 1:87, 4:142 Chaotic attractors, 2:3 Charlemagne, 3:34 Chawla, Kalpana, 1:45 Check digits, 1:120

Chefs, 1:146, 147–148 Chert, defined, 2:14 and Glossary Chess, 2:76–77 Chi-square test, defined, 4:64 and Glossary Chlorofluorocarbons, defined, 2:118 and Glossary Chords, 1:30 Chromakeys, defined, 4:134 and Glossary Chromatic aberration, 4:83 Chromosomes, 2:84–85 Chromosphere, defined, 4:72 and Glossary Chronometer, 3:67, 68 Cierva Codorniu, Juan de la, 1:89–91, 90 Circle graphs. See Pie graphs Circles angle measurement, 1:29–30 great and small, 2:96, 4:119 measurement, 1:91–94 See also Trigonometry Circumference, 1:48 circles, 1:92 defined, 4:18 and Glossary Earth, 1:30, 49–50 significant digits and, 4:14 Circumferential pattern of city planning, 1:96–97 Circumnavigation, defined, 4::18 and Glossary Circumspheres, defined, 4:19 and Glossary City of New York, Wisconsin v. (1996), 1:81–82 City planners, 1:95 City planning, 1:95–98 Clarkson, Roland, 3:146 Clients, 1:7 Clones (computer), 1:137 Closing prices (stock), 4:69 Closure, 2:57 Cobb, Jerrie, 1:41 COBOL, 2:122 Cockrell, Kenneth, 1:42 Codomain, 3:3–4 Cold dark matter, 1:157 Color, 2:157–158 Colossus computer, 1:130 .com domain, 2:146 Comas (comet), defined, 4:42 and Glossary Combinations, 2:51 Combinations and permutations, 3:116–118 Combustion, defined, 2:117, 4:51 and Glossary

193

Cumulative Index

Comets, 1:98–101 Commands, computer, 1:127 Commercial Space Act (1998), 4:39 Commercialization of space, 4:35–40 Commodities, 1:12 Communication methods, 1:101–106 Commutative property, 2:57 Compact discs (CDs), 1:106–109, 107 Compact surfaces, 3:24–25 Compasses (dividers), 1:93, 2:101 Compendium, 1:10 Compiler, defined, 2:122 and Glossary Completing the square, 3:165–166 Complex conjugate, 3:89 Complex number system, 3:88–91 Complex numbers, 2:58, 3:81, 87–92 Complex planes, 1:16, 2:67 Composite numbers, 2:51 Compound inequalities, 2:132–133 Compound interest, 2:138 Compression, 1:104 Compression algorithms, 1:108 Comptometer, 3:18 Computational fluid dynamics, 1:55 Computer-aided design (CAD), 1:109–112 Computer analysts, 1:112–113 Computer animation, 1:113–116 See also Virtual Reality Computer applications, 1:129 Computer architecture, 1:125–126 Computer chips, 1:131–132 Computer commands, 1:127 Computer graphic artists, 1:116–117, 117 See also Web designers Computer hardware, 1:125–126 Computer-human user interfaces, 4:130f Computer information systems, 1:117–120 Computer operating systems, 1:128 Computer processing and calculation, 1:127–128 Computer programers, 1:120–121 Computer programs and software, 1:128 Computer simulations, 1:121–124, 122 Computer storage devices. See Computer architecture Computers binary system and, 1:124–129

194

Boole, George, 1:63–64 comparison with human brain, 1:67–69 cryptology and, 1:160–161 evolution of, 1:129–132 future of, 1:132–135 memory and speed, 3:97 Millennium bug, 3:46–48 personal, 1:135–139 randomness and, 3:176–177 See also Internet; Virtual Reality; Weather forecasting models Concave, defined, 4:81 and Glossary Concentric objects, 1:95 Conductivity, 4:73 Cone and cylinder volumes, 4:138–139 Confidence interval, defined, 2:4 and Glossary Congreve rockets, 4:51 Congruency, equality, and similarity, 1:139–140, 3:119 Congruent, defined, 4:18 and Glossary Conic projections, 3:6 Conic sections, 1:140–141, 141f, defined, 2:47, 3:111 and Glossary Conics (Apollonius of Perga), 1:34 Connection speed to Internet, 2:143–144 Conservationists, 1:141–142, 142 Consistency, 1:142–143 Construction problems, solving, 2:101–102 Consumer data, 1:143–145 Consumer Price Index, 2:31 Consumer Reports (magazine), 1:144 Contact lenses, 4:137 Continuous data, graphing, 2:111 Continuous quantities, defined, 4:5 and Glossary Contradancing, 2:1–2 Converge, 1:115 Convex, defined, 4:81 and Glossary Cooking measurements, 1:145–148 Coordinate geometry, defined, 2:135 and Glossary Coordinate planes, defined, 2:18 and Glossary Coordinate systems Cartesian coordinate system, 2:17–19, 18f polar, 1:148–151 three-dimensional, 1:151–153 Copernicus, Nicolaus, 1:49, 4:19 Corona, defined, 4:72 and Glossary Corrective lenses, 4:136–138 Correlation, 1:35, 85, 2:89

Correlate, defined, 2:10 and Glossary Corrosivity of pollutants, 4:100 Cosine, defined, 1:89, 2:98, 4:16 and Glossary See also Law of Cosines Cosine curves, 4:110f Cosmic wave background, 4:121 Cosmological distance, defined, 1:158 and Glossary Cosmology, defined, 1:98, 4:19 and Glossary Cosmonauts, defined, 1:41, 4:48 and Glossary Cosmos, 1:153–158 Cotton gin, defined, 1:11 and Glossary Counters and cups, 1:18–19 Counting boards and tables, 1:1, 3:11–12 Counting numbers, 3:81–82 Cowcatchers, defined, 1:57 and Glossary Crab nebula, 1:155 Crash tests, 1:121 Cray-1 supercomputer, 1:135 Crazy Horse Monument, 3:59 Credit cards, 2:139 Critical density, 4:122 Crocodile Hunter (TV show), 4:87 Cross dating, 2:11 Cross-staff. See Baculum Cryptanalysis, 1:158 Cryptography, 1:102, 158, 159–160, 3:26 Cryptology, 1:158–161 defined, 1:102 and Glossary Crystals, surface minimization, 3:50 Cubes, 3:134–135, 4:139 Cubes and skew lines, 2:168 Cubit, 2:23, 3:34 Culling, 1:13 Cuneiform, 3:86 Cups and counters, 1:18–19 Curbeam, Robert, Jr., 1:42 Curvature of space, 4:120–121 Curved line graphs, 2:114–116 Curved space, defined, 1:156 and Glossary Curves, 1:34, 3:42 Cycling, measurements of, 1:162–164 Cylinder and cone volumes, 4:138–139 Cylindrical maps, 3:6

Cumulative Index

D da Vinci, Leonardo. See Leonardo da Vinci Dance, folk, 2:1–3 Dark energy. See Vacuum energy Dark matter, 1:157 Data analysts, 2:3–4 Data collection and interpretation, 2:4–7 Data Encryption Standard, 1:161 Dating techniques, 2:9–15 De Revolutionibus Orbium Caelestium (On the Revolutions of the Heavenly Spheres) (Copernicus), 4:19 Dead reckoning, 3:67 Decibels (dB), 4:34–35 Decimal fractions, 2:15–16 Decimal number system. See Base10 number system; Decimals Decimals, 2:15–17 See also Fractions; Ratios Deduction, defined, 2:43 and Glossary Deficient numbers, 3:84 Degree (geometry), 1:29–30 defined, 1:29 and Glossary Degree of significance, defined, 3:10 and Glossary Demographics, defined, 1:96, 3:7 and Glossary Dendrochronology, 2:11–12, 12 Dependent variables, defined, 4:63 and Glossary Depreciation, defined, 1:7 and Glossary Deregulation, defined, 3:8 and Glossary Derivative, defined, 1:73, 2:163, 3:180 and Glossary Descartes, René, 1:16, 2:17, 17–19, 134 Determinants, defined, 1:81 and Glossary Determinate, 2:47 Deterministic models, 3:177–178 Diameter, defined, 1:50, 3:124 and Glossary Diana, Princess of Wales, death of, 1:116 Dice and probability, 3:150 Difference Engine, 1:57, 58, 3:18–19 Differential Analyzer, 1:70 Differential calculus, defined, 1:10, 2:47, 3:75 and Glossary Differential equations, defined, 2:91 and Glossary Differentiation, 3:76

