Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
Formulae for sums of consecutive square roots
For a real number x, let [x] denote the largest integer not exceeding x.The fol lowing result might be surprising. Theorem 1.
(i) (ii) (iii) (iv)
The following formulae hold for every positive integer n.
[Vn +vn+l] = [v'4n +1] [Vn +vn+1 +vn+2] [Y9n +8] [Vn +vn+1 + vn+2 +vn+3] = [Y16n +20] [Vn +vn+1 + vn+2 +vn+3 +vn+ 4] = [Y25n +49]. =
Formula (i) is folklore; (ii) is a problem in [1]; (iii) can be found in [2, p.274]. The purpose of this note is to prove (iv) and consider related questions. Proof of (iv). For positive numbers xi= y we have vX + Vy < v'2(x +y). Using this inequality twice we get Vn +vn+1 +vn+2 +vn+3 +vn+4 = cvn + vn+4) +cvn+I +vn+3) +vn+2 < v'4n +8 +v'4n +8 +vn+2 =5vn+2 = v'25n +50. Thus Vn +vn+1 +vn+2 +vn+3 +vn+4 < v'25n +50.
(1)
Usin� the fact that vX > H(x +i)�-(x-i)�} for any real number x?: 1, we obtam Vn +vn+1 + vn+2 +Vn+3 +vn+4 >
{(
2 n +2 9 3
) (n 21 ) } " 2
-
-
" 2
•
(2)
Now we show that when n ?: 12, 2
3
{(n + 29 )" (n 2 )"} -
2
-
1
2
>
Y25n +49.
(3)
Letj(x) =�{(x +t)�- (x-i)�}-Y25x +49.Thenf(l2) > 0, limx-.ooJtx) = O,j(x) is increasing on [12, 14841/400] and decreasing on [14841/400, ) Sof(x) is positive on [12, ) and (3) is proved.Combining (1), (2), and (3), we deduce that when n ?: 12, co .
co
Y25n +49 < Vn +vn+1 +vn+2 +vn+3 +vn+4 < v'25n +50. Since no integer lies strictly between v'25n +49 and v'25n +50, we conclude that (iv) is valid for the case n 2: 12.The cases n 1, 2, ..., 11 are verified by the computer software Matlab.This completes the proof. D =
In view of Theorem 1, it is natural to suspect that for any positive integer k there is a constant c depending on k such that [Vn +vn+1 + 4
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc.
·
·
·
+v'n +k - 1]
=
[Yk2n + c]
(4)
holds for all positive integers n. This is not the case.It is shown in [2, pp. 725-727] that for sufficiently large k no such c exists.Our next result shows that 6 is the first k for which (4) cannot hold for all n. For any real number c, there is a positive integer n such that
Theorem 2.
Proof Let s(n) = ['Vn + v'n+1 + vn+2 + v'n+3 + v'n+4 + \l'n+5]. Using Matlab we find that s(1) 10 < 1 1 s(11) = 22 > 21 =
Therefore, when c 2: 85, [ Y36 X 1 1 + c].
s(l)
<
= =
[ Y36 [ Y36
[ Y36 X 1
+
x X
1 + 85], 1 1 + 85].
c], and when c::; 85,
s(l1) > 0
Prompted by the evidence in Theorems 1 and 2, I pose the following conjec ture. For any positive integer k 2: 6, no constant c depending only on k exists such that (4) is valid for all positive integers n.
Conjecture.
We also have the following related question. Question.
For which positive integers k does there exist an integer c such that
lvn + vn+1 +
·
·
·
+
Vn + k - 1 - Yk2n +
cl < 1
holds for all positive integers n? When such a c exists, determine it. Acknowledgment
The author thanks the referee for valuable comments.
REFERENCES
1 . F. David Hammer, Problem E301 0, Amer. Math. Monthly 95 (1 988), 1 33-1 34. 2. Z. Wang, A City of Nice Mathematics (in Chinese). The Democracy and Construction Press,
Beijing, 2000. Xingzhi Zhan Department of Mathematics East China Normal University Shanghai 200062 China e-mail:
[email protected]
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
5
The Matrix by Jim Demmel . l.auix!
1attix! r ad or writ ,
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In the eaehe, or byt
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Department of Bectncal Eng neenng and Computer Sciences Unrverstty o Cahfomta Berkeley, CA 94720 USA
a-mad·
[email protected] edu
6
THE MATHEMATICAL INTELUGENCER © 2005 Springer Scrence+ Business Media, Inc.
th
of th r
I ft me
CLAUDIA MASFERRER LEON AND SEBASTIAN VON WUTHENAU MAYER
Reinventing the Whee : Non Circular Whees
e set out to prove the impossibility of non-circular wheels, and surprised ourselves by proving the opposite. A vehicle rolls smoothly over a surface when the axle is placed at the centre of a circular wheel. Previous attempts to adopt non-circular wheels consisted in running the vehicle
over a non-level surface to avoid the up-and-down motion. In this paper we present a method that allows us to have non-circular wheels by modifying the shape of the axle. Before the invention of the wheel, some cultures used cylindrical rollers when moving heavy loads. The load was rolled along over the cylinders, with the disadvantage that they had to be continually replaced under the front. About 5500 years ago, an anonymous Sumerian in Mesopotamia invented what must surely be mankind's sin gle greatest technological achievement-the wheel. So sim ple, yet so ingenious! What makes this innovation so re markable is that no similar artefact exists in nature; humanity did not copy or adapt some already existing ob ject, but created it in a giant leap of imagination. Nowadays, wheels play a very important role in every day life. From agriculture to the exploration of Mars, from people transportation to massive movement of products; everything is possible due to the existence of the wheel. When included as components of a vehicle, wheels al-
low the vehicle to roll smoothly over a surface. The wheel is round because a circle is the geometric locus of points equidistant from a fixed point. An axle placed at the cen tre of the wheel will stay at a constant altitude from the ground as the wheel rotates. Most people think the circle is the only shape wheels can have. Typically, they only make reference (if any) to non-circular wheels in jokes and metaphors (for an exam ple see Figure 1). An early approach to using non-circular wheels-albeit not a practical one-consisted in modifying the shape of the surface to prevent the up-and-down motion of the cen tre of gravity of the non-circular wheel (see Figure 2). Different figures rolling over modified surfaces can be found in science exhibitions around the world (Figure 3). If we use rollers, rather than wheels on axles fixed to the vehicle, then any constant-breadth shape will do in place of the circle. Constant-breadth figures have been known for a considerable time and were called by Euler orbiforms,
This work was partially supported by the lnstituto de Matematicas at the Universidad Nacional Aut6noma de Mexico and the lnstituto Technol6gico Aut6nomo de Mexico
© 2005 Springer Science+Buslness Med1a. Inc., Volume 27, Number 4, 2005
7
�tt n
fr�
n
n
Figure 2. Motion of the centre of gravity and previous solution.
:Jt 's tiH HiptOIIIJifitlHt OH
Mathematical Background
tAt� sq�tartJ wAtJt�l.
We must begin with convex sets, their representation by a support function, and some special convex sets that will be relevant later. A set of points K is convex if it contains every line seg ment with end-points inK. If in addition, K is bounded and has interior points, its boundary (denoted by oK) is called a closed convex curve. A line L is a support line of the convex set K at point aE oK if it has the following properties: •
YtJS, It tJIIHiiHtettJS OHtJ G�tlli
Figure 1 . Improvement of the square wheel.
Latin for circular curves. They are figures that have the same width in every direction, so they work as well as cir cular rollers. Other applications of these figures include the Waenkel engine, a drill for cutting square holes, and the British 50-pence coin (see [3, 1]). In this article we present a different solution: non-cir cular figures can be used as wheels not rollers, yet allow the vehicle to run smoothly on a level surface, by modify ing the shape of the axle! Ironically, this idea was the re sult of an unsuccessful attempt to prove the following as sertion from a fascinating book [3]: Obviously a wheel must be made in the form of a circle with the hub at the centre, since any other form will pro duce an up-and-down motion. This work started in 1998 with a home-made wooden car model with constant-breadth wheels for the Mexican Na tional Science Fair (Fig. 4). Mter winning the first prize, we were invited to present the project in various forums. Since then, further work has led us to this paper. 8
THE MATHEMATICAL INTELLIGENCER
•
aEL. K is contained in the closure of one of the two open half planes into which L cuts the plane.
Note that every point on oK lies on a support line and there are exactly two support lines perpendicular to each direction. We can represent K by the set of its support lines. A support line L may be parametrized by (q,p) where q is the angle between its normal and the x-axis and p is the distance from L to a fixed interior point of K (see Figure 5). Sincepis uniquely determined by q, the set of support lines is ((q,p(q)) : q E [0, 21r)}. The support function of K is p(q). For simplicity, we will consider only convex sets whose support function p(q) is of class C2, even though everything can be extended to the case when p(q) is continuous and piecewise C2. With the last hypothesis, it can be proven that p + p" :::0: 0 is a necessary and sufficient condition for a periodic func tion p to be the support function of a convex set K. Fur thermore,p(q) + p"(q) is the radius of curvature of the con vex curve at the point of contact with the support line (q,p(q)) [4]. For any convex set K, the union of all closed disks of radius rand centres inK, denoted byKn has boundary oKr parallel to oK, in the strong sense that for any XE oK and any y E oKn r =min { dist(x,z) : z E oKr} =min { dist(y z) : z E oK}. ,
Clearly, if p(q) is the support function of K, then the sup port function of Kr isp(q) + r. Constant-breadth sets are convex sets that share with the circle the property of having the same breadth in every direction. The distance between the two parallel support
Figure 4. Wooden car model. Figure 3. Science exhibitions of modified surfaces.
lines of a convex set K that are perpendicular to the di rection q is the breadth b(q) of K. We have b(q) = p(q)
+
p(q
+
)
11' .
If b(q) is the same for all q, the set K is said to be of con stant breadth. Note that in this case, the parallel set K,. is also of constant breadth. The simplest non-circular constant-breadth convex set is the Reuleaux Triangle. It is constructed by taking an equi lateral triangle and replacing each side with an arc of a cir cle that passes through the two vertices and has the oppo-
Figure 5. Parametrization of a support line.
© 2005 Springer Science+Business Media, Inc., Volume 27, Number 4, 2005
9
Figure 6. Constant-breadth bodies.
site vertex as its centre. The same procedure can be gen eralized to regular polygons with an odd number of sides. However, constant-breadth bodies need not be regular, or even consist of circular arcs (see Figure 6). It is well known that constant-breadth sets can rotate in the interior of a square, maintaining contact at all time with all four sides (i.e., all circumscribed rectangles are congruent squares). Moreover, some convex sets can rotate inside a fixed equilateral triangle. These sets are called triangular sets. [5] The simplest non-circular triangular set is formed by the two arcs of equal radius and angle f that join two end points (see Figure 7). A point belongs to an equilateral triangle iff the sum of the distances from the point to the sides is equal to the height h of the triangle. It follows (Barbier's theorem) that a con vex set is a triangular set if its support function satisfies
(
p( (}) + p (} +
) (
)
4 2 7T + p (} + 7T = h 3 3
Let h denote the side-length of the square, a(q) the sup port function of the axle, and d (> h) a constant. For the upper side of the square to remain at constant altitude d, the support function of the wheel has to satisfy, for all q, r(q) = d - a(q + 1r) d - (h - a(q)) = (d- h) + a(q). =
It can be easily verified that r" + r 2:: 0, so r(q) is indeed a support function of a convex set. Therefore, the wheel we are looking for is a figure par allel to the axle. For example, Figure 8 shows three stages of the movement of a constant-breadth wheel when the axle is a Reuleaux Triangle. This is the structure we used in our wooden model. Triangular-based Wheels
Now, consider instead the following structure:
for all angles e. [5] Curves with constant breadth h and triangular sets in side a triangle of height h share the property of having perimeter 7Th.
•
Constant-Breadth Wheels
As before, the aim is to find the shape of the wheel such that, when it rotates, the distance from the ground to the upper side of the triangle remains constant. Let h denote the length of the triangle, a(q) the su� rt function of the axle, and d a constant greater than V 3/4h. For the upper side of the square to remain at constant al titude d, the support function of the wheel has to satisfy, for all q, r(q) d - a(q + 7T).
To employ non-circular wheels we need to ensure that when the vehicle moves, it maintains the same distance from the ground. Consider the following structure: • • •
A square attached rigidly to the vehicle (with two of its sides parallel to the ground). A constant-breadth axle that rotates inside the square, maintaining contact with its sides. A wheel that is attached rigidly to the axle.
Let us try to find the shape of the wheel such that, when it rotates, the distance from the ground to the upper side of the square remains constant.
Figure 7. Non-circular triangular set.
10
THE MATHEMATICAL INTELLIGENCER
• •
An equilateral triangle rigidly attached to the vehicle (with its upper side parallel to the ground). A triangular set as the axle that rotates inside the triangle, maintaining contact with its sides. A wheel that is attached rigidly to the axle.
=
And we need to make sure that r is a support function of a convex set. We require that r(q) + r"(q) = d - (a(q
+
1r) + a"(q + 1r)) 2:: 0
'r/q.
It follows that d has to be at least equal to any radius of curvature of the axle. It can be easily verified that r(q) is
Figure 8. The movement of a constant-breadth wheel.
h
r(e)
Figure 1 0. Notation. Figure 9. A triangular wheel.
also a triangular function and the wheel is a triangular set (see Figure
9).
Given a figure that rotates inside, the position of the triangle or parallelogram determines the shape of the wheel. For instance, if the upper side of the triangle or parallelogram is parallel to the ground, the support func
General Non-Circular Wheels Generalizing, we can attach any convex polygon vehicle, have an axle
A
P
to the
that rotates inside (touching at all
times the sides of the polygon), and seek a wheel such that, when it rotates, the polygon remains at constant altitude
a(q d - a(q + 1r)
tion of the wheel is obtained by subtracting from a constant altitude
d,
i.e.,
r(q)
=
+ 1r)
as in
the previous sections. In case the triangle is attached to the vehicle with its lower side parallel to the ground,
from the ground. Since every convex polygon possesses at least one of the following properties: •
•
it is a parallelogram, the extension of three of its sides forms a triangle that contains the polygon,
we can focus on axles that rotate inside parallelograms and triangles.
An axle that rotates inside a parallelogram has constant a(8) satisfies a(8) + a(8 + 1r) = constant.
breadth; therefore its support function
On the other hand, an axle that rotates inside a fixed tri angle (with vertices DEF) has a support function
a(8)
that
satisfies
a(8)
��� + dist(D,EF)
--
a(8
+ (1r- L EDF))
�--�--�----�
dist(E,FD)
+
a(8)
+
L FED)) . d1st(F,ED)
a(8 - (1r-
= 1,
a"(8) :=:::: 0.
The reason is that a point xis contained in the triangle DEF
if and only if
distance(x,EF) distance(D,EF)
+
distance(x,ED) distance(F,ED) +
distance(x,FD) distance(E,FD)
=
1.
For every triangle there exists at least one figure that can rotate inside: the inscribed circle. The main question is whether there exist non-circular figures that can rotate inside as well. We have found such figures for the trian
3'7T/5, 1TI5, 1TI5, and believe that there are case 2'7T/3, 1rl6, 1rl6. The general case is still
gle with angles none for the open.
Figure 1 1. Traces for constant-breadth and triangular-based wheels.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
11
It is asi r to mali ian
uar a ir 1
Ulan to g
t round a math
-Aug ·tus d Morgan
to the wheel rim, or apply rotato:ry force to the axle-just as for familiar vehicles. Non-circular wheels may be a great advance in technol ogy or just a curiosity, who knows? What remains is to study the advantages of these wheels; maybe one day cars will have "triangular" or "squared" wheels. Acknowledgments
Figure 12. Different wheels.
then the wheel has to be parallel to the axle so the tri angle remains at a constant altitude. We can also allow the case where none of the sides of the triangle or par allelogram is parallel to the ground. It can be easily shown that, for the triangle or parallelogram to remain at a constant altitude, the support function of the wheel must be r(8) = p(O) + p( (} - _( 7T - a)) sin + (8) cos + h, f3 f3 p tan a sm a
(
)
where • • •
h is the distance from the triangle or parallelogram to the
ground. a is the angle of the triangle or parallelogram at the lower vertex. f3 is the angle between the ground and one of the lower sides of the triangle or parallelogram (see Figure 10).
"Squared" and "Triangular" Wheels
is clear that at whatever speed a circular wheel moves, it will always look round. For our non-circular wheels, the path that is traced (relative to the vehicle) by a point on the wheel is no longer a circle. Therefore, the wheel while rotating may not look round. For instance, if the axle is a Reuleaux trian gle rotating inside the square, the wheel traces a path that is almost a square; the only deviation is at the comers, where there is a slight rounding. If the vehicle moves fast enough, the wheel will resemble a square. Similarly, if the wheels are triangular sets rotating inside a triangle, then the path traced resembles a triangle with rounded comers (see Figure 1 1). It
Final Remarks
In
mathematics, your preconceptions are not barriers.Now we know a vehicle can have wheels of different shapes and still move at a constant altitude from the ground (Figure 12). To move it, one can push or tow the vehicle, or apply force
12
THE MATHEMATICAL INTELLIGENCER
Chronologically, we would like to thank all the people who collaborated with us during the development of non-circu lar wheels. First of all, we are grateful to Aisha Najera for her participation in the beginning of this project. We appreciate the help offered by Luis Montejano, for mer Director of the Mathematics Institute of the UNAM, and Concepcion Ruiz, former Director of the Mexican Sci ence Museum Universum, for their help and for giving us the opportunity of presenting this in various forums. It is our pleasure to thank Jorge Urrutia for many stim ulating discussions on convexity and his guidance during the completion of the first part of the work; and Margaret Schroeder for her help with the translation of a first ver sion of this paper. We appreciate their patience. We would like to thank Hector Lomeli and Jose Luis Farah for their comments in the final editing of this paper. We would credit the creator of the cartoon in Figure 1, only we don't know who it is. Congratulations to the car toonist on having your work pass into folklore. Finally, we thank COSI Columbus, David Eppstein, and Stan Wagon for providing us the photos in Figure 3. From top to bottom: • • •
Square-wheeled car from Cleveland Science Museum ex hibit.Photo courtesy of COSI Columbus. Photo courtesy of David Eppstein from the Exploratorium at the Palace of Fine Arts in San Francisco, California Stan Wagon on his bicycle at Macalester College. Photo courtesy of S. Wagon.
REFERENCES
[1 ] M. Gardner. Mathematical games: Curves of constant width, one
of which makes it possible to drill square holes. Scientific American 208, no. 2 (1963), 1 48. [2] L. Montejano. Cuerpos de Ancho Constante. Fonda de Cultura
Econ6mica, Mexico, 1998.
[3] H. Rademacher and 0. Toeplitz. The Enjoyment of Mathematics.
Princeton University Press, Princeton, NJ, 1957. [4] L. Santa16. Integral Geometry and Geometric Probability. Addison
Wesley, Reading, 1976.
[5] I. M. Yaglom and V. G. Boltyanskii. Convex Figures. Holt, Rinehart
and Winston, New York, 1961 .
AUTHORS
SEBASTIAN YON WUTHENAU MAYEA
CLAUDIA MASFERAER LEON
!nst1tut0 Techno16gco AutOnomo de M8xico
lnstrtuto Techno16g100 AutOnomo de MexiCO
San
Angel, MeXICO City,
M9XICO
San
Angel, MexiCO C1ty, Mexico
e-mail:
[email protected]
e-mail:
[email protected] Wuthenau partiCipated in
Bom tn Mexico 1n 1980. Claudia Masferrer partiCipated tn the Mex
lnst1tute at the Weizmann lnstrtute. This experience led to his
slltute at the Weizmann Institute. This experience led to her deci
tion he worked tor McKinsey & Co. MBXJCO as a bus1ness analyst.
director of her unrversity literary joumal. She Intends to go on to
Bom In M9X1Co 1n 1980. SebastiAn
von
the Mex1can MathematiCS Olympiad, and later 1n the Summer Sci ence
decisiOn to study Apphed MathematiCS at the !TAM. After gradua He
will begin graduate stud1es
next year; hiS ma1n Interests are
geomelly and d1screte mathematiCS. tn partiCUlar cryptography. He
enjoys outdoor sports and travel.
ican Mathematics Olympiad, and later 1n the Summer Science ln
Sion to study Applied MathematiCS at the ITAM. In 2004 She was
graduate school, but at present she
IS
wor1
the social sciences for a Mexican government ministry. Her hob bles are travel, theatre, belly-danCing, and reading.
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© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
13
GEORGE W. HART
Creating a Mathematical Museum on Your Desk ravel is unnecessary to see certain objects of mathematical interest, as new technology makes it convenient to be a stay-at-home explorer of 3D mathematical models. With the right equipment, one can create and enjoy a mathematical museum collection without leaving one's office. Figure 1 shows an assortment of physical models on the shelves of my office. New technologies-commercially available or under development-allow the creation of physical models that are very compact, intricate, and accurate. These processes involve the automated robotic construction of physical objects by assembling thin cross-sections calcu lated from the designer's computer file describing the struc ture to be built. Engineers call these "rapid prototyping" fabrication processes because at present their major ap plication is for quickly making prototypes of parts that are later mass-produced. However, the more general term "solid freeform fabrication" (SFF) conveys the idea that these objects need be neither rapid-some of these mod els took a machine three days to build-nor prototypes for us they are the final result. Solid freeform fabrication technology has applications in the creation of all types of educational 3D objects, e.g., topological, algebraic, molecular, crystal, anatomical, or historical models. Remnants of older collections, such as those at Gottingen and Arizona University, speak to us from another era, when models were of considerable di dactic importance. In the late nineteenth century, plaster, wood, wire, or cardboard mathematical models were pro moted by prominent mathematicians such as Felix Klein. Their use diminished in the twentieth century, but many mathematicians consider models valuable for building in tuition and for communicating mathematical ideas to students and to the public. A "Cabinet of Curiosities" fea turing mathematical forms can serve to inspire non-math ematicians and hook students. Nothing can substitute for the visual and tactile pleasure of handling a model, spin ning it in one's hand, comparing it to another model in the 14
THE MATHEMATICAL INTELUGENCER © 2005 Springer Science +Bus1ness Media, Inc.
other hand, etc. This paper illustrates only geometric mod els, but the references include links to algebraic surface models by Carlo Sequin or Jonathan Chertok that were generated in a similar spirit. Fractals, because they are intricate and procedurally generated, are natural subjects for SFF. Figure 2 shows a model of a well-known example, the Menger Sponge. This fractal is (in the limit) all surface with no volume, and a model makes this clear by its lightness when it is picked up. The model in Figure 2 is made of nylon by a SFF process called "Selective Laser Sintering" (SLS) in which a high powered computer-controlled laser fuses nylon dust in just the places where we want solidity and leaves the dust to be vacuumed away in the places where we want voids. Another familiar fractal, the Sierpinski Tetrahedron, is shown in Figure 3. This model is also made of nylon by the SLS process. Underlying all SFF models are digital de scriptions of the forms to be constructed. A ftle describing the boundary surface as a triangulated manifold must be created as input for guiding the SFF machine. With some effort, one may adapt commercial Computer Aided Design packages (CAD software designed for engineers) to create the geometry description files. However, my interest is in writing special-purpose software that conveniently gener ates families of interesting objects. Figure 4 shows models of two beautiful uniform polyhe dral compounds with icosahedral symmetry. The first is six concentric pentagonal prisms; the second is five concentric truncated tetrahedra. They were first described in the math ematics literature by John Skilling in 1976, but I may have been the first to make a physical model of them, in 1999.
Figure 1 . Mathematical forms, mostly about 3 inches in diameter, built with various SFF machines.
