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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.

Infinity: C a n You Believe It? The P r i n c e t o n astrophysicist David Spergel w a s quoted in the C o l u m b u s (Ohio) Dispatch (April 19, 1998) as telling the reporter, "If the universe is infinite, for example, that m e a n s w e ' r e having this conversation an infinite n u m b e r o f times." I r u s h e d to s e n d him an e-mail message: "Did you really s a y this? Would you please clarify the a b o v e s t a t e m e n t ? Thanks ( p e r h a p s an infinite n u m b e r of times)." Spergel replied, "Here is tile n o t very r i g o r o u s argument. Let's c o n s i d e r all the e l e c t r o n s within the visible universe. The Heisenberg u n c e r t a i n t y principle implies that we can quantize p h a s e s p a c e (position and velocity). Since all the e l e c t r o n s are at fairly low t e m p e r a t u r e s , there is only a finite volu m e in p h a s e s p a c e that the e l e c t r o n s can occupy. There is only a finite numb e r of w a y s that I can assign N1 electrons into N2 p h a s e s p a c e cells, so there are only finitely m a n y p o s s i b l e states of the Visible universe. Thus, in a nearly uniform universe of infinite volume, t h e r e is an infinite n u m b e r of repetitions of e a c h possible combination." I told him I w o u l d replace his last s e n t e n c e with, "Thus, in a n e a r l y uniform universe of infinite volume, t h e r e is an infinite n u m b e r of r e p e t i t i o n s of NEARLY e v e r y possible combination." Paul Nevai Department of Mathematics The Ohio State University Columbus, OH 43210-1174 USA e-mail: [email protected]

EDITOR'S COMMENT:Unless some f o r m of Occam's Razor is being invoked, I would replace the last sentence with, "Thus, in a nearly uniform universe of infinite volume, there is an infinite n u m b e r of repetitions of A T LEAST ONE possible combination."

But, Paul, is even that enough to escape feeling spooky? T h o u g h t s on M i l n o r John Milnor's p a p e r in The InteUigencer (vol. 19, no. 2, 30-32) is very related to [1-2]. The t y p e of c o n s t r u c t i o n r e p o r t e d by Milnor a p p e a r e d in [1]. Specialized to t w o dimensions, [1] p r o duces a r e m a r k a b l e uniformly continuous real function, f, defined on t h e plane, that is n o n d e c r e a s i n g in e a c h variable yet 1:1 on a Borel set w h o s e c o m p l e m e n t has p l a n a r Lebesgue m e a sure zero. The "paradox" w a s noted in [2], which p o i n t s out that w h e n f i s restricted to a square, the collection o f pre-images of singletons is a disjoint family of c o n t i n u a continuously filling out this square so that t h e r e is a subset of p l a n a r Lebesgue m e a s u r e 1, meeting each pre-image in exactly one point. Stressing t h e "paradoxical," reference [2] a d d s t h a t given any q (0 -< q -< 1) one can o b t a i n a set of p l a n a r Lebesgue m e a s u r e q by selecting exactly one p o i n t f r o m each o f the continua. Milnor's analysis nicely strengthens these results: N o w one can add that f can be strictly m o n o t o n e in each variable and that the p r e - i m a g e s of p o i n t s interior to the range o f f can be a continuous family of analytic curves r a t h e r than j u s t a c o n t i n u o u s family of graphs of uniformly c o n t i n u o u s functions of one real variable. The motivation in [1] for s e e k i n g f had to do with sufficient statistics. In 1-D the interesting question was: Does there exist a c o n t i n u o u s real function on an interval, 1:1 on a set of Lebesgue m e a s u r e 1, yet m o n o t o n e on no subinterval? E x i s t e n c e w a s r e p o r t e d in [2] by showing that c o m p o s i t i o n of f with a 2-D Brownian motion, b(t), yields a function with this p r o p e r t y with probability 1. Using Milnor's setup, with his fi in place o f f , j u s t c h o o s e the launch

(Continued on p. 64)

9 1998 SPRINGER VERLAG NEW YORK, VOLUME 20, NUMBER 41 1998

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A. SHfMONY Illustrations by JONATHANSHIMONY

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)pinior

Do We Do Mathematics with Our Visual Brain. Daniel J. Goldstein

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international matheraatical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-inchief, Chandler Davis.

This is not mathematical biology, nor even biological m a t h e m a t i c s - - i t is biology of mathematics! The author, as a biologist, undertakes to study us. He has concrete conjectures about our characteristic activity, and suggests experiments to test them. We don't have to sit still under his scrutiny, though: he invites us to look at his conjectures.--EDITOR'S NOTE athematics and mathematical inventions are an intrinsic characteristic of the human species. Mathematics has been practiced by people in different cultures, and is completely communicable between cultures and between individuals. Many theorems have been invented (discovered) by independent authors, either simultaneously or in different periods, and are understood by all mathematicians. The absoluteness and universality of mathematics, and its comparative independence of cultural influences, led Alain Connes to suggest that mathematics offers a more propitious terrain than the other sciences to study the brain. [Changeux and Connes] To the extent that mathematics is shared by the whole species, it may be explained in terms of the functioning of structures of the central nervous system that are common to all members of the species. The challenge, then, is to identify the neurobiological substrate of mathematics, i.e., the part(s) of the brain that make this universal capacity of the human mind possible; to discover the material bases for the comprehension and the invention (discovery) of mathematics. I propose that high-hierarchy modules of the human Visual system are involved in the invention (discovery) of mathematics, and suggest that humanity got this capacity as an epiphenomenon of the evolution of the visual system. If my hypothesis is correct, genetic mutations that led to architectural changes in the connectivity of

M

higher modules of the visual system allowed the unconscious, automatic capabilities of visual measurement and abstraction to become conscious and deliberate activities. This in turn allowed the construction of standardized tools and the practice of recordkeeping, and revolutionized social interactions by means of symbolic and naturalistic representation. If this is so, mathematical ability is a prototypical spandrel, i.e., an initially nonadaptative side consequence of evolution [Gould and Lewontin], [Gould]. With the traits initially selected for came others that allowed conscious measurement, abstraction, and iconic representation. This accident opened the possibility of visual symbolism and proto-mathematical manipulations. The mathematical spandrel allowed the possibility of transmitting abstract internal representations between the members of the group. These mutants enjoyed unexpected evolutionary assets because their protomathematical ability allowed them to communicate knowledge universally needed for the control of the physical environment, and with the course of time mathematical ability became itself a positive selective characteristic. As I elaborate the central conjecrive, other conjectures will spin off, some of them essential to its confirmation and some not. Mathematics and Vision The three canonical activities associated with the visual system are light detection, seeing (which implies abstraction and the generation of internal representations), and measuring. In animals, these activities are automatic and unconscious. In human beings these activities can to some extent be brought selectively to consciousness. The notions of distance and space are fundamental in some mathemati c s - m e a s u r e theory stands on the "common notion" of the length of a

@ 1998 SPRINGER-VERLAG NEW YORK. VOLUME 20, NUMBER 4, 1998

5

segment--but mathematicians in all fields tend to rely on ~isual cues. Mathematical lectures usually require a blackboard and/or other xdsual equipment; people "talking" mathematics usually write what they are communicating to each other on a writing surf a c e - f r o m sand to paper or blackboard. Mathematical notations ,are icons, mid mathenmtical talk and commentaries on mathematics abound in visual metaphors. Mathematics is often excruciatingly difficult to express in conventional language. Historical advances in mathematics depended on replacing pa~k:ular everyday languages by notations

and technical vocabularies with universal meaning. This provided icons which stand for universal concepts that can be apprehended, understood, used, and built upon by any human being, irrespective of culture, nationality, and social status. [Kline] Furthermore, the concepts (mid the problems inherent in the concepts) of beauty and tri~dality in mathematics and the x4sual arts are surprisingly similar. i If mathematics and plastic representation are neurally related, the proto-mathematical ability (dexterity in expressing abslractions related to measuring) must have emerged con-

comitantly with plastic ability (dexterity in expressing abstractions in pictures and statues). Both painting and matheinatics are conscious activities heavily laden with symbolic content. They may have arisen as originally non-selected consequences of the same changes in the visual system, and have become selective characteristics only later. Proto-inathematics and pictorial abilities, both grounded on visual cues, could both have preceded the acquisition of language. We would look for archeological evidence of standardized tools, calendars, inathematical artifacts, and iconic representation in the santo period of hunmn evolution. Blind M a t h e m a t i c i a n s

Yet there are hlind mathematicians. Their existence does not invalidate my hypothesis, but raises pertinent questions. Leonhard Euler lost his sight when he was 59. Blindness did not affect his mathematical productivity. His prodigious memory and immense nmthematical culture allowed him to keep doing mathematics until he died 16 years later. Nicholas Saunderson was blind since he was one yem"old. He also had a prodigious nlemory and was an extraordinary calculator; blindness did not prevent him learning Latin, Greek, French, algebra, and geomet~. Salmderson was admi~ed to the University of Cambridge in 1709 and appointed Lucasian Professor of Mathematics at Cambridge in 1711, where he taught mathematics and optics, ,and ~Tote a book on calctflus. For teaching, Saunderson used "regular solicks cut ill wood" and two special boards, one for demonstrating geometrical flmorems and the other for calculations, that highly hnpressed [)iderot. lyon Senden] [Morgan] We may suppose that there is a visual i m p r i n t i n g period during which the brain automatically and unconsciously acquires visual information which in turn imprints the processing of inputs coining from the other senses. A brain so modified call keep doing mathematics even in the absence of visual inputs. Those who lose sight ~1 argue this in more detail elsewhere. Daniel d. Goldstein, "Visualizing the Non-Visualizable." Submitted "or publication.

6

t i l MATI EMAT!CAL INTEL[ IGI NCER

as adults remember seeing (the case of Euler), but those who become blind in early childhood do not [Magee and Milligan], and Saunderson did not. His mathematical ability could be explained by assuming that he had an exceptionally efficient visual imprinting period that allowed the visual experiences of his first year of life to inform the other senses and provide the conceptual package needed for doing mathematics without further visual inputs. (Not all children are equally gifted in the realms of perception, and we all know about the precociousness of certain artists such as Picasso and Mozart. Saunderson, like most people that have lost sight before the age of two [Magee and Milligan], did not consciously remember what it is like to see, and probably did not have visual dreams. However, this does not necessarily imply the lack of unconscious visual imprints.) Euler and Saunderson lost their sight after seeing, but other people are born blind. If there are congenitally blind mathematicians, it would be particuiarly instructive to find how they work. Could they use modules of the visual system which had never treated visual information? Do they discover (invent) new mathematics, and perceive mathematics as visually pleasurable even though they lack an operative visual receptor system? Do they do mathematics in the same way, or has (say) reliance on tactile cues given them different spatial notions? Compare the literaalre on "the Molyneux problem"--whether blind people form mental images of shapes which serve if they recover sight. [von Senden] If, however, congenitally blind mathematicians are not found, I could not claim this as evidence supporting my hypothesis. Blindness is a major handicap. Nobody really knows how many potential mathematicians are among the semi-illiterate children of any urban ghetto, or the half of the population that happens to be women, or the congenitally blind. M a t h e m a t i c s That Is Said Versus M a t h e m a t i c s That Is "Seen" Like any other science, mathematics progresses in a tumultuous mixture of

intuition and proof. There is some consensus among mathematicians that invention (discovery) of new results, and their formal proof, are two distinct, different processes. The centrality of rigor in mathematical proofs is not a permanent fixture of mathematics. [Kline] Rather, "rigorous mathematics as a going concern is a rather rare phenomenon in the 4000-year history of mathematics (at least the way w e understand the term rigor)." [Kleiner and Movshovitz-Hadar] At any rate, the way mathematics is written is one thing, and quite another thing is the way it is created. Rota writes that the "description of mathematical proof ordinarily given is true but unrealistic"; a "realistic description" of mathematical proof would "bring to the open concealed features, [and] fringe phenomena that normally are kept in the background should assume their importance." He calls for the elucidation of terms such as understanding, depth, kinds of proof, degrees of clarity, as universally used in the shop talk of mathematicians [Rota]. Let me accept the language-like nature of proof, of formal checking of algebraic identities, and (of course) of saying or writing mathematics. In these senses, mathematics is a language, and a person that learns mathematics acquires a new language, and this acquisition means understanding--the exact opposite of what happens in Searle's Chinese room paradox. [Searle] It has recently been demonstrated [Kim, et al.] that native and second languages localize in contiguous but distinct regions in the frontal-lobe language-sensitive regions of the human brain (Broca's area). It would not be surprising to find that the understanding of mathematics as a language (which includes the capacity of logical proof) localizes also in Broca's area, adjacent to the native language sector. I predict, however, that a non-language-like intuitive appreciation of mathematics (which includes the perception of new hypotheses and unproved explanations) will be found to be localized outside the Broca area, in some of the higher-hierarchy modules of the visual system.

"Non-Visual" Vision Our visual system comprises two functionally independent and anatomically separate neuronal pathways, devoted to distinct functions: perception and the visual control of action. Most analysis of vision in humans goes no farther than its provision of a unified internal representation of the external world, yet the ultimate function of vision must be, as Milner and Goodale say, "to ensure an effective and adaptive behavioral output." [Milner and Goodale] I would seek the proto-mathematical spandrel rather in the "non-visual" part of the visual system, which I think developed independently of language. This hypothesis seems concordant with Holloway's estimate, based on the fossil record and comparative anatomy, that the major restrncturation of the brain in the pongid-hominid transition occurred 3 to 4 million years ago; this involved great reduction of the primary visual striate cortex, the "receptive" visual cortical area, and expansion of parietal cortical areas involved in muiti-modal processing of sensory information. This transition occurred, therefore, well before the reorganization of Broca's area in the frontal lobe and the doubling of overall brain size. [Holloway] Archeological Evidence My conjectures suggest that archeological evidence should be found indicating simultaneous occurrences of standardized tools, calendars, mathematical artifacts, and iconic representations early in human evolution. Cave paintings, the standardization of tools by shape and size (evident in the arrays of prehistoric instruments shown in any museum), and instruments for recording natural cycles, reflect the emergence of the abilities of naturalistic and symbolic representation, measuring, recording, and analytic evaluation of performance. (Although many animals--from wild crows to chimpanz e e s - u s e tools, tool manufacture-the imposition of form to an object, following a mental representation of the tool maker--is considered to be the hallmark of humanity.) The naturalistic depictions of horses,

VOLUME 20, NUMBER 4, 1998

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Figure 1. The black cow at Lascaux. (9 Editions d'art Albert Skira, Geneva. Used by permission,)

bisons, and lions---of which nearly onethird appear to be hit by arrows or javelins--and the schematic, symbolic rendering of human beings that cover the walls of the caves of Chauvet (near the estuary of the Rhone) and Cosquer (near Marseilles), indicate that ice-age hunters living 30,000 years before present (BP) had the same mental equipment as we. [Gombrich] Painting #63 of

the Lascaux cave (20,000 BP) is special (Fig. 1). The black cow stands on two rectangular surfaces subdivided into rectangular grids (Fig. 2). The meaning of these polychromic rectangles is unknown, but they show the artists' familiarity with parallel and perpendicular lines. [Batallle] The baboon fibula marked with 29 notches found in the Lebombo moun-

tains between South Africa and Swaziland, dated to 37,000 years BP, and the bone marked with 57 notches found in ex-Czechoslovakia dated to 32,000 years BP have been interpreted as tally-marks to ease counting. The 11,000-year-old carved bone found in Ishango, a little village near Lake Edward in Congo, seems to be something much more complicated than a measuring tally stick. [Huylebrouck] In these many forms we see people measuring things and time, and communicating abstractions through images. Conceivably they could transmit mythographic as well as pictographic content even to others not sharing an oral language. [Schefer] Much older are the Middle Pleistocene deposits discovered by Thieme in the open brown-coal mines at Sch6ningen, in Germany, including three 2-meter-long throwing-spears. [Thieme] [Dennell] These hunting instruments, found among stone tools and bones of (butchered) horses, and in the vicinity of something that could have been a hearth, reflect a high degree of development of manufacturing skills. These findings suggest that 400,000 years BP there already existed a highly organized enterprise for the production of standardized artifacts. If I am right in regarding this too as manifesting "mathematical-like" thinking, then the finds at Chauvet and Cosquer must be "recent" iconography, and we may hope one day to unearth "primitive" precursors: drawings from the Middle Pleistocene. W e r e W o m e n t h e First Mathematicians?

Figure 2. Unexplained rectangular grids under the black cow's feet. (9 Editions d'art Albert Skira, Geneva, Used by permission.)

8

THEMATHEMATICALINTELLIGENCER

Gerda Lerner eloquently argues that while in the distant past alien men were killed in conflict, women were retained as reproducers and slaves, suggesting that in patriarchal societies the first slaves were women. [Lerner] It is generally accepted that women invented agriculture and proto-genetics, and developed the practice of intentional production of food. Women may also have been the inventors and developers of proto-medicine, because they manipulated natural products and could, as a matter of course, have tried to apply them for healing purposes.

Agricultural production and stockpiling require conscious measuring for planning, forecasting, and accounting. These activities would have been practiced by women, if they acquired, developed, and culturally transmitted the biological knowledge about plants and their economic exploitation. It is tempting to speculate that the first proto-mathematicians (and perhaps the first artists) may have been women. Then women captives would have had value not only for reproduction and drudge work, but for technology transfer. M a t h e m a t i c s and t h e E v o l u t i o n of C o n s c i o u s n e s s

The evolution of consciousness--what I.M. Glynn calls William James's unresolved problem--is perhaps the most important open biological question in the study of the mind. In Glynn's words, "[I]t is not just that we have evolved brains that allow us to experience conscious sensations; we must also explain the correlation between the nature of those sensations and the survival value of the activities associated with them." [Glynn] The selective advantages of a highly sensitive, integrated, and (in some sense) error-correcting Visual system are obvious, for animals operating in a landscape of horizontal and vertical lines--horizons and trees--in which predators and food moved with stealth. [Hubel] [Mart] The unexpected was the emergence of mutants able not only to use their visual system for providing the "best current interpretation of the visual scene in the light of past experience" [Crick and Koch] but also to apply these powers to make largescale reductionist interpretations of reality. My hypothesis, which is consistent with the modular structure of the visual system, would fit into one of the tentative answers offered by Glynn to James's main question; namely, that consciousness evolved because "it is a necessary feature of processes selected because of the powers of analysis and discrimination that they confer." In visual awareness, the perceived information is conununicated to parts of the brain "that contemplate, plan

and execute voluntary motor outputs." [Crick and Koch] In mathematical awareness, the visual input is further abstracted and reduced to multilevel, symbolic representations (in the Crick and Koch sense) of general structures. Selection

I have been arguing that this chance occurrence of mathematical thinking and transmission in animals relying predominantly on their Visual system began as an unselected consequence of selection and only later became itself a selected marker. [E]very characteristic has additional properties besides those initially selected for. These properties---the unselected consequences of selection--create both possibilities and new vulnerabilities, and under altered circumstances those properties themselves can become the main object of selection. Furthermore, the evolutionary significance of a characteristic can change drastically from time to time .... In the extreme case, the impossible becomes first possible and then necessary. [Lewontin and Levins] Mathematicians have long criticized the traditional explanation of the emergence of geometry as the result of the practice of measuring terrains [Levi]. Even if geometry may have developed in the quest for precise measurements, the capacity for consciously thinking in geometric terms must have appeared much before the actual construction of "applied" geometry, when a human ancestor realized her capacity of expressing and transmitting the concept of measure that is innate to the visual system and had hitherto been unconscious. Mathematics, art, and religion have in common the use of visual symbolism and the design of icons. Neural mechanisms that allowed the emergence of the visiondependent abilities--computation, abstraction, symbolic representation, and graphic mimesis--might have become crucial to the species even before the emergence of language, by leading to an increase and diversification of behavioral complexity.

VOLUME 20, NUMBER 4, 1998

9

A Minimal Experimental Research Agenda The visual hypothesis of mathematics can be experimentally tested. 1. Lesions in higher modules of the human visual system, such as the V4 and V5 regions proposed by Crick and Koch (1995) as neural correlates of visum awareness, should impair mathematical thinking. 2. The positron emission tomographic analysis (PET scans) of brain function of sighted and blind mathematicians thinking about a mathematical problem should reveal activity in higherhierarchy modules of the visual system. On the other hand, if they are thinking about proving a known theorem then activity should show up in the Broca area. 3. Standardized tests to analyze mathematical understanding and the use of mathematical notions should be designed to complement the classical neurological anamnesis. These tests, applied to patients with brain tumors and cerebrovascular accidents, should lead to the identification of areas of the brain whose integrity is essential for the comprehension and elaboration of mathematics. If the present hypothesis is correct, the areas should be localized in modules of the visual system. Conversely, patients with known disorders of the visual sphere should be tested for their mathematical ability and their capacity for grasping mathematical concepts. 4. Mathematicians of different countries should be asked to volunteer for neurological examination and followup, to seek anatomical correlates of physical and supposedly psychological events that lead to the interruption of the practice of mathematics. Conclusion The identification and anatomical localization of the region(s) of the brain responsible for the human ability to understand and invent mathematics would lead into the scientific study of the neurobiological basis of consciousness. If mathematical intuition and proof could be understood in the context of the anatomy and the physiology of brain structures, this would be a start in understanding the functional

10

THE MATHEMATICAL INTELUGENCER

logic of the brain, a topic that is essentiaily unknown now. If mathematics is critically dependent on the circuitry of the visual brain, its universality is less surprising. It is the means evolution has allowed us for (in Galileo's metaphor) reading the book of nature.

Acknowledgments Cora Sadosky's question on the relation between neural structure and the universality of mathematics started me on this inquiry. BIBLIOGRAPHY Bataille, Georges, La peinture prehistorique Lascaux ou la naissance de l'art, Geneve: Skira (1986). Changeux, J.-P. & Connes, A. Matiere Penser, Paris: Odile Jacob (1955). Crick, F. & Koch, C. Nature 375:121 (1995). Dennell, R. Nature 385:767-768 (1997). Glynn, I.M. Biological Reviews 68:599 (1993). Gombrich, E.H. The New York Review of Books, 14 November (1996). Gould, S.J. Proceedings of the National Academy of Sciences, U.S.A. 94:10750 (1997). Gould, S.J. & Lewontin, R.C. Proceedings of the Royal Society, Series B 205:581 (1979). Holloway, Ralph L. Towards a synthetic theory of human brain evolution. In Origins of the Human Brain (Jean-Pierre Changeux and Jean Chavaillon, editors). Oxford: Clarendon Press (1995). Hubel, D.H. Eye, Brain, and Vision. New York: Scientific American Library (1995). Huylebrouck, D. The Bone that Began the

Space Odyssey. The Mathematical Intelligencer 18:4:56 (1996). Kim, K.H.S., Relkin, N.R., Lee, Kyoung-Min, and Hirsch, J. Nature 338:171-179 (1997). Kleiner, I. & Movshovitz-Hadar, N. Proof: A Many-Splendored Thing. The Mathematical Intelligencer 19:3:16 (1997). Kline, M. Mathematical Thought from Ancient to Modem Times, New York: Oxford University Press (1972). Lerner, G. The Creation of Patriarchy, New York: Oxford University Press (1996). Levi, B. Leyendo a Euclides, Rosario: Editorial Argos (1942). Lewontin, R. & Levins, R. The Dialectical Biologist, Cambridge: Harvard University Press (1992). Magee, B. & Milligan, M. On Blindness, New York: Oxford University Press (1995). Marr, D. Vision, San Francisco: W.H. Freeman (1982). Milner, A.D. and Goodale, M.A. The visualbrain in action, Oxford: Oxford University Press (1995). Morgan, M.J. Molyneux's Question: Vision, Touch and the Philosophy of Perception, Cambridge: Cambridge University Press (1995). Rota, G.-C. Indiscrete Thoughts, Boston: Birkhauser (1997). Schefer, J.L. L'Art Paleolithique, Les Cahiers du Music National de I'Art Modeme 59:5-33 (1997). Searle, J. The Rediscovery of the Mind, M.I.T. Press (1994). von Senden, M. Space and Sight: The Perception of Space and Shape in the Congenitally Blind Before and After Operation, London: Methuen & Co. (1960). Thieme, H. Nature 385:807-810 (1997).