Digital and analog information, 1:25–28, 102 Digital logic, 1:62 Digital maps, 3:7 Digital orthophoto quadrangles (DOQs). See Digital maps Digital Subscriber Lines (DSLs), 2:144 Digital thermometers, 4:89 Digital versatile discs (DVDs), 1:106–109 Digits, defined, 1:2, 2:15 and Glossary Dilations, 4:1–2, 104–105 Dimensional relationships, 2:19–20 Dimensions, 2:20–23 Diophantine equation, 3:183–184 Diophantus, 1:15 Diopters, defined, 4:138 and Glossary Direct and inverse variation, 4:125–126, 126f Direct friction, 1:164 Directed distance, defined, 1:149 and Glossary Direction of aircraft, 2:62 “Discount interest,” 2:138 Discount pricing and markup, 1:145 Discourse on the Method of Reasoning Well and Seeking Truth in the Sciences (Descartes), 2:18 Discrete quantities, defined, 4:5 and Glossary Discreteness, defined, 2:1 and Glossary Distance, measurement of, 2:23–25 Distance (absolute zero), 1:3 Distance illusion in computer animation, 1:115 Distribution of primes, 3:147 Distributive property, defined, 1:22, 2:57 and Glossary Diverge, defined, 1:87 and Glossary Diversity in space program, 1:45–46 Dividend, defined, 1:22 and Glossary Dividend-to-price-radio. See Yield (stocks) Divine Proportion. See Golden Mean Diving bells, 4:116 Division algebra tiles, 1:18 algorithms, 1:22–23 complex numbers, 3:88–89 inverses, 2:148–149 significant digits, 4:14 slide rules, 4:15

by zero, 2:25–26, 4:150 Divisor, defined, 1:22 and Glossary DNA-based computers, 1:133 DNA (deoxyribonucleic acid), 2:84–85 DNA fingerprinting, defined, 1:133 and Glossary Dobson, G. M. B., 3:108 Dodecahedron, 3:134–135 Dodgson, Charles Lutwidge. See Carroll, Lewis Domain, 3:3–4 defined, 1:65, 2:72, 4:101 and Glossary Domain names, 2:142–143, 146 Doppler radar, 4:145 Double-blind experiments, 4:63 Double stars, defined, 1:47 and Glossary Dow Jones Industrial Average, 4:69 Drafting, 1:164 Drag, 1:90, 164 DSLs (Digital Subscriber Lines), 2:144 Duodecimal, defined, 3:38 and Glossary Dürer, Albrecht, 2:26 DVDs (Digital versatile discs), 1:106–109 Dynamic geometry software, 2:92–95 Dynamo theory, 3:129 Dynamometer, defined, 1:57 and Glossary

E E-mail, 2:141–143 Earth circumference, 1:30, 49–50 magnetic field, 3:129–130 radius, 1:48 Earthquakes, measurement of, 2:27–28 Eccentric, defined, 1:34 and Glossary Echelons and pelotons, 1:164 Eclipse, defined, 1:48, 4:17 and Glossary Economic indicators, 2:28–32 .edu domain, 2:146 Efficient shapes, 4:9–11 Egyptian mathematics calendars, 1:77, 3:82 cubit, 2:23, 3:34 distance measurement, 2:23 geometry, 3:31 number system, 3:28

195

Cumulative Index

Egyptian mathematics (continued) Rhind and Moscow Papyri, 3:30–31 rope-stretchers and angle measurement, 1:29 squaring the circle, 1:94 Eigenvalue, defined, 3:91 and Glossary Eigenvectors, defined, 1:81, 3:91 and Glossary Einstein, Albert, 2:32, 4:25 See also Relativity; Special relativity Elasticity, 1:163 Electronic calculators, 1:72 Electronics repair technicians, 2:32–33, 33 Elements (Euclid), 2:43, 45, 3:140–141 Ell of Champagne, 3:34 Ellipses, defined, 1:34, 2:170, 4:21 and Glossary Elliptic, defined, 4:24 and Glossary Elliptical, defined, 4:21 and Glossary Elliptical orbit, defined, 1:99, 4:53 and Glossary Elson, Matt, 1:116 Empirical laws, 1:100 defined, 1:99 and Glossary Empiricism, 4:19 Encoding information, 1:119–120 See also Analog and digital information; Cryptology Encryption. See Cryptography End of the world predictions, 2:33–36 Endangered species, measurement of, 2:36–40 Endangered Species Act (1973), 2:39 Energy and alternative fuels, 1:23–25 English system of measurement, 3:34–37 ENIAC (Electronic Numerical Integrator and Computer), 1:129–130, 130 Enigma code, 1:159, 4:112 Enlarging and reducing photographs, 3:121–122 Enneper’s surface, 3:49 Enumeration, 1:81 Epicenter, defined, 2:28 and Glossary Epicycle, defined, 1:34 and Glossary Equal likelihood, 3:152 Equal probability of selection, 3:131 Equation of graphs. See Graphing notation

196

Equations, 4:77 Equatorial bulge, defined, 3:54 and Glossary Equidistant, defined, 4:25 and Glossary Equilateral triangles, 1:29 Equilibrium, defined, 1:90 and Glossary Eratosthenes of Cyrene, 1:30, 48, 4:18 Ericson, Leif, 3:66 Escape speed, defined, 1:156 and Glossary Escape velocity, 4:53 Escher, M. C., 2:21, 40–41, 3:22, 4:91, 92 Estimation, 2:41–43, 3:145 Estimation, in baseball, 4:57 Euclid, 1:93, 2:43, 43–46, 3:85 Euclidean geometry. See Geometry Euclid’s fifth postulate, 2:45–46 Euler, Leonhard, 2:46, 46–47, 3:72–74 Euler characteristic, 3:134 Euler Walk, 3:73–74 Everest, George, 3:55 Exchequer standards, 3:35 Exergy, defined, 2:117 and Glossary Exit polls, 3:132–133 Experimental probability, 3:149–151 Exploration, space, 4:40–43 Exponential functions, 2:47, 48f Exponential growth and decay, 2:47–49 Exponential models, 3:136 Exponential power series, defined, 2:38 and Glossary Exponents, 2:47, 3:141–143, 4:7, 76 defined, 2:19 and Glossary See also Logarithms Externality, 1:24 Extrapolate, defined, 1:4 and Glossary Eye charts, 4:135 Eyeglasses. See Corrective lenses

F F-stop, 3:123 Factorials, 2:51–52 Factors, 2:52–53 Fahrenheit temperature scale, 1:4, 4:89 Fantasy 5 (lottery), 2:173 Far and near transfer of learning, 1:122 Farsightedness, 4:136–137

See also Presbyopia Faster-than-light travel, 2:161–162 Fathoms, 3:34 Feigenbaum, Mitchell, 1:88–89 Feigenbaum’s Constant, 1:88–89 Felt, Dorr E., 3:18 Fermat, Pierre de, 2:53, 53–54 Fermat’s Last Theorem, 2:54–55, 103, 3:158 Fertility, 3:127 Fiber-optic cable, 1:134–135 defined, 1:134 and Glossary Fibonacci, Leonardo Pisano, 2:55–56, 56 Fibonacci sequence, 3:63–64, 112, 139, 4:8 Fibrillation, defined, 1:86 and Glossary Fidelity, defined, 1:27 and Glossary Field properties, 2:56–58 Final Fantasy: The Spirits Within (film), 1:114 Financial planners, 2:58–59 Finite, defined, 2:66 and Glossary Firing (neurons), 1:68 Fire, defined, 1:68 and Glossary First in, first out, 3:182 First law of hydrostatics, 1:36 Fish population studies and chaos theory, 1:87–88 Fission, defined, 2:14 and Glossary Five postulates (Euclid), 2:45, 3:141 Fixed terms, 2:139 Fixed-wing aircraft, defined, 1:90 and Glossary Flash point, 4:100 Flatland (Abbott), 2:22 “Flatness problem,” 4:122 Flexible software, 1:137–138 Flight, measurement of, 2:59–64 Floating-point operations, defined, 1:135 and Glossary Floods, 4:147–148 Florida State Lottery and Lotto, 2:173 FLOW-MATIC, 2:122 Fluctuate, defined, 2:30 and Glossary Fluid dynamics, defined, 3:26, 4:142 and Glossary Fluxions, 1:75 Focal point and focal length, 2:158, 4:136 focal length, defined, 4:136 and Glossary Foci, 1:141 Focus (ellipse), defined, 4:20 and Glossary

Cumulative Index

Folk dancing, 2:1–3 Forbidden and superstitious numbers, 3:92–95 Form and value, 2:64 Formula analysis, defined, 1:158 and Glossary Four-color hypothesis, 3:71 Fourier series, defined, 2:38 and Glossary Fractal forgery, defined, 2:69 and Glossary Fractal geometry, 3:1 Fractals, 2:64–70 defined, 1:89, 2:91, 3:170 and Glossary Fraction operations, 2:70–71 Fractional exponents, 3:142–143 Fractions, 2:15–16, 71–72 See also Decimals; Ratios “Free-fall,” 4:53 Free market economy, 1:143–144 Frequency (light waves), 2:157 Frequency (sound waves), 4:33, 34 Friedman, Alexander, 4:121 Fringed orchids, 2:37 Frustums, 3:30–31 Fujita scale, 4:145–146 Functions, 2:112, 164–165, 4:77 block modeling and, 1:18–19 equations and, 2:72–73 mapping, 3:2–5 See also Limit (calculus); Step functions Furnaces, 2:119 Futures exchange, defined, 1:14 and Glossary Fuzzy logic thermostats, 2:119