These are made of plaster by the "3D Printing" SFF method, which uses in�et printer technology to squirt water selec tively in the places where the plaster dust is to be hardened, leaving the unmoistened plaster dust to be vacuumed away. The object in Figure 5 is composed of seven different nested polyhedra, each of which is free to rotate indepen dently of the others. Each layer is a polyhedron with twelve pentagons and some number of hexagons, arranged with icosahedral symmetry. These are the examples with 42 to 192 faces from an infinite family of such forms described by Michael Goldberg in the 1930s. Goldberg described their topology, and I have explored various ways to realize them geometrically with planar faces. The model consists of only their edges, so one can see through the face openings into the progressively smaller inner polyhedra. This model is made of ABS plastic by a SFF technique called "Fused De position Modeling" (FDM). It works essentially like a hot-
melt glue gun on a robot arm, squirting molten plastic that cools to make the solid model. The 120-cell is a regular four-dimensional polytope com posed of 120 regular dodecahedra, three around each edge, discovered by Ludwig Schliifli in the 1850s. Figure 6 shows a three-dimensional shadow of the 120-cell, via a perspec tive transformation, which results in 119 progressively flat tened dodecahedra packed into an outer regular dodeca hedron. Wire and thread models of this form were designed by Victor Schlegel and sold commercially in the 1880s through catalogs of mathematical models. Here I have recreated the form and had it fabricated of sintered stain less steel infused with bronze by the "Prometal" SFF process. The "truncated 120-cell" is a more complex four-dimen sional polytope, analogous to an Archimedean polyhedron. First described in a 1910 paper by Alicia Boole Stott, it con-
Figure 2. Model of Menger Sponge, 3 inches.
Figure 3. Model of Sierpinski Tetrahedron, 8 inch.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
15
Figure 4. Compounds: Six pentagonal prisms, Five truncated tetrahedra, 3 inch diam.
sists of
120
truncated dodecahedra and
600
regular tetra
hedra. They are arranged so that the symmetry group is transitive on its 2400 vertices, each equivalently surrounded by three truncated dodecahedra and one tetrahedron. It can be projected via an orthogonal projection to the intricate
3D form shown in
close-up in Figure
"back" coincide, halving its
3D
7. The 4D
"front" and
complexity, which is help
ful when one is learning to visualize such forms. This model is made of an epoxy polymer by the "stereolithography"
SFF
process.
A
computer-controlled ultraviolet laser is
used to catalyze the hardening of a liquid photopolymer in just the places where one wants solidity. In summary, new SFF technologies allow the fabrication of mathematical models that would be very difficult to pro duce by any other means. I find a great pleasure in han dling a real
3D
object that I receive from these machines
after previously visualizing it, designing its coordinates, and Figure 5. Seven nested Goldberg polyhedra.
Figure 6. Shadow of 40 120-cell, 3 inches.
16
THE MATHEMATICAL INTELLIGENCER
Figure 7. Shadow of 40 truncated 120-cell, 6 inches, detail.
viewing it only on the computer screen. Although relatively expensive at present, SFF fabrication costs are decreasing as the technology develops. Machines are generally avail able at manufacturing design companies, at research uni versities, and at commercial "rapid prototyping" service bu reaus. Files for the above objects are freely available on my Web site and can be sent to a variety of machines for fab rication. In the future, I expect all schools will have SFF capability, and textbook publishers and other educational sources will provide files and software. Teachers will fab ricate what is relevant to their classes and pass the mod els around the room for students to examine. Anyone will be able to create a "cabinet of curiosities" which can ex cite a sense of wonder about mathematics. REFERENCES
Details of generating SFF files for mathematical forms:
G. Hart, "Solid-Segment Sculptures," Proceedings of Colloquium on Math and Arts, Maubeuge, France, 20-22 Sept 2000, and in Math ematics and Art, Claude Brute ed. , Springer-Verlag, 2002.
George W. Hart, "Rapid Prototyping of Geometric Models," Proceed ings of Canadian Conference on Computational Geometry, August 2001. George W. Hart, "In the Palm of Leonardo's Hand," Nexus Network Journal, vol. 4, no. 2, Spring 2002; reprinted in Symmetry: Culture and Science, vol. 1 1 , 2000 (appeared 2003), pp. 1 7-25.
George W. Hart "40 Polytope Projection Models by 30 Printing," to ap pear in Hyperspace. Jonathan Chertok, http: //www.oliverlabs.net/ Carlo Sequin, http: //www.cs.berkeley, edu/�sequin/
AUTHOR
Mathematical models
Gerd (Gerhard) Fischer, Mathematische Madelle (Plates) and Mathe matical Models: From the Collections of Universities and Museums
(English Commentary), Vieweg, Braunschweig, 1 986. Peggy Kidwell, "American Mathematics Viewed Objectively: The Case of Geometric Models," in Vita Mathematica: Historical Research and Integration with Teaching, Ron Calinger ed . , MAA, 1 996, pp.
1 97-208. William Mueller, "Mathematical Wunderkammern," American Mathe matical Monthly 1 08, (200 1 ) , 785-796.
Angela Vierling, list of online model collections: http://www.math. harvard. edu/� angelavc/models/locations. html Mathematical sources for the illustrated objects
George W. Hart, SFF files, http://www.georgehart.com/rp/rp. html H.S.M. Coxeter, Regular Polytopes, 1 963 (Dover reprint, 1 973). Michael Goldberg, "A Class of Multi-Symmetric Polyhedra," Tohoku Mathematics Journal 43, (1 937), 104-108.
GEORGE W. HART
Benoit Mandelbrot, The Fractal Geometry of Nature, Freeman, 1 982.
Department
J. Skilling, "Uniform Compounds of Uniform Polyhedra," Mathematical
Proceedings of the Cambridge Philosophical Society 79 (1976),
Stony Brook. NY 11794
e-maol:
[email protected]
447-457. Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings,"
Verhandelingen der Koninklijke
of Computer Science
SUNY at Stony Brook
George W. Hart, alter gett1ng a B.S. 1n Mathematlcs and a on
Electrical Engineering and Computer Science at MIT,
Akademie van Wetenschappen te Amsterdam, (eerste sectie), 1 1 ,
Ph.D.
No. 1 (1 9 1 0) 1-24 plus 3 plates.
was a professor at Columboa before He
SFF manufacturers
Bathsheba Grossman, http: //www.bathsheba.com/math/
is
moving to Stony Brook.
the author of Multidimensional Analysis (1995), and (woth
Henri Picciotto) of Zome Geometry (2001). He Is also a sculp tor, and his art, like h1s geometrical models. draws on
com
30 Systems, http: //www.3dsystems.com/
puter des1gn and fabrication. His sculpture has been widely
Extrude Hone Prometal, http://www.prometal-rt.com/
dosplayed. If you have m1ssed h1s shows, try http://www.
Stratasys Corporation, http: //www.stratasys.com/
georgehart.com.
Z Corporation, http: //www.zcorp.com/
© 2005 Springer Science +Busine::;s Media, Inc., Volume 27, Number 4, 2005
17
FILIZ DOGRU AND SERGE TABACHNIKOV
Dua Bi iards he first volume of The Mathematical Intelligencer contains an article by Jurgen Moser ''Is the solar system stable?" [24]. As a toy model for planetary motion, Moser proposed the system illustrated in Figure 1 and called it dual (or outer) billiards. The dual billiard table P is a planar oval. Choose a point x outside P. There are two tangent lines from x to P; choose one of them, say, the right one from x's viewpoint, and reflect x in the tangency point z. One obtains a new point, y, and the transformation T : x >-..,) y is the dual billiard map. Like the planetary motions, the dual billiard dynamic is easy to define but hard to an alyze; in particular, it is difficult to reach conclusions about its global properties, such as boundedness or unbounded ness of orbits. In this article we survey results on the dual billiard prob lem obtained since the publication of Moser's article. We hope that the reader will share our fascination with this beautiful subject. We do not assume familiarity with a much better studied subject of the conventional, inner billiards; an interested reader is referred to [13, 17, 28]. The definition of the dual billiard map has a shortcom ing: T is not defined if the tangency point z is not unique. This is the case if the dual billiard curve y, the boundary of P, contains a straight segment-for example, if y is a polygon. The dual billiard map and its iterations are not de fined for the points on the extensions of straight segments of y and their preimages under T. This set is a countable collection of lines and therefore a null set; hence one has This work is supported in part by the National Science Foundation (S.T.).
18
THE MATHEMATICAL /NTELLIGENCER © 2005 Springer Science+ Business Media, Inc.
ample room to play the game of dual billiards. The situa tion resembles inner billiards: if a billiard ball hits a comer of the billiard table then its motion is not defined beyond that point. J. Moser also considered the dual billiard system in his influential book [23). He learned about dual billiards from B. Neumann, whose 1959 address on the subject to the Man chester Mathematical Colloquium, titled "Sharing ham and eggs," appeared in [25]. By now, there exists a substantial literature on dual billiards listed in the bibliography. Let us consider examples.
Y= T(x)
z
p
Figure 1. Defining the dual billiard map.
X
'Y
3
3
3
2
3
3
2
1
2
3
2
1
1
2
3
2
1
2
3
3
2
3
3
3 Figure 2. Dual billiard about a square.
Example 1. If the dual billiard table is a circle then every point moves along a concentric circle, that is, the concen tric circles are invariant curves of the dual billiard map. Thus the dual billiard map about a circle is integrable (i.e., there is a conserved quantity): the radius of the circles is an in variant function. Since the dual billiard map commutes with affine transformations of the plane, the dual billiard system about an ellipse is integrable as well. An outer version of the celebrated Birkhoff conjecture (concerning inner billiards) states that the only integrable dual billiard system is the el liptic one. Like its inner counterpart, this conjecture is open. Example 2. If the dual billiard table is a square then the motion of every point is periodic. The structure of orbits is depicted in Figure 2, where every point of a tile marked n makes one visit to all other tiles with the same marking (there are 4n of them) before returning back. One can sim ilarly describe the dynamics of dual billiards about a tri angle or an affine-regular hexagon. Example 3. Let the dual billiard table be a regular penta gon. This example was analyzed in [29, 32]; see also [28]. The set of full measure, made of regular pentagons and decagons,
Figure 3. Dual billiards about regular pentagon and octagon.
Figure 4. Dual billiards about a semicircle.
consists of periodic orbits. In addition, there exist infinite or bits. One such orbit, or rather its closure, is shown in Figure 3 on the left. One cannot help noticing self-similarity of this set. Its Hausdorff dimension is equal to ln 6!ln(v5 + 2) 1.24. . . . Computer experiments show a similar behavior for other regular n-gons (except n =3, 4, 6), but so far a rigorous analy sis is available only in the cases n = 5, 8; see Figure 3 and the cover for the case of a regular octagon. See [1, 3, 12, 19, 21, 22] for related study of piecewise rotations. Example 4. An interesting example of a dual billiard table is a semi-circle. A numerical study of this case reveals a very complicated behavior: periodic trajectories and sur rounding elliptic "islands" (large white ovals in Figure 4) coexist with chaotic orbits (black set). There is strong com puter evidence that some orbits, and even domains, escape to infinity; these escaping domains are seen in Figure 4 as small white ovals, positioned between large elliptic islands. We finish this section with a mechanical interpretation of the dual billiard system as an impact oscillator, due to Ph. Boyland [4]. Consider a harmonic oscillator on the line, that is, a particle whose coordinate, as a function of time, satis fies x''(t) + x(t) = 0. Now let there be a 27T-periodi cally moving massive wall to the left of the particle whose position p(t) satisfies the differential equation p"(t) + p(t) = r(t), where r(t) is a non-negative periodic function which is L2 orthogonal to sin t and cos t. 1 When the particle collides with the wall, an elastic reflection occurs, so that the veloc ity of the particle relative to the wall instantaneously changes sign. This is illustrated in Figure 5, borrowed from [4]. This mechanical system will be proved isomorphic to the dual billiard system about a closed convex curve y(t), parametrized by the angle made by its tangent line with the horizontal direction, whose curvature radius is r(t) . Choose an origin 0 inside 'Y and let p (t) be the support function, that is, the distance from 0 to the tangent line at y(t). El ementary differential geometry tells us that p" (t) + p(t) = r(t), see, e.g., [26].
1The reader will easily show that this condition is necessary, because for any p one has
==
J�n (p"(t)
+
p(t)) sin t dt
=
J�n (p"(t)
+
p(t)) cos t dt
=
0.
© 2005 Springer Science+Business Media, Inc., Volume 27, Number 4, 2005
19
A'
B'
Figure 7. Area-preserving property of the dual billiard map.
p(t) Figure 5. Impact oscillator.
x(t,)
Let us compute the areas of the two quadrilaterals mod ulo t? and fl-. One has Area AYA ' &r2!2; Area CYC' = S(r - 8)2 /2 &r2/2 - Ser, =
Let x be a point outside of y, and let the whole plane ro tate clockwise with constant angular speed about the ori gin. Consider the projections of x and y on a horizontal line. The projection of the point x is a harmonic oscillator on the line, and the right end point of the projection of y is "the wall" p(t). We say the oscillator and the wall collide when the tangent line from x to y is vertical; the elastic re flection occurs in the projection because the point x gets replaced by its reflection y in the tangency point-see Fig ure 6. One computes that the horizontal component of y's velocity and that of x satisfy the relation prescribed for the impact oscillator. The Area-Preserving Property
Irrespective of the shape of the dual billiard table, the dual billiard map enjoys the fundamental area-preserving prop erty. Let us explain why. Choose infinitesimally close points X and X' on the dual billiard curve. For a positive number r, consider the tan gent segments to y of length r. The end points of these seg ments trace the curves AA' and BB', see Figure 7. The dual billiard map T takes AA' to BB'. Now repeat the construc tion replacing by 8 where 8 is an infinitesimal.We ob tain two infinitesimal quadrilaterals AA ' C'C and BB'D'D and the map T takes one to another. Let 8 be another in� finitesimal, the angle between AB and A'B'. r
r -
=
and hence Area AA'C'C Ser. Likewise, Area BB'D'D = and the area-preserving property follows. This property has numerous consequences. The phase space of the dual billiard map T is the exterior of the oval P: topologically, a cylinder.This cylinder is foliated by the positive tangent half-lines to y. The map T is an area-pre serving twist map; the latter means that the differential dT rotates the tangent vectors to the leaves in the positive sense, see Figure 8. The theory of area-preserving twist maps is well devel oped, see, e.g., [18]. One of the consequences of this the ory concerns periodic orbits of the dual billiard map. Such an orbit is an n-gon, circumscribed about y, whose sides are bisected by the tangency points, see Figure 9. A peri odic trajectory has a topological characteristic called the rotation number, the number of turns made by the respec tive circumscribed polygon around the dual billiard table. A dual billiard version of the celebrated Birkhoff theorem asserts that for every n 2:: 3 and every integer rotation num ber 1 r n/2 there exist at least two distinct n-periodic orbits with the rotation number In fact, periodic trajectories of the dual billiard map cor respond to circumscribed polygons of extremal area.This is illustrated in Figure 10: if the side AB is not bisected by the tangency point then an infinitesimal rotation of the seg ment to the new position A 'B' changes the area in the lin ear approximation (this is essentially the same argument as in Figure 7). One of the n-periodic orbits with a fixed =
Ser,
:;::;
:;::;
r.
X
Figure 6. Dual billiards a s an impact oscillator.
20
THE MATHEMATICAL INTELLIGENCER
Figure 8. Twist condition for the dual billiard map.
r
r
Figure 9. A 5-periodic orbit of the dual billiard map with the rotation
Figure 1 1. Area and string constructions.
number 2.
rotation number corresponds to the circumscribed n-gon of minimal area; the second one is of mini-max type. Suppose that the dual billiard map has an invariant curve, say, r. Can one recover the dual billiard curve y from f? The following construction does the job. Consider the 1-parameter family of lines that cut off a segment of fixed area c from r, and let y be the envelope of this family. This envelope may have singularities-generically, semi-cubical cusps (see, e.g., [9] for a study of these singularities); as sume however that y is smooth. Then the dual billiard map about 'Y preserves the curve r; a proof of this fact is, es sentially, in Figure 7. Note that this area construction de pends on the area c: there is a 1-parameter family of dual billiards with a given invariant curve.2 Note also that the area construction resembles a more clas sical string construction for inner billiards: to recover a billiard table y from an invariant curve r of the billiard map one wraps a closed non-stretchable string around the curve and moves it around as shown in Figure 1 1 on the right (see, e.g., [28]). Ap plied to an ellipse, the string construction produces a confo cal ellipse; this fact is lmown as the Graves theorem [2].
Figure 10. Periodic orbits correspond to area extrema.
The invariant curve r does not have to be smooth. For example, one can start with a square; then the dual billiard curve will consist of four arcs of hyperbolas. D. Genin [ 10] discovered recently that if the area parameter c is small enough then this dual billiard system exhibits a hyperbolic behavior inside this invariant square r (and outside of four 4-periodic regular octagons). For inner billiards, numerous examples of convex domains are lmown that enjoy hyper bolic dynamics, starting with the celebrated Bunimovich stadium, see [28] for a survey. Duality between Inner and Outer Billiards
The reader has noticed a duality of sorts between the in ner and outer billiards. For example, a periodic billiard tra jectory is a polygon of extremal perimeter length, inscribed in the billiard curve (see, e.g., [28]), while a periodic dual billiard trajectory is a circumscribed polygon of extremal area. Another manifestation of this duality is shown in Fig ure 1 1 . How does one explain this length-area duality?3 The situation becomes more clear if one replaces the plane by the unit sphere. On the sphere, one has duality be tween points and oriented lines (i.e., great circles): to a pole there corresponds its oriented equator, see Figure 12. Note that the spherical distance AB equals the angle between the lines a and b. Duality preserves the incidence relation: if a point A lies on a line b then the dual point B lies on the dual line a. Du ality extends to smooth curves: a curve y determines a 1parameter family of tangent lines, and each line determines the dual point. The resulting 1-parameter family of points is the dual curve y*. If one applies this construction to y*, then one obtains the curve that is antipodal to y. Consider an instance of the billiard reflection in a curve y, see Figure 13. The law of billiard reflection says: the an gle of incidence equals the angle of reflection. In terms of the dual picture, this means that AL = LB, and hence the dual billiard reflection about the dual curve y* takes A to B. Thus the inner and outer billiards are conjugated by the spherical duality. We can also explain the length-area duality. Consider a polygon of extremal perimeter inscribed in a curve y. The dual polygon is circumscribed about the dual curve y* and
2This construction is also known in flotation theory, where a segment of constant area represents the submerged part of a floating body·' the constant c is the density·
See [1 1 l on flotation theory. 3Which justifies the term "dual billiards. "
© 2005 Springer Science+ Bus1ness Media, Inc., Volume 2 7 , Number 4 , 2005
21
D
oo
r
Figure 14. Trajectories of the dual billiard map at infinity.
parabolas intersecting at right angles; it corresponds to a semi-circle y, cf. Example 4 in the first section. To explain these observations, assume that
y(t)
is a pa
y(t). y(t); let v(t)
rametrized smooth curve. Consider the tangent line to There is another tangent line, parallel to that at
Figure 12. Spherical duality.
be the vector that connects the tangency points of the for mer and the latter (see Figure 15).
has an extremal sum of angles. The latter is related to the area of the polygon via the Gauss-Bonnet theorem: the sum of the exterior angles of spherical-polygon equals
21r mi
For points very far away from the dual billiard table, the angle at vertex
B
in Figure 15 is very small, and the tan
gent direction to the trajectory at infmity rc t) is parallel to
nus its area. This explains why area extrema characterize
the vector v(t). Thus we need to solve the differential equa
dual billiard periodic trajectories. One may consider the
tion
plane as a sphere of infinite radius. In this limit, the sum
mothety. In fact, one can solve the equation explicitly:
f'(t)
�
v(t).
If a solution exists, it is unique, up to ho
of angles of a polygon becomes a constant, but the area re
f(t)
tains its role as the generating function of the dual billiard
_
map, whose extrema correspond to periodic orbits. where
X
v' (t) v(t) X v' (t)
denotes the cross-product, that is, the determi
Behavior at Infinity; Rational and
nant of two vectors. Indeed, a straightforward computation
Quasi-rational Polygons
(left to the reader) reveals that r, defined by the above for
A
mula, satisfies
property that is peculiar to dual billiards in the plane is
a simple limiting motion far away from the table, observed in computer experiments.
A
bird's-eye view of a dual bil
liard curve y is almost a point, and the map
T is almost the
reflection in this point. More precisely, after rescaling, the distance between a point
x and
its second iteration
T2(x)
is very small, and the evolution of a point under the sec ond iteration
T2 appears a continuous motion. This motion
follows a piecewise smooth centrally symmetric curve r
v X f'
=
0,
An explicit formula for
as needed.
f(t)
makes it possible to explain
f is v(t), v(t) X f(t) =
the Kepler law: the velocity of the motion along and the rate of change of the sectorial area is
1; of course, the value of the constant does not make much sense since everything is defmed only up to scaling.
The reader is challenged to prove that if y is centrally
symmetric, then the correspondence 'Y
�
r is a duality
that is, applied twice, it yields the original curve y.
and satisfies the second Kepler law: the area swept by the
We see that the dual billiard dynamics at infinity is ap
position vector of a point depends linearly on time (the unit
proximated by a continuous motion along curves homo
of time being one iteration of the map
T2).
Figure 14 fea
tures some dual billiard curves y and the respective tra
thetic to r. This motion has an integral (a conserved quan tity): a homogeneous function whose level-curves are these
jectories "at infinity" r. The last curve r is made of two
Figure 13. Duality between inner and outer billiards.
22
THE MATHEMATICAL INTELLIGENCER
B
Figure 15. Explaining the behavior at infinity.
curves, homothetic to f. Thus the dual billiard map at in finity is a small perturbation of an integrable mapping. As sume that y is sufficiently smooth (C 5 will do) and has everywhere positive curvature. Then one has a KAM (Kol mogorov, Arnold, Moser) theory type theorem that the dual billiard map has invariant curves arbitrarily far from y. This result was described by Moser in [23, 24]; a detailed proof was given by R. Douady [8]. A T-invariant curve serves as a wall that no orbit of the dual billiard map can cross, and hence all its orbits stay bounded. It is unknown whether this remains true for dual billiard curves that are less smooth or whose curvatures have zeros. If the dual billiard curve y is a polygon, then the trajec tory at infinity r is a centrally symmetric 2k-gon, and the vectors ±v 1 , . . . , ±vk are diagonals of y. To every side of r there corresponds "time," the ratio of the length of this side to the magnitude of the respective vector v. One ob tains a collection of "times" (t1 , . . . , tk), defined up to a common factor. The polygon is called quasi-rational if all these numbers are rational multiples of each other. An example of a quasi-rational polygon is a lattice polygon whose vertices have integer coordinates. Another example is a regular polygon: the numbers ti are all equal in this case. A partial answer to Moser's question [24] is given by the following theorem [27, 20, 15]: if the dual billiard table is a quasi-rational polygon, then every orbit of the dual billiard map T is bounded. In this situation one has an analog of invariant curves: these are T-invariant necklaces of poly gons around the dual billiard table connected to each other at their common vertices. In Figure 3, one can see the first such necklace: it consists of 5 regular decagons surround ing the fractal "pentagram" for the left table (so the fractal pentagram is the interior of the necklace), and of 8 regular octagons around the fractal 8-ended star for the right one. A corollary of this theorem is that if y is a lattice poly gon, then all dual billiard orbits are periodic. Indeed, the orbit of a point is discrete and, by the above theorem, bounded. One would expect an easy proof of this property of lattice polygons; we are not aware of one. In conclusion of this section, let us mention that, until very recently, it was not known whether the dual billiard system about a polygon always has a periodic orbit. In sum mer of 2004, a participant of the Penn State REU program, C. Culter, proved that, for every polygonal dual billiard sys tem, periodic orbits exist and, moreover, as far as measure is concerned, periodic points constitute a positive portion of the whole plane [5].4
infinity"); straight lines by the chords of this circle; and the distance between points x and y is given by the formula
d(x,y)
=
ln[a,x,y,b],
where a and b are the intersection points of the line xy with the circle and
a) Cb - x) ( -= - "-'c...--=-:.L [a,x,y,b] = -"y?_(x - a)(b - y) is the cross-ratio. The first steps in the study of dual billiards in the hy perbolic plane are made in [7, 30, 35]. In this case, the dual billiard map T extends to a continuous map t : S1 � S1 of the circle at infinity. This circle map contains much infor mation about the dual billiard map. Let pE R/Z be the Poincare rotation number of t (see, e.g., [ 18] for a defini tion and main properties). The rotation number p depends continuously on the dual billiard table. Assume first that the dual billiard curve is sufficiently smooth and strictly convex. As we saw, in the Euclidean plane this would imply that all dual billiard orbits stay bounded. In the hyperbolic plane, the situation is quite dif ferent. Assume that p is rational and the circle map t has a hyperbolic periodic orbit. Then there exists a domain in H2 that escapes to infinity under the dual billiard map-more specifically, is attracted to a hyperbolic periodic orbit at in finity. Moreover, this behavior is stable with respect to small perturbations of the dual billiard table. If the dual billiard curve is a circle in the hyperbolic plane, then the dual billiard map is integrable: its invariant curves are concentric circles, just as in Example 1 at the beginning of this article. What about an elliptical dual bil liard curve y? The dual billiard map is still integrable, and the invariant curves are ellipses from the pencil of conics generated by y and the circle at infmity. 5 One can derive the classical Poncelet porism of projective geometry from this integrability, see [30]. Consider now the case of polygonal dual billiards in the hyperbolic plane. Let y be a convex n-gon. One can prove
Dual Billiards in the Hyperbolic Plane
We have discussed dual billiards in the Euclidean plane and on the sphere. One can equally well consider dual billiards in the hyperbolic plane H2• It is convenient to use the Klein Beltrami (or projective) model of hyperbolic geometry. Then H2 is represented by the interior of the unit circle ("circle at
Figure 16. A large quadrilateral.