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segment--but mathematicians in all fields tend to rely on ~isual cues. Mathematical lectures usually require a blackboard and/or other xdsual equipment; people "talking" mathematics usually write what they are communicating to each other on a writing surf a c e - f r o m sand to paper or blackboard. Mathematical notations ,are icons, mid mathenmtical talk and commentaries on mathematics abound in visual metaphors. Mathematics is often excruciatingly difficult to express in conventional language. Historical advances in mathematics depended on replacing pa~k:ular everyday languages by notations

and technical vocabularies with universal meaning. This provided icons which stand for universal concepts that can be apprehended, understood, used, and built upon by any human being, irrespective of culture, nationality, and social status. [Kline] Furthermore, the concepts (mid the problems inherent in the concepts) of beauty and tri~dality in mathematics and the x4sual arts are surprisingly similar. i If mathematics and plastic representation are neurally related, the proto-mathematical ability (dexterity in expressing abslractions related to measuring) must have emerged con-

comitantly with plastic ability (dexterity in expressing abstractions in pictures and statues). Both painting and matheinatics are conscious activities heavily laden with symbolic content. They may have arisen as originally non-selected consequences of the same changes in the visual system, and have become selective characteristics only later. Proto-inathematics and pictorial abilities, both grounded on visual cues, could both have preceded the acquisition of language. We would look for archeological evidence of standardized tools, calendars, inathematical artifacts, and iconic representation in the santo period of hunmn evolution. Blind M a t h e m a t i c i a n s

Yet there are hlind mathematicians. Their existence does not invalidate my hypothesis, but raises pertinent questions. Leonhard Euler lost his sight when he was 59. Blindness did not affect his mathematical productivity. His prodigious memory and immense nmthematical culture allowed him to keep doing mathematics until he died 16 years later. Nicholas Saunderson was blind since he was one yem"old. He also had a prodigious nlemory and was an extraordinary calculator; blindness did not prevent him learning Latin, Greek, French, algebra, and geomet~. Salmderson was admi~ed to the University of Cambridge in 1709 and appointed Lucasian Professor of Mathematics at Cambridge in 1711, where he taught mathematics and optics, ,and ~Tote a book on calctflus. For teaching, Saunderson used "regular solicks cut ill wood" and two special boards, one for demonstrating geometrical flmorems and the other for calculations, that highly hnpressed [)iderot. lyon Senden] [Morgan] We may suppose that there is a visual i m p r i n t i n g period during which the brain automatically and unconsciously acquires visual information which in turn imprints the processing of inputs coining from the other senses. A brain so modified call keep doing mathematics even in the absence of visual inputs. Those who lose sight ~1 argue this in more detail elsewhere. Daniel d. Goldstein, "Visualizing the Non-Visualizable." Submitted "or publication.

6

t i l MATI EMAT!CAL INTEL[ IGI NCER

GIAN-CARLO ROTA

Geometric Probability* 9 am very happy to be here before you as the Colloquium Lecturer for this year, and I feel ~

deeply honored to be given this great opportunity to share with you some of the mathematics we love.

As the Council of the A m e r i c a n Mathematical Society had d e c i d e d that the colloquium lectures m u s t deal with t h r e e i n d e p e n d e n t and u n r e l a t e d topics, t h e r e b y allowing any m e m b e r of the a u d i e n c e to skip one or m o r e lectures w i t h o u t missing anything, I have no possibility to m o d e l m y talks on any previous colloquium lecturers. I d e c i d e d to s c a n the list of p r e v i o u s colloquium speakers, b u t looking for n a m e s of m a t h e m a t i c i a n s w h o had n o t b e e n chosen for this honor. Sure enough, one n a m e was conspicuously missing: that of H e r m a n n Weyl. I h o p e you will forgive m e if I digress with s o m e personal reminiscences. In the fall of 1950, I enrolled as a f r e s h m a n at Princeton, having g r a d u a t e d a few m o n t h s before from the A m e r i c a n High School of Quito, Ecuador. The principal o f the A m e r i c a n High School of Quito w a s a Princeton graduate, a n d he s t e e r e d me t o w a r d P r i n c e t o n University. In N o v e m b e r 1950, I l i s t e n e d to m y first m a t h e m a t i c s lectures. These w e r e the t h r e e Vanuxem lectures, delivered by H e r m a n n Weyl and b e a r i n g the generic title "Symmetry." T h e s e lectures were an unforgettable experience. The lectures t o o k place in the old c h e m i s t r y auditorium, p a c k e d with an e x p e c t a n t public. The first lecture began with a lengthy quotation in Greek, which no one in the a u d i e n c e u n d e r s t o o d e x c e p t Luther Pfahler Eisenhart. This brilliant start was followed by a disp l a y of slides portraying charming w o m e n wearing the longb r i m m e d hats fashionable at the time, and later b y m o r e slides showing the A l h a m b r a and the Pentagon. Not a w o r d of mathematics. The a u d i e n c e was left wondering w h e r e such a sparkling display of "Kultur" was leading. Not m u c h m o r e mathematics was m e n t i o n e d in the s e c o n d lecture, w h e n m o r e slides were s h o w n of physics experiments, for which the lecturer p r o v i d e d a learned oral c o m m e n t a r y . Only in the last lecture did s o m e group t h e o r y m a k e a modest appearance. By that time, the audience, which h a d not dwindled, was enthralled with the subject and did n o t mind the fact that the s p e a k e r h a d said very little a b o u t mathe-

matics; actually, he had said very little a b o u t anything at all. What is m o r e remarkable, the a u d i e n c e s e e m e d to be thankful to the s p e a k e r for making the c o n t e n t s of the three lectures i n d e p e n d e n t of one another. As I recall this distant episode, I realize that the injunction a b o u t t h e independence o f the p r e s e n t colloquium lectures is a wise one, all the m o r e so w h e n the s p e a k e r is not H e r m a n n Weyl. The title of this lecture is "Geometric Probability." A definition of g e o m e t r i c p r o b a b i l i t y might run as follows: geometric p r o b a b i l i t y is the study o f invariant measures. Like all definitions, this does not tell us anything until w e are s h o w n s o m e typical examples, a n d t h e s e e x a m p l e s are the c o n t e n t o f this lecture. A b o u t 100 y e a r s ago, the p r o p e r t i e s that underlie s u c h notions as length, area, and volume, a s well as the p r o b a bility of events, w e r e a b s t r a c t e d u n d e r the b a n n e r of the w o r d "measure." Let us review the definition of measure, as w e will be using this def'mition in an unusual way. A m e a s u r e /~ is a function defined on a family of subsets of a set S, w h i c h t a k e s real values, n o t necessarily positive. The family o f sets on which a m e a s u r e is defined is closed u n d e r unions and i n t e r s e c t i o n s and contains t h e empty set. Let us t a k e a minute to review the axioms. A x i o m 1. ~(|

= 0,

where Q is the e m p t y set. A x i o m 2. If A and B are two m e a s u r a b l e sets, then /~(A U B) = / ~ ( A ) + / ~ ( B ) - / z ( A N B). The m e a n i n g o f this s e c o n d a x i o m is clear. The a x i o m states that m e a s u r e is additive. ]n particular, if we have two disjoint sets A and B, then

ff(A u B) = if(A) + if(B).

*The first of three Colloquium Lectures delivered at the Annual Meeting of the American Mathematical Society, Baltimore, January, 1997. The remaining lectures will follow in coming issues.

9 1998 SPRrNGER-VERLAGNEWYORK, VOLUME20, NUMBER4, 1998

11

More generally, for any finite family F whose m e m b e r s are sets and for which a n y two m e m b e r s are disjoint, we have

ei(xl,

en

We most emphatically do n o t assume that a measure is countably additive. The b e s t - k n o w n e x a m p l e of a m e a s u r e is the v o l u m e txn(A) of a solid A in o r d i n a r y n - d i m e n s i o n a l E u c l i d e a n space. The v o l u m e ~ ( A ) of a solid A satisfies Axioms 1 a n d 2, but A x i o m s 1 and 2 do n o t c h a r a c t e r i z e v o l u m e a m o n g all p o s s i b l e m e a s u r e s . It is p o s s i b l e to characterize v o l u m e a m o n g all m e a s u r e s by a d d i n g to Axioms 1 and 2 two a d d i t i o n a l intuitive axioms, n a m e l y the following:

A x i o m 3. The volume of a set A is i n d e p e n d e n t of the position of A. If a set A in n - d i m e n s i o n a l Euclidean space can be rigidly m o v e d onto a set B, t h e n A and B have the same volume. In other words, v o l u m e is invariant u n d e r the group of Euclidean motions. Finally, we must prescribe a normalization, as physicists say. This is done by taking a parallelotope P with orthogonal sides of lengths Xl, x2, 9 9 Xn, and setting

x 2 , . . . , X n ) = X l + X2 + "'" + X n ,

e2(xi, x 2 , . 9 , x ~ ) = x l x 2 + X l X 3 ~-

ioci,X2,

"'" ~- X n - l X n ,

. . . ,Xn) =

X 2 X 3 . . . X n ~- X l X 3 X 4 . . . X n

-~- . . . + X l X 2 " " X ~ _

D

e n ( X l , X2, 9. . , X n ) = X l X 2 . . . X n .

Observe an interesting coincidence. The last of these n symmetric functions is also the formula for the volume of a parallelotope. Axiom 4 c a n be rewritten as

A x i o m 4. /~n(P) = en(xl, x 2 , . . . ,

Xn).

Let us try an experiment, a n d replace the n t h symmetric f u n c t i o n by the (n - 1)st symmetric function. Let us first take n = 3, that is, three-dimensional space, so that we c a n better visualize what is occurring. Let us see w h e t h e r we can define a m e a s u r e on subsets of 3-dimensional space by keeping three of the above axioms, b u t by replacing the normalization Axiom 4 by using a n o t h e r symmetric function instead of e 3 ( x l , x 2 , x 3 ) which gives the volume. Let us first replace the symmetric function e3 by the symmetric function e2, t h e r e b y changing Axiom 4 to

A x i o m 4'. A x i o m 4.

~ t 2 ( P ) = X l Z 2 -t- X l X 3 + X 2 X 3. p~,~(P)

=

xlx2""x~.

These axioms, together with suitable continuity conditions, uniquely determine the volume of solids in Euclidean n-space. Starting from these four axioms, by a limiting process such as one finds in an a d v a n c e d calculus textbook, one establishes the fact that the v o l u m e of a ball S r of radius r in n - d i m e n s i o n a l space is given by the following formulas: 7rn/2rn

(n/2)!

Does this axiom define a measure? Of course, it does. The right-hand side is the formula for the surface area of the parallelotope P, divided by 2. Again, we will find in any a d v a n c e d calculus textbook the explanation of the fact that axioms 1, 2, 3, and 4', together with some continuity considerations, completely d e t e r m i n e an invariant m e a s u r e which is the surface area of solids in ordinary space. For example, the following w e l l - k n o w n formula for the surface area of a ball S r of radius r in three dimensions is o b t a i n e d from these axioms: ]_t2(Sr) -~ 4 7 r r 2.

if the d i m e n s i o n n is even a n d 2 n l r (n

~(s~)

=

1)/2((n- 1)/2)!r n n!

if the d i m e n s i o n n is odd. It is still widely believed that volume is the only invariant measure in Euclidean n-space. But, in p o i n t of fact, there are other invariant measures, defined on all reasonable subsets of Euclidean n-space, which have a notable geometric significance. My objective is to describe all such invariant measures. What h a p p e n s if we keep the first three axioms but tamper with the fourth axiom, the normalization axiom? Will we get something interesting, or will we get nothing new? To answer this question, we will appeal to the basic tools of combinatorial mathematics. The basic tools of combinatorial m a t h e m a t i c s are the elementary symmetric functions, to wit, the following polynomials in n variables:

'12

THE MATHEMATICAL INTELLIGENCER

Let us take the next step. E m b o l d e n e d by our success with two symmetric functions, we n o w replace Axiom 4 by yet a n o t h e r axiom, using a n o t h e r symmetric function. Let us set

A x i o m 4". ~I(P) = e l ( x l , x 2 , x 3 )

= xl

+ x2 + x3.

The n e w m e a s u r e ~1 will satisfy Axioms 1, 2, a n d 3, and, in addition, it will satisfy Axiom 4". The symmetric function of degree 1 plays the role that in the previous two examples was played by the other two symmetric functions. But wait a minute: is this definition consistent? To realize that the definition of the n e w measure/~1 is c o n s i s t e n t (i.e., that/~1 as defined by Axioms 1, 2, 3, a n d 4" really exists and is not a d r e a m of reason), look at two parallelotopes P1 and/)2 that have a face in common. The

first parallelotope has sides equal to xl, x2, a n d x 3 , and the second parallelotope has sides equal to Xl, x2, a n d y. The two parallelotopes have a c o m m o n face with sides equal to x l and x2. The m e a s u r e iLl(P1 U P2) of the parallelotope P1 U / ' 2 can be c o m p u t e d using the left side of Axiom 2, or using the right side, and the two c o m p u t a t i o n s had better yield the same answer; in symbols: tel(P1 U P2) = tzl(P1) + iLl(P2) - t~l(Pl A P2). Let us check this. The left side is c o m p u t e d by observing that the parallelotope P1 U P2 has sides equal to Xl, x2, and x3 § y. Therefore, Axiom 4" tells us that

/tl(P1

U P2) = x l + x2 + x3 § y.

Now, let us compute the right side. We have /zl(P1) = Xl + x2 + x3, Pal(P2) = Xl + x2 + Y, ~I(PI A P2) = Xl + x2, again b y Axiom 4" applied to P = P1 n P2, as o n e side equals zero when the parallelotope is a flat, (a rectangle). Therefore, the right side of Axiom 2 equals tZl(Pt) + /~1(P2) - ~I(P1 n / ) 2 ) = xt + x 2 + x 3 + x l + x 2 + y - ( x l

t ~ k ( P ) = e k ( X l , X2, . . . , a n ) ,

+x2)

= x~ + x2 + x3 + y,

and the two sides of o u r equations agree, t h e r e b y convincing us that the definition may well be consistent. Actually, the defmition of ttl(P) for a parallelotope P has a simple geometric interpretation. When multiplied by 4, it equals the perimeter of the parallelotope P (i.e., the s u m of the lengths of all the edges). Just as happens for volume a n d area, it can be s h o w n by continuity considerations that the m e a s u r e ill can be e x t e n d e d to all reasonable solids in ordinary space, for example, to all convex sets, a n d to all polyhedra, convex or n o n c o n v e x . But, one m a y object, ~ l ( P ) makes sense for a parallelotope P, b e c a u s e a parallelotope has a well-defined perimeter. What if A is a solid that does not have a well-defined perimeter, a sphere for example? The definition of the measure/~I(A) for such a solid flies in the face of c o m m o n sense. Einstein wrote: " C o m m o n sense is the residue of those prejudices that were instilled into us before the age of seventeen." C o m m o n sense must constantly re-adjust to reality. The n e w measure t~t that we obtain in this way is called the m e a n w i d t h , a m i s n o m e r that has b e e n kept for historical reasons. The m e a n width of a solid in space is completely characterized by axioms 1, 2, 3, and 4". In particular, it is invariant, that is, it does not depend on position. For example, the form u l a for the m e a n width of a sphere of radius r is comp u t e d to be /.~l(Sr)

volume a n d area. The third, the m e a n width, is at present almost totally u n k n o w n . I k n o w of n o person who has a n intuitive feeling for the mean width, similar to the intuitive feeling we have for volume a n d area. We await a possible application of the m e a n width. A potato grower knows that a potato's volume is important because it d e t e r m i n e s the nutritional c o n t e n t of the potato. The potato grower also knows that the surface area of a potato is i m p o r t a n t because it is r u m o r e d that the vitamins in a potato are c o n c e n t r a t e d in the skin. We m a y conjecture that as soon as the potato grower b e c o m e s aware of the m e a n width, he or she will find a nutritional interpretation of the m e a n width of a potato. I a m indebted to Steve Schanuel for this example. A similar kind of reasoning works in n dimensions. We discover n different invariant measures, each of them well defmed on all polyhedra and on all finite unions of compact convex sets. Each of the n elementary symmetric functions of n variables leads to the defmition of a new invariant measure which is a different generalization of the notion of volume. These n measures are called the i n t r i n s i c v o l u m e s . The intrinsic volumes are first defmed on an orthogonal polytope P whose sides equal x l , x2, 9 9 X n by setting

= 4r.

Thus, we see that in three d i m e n s i o n s each of the three e l e m e n t a r y symmetric functions of three variables leads to a n invariant measure that enjoys equal rights with volume. The first two of these m e a s u r e s are well known, n a m e l y

where e k ( x l , x 2 , . . . , X n ) is the kth e l e m e n t a r y symmetric function. One then proceeds to extend the defmition of the intrinsic v o l u m e s to more general sets, by a technique which we will shortly see. The intrinsic v o l u m e s are i n d e p e n d e n t of each other, except for certain as-yet-unknown inequalities a m o n g them. These inequalities generalize the classical isoperimetric inequality that relates volume to area. At present, we k n o w very little a b o u t the intrinsic volumes; they have not b e e n around for long a n d very little research has b e e n done on them. We do n o t even know the f o r m u l a for the intrinsic volumes of a n n-simplex. Now you are thinking: this is all fine a n d dandy, but how is the e x t e n s i o n of the intrinsic volumes from parallelotopes to more general sets carried out? And besides, isn't there any intuitive interpretation we can give the intrinsic volumes? I will a n s w e r both these questions simultaneously. Let us go back to three-dimensional space. You k n o w that the set of all straight lines in s p a c e - - n o t necessarily through the o r i g i n - - f o r m s a nice algebraic variety, called the Grassmannian. The group of all Euclidean rigid motions acts on the Grassmannian, and there is an invariant measure on the G r a s s m a n n i a n u n d e r the action of the group of Euclidean motions. This invariant measure is unique except for a constant factor. A similar statement may be made about the set of all planes, a n d more generally for the set of all linear varieties of dimension k in Euclidean space of dimension n. In the practice of mathematics, c o m p u t a t i o n with invariant m e a s u r e s o n G r a s s m a n i a n s is rare; most mathematicians w o u l d be hard put even to recall an explicit formula for the invariant measures on Grassmanians. Let us take a few m i n u t e s to get a feeling for the invariant measure on the set of all straight lines in 3-space. Let us call

VOLUME 20, NUMBER 4, 1998

"[,~

the m e a s u r e A3; the upper index 3 stands for threedimensional space, a n d the lower index stands for the dim e n s i o n of a line, namely, one. Consider a rectangle R placed a n y w h e r e in space, and consider the set of all straight lines that m e e t the rectangleR. Can we c o m p u t e the measure of this set of lines without knowing the formula for the invariant m e a s u r e on the G r a s s m a n n i a n of all lines in 3-space? Of course we can. A straight line m e e t s the rectangle R either at a point or not at all; 1 therefore, the value of the m e a s u r e of the set of all lines meeting R d e p e n d s only on the area/~2(R) of the rectangle R. If we take a n o t h e r rectangle R ' whose area is double the area of R, t h e n the measure of the set of all lines meeting R ' is double the measure of the set of all lines meeting R. Proceeding along these lines, we get to Cauchy's functional equation, and we infer that the m e a s u r e of the set of all straight lines meeting a rectangle R equals a constant times the area /~2(R). Because we are at liberty to choose a normalization of the measure, let us agree to set this c o n s t a n t equal to one. But instead of working with a rectangle, we could have worked with a n y p l a n a r figure C whatsoever, placed in an arbitrary position in space. The m e a s u r e of the set of lines meeting C equals the area/~2(C), by the same reasoning. I stress the a s s u m p t i o n that C must lie in a plane. To conclude: even w i t h o u t knowing the formula for the invariant m e a s u r e A3, we can nevertheless c o m p u t e the value of such a measure on certain sets of lines. Let us n o w take a more sophisticated set of straight lines. We take a set D in 3-space that is the u n i o n of disj o i n t sets C1, C 2 , . . . , Cn, where each of the Ci is contained in a different plane, a n d we ask for the m e a s u r e of the set of all straight lines meeting D. Such a c o m p u t a t i o n can be carried out, b u t it is a combinatorial nightmare, so m u c h so that we are forced to do what m a t h e m a t i c i a n s do when confronted with combinatorial nightmares: they change the problem ever so slightly. In this case, we take a hint from the way probabilists work. Let XD(W) equal the n u m b e r of times the straight line w meets the set D. Instead of computing a measure, let us compute the integral

f xo(~) dA~(~), where w ranges over the Grassmannian, that is, over the set of all straight lines in space. We will see that we can compute this integral without knowing the m e a s u r e Ant on the Grassmannian. Because D = 0

C~,

i--1

a n d the C,i are disjoint, we have

f xc~(~) gAnt(~o) ~2(ci) =

a n d therefore

f XD(w) dA3(w) = ~, ~2(Ci). i--1

What is this identity telling us? The right-hand side equals the area of the surface D. Nothing stops us from passing to the lintit and making the following assertion. Let E be "any" surface in space, and let XE(W) be the n u m b e r of times the straight line w meets the surface E. Then, the integral

f XE(W) dAnt(w), ranging over all straight lines w, equals the surface area of E. In probabilistic language: the average n u m b e r of times a r a n d o m l y chosen straight line meets the surface E equals the surface area of E. Let us n o w retrace our steps and repeat the same reasoning taking the set of all p l a n e s in space, instead of the set of all straight lines. The invariant measure o n this G r a s s m a n n i a n is denoted by A3, where, again, the u p p e r index stands for three-dimensional space, and the lower index for the dimension of a plane. Because a plane meets a straight-line segment either at a point or not at all,1 the same a r g u m e n t shows that the m e a s u r e of the set of all p l a n e s that meet a line segment L equals/~t(L), namely the length of the s e g m e n t L; more generally, if F is any curve "whatsoever" in space and if XF(w) equals the n u m b e r of times the plane w meets the curve F, t h e n repeating the a r g u m e n t we used for straight lines, we infer that the integral

f XF(w) dA32(w) equals the length of the curve F. The variable of integration w n o w ranges over planes, not over straight lines. Here, again, we compute an integral without knowing the measure. We are n o w very close to getting an intuitive interpretation of the m e a n width. Recall the parallelotope P with sides equal to xl, x2, and x3. To m e a s u r e the planes meeting the parallelotope P, we first consider a family of parallel planes, all sharing the same fixed unit normal u. In other words, consider the set of all planes parallel to the plane u% Without loss of generality, place the parallelotope in space so that one of the vertices of P is at the origin and so that the vector u lies in the octant of space opposite to the parallelotope P. (We can do this generically.) Denote the edges of P that meet the origin by Xl, x2, and x3. Given the fixed unit vector u and its fmnily of normal planes, let us take the curve F to be a path along the edges (line segments) [0, Xl],

i-1

But we have c h o s e n each of the sets C~ to lie in a plane, so that a straight line meets C~ either once or n o t at all. 1 It follows that

[xl, (xl + x2)],

THE MATHEMATICALINTELLIGENCER

[(Xl + x2), (Xl +

x2 + x3)],

in that order. A plane parallel to u ~- meets the parallelotope P if and only if it meets the curve F on the parallelotope at exactly one point. Therefore, the measure of the set of all planes parallel to u ~ that meet the parallelotope P is pro-

1Such statements are to be interpreted modulo a set of exceptional lines (or planes) which has measure O.