G g (acceleration measurement), defined, 4:44 and Glossary Galilean thermometer, 4:88 Galilei, Galileo, 2:75, 75–76 pendulums, 4:96 solar system geometry, 4:21 telescopes, 4:81, 82 trial of, 3:95 Gallup polls, 3:130–131 Gambling. See Gaming Game theory, 3:26 Games, 2:76–77, 3:152–153 Gaming, 2:76–81 Law of Large Numbers and, 3:149 techniques in simulation, 1:122 See also State lotteries Gardner, Martin, 2:81, 81–84

Gargarin, Yuri, 4:48 Gates, Bill, 1:136 Gauss, Karl Freidrich, 4:25 GDP (Gross Domestic Product), 2:30–31, 4:40 Gears and chain drives, 1:163 Gemini spacecraft, 4:49 Generalized inverse, defined, 1:81 and Glossary Genes, 2:85 Genome, human, 2:84–86 Genus, defined, 2:11 and Glossary Geoboards, defined, 1:19 and Glossary Geocentric, defined, 4:19 and Glossary Geocentrism, 4:18–19 Geodeditic, defined, 3:54 Geographic matrix. See Geography Geographic poles. See Magnetic and geographic poles Geography, 2:87–92 Geometers, 1:38 Geometic cross. See Baculum Geometric equations, 1:11 Geometric sequence, defined, 1:64 and Glossary Geometric series, defined, 1:65 and Glossary Geometric solid, defined, 1:36 and Glossary Geometry, 3:70–71, 4:70 analytical, defined, 2:18 and Glossary coordinate, 2:135 defined, 1:72, 2:18, 3:2, 4:21 and Glossary dynamic software, 2:92–95 fractal, 3:1 hyperbolic, 4:25 plane, 1:34, 2:167 of solar system, historical, 4:17–22 of solar system, modern, 4:22–26 spherical, 2:95–100 tools of, 2:100–102 of the universe, 4:119–123 See also Euclid; Non-Euclidean geometry Geostationary orbits, defined, 4:37 and Glossary Germain, Sophie, 2:103 German rocketry, 4:46–47 Geysers, defined, 1:5 and Glossary Glenn, John, 4:48 Glide reflection, defined, 4:92 and Glossary Glide symmetry, 4:90–91

Global Positioning System (GPS), 2:103–106, 3:57, 69, 4:106 GNP (Gross National Product), 2:30 Goddard, Robert, 4:52 Goldbach’s Conjecture, 3:147 Golden Mean, 1:38 Golden Number, 3:64–65, 94–95 Golden Ratio. See Golden Number Golden ratios. See Golden sections Golden sections, 2:106–108 Goldenheim, David, 2:92 .gov domain, 2:146 GPS (Global Positioning System), 2:103–106, 3:57, 69, 4:106 Grades, highway, 2:108–109 Gradient, 1:30, 91 Graphical user interface (GUI), 1:137 Graphing inequalities, 2:132–133, 132f Graphing notation, 2:112 Graphs, 2:109–113 Graphs and effects of parameter changes, 2:113–116 Gravitation, theory of, 2:161 Gravity, 3:186–187, 4:23, 44 See also Microgravity Great and small circles, 2:96, 4:119 “The Great Arc,” 3:54 Great Pyramid, 2:23 “The Great Trigonometrical Survey of India,” 3:54–55 Greatest common factor, 2:53 Greek mathematics angle measurement, 1:30 calendars, 1:77 counting tables, 3:11–12 distance measurement, 2:23 efficient shape problem, 4:10–11 fathoms, 3:34 fractions, 2:72 golden sections, 2:107 perfect numbers, 3:84–85 squaring the circle, 1:94 See also specific mathematicians Greenhouse effect, 1:23 Greenwich Mean Time, defined, 1:15 and Glossary Gregorian calendar, 1:78 Grid pattern of city planning, 1:96 Gross Domestic Product (GDP), 2:30–31 defined, 4:40 and Glossary Gross National Product (GNP), 2:30 Growing old in space, 4:44–46

197

Cumulative Index

Growth charts, 2:123–124 Gunter, Edmund, 3:45–46 Gunter’s Line of Numbers, 3:14 Guth, Alan, 4:122 Gyroscope, 2:62 defined, 2:61 and Glossary

H Hagia Sophia, defined, 1:37 and Glossary Half-life, 2:12 Halley, Edmund, 1:99–100 Halley’s Comet, 1:99–101, 100 Hamilton Walk, 3:74 Hamming codes, 1:105–106 defined, 1:105 and Glossary Harriot, Thomas, 4:81 Harrison, John, 3:68 Headwind, defined, 2:61 and Glossary Hearst Castle. See San Simeon Heating and air conditioning, 2:117–119 Heisenberg uncertainty principle, defined, 2:162 and Glossary Helicoid, 3:49 Heliocentric, defined, 4:19 and Glossary Helium and lowering of temperature, 1:6 Herschel, William, 4:84 Heuristics, defined, 1:112 and Glossary Hewish, Antony, 1:155 Hexagons, regular, 1:29 High-speed digital recording, 1:53–54 Highway grades, 2:108–109 Highways and skew lines, 2:168 Himalaya Peak XV, 3:56 Hindu-Arabic number system, 2:15 Hipparchus, 1:51 Hisarlik (Turkey), excavation of, 2:10 HIV/AIDS Surveillance Database, 3:138 Hollerith, Herman, 2:119–121, 120 Hominids, 2:14 Hook, Robert, 4:88 Hopper, Grace, 2:121, 121–123 Horology, 4:94 HTML (Hyptertext Markup Language), 4:148 Hubble Space Telescope, 4:85 Huffman encoding, defined, 1:104 and Glossary

198

Human body, measurement of growth, 2:123–126 Human brain, comparison with computers, 1:67–69 Human genome, 2:84–86 Human Genome Project, 2:86 Hurricanes, 4:146–147 Huygens probe, 4:41 Hydrocarbon, 2:117 Hydrodynamics, 1:62, 2:46 Hydrographs, defined, 1:97 and Glossary Hydroscope, 2:126 Hypatia, 2:126 Hyperbola, 1:140–141 defined, 4:54 and Glossary Hyperbolic geometry, 4:25 Hypermedia, 4:130–131, 131f Hyperopia. See Farsightedness Hyperspace drives. See Warp drives Hypertext, defined, 1:70 and Glossary Hypertext Markup Language (HTML), defined, 4:148 and Glossary Hypotenuse, defined, 1:31, 3:161, 4:105 and Glossary Hypotheses, defined, 4:4 and Glossary

I IBM personal computers, 1:137 Icosahedron, 3:134–135 Ideal gases, 1:4 Identity, 2:57 Ignitability of pollutants, 4:100 iMac computer, 1:138 Image, 3:4 Image formula, 4:102 Imaginary number lines, 3:100 Imaginary numbers, 3:81, 87–88 IMAX technology, 2:127–129, 128 Imperial Standard Yard, 3:35–36 Imperial Weights and Measures Act of 1824, 3:35 Implode, defined, 1:154 and Glossary Impossible mathematics, 3:21–25 Incan number system, 3:29 “Incommensurables.” See Irrational numbers Independent variables, defined, 4:63 and Glossary Indeterminate algebra, 2:47 Index numbers, 3:173–174 Indexes (economic indicators), 2:29–30

Indirect measurement, 1:30 Induction, 2:130–131 Inductive reasoning, defined, 4:61 and Glossary Industrial Revolution, defined, 1:57 and Glossary Industrial solid waste, 4:26 Inequalities, 2:131–136 Inertia, 3:186 Inferences, defined, 4:61 and Glossary Inferential statistics, defined, 2:4 and Glossary Infinite, defined, 2:65 and Glossary Infinitesimal calculus, 3:75 Infinitesimals, defined, 4:21 and Glossary Infinity, 2:25, 133–136, 3:158–159 Inflation, 2:31 Inflation theory, 4:122–123 Information databases, defined, 2:3 and Glossary Information error detection and correction, 1:105–106 Information theory, defined, 1:63 and Glossary Infrared wavelengths, 2:158 Infrastructure, defined, 1:24 and Glossary Initial conditions, defined, 1:86 and Glossary Input-output devices. See Computer architecture Inspheres, defined, 4:19 and Glossary Instantaneous rate of change, 3:178–180 See also Functions Insurance agents, 2:136–137 Integers, 2:137, 3:80 defined, 1:17, 2:51, 3:83, 4:7 and Glossary See also Factors Integral calculus, defined, 1:10 and Glossary Integral numbers, 2:54 Integral solutions, defined, 2:126 and Glossary Integrals, 1:74–75, 163 defined, 1:73 and Glossary Integrated circuits, 1:131–132 defined, 1:125 and Glossary “Integrating machine.” See Tabulating machine Intensity (sound), 4:34–35 Interest, 2:138–140, 138f, 3:116 See also Rates Interior decorators, 2:140–141