4For inner polygonal billiards, even for obtuse triangles, the existence of periodic trajectories is an open problem. A remarkable advance has been recently made by R. Schwartz who proved the existence of periodic billiard trajectories in all triangles with the obtuse angle not greater than 1 ooo.
5A pencil consists of conics passing through four fixed points; in the case at hand, these points are complex.
© 2005 Springer Science +Business Media, Inc., Volume 27, Number 4 , 2005
23
that p :::::: 1/n. An n-gon is called large if p = lin and the cir cle map t has a hyperbolic n-periodic point; see Figure 16 for an example. The set of large polygons is open in the natural topology. As far as the stability properties of the dual billiard or bits, large polygons in the hyperbolic plane are on the op posite end of the spectrum from smooth strictly convex curves in the Euclidean plane: it is proved in [7] that every dual billiard orbit about a large polygon escapes to infinity. The class of large triangles can be described explicitly. Consider a triangle with sides a 1 , a2 , and as and semi perimeter s. This triangle is large if and only if Vsinh s sinh(s - a 1) sinh(s - a2) sinh(s - as) > _!_ . 2 The left-hand side of this formula resembles Heron's for mula for the area of a Euclidean triangle. Example 5. Let the dual billiard table P be a regular n gon with right angles (n :::::: 5). Such polygons tile the hyper bolic plane, see Figure 17. Similarly to Example 2 in Section 1, all orbits of the dual billiard map T are periodic: T cycli cally permutes the tiles that form concentric "necklaces" around P. The rotation number is given by the formula: P) = n - Vn(n - 4) 2n
P(
Multi-dimensional Dual Billiards
Inner billiards are defined in any dimension. Dual billiards can be defined in any even-dimensional space (the plane is even-dimensional, after all). Identify R2n with en and let J be the operator of multiplication by v=l. A dual billiard table is a bounded convex domain with smooth boundary M2n- 1, the dual billiard hypersurface. One would be able to define the dual billiard map if there were a unique tan gent line at every point of M. The problem is, there are too many such tangent lines. This difficulty is resolved as follows. Let N be the outer normal direction to M at point z. Then J(N) is tangent to M at z, and we obtain a well-defined oriented tangent line C(z) at every point z E M. One can prove that through every point x outside of M there pass exactly two such tangent lines to M, one oriented from M the other towards M, just as in the plane. The dual billiard map is defined as follows: find a point z E M so that C(z) passes through x and reflect x in z to obtain a new point y = T(x), cf. Figure 1. As an indication that this is "the right" definition, one has an analog of the area-preserving property. The space en carries a symplectic structure, a non-degenerate skew symmetric bilinear form, given by the formula
·
(In a sense, this formula holds for n 4 as well: a square tiles the Euclidean, not the hyperbolic, plane, and the dual billiard map "at infrnity" is just a central symmetry with the rotation number 1/2.) We do not know whether there exist polygons in the hy perbolic plane for which all orbits of the dual billiard map are bounded but not all orbits are finite. Such polygons would be analogs of quasi-rational, but not lattice, polygons in the Euclidean setup. =
w(u, v) = J(u) v, ·
where u and v are tangent vectors. The above-defrned line e(z) is the symplectic orthogonal complement to the tan gent hyperplane TzM (it is called the characteristic direc tion at point z). For every dual billiard table, the dual billiard map T pre serves the symplectic structure. As in the plane, this has numerous consequences, for example, the existence of pe riodic orbits; see [28, 29, 32, 36]. However, by and large, multi-dimensional dual billiards remain terra incognita. REFERENCES
[I ] R. Adler, B. Kitchens, C. Tresser. Dynamics of nonergodic piece wise affine maps of the torus. Ergod. Theory Oynam. Syst. 21 (2001), 959-99. [2] M. Berger. Geometry. Springer-Verlag, 1 987.
[3] M. Boshernitzan, A Goetz. A dichotomy for a two-parameter piece wise rotation. Ergod. Theory Dynam. Syst. 23 (2003), 759--770. (4] Ph. Boyland. Dual billiards, twist maps and impact oscillators. Non linearity 9 (1 996), 1 41 1 -1 438.
[5] C. Culler. Work in progress. [6] M. Day. Polygons circumscribed about closed convex curves. Trans. AMS 62 (1 947), 3 1 5-31 9.
[7] F. Dogru, S. Tabachnikov. On polygonal dual billiard in the hyper bolic plane. Reg. Chaotic Dynamics 8 (2003), 67-82. [8] R. Douady. These de Troisierne Cycle. Universite de Paris 7, 1 982. [9] D. Fuchs, S. Tabachnikov. Segments of equal area. Quantum 2 (March-April 1 992). [I O] D. Genin. Work in progress. [I I ] E. Gilbert. How things float. Amer. Math. Monthly 98 (1 991 ), Figure 1 7. Tiling of the hyperbolic plane by regular right-angled pen tagons.
24
THE MATHEMATICAL INTELLIGENCER
201 -2 1 6.
[1 2] A Goetz, G. Poggiaspalla. Rotation by 7T/7. Nonlinearity 1 7 (2004), 1 787-1 802.
AUTH O RS
SERGE TABACHNIKOV
FILIZ DOGAU
Department of MathematiCs
Department o! Matnemahcs
Penn State University
Grand Valley State UniVBrSity AJlendale. Ml 49401
UI'1MlfSity Park. PA 1 6802
USA
USA
e-mail·
[email protected]
e-1Tl31L
[email protected]
Serge Tabachn1kov got his degree at Moscow S1ate University In
Rfiz Dogru. after studtes 1n Ankara (Turkey) and Brown Univ9f'Sity
many pubhcahons, 11 IS well to mention espec�alty has book tenta
Un1verstty under the supervision of Serge Tabachn ov. Her re
1 987; since 1990 he has been teaching 1n the USA. Among his
tively titled
Geometry and Billtards, forthCOming from the American
(Rhode Island, USA), did her doctoral work at Pennsylvania State
search as on dynamiCal systBfTIS-In partiCUlar, polygonal dual
MathematiCal Soctety.
billiards in the hyperoohc plane. She feels lucky to be doing some
renowned magazme Kvant on ph}'SICS and mathematics, d1rected
which she realty enjoys. But she also enjoys reading and traveling:
In 1 988- 1 990 he was 1n charge of the Mathematics part of the
to high-school students and teachers. He is active i n special edu cation proJects. one of which was described at length in The ln
telllgencer
24
(2002).
no.
thing for a living -teaching undergraduates and doing research she hopes one day to have visited every country 1n the world.
4. 5o-56.
[1 3) E. Gutkin. Billiard dynamics: a survey with the emphasis on open problems, Reg. Chaotic Dynamics 8 (2003). 1 -1 3. [1 4) E. Gutkin, A. Katok. Caustics for inner and outer billiards. Comm.
[24) J. Moser. Is the solar system stable? Math. Intel/. 1 (1 978), 65-7 1 . [25] B. Neumann. Sharing ham and eggs. Iota, Manchester University, 1 959.
Math. Phys. 1 73 (1 995), 1 01-1 34.
[26) L. Santal6. Integral geometry and geometric probability. Addison
namics. Comm. Math. Phys. 1 43 (1 991 ) , 431 -450.
[27] A. Shaidenko, F. Vivaldi. Global stability of a class of discontinu
[ 1 5] E. Gutkin, N. Simanyi. Dual polygonal billiards and necklace dy [1 6) E. Gutkin, S. Tabachnikov. Complexity of piecewise convex trans formations in two dimensions, with applications to polygonal bil· liards. Preprint ArXiv math.DS/041 2335. [1 7] A.
Katok. Billiard table as a mathematician's playground. Student
colloquium lecture series, v. 2, Moscow MCCME (200 1 ) , 8-36 (In Russian6). [1 8) A. Katok, B. Hasselblatt. Introduction to the modern theory of dy namical systems. Camb. Univ. Press, 1 995.
[1 9) B. Kahng. Dynamics of kaleidoscopic maps. Adv. Math. 1 85 (2004), 1 78-205. [20] R. Kolodziej. The anti billiard outside a polygon. Bull. Pol. Acad. Sci. 37 (1 989), 1 63-168. [21 ) K. Kouptsov, J. Lowenstein, F. Vivaldi. Quadratic rational rotations of the torus and dual lattice maps. Nonlinearity 1 5 (2002), 1 795-1 842.
[22) J. Lowenstein, K. Kouptsov, F. Vivaldi. Recursive tiling and geom etry of piecewise rotations by TT/7 . Nonlinearity 1 7 (2004), 371 -395.
[23) J. Moser. Stable and random motions in dynamical systems. Ann. of Math. Stud. , 77, 1 973.
Wesley, 1 976.
ous dual billiards. Comm. Math. Phys. 1 1 0 (1 987), 625-640. [28] S. Tabachnikov. Billiards. SMF "Panoramas et Syntheses", No 1 , 1 995. [29] S . Tabachnikov. Outer billiards.
Russ. Math. Surv. 48, No 6 (1 993),
8 1 - 1 09. [30] S. Tabachnikov. Poncelet's theorem and dual billiards. L 'En seignement Math. 39 (1 993), 1 89-1 94.
[31 ] S. Tabachnikov. Commuting dual billiards. Geom. Dedicata 53 (1 994), 57-68. [32) S. Tabachnikov. On the dual billiard problem. Adv. in Math. 1 1 5 (1 995), 221 -249. [33] S. Tabachnikov. Asymptotic dynamics of the dual billiard transfor mation. J. Stat. Phys. 83 (1 996), 27-38. [34) S. Tabachnikov. Fagnano orbits of polygonal dual billiards. Geom. Dedicata 77 (1 999), 279-286.
(35] S. Tabachnikov. Dual billiards in the hyperbolic plane. Nonlinearity 1 5 (2002), 1 051-1 072. [36] S. Tabachnikov. On three-periodic trajectories of multi-dimensional dual billiards. A/g. Geom. Topology 3 (2003), 993-1 004.
6English translation available at A. Katok's Web site.
© 2005 Springer Sctence +Bu'"""ss Media, Inc., Volume 27, Number 4. 2005
25
M a t h e m a tic a l l y Bent
C o l i n Ada m s , Editor
Phone Interview Colin Adams The proof is in the pudding.
Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am /?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College,
Williamstown, MA 01 267 USA
e-mail:
[email protected]
26
John: Hello? Dick: Yes, hello, is this John? John: Yes, it is. Dick: Hello, John, this is Dick Der mott calling back I'm here with the hir ing committee on the speaker phone. Let me introduce them. John: Sure. Dick: Seated to my right is Angela Ambertrout. She's a number theorist. Angela: Hello, John. John: Hi. Dick: And to her right is Eric En ders. He is a logician. We like to kid him that for a logician, he is surpris ingly illogical. Eric: Ha, ha, yes, Dick's a kidder all right. Nice talking to you, John. John: Hi. Dick: And fmally, last but certainly not least is Bob Klakity, sitting to my left. He was a geometer, but now, he's a muckety-muck administrator, isn't that right, Bob? Bob: Ha, yes, I'm now the Dean of Arts and Sciences, but I check in with these jokers once in a while, just to make sure they haven't destroyed the department. John: Nice to meet you. Dick: And as you know, John, I'm chair of the department and my spe cialty is algebraic geometry. Of course, we are all familiar with your research, having read through the details of your file. But perhaps you could explain in more detail what you are working on to the committee. We're a diverse bunch, so please speak in monosylla ble words only. John: Ha, ha. Okay, sure. Well, I am interested in dihedral submonomor-
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc.
phoids defmed over bilateral Bernoulli shifts. Although factorization theories for laterally subcutaneous rungs have existed since the early days of fibroid extensions, it is only in the last few years that lifts to hyperextended lower centralized series have allowed a com plete classification of Alexandroff polyhedra in the category of ramified idempotents . . . Eric: Hey, John, this is Eric. Sorry to interrupt, but are there any faces here? John: Well, yes, the Alexandroff polyhedra have faces. Eric: Can you lift them? John: Well, yes, assuming that the fundamental group is locally extend able. Eric: You know what that would be? A face lift. Get it? Oh god, that's good. John: Ummm, yes. That is funny. Dick: Don't mind Eric, John. He's a bit of a joker. Please go on. John: Yes, well, okay, as I was say ing. It is only in the last few years that lifts to hyperextended lower central ized series . . . Bob: Angela, didn't you hyperextend your thumb a couple years back? Angela: Yes, Bob, I did. Hurt like a bitch. Had to wear a splint for a month. Bob: Hey, John, maybe you should consider a splint for your hyperex tended thingamajig. Is there a mathe matical object called a splint? John: Ummm. Not that I know of. Bob: Well, we could make one up. Maybe we could do some joint work on that, publish a paper. John: Ummm. Well, maybe . . . Eric: Bob wrote three papers with our last junior guy, what's-his-name, the one who didn't get tenure. Dick: The skinny guy who always looked depressed? I can't remember. Started with an s, I think Bob: It was Shoemaker, or Shoe string or something about shoes. All I cared about was that it came after Klakity in the alphabet, hah hah.
Angela: Hey, John. I think we've heard enough about your research. Let's talk about your teaching. John: Oh, okay. Angela: Would you like to teach multivariable calculus? John: Sure. I really enjoy that mate rial. Angela: Oops. Wrong answer, John. Eric has a lock on that course. Nobody teaches it but him. Dick: Now Angela, we talked about this. Nobody has a lock on any course. Eric: Of course, I would be happy to let someone else teach the course. The minute you let me teach probability, Dick Dick: That's my course. I created it. Eric: But I thought no one had a lock on a course. Dick: It's different if there is only one person competent to teach it. Angela: Just because you go to Las Vegas doesn't make you a probabilist. Bob: Okay, gang, we don't want to give our candidate the wrong impres sion of our happy family. John, this is Bob again, the muckety-muck admin istrator. I'm wondering what kind of supplies you need for your research. John: You mean computer facilities? Bob: God, no. I mean pens and pen cils, paper clips, pads and such. John: Oh, ummmm, I need some of those. Bob: Which ones? John: All of them, I think Bob: Oh. Well, of course, if you ab solutely need all of them, we could put in a request with the Start-Up Com mittee. But you might want to think about bringing whatever you need from your current institution. They won't miss it. We have a bit of a bud get crunch here. Angela: Bit of a budget crunch? Our department is housed in the basement of the Heating Plant. We haven't had a raise since South America shared a tec tonic plate with Africa. The average class size has just past the centennial mark Yes, we do have a bit of a bud get crunch. Dick: Angela, we are on the phone here. John, let's get back to talking about your teaching. According to your file, you have been quite successful in the classroom. Tell me this. What
would you do if a student put you in a half-nelson? John: What? Dick: A half-nelson, you know where he has you from behind, with his arm hooked around your right arm and then back up behind your neck John: I, ummm, I have never thought about it. I hope I never find myself in that position. Dick: Well, sure. We all hope that. But what would you do? John: I would scream for security? Eric: Oops, another wrong answer. Security doesn't have time to come running every time a faculty member finds himself in a half-nelson. You sim ply twist to the right, hooking your right leg through his legs. Then grab him at his belt with your left hand, pull hard and voila, he's flat on his back with you on top. John: Oh. I see . . . Dick: John, exactly how much teaching experience do you have? John: I taught recitation sections throughout my graduate career. Calcu lus mostly. Then as a post-doc, I taught my own classes for two years. Bob: Of course, the students you get there are quite different from our stu dents. John: In what sense? Bob: They look different. They wear different clothes. They are sometimes shorter and sometimes taller than our students. John: Umm, yes, but your students are sometimes taller or shorter than the students here. Bob: Yes, now you seem to be get ting it. John: No, I don't think I am. How do the individual differences in height im pact teaching? Bob: All I am saying, young man, is that you have not taught our students. You have taught some other students. And the techniques that work on those other students may not work on our students. John: Okay, you mean like your technique of how to get out of a half nelson might not work as well here, since the student might be of a differ ent height. Bob: Are you patronizing me? Dick: Urn, John, I'm going to change
the subject a bit. As you probably know, we are not allowed to ask you about your marital status. John: Yes, I am aware of that. Dick: Yes, so the only way we can find out about it is if you just tell us about it, without us asking. John: Yes. . . . Dick: Right, so if you want us to be able to tell you about opportunities for a spouse, or a partner, you would need to fill us in on that spouse or partner, with out us asking you for the information. John: I see. . . . (Pause) Angela: I don't think you do, John. Let's try something else. I give you an answer and you give me the corre sponding question. John: What? Angela: Don't start yet. Wait until I give you the answer. John: Ummm . . . Angela: The answer is "I am mar ried." John: Ummm, the question is, "What is your marital status?" Dick: That's right. Of course, we never asked it. You asked it. John: Look, I'm not married, not that it is any of your business. Dick: Oh, that is interesting unso licited information. Eric: Hey, I liked that game. Let's play more. Angela: Okay. John, here is your next answer. Thirty-two students and a duck John: Excuse me? Angela: I said thirty-two students and a duck. John: And I'm supposed to come up with a question that has that as an an swer? Angela: Right. Go ahead. You have 15 seconds. John: This is crazy. I don't know. What does this have to do with the job? Angela: Time's up. The question was, "What is the enrollment in fluid mechanics?" Bob: Good god, that's funny, Angela. John: I don't know what to say. Dick: No need to say anything. But answer me this. If you were a Muppet, which Muppet would you be? John: Are you kidding? Bob: It's not a hard question.
© 2005 Spnnger Science+ Bus1ness Media, Inc., Volume 27, Number 4, 2005
27
John: I don't see how this is rele vant. Eric: Just say Kermit. Then we'll be lieve you have leadership potential. Angela: Don't give away the an swers, Eric. John: Doesn't anybody want to talk about my qualifications for the job? Dick: No need, John. I think we have a good sense of what you have to of fer. Do you have any questions for us? John: Well, yes, I have one. Are you actually loony, or do you just put on an amazingly accurate portrayal? Dick: Hmmm, Bob, do you want to field that one? Bob: My experience is that it's no portrayal. It's real as real can be. Eric: Ha ha. That's good.
John: I think I am going to hang up now. Dick: Wait. Don't hang up yet. I want to make you the job offer. John: What? Dick: Once in a while, a candidate comes along who impresses us so much, we don't need time to debate. In fact, during this entire conversation, the four of us have been signaling each other with a variety of nonverbal cues, cues that say, "Hire this guy." Eric: That's right. Angela: We would love to have you here. Bob: I have seen some vigorous ges ticulations in support of candidates over the years, but none as vigorous as what I have seen here today in support of you.
Sc entificWo r kPiace· ·
Mathem atical Word Processing
•
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John: Are you kidding? Dick: Hardly. And if you come, we'll teach you the secret hand signals. Eric did a shadow thing with his hands where he reenacted the entire scene of you struggling to escape a half-nelson. It was truly hysterical. Eric: Oh, no. It wasn't any better than most of the other hand signals be ing flashed around here. Dick: I will send along an official letter, and then we can get to the ne gotiations about the paper clips. Nice talking to you, John. You take care, now. Angela: Bye, John. Eric: So long. Bob: We'll be in touch. (Click)
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28
THE MATHEMATICAL INTELLIGENCER
lt%Mj.i§j.@iil£11§%§4flhi l§.id ..
Meekness in Ornation: How the Weirdoes Collude
M ichael Kleber and Ravi Vaki l , Editors
T
he answers to the four italicized clues were the eight-letter words THEOREMS, CALLIOPE, SCHOOLED, and couNTESS. Each is the result of a per fect riffle shuffle of two four-letter words-THE oREM s, for example. The constituent four-letter words appeared as the unclued entries, artfully arranged so that each's uncrossed let ter was ambiguous.
The italicized words in the instruc tions, WEIRDOES and COLLUDES, share this property. Thanks to David Miller and Thomas Colthurst for suggestions. The Entertainments editors welcome submissions of crosswords or other puzzles with similar appeal. They should specifically target the mathematically in clined audience of this publication, but otherwise should be broadly accessible.
Michael Kleber
Solution to the Weirdoes Puzzle published in val. 2 7 no. 3
This column is a place for those bits of
contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
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Department of Mathematics, Bldg. 380, Stanford, CA 94305-21 25 , USA
e-mail:
[email protected]
© 2005 Springer Science+Business Media, Inc., Volume 27, Number 4, 2005
29
Brouwer's grave and the glass plate with inscriptions.
30
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+Bus1ness Media, Inc,
liiJ$•1,ijj,J§i.£hi.JihtJI i%11
A Blaricum Topology for Brouwer Dirk van Dalen
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made,
D i r k H uylebrouck, Editor
ne hundred years ago, L. E. J. Brouwer settled in one of the most attractive Dutch villages, Blaricum. The village had a reputation for undis turbed landscapes, for artists, and for experiments in social communes. One of these communes, the Christian An archists, was led by the charismatic Professor Van Rees. When this com mune fell apart, Brouwer bought part of the land and asked his friend Rudolph Mauve (son of the famous painter Anton Mauve) to design a small cottage for him. The cottage, called "the hut," was ready in 1904. In that year Brouwer and his bride Lize moved in; they remained faithful to the hut and Blaricum for the rest of their lives. The property contained some rem-
O
I
nants of the old commune (e.g., a ro tating "tuberculosis" hut). In the course of time Brouwer added some small buildings (e.g., the Padox). In the 1920s he bought a neighboring villa, De Pim pernel at the Torenlaan. In 1925-26 the Hut and De Pimpernel were the center of the Dutch topological school, with Alexandrov, Menger, Newman, Vi etoris, and even Emmy N oether as short-term visitors. The village, and the whole area, called Het Gooi, were for a long time the home of a rich variety of artists (e.g., Piet Mondriaan); it also attracted the attention of well-known Dutch ar chitects. Even today, the village offers a panorama of original (small) farm houses, interesting eccentric houses,
the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels?