14

and

portional to the length of the curve F. Averaging over all unit vectors u (and hence, over all families of parallel planes), we conclude that the m e a s u r e of the set of all planes meeting a parallelotope equals the m e a n width of the parallelotope, except for a constant factor which we will again set to be 1. In view of this realization, we can immediately see how to define the mean width of any closed convex set: it equals the measure of the set of all planes that meet the convex set. Thus, we have s h o w n that the m e a n width may be extended to all closed convex sets in space. We are n o w in a position to give a probabilistic interpretation of the m e a n width of a convex set. Take two compact convex sets A and B in three-dimensional Euclidean space, and suppose that A is contained in B. Let m e begin by belaboring the obvious. Suppose that we take a p o i n t at r a n d o m belonging to the larger set B. What is the probability that the point belongs to the smaller set A? The answer is clear: such a probability equals the ratio of the volu m e of A to the volume of B. Instead of choosing a point at random, let us choose a straight line at random in space. Assuming that such a straight line meets the larger set B, what is the probability that such a straight line will also meet the smaller set A? We have already computed the answer to this question, albeit implicitly. Such a probability equals the surface area of the set A, divided by the surface area of the set B. You can tell what is coming next. We now take a random plane in space. Assuming that the plane meets the larger set B, what is the probability that it will also meet the smaller set A? The answer is the mean width of A, divided by the mean width of B. In Euclidean n-space, by m u c h the same reasoning we obtain interpretations of the intrinsic volume /~k(C) of a c o m p a c t convex set C as the G r a s s m a n n i a n measure of the set of all linear varieties of dimension n - k that m e e t the convex set C, and a similar probabilistic interpretation holds. What comes next? There are at least two questions still open. First, are there any other invariant m e a s u r e s besides the intrinsic volumes, and second, how can the definition of the intrinsic volumes be e x t e n d e d to more general subsets of n-space than c o n v e x sets. The answers to these questions are closely related. The answer to the first question is that we are missing one measure. To discover it, I will engage for a m i n u t e in the kind of m a t h e m a t i c a l reasoning that physicists find u n b e a r a b l y pedantic j u s t to show physicists that such reasoning does pay off. Let us ask ourselves the question: what is the value of the symmetric function of order zero of a set of n variables xl, x 2 , . . . , Xn, say eo(xu x 2 , . . . , Xn)? I will give you the ans w e r a n d will leave it to you to justify this a n s w e r after the lecture is over. The a n s w e r is, eo = 1 if n > 0 (i.e., if the set of variables Xl, x 2 , . . 9 x . is nonempty), and eo = 0 if the set of variables is empty. We are led to believe that there may exist a n invariant m e a s u r e in n-space associated with the symmetric function of order zero. We set t~o(C)

=

]

if c is any nonempty compact convex set, and, of cotarse,

~0(;27) = 0. Does such a measure exist? It does indeed, and the fact that it exists is, in my opinion, one of the most remarkable discoveries ever made in mathematics. We will prove that such a measure is well defined on any set which is a finite u n i o n of compact convex sets. We do this by employing a classical device borrowed from fimctional analysis: instead of defining a measure, we define a linear functional on all simple functions, that is, on all real functions f(~o) defined for r E R ~ which are linear combinations of indicator functions of compact convex sets. Let us first begin with the case n = 1; that is, let oJ range over points on the line. Define a linear functional X1 on simple functions as follows: Xl(.f) = Z ( f ( w ) - f ( w + ) ) , where the s u m ranges over all real n u m b e r s w. The meaning of the plus sign is best gleaned from an example. L e t f be the indicator function of the closed s e g m e n t [a, b]. Then, f ( w ) - f ( w + ) = 0 for all w except oJ = b, b e c a u s e we have f ( b ) = 1 b u t f ( b + ) = 0. Thus, we see that x l ( f ) = 1 i f f i s the indicator f u n c t i o n of an interval [a, b]. Now go over to n dimensions, proceeding by induction. Do n o t worry, this w o n ' t take long. Take a straight line L, and for every point w in L let H~ be the hyperplane through the point oJ p e r p e n d i c u l a r to the line L. I f f i s a simple function defined in n-space and if w is a p o i n t on the straight line L, let f ~ be the restriction o f f to the hyperplane H~o. Define a linear functional Xn as follows: ~Yn(f) = Z

(X:n-l(fo~) - Xn l(f~o+)),

where the s u m ranges over all points oJ o n the fine L. There is only a fmite set of w's for which the s u m m a n d is nonzero. When f is the indicator function of a n o n e m p t y compact convex set, then an argument similar to the preceding shows that x n ( f ) = 1. Thus, we m a y define a measure po(G) = x~(f), where G is any finite u n i o n of compact convex sets a n d f i s the indicator function of the set G. We have thus proved the existence of a m e a s u r e / ~ which is defined on all finite u n i o n s of compact convex sets and which takes the value 1 o n all n o n e m p t y compact convex sets. This measure has a long history: it is the Euler characteristic. Now you are thinking: if this is the Euler characteristic, then it is up to you to show that it coincides with what we ordinarily believe to be the Euler characteristic. Let me conclude this lecture by deriving the formula of Enler, Schlgfli, a n d Poincar6 for polyhedra. As a m a t t e r of fact, this formula can be encapsulated into a simpler formula, one that is easy to remember: Let C be a n o n e m p t y compact c o n v e x polytope of d i m e n s i o n n a n d let int(C) be the interior of C. Then, we have the following f u n d a m e n t a l formula for the Euler characteristic of int(C): /~o(int(C)) = ( - 1) n. Indeed, i f f i s the indicator function of the set int(C), we have p~(int(C)) = Z (X~-I (f~) - X n - l ( f ~ + ) ) , where the s u m ranges over all points w on the line L as above. But, by induction, we see that every term on the right-hand side equals zero, except w h e n ~0is the first point

VOLUME 20, NUMBER 4, 1998

'15

o n the line L for which the intersection C A H~ is not empty. If we is such a first point, then we have )dr,-t (f~e) = 0 because the point we is on the b o u n d a r y of C, and )dr, 1 (f~oe+) = ( - 1 ) n-1 by the induction hypothesis, because f~,e+ is the indicator function of the set int(C) rh H~,e+, which is the interior of a convex p o l y h e d r o n one dimension lower. Putting all this together, we obtain /xo(int(C)) = ~

(X.-t(f~e)

-

)dn-l(fwr

=

- ( - 1 ) n-1 = ( - 1)*', as desired. We are now in a position to state the famous Euler formula for polyhedra. What is a polyhedron? A polyhedron is a finite union of convex polyhedra. Given a polyhedron, we must define a system of faces (of all dimensions, ranging from dimension 0 (a point) to dimension n). We will say that a set F of convex polyhedra is a system of faces for an arbitrary polyhedron K when the elements of F, called faces, are nonempty compact convex sets F w i t h disjoint interiors such that g

=

h

int(F).

M a i n T h e o r e m o f Geometric Probability. The n + 1 in: t r i n s i c v o l u m e s P.o, b~l,. 9 9 t&~ are a basis o f the space o f all c o n t i n u o u s i n v a r i a n t m e a s u r e s defined on all f i n i t e u n i o n s o f compact convex sets. The first proof of this theorem is due to Hadwiger; the first e l e m e n t a r y proof was published last year by Dan Klain of Georgia Tech. In closing, let me try to a n s w e r the question you are a b o u t to ask: what has this got to do with geometric probability, anyway? I will attempt a sketchy answer. Consider two c o m p a c t convex sets A a n d B. We imagine B to be fixed in n-space and that we "drop" the rigid set A at random. What is the probability that A meets B? We a n s w e r this question in three steps. First, we realize that by keeping B fixed and varying A by the group of Euclidean motions, we define an invariant m e a s u r e on convex sets B. Second, we apply Hadwiger's t h e o r e m and infer that such an invariant measure equals a linear c o m b i n a t i o n of the n + 1 intrinsic volumes, with coefficients depending on A a n d n o t o n B. Third, we d e t e r m i n e these coefficients by taking suitable B's. The end result is an identity which is k n o w n as the kinematic formula, which has b e e n the object of m u c h research in this century, still going on today. T h a n k you for your attention.

F~F

Caution: The interior of a face of dimension k is to be taken relative to the linear space of dimension k that contains the face, and the interior of a point is that point. U n d e r these conditions, we may take the Euler characteristic of both sides, a n d (because any two interiors of faces are disjoint, so their measures add) we obtain po(K) = ~ . t~0(int(F)) = f 0 - f l + f 2 . . . .

+ '",

FCF

where f i equals the n u m b e r of faces of d i m e n s i o n i. This is Euler's formula. We can n o w a n s w e r the second of the questions left open: how to extend the definition of the intrinsic volumes from compact c o n v e x sets to all finite u n i o n s of compact convex sets. If G is such a finite union, t h e n we set

tZk(G) = f po(G A w) d)tn_k(W), where o~ ranges over all linear varieties of dimension n - k in n-space. The left-hand side defines a measure, and w h e n G is a c o m p a c t convex set, it agrees with the definition we have already given. It is therefore the desired extension. The Euler characteristic does all the work for us. I am n o w in a position to state the m a i n t h e o r e m of geometric probability. We will say that an invariant m e a s u r e t* on Euclidean n-space, defined on all finite u n i o n s of compact convex sets, is c o n t i n u o u s w h e n lint tz(Cj) = tz(C)

Cj -+C

for all sequences Cj of c o m p a c t convex sets converging to the compact convex set C. We have the

16

THE MATHEMATICAL INTELLIGENCER

BIBLIOGRAPHY

D.A. Klain and G.-C. Rota, Introduction to Geometric Probability (Lezioni Lincee), Cambridge: Cambridge University Press (1997).

9 ,'~ R I I , [ ~ , , P - ~ I

[.-1~. I I , * z , ] ,,~ , , = , m i ~ i l [:-i--1 M a r j o r i e

Oberwolfach, 1944-1964

This column is a foram for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of aU kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA; e-mail: senechal@minkowskismith'edu

Senechal,

Editorq

hese days, mathematical conferences take place on mountain peaks and in valleys, in the city and in the country, on every continent and in cyberspace, and research institutes spring up like mushrooms after a rain. But the Mathematisches Forschungsinstitut at Oberwolfach is still special. Idyllically and improbably nestled into the side of a mountain deep in Germany's Black Forest, it is host to 50 week-long research conferences each year, ranging over all the fields of modern mathematics. The participants, both senior and junior specialists, are invited by the Institute; they come from all over the world. The length and pace of the conferences, devices such as assigned seating at lunch and dinner and group hikes on Wednesday afternoons, and especially the Institute's buildings, are all designed to foster interaction and companionship among the Visitors. Week after week, Oberwolfach demonstrates its raison d'etre: the creation of mathematical communities and, thereby, mathematics. How did it happen that a project conceived in the ashes of the Third Reich became the very symbol of the international spirit of this most international subject?

T

Wilhelm $ i i s s As a preoccupied researcher, you may not even notice, much less meet, the staff that work year-round behind the scenes to make your visit possible: the Institute's Director, Professor Matthias Kreck; the administrative staff; the housekeeper and the people who work in the kitchen and maintain the grounds. The vision, dedication, diplomacy, hard work, and good luck that make Oberwolfach possible are themselves invisible. If you do ask how such a remarkable Institute exists, you will be told that almost everything about it

was the inspiration of Professor Wilhelm Stiss, the Institute's founder. Although modern steel and glass buildings replaced the Lorenzenhof--the "castle"--some 30 years ago, everything else, from the location and the informality of the conferences to the hikes and even the occasional evening concerts and Weinabende, seems to have been his idem Who was Wilhelm Stiss? The inscription on the bronze relief at the entrance to the Institute states only his name. It is noteworthy that none of the many articles describing how Stiss created the Institute almost single-handedly has very much to say about Siiss himself, although he was a productive research mathematician. This may be because Stiss was not only a researcher, he was also the President of the German Mathematical Society [Deutsche Mathematiker-Vereinigung, or DMV] and the Rector of the Freiburg University throughout tile Second World War. These roles eclipsed his research career and have themselves been eclipsed by the sensitive and complicated issues surrounding them. Stiss himself never spoke of the war years afterward, and some important documents concerning the DMV and Oberwolfach in the Nazi era have only recently been made available to scholars*. The picture of Stiss that will emerge from them will surely be complex. "Stiss 'did business' with the Nazis, but often in the interests of those despised by them," notes Sanford Segal [1], who is completing a major study of mathematics and the Nazis [2]. Oberwolfach, Stiss's greatest contribution to mathematics, was founded in difficult circumstances. Wilhelm Stiss was born in Frankfurff Main in 1895, and graduated from the Goethe-Gymnasium there in 1913. He began his university studies in

*With the agreement of their owners, the DMV and Freiburg University, Professor Kreck has deposited the papers in the archives at Freiburg University.

9 1998 SPRINGERVERLAGNEWYORK,VOLUME20, NUMBER4, 1998

17

Wilhelm S/iss.

Freiburg, and c o n t i n u e d t h e m in GSttingen and Frankfurt; w h e n in 1915 he w a s called into military service, he carried Hilbert's "Grnndlagen d e r Geometrie" with him. After the war, Stiss c o m p l e t e d his thesis, on polyhedra, under Ludwig Bieberbach, and then w o r k e d as B i e b e r b a c h ' s assistant in Berlin. In 1922 Stiss w e n t to Japan, w h e r e he s u p e r v i s e d the teaching of G e r m a n language a n d literature at Kagoshima University. He c o n t i n u e d

The Lorenzenhof.

18

THE MATHEMATICALINTELLIGENCER

his geometrical r e s e a r c h on the side; b y the time he r e t u r n e d to G e r m a n y in 1928, he had p u b l i s h e d 36 papers. (He w o u l d eventually publish 50 more.) In 1934 Stiss w a s called from Greifswald to Freiburg, w h e r e he was s o o n elected Rector and, n o t long aft e r that, President o f the DMV. Although he was a m e m b e r o f the Nazi Party, and ipso facto a Nazi l e a d e r by virtue of these offices, "after the war, m a n y m a t h e m a t i c i a n s c a m e to his defense (ultimately successful) in the denazification p r o c e d u r e s , to talk a b o u t his aid and rescue for colleagues in trouble" [1]. Stiss w o r k e d t h r o u g h o u t the w a r to k e e p the flames of m a t h e m a t i c a l res e a r c h alive in Germany, and the Institute that he founded at Oberwolfach was his principal means to that end. F o r the rest o f his life, Stiss was Oberwotfach and Oberwolfach was Siiss. "The w a y w e worked, the w a y w e lived together, the structure of the days, w e r e designed by Wilhelm Stiss. It is difficult to find the w o r d s to e x p r e s s it. We

think for e x a m p l e o f the casual lectures on the t e r r a c e in good weather, of his p e r s o n a l greetings and farewells to the participants, and the philosophical d i s c u s s i o n s walking on the p a t h s near the house" [3]. Oberwolfach retains Stiss's indelible s t a m p to this day. In 1957 Stiss w a s again e l e c t e d R e c t o r of Freiburg, but illness prevented him taking office. He died on May 21, 1958. In w h a t follows, w e will h e a r a b o u t O b e r w o l f a c h ' s founding and early dev e l o p m e n t from s o m e of the p e o p l e w h o p a r t i c i p a t e d in that story. The W a r Y e a r s in the Lorenzenhof Stiss's wife I r m g a r d (1894-1989) w a s the m a i n s t a y of Oberwotfach throughout its early years, s i m u l t a n e o u s l y playing t h e roles of hostess, housekeeper, a n d jane-of-all-trades. She w a s also a t a l e n t e d writer and an a m a t e u r artist of s o m e skill; h e r u n d a t e d m e m oir o f the early history of the Institute was p u b l i s h e d privately [4]. The fol-

lowing account consists of excerpts from this history. Irmgard Stiss was the daughter of an American mother and a German father; the English, including the punctuation and the use of the third person voice, is hers. During the first decade of its existence [1944-54], when its [permanent] location in the Black Forest was not yet definitely decided, the mathematicians called their institute simply by the name of the house, "Lorenzenhof". "Lorenzenhos just happened to sound dignified and vague enough for the name to be accepted as an inheritance from the valley farm. The present Lorenzenhof [i.e., the "castle," which was still extant when the memoir was written] was the hunting-lodge of one Baron Stoesser, Hessian Minister of State. Baron Stoesser did not come back from the world-war in 1918. A Belgian banker, a Mr. Hildesheim, bought the country-seat with its forests in 1928, leaving only the woods on the opposite side of the valley in possession of Baron Stoesser's heirs. The Hitler-period then broke out. A Black Forest dealer in timber, Mr. Rothfuss, could acquire possession. For some time he lived in it with his wife and five children, but later moved into the gardener's cottage, turning the big house with a surrounding area of land into money [by renting it out], and keeping the wide forests for his lumber-trade. The purchaser who entered the stage now, again corresponding to the course of history, was the State. The Baden Ministry of Education contemplated the purchase of Lorenzenhof. The question, too, arose whether it might be useful as a dependency, possibly to the Freiburg University. To the great disappointment of Mr. Siiss in his position as universityrector, however, he did not succeed in winning the place for Freiburg University. A training camp for Alsacien teachers, first for men, later for young girls, to be imbued with the principles of National Socialism, that is what the house had come to be.

Once again a new day dawned. The National Research Council (Reichsforschungsrat, NRC), was founded (1942/43) to organize science so as to win the war. "Their laboratory, their writing desk is the place where the gentlemen of the University belong", so Rust, Minister of Education in the Government in Berlin, told Stiss in a private talk. "When I, as assistant professor in Greifswald, saw you for the first time", Sttss replied, "it sounded different. You said 'March, gentlemen, march! ..... I had to speak like that," said Rust, "at that time, in order to save you. None of you imagined the size of the inuninent danger. There was such a storm of hatred against the intellectuals raging through Germany, universities would have been simply swept away if you had not got into line." What essentially mattered to Siiss was, firstly to be entitled to confer on mathematicians assignments describable as important for warfare. Siiss was even successful in having fundamental research being declared of military importance; this enabled the mathematicians to pursue their own mathematical problems. There was a second purpose though, in Siiss's activity. It's underhand aim was, to rescue and save for the dark German future scientific qualification and brain-potential as a capital fund for starting anew after the catastrophe. Siiss had the possibility of systematically fetching back from the front of any non-mathematical employment mathematicians of proven ability in research work. The fact that Siiss was in the exceptional position of being President of the DMV as well as Rector of Freiburg University for more than the usual number of years, now gave him the lever to get his project of an institute in motion. To Stiss, it went without saying that G6ttingen must be upheld as the German stronghold of mathematics. So, to him, quite evidently G6ttingen was the given place to establish the institute. [But] the connection with G6ttingen was destroyed in the universal ruin. Michel Fuhs [a high school teacher

in the Baden Ministry of Education] came out with the idea what Stiss would think about Lorenzenhof if one could wrench it from political Party-schooling. Since Stiss already knew this house from the time of its purchase, this proposition quickly decided its future. On September 1, 1944, the transformation of the maidens' boarding school into a place for mathematical work could be started. It was most pleasant that Prof. Hellmuth Kneser appeared on the scene to have a look at the developing Institute. That meant help at the right moment, for it was rather hard for Mrs. Stiss and Hilo [the Stiss's daughter] alone to dismount the numerous beds in the dormitories and to distribute them into all the different rooms. But being three working together, it was fun. The order, issued from Berlin regarding removal of university libraries to air-raid proof places, applied to Strassburg too. Oberwolfach was designated as an appropriate place for safely stowing away the Strassburg mathematical library. In the beginning there was a general coming and going between Freiburg and Oberwolfach. The destruction of the town and university of Freiburg, on November 27, 1944, deprived them [the mathematicians at Freiburg], at one blow, of their working possibilities there. They all took refuge under the roof of Lorenzenhof, bringing their assistants and secretaries with then], rendering superfluous further plans about configuration of the Institute. Of his own responsibility, Mr. Siiss offered a home there as well to his French colleague Roger, who otherwise would have had to return to a prisoner's camp. The well-furnished library in the house afforded the best conditions imaginable for study in war-time. Even though no one believed in the pretended meaning of his work as labeled of importance to the war, to each of the members, keeping science going was a valuable activity beyond doubt in every other respect. Here was the tranquillity they needed, and even if the air planes

VOLUME 20, NUMBER 4, 1998

19

John Todd today.

d r o n e d above, p a s s i n g b y for s o m e air-raid in the evening dark, there w a s s c a r c e l y any n e e d to hurry into the cellar. The food supply, o f course, w a s one o f the m o s t difficult p r o b l e m s . It c a u s e d us considerable a n x i e t y to h e a r that the Party Staff had m o v e d from Strassburg to Rippoldsau. To have t h e m as neighb o u r s was m o s t alarming. One still h e a r d of e x e c u t i o n of 'defeatists'. The dreaded day came, when the army wanted to o c c u p y the house. But Mr. Stiss upheld the view that he was not yet entitled to give up the Institute. Dissolution was avoided by a halr's breadth . . . . As a matter of fact, the passing of the front brought a loss of scarcely a fortnight to science. It reminds one of Archimedes' "noli turbare", seeing Mr. Seifert start a series of lectures again on April 26th. But it r e m a i n e d uncertain what the occupation-army would decide to do. O b e r w o l f a c h is S a v e d

Irmgard's Sfiss's c o n c e r n for w h a t the "occupation-army" might do was not unfounded: O b e r w o l f a c h w a s in the F r e n c h o c c u p a t i o n zone, and barely esc a p e d being requisitioned. But an err o r on the p a r t o f British Intelligence led to O b e r w o l f a c h ' s d r a m a t i c rescue b y the m a t h e m a t i c i a n J o h n Todd. In World War II, Professor Todd, a native of Ireland, served in the British Navy. Although his official title is now

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THE MATHEMATICALINTELLIGENCER