Cumulative Index

International Geophysical Year 1957-1958, 3:108 International migration and population projection, 3:138 International Space Station, 4:36 International System of Units (SI). See Metric system Internet, 2:141–145 connection and speed, 3:98 reliability of data, 2:145–148 searching and Boolean operators, 1:118–119 See also Computers; Web designers Interplanetary, defined, 1:5 and Glossary Interpolation, defined, 1:111 and Glossary Intraframe, defined, 1:108 and Glossary Inverse, 3:173 Inverse square law, 1:52 Inverse tangent, defined, 1:33 and Glossary Inverse variation. See Direct and inverse variation Inverses, 2:57, 148–149 Ionization, 1:99 IP packets, 2:142–143 Irrational numbers, 2:16, 3:79, 80, 96, 160, 174–175 See also Pi Irregular measurements, 3:40–44 Ishango Bone, 3:28 Islamic calendar, 1:78 Islands of Truth (Peterson), 3:21 Isosceles triangles, 1:98 Isotope, defined, 1:6, 2:12 and Glossary Iteration, 1:88, 2:65 Ivans, Marsha, 1:42

J Jack, Wolfman, 3:175 Jackiw, Nicholas, 2:93–94 Jacob’s Ladder quilt pattern, 3:168, 169f Jobs, Steve, 1:138 Jones, Thomas, 1:42 Julia Set, 2:67–68 Julian calendar, 1:77–78 Jupiter C Missile RS-27, 4:47

K Kaleidoscope method of tessellation, 4:93 Kamil, Abu, 1:15

Kangchenjunga, 3:56 Keck Telescope, 4:85 Kelvin scale, 1:5, 4:89 Kepler, Johannes, 1:49, 4:19, 82 See also Kepler’s laws Kepler’s laws, 1:99, 4:20–21, 72 Kerrich, John, 3:149 Kilogram, 3:39 Kinetic energy, defined, 1:5, 4:23 and Glossary Kinetic genes, 2:85 Kinetic theory of gases, 1:62 Klein bottle, 3:23–24, 24f, 4:98 Klotz, Eugene, 2:93–94 Knots (speed measurement), 1:15, 2:60, 3:45 Knuth, Donald, 2:151–152, 152 Kobe (Japan) earthquake, 2:27 Koch Curve, 2:65–67 Königsburg Bridges, 3:72–74 Kovalesky, Sofya, 2:152–153, 153

L Lagging economic indicators, 2:29 Lagrange, Joseph-Louis, 4:23 Lagrange points, defined, 4:36 and Glossary Lambton, William, 3:54 Landfills, 4:27 Landing (flight), 2:63–64 Lands and pits, 1:107, 108 Landscape architects, 2:155 Latitude, 1:97, 3:5, 67–68 Launch vehicles, 4:36–37 Law (scientific principle), defined, 1:16 and Glossary Law of conservation of energy, 3:187 Law of cosines, 4:106, 108 defined, 1:81, 4:128 and Glossary Law of Large Numbers and probability, 3:148–149 Law of sines, 4:106, 108 defined, 4:128 and Glossary Law (scientific principle), 1:16 Laws of Motion. See Newton’s laws Laws of Planetary Motion. See Kepler’s laws Laws of probability, 3:20 Le Verrier, Urbain, 4:24–25 Leading economic indicators, 2:29–30 Leavening agents, 1:147 Leibniz, Gottfried, 3:17 Lenses and refraction, 2:158 Leonardo da Vinci, 2:156–157 Leonardo’s mechanical man, 2:156

Letters, numeric value of, 3:93 Level flight, 2:59–60 Leviathan telescope, 4:84 Libby, Willard F., 2:12 Liebniz, Gottfried Wilhelm, 1:75 Lift, 1:90 Light, 2:24, 157–159 Light speed, 2:157, 159–163 Light-years, 2:159–160 defined, 1:51, 4:122 and Glossary Lighthouse model of pulsars, 1:156 Limit (calculus), 1:74, 2:163–165, 165f, 3:178, 180 See also Functions Limit (interest), 2:139 Limit of the sequence, 2:196–197 Linear equations, 2:126 Linear functions, 2:73 Lines parallel and perpendicular, 2:166–167 skew, 2:167–169 Lipperhey, Hans, 4:81 Liquid presence, 4:132–133 Lisa computer, 1:137 Litchfield, Daniel, 2:92 Litmus test, defined, 4:24 and Glossary Live trapping, 2:37 Livestock production and mathematics, 1:13 Loans, 2:139–140 Lobachevsky, Nicolai, 4:25 Locus, 2:53, 169–170 Logarithmic scale, 4:34 Logarithms, 2:170–171, 3:14 defined, 1:57, 4:15 and Glossary Logic, 3:154 Logic circuits, defined, 1:67 and Glossary Logic gates. See Logic circuits Logistic models, 3:137 “Long count” (Mayan calendar), 2:34–35 Longitude, 3:67, 68 defined, 1:15, 3:5, 4:18 and Glossary Longitudinal sound waves, 4:33 Lorenz, Ed, 1:87 Lotteries, state, 2:171–175 Lottery revenues, 2:174 Lovelace, Ada Bryon, 2:175 Low temperature measurement, 1:5 Lowell, Percival, 4:84 Lucid, Shannon, 4:45 Luna 1, 2, and 3 spacecraft, 4:48 Lunisolar calendars, 1:77

199

Cumulative Index

M Machine code, defined, 1:62 and Glossary Machine language, defined, 1:128, 2:122 and Glosssary Magentosphere, 4:41 Magic Circle of Circles, 3:94f Magic squares, 3:93–94, 94f Magnetic and geographic poles, 3:127–130 Magnetic declination, 3:129 Magnetic reversals, 3:130 Magnetic trap, 1:6 Magnetosphere, defined, 4:41 and Glossary Magnification factor calculation, 3:122f Mainframe, defined, 1:124 and Glossary Malleability, defined, 4:73 and Glossary Mandelbrot, Benoit B., 3:1, 1–2 Mandelbrot set, 2:68, 68–69, 3:2 Mapmakers. See Cartographers Mapping, mathematical, 3:2–5, 3f Maps and mapmaking, 3:5–7, 69, 4:2 See also Cartographers; Navigation Margin of error, 2:6–7 defined, 2:2 and Glossary Mark-1 computer, 1:131 Marketers, 3:7–8 Marketing companies, use of TV ratings, 4:87 Markup and discount pricing, 1:145 Mars 2001 Odyssey, 4:41–42 Mars orbit, 1:49 Masonry. See Stone masons Mass media and mathematics, 3:8–11 Massive numbers, 3:96–98 Mathematical devices ancient, 3:11–15 mechanical, 3:15–19 Mathematical mapping, 3:2–5, 3f Mathematical operations. See Addition; Division; Multiplication; Subtraction Mathematical predictions, 3:143–145 Mathematical symbols, 4:75–77, 4:76f See also Variables Mathematics ancient, 3:27–32 definition, 3:19–21 impossible, 3:21–25

200

mass media and, 3:8–11 nature and, 3:63–65 new trends in, 3:25–27 population and, 3:135–139 Pythagoreans’ contributions to, 3:160–161 of spaceflight, 4:51–54 See also Algebra; Babylonian mathematics; Calculus; Egyptian mathematics; Geometry; Greek mathematics; Trigonometry Mathematics teachers, 3:27 Matrices, 3:32–33 Matrix, defined, 1:111, 2:2, 4:105 and Glossary Maximum height, 1:65 May, Robert, 1:87–88 Mayan mathematics calendar, 2:34–35 number system, 3:29 Mean, 1:84–85, 3:9, 4:62 defined, 2:7, 3:155, 4:58 and Glossary Measure of dispersion. See Standard deviation Measurement accuracy and precision, 1:8–10 angles, 1:28–31 angles of elevation and depression, 1:31–33, 32f astronomy, 1:47–52 bouncing balls, 1:64–67 circles, 1:91–94 cooking, 1:145–148 cycling, 1:162–164 distance, 2:23–25 earthquakes, 2:27–28 endangered species, 2:36–40 English system, 3:34–37 flight, 2:59–64 human growth, 2:123–125 irregular, 3:40–44 metric system, 3:37–40 Mount Everest, 3:54–57 Mount Rushmore, 3:57–60 nautical and statute miles, 3:44–46 rounding, 3:187–188 scientific method and, 4:4–7 solid waste, 4:26–31 temperature, 4:87–90 time, 4:94–97 toxic pollutants, 4:99–101 vision, 4:135–138 Mechanical calculator, 3:15–16 Mechanical clocks, 4:95–96