Huizerweg 526)
If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor,
Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium
Map of the center of Blaricum, showing the cemetery (above) and
e-mail:
[email protected]
the Torenlaan (below), where Brouwer lived.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
31
Red Brouwer (t)
Save Brouwer ft ) Translat ion of De Volkskrant arti ·le
90
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Een doodgewoon graf is het, op de Gemeentelijke Begraalplaats aan de Woensbergweg in het noorden van Blaricwn. L. E. J. Brouwer staat er op de sobere steen, 18811966, emaast het graf van de in 1959 overleden mevrouw Brou wer-de Holl. Oat hier volgens de kenners de grootste Nederlandse wiskundige sinds Christiaan Huy gens rust, blijkt nergens wt Tach, zegt Brouwers biograaf prof. dr. Dirk van Dalen, filosoof te Utrecht, is dit graf een beschei den bedevaartsoord vooi ingewij den, buitenlandse wiskundigen
vooral, die weten hoe de piepjon
ge Nederlander begin twintigste eeuw een born onder de gevestig de wiskunde legde. Sinds vorig jaar op Torenlaan 70 zonder par don Brouwers oude houten woon huis ('de hut') is gesloopt - om plaats te maken voor een forse nieuwbouwvilla - is het graf het enige spoor dat nog van de grote wiskundige re.'t. En dat, meldt de hevig veront ruste Van Dalen, kan binnenkort ook wei eens verleden tijd zijn. Bij navraag ontdekte ,hij onlangs dat beide graven van de Brouwers aan de gemeente Blaricwn zijn verval len omdat de grafrechten al lang niet meer waren voldaan. 'De gra ven kunnen zo geruimd worden. Oat zou na het verlies van de hut verschikkelijk zijn.' Van Dalen heeft de zaak-Brou wer met de burgemeester per soonlijk opgenomen. Die bleek oud-wiskundeleraar en had dus van nature enige voeling met de uak. Van Dalen: 'lruniddels heeft hij me gezegd de zaak wei te wit
len proberen te redden.' Uiteindelijk is het echter de ge-
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THE MATHEMATICAL INTELLIGENCER
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cultuurhistoriscbe monument voor Nederland te behouden'. Maar hoe dat precies zou moeten, ook linancieel, weet de Utrechtse filosoof nog niet. 'Het gaat er nu om een eventuele ruiming te blok keren', zegt hij. Hoofd Waalman van de Buite!l' dienst van de Gemeente Blari cwn, tevens verantwoordelijk voor de Gemeentelijke Begraaf plaats, haalt met hoorbare tegen zin de betreffende kaart uit de kaartenbak. Brouwer, L. E. J., dat is inderdaad perceel J 32. Het graf is al in 1994 vervallen aan de ge meente, leest hij voor. Er kan dus inde.rdaad geruimd worden. Maar een ruimingsbesluit is er niet en de dienst Buitendienst heeft momen tcel ook geen plannen in die rich1 ting. En dan nog zou de gemeente raad die eerst moeten goedkeu ren. Vandaar, waarschijnlijk, dat hij als verantwoordelijke de burge meester nog niet over de graf kwestie heeft gehoord. 'Oat zal dan nog we! komen.'
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meentetaad die beslist. En dus heeft Van Dalen deze week een brief naar de Gooise gemeente op de post gedaan waarin hlj nog eens bet cultureel-wetenschappe lijke belang van de in 1966 overle den Blaricummer uiteenzet. 'Bia ricwn', besluit hlj, 'kan terecht trots zijn een geleerde van het for maat van een Newton of Gauss onder haar bewoners te hebben geteld. Het is ondenkbaar dat Cambridge of Gottingen de gra ven van Newton of Gauss zou op geven.' Van Dalen vraagt in de brief de
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A cottage designed in 1904 by Brouwer's friend Rudolph Mauve. It was always referred to as the hut.
Brouwer at work in his Blaricum place (Brouwer Archive).
© 2005 Springer Science+ Business Media, Inc. . Volume 27, Number 4 . 2005
33
"Pimpernel," the villa in Blaricum, adjacent to the hut.
A small structure on a rotating base that could follow the sun. These little houses were often used by tuberculosis patients, hence the name "TBC hut."
34
THE MATHEMATICAL INTELL IGENCER
"The Padox," a prefab house, used for guests.
and opulent villas in the pre-war style. In 2000, Brouwer's hut and the other small buildings fell victim to property developers; fortunately De Pimpernel escaped the demolition crews. The fate of the hut raised fears that Brouwer's grave, for which the lease had run out, could also be cleared out. The national press voiced its concern (see inset), and the town of Blaricum acted with a great sense of responsi bility; it decided to preserve the graves of Lize and Bertus Brouwer and to care for the graves. The Dutch mathematical community (repre sented by the Royal Dutch Mathemat-
ical Society) and the University of Amsterdam acted fittingly by placing a modest memorial-a glass plate etched with the text "Luitzen Egber tus Brouwer, Mathematician-Philoso pher. Father of the New Topology. Founder of Intuitionism," followed by a text in Brouwer's handwriting etched into the glass plate. A bus from the train station in Hil versum takes the visitor to the center of Blaricum, from which it is a 10minute walk to the cemetery. For hikers, there is a path round the IJsselmeer, the Zuiderzeepad, which passes by the cemetery (see map).
More historical information on Brouwer can be found in my biography Mystic, Geometer and Intuitionist: The Life of L.E.J. Brouwer. Vol l. The Dawning Revolution; vol. 2, Hope and Disillusion, Oxford University Press, 1999 and 2005, resp. The unique pho tographs that accompany this contri bution are by Dokie van Dalen.
Department of Philosophy Utrecht University 3508 TC Utrecht The Netherlands e-mail:
[email protected]
© 2005 Springer Science +Business Media, Inc., Volume 27, Number 4, 2005
35
la§'jl§l.lfj
Osmo Pekonen , Editor
I
Cogwheels of the M ind. The Story of Venn Diagrams by A. W. F. Edwards with a foreword by Ian Stewart THE JOHNS HOPKINS UNIVERSITY PRESS, BALTIMORE, MARYLAND, USA. 2004, xvi+ 1 1 0 pp. $25.00 ISBN:
0-8018-7434-3.
REVIEWED BY PETER HAMBURGER
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book ofyour choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
Column Editor: Osmo Pekonen, Agora Center, University of Jyvaskyla, Jyvaskyla,
40351 Finland
e-mail:
[email protected]
36
his book purports to be on recre ational mathematics. It purports to popularize an area of discrete geome try, namely Venn diagrams. Unfortu nately, the text is confusing and fl.lled with historical and mathematical mis takes; many of the figures are useless or meaningless; many statements are pretentious or marred by the author's seeming vendetta against mathemati cians. I will detail some of these com plaints below and also try to set straight a record the author has misreported. The famous three-circle Venn dia gram, which is known to most people, had already been used by Euler. Venn himself calls this diagram "Euler's fa mous circles." So why do we speak of Venn diagrams and not Euler diagrams? I believe there are two reasons. It was John Venn who first gave a rigorous de finition of the notion (though he did not always follow it consistently); and he was the first to prove that the desired diagrams exist for any number of sets. A modem definition is this. A planar Venn diagram is a set of n closed non-self-intersecting continuous planar curves, intersecting each other in iso lated points, and such that the con nected components of the complement (which are bounded by unions of arcs of these curves) are 2n in number. Then these regions can be assigned distinct binary codes, in the following manner. Label the curves 1, 2, . . . , n. If a region is inside the curve i, then write 1 in the ith place in its binary code, otherwise write 0. As the n-digit binary codes are
T
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc.
exactly 2n in number, the definition of Venn diagram means that they allow all the codes to be assigned to regions. Branko Griinbaum wrote the fol lowing [8]: "Venn diagrams were intro duced by J. Venn in 1880 [ 12] and pop ularized in his book [13]. Venn did consider the question of existence of Venn diagrams for an arbitrary number n of classes, and provided in [12] an in ductive construction of such diagrams. However, in his better known book [13], Venn did not mention the con struction of diagrams with many classes; this was often mistakenly in terpreted as meaning that Venn could not fmd such diagrams, and over the past century many papers were pub lished in which the existence of Venn diagrams for n classes is proved."
H is "cogwheels of the mind" have a mon key wrench in the works .
It is unsettling to find, in this book's introduction by an eminent mathemati cian, the mistake perpetuated: "But what of five, six . . . any number of sets? On this question, the great man was silent" (p. xi). The author, though denying that he means to make priority claims (p. xv), seems to do so when he says (falsely) that the possibility of drawing Venn di agrams on the sphere was ignored "for more than a century until it indepen dently resurfaced and inspired the gen eral solution. It is as though Venn had no geometric insight . . . " (p. 15). Again, in the introduction the author enthusi astically declares that Edwards "has added further conditions to the shapes he seeks-conditions like symmetry. He has become a world expert on Venn di agrams." But no: it was Henderson [9] who posed the problem of symmetric Venn diagrams; he found some with five or seven curves, as did Griinbaum,
Schwenk, and Gointly) Savage and Winkler, all earlier than Edwards. Edwards knows that Griinbaum showed in 1975 [8] the possibility of constructing Venn diagrams with any number of convex curves, and he recognizes that this was a "re markable advance," but he gratuitously disparages others' figures in comparison to his own, and he misrepresents the history. This development began with another remarkable advance by A. Renyi, V. Renyi, and J. Suranyi in 1951 [10]. Edwards must have been aware of the paper [ 10], the start ing point for all modem study of Venn diagrams, for it is given its due in [8]. Griinbaum's result is stronger than Ed wards quotes: not only may all the n curves be chosen so that they are convex, but also so that the 2n - 1 interior in tersection regions, and also their union, are convex. A Venn diagram is called reducible if there is some one of its curves whose deletion results in a Venn diagram with one less curve. It is called simple if at every intersection at most two curves meet. It is known that there are irreducible Venn diagrams, and Edwards refers to this "counterintuitive property" (p. 23)-diagrams with 5 curves can even be sim ple irreducible-but Edwards says falsely (p. 43) that if a Venn diagram can be built up by adjoining n curves one by one, that determines its topological (graph-theoretic) struc ture uniquely. Some of the reducible structures can be re alized by curves all of which are convex, and some can not; even among those which can, there are many graphically different ones. This richness is one of the attractions of the subject to the geometer. In Chapter 6 Edwards discusses the dual graph, stating, "The dual graph of a Venn diagram is a maximal planar sub graph of a Boolean cube" (p. 77). He says he realized this in 1990 but his paper on the subject was rejected. (Well may it have been! The dual graph is always a planar sub graph, but it need not be a maximal one unless the given Venn diagram was simple.) He cites a 1996 paper [2] but says "the proof, though trivially short, assumes a knowl edge of graph theory and is therefore omitted here" (p. 83). Who can be the intended reader? Someone who would be daunted by a trivial proof in graph theory and yet can cope with maximal planar subgraphs of the hypercube ! One of the most disturbing mistakes in the book is when Edwards presents an induction argument to prove a state ment, namely, every Venn diagram can be colored with two colors such that no regions with common arc boundary have the same color, p. 23. This statement does not need inductive proof, and the proof offered is incorrect. It is one of the types of "proof' that college instructors have diffi culty explaining to their students why it is incorrect; Johns Hopkins University Press by publishing this book under mines their teaching efforts. After publication of the fundamental [ 10], there was a pause before the study of Venn diagrams was revived by Griinbaum [6], [7], [8] and Peter Winkler [ 14]. Their deep understanding and challenging conjectures have motivated more recent work. Let me mention two advances here. In [2] the authors show that it is possible to extend any pla nar Venn diagram to a planar Venn diagram with one more
curve. In [5] the authors show that for every prime number p there is a planar Venn diagram with p curves and p-rota tional symmetry. In both problems, it remains unknown whether the Venn diagrams can be chosen simple. The lat ter of these problems is surveyed by Barry Cipra [3]. Read ers may consult an online, regularly updated survey [ 1 1]. An accurate essay by M. E. Baron [1] gives the history of representations of logic diagrams up to the time of Venn. The author's sniping at mathematicians reflects a pro found ambivalence, shown explicitly in this passage (pp. xx-xvi):
Mathematical discovery is perhaps the most delightful ex perience which Academic life has to offer. The pure math ematician G. H. Hardy (1877-1947) wrote in A Mathe matician's Apology, "It will be obvious by now that I am interested in mathematics only as a creative art, " but Hardy was a mathematician's mathematician and most of us cannot appreciate his work. One of thejoys of work ing with Venn diagrams is that there have been simple delights still to be uncovered that can be appreciated by the far wider audience of amateur mathematicians (amongst whom I count myself, for my Cambridge col lege, Trinity Hall, declined to admit me to read the math ematical tripos, for which I am grateful because it meant I became a scientist instead). Hardy created beautiful mathematics, but working with Venn diagrams has been much more of a voyage of discovery. Though Hardy felt his research to be a creative en deavor, he surely regarded it as discovery! One does not begrudge Edwards his post-retirement hobby of venturing in our domains, and if he takes satisfaction in being a sci entist rather than a mere mathematician, let him. But his lofty position as College Dean and status as scientist (not to be undermined, I hope, by any humorless book reviewer) do not entitle him to publish his dabblings without bring ing them up to the standards of our science. Truly his "cog wheels of the mind" have a monkey wrench in the works! REFERENCES
[1 ] M. E. Baron, "A Note on the Historical Development of Logic Di agrams: Leibniz, Euler and Venn, Mathematical Gazette 53 (1 969), 1 1 3-1 25. [2] K. B. Chilakamarri, P. Hamburger, R. E. Pippert, "Hamilton Cycles
in Planar Graphs and Venn Diagrams," Journal of Combinatorial The ory Series B 67 (1 996), 296-303.
[3] B. Cipra, "Venn Meets Boole in Symmetric Proof, " SIAM News 37,
no. 1 (January/February 2004).
[4] L. Euler, Lettres a une Princesse d'AIIemangne, St. Petersburg, 1 768. English translation: H. Hunter, Letters to a German Princess,
London, (1 795). [5] J. Griggs, C. E. Killian, C. D. Savage, "Venn diagrams and sym
metric chain decompositions in the Boolean lattice," The Electronic Journal of Cornbinatorics (2004) [6] B. Grunbaum, "Venn Diagrams and Independent Families of Sets," Mathematics Magazine 48 (1 975), 1 2-22.
[7] B. Grunbaum, "The construction of Venn diagrams, " College Math ematics Journal 1 5 (1 984), 238-247.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
37
[8] B. Gri.inbaum, "Venn diagrams 1," Geombinatorics 1 (1 992), 5-1 2 . [9] D. W. Henderson, "Venn Diagrams for More Than Four Classes, " American Mathematical Monthly 7 0 (1 963), 424-426.
[1 O] A. Renyi, V. Renyi, and J. Suranyi, "Sur l'lndependance des Do maines Simples dans I'Espace Euclidien a n-dimensions," Collo quium Mathematicum 2 (1 95 1 ) , 1 30-1 35.
[1 1 ] F. Ruskey, M . Weston, The Electronic Journal of Combinatorics, www.combinatorics.org/SurveysNennEJC.html
[1 2] J . Venn, "On the diagrammatic and mechanical representation of propositions and reasonings," The London, Edinburgh, and Dublin Philos. Mag. and J. Sci. 9 (1 880), 1 -1 8.
[1 3] J. Venn, Symbolic Logic, Macmillan, London, 1 881 , second edi tion 1 894. [1 4] P. Winkler, "Venn diagrams: some observations and an open prob lem," Congressus Numerantiurn 45 (1 984), 267-274. Department of Mathematical Sciences Indiana University- Purdue University Fort Wayne, Fort Wayne, IN 46805 USA e-mail:
[email protected]
In the Light of Logic b y Solomon Fejerman OXFORD U NIVERSITY PRESS, 1 998, 352 PP. $ 60.00 US, ISBN 01 95080300
REVIEWED BY ANDREW ARANA
P
oincare famously compared the logician's understand ing of mathematics to the understanding we would have of chess if we were only to know its rules. "To understand the game," Poincare wrote, "is wholly another matter; it is to know why the player moves this piece rather than that other which he could have moved without breaking the rules of the game. It is to perceive the inward reason which makes of this series of moves a sort of organized whole." (P, pp. 217-218] The Dutch mathematician L. E. J. Brouwer took a position similar to Poincare's: genuinely mathematical rea soning is not simply a matter of logical inference. It is, as Poincare put it, a matter of mathematical insight. Despite those views concerning logic, Poincare and Brouwer believed in studying the foundations of mathe matics, and indeed they carried out fundamental work in this area. This might strike contemporary minds as a bit odd, but it is a consistent view. Mathematical logic and the foundations of mathematics are frequently lumped to gether, as though they are the same. They are not. Mathe matical logic is a mature mathematical subdiscipline, with its own problems generated by reflecting on what is known from other logic problems and solution attempts. Like any mature mathematical subdiscipline, what counts as a good problem is largely determined by factors "internal" to the subdiscipline, such as how the problem contributes to other work in progress and to what is already known. Foun dations of mathematics, on the other hand, has a different 38
THE MATHEMATICAL INTELLIGENCER
standard. It raises questions about the objects and struc tures of mathematics: what are they, and how do we know anything about them? It raises questions about mathemat ical statements: how should we go about discovering and justifying them? It raises questions about mathematical proofs: what is a proof, what kinds of proofs do we prefer, and for what reasons? Foundations of mathematics is there fore not a mathematical subdiscipline at all, but rather a body of reflections on mathematics itself. A striking insight reached by David Hilbert and others in the early twentieth century was that the foundations of mathematics could be studied by the application of math ematical logic. By taking mathematical objects and struc tures to be described by axioms in formal languages, these axioms and their consequences could be studied using mathematical logic. In this way, contra Poincare and Brouwer, logic could be used to shed light on the founda tions of mathematics, the light of logic to which the title of Feferman's excellent book refers. Ofthose who have shed light on the foundations of math ematics using logic, there is one figure whose influence and views tower over the rest: Kurt Godel. His incompleteness theorems both answered existing questions and raised many new ones, thereby deepening considerably the study of the foundations of mathematics. On account of that, his specter haunts almost every page of Feferman's book Feferman classifies the essays (all previously published) of the book into five parts based on their topics, and for each topic, Godel's work and views are of utmost importance. In Part I, Feferman raises as a problem the role of transfinite set theory in mathematics. Because transfinite sets are sup posed to be infinite objects about which facts are true inde pendent of our abilities to verify them, it seems that these ab stract entities must exist independent of human thoughts or constructions. This family of beliefs about sets is frequently called platonism. Feferman finds platonism philosophically unsatisfying, and thus he presents the three projects aimed at avoiding platonism: L. E. J. Brouwer's "intuitionism," David Hilbert's "finitism," and Hermann Weyl's "predicativism." Feferman characterizes Brouwer's solution as excessively radical, leaving Hilbert's and Weyl's as acceptable options. Feferman believes that Godel's incompleteness theorems cast doubt on the viability of Hilbert's project, as is commonly (but not universally) thought. This leaves Weyl's predica tivism as Feferman's preferred alternative to platonism. I will return to predicativism shortly. The discussion in Part I sets the agenda for the rest of the book Finding an acceptable alternative to platonism emerges as one central theme. Another is the question of whether there is any justification for new axioms for set theory. These two themes are tied together by GOdel's view that platonism could be used to justify new axioms for set theory. These new axioms assert the existence of sets which Godel thought the platonist had every reason to be lieve in, on account of their uniformity with sets already believed to exist, and on account of a sense-perception-like faculty he thought we possess for experiencing mathemat ical objects. In addition, he supported new axioms for set
theory because he thought they would eventually be used
sis, are just sets; but this requires that we justify our use
to solve open mathematical problems, just as they can be
of sets. Feferman is critical of platonist attempts to justify
used to prove the arithmetically unprovable sentences that
set theory, and offers instead a predicativist view.
he had studied in his work on the incompleteness theo
I think there are four main reasons why Feferman thinks
rems. We may justly view Feferman's book as a wrestling
that predicatively definable sets are justifiable, as follows.
match with Godel, the arch-platonist. It is unsurprising,
(1) Consider Grelling's paradox. Suppose we define a word
therefore, that Feferman dedicates one of the book's five
as being heterological if it does not describe itself. The word
parts-Part III-to essays on Godel's life and work
"heterological" is heterological if and only if it is not het
Though these central themes are explored in every part
erological. This definition is infelicitous, since it does not
V to his preferred
determine whether "heterological" is heterological. Pre
alternative to platonism, predicativism. Here Feferman
dicative definitions avoid these vicious circles, as follows.
argues first against attempts to show that transfinite set the
We typically define sets as consisting of all objects satis
of the book, Feferman returns in Part
ory is
necessary
for ordinary finite mathematics. Respond
fying some condition. In predicative definitions, the satis
ing to arguments of Godel and Harvey Friedman, Feferman
faction of this condition for all objects is determined inde
concludes that "the case remains to be established that any
pendently of the set being defined. Hence, there are no
the mathematics of the finite in the everyday sense of the
finable sets entails commitment to whatever is needed for
word" (p. 243). Instead, he supports a much more restricted
Peano Arithmetic, presumably just countably infinite sets.
use of the Cantorian transfinite beyond � 0 is necessary for
vicious circles.
(2) Our commitment to predicatively de
view on the transfinite, maintaining that only predicatively
(3) Predicatively definable sets suffice for doing all scien
definable sets should be
admitted. A set is predicatively de
finable if it is defined by way of the system of natural num
tifically applicable mathematics, so working with just them is adequate for the applicability of mathematics. (4) Pre
bers, or by way of predicatively definable sets that have al
dicatively definable sets suffice for doing all ordinary finite
ready been defined. Sets defined by way of a collection of
mathematics, perhaps the minimum part of mathematics
sets that includes the set to be defined are thereby excluded,
for which any reasonable foundation must account.
such as the "set" of all sets that do not contain themselves,
I will comment briefly on these four reasons.
as used in Russell's paradox. Feferman explains how he used methods from modem logic to develop Weyl's pre
1. The avoidance of vicious circle paradoxes does not en
dicative set theory, yielding a system in which, he argues,
sure the consistency of predicative mathematics. Pre
This system is up to such a task, he argues, because analy
dicative mathematics, but that does not mean that it is
sis, both classical and modem, can be formalized within it.
perfectly secure. Indeed, as Feferman showed in
Yet any (first-order) truth that can be proved in this system
the consistency of predicative analysis cannot be proved
can be proved from the (first-order) Peano Arithmetic ax
predicatively, though it can be proved impredicatively
all "scientifically applicable mathematics" can be proved.
dicative mathematics may be more secure than impre
1964,
1-30] . Furthermore, this characterization of the
ioms, which formalize elementary number theory. Feferman
[F, pp.
argues that this vindicates his view that the predicativist
value of predicativity leaves it open whether predicative
need not admit any transfinite sets beyond the countably in
definitions have any other value. One reason to be wor
finite, because, he maintains, commitment to Peano Arith
ried about this is that there are many sets that can be
metic entails commitment only to the countably infinite.
defined predicatively, but whose impredicative defini
In Parts II and IV of the book, Feferman discusses how
tions mathematicians find more natural. For instance,
logic can be used to shed light on aspects of mathematical
the closure of a set in a topological space is naturally
practice besides that part already formalized within set the
defined as the intersection of all closed sets containing
ory. He critically examines Imre Lakatos's views on math
the set, but this is impredicative. Mathematicians typi
ematical discovery, comparing them with George P6lya's
cally find this definition unproblematic because the ex
views on discovery. He explains how logic can help clarify
istence of the sets involved follows from set-theoretic
vague mathematical concepts such as
construction, infin
axioms such as ZFC. Predicativists reject existence-in
itesimal,
In particular, Fefer
ZFC as sufficient for set existence, demanding instead a
man uses his expertise in proof theory, a branch of math
description (in some weaker axiomatic system, perhaps)
and
natural well-ordering.
ematical logic, to emphasize its utility for understanding
of how a set may be generated from other sets already
As he explains, proof theory can be used to
known to exist. Consider also the following example:
clarify what parts of mathematics can be reduced to other
given a homeomorphism of a compact space, there is al
mathematics.
parts, and in what ways. Feferman's moral is that logic is
ways a "minimal" non-empty closed invariant subset.
useful for more than just the systematic organization of pre
The standard proof uses Zorn's lemma and intersections,
existing, well-understood bodies of mathematics-though
and is thus impredicative. There is a predicative proof,
it is useful for that too.
but it is more involved than the standard proof [BHS, p.