Professor Emeritus (after a long career We first went to Magdeburg [Reuter's as Professor of Mathematics at the birthplace], and then we t r a v e l e d from n e a r b y California Institute o f Technol- p l a c e to p l a c e and we e n d e d up at ogy), his home in Pasadena, stuffed with Oberwolfach. mathematics books and papers, belies The G e r m a n m a t h e m a t i c i a n s w e any suggestion that its o c c u p a n t has re- met k n e w t h e y h a d not b e e n able to tired. In the following transcription of keep up with the latest m a t h e m a t i c a l our conversation (October, 1997), I in- d e v e l o p m e n t s . For example, in tersperse, with Professor Todd's per- GSttingen w e found Wilhelm Magnus, mission, some remarks from his pub- w h o w a s trying to do good, acting as a lished reminiscences [5]. sort o f i n t e r p r e t e r in the p r i s o n e r o f S e n e c h a l . How did you f i r s t learn w a r camps. When I m e t him he w a s tryabout Oberwolfach? ing to s t o p a fight and he was all bloody. T o d d . As the war w a s d r a w i n g to an And so I a s k e d him, "What can I do for end, plans were m a d e to collect infor- you?", thinking I could give him s o m e m a t i o n and scientists, e.g., in r o c k e t r y cigarettes or something, b e c a u s e I and atomic weapons. A m o n g the tar- don't smoke, you see. And he said, gets was a Dr. Hellmuth Walter, an ex- "Send m e Mathematical Reviews!" I p e r t on r o c k e t engines. It c a m e to p a s s thought this might be difficult, so I said, that Dr. Alwyn Walther w a s b r o u g h t to "Can w e c a p t u r e you?" and w e h a d him L o n d o n b y mistake. When the brought to London for interrogation. A m e r i c a n s i n t e r r o g a t e d t h e "Walter" We lived quite close, so he c a m e to o u r t h e y had c a p t u r e d a b o u t gas dynamics, house, in principle to be i n t e r r o g a t e d he w o u l d n ' t reply to them. He said he by me, b u t also to r e a d Mathematical d i d n ' t k n o w - - b e c a u s e he d i d n ' t ! - - b u t Reviews. S o m e b o d y drove him a n d I t h e y thought he w a s s e c u r i t y con- signed the receipt for him. Later he scious. My wife Olga [Taussky Todd], came to NYU [New York University[. w h o was from Vienna, w a s brought in Anyway, Reuter and I r e a c h e d to interview him. She i n t e r r o g a t e d him Lorenzenhof, in the Black F o r e s t a b o u t mathematics in G e r m a n y and he a b o v e Wolfach, in the first w e e k of July told h e r about Oberwolfach. She told 1945. We w e r e e x h a u s t e d after being m e and I told my colleague Harry on the r o a d for a b o u t a m o n t h a n d Reuter, and so, very quickly, we de- a s k e d if w e could rest there for t h e c i d e d to go there. This w a s at the end w e e k e n d . F o r safety, we p o s t e d noo f the war. We p r e p a r e d a p r o p o s a l to tices on the m a i n entrance to the effect o u r superiors to investigate the situa- that the building w a s u n d e r the protion of mathematics in G e r m a n y - - w e tection o f the British Navy! w e r e c o m m i s s i o n e d in the N a v y - - a n d S e n e e h a l . Did you know Professor w e w e r e given open orders: w e could Si~ss ? do anything that s e e m e d interesting [6]: Todd. I h a d h e a r d o f him before. I w a s i n t e r e s t e d o n c e in the F o u r Vertex By C o m m a n d of the C o m m i s s i o n e r s Theorem, on w h i c h he had w o r k e d . I for executing the Office o f Lord High didn't k n o w him personally, b u t I k n e w A d m i r a l of the United Kingdom, &c. that much. S e n e e h a l . What was the Lorenzenhof To Lieutenant C o m m a n d e r J o h n like when you arrived there? Todd R.N.V.R. Todd. It w a s a castle, a hunting lodge. The Lords C o m m i s s i o n e r s of the The thing I r e m e m b e r is that every Admiralty h e r e b y a p p o i n t you morning at b r e a k f a s t you h a d to s h a k e T e m p o r a r y Lieutenant C o m m a n d e r h a n d s with e v e r y o n e before eating. Of RNVR (Special Branch) o f His course, t h e y h a d trouble getting raMajesty's Ship PRESIDENT additions, b u t t h e y k e p t a lot of hens, it's tional for duty outside A d m i r a l t y with in the country, and it had a nice r o s e D i r e c t o r of Scientific R e s e a r c h (for garden a r o u n d it. time only). Your a p p o i n t m e n t is to The n e x t day [after we arrived], t a k e effect l l t h J u n e 1945. Reuter w e n t off to Heidelberg to fill By C o m m a n d of their Lordships, our gas t a n k a n d get rations, and I w a s H. V. Markham. left alone. We w e r e having a d i s c u s s i o n

in the rose garden w h e n t h e r e a r o s e a c o m m o t i o n among the servants. It was c a u s e d b y a foraging p a r t y of M o r o c c a n t r o o p s w h o w a n t e d to occ u p y the building. I quickly got into p r o p e r dress with h a t a n d in m y b e s t F r e n c h p e r s u a d e d t h e m to leave the mathematicians and "m6me les poules" undisturbed. We s t a y e d a few days, and then I s a w the local military g o v e r n o r who c a m e in, and I per-

s u a d e d him that this should b e left alone. T h e y w o u l d have b u r n e d the books, y o u see, for fuel. It w o u l d have b e e n a disaster. I told the military gove r n o r t h a t they h a d b e f r i e n d e d s o m e F r e n c h m a t h e m a t i c i a n s who h a d b e e n p r i s o n e r s of war, had got t h e m o u t of the c a m p s and p u t t h e m in this place, and t h e r e w a s nothing military being done there. And w h e n I w e n t b a c k to England after this visitation, I con-

t a c t e d the F r e e F r e n c h scientific delegation, which included J a c q u e s H a d a m a r d and Szolem Mandelbrojt (the uncle of the fractal Mandelbrot). Later ! w e n t to Paris, saw Mandelbrojt, and the p r o t e c t i o n was m a d e official. S e n e e h a l . A n d so it w a s that you bec a m e "The S a v i o r o f Oberwolfach." T o d d . Yes. This is my b e s t w o r k for mathematics. S e n e c h a l . A n d after that, it s e e m s that the r e c o n n e c t i o n w i t h the rest o f the world w a s m a d e as i f n o t h i n g had ever happened! H o w did things get p u t back together so quickly, w i t h o u t hard f e e l i n g s ? T o d d . People w a n t e d to go b a c k to mathematics, in w h a t e v e r way t h e y could do. M a t h e m a t i c s c a m e first. F o r exan]ple, Olga w a s Jewish, and in G e r m a n y b e f o r e the war, before t h e Nazi main activities, and also "after the war, m a n y of t h e Nazi people, like Bieberbach, and tiasse, and m a y b e others, w e r e v e r y kind to her. She w a s always s u r p r i s e d at that; you might have e x p e c t e d t h a t they w o u l d n ' t have talked to her. S e n e e h a l . D i d she have m i x e d feelings about t h e m ? T o d d . No, this w a s mathematics. On a p e r s o n a l level t h e r e w e r e feelings, a s one w o u l d expect, but on a professional level, p e o p l e w a n t e d m a t h e m a t ics to c o m e first. S e n e c h a l . When w a s the f i r s t t i m e y o u returned to Oberwolfach f o r a m a t h e m a t i c s conference? T o d d . Very soon. The visit I have b e e n describing w a s in 1945, w h e n the w a r was over. We left for A m e r i c a in '47, to c o m e to the National Bureau o f Standards, then w e w e n t b a c k to London for a y e a r . . . . certainly w e w e r e in O b e r w o l f a c h during the y e a r 1948-49. S e n e c h a l . You have seen O b e ~ o l f a c h develop over the years. Did you t h i n k that they w e r e m a k i n g the r i g h t choices as they w e n t along? T o d d . Yes, it w a s the natural thing to do. If you have a c o n f e r e n c e in New York, or London o r Los Angeles, people always have relatives there or o t h e r things t h e y w a n t to do. But at Oberwolfach you are so isolated, the only thing y o u can do, is do mathematics and w a l k around.

VO-UME 20, NUMBER 4, 1998

21

From I n s t i t u t e to I n s t i t u t i o n Oberwolfach, as w e k n o w it today, t o o k shape gradually in the p o s t - w a r years.

Excerpts from an unclassified Technical Report by F. J o a c h i m Weyl, 1952 [7]. The S y m p o s i u m on F u n c t i o n Theory, h e l d during the last w e e k o f October, 1951, at the Mathematical Research Institute, Oberwolfach, was the last of a series of similar meetings on Geometry and Algebra which had taken place in the course of the summer. It was attended by about 45 mathematicians who came from German and Swiss universities. Also F r e n c h representatives had been invited but were unable to come because of belatedly scheduled fall examinations . . . . The part played by the mathematical Research Institute at Oberwotfach in the revival of German mathematical life during the post-war years is considered of sufficient interest to warrant a brief outline of the Institute's history. This will at the s a m e time lead to a better understanding of its c h a r a c t e r and m o d e of o p e r a t i o n as reflected in the proceedings of the colloquium itself.... In spite of the vicissitudes which b e s e t the Institute during the winter of 1944/45, it b e c a m e a kind of mathematical Shangri-La, and the log b o o k of t h e s e m o n t h s s h o w s that rather carefully p r e p a r e d , yet informal presentations, a kind of shirtsleeved colloquia given by individual m e m b e r s of the current work, constituted an early developed group activity. . . . F o r the next few months [after John Todd's visit] and, as a matter of fact, for most of the following two years, the Institute at Oberwolfach b e c a m e a refuge for mathematicians in distress. The first to arrive were the German mathematicians who had worked, or even held a position, at the University of Strassburg; later, refugees from East German universities c a m e w h o had either fled on their own or had b e e n brought to H e i d e n h e i m and apparently forgotten; finally also a num-

22

THE MATHEMATICAL INTELLIGENCER

b e r of m a t h e m a t i c i a n s from the Western Zone sought refuge t h e r e for longer or s h o r t e r periods, w h e n the misery of the t i m e s a n d the rigors of denazification p r o c e d u r e s p r o v e d too much for them. The pattern was always the same. They arrived in various states of physical and psychological exhaustion, were given such food and shelter as was available, and encouraged to particip a t e in the work. Companionship and the detached a t m o s p h e r e re-awakened their interest in mathematics, and they left grateful and r e a d y once more to resume their o w n research. With but rare exceptions on political grounds, this hospitality was extended to many who h a d b e e n known as Nazis, a fact which has occasionally been criticized. There is no reason to doubt, however, that this was done in a spirit of toleration and personal loyalty rather than out of solidarity with the views that these people had professed . . . . The y e a r 1947/48 w a s a quiet one at the Lorenzenhof. In the meantime, however, the t r a n s i e n t presence of refugees and FIAT-Review writers alike had t r a n s f o r m e d the shirt-sleeve colloquia of the established group into shirt-sleeve symposia, held on topics o f their selection b y those who h a p p e n e d to be present. The log book, c o n t i n u e d in c o m p l i a n c e with the r e s e a r c h control act, gives e v i d e n c e that the characteristic style o f p r e s e n t a t i o n , inviting group p a r t i c i p a t i o n - - w e l l p r e p a r e d yet showing clearly all loose e n d s , - - h a d on the w h o l e b e e n preserved. Out of this tradition has grown the activity of the Institute during the last three years, in the c o u r s e of w h i c h it has b e e n the m e e t i n g p l a c e for numerous gatherings, organized around one theme or another, either of a mathematical or regional character. The series began with a meeting of French and G e r m a n mathematicians in the s u m m e r of 1948 for whose organization the assistance of both the French Military Government and the Land Baden had been enlisted. Fifteen of the Bourbaki (a guild-like association of younger

F r e n c h mathematicians interested in m o d e r n abstract mathematics) and fifteen German mathematicians of corresponding interests could be invited with expenses paid, and a dozen m o r e attended without financial assistance. Since then, conferences under such titles as "Topology" (Spring, 1949), "Young Mathematicians' Meeting" (Autumn, 1949), "Mathematics around the Rhine" (Autumn, 1950), and this summer's series have brought together at Oberwolfach mathematicians from the surrounding countries, generally with s o m e assistance from the French authorities who have helped not only financially but also in matters of victuals and transportation. In order to insure the presence also of promising young people at these gatherings, the invited outstanding scientists all receive a certain n u m b e r of blank invitations allowing them to bring along their best assistants. International c o n t a c t s are cons c i o u s l y furthered not only in the scientific p r o c e e d i n g s b u t also b y s u c h m e a n s as the a s s i g n m e n t o f roommates .... U n d o u b t e d l y the F r e i b u r g mathe m a t i c i a n s and especially P r o f e s s o r Stiss will m a k e every effort to pull the O b e r w o l f a c h Institute t h r o u g h for a n o t h e r y e a r by having it serve as a m e e t i n g p l a c e for m a t h e m a t i c a l conferences, in the hope of ultim a t e l y securing for it s o m e kind o f p e r m a n e n t existence in t e r m s o f a r e e s t a b l i s h e d r e s e a r c h activity . . . . At the s a m e time the i d e a of small t o p i c a l conferences, as e x e m p l i f i e d by the s y m p o s i u m on function theory u n d e r review, has p r o v e d a very fruitful one, and such m e e t i n g s w o u l d certainly be c o n t i n u e d even if the Institute should a s s u m e again a m o r e p e r m a n e n t character. The death of Stiss, in 1958, r o b b e d the Institute not only of its founder but of a skilled and inventive diplomat who had s o m e h o w m a n a g e d to find the funds e a c h y e a r to keep it going. But such improvisation was no longer desirable, even if it were possible. In 1959, a group of 15 mathematicians f o r m e d the Mathematical Research Society

(Gesellschaft ftir mathematische Forschung) to provide a legal basis for the Mathematisches Forschtmgsinstitut Oberwolfach [8]: The a i m of the Gesellschaft fiir mathematische F o r s c h u n g is to p r o v i d e rsearch facilities for m a t h e m a t i c s in G e r m a n y similar to those prov i d e d b y various institutions The Institute transfigured. in o t h e r countries and following the tradition of S e n e c h a l . How did they do that? m a t h e m a t i c a l c o m m u n i c a t i o n and B a r n e r . They invited them to small cone x c h a n g e s t a r t e d b y Wilhelm Stiss ferences. This began around 1947. You in Oberwolfach. The Society shall see, G e r m a n m a t h e m a t i c i a n s had had form a center of scientific c o o p e r a - no chance for the last four years, if not tion among m e m b e r s of different twelve years, for c o n t a c t with t h e ingenerations and facilitate the ex- t e r n a t i o n a l c o m m u n i t y of m a t h e m a t i change of ideas with foreign re- cians. They n e e d e d to do s o m e t h i n g a b o u t their l a c k of lmowledge. The persearchers. son from the Bourbaki group w h o was The Volkswagen Foundation and the lo- the central contact was [Jean-Pierre] ca] government (now the State of Baden- Serre. He was very young then. Stiss Wiirttemberg) pledged, and still provide, first got s o m e money from Karlsruhe most of the Institute's financial support. for the old castle; later he got s o m e Martin Barner was the Director of m o n e y from Adenauer [the post-war O b e r w o l f a c h from 1964 to 1994 and Chancellor of the Federal Republic of P r o f e s s o r of M a t h e m a t i c s at Freiburg Germany] and he met some p e o p l e in Baden-Baden. At that time there were University; under his leadership, the He was a great Institute evolved into the international two local states . . . . c e n t e r that it is today. Ludwig Danzer, diplomat. He also had a discretionary fund, but that was just enough to have P r o f e s s o r Emeritus at D o r t m u n d University, was a "pre-doc" assistant to two or t h r e e conferences. Every y e a r it Stiss from 1956 to 1959. We s p o k e at was a question of whether the Institute would survive. When Stiss died in 1958, O b e r w o l f a c h in May, 1997. the p r o b l e m was not keeping the castle, S e n e e h a l . How did Oberwolfach it w a s w h e t h e r we could really use it, make the transition f r o m its associaw h e t h e r t h e r e w o u l d be m o n e y to ortion w i t h the w a r to a highly respected ganize conferences. There was no forinternational center of mathematical mal contract, no security. research? S e n e c h a l . How did Oberwolfach beB a r n e r . That was Siiss's idea. He had a come truly international, on a scale chair at Freibttrg, and in that position greater than regional? he tried to have c o n t a c t s at Strasbourg, Nancy, and Basel. Ehresmann, at B a r r i e r . The i d e a was there f r o m the Strasbourg, w a s one of the key people. start, a n d it grew in a natural way, as T h e y had good relations with Swit- the f r e q u e n c y of the c o n f e r e n c e s inzerland and they began to make con- creased, a n d also the n u m b e r s of participants. There were never any rules tact with Bourbaki also.

about the percentage of p a r t i c i p a n t s from one c o u n t r y or another. We j u s t invited the leading p e o p l e in the field of e a c h specialty. Senechal. Was there a reluctance on the part of J e w i s h mathematicians to come, at first? B a r n e r . We n e v e r m a d e distinctions among mathematicians, b u t I did try to r e a c h out to Jewish mathematicians, especially those w h o had emigrated from Germany. I helped t h e m to m a k e c o n t a c t s again, a n d tried to att r a c t t h e m to Oberwolfach. S o m e t i m e s they didn't w a n t to come; that is understandable. However, once they did c o m e to Oberwolfach, t h e y p r a i s e d it, as e v e r y b o d y did. S e n e c h a l . Had Si~ss thought of the Institute as a place where people would come to stay f o r long periods, or was it always intended to be a conference center? B a r n e r . His i d e a w a s to have s o m e p e r m a n e n t p e o p l e here doing research. But he n e v e r got the n e c e s s a r y m o n e y for that, so t h e r e were c o n f e r e n c e s m o s t of the time, e x c e p t at the end o f the war. Maybe he d r e a m e d of s o m e thing like the Institute for A d v a n c e d Study in Princeton, b u t there was no chance to realize it. S e n e c h a l . So Oberwolfach became a conference center f o r f i n a n c i a l reasons. Somewhere along the line, did people realize that this was a good thing in itselj? D a n z e r . I think it w a s a c o n t i n u o u s development. When I c a m e here in 1956, as an assistant to Stiss, we had 12 conferences. When I left in 1959 there w e r e already 20 conferences. But it only became a plan in 1964, w h e n P r o f e s s o r Barner b e c a m e director; that was his achievement. B a n t e r . I refused to b e c o m e director unless there would be a long-range plan and a budget. And the first thing I did

VOLUME 20, NUMBER 4, 1998

23

able to me, to Christoph Scriba and Sanford Segal for information a b o u t Stiss, to J a m e s Callahan, Matthias Kreck, Doris Schattschneider, Christoph Scriba, and Sanford Segal for their c o m m e n t s on various drafts of this article, and to Ludwig Danzer for assistance in w a y s too n u m e r o u s to mention.

REFERENCES

Diners are assigned places at the table.

was to build the residence hall. Before that, people had to stay in the castle. There were only b e t w e e n 20 and 25 beds. D a n z e r . W h e n e v e r t h e r e w a s a conference, that w a s a job! I h a d to go d o w n to the village to the hotels and tell t h e m w e w e r e e x p e c t i n g five m o r e people, and s o m e t i m e s I w o u l d even have to go to the inhabitants of Oberwolfach to a s k if t h e y h a d a r o o m for a guest, a n d t h e n if the guest didn't c o m e . . . . So that w a s a p r o b l e m until the r e s i d e n c e hall w a s built. Barrier. Later w e d e c i d e d to replace the castle with this n e w building. The m a t h e m a t i c i a n s w e r e o p p o s e d to this, b e c a u s e they w e r e in love with that old castle. But it w a s n e c e s s a r y to r e p l a c e it. The castle w a s d a m p in the winter. It had b e e n built as a s u m m e r residence originally. D a n z e r . The m a i n lecture r o o m in the n e w building (which o p e n e d in 1967) was e r e c t e d p r e c i s e l y at the s a m e place as the old lecture r o o m had been; even the p l a c e m e n t of the b l a c k b o a r d is the same, on the east walt. The spirit of this building is p r e c i s e l y the spirit of the old building, although it looks quite different. Very s o o n everyone agreed that it was wonderful. The architect (Erich R o s s m a n n ) w o n a prize for it. He m a n a g e d to p r e s e r v e not only the

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THE MATHEMATICAL INTELLIGENCER

spirit but the w a y that things are handled here.

Postscript The continuing story of O b e r w o l f a c h - an internal history of a very busy, wellorganized, world-famous institute---was published by the Gesellschaftfi~r mathematische Forschung on the occasion of the 40th anniversary of the Institute, in 1984, and the 25th anniversary of the Gesellschaft. It describes the solid organization, the stable finances, the sop h i s t i c a t e d buildings, a n d the detailed w o r k of the institute; it e x p l a i n s h o w c o n f e r e n c e p r o g r a m s are planned, h o w invitations are issued a n d h o w a typical c o n f e r e n c e p r o c e e d s [8]. Reading this confident document, the painful and u n c e r t a i n days of Wilhelm Stiss s e e m b e y o n d recall. But his "march on the knife's edge," as Christoph Scriba has d e s c r i b e d it [9], is n o t ancient history: w e live in a time o f opening of files. The s t o r y of O b e r w o l f a c h is w o r t h retelling, and remembering, for a n o t h e r r e a s o n too: it is the story of h o w s o m e m a t h e m a t i cians b e h a v e d t o w a r d one another, in d a r k times, for m a t h e m a t i c s ' sake.

Acknowledgments I a m grateful to John Todd for making several declassified d o c u m e n t s avail-

1. Sanford Segal, private communication. 2. Sanford Segal, Mathematics and German politics: the National-Socialist Experience, in preparation. 3. Helmuth Gericke, "Wilhelm S0ss, der Gr0nder des Mathematischen Forschungsinstitutes Oberwolfach", Jahresbericht der Deutschen Mathematiker-Vereinigung 69, no. 4 (1968). 4. Irmgard S0ss, "Beginnings of the mathematical research institute Oberwolfach at the country-house Lorenzenhof" and "The Mathematical Research Institute Oberwolfach through critical times," published privately; available in the library of the Mathematisches Forschungsinstitut. 5. John Todd, "Oberwolfach--1945", in General Inequalities 3, edited by E. F. Beckenbach and W. Walter, ISNM64, BirkhAuser, 1983. 6. "Applied Mathematical Research in Germany, with Particular Reference to Naval Applications, British Intelligence Objectives Subcommittee," by Lt. Cdr. John Todd, Lt. G.E.H. Reuter, Lt. F.G. Friedlander, Cdr. D. H. Sadler, Lt. Cdr. A. Baxter, and Lt. Cdr. F. Hoyle. Report on an investigation of intelligence targets in Germany, carried out by CIOS parties 382 and 482a in June, July, and August 1945. 7. Unclassified Technical Report, ONRL-1252, prepared for the (U.S.) Office of Naval Research in London, by F. Joachim Weyl, Scientific Liaison Officer, on "A Symposium at the Mathematical Research Institute, Oberwolfach," February 5, 1952. 8. Mathematisches Forschungsinstitut Oberwolfach, Information on the Work, Organization, and History of the Mathematical Research Institute Oberwolfach on the Occasion of its Anniversary 1984, published by Gesellschaft for mathematische Forschung e. V., Freiburg/Br. in cooperation with Stiftung Volkswagenwerk, Hannover, 1984. 9. Christoph Scriba, private communication.