Mechanical crossbow, 2:156f Mechanical mathematical devices, 3:15–19 Mechanical multiplier, 3:17 The Mechanism of Heavens (Somerville), 4:32 Median, 1:83–84, 3:10, 4:62 Mega Money lottery, 2:173 Megabyte, defined, 1:108 and Glossary Meissner effect, 4:74 Mercalli Intensity Scale, 2:27–28 Mercator, Gerardus, 3:6 Mercury 13 Program, 1:41 Mercury Seven astronauts, 1:40 Meridian, defined, 1:15, 3:5, 4:97 and Glossary See also Prime Meridian Mersenne prime numbers, 3:85 Mesocyclones, 4:144 Metal thermometers, 4:88–89 Metallurgy, defined, 1:52 and Glossary Meteorologists, defined, 1:87, 4:141 and Glossary Meter, 2:24, 3:38 Meterology, 4:88 Methanol, defined, 1:23 and Glossary The Method of Fluxions and Infinite Series (Newton), 3:76 Metius, Jacob, 4:81 Metric measurements in cooking, 1:147 Metric prefixes, 3:38f Metric system, 3:36, 37–40, 96 Microchips, 1:132 Microcomputers, 1:124 Microgravity, 4:44–45 defined, 1:43, 4:36, 44 and Glossary See also “Free-fall”; Gravity Microkelvin, defined, 1:6 and Glossary Miles, nautical and statute, 3:44–46 Military time, 4:97 Milky Way, 1:157 Millennium bug, 3:46–48 Miller, William, 2:36 Minicomputers, defined, 1:124 and Glossary Minimum perimeter, 4:10 Minimum surface area, 3:48–50, 4:10 Minimum viable population, 2:36, 38–39 Mining in space, 4:38–39 Minorities and U.S. Census, 1:82–83

Cumulative Index

Mirages, 2:158 Mirror symmetry, 4:78, 90 Mirrors and reflection, 2:159 Mission specialists (astronauts), 1:43, 44 defined, 1:41 and Glossary Mitchell, Maria, 3:51 Mixed reality, 4:131 Möbius, August Ferdinand, 3:51–53 Möbius strip, 3:23, 52, 4:98 Mode, 1:83, 1:84, 3:10, 4:62 Modems, 1:137, 2:143 See also Cable modems Modern telescopes, 4:84–85 Monomial form. See Form and value Moon landing, 4:40, 41, 49–50 Morgan, Julia, 3:53–54 Morse code, defined, 1:102 and Glossary Mortality, 3:137–138 Mortgages, 2:140 Moscow Papyrus, 3:30–31 Motion, laws of. See Newton’s laws Motion picture film, 2:128–129 Motivation enhancement by simulations, 1:122 Mount Everest, measurement of, 3:54–57 Mount Rushmore, measurement of, 3:57–60, 4:3–4 Mountains, measuring, 3:56–58 Mouse (computer), defined, 1:136 and Glossary Mouton, Gabriel, 3:37 Movement illusion in computer animation, 1:115–116 Moving average, defined, 1:14 and Glossary Moving Picture Experts Group audio Layer-3. See MP3s MP3s, 1:106–109 Multimodal input/output, defined, 4:130 and Glossary Multiplication algebra tiles, 1:17 algorithms, 1:22 complex numbers, 3:88–89 factorials, 2:51–52 inverses, 2:148–149 power, 1:20, 2:15 precision, 1:9 significant digits, 4:14 slide rules, 4:15 Multiplicative identity, 3:88–89 Multiplicative inverse, 4:76 Multiprocessing, 1:128 Municipal solid waste, 4:26–27, 28f

See also “Pay-as-you-throw” programs; Reuse and recycling MUSES-C spacecraft, 4:42 Music recording technicians, 3:60–61 Myopia. See Nearsightedness

N Napier, John, 2:35–36, 171, 3:13–14 Napier’s bones, 3:13–14 NASA Aerodynamics and Sports Technology Research Program, 1:53 NASA and space commercialization, 4:38, 39 NASDAQ, 4:69 Natural numbers. See Counting numbers Nature and mathematics, 3:63–65 Nautical and statute miles, 2:99, 3:44–46 Nautilus shells, 3:65 Navigation, 2:98–99, 3:65–70 See also Maps and mapmaking Navigation, space, 4:52–53 NEAP (Near Earth Asteroid Prospector), 4:38–39 Near and far transfer of learning, 1:122 Near point, 4:136 Nearsightedness, 4:137 Negative discoveries, 3:70–72 Negative exponential series, defined, 2:38 and Glossary Negative exponents, 3:142 Negative numbers, 3:82–83 Neptune, discovery of, 4:24 .net domain, 2:146 Net force, defined, 4:51 and Glossary Nets (geometry), 3:72–75, 135 Neurons, 1:67–68 Neutron stars, 1:155 Neutrons, defined, 1:154 and Glossary New York Stock Exchange (NYSE), 4:68, 69 News coverage and frequency of events, 3:9 Newton, Sir Isaac, 1:75, 148, 3:75, 75–76, 4:22–23, 83–84 Newtons, 1:164 Newton’s laws, 3:144, 4:51 Nicomachus’ treatise, 3:84 Nielsen ratings. See Television ratings Nominal scales, defined, 4:64 and Glossary

Non-Euclidean geometry, defined, 2:40, 3:71, 4:25 and Glossary Nonorientable surfaces, 3:23–24 Nonsimilarity, 3:119–120 Nonterminating and nonrepeating decimals, 2:16 North Pole expeditions, 3:127–128 North pole of balance, 3:128 Nova Cygni, 1:154 Novae and supernovae, 1:153–155 Nuclear fission, defined, 4:42 and Glossary Nuclear fusion and stars, 1:154 Nuclear fusion, defined, 1:154 and Glossary Nucleotides, 2:85 defined, 1:133 and Glossary Nucleus, defined, 1:6 and Glossary Null hypotheses, defined, 4:63 and Glossary Number lines, 3:76–78 Number puzzles, 3:157–159 Number rods, 3:82 Number sets, 3:78–81 Number theory, defined, 1:102, 2:53 and Glossary Numbers, 3:80f abundant, 3:83–84 amicable, 3:85 binary, 1:25, 102–103, 126 complex, 2:58, 3:81, 87–92 composite, 2:51 counting, 3:81–82 deficient, 3:84 forbidden and superstitious, 3:92–95 imaginary, 3:81, 87–88 index, 3:173–174 integral, 2:54 irrational, 2:16, 3:79, 80, 96, 160, 174–175 massive, 3:96–98 Mersenne primes, 3:85 negative, 3:82–83 perfect, 3:84–85 pi, 2:136, 3:124–127 prime, 2:51–52, 3:25, 145–148 pseudo-random, 3:177 Pythagorean philosophy, 3:160 rational, 2:16, 57–58, 3:80, 98–99 real, 1:3, 2:58, 3:78, 79, 81, 99–100 size of, 4:76 transcendental, 1:94 transfinite, 3:159 tyranny of, 3:100–103

201

Cumulative Index

Numbers (continued) whole, 1:65, 2:51, 3:79, 80, 103–104 writing and, 3:86–87 See also Base-2 number system; Base-10 number system; Base60 number system; Calendars; Integers; Roman number system; Zero Numerical differentiation, defined, 1:111 and Glossary Numerical sequences, 3:112 Numerical weather prediction, 4:142 Numerology, 3:93 Nutrient, defined, 1:11 and Glossary Nutritionists, 3:104–105 NYSE (New York Stock Exchange), 4:68, 69

O Oblate spheroids, 3:54 Observation and photography in wildlife measurement, 2:37 Oceanography, 4:115 Octahedron, 3:134–135 Office of Research and Development, 1:70 Ohm’s Law, 4:73–74 Olmos, Edward James, 3:27 On the Economy of Machinery and Manufactures (Babbage), 1:58 One-time pad encryption method, 1:160 One-to-one functions, 3:4 Onnes, Kamerlingh, 4:74 Onto functions, 3:4 Öort cloud, defined, 1:99 and Glossary Operator genes, 2:85 Opinion polling. See Polls and polling Optical computers, 1:134–135 Optical discs, 1:26 Orbital flights and moon satellites, 4:48–49 Orbital velocities, 4:53 defined, 4:51 and Glossary Ordered pairs, mapping for, 3:4–5 Ordinates, 3:10 .org domain, 2:146 Organic, defined, 2:13 and Glossary Osborne I portable computer, 1:137 Oscillate, defined, 1:88 and Glossary Oughtred, William, 3:14, 4:15 Outliers, defined, 1:85, 2:124 Over the counter stocks, 4:69 Oxidants, defined, 4:51 and Glossary

202

Oxidizers, defined, 4:52 and Glossary Ozone hole, 3:107–109

P Pace, 2:23 Palindromic primes, 3:148 Pappus, 4:11 Parabolas, 1:140–141, 3:166–167 defined, 1:34, 2:73, 3:42, 4:53 and Glossary Parabolic geometry. See Geometry Paradigms, defined, 1:25, 3:140 and Glossary Parallax, 1:50–51, 98 defined, 1:50, 4:106 and Glossary Parallel and perpendicular lines, 2:166–167 Parallel axiom, 3:70–71 Parallel operations, 1:130 Parallel processing, defined, 1:68 Parallelogram, 1:93–94 defined, 1:93 and Glossary Parallelogram law, 4:128 Parameter changes and graphs, 2:113–116 Parameters, defined, 1:88, 2:38 and Glossary Parity bits, defined, 1:106 and Glossary Parsecs, 1:51 Parsons, William, 4:84 Partial sums, defined, 4:9 and Glossary Parts per million (ppm), 4:99 Pascal, Blaise, 3:16–17, 111, 111–112 Pascaline, 3:16–17 Pattern recognition, defined, 1:132 and Glossary Patterns (sequences), 3:112–114 “Pay-as-you-throw” programs, 4:28–31 Payload specialists, 1:42 Payloads, defined, 4:36 and Glossary Payoffs, lottery, 2:174–175 Peary, Robert E., 3:127 Pelotons and echelons, 1:164 Penrose tilings, 4:91 Percent, 3:114–116 See also Decimals; Fractions; Ratios Percentages and ratios in sports, 4:55 Perceptual noise shaping, 1:108–109 defined, 1:108 and Glossary Percival, Colin, 3:125