Part of accounting for mathematical practice is saying
152]. (Thanks to Jeremy Avigad for pointing out this ex
how we are justified in admitting the objects we seem to
ample to me.) Predicativity thus exacts a toll, in that it
need to do mathematics in specific areas like analysis. Fre
costs us natural definitions and proofs-leaving what is
quently this is done by saying that the objects of, e.g., analy-
natural unspecified but, I take it, uncontroversial in
© 2005 Springer Sc1ence+Bus1ness Media, Inc., Volume 27. Number 4, 2005
39
these cases. We must weigh the apparent security pur chased by requiring predicative definitions against the burden of having to abandon in many cases what we, as mathematicians, consider natural definitions. 2. It is unclear exactly what objects we are committed to when we are committed to Peano Arithmetic. There are plenty of problems in number theory whose proofs use analytic means, for instance. Does commitment to Peano Arithmetic entail commitment to whatever objects are needed for these proofs? More generally, does commit ment to a mathematical theory mean commitment to any objects needed for solving problems of that theory? If so, then Godel's incompleteness theorems suggest that it is open what objects commitment to Peano Arithmetic entails. 3. As Feferman admits, it is unclear how to account pre dicatively for some mathematics used in currently ac cepted scientific practice, for instance, in quantum me chanics. In addition, I think that Feferman would not want to make the stronger claim that all future scien tifically applicable mathematics will be accountable for by predicative means. However, the claim that currently scientifically applicable mathematics can be accounted for predicatively seems too time-bound to play an im portant role in a foundation of mathematics. Though it is impossible to predict all future scientific advances, it is reasonable to aim at a foundation of mathematics that has the potential to support these advances. Whether or not predicativity is such a foundation should be studied critically. 4. Whether the use of impredicative sets, and the un countable more generally, is needed for ordinary finite mathematics, depends on whether by "ordinary" we mean "current." If so, then this is subject to the same worry I raised for (3). It also depends on where we draw the line on what counts as finite mathematics. If, for in stance, Goldbach's conjecture counts as finite mathe matics, then we have a statement of finite mathematics for which it is completely open whether it can be proved predicatively or not. In emphasizing the degree to which concerns about predicativism shape this book, I should not overempha size it. There is much besides predicativism in this book, as I have tried to indicate. In fact, Feferman advises that we not read his predicativism too strongly. In the pref ace, he describes his interest in predicativity as con cerned with seeing how far in mathematics we can get without resorting to the higher infinite, whose justifica tion he thinks can only be platonic. It may tum out that uncountable sets are needed for doing valuable mathe matics, such as solving currently unsolved problems. In that case, Feferman writes, we "should look to see where it is necessary to use them and what we can say about what it is we know when we do use them" (p. ix). Nevertheless, Feferman's committed anti-platonism is a crucial influence on the book. For mathematics right now, Feferman thinks, "a little bit goes a long way," as one of the essay titles puts it. The full universe of sets
40
THE MATHEMATICAL INTELLIGENCER
admitted by the platonist is unnecessary, he thinks, for doing the mathematics for which we must currently ac count. Time will tell if future developments will support that view, or whether, like Brouwer's view, it will re quire the alteration or outright rejection of too much mathematics to be viable. Feferman's book shows that, far from being over, work on the foundations of mathe matics is vibrant and continuing, perched deliciously but precariously between mathematics and philosophy. REFERENCES
[BHS] A. Blass, J. Hirst, and S. Simpson, "Logical analysis of some theorems of combinatorics and topological dynamics," in Logic and Combinatorics (ed. S. Simpson), AMS Contemporary Mathematics
val. 65, 1 987, pp. 1 25-1 56. [F] S. Feferman, "Systems of Predicative Analysis," Journal of Symbolic Logic 29, no. 1 (1 964), 1 -30.
(P] H. Poincare, The Value of Science (1 905), in The Foundations of Science, ed. and trans. G. Halsted, The Science Press, 1 946.
Department of Philosophy Kansas State University Manhattan, KS 66506 USA e-mail:
[email protected]
The SIAM 1 00-Digit Challenge: A Study in High-Accuracy Numerical Computing by Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jorg Waldvogel SIAM. PHILADELPHIA, PA, USA 2004, Xll+306 PP. SOFTCOVER, ISBN 0-8987 1 -561 -X, US$57.00
REVIEWED BY JONATHAN M. BORWEIN
L
ists, challenges, and competitions have a long and pri marily lustrous history in mathematics. This is the story of a recent highly successful challenge. The book under re view makes it clear that with the continued advance of com puting power and accessibility, the view that "real mathe maticians don't compute" has little traction, especially for a newer generation of mathematicians who may readily take advantage of the maturation of computational pack ages such as Maple, Mathematica, and MATLAB. Numerical Analysis Then and Now
George Phillips has accurately called Archimedes the first nu merical analyst [2, pp. 165-169]. In the process of obtaining his famous estimate 3 + 10/71 < TT < 3 + 1n, he had to mas ter notions of recursion without computers, interval analy sis without zero or positional arithmetic, and trigonometry without any of our modem analytic scaffolding. . . . Two millennia later, the same estimate can be obtained by a computer algebra system [3].
Example 1. A modem computer algebra system can tell one that
0<
(1.1)
11
(1 - x)4x4 1 + x2
o
_
dx -
22 7
_
7f,
since the integral may be interpreted as the area under a positive curve. This leaves us no wiser as to why! If, however, we ask the same system to compute the indefinite integral, we are likely to be told that
a calculator in the last pre-computer calculations of 7f though until around 1950 a "computer" was still a person and ENlAC was an "Electronic Numerical Integrator and Calculator" [2, pp. 277-281 ] on which Metropolis and Reit wiesner computed Pi to 2037 places in 1948 and confirmed that there were the expected number of sevens. Reitwiesner, then working at the Ballistics Research Laboratory, Aberdeen Proving Ground in Maryland, starts his article [2, pp. 277-281 ] with
Then (1.1) is now rigorously established by differentiation and an appeal to Newton's Fundamental theorem of cal culus. 0
Early in June, 1949, Professor JOHN VON NEUMANN ex pressed an interest in the possibility that the ENIAC might sometime be employed to determine the value of 7f and e to many decimal places with a view toward obtaining a statistical measure of the randomness of distribution of the digits.
While there were many fine arithmeticians over the next 1500 years, this anecdote from Georges Ifrah reminds us that mathematical culture in Europe had not sustained Archimedes's level up to the Renaissance.
The paper notes that e appears to be too random-this is now proven-and ends by respecting an oft-neglected "best-practice":
lt . 0
=
_!_ t7 - � t6 + t5 - ± t3 7
3
3
+
4t
-
4 arctan (t).
A wealthy (15th-century) German merchant, seeking to provide his son with a good business education, con sulted a learned man as to which European institution offered the best training. "l.f you only want him to be able to cope with addition and subtraction, " the expert replied, "then any French or German university wiU do. But if you are intent on your son going on to mul tiplication and division-assuming that he has suffi cient gifts-then you wiU have to send him to Italy. 1 By the 19th century, Archimedes had finally been out stripped both as a theorist and as an (applied) numerical analyst, see [7].
In 1831, Fourier's posthumous work on equations showed 33 figures of solution, got with enormous labour. Thinking this a good opportunity to iUustrate the superiority of the method of W. G. Horner, not yet known in France, and not much known in England, I proposed to one of my classes, in 1841, to beat Fourier on this point, as a Christmas exercise. I received sev eral answers, agreeing with each other, to 50 places of decimals. In 1848, I repeated the proposal, request ing that 50 places might be exceeded: I obtained an swers of 75, 65, 63, 58, 57, and 52 places. (Augustus De Morgan2) De Morgan seems to have been one of the first to mis trust William Shanks's epic computations of Pi-to 527, 607, and 727 places [2, pp. 147- 161], noting there were too few sevens. But the error was only confirmed three quar ters of a century later in 1944 by Ferguson with the help of
Values of the auxiliary numbers arccot 5 and arccot 239 to 2035D . . . have been deposited in the library of Brown University and the UMT file of MTAC. The 20th century's "Top Ten"
The digital computer, of course, greatly stimulated both the appreciation of and the need for algorithms and for algo rithmic analysis. At the beginning of this century, Sullivan and Dongarra could write, "Great algorithms are the poetry of computation," when they compiled a list of the 10 algo rithms having "the greatest influence on the development and practice of science and engineering in the 20th cen tury".3 Chronologically ordered, they are: #1. 1946: The Metropolis Algorithm for Monte Carlo. Through the use of random processes, this algorithm offers an efficient way to stumble toward answers to problems that are too complicated to solve exactly. #2. 1947: Simplex Method for Linear Programming. An elegant solution to a common problem in planning and decision making. #3. 1950: Krylov Subspace Iteration Method. A tech nique for rapidly solving the linear equations that abound in scientific computation. #4. 1951: The Decompositional Approach to Matrix Computations. A suite of techniques for numerical lin ear algebra. #5. 1957: The Fortran Optimizing Compiler. Turns high-level code into efficient computer-readable code. #6. 1959: QR Algorithm for Computing Eigenvalues. Another crucial matrix operation made swift and prac tical.
1 From page 577 of The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French, John Wiley, 2000.
2Quoted by Adrian Rice in "What Makes a Great Mathematics Teacher?" on page 542 of The American Mathematical Monthly, June-July 1 999.
3From "Random Samples," Science page 799, February 4, 2000. The full article appeared in the January/February 2000 issue of Computing in Science & Engineering.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
41
#7. 1962: Quicksort Algorithms for Sorting. For the ef ficient handling of large databases. #8. 1965: Fast Fourier Transform. Perhaps the most ubiquitous algorithm in use today, it breaks down waveforms (like sound) into periodic components. #9. 1977: Integer Relation Detection. A fast method for spotting simple equations satisfied by collections of seemingly unrelated numbers. #10. 1987: Fast Multipole Method. A breakthrough in dealing with the complexity of n-body calculations, applied in problems ranging from celestial mechanics to protein folding. I observe that eight of these ten winners appeared in the first two decades of serious computing, and that Newton's method was apparently ruled ineligible for consideration.4 Most of the ten are multiply embedded in every major math ematical computing package. Just as layers of software, hardware, and middleware have stabilized, so have their roles in scientific, and espe cially mathematical, computing. When I first taught the sim plex method thirty years ago, the texts concentrated on "Y2K"-like tricks for limiting storage demands. Now seri ous users and researchers will often happily run large-scale problems in MATLAB and other broad-spectrum packages, or rely on NAG library routines embedded in Maple. While such out-sourcing or commoditization of scien tific computation and numerical analysis is not without its drawbacks, I think the analogy with automobile driving in 1905 and 2005 is apt. We are now in possession of mature not to be confused with "error-free"-technologies. We can be fairly comfortable that Mathematica is sensibly handling round-off or cancelation error, using reasonable termina tion criteria and the like. Below the hood, Maple is opti mizing polynomial computations using tools like Homer's rule, running multiple algorithms when there is no clear best choice, and switching to reduced complexity (Karat suba or FFT-based) multiplication when accuracy so de mands. Wouldn't it be nice, though, if all vendors allowed as much peering under the bonnet as Maple does! Example 2. The number of additive partitions ofn, p(n), is generated by (1.2)
q P( )
= 1
+
L P(n)qn = n
n� l
n ;:::: l
(1 - qn) � l.
Thus p (5) = 7, because
5=4+ 1 =3+2=3+ 1 + 1 =2+2+ 1 = 2 + 1 + 1 + 1= 1 + 1
+
1 + 1 + 1,
a s w e ignore "0" and permutations. Additive partitions are less tractable than multiplicative ones, for there is no ana logue of unique prime factorization nor the correspond ing structure. Partitions provide a wonderful example of
why Keith Devlin calls mathematics "the science of pat terns." Formula ( 1 .2) is easily seen by expanding (1 - qn) 1 and comparing coefficients. A modem computational tempera ment leads to �
Question: How hard is p(n) to compute-in 1900 (for MacMahon the "father of combinatorial analysis") or in 2000 (for Maple or Mathematica)? Answer: The computation of p(200) = 3972999029388 took MacMahon months and intelligence. Now, however, we can use the most naive approach: Computing 200 terms of the se ries for the inverse product in (1.2) instantly produces the result, using either Mathematica or Maple. Obtaining the re sult p(500) = 2300165032574323995027 is not much more difficult, using the Maple code
N : = 5 0 0 ; c o e f f ( s eries ( l/product ( 1 - q"n , n= l . . N + l ) , q , N + l ) , q , N ) ; Euler's Pentagonal number theorem
Fifteen years ago computing P(q) in Maple, was very slow, while taking the series for the reciprocal Q(q) = lln ;,: 1 (1 qn) was quite manageable! Why? Clearly the series for Q must have special properties. Indeed it is lacunary:
Q(q) = 1
- q - q2 + q5 + q7 - ql2 - ql 5 + q22 + q26
- q35 - q40 + q5 1 + q57 - q70 - q77 + q92 + O qlOO) . (
(1 .3)
This lacunarity is now recognized automatically by Maple, so the platform works much better, but we are much less likely to discover Euler's gem: 00
n
n= l
c1
_
qn)
=
oc
I
n= - x
c _ 1)nqn(3n + l)/2_
If we do not immediately recognize these pentagonal num bers, then Sloane's online Encyclopedia ofInteger Sequences5 immediately comes to the rescue, with abundant references to boot. This sort of mathematical computation is still in its rea sonably early days, but the impact is palpable-and no more so than in the contest and book under review. About the Contest
For a generation Nick Trefethen has been at the van guard of developments in scientific computation, both through his own research, on topics such as pseudo-spec tra, and through much thoughtful and vigorous activity in the community. In a 1992 essay "The Definition of Numer ical Analysis"6 Trefethen engagingly demolishes the con ventional definition of Numerical Analysis as "the science of rounding errors." He explores how this hyperbolic view emerged, and finishes by writing,
I believe that the existence offinite algorithms for cer tain problems, together with other historicalforces, has
41t would be interesting to construct a list of the ten most influential earlier algorithms. 5A fine model for of 2 1 st-century databases, it is available at www.research.att.com/�nJas/sequences 6SfAM News, November 1 992.
© 2005 Springer Sc1ence+Business Media, Inc., Volume 27, Number 4 , 2005
43
distracted us for decades from a balanced view of nu merical analysis. Rounding errors and instability are important, and numerical analysts will always be the experts in these subjects and at pains to ensure that the unwary are not tripped up by them. But our cen tral mission is to compute quantities that are typically uncomputable, from an analytical point of view, and to do it with lightning speed. For guidance to the fu ture we should study not Gaussian elimination and its beguiling stability properties, but the diabolically fast conjugate gradient iteration, or Greengard and Rokhlin's O(N) multipole algorithm for particle simu lations, or the exponential convergence of spectral methods for solving certain PDEs, or the convergence in O(N) iterations achieved by multigrid methods for many kinds of problems, or even Borwein and Bor wein's7 magical AGM iteration for determining 1 , 000, 000 digits of 7T in the blink of an eye. That is the heart of numerical analysis. In the January 2002 issue of SIAM News, Nick Trefethen, by then of Oxford University, presented ten diverse prob lems used in teaching modern graduate numerical analysis students at Oxford University, the answer to each being a certain real number. Readers were challenged to compute ten digits of each answer, with a $100 prize to be awarded to the best entrant. Trefethen wrote, "If anyone gets 50 dig its in total, I will be impressed." And he was. A total of 94 teams, representing 25 dif ferent nations, submitted results. Twenty of these teams received a full 100 points (10 correct digits for each prob lem). They included the late John Boersma, working with Fred Simons and others; Gaston Gonnet (a Maple founder) and Robert Israel; a team containing Carl De vore; and the authors of the book under review variously working alone and with others. These results were much better than expected, but an originally anonymous donor, William J. Browning, provided funds for a $ 100 award to each of the twenty perfect teams. The present author, David Bailey,8 and Greg Fee entered, but failed to qual ify for an award. 9 The ten challenge problems
The purpose of computing is insight, not numbers. (Richard Hamming lO) The ten problems are: #1. What is limE->0 J: x - 1 cos(x-1 log x)dx? #2. A photon moving at speed 1 in the x-y plane starts at t = 0 at (x,y) = (1/2, 1/10) heading due east. Around every integer lattice point (i, J) in the plane, a circu lar mirror of radius 1/3 has been erected. How far from the origin is the photon at t = 10?
7As
#3. The infinite matrix A with entries au = 1, a 12 = 112, a2 1 = 1/3, a 13 = 114, a22 = 115, a3 1 1/6, etc., is a bounded operator on €2 • What is I !A ll? #4. What is the global minimum of the function exp(sin(50x)) + sin(60e11) + sin(70 sin x) + sin(sin(80y)) - sin(10(x + y)) + (x2 + y2)J4? =
#5. Let .f(z) = 1/f(z), where f(z) is the gamma function, and let p(z) be the cubic polynomial that best ap proximates f(z) on the unit disk in the supremum norm ll · lloo- What is II! - Plloo? #6. A flea starts at (O,o) on the infinite 2-D integer lattice and executes a biased random walk: At each step it hops north or south with probability 1/4, east with probability 114 + E, and west with probability 114 - E. The probability that the flea returns to (0,0) sometime during its wanderings is 1/2. What is E? #7. Let A be the 20000 X 20000 matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, · 224737 along the main diagonal and the number 1 in all the positions aij with ji - jj = 1, 2, 4, 8, · · , 16384. What is the (1,1) entry of A - 1? #8. A square plate [ - 1 , 1 ] X [ - 1, 1 ] is at temperature u 0. At time t 0 the temperature is increased to u = 5 along one of the four sides while being held at u 0 along the other three sides, and heat then flows into the plate according to U t = Au. When does the temperature reach u = 1 at the center of the plate? #9. The integral I(a) = Jg [2 + sin(10a)]xa sin(a/(2 - x)) dx depends on the parameter a. What is the value a E [0,5] at which /(a) achieves its maximum? #10. A particle at the center of a 10 X 1 rectangle under goes Brownian motion (i.e., 2-D random walk with in finitesimal step lengths) till it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides? ·
=
=
=
Answers correct to 40 digits to the problems are avail able at http://web.comlab.ox.ac.uk/oucVwork/nick.trefethenl hundred.html Quite full details on the contest and the now substantial related literature are beautifully recorded on Bornemann's Web site http://www-m8.ma. tum. de/m3/bornemann/challenge book/ which accompanies The SIAM 1 00-digit Challenge: A Study In High-accuracy Numerical Computing, which, for brevity, I shall call The Challenge. About the Book and Its Authors
Success in solving these problems requires a broad knowl edge of mathematics and numerical analysis, together with
9We took Nick at his word and turned in 85 digits! We thought that would be a good enough entry and returned to other activities. 101n Numerical Methods for Scientists and Engineers, 1 962.
44
THE MATHEMATICAL INTELLIGENCER
,
·
in many cases, this eponym is inaccurate, if flattering: it really should be Gauss-Brent-Salamin.
8Bailey wrote the introduction to the book under review.
·
significant computational effort, to obtain solutions and en sure correctness of the results. The strengths and limita tions of Maple, Mathematica, MATLAB (The 3Ms), and other software tools such as PARI or GAP, are strikingly revealed in these ventures. Almost all of the solvers relied in large part on one or more of these three packages, and while most solvers attempted to confirm their results, there was no explicit requirement for proofs to be provided. In De cember 2002, Keller wrote:
To the Editor: Recently, SIAM News published an interesting article by Nick Trefethen (July/August 2002, page 1) pre senting the answers to a set of problems he had pro posed previously (January/February 2002, page 1). The answers were computed digits, and the clever methods of computation were described. I found it sutprising that no proof of the correctness of the answers was given. Omitting such proofs is the accepted procedure in scientific computing. However, in a contest for calculating precise digits, one might have hoped for more. Joseph B. Keller, Stanford University In my view Keller's request for proofs as opposed to compelling evidence of correctness is, in this context, somewhat unreasonable, and even in the long term counter productive [3, 4]. Nonetheless, the authors of The Challenge have made a complete and cogent response to Keller and much much more. The interest generated by the contest has with merit extended to The Challenge, which has al ready received reviews in places such as Science, where mathematics is not often seen. Different readers, depending on temperament, tools, and training, will find the same problem more or less interest ing and more or less challenging. The book is arranged so the ten problems can be read independently. In all cases multiple solution techniques are given; background, math ematics, implementation details-variously in each of the 3Ms or otherwise-and extensions are discussed, all in a highly readable and engaging way. Each problem has its own chapter with its own lead author. The four authors, Folkmar Bornemann, Dirk Lau rie, Stan Wagon, and Jorg Waldvogel, come from four countries on three continents and did not know each other as they worked on the book, though Dirk did visit Jorge and Stan visited Folkmar as they were finishing their manuscript. This illustrates the growing power of the collaboration, networking, and the grid-both human and computational.
Some high spots
As we saw, Joseph Keller raised the question of proof. On careful reading of the book, one may discover proofs of correctness for all problems except for #1, #3, and #5. For problem #5, one difficulty is to develop a robust interval implementation for both complex number computation and, more importantly, for the Gamma junction. While er ror bounds for #1 may be out of reach, an analytic solution to #3 seems to this reviewer tantalizingly close. The authors ultimately provided 10,000-digit solutions to nine of the problems. They say that this improved their knowledge on several fronts as well as being "cool." When using Integer Relation Methods, ultrahigh precision com putations are often needed [3] . One (and only one) prob lem remains totally intractable1 1-at press time, getting more than 300 digits for #3 was impossible. Some surprises
According to the authors, 1 2 they were surprised by the fol lowing, listed by problem: #1. The best algorithm for 10,000 digits was the trusty trapezoidal rule-a not uncommon personal experi ence of mine. #2. Using interval arithmetic with starting intervals of size smaller than 10-5000, one can still find the position of the particle at time 2000 (not just time ten), which makes a fine exercise for very high-precision interval computation. #4. Interval analysis algorithms can handle similar prob lems in higher dimensions. As a foretaste of future graphic tools, one can solve this problem using current adaptive 3-D plotting routines which can catch all the bumps. As an optimizer by background, this was the first problem my group solved using a damped Newton method. #5. While almost all canned optimization algorithms failed, differential evolution, a relatively new type of evolu tionary algorithm, worked quite well. #6. This problem has an almost-closed form in terms of el liptic integrals and leads to a study of random walks on hypercubic lattices, and Watson integrals [3, 4, 5] . #9. The maximum parameter is expressible in terms of a MeijerG junction. While this was not common knowl edge among the contestants, Mathematica and Maple both will figure this out. This is another measure of the changing environment. It is usually a good idea-and not at all immoral-to data-mine13 and find out what your favourite one of the 3Ms knows about your cur rent object of interest. For example, Maple tells one that:
1 1 1f only by the authors' new gold standard of 1 0,000 digits. 1 2Stan Wagon, private communication.