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Alexander

Shen,

Editor

This column is devoted to mathematics

Two More Probabilistic Arguments

for fun. What better purpose is there

After the column a b o u t probabilistic arguments was finished, I came across two problems (both from high-school mathematical competitions in Russia) that may be easily solved using nice probabilistic arguments, a n d I'd like to share these. 1. The sets $1, $ 2 , . . . , Sk are different subsets of a set S that has 200 elements. Moreover, St r Sj for any 200~ i r j. Prove that k -< f~100JHere is the solution. Consider the following process: We start with an empty set and add r a n d o m elements of S one by one until (after 200 steps) we get the whole set S. For a fixed subset A, let us compute the probability Pr[A] that A will appear during this process. For example, Pr[Q] = Pr[S] = 1; for a n y s E S, the probability Pr[{s}] is equal to 1/200 (all elements of S can be chosen and added to • with equal probabilities). Moreover, any subset A C S of a given cardinality a has the same chance to appear during this process, and only one subset of cardinality a may appear, so Pr[A] = 1/(2a~176 Consider k r a n d o m variables ~1, 99 9 ~rk; the value of ~i is equal to 1 if the given set S,i appears during the process; otherwise, ~r~is equal to 0. The expected value of ~i is 1/( 2OO s~), where st is the n u m b e r of e l e m e n t s in S~, so 200 this expected value is at least 1/(100) (each row in the Pascal triangle has a m a x i m u m in the center). Now, consider the r a n d o m variable = ~rl + "" § ~k. This s u m c a n n o t exceed 1, as two different sets Si and Sj m a y not appear in the process (if S,~ precedes Sj in the process, then Si r Sj). So, the expected value of ~r does not exceed 1, and each term has ex200 Therefore, pected value at least 1/(100). the n u m b e r of terms k does n o t exceed

for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions from readers--either new notes, or replies to past columns.

Please send all submissions to the Mathematical Entertainments Editor, Alexander Shen, Institute for Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:[email protected]

30

( 200"~ 1002.

THE MATHEMATICALINTELUGENCER@ 1998 SPRINGER-VERLAGNEW YORK

I

R

F i g u r e 1. R o b o t in

the labyrinth

2. A r o b o t R placed in the labyrinth (as in Fig. 1) is equipped with a program. The labyrinth is a square n x n where some walls are placed b e t w e e n cells (in addition to the external walls around the square). The program is a sequence of c o m m a n d s 1 e f t ~ r i 9 h t u p , and d o u n (no loops or branches). Executing each command, the r o b o t moves in the prescribed direction if possible (and does nothing w h e n there is a wall in this direction). Prove that for a n y n, there exists a program that works correctly for all labyrinths of size n )< n (independently of the positions of walls inside the square and the robot's initial position). Here, "works correctly" m e a n s that the robot visits all reachable cells. To solve this problem, we prove that a sufficiently long r a n d o m program will w o r k with positive p r o b a bility. F o r each n x n labyrinth, t h e r e is a p r o g r a m of size 4 n 3 that w o r k s for it, as each cell is r e a c h a b l e in at m o s t 4 n steps (round-trip) a n d there are at m o s t n 2 admissible cells. Therefore, a r a n d o m p r o g r a m of size N = 4 n 3 will work with p r o b a b i l i t y at least ~ = (1/4) 4n3 a n d fail with probability at m o s t 1 - s . A r a n d o m pro-

g r a m of size 2N will fail with p r o b a bility at m o s t (1 - e)9; a r a n d o m prog r a m of size kN will fail with p r o b a bility at m o s t (1 - s)k. This p r o b a b i l i t y is c o m p u t e d for a fixed labyrinth; if k is large enough, (1 - s )k is s m a l l e r t h a n 1 divided by the n u m b e r of different labyrinths of size n x n, and a r a n d o m p r o g r a m o f size k N w o r k s for all of t h e m with positive probability. Q.E.D. P o n c e l e t T h e o r e m Revisited C o n s i d e r two circles C1 a n d C2 (Fig. 2). The well-known P o n c e l e t t h e o r e m g u a r a n t e e s that if t h e r e exists a triangle i n s c r i b e d in C1 a n d c i r c u m s c r i b e d a r o u n d C2, then t h e r e are infinitely m a n y triangles with this p r o p e r t y . P o n c e l e t ' s t h e o r e m c a n be reform u l a t e d as follows. C o n s i d e r the mapping f : Ci ~ Ci defined a s s h o w n in Figure 3.

Therefore, if w e define p(X) f o r X E C1 as the r e c i p r o c a l of the length o f the tangent f r o m X to the circle C2, a r c s A1 and A2 will have equal m e a s u r e s , and w e are done. What p r o p e r t i e s of curves Ci a n d C2 w e r e u s e d in this proof?. F o r C2, w e n e e d to k n o w that two tangents to C2 going f r o m the s a m e point X are equal (Fig. 5). If the tangents were of different lengths, the density p(X) w o u l d n ' t be well defined.

If f ( f ( f ( A ) ) ) = A for s o m e p o i n t A on Cb then f ( f ( f ( X ) ) ) = X for any p o i n t X on C1. Figure 2. Two circles and triangles.

There is a nice p r o o f o f this statem e n t (it is explained, for example, in P r a s o l o v and T i k h o m i r o v ' s t e x t b o o k on geometry): one can define a measure on C1 in such a w a y t h a t the measure of the arc X - f ( X ) is a c o n s t a n t that does not d e p e n d on t h e choice of X. T h e n , f ( f ( f ( A ) ) ) = A m e a n s that this c o n s t a n t equals one-third o f the measure o f C~. The s a m e a r g u m e n t allows us to p r o v e the Poncelet t h e o r e m n o t only for triangles but for a r b i t r a r y n-gons [if f(~O(A) = A, then this c o n s t a n t equals (1/n)th fraction of the m e a s u r e of C1 a n d f ( ' 0 ( X ) = X for any X]. OK, b u t why should s u c h a m e a s u r e exist? After we d e c i d e to l o o k for it, finding such a m e a s u r e is r a t h e r easy. A s s u m e that the m e a s u r e is p(X)ds, w h e r e P(J0 is s o m e (yet u n k n o w n ) d e n s i t y function and s is the natural parameter. To find c o n d i t i o n s on p that g u a r a n t e e the d e s i r e d p r o p e r t y , consider two infinitesimally close tangents to C2. The m e a s u r e s of infintesimal a r c s A1 and A2 cut b y t h e s e lines are to b e m a d e equal (Fig. 4). The lengths of arcs A1 a n d A~ are p r o p o r t i o n a l to the s e g m e n t s 11 and 12.

F o r C1, we n e e d a n o t h e r p r o p e r t y o f a circle: any line intersecting a circle at two points, f o r m s equal angles with the circle in b o t h intersection p o i n t s (Fig. 6). This p r o p e r t y g u a r a n t e e s that t h e arcs A i a n d A2 (Fig. 4) are p r o p o r t i o n a l to 11 and 12 (infinitesimal triangles a r e similar). The P o n c e l e t t h e o r e m is valid n o t only for circles b u t for any conic sections. However, this p r o o f s e e m s to b e not applicable in the general case. Prasolov and T i k h o m i r o v s a y (after explaining the p r o o f for the case of t w o circles), "We w o n ' t prove this t h e o r e m in the general c a s e since all k n o w n proofs are complicated." However, the M o s c o w mathematician A.A. P a n o v found that this p r o o f can be generalized. His argument is explained below. T h e inspiration c o m e s from classical mechanics, so let us recall s o m e facts. It is well k n o w n that there is no gravity inside t h e sphere. A similar two-dimensional s t a t e m e n t is also true if the gravitational force is p r o p o r -

~)

X

Figure 4. Two infinitesimal arcs should be

Figure 3. Poncelet mapping.

equal.

Figure 5. Two equal tangents to C2:I1

-/2.

Figure 6. Two equal intersection angles.

VOLUME 20, NUMBER 4, 1998

31

A2

C]

~ J Figure 7. Two elliptic a r c s h a v e the s a m e measure.

tional to the inverse distance (not the squared inverse distance, as in the three-dimensional case). To see why, l o o k again at Figure 4: forces coming from arcs A1 a n d A2 c o m p e n s a t e e a c h other, b e c a u s e d i s t a n c e s are p r o p o r tional to masses. Now, w h a t c a n b e said about the gravity inside an ellipsoid? Or inside an ellipse in the t w o - d i m e n s i o n a l case? Of course, the a n s w e r d e p e n d s on the m a s s distribution. I will s h o w that there exists a distribution that guaran-

t e e s the a b s e n c e of gravity inside the ellipse. Indeed, imagine that a cir" ~ cle is d r a w n on a weightless ) elastic film using h e a v y ink, and then this film is s t r e t c h e d together with the circle (so the circle b e c o m e s an ellipse). Then, the gravity is still a b s e n t inside the ellipse. Here is why. Although the lengths 11 a n d 12 in Figure 4 do c h a n g e w h e n w e s t r e t c h the film, their ratio r e m a i n s the same, as do the m a s s e s on a r c s A1 and A2, so the gravitational f o r c e s from A1 a n d A2 still c o m p e n s a t e e a c h other. Thus, w e have c o n s t r u c t e d a distribution of m a s s e s along the ellipse (we den o t e this distribution by d~p in the sequel) that generates no gravity inside the ellipse. Returning to P o n c e l e t ' s theorem, let us p r o v e it for the c a s e w h e n C1 is an ellipse and C2 is a circle. C o n s i d e r a distribution d~/l(x) on C1, w h e r e l(x) is the length of the t a n g e n t from x E C1 to the cirlce C2 (Fig. 7).

The s a m e argument as b e f o r e s h o w s t h a t the m e a s u r e s of a r c s A1 a n d A2 are equal. Therefore, all t a n g e n t s to the circle C2 cut the same fraction of ellipse C1 (when m e a s u r e d a c c o r d i n g to the distribution d~/l(x)), a n d the P o n c e l e t t h e o r e m is proved. What if b o t h C1 and C2 are ellipses? Then, w e stretch the picture to c o n v e r t C2 into a circle. The s t a t e m e n t o f the P o n c e l e t t h e o r e m is invariant u n d e r affine transformations, so w e are done. It is also invariant u n d e r p r o j e c t i v e transformations, so the s t a t e m e n t is true for a n y conic sections. Remark: As A.A. Panov p o i n t s out, in fact any two conic sections c o u l d b e t r a n s f o r m e d to circles by one projective transformation; this o b s e r v a t i o n gives us a n o t h e r w a y to p r o v e P o n c e l e t ' s t h e o r e m for any t w o conic s e c t i o n s after w e have p r o v e d it for circles. I close with an "archaeological discovery" from David Gale of Berkeley.

Euclid's Last (or Lost) Theorem by David Gale In a triangle called ABC, Pick a p o i n t on AB, call it P. Pick a Q on BC, Where BQ is BP. Ah the j o y s of p u r e geo-me-tree! On CA p i c k an R, oh p l e a s e do, Where CR is e x a c t l y CQ, And n o w p i c k an S

On AB, m o r e o r less, So that "AS is AR" is true. On BC the next l e t t e r is T, Where BT is BS, d o n ' t y o u see. On CA pick a U, A n d you'll k n o w w h a t to do, Next w h a t ' s this? w e ' v e arrived b a c k at P!

N o w s o m e p r o o f s were s o o n f o u n d c l o s e at hand, But it didn't turn out quite as planned, F o r t h o u g h n o t very large (They w o u l d fit in the margin) regrettably, none o f t h e m scanned.

C

T

B 32

THE MATHEMATICALINTELLIGENCER

U

P

S

A

HARLAN J. BROTHERS AND JOHN A. KNOX

New Closed-Form

Approximations to the Logarithmic Constant

~

f'h k.l

ecently, the d e t e r m i n a t i o n of n digits o f ~ has become something of an ind u s t r y [3; 10, pp. 62-63]. B y contrast, however, f e w m a t h e m a t i c i a n s seem interested i n calculating the logarithmic constant e to comparable precision [7]. This area is underexplored, perhaps because i n the case of e there is a straight-

f o r w a r d Maclaurin s e r i e s s u m m a t i o n that is quite accurate. In this article, we demonstrate that there exist alternative approximations to e that are also very accurate. We have found over 20 such approximations, all of which are elegant closed-form expressions obtainable using elementary calculus. We have used s o m e o f these approximations to calculate e to tens of thousands o f decimal-place accuracy using commercially available software. Our most impressive result is a class of closed-form approximations with extremely rapid convergence that should outperform the familiar maltiterm Maclaurin series approximation. Having b e e n unable to find these approximations in a search of the published and electronic literature, we elaborate upon them here. Traditional Approximations to e The calculation of e has intrigued m a t h e m a t i c i a n s for centuries. J o o s t Biirgi a p p e a r s to have f o r m u l a t e d the first app r o x i m a t i o n to e a r o u n d 1620 [5, p. 31], obtaining threedecimal-place accuracy. Isaac Newton, in his De Analysi of 1669 [8, p. 235], p u b l i s h e d the first version o f w h a t is

n o w k n o w n as the Maclaurin series e x p r e s s i o n for e z, which for z = 1 is equal to Direct:

~1

1 •

~=0~.,=1+1+~.,+3!

1

u

(1)

Equation (1) is a "simple, direct a p p r o a c h [that] is the b e s t way of calculating e to high a c c u r a c y " [2, p. 313]. Today, numerical values of e are derived using either optimized versions of this Maclaurin series [7, p. 157; S. Plouffe, personal communication] or the continued-fraction e x p a n s i o n a p p r o a c h p i o n e e r e d by Euler [11, p. 1019]. An alternative a p p r o a c h to a p p r o x i m a t i n g e e m p l o y s the Maclaurin series e x p r e s s i o n for ln(1 § x). This series w a s first d i s c o v e r e d i n d e p e n d e n t l y b y N e w t o n in a b o u t 1665 [6, p. 354] a n d Nicolaus Mercator in 1668 [7, pp. 38 and 74] and is valid on the interval - 1 < x ~ 1: x2

x~

ln(1 + x) = x - - - + 2 3

x4

4

+

x 5

x 6

5

6

x 7

+

7

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 4, 1998

(2)

25

Equation (2) can be u s e d to obtain closed-form approxim a t i o n s to e that require the calculation of a single exp r e s s i o n i n s t e a d of a sum of n + 1 t e r m s involving factorials as in Eq. (1). The only e x a m p l e of this alternative a p p r o a c h w e have found in the literature sets x = 1/x in Eq. (2) and multiplies the result by x to o b t a i n

xln(l+l)= 1

1 -~

1 1 1 + 3x--g- 4 7 + 5 7 -

1 ~6x

1 + ~7x

....

(3)

Exponentiating a n d using the Maclaurin series for ex leads to an a p p r o x i m a t i o n to e valid on the intervals x < - 1 and x - > 1, one t h a t h a s b e e n k n o w n b y m a t h e m a t i c i a n s and b a n k e r s alike since the early s e v e n t e e n t h century:

(1/x [ 1+11

Classical:

1+

= e 1

2x

CCM: +

1-

=e

2447 959 + 5760~ + ~ 2304x

9 5 9 + 238 043 2304x 5 580 608x 6

= SeriesIx

.|.

J

LogIl+l/x]

x-~ l/y, Collect

[E^classical,

2. Complementary Addition Method (CAM): A s i m p l e i m p r o v e m e n t results f r o m adding the Classical m e t h o d a n d CCM, and dividing the s u m by 2:

CAM: 2[(1 + 1)x+ (1 1)-xJ

E]

/.

/. y,0,6]

y-~

1/x

~ 2.71826 82372.

3. Mirror-Image Method (MIM): An a p p r o a c h similar to that u s e d to derive CAM c a n also be used to c r e a t e a distinct, and even m o r e accurate, a p p r o x i m a t i o n to e. By replacing x with 2x in Eq. (3), adding this to Eq. (3) in w h i c h x is r e p l a c e d with - 2 x , dividing b y 2, a n d t h e n exponentiating, we o b t a i n \( -22x~+-l~/ x- ]

(4')

F r o m Eq. (4'), it is clear that for x = 100,000, the closedform left-hand side o f Eq. ( 4 ) - - w h i c h w e call the Classical m e t h o d - - y i e l d s an a p p r o x i m a t i o n to e that is a c c u r a t e to four decimal places. In c o m p a r i s o n to the Direct method, however, this is small potatoes; for example, Eq. (1) with N = 16 p r o v i d e s 14-decimal-place a c c u r a c y without too much m o r e c o m p u t a t i o n a l overhead. As a result, perhaps, closed-form a p p r o x i m a t i o n s to e have r e c e i v e d scant attention outside o f the obligatory i n t r o d u c t o r y - c a l c u l u s discussion of Eq. (4) (e.g., [11, p. 5581). Seven New Ways of Looking at e A p p r o x i m a t i o n s to e far m o r e a c c u r a t e than the Classical m e t h o d can be o b t a i n e d via very similar methods. Below, we describe, in o r d e r of increasing accuracy, seven distinct algebraic e x p r e s s i o n s that a p p r o x i m a t e e for all x > 1. In each of Eqs. (4)-(14), it is the left-hand e x p r e s s i o n that is being p r o p o s e d as an a p p r o x i m a n t to e.

2447 238 043 + 5 7 6 0 ~ {- 580 608x 6

Note that this is, by the s e r i e s analysis, the equivalent of the Classical m e t h o d with all of the odd p o w e r s of x eliminated.

MIM:

1 /,00,000 1 + 100,000]

= e 1+ ~

(4)

Other series e x p a n s i o n s in this article can b e d e t e r m i n e d in like manner.] To d e m o n s t r a t e h o w this a p p r o x i m a t i o n works, w e insert x = 100,000 in Eq. (4) and obtain

(

238043 + . . . ] (5) 580 608x 6 "

24x 2

[In Eq. (4) a n d all similar equations later in this article, the right-hand side is the p r o d u c t of e a n d the b r a c k e t e d quantities. The series e x p a n s i o n in Eq. (4) can be obtained in Mathematica using the following c o m m a n d s : classical

+

+ 24x 2

CCM p o s s e s s e s virtually the s a m e rate of convergence as the Classical m e t h o d , but it a p p r o a c h e s e f r o m above, not below. Therefore, CCM can be c o m b i n e d with the Classical m e t h o d to c r e a t e n e w a p p r o x i m a t i o n s to e that converge m u c h m o r e rapidly than either form b y itself (see below).

11 7 + 244_____~7 16x 3 5760x 4

l+~-x

1 = e [ 1 + ~12x

23 1223 +O(~)+ + 1440~ + 362 880x 6

"']

(7)

Like CAM, this e l i m i n a t e s all o d d p o w e r s of x f r o m the right-hand side (RHS), b u t MIM's coefficient for 1/x2 is s m a l l e r than in CAM. The derivation of MIM b e a r s a striking r e s e m b l a n c e to Gregory's series e x p a n s i o n for In [(1 + x)/(1 - x)] [1, p. 661] and also to the series exp a n s i o n for coth lx [2, p. 310]. To our knowledge, however, this a p p r o x i m a t i o n to e has never a p p e a r e d in the literature. . Power Ratio Method (PRM): The P o w e r Ratio Method w a s arrived at numerically b y investigating the behavior of n u m b e r s that have b e e n raised to their o w n power. E x a m i n a t i o n of the rate o f change of the ratio b e t w e e n a d j a c e n t integer values of x that have been r a i s e d to the x p o w e r leads to the following a p p r o x i m a t i o n to e: PRM:

(x + 1)x+1 :c~

(x-

xx 1)x 1

=--(X + I)(I + I ) X - - ( X - - 1 ) ( 1 - - 1 ) -x 1. Complementary Classical Method (CCM): The Classical m e t h o d has a c o m p l e m e n t a r y form that results from letting x = - x in Eq. (4):

26

THE MATHEMATICALINTELLIGENCER

[ 1 = e 1+ ~ +

11 5525 + O(7~)+ 6 ~ x 4 + 580608x 6

...]

(8)

As x4~>>x! for integer values o f x > 1, the RHS o f Eq. (12) i m p l i e s that BK should converge to e much m o r e quickly t h a n the Direct m e t h o d w h e n the t w o m e t h o d s are c o m p a r e d using N = x in Eq. (1). Furthermore, the c o m p a r a t i v e advantage of BK v e r s u s the Direct m e t h o d will only w i d e n for increasing x.

As with CAM a n d MIM, PRM eliminates all o d d p o w e r s o f x and yields a rate o f convergence of O(1/x2).

5. C A M - M I M - P R M A m a l g a m Method (CMPAM): Using the series e x p a n s i o n s as a guide, a s t r a i g h t f o r w a r d combination of CAM, MIM, a n d PRM can be c r e a t e d that achieves a c c u r a c y to O(1/x6), five o r d e r s of m a g n i t u d e b e t t e r than the Classical method: 1 1~(1941

CMPAM:

= e 1

P R M - 679 MIM - 53 CAM)

83 566 080x 6

7. Hyperexponentiated Brothers-Knox Method ( B I ~ ) : Obviously, the BK method can be generalized for a = x; x r e p l a c e d by 2x, and an arbitrary n u m b e r of e x p o nentiations:

"

A variety of o t h e r forms with better-than-Classical acc u r a c y may be f o r m e d in this manner, but usually at the c o s t of algebraic e l e g a n c e and c o m p u t a t i o n a l time; we e x a m i n e t h e m elsewhere. Below, we e x p l o r e m o r e a d r o i t w a y s to i n c r e a s e the a c c u r a c y of t h e s e approximations.

BK~:

in w h i c h n indicates the o r d e r o f the exponentiation. BK ~ is, in fact, a generalization of o t h e r a p p r o x i m a t i o n s p r e s e n t e d here; for example, BK in the form of Eq. (12) c o r r e s p o n d s to BK ~ with n = 1; MIM c o r r e s p o n d s to t h e derivation o f BKn with a = x u2, and n = O. The m o s t rapidly converging e x a m p l e o f this class o f a p p r o x i m a t i o n s results w h e n n is set equal to x in BK n (or equivalently, w h e n a = x and x is r e p l a c e d by 2x x in BK):

6. B r o t h e r s - K n o x Method (BK): Drawing on t h e ideas i n h e r e n t in MIM a n d PRM, an e x t r e m e l y rapidly conv e r g e n t class of a p p r o x i m a t i o n s to e can be c r e a t e d b y substituting a x for x in MIM and then dividing t h e num e r a t o r and d e n o m i n a t o r by ar:

(, + a- )ox BK:

~

)

[ =e

+ ~1440a

1 ~

+ "" " (10)

= e 1 + - 3(2 ~x)

+ 90(24~. ~

+ 5670(26x ) +

~-

+ ....

(11)

A n o t h e r special c a s e of BK, in which a = x and x is r e p l a c e d b y 2x, p r o v i d e s exceptionally r a p i d convergence to e: (2x~ + x-~/xZ~ [ 1 23 ~-~- x-Z] = e 1 + ~12(x4X + 1440(xSX ) 1223 + 362 880(x ~2~) +

O( ] )1 ~

+ "'" " (12)

1 + 12(x-4xx)

+.-- ,

(14)

The c o n v e r g e n c e of BKx is astonishingly rapid; it w o u l d a p p e a r to p o s s e s s greater-than-quadratic convergence even for small x [4, pp. 70-71]. The ultimate e x a m p l e o f h y p e r e x p o n e n t i a t i o n w o u l d e m p l o y in BK the relations a = x a n d x = x 1' 1' x, w h e r e o u r n o t a t i o n follows Knuth's [9, p. 38] a n d indicates t h a t x is to b e raised to the x p o w e r x n u m b e r o f times. We do not p u r s u e this here. Although this a p p r o a c h of hyp e r e x p o n e n t i a t i o n closely r e s e m b l e s w o r k on infinite ite r a t e d e x p o n e n t i a l s [http://www.mathsoft.com/asolve/ c onstant/itrexp/itrexp.html], w e a r e not a w a r e of any application of the latter to the calculation of e.