Perfect numbers, 3:84–85 Perimeter, 2:66 Peripheral vision, defined, 2:127 and Glossary Permutation, 1:158, 2:2 Permutations and combinations, 3:116–118 Perpendicular, 1:151, 3:12 defined, 1:29, 2:37 and Glossary See also Parallel and perpendicular lines Personal computers, 1:135–139 Perspective, 2:26 Perturbations, defined, 4:23 and Glossary Pets, population of, 3:139–140 pH scale, 4:100 Pharmacists, 3:118 Phi. See Golden Number Phonograph, 1:102 Photocopiers, 3:119–123 Photographers, 3:123–124 Photosphere, defined, 4:72 and Glossary Photosynthesis, defined, 2:13 and Glossary Physical stresses of microgravity, 4:44–45 Pi, 2:136, 3:124–127 Pictorial graphs, misuse of, 3:10–11 Pie graphs, 2:110 See also Bar graphs Pilot astronauts, 1:43, 1:44–45 Pitch (sound), 4:34 Pitch (steepness). See Slope Pits and lands, 1:107, 108 Pixels, defined, 1:111, 3:7 and Glossary Place value, defined, 1:2, 2:15 and Glossary Place value concept, development of 3:82 The Plaine Discovery of the Whole Revelation of St. John (Napier), 2:35–36 Plane geometry, defined, 1:34, 2:167 and Glossary See also Geometry; Non-Euclidean geometry Plane (geometry), defined, 1:28, 2:166 and Glossary Planetary, defined, 1:5 and Glossary Platonic solids, 3:134–135, 4:19 Plimpton 322 clay tablet, 3:29 Plumb-bobs, 3:58 Plus/minus average, computing, 4:56–57 Pluto, discovery of, 4:24

Cumulative Index

Pneumatic drills, 3:59 Pneumatic tires, defined, 1:162 and Glossary Poincaré, Henri, 1:86–87, 87 Point symmetry. See Rotational symmetry Polansky, Mark, 1:42 Polar axis, 1:148 Polar coordinate system, 1:148–151 Poles, magnetic and geographic, 3:127–130 Pollen analysis, defined, 2:11 and Glossary Polls, defined, 2:7 and Glossary Polls and polling, 2:7, 3:130–133, 144–145 Pollutants, measuring, 4:99–101 Polyconic projections, 3:6 Polygons, 3:133–134 defined, 1:139, 3:5, 4:1 and Glossary Polyhedrons, 3:133–135 Polynomial functions, 2:47 Polynomials, 1:124, 3:89, 4:77 defined, 1:16 and Glossary Population density, 1:96–97 Population mathematics, 3:135–139 Population of pets, 3:139–140 Population projection. See Population mathematics Position-tracking, defined, 4:130 and Glossary Positive integers, 2:120 See also Whole numbers Postulates, 3:140–141 Potassium-argon dating, 2:14 Power (multiplication), defined, 1:20, 2:15 and Glossary See also Exponents Precalculus, defined, 1:10 and Glossary Precision and accuracy, 1:8–10 Predicted frames, defined, 1:108 and Glossary Predicting population growth and city planning, 1:96 Predicting prime numbers, 3:146 Predicting weather. See Weather, violent; Weather forecasting models Predictions, mathematical, 3:143–145 Predictions of the end of the world, 2:33–36 Prefixes massive numbers, 3:97 metric, 3:38f Presbyopia, 4:136–137 See also Farsightedness

Pressure-altitude, 2:60 Price-to-earnings ratio, 4:69 Prime Meridian, 3:68 defined, 1:15, 3:5 and Glossary Prime numbers, 2:51–52, 3:25, 145–148 Prime twin pairs, 3:147 Primitive timekeeping, 4:94–95 Privatization, 3:8 Probability and probability theory defined, 1:35, 2:53, 3:111 and Glossary density function, 2:38 experimental, 3:149–151 gaming and, 2:76–81 Law of Large Numbers and, 3:148–149 laws of, 3:20 lotteries and, 2:172–174 randomness and, 3:177–178 theoretical, 3:151–153 village layout and, 2:91–92 Problem solving, 3:153–155 Processors, 1:127 Profit margin, defined, 1:13 and Glossary Programming languages, 1:129 Projections, 3:5–7 Proofs, 3:140–141, 155–157 Properties of zero, 4:150 Proportion, 3:181–182 Proportion of a Man (Dürer), 2:26f Protractors, defined, 1:31, 2:101, 3:58 and Glossary Pseudo-random numbers, 3:177 Ptolemaic theory, defined, 4:82 and Glossary Ptolemy, 3:5, 4:18 Pulsars, 1:155–156 Pulse Code Modulation, 1:107 Puzzles of prime numbers, 3:145–148 Pyramids, 3:31, 31–32 Pythagoras, 3:93, 159, 159–162 Pythagorean theorem, 3:79, 161, 4:105 defined, 3:31, 4:18 and Glossary Pythagorean triples, 2:56, 3:157 Pythagoreans. See Pythagoras

Q Quadrant (navigational instrument), 3:67 Quadratic equations, 3:163–167 defined, 1:15, 3:30 and Glossary Quadratic form, 1:81 Quadratic formula, 3:163–167

Quadratic models of bouncing balls, 1:65–66 Quantitative, defined, 2:145 and Glossary Quantum, defined, 1:7, 2:32 and Glossary Quantum computers, 1:134 Quantum theory, 1:134 Quartz-crystal clocks, 4:96 Quasars, 1:157–158 Quaternion, defined, 1:16, 2:3, 4:32 and Glossary Quilting, 3:167–171 Quipu, 3:29 defined, 1:101 and Glossary

R R-7 rockets, 4:47 Rabin, Michael, 1:161 Radial spokes, 1:162 Radians, 1:91–92, 149 defined, 1:51, 4:109 and Glossary Radical signs, 3:173–175 See also Square roots Radicand, defined, 3:174 and Glossary Radio disc jockeys, 3:175–176 Radiocarbon dating, 2:12–14 Radiometric dating methods, 2:12 Radius, defined, 1:29, 4:17 and Glossary Radius of Earth, 1:48 Radius vectors, defined, 4:20 and Glossary Random, defined, 1:122 and Glossary Random digit dialing, 3:131 Random sampling, 3:130–131, 4:61–62 Random walks, defined, 3:177 and Glossary Randomness, 3:176–178 Range, 2:72, 3:4, defined, 4:101 and Glossary Rankine cycle, 2:117–118 Rankine scale, 1:5 Ranking, in sports, 4:54 Raster graphics, 1:111 Rate, defined, 2:139 and Glossary Rate of change, average, 3:178–179, 179f, 180f See also Functions Rate of change, instantaneous, 3:178–180 See also Functions Rates, 3:181–182 See also Interest

203

Cumulative Index

Ratio of similitude, defined, 2:20 and Glossary Rational numbers, 2:57–58, 3:80, 98–99 defined, 2:16 and Glossary Rations, defined, 1:13 and Glossary Ratios, 3:181–182 See also Decimals; Fractions Rays (geometry), 1:28 Reactivity of pollutants, 4:100 Real estate appraisal and city planning, 1:97 Real number sets, defined, 2:73 and Glossary Real number system, 3:81–83, 174 Real numbers, 3:78, 79, 81, 99–100 defined, 1:3, 2:58 and Glossary See also Irrational numbers Realtime, defined, 4:130 and Glossary Reapportionment, defined, 1:81, 2:4 and Glossary Recalibration, defined, 3:35 and Glossary Reciprocals, 2:149 Red giants, 1:154 Red shift, defined, 1:157 and Glossary Redstone rocket, 4:47 Reducing and enlarging photographs, 3:121–122 Reduction factor calculation, 3:121f Reflecting telescopes, 4:83–84 Reflection (light), 2:159 Reflection symmetry. See Mirror symmetry Reflections (of points), 4:102 defined, 4:93 and Glossary Refraction and lenses, 2:158 defined, 3:56 and Glossary Refrigerants, 2:117–118 defined, 2:117 and Glossary Regular hexagons, 1:29 Regulatory genes, 2:85 Relative dating, 2:9–11 defined, 2:9 and Glossary See also Absolute dating Relative error. See Accuracy and precision Relative speed, 2:160–161 Relativity, 2:32, 4:25, 96, 120 defined, 4:43 and Glossary See also Special relativity Reliability of Internet data, 2:145–148 Remediate, defined, 3:7 and Glossary Restaurant managers, 3:182