13By its own count, Wai-Mart has 460 terabytes of data stored on Teradata mainframes, made by NCR, at its Bentonville headquarters. To put that in perspective, the
Internet has less than half as rnuch data . . . ," Constance Hays, "What Wai-Mart Knows About Customers' Habits," New York Times, Nov. 1 4, 2004. Mathematicians also need databases.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4 , 2005
45
The Me i j er G func t i on i s de f ined by the inverse Laplace trans f o rm Me i j erG ( [ as , bs ] , [ c s , ds ] , z ) I
_ ---------�-------2 Pi I
I
I
I
where as = bs = cs = ds =
[ al , [bl , [ cl , [ dL
GAMMA ( 1 - a s + y ) - -
-
- ----
_ _
_
-
-
- ---
, am ] , , bn ] , , cp ] ' . dq ] '
GAMMA ( l - a s + y ) = GAMMA ( l - a l + y ) GAMMA ( b s -y ) = GAMMA ( bl -y ) GAMMA ( c s - y ) = GAMMA ( c l -y ) GAMMA ( l -ds + y ) = GAMMA ( l - d l + y )
Two big surprises
I finish this section by discussing in more detail the two problems whose resolution most surprised the authors_ The essay on Problem #7, whose principal author was Bornemann, is titled: "Too Large to be Easy, Too Small to Be Hard." Not so long ago a 20,000 X 20,000 matrix was large enough to be hard. Using both congruential and p-adic methods, Dumas, Turner, and Wan obtained a fully symbolic answer, a rational with a 97,000-digit numerator and like de nominator. Wan has reduced the time to obtain this to about 15 minutes on one machine, from using many days on many machines. While p-adic analysis is susceptible to parallelism, it is less easily attacked than are congruential methods; the need for better parallel algorithms lurks below the surface of much modern computational mathematics. The surprise here, though, is not that the solution is ra tional, but that it can be explicitly constructed. The chap ter, like the others, offers an interesting menu of numeric and exact solution strategies_ Of course, in any numeric ap proach ill-conditioning rears its ugly head, while sparsity and other core topics come into play. My personal favourite, for reasons that may be appar ent, is: Problem #10: "Hitting the Ends." Bornemann starts the chapter by exploring Monte-Carlo methods, which are shown to be impracticable. He then reformulates the prob lem deterministically as the value at the center of a 10 X 1 rectangle of an appropriate harmonic measure of the ends, arising from a 5-point discretization of Laplace's equation with Dirichlet boundary conditions. This is then solved by a well-chosen sparse Cholesky solver. At this point a reliable numerical value of 3.837587979 10-7 is ob tained. And the posed problem is solved numerically to the requisite 10 places. But this is only the warm-up. We proceed to develop two ·
· · · ·
GAMMA ( l -am+y ) GAMMA ( bn-y) GAMMA ( cp-y) GAMMA ( l - dq + y )
analytic solutions, the first using separation of variables on the underlying PDE on a general 2a X 2b rectangle. We learn that (3.4)
THE MATHEMATICAL INTELLIGENCER
z dy
L
Another excellent example o f how packages are chang ing mathematics is the Lambert W function [4], whose properties and development are very nicely described in a recent article by Brian Hayes [8] , Why W?
46
Y
GAMMA ( c s -y )
-c;� ( b� =; ; GAMMA ( 1 =d� + � ;
-
p(a,b) =
4
71'
�0 oo
( - 1 )n sech 2n + 1
(
7T(2n + 1) 2
)
p
where p : = alb. A second method using conformal map pings yields (3.5)
arccot p
=
; + arg K(eip(a,b)1T),
p(a,b)
where K is the complete elliptic integral of the first kind. It will not be apparent to a reader unfamiliar with inver sion of elliptic integrals that (3.4) and (3.5) encode the same solution; but they must, as the solution is unique in (0, 1); each can now be used to solve for p = 10 to arbitrary pre cision. Bornemann fmally shows that, for far from simple rea sons, the answer is (3.6)
p=
2 . arcsm (kwo), 71'
where
a simple composition of one arcsin and a few square roots. No one anticipated a closed form like this. Let me show how to finish up. An apt equation is [5, (3.2.29)] showing that (3.7)
f --( - 1) sech 2n + 1
L
n
n�o
(
7T(2n + 1) 2
)
p
=
. -21 arcsm k '
exactly when k = kp2 is parametrized by theta functions in terms of the so-called nome, q exp( - 7rp) , as Jacobi dis covered. We have =
(3.8)
e2 ( )
kp2 - � e�(q) -
I� � q Cn+ 112)2 I�� -oo qn2 -co
�����--
Comparing (3. 7) and (3.4), we see that the solution is
kwo = 6.02806910155971082882540712292 . . . . 10 - 7,
as asserted in (3.6). The explicit form now follows from classical nineteenth-century theory as discussed in [1, 5] . In fact k21o is the singular value sent by Ramanujan to Hardy in his famous letter of introduction [2, 5]-if only Trefethen had asked for a V210 X 1 box, or even better a V15 X Vi4 one!
> > > >
{ "D ( 4 ) [x] 2 " , " E ( 8 ) : 2 " } , " + " , 1 6 , { " 4 5 ) ( 6 7 ) " , " (1 8) (2 3 ) (4 5 ) ( 6 7) " , " (2 8 ) ( 1 3 ) (4 6 ) (5 7) " }
which finds the minimal polynomial for k 100, checks it to 100 places, tells us the galois group, and returns a latex ex pression 'p' which sets as:
p(_X) = 1 - 1658904 _X - 3317540 X 2 + 1657944 X3 + 6637254 _X 4 + 1657944 X5 - 3317540 _X6 - 1658904 x7 + xs _
_
_
_
_
,
and is self-reciprocal: it satisfies p(x) = x8p(l!x). This sug gests taking a square root, and we discover that y � satisfies =
1 - 1288y
+
20y2 - 1288y3 - 26y4
+
1288y5 + 20y6 + 1288y7
+
y8.
Now life is good. The prime factors of 100 are 2 and 5, prompting subs ( _X= z , [ op ( ( ( factor ( p , { sqrt ( 2 ) , sqrt ( 5 ) } ) )
) ) ] )
)
This yields four quadratic terms, the desired one being = z2
Example 3. Maple allows the following
Digi t s : = 1 0 0 : wi th ( Po lynomial Too l s ) : k : = s - > eva l f ( E l l ip t i cModulus ( exp ( - P i * sqrt ( s ) ) ) ) : p : = latex ( Minimal Polynomi al ( k ( 1 0 0 ) , 1 2 ) ) : ' Error ' , f s o lve ( p ) [ 1 ] - eval f ( k ( 1 0 0 ) ) ; ga l o i s ( p ) ; Error , 4 1 0 - 1 0 6
" 8T 9 " ,
q
Alternatively, armed only with the knowledge that the singular values are always algebraic, we may finish with an au courant proof: numerically obtain the minimal polyno mial from a high-precision computation with (3.8), and re cover the surds [4] .
+
+
322 z - 228 z\12 + 144 zV5 - 102 zV2V5 323 - 228 V2 + 144V5 - 102V2V5.
For security, w : = s o lve ( q ) [ 2 ] :
eva l f [ 1 0 0 0 ] ( k ( 1 0 0 ) -wA 2 ) ;
gives a 1000-digit error check of 2.20226255 10 - 998. We leave it to the reader to find, using one of the 3Ms, the more beautiful form of k 100 given above in (3.6). D ·
Considering also the many techniques and types of math ematics used, we have a wonderful advertisement for multi field, multi-person, multi-computer, multi-package collabo ration. Concrete Constructive Mathematics
Elsewhere Kronecker said "In mathematics, I recognize true scientific value only in concrete mathematical truths, or to put it more pointedly, only in mathemati cal formulas. " . . . I would rather say "computations"
" (4 8) (1 5) (2 6) (3 7) " ,
than 'jormulas, " but my view is essentially the same. (Harold M. Edwards [6, p. 1]) Edwards comments elsewhere in his recent Essays on Constructive Mathematics that his own preference for con structivism was forged by experience of computing in the fifties, when computing power was, as he notes, "trivial by today's standards." My own similar attitudes were ce mented primarily by the ability in the early days of personal computers to decode-with the help of APL-exactly the sort of work by Ramanujan which finished #10. The SIAM 1 00-Digit Challenge: A Study In High-accu racy Numerical Computing is a wonderful and well-writ ten book full of living mathematics by lively mathemati cians. It shows how far we have come computationally and hints tantalizingly at what lies ahead. Anyone who has been interested enough to finish this review, and had not yet read the book, is strongly urged to buy and plunge in-computer in hand-to this fine advertisement for constructive math ematics 21st-century style. I would equally strongly suggest a cross-word solving style-pick a few problems from the list given, and try them before peeking at the answers and extensions given in The Challenge. Later, use it to illustrate a course or just for a refresher; and be pleasantly reminded that challenging problems rarely have only one path to so lution and usually reward study. REFERENCES
[ 1 ] Folkmar Bornemann, Dirk Laurie and Stan Wagon, Jorg Waldvogel, The SIAM 1 00-0igit Challenge: A Study In High-accuracy Numeri cal Computing, S I AM 2004.
[2] L. Berggren , J.M. Borwein and P.B. Borwein, Pi: a Source Book, Springer-Verlag, 1 997. Third Edition, incorporating A Pamphlet on Pi [CECM Preprint 2003 : 2 1 0] , June 2000.
[3] J.M. Borwein and D.H. Bailey, Mathematics by Experiment: Plausi ble Reasoning in the 2 1 st Century, AK Peters Ltd, 2004.
[4] J . M . Borwein, D . H . Bailey and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, AK Peters Ltd,
2004.
© 2005 Springer Science+Business Media, Inc., Volume 27, Number 4, 2005
47
[5] J.M. Borwein and P.B. Borwein, Pi and the AGM: A Study in Ana lytic Number Theory and Computational Complexity, John Wiley,
New York, 1 987. [6] Harold M. Edwards, Essays on Constructive Mathematics, Springer
employ a broad brush and are not always sufficiently de veloped. Nevertheless, thanks to the exceptional standing of the protagonists, the debate manages to be compelling and relevant.
Verlag, 2005. [7] H. H. Goldstine, History ofNumericalAnalysis from the 1 6th Through the 19th Century. Springer-Verlag, 1 977.
[8] Brian Hayes, "Why W?" American Scientist, 93 (2005), 1 004-1 008. Faculty of Computer Science Dalhousie University Halifax, Nova Scotia B3H 1W5 Canada e-mail:
[email protected]
Conversations on M ind, Matter, and Mathematics by Jean-Pierre Changeux & Alain Cannes edited and translated by M. B. DeBevoise PRINCETON UNIVERSITY PRESS, PRINCETON, NJ, 1 995. PAPERBACK: ISBN 0-691 -00405-6, 260 PP. US$ 22.95.
REVIEWED BY JEAN PETITOT
W
hat exactly is the type of reality of mathematical ideal entities? This problem remains largely an open question. Any ontology of abstract entities will encounter certain antinomies which have been well known for cen turies if not millennia. These antinomies have led the var ious schools of contemporary epistemology increasingly to deny any reality to mathematical ideal objects, structures, constructions, proofs, and to justify this denial philosoph ically, thus rejecting the spontaneous nai:ve Platonism of most professional mathematicians. But they throw out the baby with the bath water. Contrary to such figures as Poin care, Husserl, Weyl, Borel, Lebesgue, Veronese, Enriques, Cavailles, Lautman, Gonseth, or the late Godel, the domi nant epistemology of mathematics is no longer an episte mology of mathematical content. For quite serious and pre cise philosophical reasons, it refuses to take into account what the great majority of creative brilliant mathematicians consider to be the true nature of mathematical knowledge. And yet, to quote the subtitle of Hao Wang's (1985) book Beyond Analytic Philosophy, one might well ask whether the imperative of any valid epistemology should not be "do ing justice to what we know." The remarkable debate Conversations on Mind, Mat ter, and Mathematics between Alain Connes and Jean Pierre Changeux, both scientific minds of the very first rank and professors at the College de France in Paris, takes up the old question of the reality of mathematical idealities in a rather new and refreshing perspective. To be sure, since it is designed to be accessible to a wide audience, the de bate is not framed in technical terms; the arguments often
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Jean-Pierre Changeux's Neural Materialism
Let me begin by summarizing some of Jean-Pierre Changeux's arguments. Because mathematics is a human and cognitive activity, it is natural first to analyze it in psychological and neuro cognitive terms. Psychologism, which formalists and logi cists have decried since the time of Frege and Husserl, de velops the reductionist thesis that mathematical objects and the logical idealities that formulate them can be re duced-as far as their reality is concerned-to mental states and processes. Depending on whether or not mental representations are themselves conceived as reducible to the underlying neural activity, this psychologism is either a materialist reductionism or a mentalist functionalism. J-P. Changeux defends a variant of materialist reduc tionism. His aim is twofold: first, to inquire into the nature of mathematics, but also, at a more strategic level, to put mathematics in its place, so to speak. He has never con cealed his opposition to Cartesian or Leibnizian ratio nalisms that have made mathematics the "queen" of the sci ences. In his view, mathematics must abdicate its overly arrogant sovereignty, stop laying claim to universal valid ity and absolute truth, and accept the humbler role assigned to it by Bacon and Diderot-that of "servant" to the natural sciences (p. 7). And what better way to make mathematics surrender its prestigious seniority than to demonstrate sci entifically that its claims to absolute truth have no more ra tional basis than do those of religious faith? Pursuing his mission with great conviction, Changeux revisits all the traditional touchstones of the empiricist, ma terialist, and nominalist critiques of Platonist idealism in mathematics. He cites an impressive mass of scientific data along the way, including results from neurobiology and cog nitive psychology in which he has played a leading role. It is this aspect of his approach which commands attention. 1. The empiricist and constructivist theses hold that mathematical objects are "creatures of reason" whose re ality is purely cerebral (p. 1 1). They are representations, that is, mental objects that exist materially in the brain, and "corresponding to physical [i.e., neural) states" (p. 14). Mental representations-memory objects-are coded in the brain as forms in the Gestalt sense, and stored in the neurons and synapses, despite significant variability in synaptic efficacy (p. 128). Their object-contents are reflexively analyzable and their properties can be clarified axiomatically. But that is possible only because, as mental representations, they are endowed with a material reality (pp. 1 1-15). What's more, the axiomatic method of analysis is itself a "cerebral process" (p. 30). 2. One might try to salvage an autonomy for the formal logical and mathematical levels by admitting, in line with
the functionalist theses of computational mentalism in the style of Johnson-Laird, Fodor, and Pylyshyn, that the algo rithms of psychological "softwares" are independent of the neural "hardware" that implements them: mental repre sentations would then constitute, as they do for Fodor, an "internal language of thought" possessing all the charac teristics of a formal language (symbols, symbolic expres sions, inference rules, etc.). But, according to Changeux, such theses run into a "real epistemological obstacle" be cause they assume that
tures of reason" (p. 104), not to mention the neural as semblies that code the cognitive acts of "understanding" (what is called "population coding"). This hierarchical com plexity, of which we are beginning to get a pretty good grasp, obviously plays a fundamental role in the progres sive structuring of the mathematical universe. 5. The evolutionist conception of mathematical episte mology leads to a "mental Darwinism" which J-P. Changeux develops in detail as a "new idea." This idea is, let me say once again, that the brain is a natural evolutionary machine
it's possible to identify a mathematical algorithm with a physical property of the brain (p. 167).
[that] evolves in a Darwinian fashion, simultaneously at several different levels and on several different time scales (p. 168).
The brain cannot be a biological computer because the brain's program and machine [ . . . ] exhibit from the first stages of development a very intricate interplay (p. 168). In that sense, the brain is an evolutionary Darwinian ma chine. 3. Even though they can be identified with mental processes and represen-tations, mathematical objects, structures, and theories are not of a purely private and sub jective nature. That would lead to solipsism. They are com municable, public, historical, and cultural representations and, for this reason, "secular" and "contingent" (p. 18). They are selected by a contingent evolutionary process. They are cultural objects, ( . . . ) public representations of men tal objects of a particular type that are produced in the brains of mathematicians and are propagated from one brain to another (p. 35). Mathematics constitutes a language and must therefore be approached cognitively, like any other language, taking off from cognitive theories of concept formation, abstraction, symbolic coding, reasoning, procedures, learning, etc. It fol lows that there can be no ontology of mathematics: here evolutionist historicism (where chance becomes "neces sity" through selection) takes the place of ontological ne cessity. The reality, existence, coherence, and rigidity of mathematics are "a posteriori results of evolution" (p. 36). The science of the "why?" isn't theology, it's evolution ary biology. And the "why?" of the existence of mathe matics has as much to do with the evolution of our knowledge acquisition apparatus-our brain-as it does with the evolution of mathematical objects themselves (p. 40). 4. There exist several different levels of cognitive orga nization, from the most concrete (the perceptive) to the most abstract (the symbolic). They are realized in the neural architecture, from elementary neural circuits of the spinal cord, the brain stem, and ganglions (p. 98) all the way to the frontal cortex, the seat of "the neural architec-
The general model of Darwinism combines, as we know, a generator of diversity with a system for selection. At a certain level of organization (itself rooted in lower levels), elements functioning as "matter" combine to generate the "forms" ("Darwinian variations") of the next level. Some of these forms are stabilized through selection on the basis of their functional efficacy. In this sense, the function feeds back into the "variation-form" transi tion (p. 108). Changeux was one of the first, along with Antoine Danchin and Philippe Courrege, to propose a detailed model of the fundamental mechanism of epigenesis through selective stabilization of synapses. This explains how neural Dar winism naturally extends into a psychological Darwinism pertaining to the generation/selection of representations. 6. This purely representational and communicational, cognitive, neural, and Darwinian reality of mathematical ac tivity is then used to justify a materialism denouncing any Platonism as an irrational belief. The Platonic realism which holds that mathematical objects exist "somewhere in the universe," independently of all material and cerebral support (p. 18) is, according to Changeux, the "mythic residue" (p. 25) of a bygone magico-theological age, an irrational belief that must be eliminated through the intellectual ascetic rigor of the materialist (p. 25). The materialist epistemology which, since Galileo, has been the "victim of a special form of in tolerance" (p. 26) is the best one available to the informed scientist [who is honest with himself] (p. 26). Mathematical objects cannot exist in nature. They are not natural objects. Where, then, could they exist? For "to exist" means, and can only mean, to exist in nature, to "ex ist in the universe prior to [their] existing in the brain of the mathematician" (p. 41), in short, to exist materially as an independent substance outside the mind. Mathematics
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can therefore be nothing but a series of mental construc tions, which, if Changeux is to be believed, is what Kant al ready said: the ultimate truth of mathematics lies in the possibility of its concepts being constructed by the human mind (p. 40). Thanks to a subtle rhetoric, Galileo, who was condemned for having elevated mathematics to the rank of an objec tive reality, and Kant, who never ceased to assert the absolutely irreducible role of pure mathematics in achiev ing the objectivity of the true sciences, thus find them selves enlisted in the service of an anti-mathematical ma terialism. 7. Such a conception of the reality of mathematical ide alities obviously leads to an "extremely concrete and prag matic" (p. 64) conception of their applicability. Mathemat ics is not the "organizing principle of matter." It is only "a rough language" for describing matter. To be sure, there exist regularities in nature, but these are "properties in trinsic to matter" and not mathematical laws (p. 46). Math ematics confines itself-and should confine itself-to pro viding models (foreign to nature), which are selected by the scientific community on the basis of what "fits the best with the real world" (p. 64). Moreover, as several examples demonstrate, a mathematical equation (such as that of Hodgkin and Huxley for the nerve impulse, for example)
Alain Connes's S�ructural Objectivism
Evolutionist biological materialism and neural Darwinism are certainly positions with a great deal of validity. They should ultimately lead to a complete rethinking of the foun dational problems of mathematics. If I may offer a bit of personal testimony, I am myself involved in theorizing the neural bases of space using models of the functional ar chitecture of visual areas and of the kinesthetic coupling of perception and action, and I have witnessed the extent to which the question of the foundations of geometry is thereby transformed (see Petitot [2003]). But, for all that, neural Darwinism does not make it possible to "psycholo gize" mathematics. Indeed, a classic difficulty encountered by reductionist materialism derives from its identifying objects with the cognitive acts that provide access to them. It maintains that mathematical idealities cannot exist because • •
existence is equivalent to a sort of ontological indepen dence, what phi-losophers call "transcendence," and no ontological transcendence could arise out of the immanence of cognitive acts.
The conclusion reached from these premises is that the re ality content of mathematical objects must be reduced to the conditions of epistemic access to them. Alain Connes rejects this conclusion outright, for, in his view, mathe matical reality is fundamentally different from the manner in which it is apprehended (p. 14).
describes a function. It allows us to grasp a certain be havior, but not to fully explain the phenomenon (p. 60). An
explanation would require the identification of the un derlying structure (in the case at hand, the biological struc ture of the channels for sodium and potassium ions in the axon membrane) (p. 60). But the argument here is hardly self-evident. The math ematical models of physical theories always operate at a certain level of reality. The Navier-Stokes equations are called upon to determine the flow dynamics of liquids and not their molecular structure, which, for its part, will be marvelously described by the equations of quantum me chanics. The equations of Newton and Einstein are called upon to determine gravitational interactions and not the chemistry of planets, etc. As to the explanation of a phe nomenon by underlying structures, it is clearly no longer possible at the level of fundamental physics, even though this level universally constrains all the other levels of reality. 8. Given this set of "self-evident truths," epistemologists who refuse to identify the "apparatus of knowledge" with the brain can do so only through their "ignorance of neu roscience" (p. 25). Here Jean-Pierre Changeux takes ex plicit aim at Jean-Toussaint Desanti, the leading French philosopher of mathematics of the post-war era, who, in his authoritative work Les idealites mathematiques, took up and developed many important theses first advanced by Edmund Husserl and Jean Cavailles. 50
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The materialist thesis is of course perfectly defensible. But if he adopts it, a "well-informed scientist who is hon est with himself " must adopt itfully and accept all the con sequences of his refusal of any realism where abstract en tities are concerned. Ever since the medieval controversies between realists and nominalists over this question, which is nothing other than the question of "universals," human ity has devoted a great deal of reflection to it. Now, among all the consequences, there is one in particular, traditional but formidable, which, from Berkeley to Husserl and Quine, philosophers have analyzed in all its facets. Materialism, and the nominalism that goes with it, presupposes an in dependent reality composed of individualized and "sepa rate" substances. But how do we obtain access to this tran scendent material reality? Through the objects of our perception (aided by all the measuring devices one might want), that is, through phenomena. But the phenomena which are the objects of perception are a prototype of cog nitive construction. They are constituted out of sensory data and, insofar as they are constituted, they are just as abstract as pure numbers. In other words, the anti-realist thesis concerning math ematics must then be extended to perception itself, and that leads, with no hope of escape, to a radical solipsism. If one adopts an ontological realism with respect to exter nal material reality, one is of course able to justify an anti Platonism where mathematical entities are concerned, but one finds oneself equally obliged, reluctantly but in-
eluctably, to reject the reality of perception and thus to in vert ontological realism into subjective idealism. Hilary Putnam has studied very discerningly the conflict between physicalist realism and commonsense realism which runs through our modem conception of reality. In his 1987 essay The Many Faces ofRealism (The Paul Cams Lectures), he recalls the genesis of the dualism between, on the one hand, the ontology of a transcendent, indepen dent external reality existing in itself and, on the other hand, the cognitive reconstruction of the perceived world through sensory data, and he shows to what extent this du alism is detrimental. For, if they cannot be expressed in the language of physics, how are we to think about the qualitative struc tures of the phenomenologically manifested world? Ac cording to Putnam, we must call into question the com monly accepted opposition between properties that are intrinsic (i.e., transcendent and independent of the mind, of perception, and of language) and properties that are extrinsic, apparent, projected, and dispositional. As he puts it, to explain the features of the commonsense world, in cluding color, solidity, causality [ . . . ] in terms of a men tal operation called "projection" is to explain just about every feature of the commonsense world in terms of thought (p. 12). The immediate result is that, in practice, realism reverses itself into a pure subjective idealism: So far as the commonsense world is concerned [ . . . ] the effect of what is called "realism" in philosophy is to deny objective reality, to make it all simply thought (p. 12). Putnam goes on to explain that if one wishes to develop a physicalist monism on these bases, one is obliged to in terpret mental phenomena as complex and derived physi cal phenomena. But, as the theses offunctionalism make explicit, there is no necessary and sufficient condition (NSC) characterizing mental contents and propositional at titudes that can be formulated in physical language. Such an NSC would in fact be infinite and lack effective rules of construction. The intentionality of consciousness remains; it seems not reducible to the physical and the computa tional levels. But then it should itself be conceived as a pro jection, which is absurd. In his debate with Changeux, Con-nes has made a very good case for this point. Apart from an irrational belief in the reality of the external material world, what proves this reality if not the coherence of perceptions? If mathematics were reduced to nothing but a language and if one denied any reality to it, then there would be no reason not to deem perceived real objects to be merely a mental construction useful for explaining certain vi sual phenomena (p. 23).