One special case of BK s e e m s especially well s u i t e d to c o m p u t a t i o n a l analysis: ---2 -x ]

=e

BKX:

1 + ~12a2

+ 3 6 2 8 8 0 a 6x

{~+---x-(X")lx(2X~)=e[l+l~ ...], (13) \2x n ~ ] 12(x4Xn ) +

Numerical Computations F o r the visually inclined, we p r o v i d e t w o figures and a table which illustrate the utility of o u r n e w closed-form a p p r o x imations to e. Figure 1 c o m p a r e s the Classical a n d Direct m e t h o d s to 3

9 ~"

MIM

""-

CAM

2.i3 o o

2.6

E O

Figure 1. A c o m p a r i s o n of the new approximations C A M , M I M , and PRM versus the Classical and Direct methods for I -< x < 4. All meth-

2.4

< 2.2

r

Classical

ods are defined in the text. The Direct method is calculated u s i n g Eq. (1) in w h i c h N = x and, t h e r e f o r e , is t h e s u m o f x + 1 t e r m s ; all other methods are closed-form a p p r o x i m a t i o n s . A 15-decimal-place-

2

3

4

accurate approximation to e is plotted for visual reference.

VOLUME 20, NUMBER4, 1998

27

2.8

2.75 o o

2.7

.E_

//CMPAM /

e~ <

2.65

Direct

I t I t I

4

F i g u r e 2. A c o m p a r i s o n o f t h e n e w a p p r o x i m a t i o n s C M P A M , B K , a n d B K n, w h e r e n = 2, v e r s u s t h e D i r e c t m e t h o d f o r 1 -< x -< 4. A 1 5 - d e c -

i

2.6

4

2

CAM, MIM, and PRM for small x. The direct m e t h o d (1) is c o m p u t e d w h e r e N = x. It is n o t e w o r t h y that all of the new a p p r o x i m a t i o n s s h o w n h e r e tend to e f r o m above, w h e r e a s the Direct a n d Classical m e t h o d s a p p r o a c h e from below. The s u p e r i o r c o n v e r g e n c e of CAM, MIM, a n d PRM versus the Classical m e t h o d is obvious; however, for x -> 5, the Direct m e t h o d is m o r e a c c u r a t e than t h e s e t h r e e n e w approximations. In Figure 2, t h e Direct m e t h o d is c o m p a r e d to CMPAM, BK, and BK n for small x. The Direct m e t h o d is calculated as in Figure 1; BK is calculated using the special case (12); and BKn is Eq. (13) with n = 2. The e x t r e m e l y rapid convergence of CMPAM, BK, and BKn clearly o u t p a c e s that of the Direct m e t h o d for small x. Table 1 p r e s e n t s a c o m p a r i s o n o f all t h e a p p r o x i m a tion m e t h o d s for l a r g e r v a l u e s of x t h a n t h o s e s h o w n in the figures. In T a b l e 1, BK and BK n a r e c a l c u l a t e d as in Figure 2. In addition, w e have m a d e a very m o d e s t foray into the realm of high-precision calculations of e. We have used MIM for this purpose, as it involves very few arithmetic operations and p e r f o r m s well without optimization. Running Mathematica 2.2 on an IBM RS/6000 computer, we calculated MIM in the form [(x + 1)/(x - 1)]x/2. This form of MIM is chosen to mitigate the toss of precision that occurs if the calculation is done with MIM as shown in Eq. (7). F o r the same reason, w e manually typed in x as 1015,0~176 with 30,000 decimal places; in M a t h e m a t i c a , t h e r e is loss of precision if one defines x as x - - ) N [ ' I , O ~ 3 0 0 0 0 ] ^ 1 5 0 0 0 instead. (Manual typing o f 30,000 decimal p l a c e s is n o t an o n e r o u s t a s k using M a t h e m a t i c a ' s p a s t e option.) Once x w a s defined, we e m p l o y e d MIM to calculate 29,999 c o r r e c t d e c i m a l p l a c e s of e in 15 s. This c o m p a r e s v e r y favorably to the 12-s runtime on the s a m e h a r d w a r e

i

imal-place-accurate

approximation

t o e is p l o t t e d f o r v i s u a l r e f e r -

e n c e . N o t e t h a t the o r d i n a t e s c a l e is n o t t h e s a m e a s in F i g u r e 1.

x

and s o f t w a r e n e e d e d to calculate an equivalent n u m b e r of c o r r e c t decimal places using the N Ir E x p [ 1 ] , 3 0 0 0 0 ] c o m m a n d in Mathematica. This is b y no m e a n s a r i g o r o u s t e s t o f the c o m p u t a t i o n a l s p e e d and a c c u r a c y o b t a i n a b l e with o u r n e w a p p r o x i m a t i o n s using this s o f t w a r e / h a r d w a r e configuration, but is p r e s e n t e d to give some c o m p a r i s o n b e t w e e n existing and n e w m e t h o d s . Given the extraordinarily r a p i d convergence o f BKn, w e believe that it m a y be p o s s i b l e to use our m e t h o d s to comp u t e e to u n p r e c e d e n t e d accuracy. As an e x t r e m e example, w e estimate that for x = 10, the BKx a p p r o x i m a t i o n w o u l d yield e accurate to 40 billion decimal places, although obviously that c o m p u t a t i o n a l t a s k would be formidable. However, the computational potential of the BK n class of a p p r o x i m a t i o n s calmot be thoroughly evaluated until optimization of the calculation of x (x') using F a s t F o u r i e r Transform ( F F F ) m e t h o d s is p e r f o r m e d [S. Plouffe, personal communication]. We leave these e x p e r i m e n t s to the e x p e r t s on this subject. Discussion We have identified and formally e s t a b l i s h e d the e x i s t e n c e o f n e w closed-form a p p r o x i m a t i o n s to e. Six of the n e w app r o x i m a t i o n s d i s c u s s e d h e r e i m p r o v e u p o n the classical closed-form approximation. In particular, the BK n class o f a p p r o x i m a t i o n s converges to e m u c h m o r e rapidly than even the direct Maclaurin series method. Therefore, o u r w o r k m a y have practical application. The impressive n u m e r i c a l a c c u r a c y of these n e w app r o x i m a t i o n s should n o t cloud our eyes to an even m o r e e x t r a o r d i n a r y aspect: the elegance and simplicity of the exp r e s s i o n s for e, particularly MIM. C o m p a r e d to m a n y o t h e r m e t h o d s for computing classical constants, MIM is breathtaking. Only one addition, one subtraction, one multiplica-

r_,1:t q =i5 IqK~[,],.. m. ~,], [.] dr.c~. h ,m tf;~m('~. t ~ , I r. [~*~., r r - ] . [.], ~r.I,J.] |,til I li ~11p] i F | ~,|

X (=N)

Classical

CAM

MIM

PRM

CMPAM

Direct

10

0

1

2

2

7

7

100

1

3

4

4

13

159

1000

2

5

6

6

19

2570

BK

BK"

-40

400

-800

-80,000

--12,000

-12,000,000

Note: T o i l l u s t r a t e this c o m p a r i s o n , in the flint r o w w h e r e x = N = 10, th e C l a s s i c a l m e t h o d = (1 + 1/10) w, MIM = (21/19) 1~ a n d th e D i r e c t m e t h o d = ~~

28

0 (l/k!). BW' is c a l c u l a t e d w i t h n = 2. The - sign i n d i c a t e s a t h e o r e t i c a l e s tim a te .

THE MATHEMATICALINTELLIGENCER

tion (employed twice), one division, and one exponentiation are required to approximate e to tens of thousands of decimal places. The mathematical knowledge required to understand it is provided in introductory calculus, but the end result can be grasped and computed by an elementaryschool student. The logarithmic constant e is famous for turning up whenever natural beauty and mathematical elegance commingle. Our work provides a new glimpse of its austere charm. Acknowledgments Thanks to the Editor and to the anonymous referees for their constructive comments. Thanks also to Professor Richard Askey of the University of Wisconsin-Madison, Professor Lee Mohler of Saint Martin's College, and Dr. David Ortland of the University of Michigan for helpful discussions on this work; Simon Plouffe for an alacritous description of his numerical calculations of e; Chris Genly for his help with floating-point and arbitrary precision method implementations; and Steven Krantz for some guidance during the early stages of this project. Mathematica was used for the numerical and algebraic work shown here and is a registered trademark of Wolfram Research, Inc.

REFERENCES [1] H. Anton, Calculus, New York: John Wiley and Sons (1980). [2] G. Arfken, Mathematical Methods for Physicists, 3rd ed., New York: Academic Press, 1985. [3] D.H. Bailey, P. Borwein, S. Plouffe, On the rapid computation of various polylogarithmic constants, Math. Computat. 66(218) (1997), 903-913. [4] R.L Burden and J.D. Faires, NumericatAnalysis, 5th ed., Boston: PWS Publishing Company, 1993. [5] H.T. Davis, Tables of the Mathematical Functions, VoI. I, San Antonio, TX: Principia Press of Trinity University, 1963. [6] M. Kline, Mathematical Thought from Ancient to Modem Times, New York: Oxford University Press, 1972. [7] E. Maor, e: The Story of a Number, Princeton. NJ: Princeton University Press, 1994. [8] I, Newton, The Mathematical Papers of Isaac Newton, Vol. II 1667-1670 (D.T. Whiteside, ed.), New York: Cambridge University Press, 1968. [9] B. Rotman, The truth about counting, The Sciences 37(6) (1997), 34-39. [10] S. Wagon, Review of Mathematica 3.0 and The Mathematica Book, 3rd Edition, Math. Intell. 19(3) (1997), 59-67. [11] S. Wolfram, The Mathematica Book, 3rd ed., New York: Wolfram Media/Cambridge University Press, 1996.

VOLUME 20 NUMBER 4 1998

29

JIM HOSTE, MORWEN THISTLETHWAITE, AND JEFF WEEKS

The irst 1,701,936 Knots he history of knot tabulation is long established, having begun over 120 years ago. In m a n y ways, the compilations of the f i r s t knot tables marked the beginning of the modern study of knots, and it is perhaps not surprising that as knot theory and topology grew, so did the knot tables. Over the last f e w years, we have extended the tables to include all prime k n o t s with 16 or fewer crossings. This r e p r e s e n t s m o r e than a 130-fold increase in the n u m b e r of t a b u l a t e d knots since the last b u r s t of t a b u l a t i o n t h a t t o o k p l a c e in the early 1980s. With m o r e than 1.7 million knots n o w in the tables, w e h o p e that the census will serve as a rich s o u r c e of e x a m p l e s a n d c o u n t e r e x a m p l e s a n d as a general testing g r o u n d for o u r collective intuition. To this end, w e have written a UNIX-based c o m p u t e r p r o g r a m called KnotScape w h i c h allows easy a c c e s s to the tables. The account of o u r m e t h o d o l o g y is p r e f a c e d b y a b r i e f history of knot tabulation, concentrating m o s t l y on events taking p l a c e within the last 30 years. The survey article [Thil] contains further details on the w o r k of the nineteenth-century tabulators, but, above all, the r e a d e r is e n c o u r a g e d to consult the original sources, in p a r t i c u l a r the excellent series o f p a p e r s by Tait [Tail. K i r k m a n ' s pap e r s m a k e fascinating reading, as they a b o u n d with original ideas and ornate l a n g u a g e - - h i s definition of t h e t e r m

"knot' is a single sentence of 101 words. Conway's landm a r k p a p e r [Con] is also highly r e c o m m e n d e d . An i m p o r t a n t feature of our p r o j e c t is that we have w o r k e d in t w o completely s e p a r a t e teams, producing two t a b u l a t i o n s w h i c h w e r e kept s e c r e t until after they w e r e complete. 1 Although it w o u l d be f o o l h a r d y to claim with absolute certainty that our tables a r e correct, we must rep o r t the gratifying e x p e r i e n c e of fmding that our lists o f 1,701,936 k n o t s w e r e in c o m p l e t e agreement! Moreover, w e did n o t use e x a c t l y the s a m e m e t h o d s , the p r i m a r y difference being the use of hyperbolic g e o m e t r y by Hoste a n d Weeks and the c o m p l e t e a b s e n c e o f hyperbolic invariants in Thistlethwaite's approach. Nevertheless, our overall programs are similar in spirit and differ little from the m e t h o d s of most tabulators who p r e c e d e us. As part of our tabulation, using Weeks's program SnapPea w e were able to compute the symmetry groups of the knots; we have included a short introduction to this beautiful and intriguing topic.

~At a recent conference, people who were aware of our project jokingly scolded us for conversing together.

9 1998 SPRINGERVERLAGNEW YORK, VOLUME20, NUMBER4, 1998

33

Obviously, we n o w have a great deal of data, and reporting on every aspect of the tabulated knots is not possible. Instead, in the first of three appendices we present a statistical s u m m a r y of the census. The s e c o n d appendix contains lists of hyperbolic knots with selected symmetries, and the final appendix contains brief descriptions of K n o t S c a p e and Weeks's program S n a p P e a which figures prominently in our work. We describe the hardware and software requirements of these programs, where to obtain them, and their capabilities. A Brief History of Knot Tabulation In the late 1860s, the great Scottish physicist William Thomson (Lord Kelvin) suggested that atoms were knotted vortices in the ether. If only we could better understand knots, we could unravel the secrets of the atom and of matter itself! Inspired by this theory, T h o m s o n ' s countryman and fellow physicist, Peter Guthrie Tait, embarked on a major investigation of knots which included production of the first knot tables. By a k n o t we mean a smoothly embedded circle in 3-dimensional Euclidean space R 3. A knot d i a g r a m is a projection of a knot into a plane containing only transverse double points and, furthermore, drawn with c r o s s i n g s at each double point so that the embedding can be recovered from the diagram (Fig. 1). We will consider different knots to be equivalent if there is a homeomorphism of ~3 to itself taking one knot to the other. Thus, a single knot can be represented by infinitely many diagrams, but only a finite number of diagrams will have a minimal number of crossings, and it is with respect to this c r o s s i n g n u m bet" that Tait organized his table. The trivial knot, or unknot, can be drawn with no crossings, the trefoil knot with three, the figure-eight knot with four, and so on (Fig. 2). The strategy employed by Tait, and still used today in our tabulation, is simple: enumerate all possible diagrams up to a given crossing number and then group together those diagrams that represent the same knot type. To begin this process, Talt invented a scheme for encoding knot diagrams. Many years earlier, Gauss and his student Listing had studied knots and invented their own notations for this purpose [Lis]. Although initially he was unaware of their work, Tait's scheme is similar. Our own notation, frost used by Dowker and Thistlethwaite [DT, Thil], is a further refinement. It allows any knot diagram with N crossings to be encoded as a sequence of N (signed) even integers al, 99 9 aN, where the sequence of absolute values is a rearrangement of 2, 4 , . . . , 2N. The encoding scheme is described in Figure 1, and its subtleties and limitations are discussed in the third section. 2 Tait considered all such sequences up to seven crossings and successfully grouped them together by knot type. In 1876, he published his first table, containing the knots through seven crossings and all their minimal diagrams. (Figure 2 illustrates these 15 knot types in the order in which they are listed in our table supplied with the software package K n o t s c a p e , v i z . Appendix III). But, daunted by the combinatorial explosion of sequences for larger crossing number,

1 6

3 5 8 -12

7 2

9 14

11 16

13 -4

15 10

Figure 1. To encode a diagram, choose a basepoint and orientation of the knot; in the above figure, the chosen basepoint is indicated by a black dot at the overpass labeled 1. Traveling from this basepoint in the given direction, label points on the knot curve lying directly above or below crossings with consecutive integers 1,2, 3 , . . . . Each crossing thus receives two labels, one even and one odd, and this defines a one-to-one correspondence between the set of odd labels and the set of even labels. The overcrossing-undercrossing structure is then captured by associating a minus sign to each even integer which is the label of an overpass; thus, a diagram is alternating (i.e., --over--under-over-under-) if and only if all even integers have the same sign. The data for this example are displayed in the table immediately below the figure; because the odd numbers have been written in their natural order, all the information is contained in the sequence of signed even numbers. Therefore, for the given choice of basepoint and direction, the code for this diagram is 6 8 - 12 2 14 16 - 4 10. We choose the standard code for the diagram to be the sequence which is minimal over all choices of starting point and direction, with respect to a suitable ordering of sequences.

Tait stopped at seven crossings. It is important to remember that Tait had no theorems from topology to enable him to distinguish different knots. In fact he wrote, "... though I have grouped together many widely different but equivalent forms, I cannot be a b s o l u t e l y certain that all those groups are essentially different one from another." Indeed, it is the task of grouping the diagrams together by knot type rather than enumerating all possible diagrams that remains to this day the most difficult part of knot tabulation, for producing all possible diagrams is algorithmic and therefore, at least theoretically, trivial. However, for a large crossing number, the sheer number of possible combinations is so huge that

2A mild refinement of this notation [DH] can be used to encode link diagrams. (,4 link of k components is the union of a family of k disjoint simple closed curves in 1;13.)

34

THE MATHEMATICALINTELLIGENCER

even with today's high-speed computers, the task of enumerating all possible diagrams remains difficult in practice. To aid in the comparison of different diagrams, Talt invented a certain diagrammatic transformation which preserves crossing number, now known as the flype (Fig. 3). 3 He also classified crossings as left-handed with associated sign -1, or right-handed with associated sign + 1 (Fig. 4); the writhe of the diagram 4 is then defined as the sum of the signs of the crossings. He further declared that a crossing is nugatory ("worthless") if there is a circle in the projection plane meeting the diagram transversely at that crossing, but not meeting the diagram at any other point (Fig. 4). Nugatory crossings can obviously be removed by twisting, so they cannot occur in a diagram of minimal crossing number. A diagram is reduced if none of its crossings is nugatory. Tait set forth a number of conjectures concerning alternating knots, none of which was resolved until the advent of the Jones polynomial in 1984. He conjectured (i) that reduced alternating diagrams had minimal crossing number, (ii) that any two reduced alternating diagrams of a given knot had equal writhe, and (iii) that any two reduced alternating diagrams of a given knot were related via a sequence of flypes. The third of these conjectures implies the second, because flypes preserve writhe. The first two conjectures have been proved in various ways [Kau, Murl, Thi2, Mur2, Thi3], but all proofs use properties of the Jones polynomial or the Kauffman two-variable poly-

nomial. A solution of the third conjecture is given in [MT]; the proof is mostly geometric, but, again, it relies in an essential way on properties of the Jones polynomial. The confirmation of these conjectures has significantly lightened the task of tabulating alternating knots. After Tait's first paper appeared, he learned of the work of the Reverend Thomas P. Kirkman [Kirl, Kit2] who had himself set out to enumerate knot projections. Kirkman had used a method quite different from Tait's; he started with a relatively small set of "irreducible" projections and then produced complete lists of knot projections by inserting crossings in a systematic way. Nearly a century later, Conway used a modification of Kirkman's method with great success [Con]. Using Kirkman's projections Tait went on to produce tables, in 1884 and 1885, of alternating knots (and all their minimal diagrams) through 10 crossings. Just before going into print, Tait learned of another census of knots through 10 crossings produced by the American C.N. Little [Litl]. Comparing their work, Tait noted one duplication in his own table and one duplication and one omission in Little's, and promptly corrected his own table prior to publication. With Tait's encouragement, Little went on to tabulate the l 1-crossing alternating knots, starting from the polyedral 5 (sic) diagrams of Kirkman [Lit3]. Little also undertook the more difficult task of tabulating the nonalternating knots, ones which admit no alternating diagram. These

Figure 2. Knots to seven crossings, 3Tait had used the word flype to denote a different kind of transformation, namely a change of infinite complementary region: "flype" is an old Scottish verb whose approximate meaning is "to turn or fold back" (as with a sock). Currently, the word "flype" designates the transformation illustrated in Figure 3. 4Tait and Little used the term "twist" in place of "writhe." 5Kirkman held passionate views on the spelling of certain words.

VOLUME 20, NUMBER4. 1998 35

Reidemeister move I

Reidemeister move H

The flype

The 2-pass

The Perko move: fix lower left disk, flip the other two disks

Reidemeister move III

The (3, 2) - pass

A double 2-pass: re-route the arc with two overpasses, and also the arc with two underpasses

Figure 3. Moves on diagrams.

do not appear with fewer than eight crossings, and from Talt's first paper, it is evident that initially he did not believe that nonalternating knots were possible. In fact, the first proof of the existence of a nonalternating knot did not appear until 1930. Little states that he worked for 6 years, from 1893 to 1899, to produce his list of 43, 10-crossing nonalternating knots [Lit4]. As we shall soon see, his list had no omissions, but it did have one duplication. One obstacle to tabulating nonalternating knots is their sheer quantity. Although nonalternating knots do not predominate until 13 crossings (as mentioned earlier, they do not even appear until 8 crossings), it is plausible that the proportion of knots which are alternating tends exponentially to zero with increasing crossing number. Recently, this was proved for links by Sundberg and Thistlethwaite [ST, Thi4]. Determining the asymptotic rate of growth of the number of knots is an interesting problem [ES]; it is known [Wel, Thi4] that if Kn denotes the number of n-crossing prime knot types, then lira sup(Kn) t/n < 13.5. Another problem with nonalternating diagrams is that flypes no longer suffice to pass between all minimal diagrams of the same knot. Although this was apparent to Little, he erroneously believed that just two kinds of moves, the flype and the 2-pass, were sufficient. 6 Finally, after over 25 years of laborious handwork, Tait, Kirkman, and Little had created a table of alternating knots through 11 crossings and nonalternating knots through 10 crossings. Of course, in the absence of a rigorous theory,

they could not know whether their tables were correct; indeed, a few errors have come to light in the ensuing years. But, remarkably, the table of alternating knots through 10 crossings has stood the test of time. The era of rigorous knot theory began in the early part of this century. In 1914, the subject of topology had developed to the extent that Dehn was able to publish a proof that the right-handed and left-handed trefoils were distinct [Deh]. In 1927, using the ftrst homology groups of branched cyclic covers, Alexander and Briggs were able to distinguish all the tabulated knots through nine crossings, with the exception of three pairs [AB]. In 1932, Reidemeister completed the classification of knots up to nine crossings, using the linking numbers of branch curves in irregular covers associated to homomorphisms of the knot group onto dihedral groups [Rei]. In 1949, Schubert proved that every knot can be uniquely decomposed, up to order, as a connected sum of prime knots (Fig. 4). In close analogy with arithmetic, a knot is prime if it cannot properly be decomposed as a connected sum. In the light of Schubert's theorem, it is only necessary to tabulate prime knots; the composite knots are then easily constructed by taking connected sums. Another important consideration is that of chirality. So far, we have considered two knots to be equivalent if there is a homeomorphism of ~3 mapping one to the other. According to this defmition, any knot and its mirror image (with respect to some plane) are equivalent. But this does not tally with the layman's concept of equivalence; a piece

6Thistlethwaite used no fewer than 13 different diagrammatic moves when generating the initial raw list of 16-crossing nonaltemating knots, yet this list of 1,018,774 knots still had 9,868 duplicates, Two of the more exotic moves are illustrated in Figure 3.