204

Retrograde, 1:34 Reuse and recycling, 4:27–28, 28f, 29 Revenue, defined, 1:7 and Glossary Revenues, lottery, 2:174 Rhind Papyrus, 3:30 Rhomboids, defined, 1:115 and Glossary Rhombus, defined, 4:92 and Glossary Richardson, Lewis Fry, 4:141–142 Richter scale, 2:27 Right angles, 1:28–29, 1:31 defined, 1:28 and Glossary “Rise over run.” See Slope RNA (ribonucleic acid), 2:85 Robinson, Julia Bowman, 3:183–184 Robot arms, defined, 1:43 and Glossary Robotic space missions, 4:41–42 Roche limit, defined, 1:154 and Glossary Rockets. See Spaceflight; specific rockets and space programs Rods, 3:34–35 Roebling, Emily Warren, 3:184–186, 185 Roebling, John August, 3:184–185 Roebling, Washington, 3:185 Roller coaster designers, 3:186–187 Roller coasters, 3:186 Rolling resistance, 1:162 Roman number system, 4:149–150 Rope-stretchers and angle measurement, 1:29 Rose, Axl, 4:34 Rosette spacecraft, 4:42 Ross, James Clark, 3:127 Rosse, William Parsons, Earl. See Parsons, William Rotary cutters, 3:169–170 Rotary-wing design, defined, 1:90 and Glossary Rotation, 4:102–103 defined, 4:93 and Glossary Rotational symmetry, 4:78–79, 90 Roulette, 2:78–79 Rounding, 3:187–188 defined, 2:41 and Glossary See also Irregular measurements Royal standards of measurement, 3:34–35 Rulers (measurement), 2:100 Rules for the Astrolabe (Zacuto), 3:67–68 Rushmore pointing machine, 3:58–59

S Sacherri, Giovanni Girolamo, 3:71 Saffir-Simpson hurricane scale, 4:147 Sale pricing. See Discount pricing and markup Sample size, 3:131–132 Samples and sampling, 2:5, 88, 3:10, 20, 145, 4:61 Sample, defined, 2:5 and Glossary Sampling, defined, 2:88, 3:20 and Glossary San Simeon, 3:53 Satellites, 4:35 Saturn V rocket, 4:49 Scalar quantities, 2:62 Scale drawings and models, 4:1–4 defined, 1:35 and Glossary See also Similarity Scale factor, 4:1 Scale in architecture, 1:37–38 Scaled test scoring, 4:59 Scaling, defined, 1:67, 2:124 and Glossary Schematic diagrams, defined, 2:33 and Glossary Scheutz machine, 3:18 Schickard, Wilhelm, 3:15–16 Schmandt-Besserat, Denise, 3:86 Schoty, 1:3 Scientific method and measurement, 4:4–7 Scientific notation, 4:7–8 Search engines, defined, 2:147 and Glossary Seattle (WA), 1:96 Secant, defined, 3:6 and Glossary Second Law of Thermodynamics, 2:117 Security and cryptography, 1:159–160 Self-selected samples, 4:61 Semantics, defined, 3:9 and Glossary Semi-major axis, defined, 1:99, 4:20 and Glossary Sensitivity to initial conditions, 3:144 Sequences and series, 4:8–9 Seriation, 2:10 Set dancing, defined, 2:1 and Glossary Set notation, defined, 3:4 and Glossary Set theory, 1:118–119, 2:135 defined, 1:118 and Glossary Sets, defined, 1:28 and Glossary Sextant, 2:23

Cumulative Index

Shapes, efficient, 4:9–11 ShareSpace Foundation, 4:37 Shepard, Alan, 1:45, 4:48 Shutter speed, 3:123 SI (International System of Units). See Metric system Sierpinski Gasket, 2:66 Sieve of Eratosthenes, 3:146 Signal processors, 1:124 Significant difference, 4:63 Significant digits, 4:12–14 defined, 1:9 and Glossary Silicon in transistors, 1:130–131 Similarity, 3:119–120 Similarity mapping, 4:2 Simple interest, 2:138 Simulation in population modeling, 2:39 Simulations, computer, 1:121–124 Sine, defined, 1:89, 2:89, 4:16 and Glossary See also Law of sines Sine curves, 4:110f Single point perspective, 1:115 Sirius, 1:154 Six-channel wrap-around sound systems, 2:128 666, superstition surrounding, 3:92–93 61 Cygni, 1:51 Size of numbers, 4:76 Skepticism, defined, 2:17 and Glossary Skew, defined, 1:85, 2:5 and Glossary Skew lines, 2:167–169 Slayton, Donald, 1:40 Slide rule, 3:14, 4:15–16 Slope, 1:39, 2:135, 4:16–17 defined, 2:125, 3:42 and Glossary Slow thermometer. See Galilean thermometer Small and great circles, 2:96, 4:119 Snell, Willebrord, 1:49–50 Soap bubbles and soap film surfaces, 3:48–49, 50 Software. See Computer programs and software Solar masses, defined, 4:7 and Glossary Solar panels, 1:24 Solar power collectors, 4:37 Solar Probe, 4:42 Solar sail spacecraft, 4:42 Solar system geometry, history of, 4:17–22 Solar system geometry, modern understandings of, 4:22–26

Solar winds, defined, 1:5, 4:41 and Glossary Solid geometry. See Geometry Solid waste, measuring, 4:26–31 Solving construction problems, 2:101–102 Somerville, Mary Fairfax, 4:31–33 Somerville, William, 4:32 Soroban, 1:2, 3:13 Sound, 4:33–35 Soviet space program, 4:47–48, 49 Space commercialization of, 4:35–40 exploration, 4:40–43 growing old in, 4:44–46 See also Spaceflight Space-based manufacturing, 4:37 Space navigation, 4:52–53 “Space race,” 4:47–48 Space shuttle crews, 1:42–43 Space stations, 4:36 Space tourism, 4:37–38 SpaceDev, 4:38–39 Spaceflight history of, 4:46–50 mathematics of, 4:51–54 Spatial sound, defined, 4:130 and Glossary Special relativity, 2:160, 162 See also Relativity Spectroheliograph, 4:73 Spectrum, defined, 1:52 and Glossary Speed of light, 2:157, 159–163 Speed of light travel, 4:43 Speed of sound, 4:33–34 Spheres, 3:22–23 defined, 1:49 and Glossary Spherical aberration, 4:83 Spherical geometry, 2:95–100 Spherical triangles, 2:97–98 Spin, defined, 1:53 and Glossary Sports data, 4:54–58 Sports technology, 1:52–55 Spreadsheet programs, 1:136–137 Spreadsheet simulations, 1:123–124 Sputnik, 4:47 Square roots, 1:142–143, 3:83, 173 defined, 4:18 and Glossary See also Radical signs Square, defined, 2:56 and Glossary Squaring the circle, 1:94 Stade, defined, 4:18 and Glossary Stand and Deliver (film), 3:27 Standard and Poor’s 500, 4:70 Standard deviation, 2:7, 4:58–60 defined, 4:58 and Glossary

Standard error of the mean difference, 4:64 Standard normal curve, 4:60 Standardized tests, 4:58–60 Starfish, 4:79 Stars, brightness of, 1:51–52 State lotteries, 2:171–175 Statistical analysis, 4:60–65 defined, 2:4 and Glossary Statistical models, 3:144–145 Statistics, 82, 118 defined, 1:57, 2:4 and Glossary See also Sports data; Statistical analysis; Threshold statistics Steel bicycle frames, 1:163 Stellar, defined, 1:6 and Glossary Step functions, 4:65–66 Stereographics, defined, 4:130 and Glossary Stifel, Michael, 2:35 Stochastic models, 3:177–178 Stochastic, defined, 3:178 and Glossary Stock market, 4:66–70 Stock prices, 4:67, 68 Stock tables, 4:67 Stone masons, 4:70–71 Stonehenge, 2:34 Storage of digital information, 1:126–127 See also Compact discs (CDs); Digital versatile discs (DVDs); MP3s Storm surges, defined, 4:147 and Glossary Straight line graphs, 2:113–114 Straightedges, 2:100 Stratigraphy, 2:10 See also Cross dating Strip thermometers, 4:89 Structural genes, 2:85 Suan pan, 1:1–2, 3:12–13 Sublunary, defined, 1:98 and Glossary Subtend, defined, 1:48, 2:97 and Glossary Subtraction algebra tiles, 1:17 algorithms, 1:21–22 complex numbers, 3:88–89 inverses, 2:148–149 matrices, 3:32–33 significant digits, 4:14 slide rules, 4:15 Summary of Proclus (Eudemus), 1:47 Sun, 4:71–73 Sundials, 4:95, 96