That is why reducing [mathematics] to a mere language would be a serious mistake (p. 22). If mathematics is effectively reduced to the brain, why then not equally reduce the world to the brain through the in termediary of perception? (p. 56). In the debate, J-P. Changeux rejects this parallel between mathematics and perception as a "metaphor." But the argument carries con siderable weight. It can even be reinforced by applying it not only to the objects of perception but also to those phys ical objects which themselves constitute, for the material ist, the ultimate ontological reality. In this sense, the argu ment has been spelled out quite well by Quine. Quine has remarked that the physical objects postulated by physical theories are neither more nor less ideal than mathematical idealities themselves, and that it is therefore just as legitimate, or just as illegitimate, to accept one as to accept the other. One cannot be at the same time a re alist in physics and a nominalist in mathematics. Physical objects, too, are explanatory idealities that allow us to re duce the complexity of sensory experience to a conceptual simplicity. Platonist ontology [ . . . ] is, from the point of view of the strictly physicalistic conceptual scheme, as much a myth as that physicalistic conceptual scheme itself is for phenomenalism (Quine [ 1948]). As soon as one treats physical objects as real, one must ac cept their existence ("ontological commitment"). But then one must equally accept the existence of mathematical ide alities. One's ontological commitment must be coherent. To refuse to be coherent would amount to "intellectual dis honesty" (see Maddy [ 1989], p. 1 131; it will be noted that both Changeux and Quine appeal to our intellectual "hon esty"). Consequently, Quine criticizes the positivists who seek to exclude as nonsensical statements on the existence of abstract objects. Mathematics is part of science and we can have reasons, and essentially scientific reasons, for including numbers or classes or the like in the range of values of our variables (Quine [ 1969], p. 97). For this debate to move forward, it is philosophically necessary to change viewpoints and to realize that the prob lem is not that of an ontology of mathematics in the tradi tional sense, but rather that of its objective reality. Alain Connes clearly positions himself on this terrain when he insists on the reality of mathematical idealities, for exam ple, in the case of prime numbers, the infinity of which is a reality every bit as incontestable as physical reality (p. 13). prime numbers ( . . . ) constitute a more stable reality than the material reality that surrounds us (p. 12).
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Connes returns several times to the necessity of admitting such a mathematical reality as "raw and immutable" and not reducible to the conceptual tools employed to investi gate it, a reality every bit as constraining as physical reality, but one that's far more stable than physical reality, for it is not located in space-time (p. 28). No serious philosophical debate about modem science is possible if one fails to distinguish carefully between on tology and objectivity. But once one has done so, thus pos ing the problem of the reality of mathematical idealities in terms of their objective status rather than in terms of an ontology, existence in the spatio-temporal world is no longer the exclusive criterion of reality and it becomes pos sible to display criteria of objectivity. Alain Connes re peatedly underscores three such criteria, which are indeed fundamental. 1 . The possibility of exhaustively classifying the objects defined by an axiomatics, the axiomatics allowing classification problems to be posed for mathematical ob jects defined by very simple conditions (p. 13). For example, for every prime number p and every posi tive integer n there exists one and only one finite field of characteristic p and of cardinal pn, and one obtains in this way all the finite fields (p. 13). The complete classifica tion of locally compact fields is equally known (the field R of real numbers, the field C of complex numbers, the p-adic fields and their algebraic extensions, the fields of formal series over finite fields, p. 16). In the same way, an uninterrupted series of brilliant efforts (from Galois to Chevalley and then Feit and Thompson) have led to the classification of simple finite groups. One could cite many other examples from topology, geometry, etc. This history begins with the Greek geometers who classified the five Platonic solids. Such results manifest the existence of ob jective constraints that necessarily limit the domains of possibility. 2. The global inter-theoretical consistency and har mony of mathematical theories (p. 152). Despite being "in explicable" (p. 1 7) and constituting a crucial problem, these are incontestable and objective. They are "the very opposite of randomness" (p. 1 16). This aspect of things cannot be overemphasized. As Jean Dieudonne has ob served with respect to what the great philosopher Albert Lautman called the unity of mathematics, all the major theorems bring into play a huge number of different the ories and manifest absolutely unsuspected solidarities among apparently unrelated objects and structures. Among the examples supplied by Connes, one might sin gle out the way in which V. Jones, working in analysis on the classification of the "factors" of von Neumann alge-
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bras, used a braid group in one of his proofs, and, mak ing the link with knot theory, discovered a new invariant which since then has proved to have fundamental appli cations in quantum field theory. One could cite a signifi cant number of other examples which have brought to light unforeseeable overall consonances between appar ently quite distant areas of mathematics and which have had remarkable physical applications. This holistic con sistency is quite astonishing and shows that mathematical reality, in its very structure, its internal harmony, is an inexhaustible source of organization (p. 125). The "immediate comprehension" of it by mathematicians is essential to their creativity and to their understanding of the power of new tools (p. 152). But it remains quite enigmatic. 3. The fact that interesting mathematical theories pos sess an infinite informational content. Godel's incom pleteness theorem in its most profound formulation ( . . . ) shows that math ematics can't be reduced to a formal language (p. 159). It means that interesting structures (able to code arith metic) contain an infinite quantity of information that can not be finitely axiomatized. As it is explained in the French edition (p. 2 13) in reference to Chaitin's works: One may consider this theorem to be a consequence of constraints imposed by the theory of information, due to the finiteness of the complexity of any formal system. From this a criterion of objectivity may be deduced, for isn't that the distinguishing characteristic of a reality in dependent of all human creation? (p. 160). It will be noticed that these criteria of objectivity are not satisfied by any other cognitive symbolic system, whether one thinks of natural languages or of the various "games" (chess and other systems of rules) to which mathematics has been compared. To be able to see them for what they are-"to do justice to what we know"-a correct doctrine of objectivity is called for. The Antinomy of Mathematical Reality
The anti-Platonic theses, whether they be psychologistic, empiricist, nominalist and/or materialist, or neurocognitive (the repertory is rather vast), seem at first sight to be self evident. However, they are not nearly as self-evident as they look There are several reasons for this. 1 . First, they all rest upon a certain preconception of what physical objectivity is. They conceive external reality as founded on a substantialist ontology of autonomous ma terial things (independent of the mind, transcendent)
endowed with a sufficient structural stability and main taining relations of causality (material and efficient) and reciprocal interaction between themselves. What's more, this substantialist ontology is believed to be, if not explic able, at least describable by an appropriate scientific lan guage of description built on natural language. Different levels of organization are then introduced, and it is posited (reductionist thesis) that the lower levels causally explain the higher levels. Atoms, molecules, the genome, proteins, neurotransmitters, etc. really and truly exist in nature, whereas mathematical structures such as numbers are not supposed to exist in the same manner and will be conceived as the product of a contingent symbolic, historical, and cul tural evolution. Jean-Pierre Changeux vigorously and rigorously defends a materialism of this type, and he does so in a fashion that does not suffer from the kind of inconsistency denounced by Quine. He doubts that numbers exist: I have a hard time ( . . . ) imagining that integers exist in nature (p. 28); but he is just as dubious about the constructs of theoreti cal physics: atoms exist in nature-but Bohr's atom doesn't (p. 28). In this conception, basic material reality functions meta physically as a reality in-itself. Now, the hypothesis of a material reality existing in itself, transcendent and in dependent-and, moreover, of an independent reality sat isfying a substantialist ontology of things-is a hypothesis which is itself anti-scientific and equally based on an ir rational belief. Not that the idea of such a reality in-itself must be re jected. One might well hypothesize that it "exists" as a tran scendent "foundation" of empirical reality. The problem is that, as can be argued since Kant, this foundation is cog nitively inaccessible and therefore cannot be used in sci entific reasoning. We meet here an inescapable scientific datum: physics does not describe a substantial world of structurally stable material things, interrelated and interacting in causal fash ion. At the fundamental (quantum) level, physical phe nomena are devoid of any underlying ontology. This is a well-known theorem (von Neumann, Bell, Kochen Specker). In the very technical nature of their physico mathematical contents, the fundamental physical theories (symplectic mechanics, general relativity, quantum field theory and Feynman's integrals, gauge theory, string the ory, etc.) confirm that objectivity cannot be identified with an ontology. It must be said that here again Changeux is perfectly consistent. In the exchanges on quantum mechanics, he de fends the principle of theories with hidden parameters. To his mind, quantum theory "is bad" because it rests on pre suppositions that do not satisfy the principle that
the experimental conditions must be defined in such a way that it [the quantum phenomenon in question] be comes reproducible (p. 71). In other words, quantum mechanics is incomplete, and there remains an unexplained sublevel to which theo reticians haven't yet gained mental access (p. 71). As we know, however, it is contradictory to try to "com plete" quantum mechanics preserving the locality of inter actions. 2. The anti-Platonist materialist viewpoints under dis cussion use in a non-problematized fashion certain con cepts which, however, are fundamentally problematic. I will cite here only the simple but absolutely crucial con cept of space-time and that of continuum which under girds it. Space-time is not in itself a physical reality with which we can enter into causal interactions. As Kant was the first to explain, in his celebrated "exposition" of the Transcen dental Aesthetics, it is a form of external reality. If to ex ist means to exist materially in nature, then space-time does not exist in this sense. It, too, is, like mathematical ideali ties, a pure mental representation. Which, by the way, fits well with the hypothesis that its mathematical (geometric) determination should also be of an exclusively mental na ture. There is a catch, however. It follows that the sub stantialist ontology serving as the foundation of material ist positivism should then logically be, as in Leibniz, an a-spatial and a-temporal ontology. The problem is that the physical objectivity to which recourse is constantly made as the materialist foundation is in the last resort entirely constructed on a spatio-temporal basis. For "to exist" is taken to mean "to exist in nature" and "to exist in nature" is taken to mean "to exist in space-time." It is a recurrent paradox of materialisms and nominalisms that they refuse the reality of abstract entities in order to confme ontology to independent, individual, and "separate" substances, while simultaneously subordinating this very ontology to a space and a time which are prototypical instances of ide alities, fully as cognitive and abstract as numbers, and which, therefore, do not exist. To this it must be added that space and time are based on the continuum, and that the latter may be "arithmetized," in other words, reduced to numbers (even if that raises very difficult questions, as we shall see). And this paradox will be taken to dizzying heights by modem physics because, in its physical determinations, matter is fundamentally identified-since Riemann and Clifford-with a geometry, or, more precisely, to borrow Wheeler's term, with a "geometro-dynamics" founded on the geometry of space-time. From general relativity to con temporary gauge theories, to super-string theory and to Alain Connes's work on the physical applications of non commutative geometry, all of modem physics confirms Clifford's slogan "Physics is Geometry." Now, space-time is
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not an independent reality in itself. It is devoid of any on tological content. And yet, if one uses this to justify re ducing it to a mere appearance, a mental projection, one will be condemned to adopt a solipsistic idealism. To get out of this dilemma, one needs to understand that space-time is objective and not ontological-that it is in fact the primary form of physical objectivity. As Kant said, one must succeed in maintaining at one and the same time the thesis of the "empirical realism" of space and that of its "transcendental ideality." But once objectivity has been distinguished from ontology, there is no longer any reason to deny mathematics the same objective status as physics quite the contrary. Mathematical Idealities and Objectivity
Jean-Pierre Changeux's point of view will no doubt be ac cepted and defended by a majority of scientists. It is part of the current revival of "psychologism" powerfully fueled by the various schools of epistemology which seek to "nat uralize" the problems of the theory of knowledge by re ducing them to problems of a cognitive psychology founded on the neurosciences. Most of these schools are obliged to deny any reality to mathematical objects, structures, and theories for the following obvious reason: if to exist ob jectively means to exist materially in nature, then how can one obtain epistemic access (learning, beliefs, knowledge) to external abstract entities with no causal efficacy? Michael Resnik puts it well: if we
have no physical traffic with the most basic math ematical entities and they are not literally the products of our own minds either, how can we learn any mathe matics? How could it even be possible for us to acquire beliefs about mathematical objects? (Resnik [1988], p. 403). To salvage such a problematical mathematical ontology, one then must always introduce, in one way or another, "supernatural" cognitive faculties on the order of an intel lectual intuition (cf. Frege, Husserl with his "intuition of essences," Godel, etc.). Because that is clearly incompati ble with a naturalized epistemology, there is no choice but to fall back onto a materialist nominalism. This last point is essential. A fundamental thesis, linked to what is called the causal theory of reference, is that no knowledge of and no reference to external ab stract entities can be legitimately introduced and used in sofar as all knowledge of and all reference to an exter nal entity requires a causal interaction of the subject with that entity. Now, by definition, an abstract entity cannot sustain causal relations. As Philip Kitcher asserts, it is therefore impossible for symbolic constructions and ma nipulations to provide any type of access to abstract reality (Kitcher [ 1988], p. 527).
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Mathematics must be conceived, on the contrary, as a sym bolic activity of a logico-linguistic (and even semio-narra tive: "in certain respects, mathematics is like story-telling") nature, which allows us, through a series of successive ap proximations inscribed in the traditions, to structure our experience more and more adequately by means of ideali ties. Mathematics will have emerged, through a process of idealization, from proto-mathematical (perceptive, etc.) knowledge constrained by the structures of natural reality. Transmitted historically and socially through the scien tifico-technical legacy of humanity, it will have progressed in the same way as all of humanity's other symbolic for mations. It can therefore be understood without there be ing the least need to invoke a mysterious world of ideas to which an incomprehensible intellectual intuition would grant us access. Of course, the whole problem with such a line of rea soning is that it presupposes that we know the meaning of terms such as "external reality," "matter," "causality," etc. But it is impossible to define these terms objectively ex cept in mathematical fashion. And that is precisely where the difficulty lies. The belief in the possibility of under standing the concept of "reality" independently of any ob jective determination and constitution is a belief even more irrational and archaic than na:ive Platonism. It can thus be seen that one's conception of the reality of mathematical idealities is tightly bound up with one's conception of their applicability: the fundamental physical objects are themselves mathematical construc tions in the first place. Hence the question which for M. Resnik is one of the most important ones in the philosophy of mathematics: How can we retain the advantages of an ontology of ab stract entities for mathematics while removing its obvi ous epistemological disadvantages? (Resnik [ 1988], p. 407). The problem is clear. If, as the nominalists insist (see Hartry Field, for example, in Science Without Numbers [1980]), there do not exist mathematical idealities pos sessing the status of things, then what are the "truth-mak ers" for mathematical statements? It is consistent to posit, with the second Wittgenstein, that mathematical contents are prescriptive and not descriptive-that they are nothing but rules for the use of concepts. But as soon as one aban dons this radical position, then the problem of truth-mak ers becomes crucial. As Crispin Wright recalls, the traditional platonist answer is that the truth-condi tions of pure mathematical statements are constituted by the properties of certain mind-independent abstract objects, the proper objects of mathematical reflection and study. (Wright [1988], p. 426) Other answers are well known. For classical intuition ists, mathematical statements refer to mental constructions
that have to be investigated with a particular logic, re flecting their constructive character (but, as we know, thanks in particular to the work of F. W. Lawvere and M. Tierney, the intuitionist logic is the internal logic of uni verses of sets endowed with a certain structure, and in par ticular of Grothendieck's topoi:, that is, categories of sheaves over categories endowed with a "topology"). For formalist structuralists, mathematical statements refer to structures, etc. These questions belonging to the pure philosophy of mathematics enter into the Changeux-Connes debate with regard to the opposition between formalism and construc tivism. Jean-Pierre Changeux rightly emphasizes that many intuitionist and constructivist philosophers of mathematics agree with him on anti-Platonism. He refers in particular to Allan Calder's denunciation of realism. Alain Connes picks up the argument by maintaining that the distinction between constructivism and formalism is a methodological distinction more than anything else (p. 42), and by discussing the axiom of choice (AC), which is the prototype of a non-constructive axiom whose conse quences are omnipresent in proofs. My own opinion, if I may inject it, is that the non-con structive axioms of existence in mathematics must indeed be understood as methodological principles whose only value lies in their operational capacity. But in mathemat ics, "methodological" means a lot because the object is in this case the correlate of the method. The theory of the con tinuum provides an especially striking example (Petitot [ 1995]). A "good" theory of the continuum consists in showing that large classes of subsets of the field R of real numbers are "regular" in the sense of sharing "good" properties, such as being measurable in Lebesgue's sense or pos sessing the perfect set property. Cantor had already shown that the closed subsets of R are regular in this sense, and it was subsequently shown that it is also the case for the hierarchy of Borel subsets obtained from open and closed subsets by countable union and inter section and complementation. But there exists a more complex hierarchy, called the projective hierarchy. The projective subsets � 1n, IJln, Ll 1n = �1nnii1n are obtained from open subsets by iterating the set-theoretical opera tions of complementation, countable union, and projec tion (image under continuous mapping). It can be shown that the smallest projective class Ll1 1 is the class of all Borel subsets (Suslin's theorem). With respect to these new classes, it is also natural to raise the question of their regularity. But such a proof quickly becomes impossible in ZFC (the standard set theory of Zermelo-Fraenkel with AC), starting with the �12 and II 12 levels, in fact for meta mathematical reasons pertaining to Godel's incomplete ness theorem. Hence Godel's idea of completing the ZFC axioms.
Godel began by introducing the constructive theory of sets where all sets are "constructible." But the constructibility axiom turns out to be too constraining. It entails the exis tence of a low-level (Ll12) projective well-ordering of R and thus the existence of a non-Lebesgue-measurable Ll12 set. Now, a well-ordering of R should be highly non-con structible and undefinable. Generally speaking, the axiom of choice AC (which remains true in the universe of con structible sets) entails the existence of very complicated and very irregular sets that are nonetheless projective. These sets should be highly non-constructive. But the ax iom of constructibility forces them to exist in the hierar chy of projective sets. Hence a complete reversal of strat egy on GOdel's part. Priority is now given to being able to prove good prop erties of regularity for projective sets and to generalizing the results of Luzin and Suslin that every � 1 and every 1 II1 are Lebesgue measurable and that every � 1 displays 1 1 the perfect set property. This involves enriching the ax ioms of set theory by specifying the size of the universe. The best way of doing this is to introduce new existence axioms known as large cardinals axioms (inaccessible, measurable, etc.), which introduce higher levels of infin ity into the transfinite. From the standpoint of large car dinals axioms, it is no longer a matter of elaborating a model of the continuum that is dogmatically constrained by a constructive a priori, but rather of reconstructing as well as possible, from within mathematics itself, the "in tuitive" continuum. The fundamental result is then that the "good" structure of the continuum in a ZFC universe is the counterpart of very strong non-constructible (Platonist) axioms of exis tence for large cardinals. We may therefore consider these, as Godel and Martin proposed, to be hypotheses regarding, not a fixed and completely predetermined mathematical universe, but a universe to be determined in the most har monious way possible. We see that, if one wishes to avoid a cascade of insur mountable difficulties, one must not apply to mathemat ics, apart from the relationship between syntax and se mantics proper to the logical theory of models, the traditional and general conception of a denotative rela tionship between a language and a reality (theory of ref erence). Indeed, it is only at this point that one runs into the problem of what makes mathematical statements true (in the sense of a truth-correspondence) and of what al lows us to know that true statements are true (epistemic access to truth). The conception of denotation and of truth that one adopts will determine how one conceives the nature of proofs. For a traditional Platonist, proofs are only cogni tive auxiliaries providing access to independent truths (with ontological content). In this perspective, truth thus transcends provability. Radical finitist intuitionists like Wittgenstein and Dummett deny this thesis: for them, a mathematical truth cannot transcend the proof that deter mines it. But then the problem of the applicability of math-
© 2005 Springer Science+ Business Media, Inc .. Volume 27, Number 4, 2005
55
ematics becomes incomprehensible. For, as Crispin Wright emphasizes,
REFERENCES
[1] Cavailles, J . , 1938. Methode axiomatique et Formalisme. Essai sur le probleme des fondements des mathematiques, Paris, Hermann.
How is it possible to apply mathematics to statements which concern ordinary things, and how does the cred ibility which attaches to a pure mathematical statement as a result of proof carry over to its application? (Wright [ 1988], p. 429).
[2] Desanti, J. T., 1968. Les ldealites matMmatiques, Paris, Le Seuil. [3] Dieudonne, J., 1 981. "Bourbaki et Ia philosophie des mathema
tiques," Un siecle dans Ia philosophie des mathematiques, Archives de l ' lnstitut International des Sciences Theoriques, Bruxelles, Of fice International de Librairie. [4] Feferman, S., 1 989. "Infinity in Mathematics: Is Cantor Necessary?"
Personally, I believe that the question of the reality and the applicability of mathematical idealities should not be conceived in terms of an analogy with the relationship be tween a language and the world. In their relationship to re ality, mathematical theories do not denote, any more than do the physical theories which bring them into play. They determine-they legalize-phenomenal data, which is something else altogether. To be sure, the theory of models internalizes, in meta-mathematics, a relationship which is apparently of the "language-reality" type. But the latter is intra-mathematical and thus does not entail any relationship with an external world. Consequently, it remains foreign to the questions of reality and of applicability. To pose these questions while trying to couple this meta-mathematical re lationship with an "ontological" relationship of the "(math ematical) language-(real) world" type amounts to conceiv ing knowledge in terms of predication and denying the essential gap separating science from common sense. To conceive of knowledge in terms of predication is to cling to a classical metaphysical tradition that no longer possesses any value. It means neglecting the philosophical fact of its having been replaced by a problematics of objectivity. The problem of the reality of mathematical idealities is not that of their reality in a traditional ontological sense, but that of their objectivity, which is-one cannot emphasize this enough-something else entirely. The notion of reality is a modal category inseparable from a transcendental doctrine of constitution and not an absolute concept. Likewise, the problem of the applicability of mathematical idealities is not that of their applicability to an ontological reality of the world, but that of their entailment in physical objectivity, which, once again, is something else altogether.
Philosophical Topics, XVI I , 2, 23-45. [5] Field, H . , 1980. Science without Numbers, Princeton. [6] Kant, E . , 1 781-87. Critique de Ia Raison pure (trans. A. J. L. Dela
marre and F. Marty), Paris, Plelade, Gallimard, 1980. [7] Kitcher, P., 1 988. "Mathematical Progress," in "Philosophie des
Mathematiques" (P. Kitcher, ed.), Revue lntemationale de Philoso phie, 42, 1 67, 518-540. [8] Lautman, A., 1 937-39. Essai sur /'unite des mathematiques et divers ecrits (reprinting of books published by Hermann between 1 937
and 1939 and, posthumously, in 1946), Paris, Bourgois, 1977. [9] Maddy, P., 1 989. "The Roots of Contemporary Platonism, " The Journal of Symbolic Logic, 54, 4, 1 121 -1 144. [10] Petitot, J . , 1 995. "Pour un platonisme transcendantal, " in L 'ob jectivite mathematique. Platonisme et structures formelles (M.