36

THE MATHEMATICALINTELLIGENCER

RH c r o s s i n g Figure

LH c r o s s i n g

A nugatory crossing

A connected sum of a trefoil and a figure-eight knot

4

of rope tied as a left-handed trefoil cannot be manipulated into a right-handed trefoil. It would be more intuitive to consider knots KI and K2 to be equivalent if they are related by an ambient isotopy; by this we mean that there exists a continuously parametrized family of h o m e o m o r phisms ht: ~3 ..__>~3 (0 <-- t <-- 1) such that h0 is the identity map on ~3 and hi maps K1 onto K2. One can think of t as representing time, and one can imagine the knot moving in a viscous fluid; the isotopy describes how to move K1 to K2 and h o w the molecules of fluid are moved in the process. At first, this new definition seems to be radically different from the original one, but, in fact, it amounts merely to saying that two knots are equivalent if there is an orientationpreserving h o m e o m o r p h i s m of ~3 mapping one to the other. Under this stronger version of equivalence, which knots remain equivalent to their mirror images? Those that do are endowed with a special kind of symmetry and are called achiral or amphicheiral. When we discuss amphicheirality in more detail later in the article, we shall use the following alternative definition: a knot K is amphicheiral if there exists an orientationreversing h o m e o m o r p h i s m of R 3 mapping K to itself. To see that the definitions are equivalent, simply c o m p o s e the h o m e o m o r p h i s m of A3 given by either definition with reflection in the projection plane. Amphicheiral knots are important to chemists, who are often concerned with the right- or left-handedness of molecules. The figure-eight is amphicheiral, a fact that was known

to Listing. The trefoil certainly appears not to be equivalent to its mirror image, but without proof, this cannot be asserted as a fact. As mentioned earlier, the fn'st proof was given by Dehn in 1914 [Deh], in the early days of topology. Talt was interested in the c o n c e p t of amphicheirality but did not have an intrinsic definition; he was somewhat hampered by being tied to the projection plane and had rather artificial distinctions between "kinds" of amphicheirality. Nonetheless, he successfully identified all amphicheiral alternating knots with up to 10 crossings; he did not consider nonalternating knots, but, as it happens, there are no amphicheiral nonalternating knots with fewer than 12 crossings. He conjectured that amphicheiral knots with odd crossing n u m b e r s could not exist; this is now known to be the case for alternating knots, but our census has turned up a 15-crossing nonalternating amphicheiral knot (Fig. 5). Another w a y of refining the notion of equivalence is to consider the orientation, or direction, of the knot curve. A knot is said to be invertible (or reversible) if there is an ambient isotopy carrying the knot onto itself, but with its direction reversed. The trefoil is clearly invertible: take the diagram illustrated in Figure 2 and rotate it through half a turn about an axis in the plane. In fact, every knot illustrated in Figure 2 is invertible, and it is not immediately clear that there exists a knot which is not invertible. This question was not resolved until 1964, when Trotter discovered an infinite family of noninvertible knots, beginning at nine crossings [Tro]. The situation with a low crossing

Figure 5. The 1S-crossing amphicheiral knot.

VOLUME 20, NUMBER 4, 1998

37

number is not typical, however; inspection of Table A1 in Appendix I reveals that, overall, only a tiny proportion of the knots up to 16 crossings are invertible. It is obviously of fundamental importance to be able to show that the knots listed in any purported table of prime knots really are prime. It is also desirable to know which knots are amphicheiral or invertible. 7 During the course of several decades, researchers devoted much effort to answering these and other questions for knots tabulated in the nineteenth century, but for over 50 years, almost nothing was done to verify or extend the tables. The only exception to this was a compilation of 12- and 14-crossing amphicheiral knots undertaken by Haseman in 1917 and 1918 [Hasl, Has2]. Activity picked up again in the 1960s, when Conway invented a striking new notation for knots and links. This notation, which uses some of Kirkman's ideas, enabled Conway to enumerate all prime knots up to 11 crossings and all links up to 10 crossings. Conway's scheme was so efficient that he claims to have completed this task, by hand, in a "few hours"! Conway found 11 omissions and 1 duplication in Little's table of 11-crossing alternating knots, but his own new table of ll-crossing nonalternating knots had 4 omissions. Conway also overlooked the famous duplication in Little's table of 10-crossing nonalternating knots; this duplication was finally brought to light in 1974 by Perko, who showed that the two diagrams were related by the move which bears his name (Fig. 3). 8 This pair of 10-crossing nonalternating diagrams of the same knot have different writhes and this is probably why the duplication went undetected for so long. Indeed, Little had even published a "proof' that the writhe of a minimal diagram was a knot invariant, based on the mistaken assumption that flypes and 2-passes sufficed to pass between any two minimal diagrams of the same knot. In the late 1970s, Caudron [Can] used an alternative version of Conway's notation to retabulate knots up to 11 crossings, discovering, in the process, the 4 omissions referred to above. Meanwhile, Bonahon and Siebemann, noticing that Conway's notational system reflected deep structure properties of knots, proved a general classification theorem for the family of so-called algebraic knots [BS]. The great majority of knots through 11 crossings are algebraic, although (as with all "nice" families) they are soon overwhelmed by the nonalgebraic knots. In a tour de force, Perko [Per 2, Per 3] computed enough invariants by hand to distinguish the knots not covered by Bonahon and Siebenmann's result, thus finally completing the classification through 11 crossings. These efforts mark the end of the hand-calculation era, as all subsequent tabulations have been carried out by computer. In the early 1980s, Dowker and Thistlethwaite computerized the tabulation process and extended the table to 13 crossings. Although Conway's notation has been a major conceptual influence in knot theory and is very compact

for knots of low crossing number, it does not lend itself readily to computer programming, as it draws on a large set of symbols assembled according to a rather large set of rules, both of which grow with crossing number. Instead, Dowker and Thistlethwaite used the refinement of Tait's notation already mentioned to enumerate all possible diagrams. Flypes, 2-passes, and other moves (Fig. 3) were used to group diagrams into equivalence classes which could then be distinguished by topological invariants. The table stood at 13 crossings for about a decade until Hoste was recruited by local high-school students who had just won access to a Cray supercomputer. Together, they tabulated all alternating knots through 14 crossings and provided the first check of Thistlethwaite's list of alternating knots [Am]. At the same time, Thistlethwalte returned to the tabulation problem and extended the table even further. Most recently, Hoste and Weeks have collaborated to produce a table. Working in parallel with Thistlethwalte, we have currently tabulated all prime knots through 16 crossings and, as this article goes to press, have begun the 17-crossing list.

Our Methodology Both our tabulations, Hoste/Weeks's and Thistlethwaite's, begin by listing all prime alternating knots of a given crossing number. We do this by generating all possible alternating diagrams and then grouping them into flype equivalence classes. We both use the same encoding scheme for diagrams, namely the sequence of even integers already explained in Figure 1. We will refer to these sequences as DT sequences (for Dowker and Thistlethwalte). Using the same notation makes it easy for us to compare our results. To reach our immediate goal, namely a list of all N-crossing alternating knot types, we need to be aware os certain features of the DT encoding scheme. Our first observation is that the encoding of any particular diagram depends on the choice of basepoint and direction. There are (2N)(2) = 4N such choices, and for highly asymmetric diagrams, these choices can result in 4h/different sequences. We need to choose a distinguished member of the collection of sequences representing a given diagram; to this end, we declare that the standard sequence for the diagram is the DT sequence, which is minimal with respect to lexicographical ordering, over all choices of basepoint and direction. Indeed, we may go further and declare the standard sequence for a knot type to be the minimal sequence over all diagrams of minimal crossing number representing that knot type. A second point to consider is that not every DT sequence encodes a knot diagram. A moment's reflection should convince the reader that the objects really being encoded by DT sequences are 4walent graphs obtained by identifying pairs of points on a circle, and, clearly, such a graph need not be planar. Thus, it is not surprising that some DT sequences are not realizable. However, algorithms for deciding the planarity of a graph are well known and it is a

7For hyperbolic knots, SnapPeaeasily settles all these questions. 8To his credit, Conway had discovered more complicated instances of the Perko move among the 11-crossing knots.

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THE MATHEMATICAL1NTELLIGENCER

simple matter to test a DT sequence for realizability. Furthermore, Dowker aIld Thistlethwaite show that if a DT sequence is coming from aprime diagram, then the diagram is tmiquely determined by the sequence, up to reflection and isotopy of the extended plane [DT]. (A diagram is prime if there is no circle in the plane cutting the diagram transversely in two points, with crossings lying on both sides; an example of a diagram which is not prime is that of the connected sum of the trefoil and the figure-eight given in Figure 4.) As already stated, we are only interested in prime knots and must, therefore, exclude DT sequences representing composite knots. In general, deciding whether a given knot is prime is a nontrivial matter, as the decomposing 2-sphere of a composite knot might not be immediately visible. Fortunately, in the case of alternating knots, there is a beautiful result of Menasco [Men] that can be brought to bear. The result states that a reduced alternating diagram represents a prime knot if and only if the diagram is itself prime; informally, an alternating knot is prime if and only if "it looks as if it is prime. ''9 Deternfining whether a DT sequence represents a prime diagram is straightforward: a diagram is composite if and only if there is a proper subinterval of {1,2,..., N} such that each number In the subinterval is paired with another number in the same subinterval (the subinterval contains the points associated with crossings of one of the summands of the diagram). Note that if a diagram contains a nugatory crossing, then it also fails to be prime; therefore, once we have eliminated the composite diagrams, we no longer have to worry about nonreduced diagrams. Our strategy is n o w clear. We generate all possible DT sequences of length N in lexicographic order and immediately discard those that are nonreliazable and those that are nonprime. We also discard those sequences which are not minimal over all choices of starting point and direction. After this filtering process, we are left with a list of sequences representing the prime, N-crossing alternating diagrams. These sequences are then sorted into equivalence classes with respect to the operation of flyping; for this step, it is necessary to write a procedure which detects all possible flypes and implements them as transformations of DT sequences. The final list of alternating knot types is then obtained by recording the member of each equivalence class which is lexicographically minimal. In practice, of course, it is not necessary ever to have the complete "raw" list of diagrams; one can reject diagrams that are not minimal with respect to flyping, as they are generated. The entire process is algorithmic and straightforward to implement as computer code. Nonetheless, we should mention that this is a nontrivial exercise in computer programming, given that we wish the entire task to be completed in a reasonable amount of time! Our runtimes for N = 16 are presently of the order of 1-2 weeks, 1~ and our own experi-

ence shows that careless programming can easily increase this by one or two orders of magnitude. The trick is to rule out whole sets of sequences whose first few entries already guarantee eventual failure of one of the tests. For example, we may ignore DT sequences that start with 2, as the corresponding diagram would have a nugatory crossing labeled {1, 2}. Also, if a sequence begins with 4, primality of the associated diagram dictates that the numbers paired with 2 and 3 must differ by 1. Furthermore, any sequence of length >3 which begins 4 6 2 . . . cannot represent a prime diagram, as there will be a trefoil summand. These examples are quite simple, but hint at the possibilities. After obtaining the list of all prime, alternating, unoriented knots with N crossings, we generate the nonalternating knots with N crossings. Each N-crossing nonalternating diagram can be obtained by switching crossings of an N-crossing alternating diagram, so by switching crossings in all possible ways in each N-crossing alternating diagram, we produce all possible N-crossing nonalternating knot types. At first, it might appear that we need to use all (reduced) alternating diagrams, but, in fact, we only need to use one alternating diagram per alternating knot type. For if D and E are two alternating diagrams related by flypes and E ' is a nonalternating diagram obtained by switching crossings of E, then E' is related by flypes to a nonalternating diagram D ' obtained by switching crossings of D. To generate the nonalternating diagrams, we just insert minus signs in all possible ways into the DT sequences that represent the alternating knots. Because we are only interested in tabulating knots up to reflection, we may assume the fLrst entry of each sequence is positive. Thus, there are 2g - 1 sequences to consider for each alternating diagram. Most of these nonalternating diagrams reduce to diagrams of fewer crossings; again, with careful programming, we can avoid ever considering the myriad sequences that immediately reduce to smaller diagrams by means of a single Type II Reidemeister move (Fig. 3) or even a combination of two or three Reidemeister moves. Still, many diagrams are left, and a more careful sorting into knot types must be undertaken. Before continuing with this discussion, we describe h o w we have chosen to extend the lexicographical ordering of unsigned DT sequences of length N to signed DT sequences of length N, for, again, our ultimate aim is to settle on a unique representative for each knot type. Our first convention is that a DT sequence of positive numbers always precedes a sequence containing one or more negative numbers; this merely reflects the fact that we enumerate all alternating diagrams before proceeding to the nonalternating diagrams. Now, suppose that we have two sequences sl and s2, each with at least one negative term. For i = 1, 2, let Isil be the sequence whose terms are the absolute values of those of si. If IslJ precedes Is21 lexicographically, we declare that

9The reader can easily construct examples of (nonalternating) prime diagrams representing composite knots, and also examples of composite diagrams representing prime knots; such examples illustrate the essential difference between the two concepts "pnme knot" and "prime diagram" and highlight the significance of Menasco's result, 1~ increment of crossing number results (roughly) in a 10-fold increase in computer runtime, Our programs will accomplish Conway's enumeration up to 11 crossings in about 9 s!

VOLUME20, NUMBER4, 1998 39

Sl < s2. If, on the other hand, Is,I = is21, we look at the first position where Sl and s2 have terms of opposite sign and declare that sl < s2 if sl has a negative term in that position. It is at this p o i n t that our two m e t h o d s o f t a b u l a t i o n begin to differ significantly, as Hoste a n d Thistlethwaite apply different t y p e s o f m o v e s to the n o n a l t e r n a t i n g diagrams in an effort to eliminate r e d u n d a n t diagrams. After being left with n o n a l t e r n a t i n g diagrams that do n o t obviously reduce to fewer crossings, Hoste c o n s i d e r s all diagrams related to a given one b y flypes and 2-passes a n d in every diagram in this class looks for any r e d u c t i o n to fewer crossings given b y an (i, 3)-pass, w h e r e i > j (Fig. 3). If none is found, the lexicographically s m a l l e s t d i a g r a m in the class is retained. What is left in the e n d is a s u p e r s e t of the set of all u n o r i e n t e d prime nonalternating knots with N crossings, up to reflection. Even for N = 10, this s u p e r s e t is too big: the f a m o u s Perko pair of d i a g r a m s m e n t i o n e d earlier still r e m a i n s and the list contains 43 r a t h e r than 42 diagrams. F o r N = 16, the list has a p p r o x i m a t e l y 10% too m a n y diagrams. At this point, no further a t t e m p t s are m a d e by Hoste to eliminate duplicates by m e a n s o f Reidemeister or o t h e r d i a g r a m m a t i c moves. Instead, the list is p a s s e d to Weeks and his c o m p u t e r p r o g r a m SnapPea. We will return in a m o m e n t to a d i s c u s s i o n of h o w t h e final list of nonalternating k n o t s is found by SnapPea. Thistlethwalte, on the other hand, applies m o r e diagrammatic m o v e s to e a c h nonalternating diagram. In addition to the flypes a n d p a s s moves u s e d b y Hoste, he also e m p l o y s "double-pass" moves, the "Perko" m o v e (Fig. 3), and a few o t h e r e s o t e r i c m o v e s d e s i g n e d specifically to root out s t u b b o r n pairs of equivalent diagrams. These m o v e s p r e s e r v e crossing n u m b e r and, with j u s t one exception, suffice to eliminate all d u p l i c a t e s t h r o u g h N = 13. After arriving at an initial superset of nonalternating knots, Thistlethwalte then turns his attention to distinguishing as many knots as possible. He first computes the Jones polynomial, which places the knots in small equivalence classes, each class consisting of all knots with a given Jones polynomial. These equivalence classes are then attacked by invariants based on representations of the knot group (Perko had already used this type of invariant with notable success in dealing with l l - c r o s s i n g knots). In the case at hand, a few thousand pairs and triples of knot diagrams still resolutely refused to be distinguished, but, fortunately, it was shown that the diagrams in each stubborn pair or triple were equivalent. The method for this last step was to apply moves to increase the number of crossings of the diagrams and then to apply all the previous moves to these "expanded" diagrams. Returning to the s u p e r s e t of n o n a l t e r n a t i n g diagrams g e n e r a t e d b y Hoste, the n e x t step is the application of Weeks's p r o g r a m SnapPea. There are o v e r 1.7 million knots in our table, but, amazingly, it turns out that all but 32 are hyperbolic a n d thus susceptible to the full w e a p o n r y of hyperbolic g e o m e t r y (an e x p l a n a t i o n of the t e r m "hyperbolic" follows shortly). Several i m p o r t a n t t h e o r e m s n o w apply w h i c h lead, in the case of h y p e r b o l i c knots, to

a c o m p l e t e k n o t invariant. This invariant is t h e n used, in the c a s e of the hyperbolic knots, to remove all d u p l i c a t e s from H o s t e ' s list. The n o n h y p e r b o l i c knots are so few in n u m b e r that they are easily dealt with separately. The first important theorem, due to Gordon and Luecke, is that two knots are equivalent if and only if their comp l e m e n t s are h o m e o m o r p h i c [GL]. The second, due to M o s t o w and Prasad, states t h a t if a knot c o m p l e m e n t admits a c o m p l e t e Riemannian metric of constant Gaussian curvature - 1 , in other w o r d s the knot is hyperbolic, t h e n such a metric is unique [Pra]. Thus, two hyperbolic k n o t s are equivalent if and only if their c o m p l e m e n t s are isometric. The final result w e n e e d is the existence of a canonical triangulation of hyperbolic k n o t complements, s h o w n to exist by Epstein and P e n n e r [EP] and S a k u m a and W e e k s [Wks, SW]. The canonical decomposition is described in Figure 6. It is a complete invariant for hyperbolic knots, as two knot complements are isometric if and only if they have the s a m e canonical triangulation. Finally, it is important to note that the canonical triangulation by ideal polyhedra can be described entirely combinatorially, by designating which faces of which polyhedra must be identified. Thus, once the canonical decomposition has been found for each of the hyperbolic knots on the list of nonalternating knots, they can b e c o m p a r e d combinatorially. If two are alike, the knots are the s a m e and the redundant diagram can be d r o p p e d from the list. If two are different, the two knots are different. SnapPea takes as input Hoste's list of nonalternating diagrams and attempts to fmd the canonical decomposition for each. Through N = 16, the only nonhyperbolic knots are the 12 t o m s knots and 20 satellite knots listed at the end of this section. 11 F o r the rest, SnapPea succeeds in fmding a hyperbolic structure, which it then uses to construct the canonical decomposition. Although the basic data used to describe the hyperbolic structure are algebraic numbers and m a y eventually be recorded as such by future versions of SnapPea, they are presently rounded off and stored as floating-point numbers. This has the undesired effect that roundoff error m a y lead SnapPea to a decomposition which is not the canonical one. To understand h o w this happens, recall the imagery of Figure 6. If two adjacent triangular faces of the convex hull are found to be coplanar to an accuracy of, say, 10 -12, should SnapPea treat them as distinct triangular faces, or should they be combined to form a single quadrilateral face? If SnapPea guesses wrong, it m a y give a false negative to the question, "Are these two hyperbolic knots the same?" Fortunately, false positives are impossible, because if two decompositions are equivalent, the knots must be the same, w h e t h e r or not the decompositions are the canonical ones. Therefore, the list of knots c o m p u t e d by this method is guaranteed to be complete, but it could, in principle, contain duplications. Comparison with Thistlethwaite's results s h o w s rigorously that, in fact, no duplications are present, b e c a u s e Thistlethwalte distinguishes knots by integer invariants. We should point out t h a t unlike the tabulation o f the alt e r n a t i n g knots, our m e t h o d s for nonalternating k n o t s are

11A famous theorem of W. Thurston states that any nonhyperbolic knot is either a torus knot or a satellite knot.

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THE MATHEMATICALINTELLIGENCER

Therefore, we have p r o d u c e d a table of all prime, uno r i e n t e d k n o t t y p e s up to N crossings, w i t h o u t duplications. So far, we have ignored the i s s u e s of amphicheirality and invertibility, but these issues n e e d to b e a d d r e s s e d if w e wish to classify oriented k n o t s up to isotopy. T h e y are b e s t d i s c u s s e d in the c o n t e x t o f knot symmetries.

Figure 6. Given a fixed set of vertices, one may construct a canonical triangulation of a sphere by taking the convex hull in Euclidean 3-space and radially projecting its edges back onto the sphere itself. A similar construction in (3 + 1)-dimensional Minkowski space [EP, W k s , SW] d e f i n e s a canonical decomposition for a hyperbolic

knot complement.

n o t algorithmic. Instead, w e simply e m p l o y a collection of m e t h o d s that work for N --< 16.12 After producing the t w o tabulations, it is then a simple m a t t e r to compare our results. Because w e share the same encoding scheme and the s a m e choice of lexicographic order, the two lists should be exactly the same. In practice, we have c o m p a r e d our lists after completing the table for each crossing number. Only for N = 14 did our results differ, and then only by one of us haxdng 4 omissions that the other did not. A programming error was quickly found that a c c o u n t e d for the discrepancy. F o r N = 15 and 16, our lists have agreed on first comparison, a very satisfactory experience! At this point, the r e a d e r might w o n d e r h o w w e can be certain that all the k n o t s in our tabulation are prime. This is a nontrivial matter, but, fortunately, it is a f u n d a m e n t a l p r o p e r t y of hyperbolic k n o t s that they c a n n o t be c o m p o s ite. Therefore, in finding their hyperbolic structures, SnapPea has a l r e a d y e s t a b l i s h e d primality for all b u t 32 o f t h e k n o t s . 13 As m e n t i o n e d earlier, each of the remaining k n o t s is a torus or satellite knot. A p r o o f that t o m s k n o t s are p r i m e m a y be found in a t e x t b o o k on knot t h e o r y such as [BZ]; the satellite k n o t s all have structures w h i c h are well k n o w n and d o c u m e n t e d (Fig. 9), and their p r i m a l i t y follows from a simple g e o m e t r i c argument.

The S y m m e t r y Group of a Knot Recall that in plane geometry, a s y m m e t r y of a regular polygon of n sides is defined as a rigid m o t i o n (isometry) of the plane w h i c h m a p s the polygon onto itself; for example, a square m a y be m a p p e d onto itself b y any of four rotations about its center, b y a reflection a b o u t a diagonal of the square, or b y a reflection a b o u t a p e r p e n d i c u l a r bisector o f opposite sides of the square. These eight symmetries of a square form a group under the operation of composition, k n o w n as the dihedral group D4. More generally, a regular n-sided polygon has 2n symmetries, o f which n are rotations and n are reflections, and these form the dihedral group D,~. Informally, there are 2n distinct w a y s o f picking up the polygon and putting it b a c k onto its original location. The n rotations also form a group, called the cyclic group Zn. In the theory of knots, symmetries are defined analogously. We could merely consider a s y m m e t r y of a knot K to be a h o m e o m o r p h i s m of ~3 which maps K to itself, or, m o r e succinctly, a h o m e o m o r p h i s m of the pair of s p a c e s (~3,K). However, it is natural to regard two symmetries of K as being equivalent if there is a continuously parametrized family of symmetries at (0 <- t -< 1) s u c h that one of the symmetries in question is a0 and the other symmetry is al (i.e., a0 can be "deformed" to al through the family of symmetries {at}). Thus, it is customary to regard a s y m m e t r y of the knot K as an equivalence class of h o m e o m o r p h i s m s of (~3,K) with regard to this equivalence relation. As in plane geometry, the operation o f composition induces a group strncture on the set of equivalence classes. We can think of a symmetry of K as a way o f transforming space so that the knot is m a p p e d onto itself; w e just have to r e m e m b e r that two symmetries are "the same" if one can be deformed to the other, as above. We m a y classify knot s y m m e t r i e s into four types, depending on w h e t h e r the s y m m e t r y reverses the orientation of ~3 o r reverses the orientation of K, according to Table 1. In Figure 1, for example, if w e p e r f o r m a rotation through half a turn about a w e s t - e a s t axis in the projection plane

Symmetry type

Orientation o f •z

Orientation of K

0

Preserves

Preserves

1

Preserves

Reverses

2

Reverses

Preserves

3

Reverses

Reverses

12Of course, the lack of an algorithm does not detract from the rigor of our approach. We merely run the risk of encountering a pair of knots so awkward that we will not be able to decide whether the knots are equivalent. To date, fortunately, this has not happened. In principle, there does exist an algorithm, due to Haken and Hemion [Hak, Hem], for deciding whether or not two given knots are equivalent. However, to the best of our knowledge, this algorithm has not been implemented except in a few isolated cases. 13Thistlethwaite has different techniques for demonstrating primality, but SnapPea's method is much more efficient.