205

Cumulative Index

Sunspot activity, defined, 2:11, 4:72 and Glossary Supercell thunderstorms, 4:144 Superconductivity, 4:73–75 defined, 4:74 and Glossary Supernovae and novae, 1:153–155 Superposition, defined, 2:10 and Glossary Superstitious and forbidden numbers, 3:92–95 Supressor genes, 2:85 Surface illusion in computer animation, 1:115 Surfaces, 3:22–25 Suspension bridges, defined, 3:184 and Glossary Symbols, mathematical, 4:75–77, 76f Symmetry, 4:77–79 See also Tessellations Symmetry group, 4:91 Synapse, defined, 1:68 and Glossary Systems analysts. See Computer analysts

T T1 and T3 lines, 2:144 Tabulating machine, 2:120 Tactile, defined, 1:133 and Glossary Tailwind, defined, 2:61 and Glossary Take-off (flight), 2:62–63 Tall ships, 3:66 Talleyrand, 3:37–38 Tally bones, 3:27–28 Tangent, defined, 1:73, 3:6, 4:16 and Glossary Tangent curves, 4:110f Tangent lines, 1:73, 73f Tangential spokes, 1:162 Tectonic plates, defined, 2:27 and Glossary Telepresence, 4:132f Telescopes, 4:81–85 Television ratings, 4:85–87 Telomeres, 4:46 Temperature measurement, 4:87–90 Temperature of the sun, 4:72 Temperature scales, 1:4, 5 Tenable, defined, 4: 21 and Glossary Tenth Problem, 3:183–184 Terraforming, 4:39 Terrestrial refraction, definition, 3:56 and Glossary Tessellations, 4:90–92, 90f defined, 3:170 and Glossary See also Symmetry

206

Tessellations, making, 4:92–94 Tesseracts, defined, 2:22 and Glossary Tetrahedral, defined, 3:48 and Glossary Tetrahedrons, 3:134–135 Thales, 2:23 Theodolite, 2:24 Theorems, 3:140–141 defined, 2:44 and Glossary Theoretical probability, 3:151–153 Theory of gravitation, 2:161 Theory of relativity. See Relativity; Special relativity Thermal equilibrium, 4:88 Thermograph, 4:89 Thermoluminescence, 2:14 13, superstitions surrounding, 3:92 Thomas, Charles Xavier, 3:17 Threatened species, 2:39 defined, 2:36 and Glossary Three-body problem, 1:86–87 3-D objects, display of, 1:111–112 Three-dimensional coordinate system, 1:151–153 Threshold statistics, 4:56 Thrust, 1:90 Thunderstorms, 4:144–145 Thünen, Johan Heinrich von, 2:89 Tic-Tac-Toe, 2:76 Ticker symbols, 4:68 Tilings. See Tessellations Time dilation, defined, 1:156, 2:162 and Glossary Time measurement, 4:94–97 See also Calendars Time zones, 4:97 Titanium bicycle frames, 1:163 Tito, Dennis, 4:37–38 Tools of geometry, 2:100–102 Topography, 1:97 Topology, 4:97–99 defined, 2:40, 3:22 and Glossary Topspin, defined, 1:53 and Glossary Tornadoes, 4:145–146, 146 Torus, 3:23 TOTO (TOtable Tornado Observatory), 4:145 Toxic pollutants, measuring, 4:99–101 Toxicity Characteristic Leaching Procedure, 4:100–101 Toxicity of pollutants, 4:100 Toy Story (film), 1:114 Traffic flow and city planning, 1:97 Trajectory, defined, 2:3, 4:52 and Glossary

Transcendental numbers, defined, 1:94 and Glossary Transect, defined, 2:36 and Glossary Transect sampling, 2:37–38 Transfer of learning, 1:122–123 Transfinite numbers, 3:159 Transformations, 4:101–105 defined, 2:65 and Glossary Transistors, 1:125–126, 130–131 defined, 1:62 and Glossary Transits (tool), defined, 1:28 and Glossary Translation, 4:90, 103 Translational symmetry, 4:90 Treaty of the Meter (1875), 3:36, 37 Tree-ring dating. See Dendrochronology Trees (pattern type), defined, 1:104 and Glossary Triangles, 4:105–106 in astronomy, 1:46 equilateral, 1:29 isosceles, 1:98 similarity, 3:120 spherical, 2:97–98 See also Trigonometry Triangulation, 2:104–105 Trichotomy Axiom, 3:154 Trigonometric ratios, defined, 1:49, 4:107 and Glossary Trigonometry, 4:105–106, 107–111 defined, 1:31, 3:6, 4:18 and Glossary Triple point of water, 4:88 Triskaidekaphobia, 3:92 Triton, 1:5, 6 Tropical cyclones. See Hurricanes Troposphere, ozone in, 3:107 Troy, excavation of, 2:10 True-altitude, 2:60 Turing, Alan, 4:111–113, 112 Turing machines, 4:112 Turnovers, computing, 4:57 Twenty 21 (card game). See Blackjack 2000 U.S. presidential election, 3:132–133 Tyranny of numbers, 3:100–103

U Ultimate strength, 1:163 Ultraviolet wavelengths, 2:158 Unbiased samples, defined, 2:5 and Glossary Undercounting, U.S. Census, 1:81–82 Undersea exploration, 4:115–119

Cumulative Index

Underspin, defined, 1:53 and Glossary Unemployment rate, 2:29 Unicode, defined, 1:103 and Glossary Union, defined, 1:28 and Glossary Unit pricing, 1:144–145 Unit vectors, 4:127 Universal Law of Gravity. See Gravity Universal Product Codes, 1:119 Universe, geometry of, 4:119–123 Unlucky numbers. See Forbidden and superstitious numbers Upper bound, defined, 1:87 and Glossary Uranus, discovery of, 4:24 U.S. Census, 1:81–83, 2:4 U.S. Survey Foot, 3:37 Usachev, Yuri, 4:45

V V-2 rockets, 4:47, 52 Vacuum energy, 4:123 Validity and mathematical models, 3:143–144 Value and form, 2:64 Vanguard launch rocket, 4:48 Vapor-compression refrigeration cycle. See Rankine cycle Variable stars, defined, 1:47 and Glossary Variables, 1:123 defined, 1;15, 2:47 and Glossary See also Variation, direct and inverse Variation, direct and inverse, 4:125–126, 126f Vector graphics, 1:111 Vector quantities, 2:62 Vectors, 4:127–129 See also Cartesian coordinate system Velocity, 1:72–73, 4:128–129 defined, 1:72, 2:63, 3:178, 4:21 and Glossary Vernal equinox, defined, 1:78 and Glossary

Vertex, defined, 1:98, 2:18, 3:55 and Glossary Viable population. See Minimum viable population Video games and topology, 3:23 Vikings, navigation, 3:65–66 Vinci, Leonardo da. See Leonardo da Vinci Violent weather, 4:144–148 Virtual Reality, 1:111, 133–134, 4:129–135, 134 See also Computer animation Visibility in water, 4:116 Vision, measurement of, 4:135–138 Visual sequences, 3:112 See also Fibonacci sequence Vivid Mandala system, 4:134 Volume equations for stored grain, 1:14f Volume of cones and cylinders, 4:138–139 Volume (stocks), 4:69 Von Neumann, John, 4:142 Voskhod spacecraft, 4:48–49

W Warp drives, 2:162 Water and temperature scales, 4:89 Water pressure, 4:115–116 Water temperature, 4:116 Watts, 4:34 Waugh, Andrew, 3:56 Wavelength, 2:157–158, 4:33 Wearable computers, 1:133 Weather, violent, 4:144–148 Weather balloons, 4:141 Weather forecasting models, 1:87, 4:141–144 Web designers, 4:148 See also Computer graphic artists; Internet Weighted average, 4:56 Wheel spokes, 1:162 Wheelchair ramps, 2:109 White dwarfs, 1:153–154 White holes, 2:161 Whole numbers, 3:79, 80, 103–104 defined, 1:65, 2:51 and Glossary

Wiles, Andrew, 2:54, 54–55 Wind resistance. See Drag Wind tunnels, and athletics technology, 1:55 Wisconsin v. City of New York (1996), 1:81–82 “Witch of Agnesi,” 1:10 Women astronauts, 1:41–42, 45 Working fluids in air conditioning. See Refrigerants World Trade Center attack, affect on stock market, 4:69 World Wide Web, 2:145 defined, 4:148 and Glossary Wormholes, 2:161–162 Wright, Frank Lloyd, 4:3 Writing and numbers, 3:86–87

X X chromosome, 2:85

Y Yaw, 2:62 Yeast, 1:147 Yield (stocks), 4:68–69 Yield strength, 1:163 Y2K (Year 2000). See Millennium bug

Z Zacuto, Abraham, 3:67–68 Zaph, Herman, 2:152 Zenith, defined, 3:108 and Glossary Zenith angle, defined, 3:108 and Glossary Zenodorus, 4:10–11 Zero, 3:82, 4:149–151 division by, 2:25–26, 4:150 inverses, 2:148, 149 Zero exponents, 3:142 Zero pair, defined, 1:17 and Glossary Zero product property, 3:163 Ziggurats, defined, 1:37 and Glossary

207