Panza, J.-M. Salanskis, eds.), Paris, Masson, 147-78. [11 ] Petitot, J . , 2003. "The neurogeometry of pinwheels as a sub
Riemannian contact structure," Journal of Physiology-Paris, 97, 265-309. [12] Putnam, H . , 1987. The Many Faces of Realism, Lasalle, Illinois,
Open Court.
[13] Quine, W. V. 0 . , 1948. "On what there is," in From a Logical Point of View. Cambridge, Harvard University Press, 1 961.
[14] Quine, W. V. 0., 1 969. "Existence and Quantification," Ontologi cal Relativity and other Essays, 91-1 13, New York, Columbia Univ.
Press. [15] Resnik, M . D . , 1 988. "Mathematics from the Structural Point of
View" in "Philosophie des Mathematiques" (P. Kitcher, ed.), Revue lnternationale de Philosophie, 42, 167, 400-424. [16] Shanker, S.G., 1 987. Wittgenstein and the Turning-Point in the Philosophy of Mathematics, State University of New York Press. [17] Wang, H . , 1 985, Beyond Analytic Philosophy, Cambridge, M . I.T.
Press. [18] Wang, H . , 1 987. Reflections on Kurt G6del, Cambridge, M . I .T.
Press.
Conclusion
The debate between Jean-Pierre Changeux and Alain Connes is one of the most interesting to take place in recent years. It re-frames in a very up-UH:late context a whole series of tra ditional and difficult questions from the standpoint of the knowledge and experience of two of the leading protagonists of contemporary science. To the choice presented by the neu robiologist between a Platonist ontology and a neurocogni tive psychology of mathematical activity, the mathematician replies with a conception that is objective (neither ontologi cal nor psychological) of the thoroughly consistent universe of mathematical idealities. It is indeed in this three-sided arena that the major difficulties play themselves out. One ofthe great virtues of the book is to cast a spotlight on this confrontation.
[19] Willard , D . , 1984. Logic and the Objectivity of Knowledge, Ohio
University Press. [20] Wittgenstein, L., 1 956. Bemerkungen uber die Grundlagen der Mathematik, Remarks on the Foundation of Mathematics (G. H. von
Wright, R. Rhees, G.E.M. Anscombe, eds.), Oxford, Blackwell, 1978. [21 ] Wright, C., 1 988. "Why Numbers Can Believably Be: A Reply to
Hartry Field," in "Philosophie des Mathematiques" (P. Kitcher, ed.), Revue lnternationale de Philosophie, 42, 1 67, 425-73.
CREA, Ecole Polytechnique, 1 rue Descartes 75005 Paris, France.
Translated by Mark Anspach 56
THE MATHEMATICAL INTELLIGENCER
e-mail:
[email protected]
ries, most of those considered converge geometrically. For
Mathematics by Experiment. Plausible Reasoning in the 11 sf Century
the others, a suitable use of the Euler-McLaurin summation formula is often sufficient, and this is explained in Section
7.5 of the second volume. The only two points to be noted here are that some series (for instance for
by Jonathan Borwein and David Bailey
1r)
give
spigot
algorithms, in other words allow the computation of deci
NATICK, MA A K PETERS, 2003, 288 PAGES. $US 45.00 ISBN: 1 -5688 1 -21 1 -6
mals one by one, but more importantly that many numbers have a BBP (Bradley-Borwein-Plouffe) expansion in a suit able base, which allows the computation of a number of
Experimentation in Mathematics. Computational Paths to Discovery
digits in that base without knowing the preceding ones. In Section 7.4 of the second book the authors give a brief account of numerical quadrature methods. I had consid ered these methods quite boring and reserved to numeri
by Jonathan Borwein, David Bailey, and
cal analysts. However, the brief description of the "doubly
Roland Girgensohn
exponential" tanh-sinh method given in the book led me to study in more detail this type of numerical quadrature
NATICK, MA A K PETERS, 2004, 350 PAGES. $US 49.00 ISBN: 1 -56881 - 1 36-5
methods, and it has completely changed my point of view:
REVIEWED BY HENRI COHEN
these doubly-exponential methods are amazingly efficient, especially for people who, like us, need several thousand
I
emphasize from the start that the aim of these two books
decimals, since we often need to recognize constants. On
is the enj oyment of the reader, together with the oppor
the other hand, for standard numerical analysis work
(28
tunity to learn new results, techniques, and ideas. The first
decimals at most), the usual quadrature methods are suffi
book contains a large number of miscellaneous results,
cient. I strongly recommend looking at the (unfortunately
problems, techniques, comments, and many of the authors'
sparse) literature on the subject. Note that the actual im
philosophical beliefs on the notion of "experimental math
plementation of these methods is also extremely simple.
ematics" (please note that the reviewer himself belongs to
Once a constant is computed, we can try to recognize
a lab where the name "experimental" occurs). The second
it. The authors' tool for doing so is a very efficient imple
book is a little more systematic, and it can be considered
mentation of Ferguson's PSLQ algorithm, which is de
a sequel or a complement to the first.
scribed in detail in Section
6.3
of the first book. I would
The main idea surrounding these books, well known to
like to make two comments concerning this. On the one
some parts of the mathematical community, but which the
hand, nobody doubts that the authors' (or similar) imple
authors want to promote, is that theory and experimenta
mentation of constant recognition is one of the best exist
tion
ing ones. On the other hand I am not convinced that a sim
are
intimately
linked
in
the
process
of
math
ematical discovery. Of course some subj ects are more sus
ilar highly tuned multistage version of the LLL algorithm
ceptible to experimentation than others, but still, the
itself, perhaps adapted to constant recognition, would not
amount of theoretical insight to be gained through experi
give similar results, up to a few percent.
ments is enormous.
serious scientific comparison has been made between the
It
seems that no
For this dialectical exchange to take place fruitfully, a
two. My second comment is that, as given, the PSLQ algo
number of conditions must be met. In particular, re
rithm does not seem to be able to recognize complex con
searchers must have at their disposal a large number of im
stants directly (in other words without separating real and
plementations of some basic algorithms, which may or may
imaginary parts), whereas this is automatic with LLL-based
not be easily found in standard computer algebra systems.
algorithms. Note that modem LLL implementations use LQ
Although other topics are considered, the main objects
based orthogonalization instead of Gram-Schmidt, so that
of study in these books are sequences, infinite series, and
numerical stability is no longer an issue.
definite integrals. One important point is the following: we
At this point, it is time to include a small caveat. Thanks
must be able to compute such quantities sometimes to very
to the numerical summations, and especially the doubly
high accuracy (several thousand digits for instance), in or
exponential quadrature methods, I have checked a large
der to apply a constant recognition algorithm to determine
number (around
experimentally (if possible) the value of the given quantity.
in the text or in the exercises. I must warn the reader that
This experimental part is then followed by a more theo
I have found an abnormally large number of errors ("mis
retical part, where one tries to prove the experimentally
prints"?) which do not all appear to be typographical. This
found identity.
200)
of constant evaluations given either
is not the place to give errata, but the reader should be
The authors do not spend much time explaining how to
aware that while a given result or answer may be in prin
compute limits of sequences or sums of infinite series: in
ciple correct, the exact value may be different from the
deed, most of the sequences considered have at least lin
printed one. Because the authors are distinguished exper
ear convergence (a linear number of digits accuracy gained per iteration), but most often quadratic or better conver gence, such as the AGM-type iterations.
As
for infinite se-
imentalists and careful writers, I do not understand how
this came about, and I hope that an errata list will be made
available on line. In addition, certainly because of a macro
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
57
error, a number of bibliographical numbers are off by 1 or 2, but this is usually easy to spot. I now summarize the contents with some comments. The first volume, by Jonathan Borwein and David Bailey, is ti tled "Mathematics by Experiment: Plausible Reasoning in the 2 1st Century." In Chapter 1 the authors explain their views on experi mental mathematics, including a number of examples and many challenge problems, ending with a large list of Inter net-based resources. I was surprised to see that the popu lar Pari/GP software which I developed and is one of the most commonly used in number theory is not even men tioned! In Chapter 2, ten miscellaneous but highly inter esting examples of experimental mathematics in action are discussed, and 50 additional examples are given more briefly. Chapter 3 is devoted entirely to 1T and similar con stants-how to compute them using AGM-type or BBP-type algorithms, and so on. Chapter 4 is more theoretical and discusses normality of expansions of numbers. As is well known, this is essentially a hopeless subject, but the au thors give their views about it, including possible links with BBP algorithms. Chapter 5 also gives a number of miscel laneous results, the philosophy being to concentrate on constructive proofs as opposed to abstract ones. Chapter 6 is the first chapter devoted to the explicit numerical al gorithms used in experimental work: Fourier transforms (FFI' and DFI'), multiprecision arithmetic, including fast high-precision evaluation of exp and log using the AGM and Newton, and constant recognition (PSLQ). Chapter 7 is a concluding chapter containing little information. The second volume, by Jonathan Borwein, David Bailey, and Roland Girgensohn is titled "Experimentation in Math ematics: Computational Paths to Discovery." Chapter 1 deals with miscellaneous results and proofs on sequences, series, products, and integrals. The structure of this chapter is similar to that of Chapter 2 of the first volume: first 10 examples are described in detail, then al most 60 additional examples are offered as problems with indications. This chapter makes for very enjoyable reading, and although most of the problems are classical, there are some real gems. Chapter 2 contains a serious and quite clas sical exposition of Fourier series, Fourier integrals, and summation kernels. The most amusing part of this chapter is probably Section 2.5, dealing with sine integrals (sinc(x) = sin( 17X)/( 11X)), together with several of the ad ditional examples. Chapter 3 is devoted entirely to zeta functions and multizeta values, essentially sums of the form
�(s b
Sz,
.
.
. , sk) =
I
n1>n2> .. .nk� l
THE MATHEMATICAL INTELLIGENCER
M= and other examples. Chapter 5 is devoted to a number of miscellaneous subjects: prime number conjectures, repre sentation of integers by xy + yz + zx, Grobner bases and metric invariants, spherical designs, and many others. Chap ter 6 is a sequel to Chapter 5 of the first volume: its intent is to illustrate that very classical undergraduate theorems of real but especially complex analysis are amenable to prac tical computation, and in fact help to solve computational problems. The final Chapter 7, which is a sequel to Chapter 6 of the first volume, gives a number of other numerical tech niques. The authors have little to say on the Wilf-Zeilberger algorithm of creative telescoping, primality testing, comput ing complex roots of polynomials, or the use of Euler MacLaurin for infinite series summation. Nevertheless, Sec tion 7.4 contains a description of numerical quadrature methods, including the remarkably efficient doubly-expo nential methods such as the tanh-sinh method. Once again I urge the reader to pursue the study of these methods. To conclude, these two books contain a wealth of di verse examples (I did not count, but it may reach 1000), al though the reader must be warned that there are many mathematical misprints. The two volumes are very enjoy able reading and belong on the bookshelves of any math ematician or graduate student who does mathematics for pleasure (which one hopes is the case for most of them!). Laboratoire A2X UFR de Mathematiques et lnformatique Universite Bordeaux I 33405 Talence Cedex France e-mail:
[email protected] .fr
lndra's Pearls. The Vision of Felix Klein b y David Mumford, Caroline Series, and David Wright
1
where the si are positive integers. The latter have been in recent years the object of a vast literature, in number the ory and analysis of course, but also in knot theory, combi natorics, and theoretical physics. These values are espe cially well suited to experimentation, because they can be easily computed to hundreds or thousands of decimals (al though the algorithms for computing them are not com pletely trivial), and constant recognition algorithms such 58
as PSLQ allow the experimenter to find remarkable rela tions between these values. One such experimentally dis covered relation by Zagier gives the value of �(3, 1, 3, 1 . . . , 3, 1) as a rational multiple of a power of 1r, and the authors include a proof of this relation. Chapter 4 is devoted to partition functions, special values of theta functions, Madelung's constant
CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE, 2002 396 PAGES, £33.00 HARDCOVER ISBN-1 0: 0521 352533 -ISBN-13: 9780521 352536)
REVIEWED BY LINDA KEEN
I
recently received this e-mail message from the distraught mother of a twelve-year-old girl: Dear Mrs. Keen, I left you a message this afternoon requesting your help. I realize this is an inconvenience for you but my 12 year
old daughter, Megan, has been assigned to write about
the text from becoming too cluttered. There are also very
your life and to list one of your theories and to explain
informative boxes giving biographical information on vari
As you can imagine this is difficult to do. I've attempted to read 4 of your papers and was
ous (dead) mathematicians who have made substantial
lost after the intros! I passed calculus with only a B and
projects for readers, involving both computer programs and
this is WAY over my head! Would you be able to pick
colored pencils and paper.
it in her own terms.
contributions. In addition, there are many well-thought-out
one theory and briefly explain it in layman's language.
The book begins with a discussion of symmetry as the
She needs to report what the theory is and how is it used.
basis of geometry as propounded by Felix Klein. In his
I really appreciate your help
view, geometry is the study not only of objects, triangles
I replied that Megan's teacher had given both her and
are just affine motions, but Klein's motions were more
me an impossible task I then suggested that they think
general. The key here is that the motions that preserve
and such, but also of motions. In Euclidean geometry these
about a curve that was so crinkled that no matter how much
symmetrical objects form a group. All the essential infor
it was magnified, it looked as crinkled as ever, and I indi
mation is encoded by the group. Thus, the first chapter is
cated how they might construct it. Megan wrote back
devoted to a discussion of symmetries of the plane, that
thanking me and telling me that the teacher read my e-mail
is, tilings of the plane by regular shapes. A group is de
to the class and they tried to work out the construction and
fined by the properties of the set of transformations
found it interesting. Megan and her mother were going to
needed to move one tile onto another. There are lots of
spend some more time on it over the weekend.
illustrations, both hand-drawn and computer-drawn. The
This incident points out the problem we, as individual
authors talk about their programming techniques and in
mathematicians, and as a community, have. How can we
troduce the first pseudo-code programs to create the tiling
convey to others what it is about mathematics that excites
pictures.
us and makes us get up in the morning? We even have trou ble telling other mathematicians what we do.
The next chapter tackles the next basic concept-com plex numbers. After a bit of history, the authors discuss the
There are some problems in mathematics whose state
arithmetic operations of complex numbers and computer
ments, at least, are accessible to laymen-for example, the
programs to implement them. They quickly progress to the
twin primes conjecture. But, how do we develop the ideas
Riemann sphere and stereographic projection. The impor
and the language that goes with them to tell anyone but a
tant motions here, of course, are inversions. The third chap
few specialists what we are working on?
ter completes the presentation of the basic material by dis
Since the late 1970s, as computers have become more
cussing Mobius transformations of the plane. These are
powerful and their graphics more sophisticated, mathemati
compositions of an even number of inversions. In addition
cians have been able to draw mathematical pictures that ap
to good pictorial descriptions of how these maps act, com
pear beautiful, even to non-specialists. Just as a nonmusi
puter programs are given that find the image of a circle un
cian can listen to a Haydn symphony and enjoy the music
der such a map.
without being able to articulate what it is about the under
Having assembled the mathematical and programming
lying structure that appeals, nonmathematicians can look at
tools, the authors move on to the groups of symmetries that
many of these pictures and fmd them pleasing.
they are really interested in. They start with two pairs of cir
The book under review,
Indra 's Pearls,
is on one level
an attempt to tell a broad audience something about math
cles in the plane, CA, Ca and CB, cb, and two Mobius maps
a
and b, where a maps the exterior of CA to the interior of Ca
b.
ematics. The book grew out of all three authors' enchant
and similarly for
ment with the computer pictures they have been making
pairs determines the symmetrical pattern for the group
The initial arrangement of these circle
a, b
for the last thirty years in their study of discrete groups of
formed by applying
Mobius transformations, otherwise known as Kleinian
binations to the pairs of circles. In the simplest case, the
and their inverses in various com
4 mutually disjoint disks, and the basic tile
groups. The pictures reminded them of the ancient Bud
circles bound
dhist dream of Indra's net. The infinite net, stretching
for the pattern is their common exterior. Tiling, using the
across the heavens, is made from diaphanous threads, and
group elements, covers the whole plane with the exception
at each intersection there is a reflecting pearl. In each pearl,
of a Cantor set-which the authors call "Fractal Dust"
all the other pearls are reflected, and in each reflection
that is invariant under any element of the group.
there are again infinitely many reflected pearls. They
To describe how to plot the fractal dust, the authors take
wanted to share the pictures with a broad audience of math
an excursion into the realm of search algorithms for trees.
ematicians and non-mathematicians alike. Even more, they
I have taught this often from computer science textbooks,
wanted to enable the reader to write programs to recreate
and this is the best description of these algorithms I have
the computer pictures.
seen.
To this end, the book is written in a chatty informal style
In the next two chapters, the initial arrangement of the
and begins at the beginning. The illustrations are both hand
circle patterns is changed. First, the circles are moved in
and computer-drawn. They are well chosen, and the cap
the plane until they just touch and form a "necklace"-CA
tions, set in the margins, give a good description of them.
is tangent to CB, CB is tangent to Ca, Ca is tangent to cb,
There are boxes containing calculations and asides to keep
and cb is tangent to CA. The map
a
sends CA to Ca and
© 2005 Spnnger Science+ Business Media, Inc., Volume 27, Number 4, 2005
59
sends the other three circles, now thought of as beads, to
relations, so each family depends on one complex parame
three smaller beads inside
Ca. The "necklace" is now made cb, CA, CB, and the three inside Ca. The
ter.
up of the six beads
variant sets move, but retain their basic characteristics: for
A, B,
b,
As the parameter varies in the complex plane, the in
act similarly, replacing each
example, the number of components of the complement of
of the original beads with three smaller beads. Iterating this
the fractal invariant. At some points in the plane, these char
other three maps,
and
process indefinitely, the beads of the necklace become
acteristics change: circles may appear in the fractal creating
smaller and smaller and more and more numerous, and
new components, components may disappear, etc. Such
form "Indra's necklace," a fractal continuum invariant un
points form the boundary for the family.
der the group generated by the maps
a
and
b.
In the Riley and Maskit families, there seem to be round
Then the initial arrangement is changed again, so that in addition to the tangencies above, the circles
CA and Ca are
also tangent. The circles no longer form a necklace. Nev
disks, albeit overlapping, in the simply connected invariant component. These disks form circle chains with discernible patterns.
As the parameter is varied appropriately, the
ertheless, the process of forming the necklace, applying the
chains persist until, at points called "accidentally parabolic
group transformations to the circles again and again, re
points, " they become chains of tangent disks. These are
sults in an invariant set for the group that is recognizable
boundary points of the family. The existence of the circle
as the classical Apollonian gasket.
chains reflects the relationship between the elements of the
After this, the going gets tougher and the mathematics
group as words in the generators and continued fractions.
becomes very deep. The groups are divided into families
The parameter spaces for these families of groups can
based on the pattern of the four initial circles. In each fam
also be drawn by computer. It is possible to get the com
ily, the invariant set has certain characteristics. These fam
puter to find the parameter values of enough accidentally
ilies of groups have historically been named after individ
parabolic boundary points to draw the boundary and see
uals-which is unfortunate because it makes it even more
that it is fractal and has an interesting structure of its own.
difficult to keep straight which is which. There are Classi
The book ends with a discussion of the relationship of the
cal Schottky groups whose invariant set is fractal dust,
material to three-dimensional topology and Thurston's work
Fuchsian groups whose invariant set is a circle, and Quasi
So, have the authors succeeded in their attempt to make
fuchsian groups whose invariant set is a closed fractal
this mathematics accessible to a broad audience? Could
curve. There are also Riley groups and Maskit groups,
Megan's mother understand the book? I doubt it, but she
whose fractal invariant sets divide the plane into one sim
might get something from it. The book is written to be read
ply connected invariant component with fractal boundary
on many levels, and a given reader will have to find his own.
and infinitely many components whose boundary is a round
An undergraduate who has taken some algebra and complex
circle.
analysis can certainly get something out of the book, espe
These families of groups depend on parameters, the triple
ta, tb, tab.
cially the material and projects in the first three chapters; a
subject to
sophisticated mathematician can get the flavor of the subject;
certain relations depending on the family. For Fuchsian
and a graduate student can work her way from the beginning
of traces of the Mobius transformations,
As
groups, the parameters are real and there is one relation, so
to interesting research problems in some of the projects.
the family depends on two real parameters. For the Riley and
an "expert," I er\ioyed it very much-and in fact, I learned
Maskit groups the parameters are complex and there are two
some new things.
Grothendieck on Triviality Alexand r Grothendi ck was again ·t "I I told him of a <' mment
to
uppose that a llu'orem i
mathematic
was
and on anoth r dark."
UmverStty of Wales,
Bangor Bangor, Gwynedd LL57 2DG Wales
60
THE MATHEMATICAL INTELLIGENCER
and
·
l li\ial b
au
th
pr of i
for lack of th
w
n bbi ·hn
d l ight d wh n
trivial." H
"tli\ial" con
of th
young
wr t
pl of
that z
ro;
ion of "the difficulty of bringing new con ·ept.o.; out of the
Ronme Brown
e-mail:
,"
·
held up for centmie a!
nobc m
f Heruy Whit h ad: "It i th
r
[email protected]
41fii.i.CQ.h.i§i
R o b i n W i l son
The Philamath' s Alphabet-JK
I
and others
(not
commemorated on
value for sin
1°, from which many other
stamps), Indian mathematicians and as
trigonometrical values can be calcu
tronomers became interested in practi
lated.
cal astronomy, building magnificent ob
Kepler: Using Tycho Brahe's exten
servatories such as the Jantar Mantar in
sive observational records, Johannes
(1571-1630) was led to his three
Jaipur.
Kepler
Jefferson: Thomas Jefferson (17431826), the third president of the United
laws of planetary motion-in particu lar, that the planets move in elliptical
States, extolled the virtues of science,
orbits with the sun as focus (a word he
wrote of the importance of calculation
introduced), and that the line from the
(extracting square and
sun to a planet sweeps out equal areas
cube
roots,
solving quadratic equations, and using
in equal times. Kepler also discovered
acob's staff: The Jewish scholar Levi ben Gerson ( 1288-1344), a
logarithms) , and strongly advocated
the cuboctahedron and the antiprisms,
decimalising the American coinage. In
and foreshadowed the integral calcu
mathematician, philosopher, and as
terested in the theory and practice of
lus by calculating the volumes of over
tronomer, invented the Jacob's staff, or
classical architecture, he designed his
J
cross-staff, for measuring the angular
home, Monticello, and the central ro
separation between two celestial bod
tunda of the University of Virginia.
ies. Although widely used, it had the
Kashani: The Persian mathematician
90 solids of revolutions. Kovalevskaya: The mathematician and novelist Sonya Kovalevskaya (18501891) made valuable contributions to
disadvantage that to measure the angle
and astronomer Jamshid al-Kashani
mathematical analysis and partial dif
between the sun and the horizon one
(or al-Kashi) (d.
1429) made extensive
ferential equations. Barred by her gen
had to look directly at the sun. Later
calculations
fractions
der from studying in Russia, she went
modifications, such as the back-staff,
and established a notation for them, us
with
decimal
to Heidelberg and Berlin, later becom
solved this problem.
ing a vertical line to separate the inte
ing the first female professor in Stock
Jantar Mantar: Following the great
ger and fractional parts. A prodigious
holm. She won the coveted Prix Bordin
developments in arithmetic and algebra
calculator, he determined
by Aryabhata, Brahmagupta, Bhaskara,
imal places and obtained a very precise
7T
to
16 dec
of the French Academy for a memoir on the rotation of bodies.
N E D E Jacob's staff Jefferson Kepler
Jantar Mantar
Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes, MK7 6M, England
e-mail:
[email protected]
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Kovalevskaya Kashani
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+Business Media, Inc.