VOLUME 20, NUMBER4, 1998 4.1

Class

Symmetries contained in group

Symmetry type of knot

c

Type 0 symmetries only

Chiral, noninvertible

i

Type 0 and 1 symmetries only

Chiral, invertible

+

Type 0 and 2 symmetries only

+ Amphicheiral, noninvertible

-

Type 0 and 3 symmetries only

- AmDhicheiral, noninver~ible

a

Type 0, 1, 2, 3 symmetries

Fully amphicheiral, invertible

passing through the center of the diagram, the knot is mapped to itself with its orientation reversed; therefore, this rotation is a symmetry of type 1. We can also achieve a symmetry of type 0 for tiffs knot, by rotating through half a turn about an axis normal to the projection plane. We can n o w classify symmetry groups according to which types of symmetry they contain. We observe that if a symmetry group contains two of the three types 1, 2, and 3, then it must contain the third one as well: simply take the product. Every symmetry group contains the identity, which is of type 0. Therefore, there are exactly five classes of symmetry groups, which we present in Table 2. A handy feature of SnapPeais that it is able to compute the symmetry group of a hyperbolic knot and also the type of each symmetry within the group. It accomplishes this by fmding all the ways of mapping the canonical triangulation onto itself. An obvious question is the following: which groups can be symmetry groups of knots? In the case of hyperbolic knots, it turns out that the group has to be fmite and, moreover, has to be cyclic or dihedral [Ril, KS]. For nonhyperbolic knots, the characterization of symmetry groups is slightly more complicated. It so happens that all knots with crossing number -< 8 are either amphicheiral or invertible; however, the hordes of "random" knots soon begin to take over, and as Table A2A shows, at the level of 15 or 16 crossings almost all knots are completely asymmetric. We round off this section with a look at two 16-crossing exceptions to this trend toward chaos: (i) the "most symmetric" hyperbolic knot in the table, with symmetry group D16 and (ii) the only hyperbolic knot in the table with symmetry group D 9 (Fig. 7). Although the diagram of the D16 knot in Figure 7 displays its entire symmetry group in an obvious way, the diagram of the D9 knot looks most unremarkable. The fact that the symmetry group of the D16 knot is so visible is no coincidence; it is proved in [MT] that any symmetry of a prime alternating link must be visible, up to flypes, in any alternating diagram of the link.14 Even if the alternating diagram admits nontrivial flypes, it is a relatively simple matter to determine the symmetry group of the link by inspection of the diagram, although this procedure does require a modest amount of practice. But the minimal diagram of the D16 knot does not admit any nontrivial flype, so the symmetries of this knot must be immediately apparent, in the sense that they correspond to symmetries of the diagram. The symmetries of the D 9 knot, by w a y of contrast, are not remotely visible in the 16-crossing diagram of Figure

J The D16 knot

The D9 knot Figure 7

7. However, one way of revealing the symmetries is to construct the knot by cyclically pasting together nine identical pieces, as in Figure 8. E a c h piece, shown dark gray in the picture, is a "bundle" of five arcs, and two adjacent pieces differ by a "twist" of a third of a turn.

The Nonhyperbolic Knots As already mentioned, there are 32 nonhyperbolic knots with 16 or fewer crossings, of which 12 are torus knots and the remaining 20 are satellites of the trefoil. A t o m s knot is a simple closed curve sitting on a standardly embedded toms. When we speak of a "(p,q)-torus knot," we mean a simple closed curve on the torus which wraps around p times meridianally and q times longitudinally. The integers p and q are necessarily relatively prime: otherwise, we would have a torus link. In Figure 2, the second knot in the first row is a (3, 2)-torns knot, the last knot

~4A similar result was obtained by Bonahon and Siebenmann for "algebraic" links [BS].

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THE MATHEMATICALINTELLIGENCER

Figure 8. A symmetric representation of the Dg knot. It was rendered using Larry Gritz's Blue Moon Rendering Tools.

in the first row is a (5, 2)-toms knot, and the last knot in the third row is a (7, 2)-toms knot. From this, the reader can easily produce analogous diagrams of (p, q)-toms knots for other values o f p and q and will observe that each diagram thus constructed has p ( q - 1) crossings. It is not hard to show that for a given p and q, a (p, q)-torus knot is isotopic to a (q, p)-torus knot and that t o m s knots are classified up to unoriented equivalence by the unordered pair of positive integers p and q. Henceforth, without fear of ambiguity, we shall speak of the (p, q)-toms knot. Because interchanging p and q does not alter the knot type, it follows that the (p, q)-toms knot has two "natural" diagrams, one with p ( q - 1) crossings and the other with q(p - 1) crossings. It is proved in [Wil, MP] that the crossing number is the smaller of these two numbers. From this result on crossing number and the fact that p and q must be relatively prime, the values of (p, q) giving a torus knot with 16 or fewer crossings are (3, 2), (5, 2), (7, 2), (9, 2), (11, 2), (13, 2), (15, 2), (4, 3), (5, 3), (7, 3), (8, 3), and (5, 4). It is well known [Sch] that torus knots are all chiral and invertible and that each has symmetry group D1. Therefore, our discussion of t o m s knots is complete, and we turn to satellite knots. If a knot K is placed inside a solid t o m s V, and V is itself knotted in E3, then K is called a satellite k n o t (certain mild restrictions must be imposed on the way in which K i s placed in V, to avoid trivial cases). The core C of V (i.e., the knot traveling around the center of V) is called a c o m p a n i o n of K, and we say that K is a satellite o f C. Figure 9 illustrates a satellite knot 15 of the trefoil (i.e., the knot is sitting inside a solid t o m s which is a thickened trefoil). Let us suppose that the companion knot has crossing n u m b e r k and that the satellite wraps (or ravels) m times longitudinally around the solid toms; for example, in Figure 9, we have k = 3 and m = 2. There is an obvious diagram

of the satellite where at each crossing of the companion we see an m x m "grid" of crossings of the satellite. Such a diagram has at least k m 2 crossings, and it is an unproven "factoid" of knot theory that the satellite cannot be projected with fewer than k m 2 crossings. Because the trefoil has 3 crossings, any satellite of it appearing in our table should have wrapping number 2 (a higher wrapping number should entail at least 27 crossings). Moreover, we should not expect any satellite of the figure-eight knot to have fewer than (4)(22) + 1 = 17 crossings, where + 1 refers to an additional crossing needed to produce a knot rather than a link of two components. Our table does not contain any counterexample to the conjecture: the satellites through 16 crossings all wrap twice around the trefoil, as in Figure 9. Because the trefoil is chiral, it follows from a standard result of knot theory that these 20 satellites are also chiral. Moreover, it is not hard to see that they are all invertible: if we rotate through half a turn about a "north-south" axis in the projection plane passing through the substituent tangle, the tangle is flipped over and the rest of the satellite knot is mapped onto itself with reversed orientation. But each of the 10 tangles is 2-bridged, and it is well known that 2-bridged tangles are invariant (up to isotopy) under such a rotation. The s y m m e t r y groups of the satellite knots were computed: the four cable knots 16 (i.e., the knots with substitutent tangles ~f'..~, ~ , / ~ , ~x_.'-~) have symmetry groups D3, D5, D7 and D9, respectively, whereas each of the remaining 16 satellite knots has infinite dihedral symmetry group. In general, the symmetry group of a satellite knot need not be cyclic or dihedral.

Figure 9. Each satellite knot with -<16 crossings is obtained by substituting one of these tangles, or its reflection, into the shaded disk.

15More precisely, it is a template: it becomes a knot when 1 of the 10 tangles in the box underneath is substituted for the shaded disk. 16A satellite knot K is a cable knot if it lies on the boundary of the companion solid torus V [BZ].

VOLUME 20, NUMBER4, 1998

43

Appendix

h

Summary

# Crossings

Data

# Knots

# Torus

o

1

3a

1

4a

1

5a

2

6a

3

7a

7

# Sat.

c

+

-

i

a

1 1

1

1

2

1

7

1

2

8a

18

8n

3

1 1

1

13

9a

41 8

lOa

123

1On

42

1

6

36

1

123

244

64

121

2

8

11a

367 185

12a

1,288

637

12n

888

466

13a

4,878

13n

5,110

14a

19,536

14n

27,436

1

15a

85,263

1

15n

168,030

1

16a

379,799

16n

1,008,906

1

2

c

+

Z4

c

+

597

.16

3

418

1

1,617 1,452

15,229

5

183

4,084

35

22,656

1

44

4,729

6

10

~

1

15,672

350,260

40

927

28,490

82

958,189

25

434

50,227

31

~i

D1 -

11,o40

152,357

6

7

37

74,223

1

Z3

1

89

3,658

2

Z2

6

3,261

n i~ .~rl.l..l,_ll , i ~ 111..-i1t,i,ll,l li.~l,t,,i. [:t~ . , l i ~,l.i ,=.-i[.i i ,i,t,lq i t ~ ~

c

39

21

11n

# Crossings

4

3

9n

Z1

1

D3

D2 i

i

a

D4

i

D5 a

i

De

D~ a

i

De i

DI~ D14

D9 a

i

a

a

Die a

4a

5a

1

6a

2

7a

4

8a

1

8n

44'

2

11

1

1

9a

91

!2

1

9n

4

1

2 1

lOa

19

1On

5

11a 11n

35

;3

25

9

116

120

1 2

57

97

2

1

12a

6O2

12n

445

21

13a

3,154

13n

3,555

14a

14,851

37

14n

22,326

33

15a

73,185

1,037

15n

151,126

1,231

16a

347,111

3,149

33

16n

954,381

3,808

21

1

3

350

22

1 2 1

344

iO

107

38O

5 1

10

311

16

998

1,0 4

11

356

38

4

THE MATHEMATICALINTELLIGENCER

2

4

1 1

17 42

1

10

2

1

4

4

13

21 1

1 23

4

6

2

1 2

395

2,2 5

16

51

1

1

821

80

33

2

2

7

891

725

4,7 4

25

2

3

54

4

423

224

1,9 1

24

13

27

5

2

2 2 1

2 5

1

1

1

table A2B. Distribution of s y m m e t r y groups of s a t e l l i t e knots

# Crossings

D3

D5

D7

D9

D~

i

i

i

i

i

1

1

13n 14n

2

15n

1

1

4

16n

10

Appendix Ira: Tables of Hyperbolic Knots with Selected Symmetries For compactness, we have adopted the following alphabetic encoding of the DT notation. a

b

c

d

e

f

g

h

i

j

k

I

m

n

o

p

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-22

-24

-26

-28

-30

-32

Knots with + amphicheiral symmetry beifhakdlcgj bfjgiaeldncmhk bgihkmjacnldfe cflnhkbjmegadi dfgljabkmnechi dfgljbakmnecih bdganhmcjloifpek bdganpmcjloifhek begnajpdlofmichk belfiaojdhpmckgn bfglhkacmnepdoij bfgljapdmoecnihk bfhlnjaicgpdkoem bfkhjgaendpcoim] bfkhojalndpgcime bfkohjaendgpciml bfighjadneopcimk bflhgjadnepocimk

Z2 Z2 Z4 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z4 Z2 Z2 Z2 Z2 Z2

bgempkajnlfhdoic bgihlnkajcpmdofe bglmhjkaoefpcdin bglmhjkaofepdcin bhijkmpoanedflcg bhjilknapcomegfd bhjilonagcpmekfd cdganhmbkloifpej cdganpmbkloifhej cegnahlbkmofipdj cegnhblakmofpjdi cfglhkbanoepdjim cfmjohlaknebgpdi cgenpblakomfhjdi cgfjiplmkonbahde cgf]hkbaonepdjim cglnipbmkodfahje cgmnhibdkoefpajl

Z2 Z4 Z2 Z2 Z4 Z4 Z4 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2

cgmpniblkoehfajd Z2 cgnmhidbkofepalj Z2 chjmglnakpbeodfi Z4 chlnoibmkpdfgaje Z2 chmnoilbkpefgadj Z2 chmonilbkpegfadj Z2 clfgibpmkdnoajhe Z2 clfoipbmkdngahje Z2 dehnaojclmpfigbk Z2 fhkmjloanpcebdgi Z4 bfhgIlacKMDnJe Z4 bdFJaILmCEnPOkhG Z4 bdGaNHMCjIOiFPEK Z2 bdGaNPMCjIOiFHEK Z2 bdGaNhCmJLolfPkE Z2 bdGanHCMJLoIFpke Z2 bdJFaILOCEnHPkmG Z4 bdgaNhcmJLOIfPKE Z2

bdganHcMJLOIFpKe Z2 beGNaJPDIOFmiCHK Z2 begnaJdLPFIMHocK Z2 bfKHOJaLnDPGCimE Z2 bfLgjhadNeOCPIMK Z2 bfhgKnaicLoDpemj Z4 bfhlgJadNPEOcIMK Z2 bglhkNajoPledFMC Z4 bhEMGkCaOlfnDjPI Z2 bheMgkcaOLfNDJPI Z2 cEnoHBIDKmFGpjal Z2 ceghNblaKMOPfJDI Z2 cegnaLKbFOPHIJdM Z2 cehNgblaKMPfOJDI Z2 ceoHnbIDKMGpFJal Z2 cfmoHIbDKNEGpaJl Z2 cgnmHIbDKOFEpaJl Z2 cnoHabMDKFGpIJel Z2

Knots with Z3, 24 symmetry dfholikmebnacjg bgihkmjacnldfe bfhlnjaicgpdkoem bgihlnkajcpmdofe

c + + +

Z3 Z4 Z4 Z4

bhijkmpoanedflcg bhjilknapcomegfd bhjilonagcpmekfd chjmglnakpbeodfi

+ § + +

Z4 Z4 Z4 Z4

fhkmjloanpcebdgi + bfhgIlacKMDnJe + bdFJaILmCEnPOkhG + bdJFaILOCEnHPkmG +

Z4 Z4 Z4 Z4

bfhgKnaicLoDpemj + Z4 bgIhkNajoPledFMC * Z4

Knots with Ds symmetry cemfhaikdlngobj i deflijcKabGH i

beKFhaJmdLCGI i beKGaHjDmFLCni i

bfINhaDKeMGicJi cEMfHAiKDINGoBJi

defLhmjkaCnoIbgi

VOLUME 20, NUMBER 4, 1998

'#,'5

Knots with D6 symmetry chgfbaied dfhgibace befgiahcjd eghijkbadcf bfghikacjdle bfhgjalicked cfehgjilkbad cjelgbidkfah

bfgilaejchmdk fhijklmacbedg bghljkmacdlenf bghlklanjdcmfe bfjhmgaokciendl bghiknafcldjoem eimkljncabodhfg eimlkjncbaodhgf

i i i i i i a a

i i i i i i i i

fijlmknocbaedhg fjimlkoncbaedhg gijklmnoabdcfeh bfhjngaelkcipodm bhijklmoacdenfpg bhljkmnapcledogf chljklnobamedpgf cihjlkonbamedpgf

i i i i i i i i

beGFhaCid bdJFalKCEHLG cFEhgJllkBAd cJElgBldkFAh ceFhGjIKIBaD cEfjhkmdlgaBi beiHLajNOgfMDKC chiKLJmnbaEDogf

i i

i i a i i i

cihLKJnmbaEDogf i eImKLjNcABoDhFG i ehGkinmLaOdcBfJ i beNGaHJDlFmpiOCk i bfKhIaLoNGPEJCdM a cENfhAJdKMGolpBI i CGKhJNLAmOFBDpEI i cfglijkPmnbaoedH i

Knots with D7 - D16 symmetry efgIOklmCNpabhDj cdefghab bfghjlaickdme fhijlkmbadceg bghijlnackdmeof gijklmnobadcfeh cnepgbidkfmhojal

i a i i i i a

D7 D8 D8 D8 D8 D8 D8

dkfmhojalcnepgbi a bfHGikaCjdle i bgIHJLKaCEDF i bfHKNapICGLDJOEM i bgihkNapjcledOFM i cEonhBIdKmGFPjaL i cNEpgBIdkFMhoJAl i

D8 D8 D8

D8 D8 D8 D8

Appendix IIh Knotscape and SnapPea Knotscape is primarily a graphical interface to the knot tables. It is currently still in the development phase, but it already allows the user to browse through the knot tables and locate user-supplied knots in the tables. It will display a picture of the knot currently selected and will compute polynomial invariants and a few other invariants. The graphical part of the program is written in Tcl7.4/Tk4.0, and the computational modules are written in C. It has been tested on Linux systems and on Sun systems. The program is available for download from http://www.math.utk.edu/ -morwen. SnapPea is an interactive c o m p u t e r p r o g r a m for creating and studying hyperbolic 3-manifolds. At present, the most full-featured version runs on a Macintosh and is available for free from www.geom.umn.edu. (For current information about other platforms, please contact [email protected].) SnapPea w o r k s with arbitrary closed and c u s p e d hyperbolic 3-manifolds. Initially, the user specifies a manifold by drawing a knot or link and asking SnapPea to construct its complement, or by selecting a manifold from SnapPea's built-in databases, or by some other method. Thereafter, the user can create new manifolds from old ones by taking finite-sheeted covers (or b r a n c h e d covers) by drilling out closed geodesics to create new cusps, or by doing Dehn fillings to seal off old cusps. F o r all manifolds, SnapPea computes a wide variety of numerical, algebraic, and graphical invariants.

46

THE MATHEMATICALINTELLIGENCER

dKFmhOJa|CNepGBi i D8 deigkNjPlmaocFbH a D8 dJiIKnAcOMFpgBeH i D9 defghijabc a DIO dinglbjoemchafk i DIO gikjlnmobacedfh i DIO bghikmoajcldnepf i DIO

bglHjlnaCkdmeof bhJIKMLnaCEDFog efghijklmnabcd fghijklmnopabcde

i i a a

DIO DIO DI4 DI6

REFERENCES [AB]

J.W. Alexander and G.B. Briggs, On types of knotted curves, Ann. Math. 28 (1927), 562-586. [Arn] B. Arnold, M. Au, C. Candy, K. Erdener, J. Fan, R. Flynn, J. Hoste, R.J. Muir, and D. Wu, Tabulating alternating knots through 14 crossings, J. Knot Theory Ramifications 3(4) (1994), 433-437. [BS] F. Bonahon and L. Siebenmann, The classification of algebraic links, unpublished manuscript. [BZ] G. Burde and H. Zieschang, Knots, New York: de Gruyter (1985). [Cau] A. Caudron, Classification des noeuds et des enlacements, Prepublication Math. d'Qrsay, Qrsay, France: Universitr ParisSud (1980). [Con] J . H . Conway, An enumeration of knots and links, Computational Problems in Abstract Algebra (Leech, ed.), New York: Pergamon Press (1970), 329-358. [Deh] M. Dehn, Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), 402-413. [DH] H. Doll and J. Hoste, A tabulation of oriented links, Math. Computat. 57(196) (1991), 747-761. [D-FJ C.H. Dowker and M.B. Thistlethwaite, Classification of knot projections, Topoi. Appl. 16 (1983), 19-31. [EP] D.B.A.Epstein and R.C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Diff. Geom. 27 (1988), 67-80. [ES] C. Ernst and D.W. Sumners, The growth of the number of prime knots, Math. Proc. Camb. Phil. Soc. 102 (1987), 303-315. [Gq C. Gordon and J. Luecke, Knots are determined by their complements, J. Am. Math. Soc. 2 (1989), 371-415.

[Hak]

W. Haken, Theorie der Normalfl&chen, Acta Math. 105 (1961), 245-375. [Has1] M.G. Haseman, On knots, with a census of the amphicheirals with twelve crossings, Trans. Roy. Soc. Edinburgh 52 (1917), 235-255. [Has2] M.G. Haseman, Amphicheiral knots, Trans. Roy. Soc. Edinburgh 52 (1918), 597-602. [Hem] G. Hemion, On the classification of homeomorphisms of 2manifolds and the classification of 3-manifolds, Acta Math. 142 (1979), 123-155. [Kau] L.H. Kauffman, State Models and the Jones Polynomial, Topology 26 (1987), 395-407. [Kirl] T.P. Kirkman, The enumeration, description and construction of knots of fewer than ten crossings, Trans. Roy. Soc. Edinburgh 32 (1885), 281-309. [Kir2] T.P. Kirkman, The 634 unifilar knots of ten crossings enumerated and defined, Trans. Roy. Soc. Edinburgh 32 (1885), 483-506. [KS] K. Kodama and M. Sakuma, Symmetry groups of prime knots up to 10 crossings, in Knots 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka, Japan, 1990 (A. Kawauchi, ed.), Berlin: de Gruyter (1992), 323-340. [Lis] J.B. Listing, Vorstudien zur Topologie, GOttingen Studien, University of GOttingen, Germany (1848).

[Lit1] [Lit2] [Lit3] [Lit4] [Lit5] [Men]

[MP]

[M-rJ [Murl] [Mur2]

[Per1]

C.N. Little, On knots, with a census of order ten, Trans. Connecticut Acad. ScL 18 (1885), 374-378. C.N. Little, Non alternate +_ knots of orders eight and nine, Trans. Royal Soc. Edinburgh 35 (1889), 663-664. C.N. Little, Alternate -+ knots of order 11, Trans. Roy. Soc. Edinburgh 36 (I 890), 253-255. C.N. Little, Non-alternate + knots, Trans. RoyalSoc. Edinburgh 39 (1900), 771-778. C.N. Little, Knots, with a census for order ten, Ph.D. Thesis, Yale University (1885). W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23(1) (1984), 3744. K. Murasugi and J. Przytycki, The skein polynomial of a planar star product of two links, Math. Proc. Camb. Phil. Soc. 106(2) (1989), 273-276. W. Menasco and M.B. Thistlethwaite, The classification of alternating links, Ann. Math. 138 (1993), 113-171. K. Murasugi, The Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187-194. K. Murasugi, Jones polynomials and classical conjectures in knot theory II, Math. Proc. Camb. Phil. Soc. 102 (1987), 317-318. K. Perko, On the classification of knots, Proc. Am. Math. Soc. 46 (1974), 262-266.

VOLUME 20, NUMBER 4, 1998

4,7

[Per2]

K. Perko, Invariants of l 1-crossing knots, Prepublications

[Per3]

Math. d'Orsay (1980) K. Perko, Primality of certain knots, Topology Proc 7 (1982),

[Pra]

109-118. G. Prasad, Strong rigidity of Q-rank 1 lattices, Invent. Math. 21

[Rei]

[Ril]

[sw] [Sch] [ST]

t-rail

(1973), 255-286. K. Reidemeister, Knotentheorie, Berlin: Springer-Verlag, 1932. R. Riley, An elliptical path from parabolic representations to hyperbolic structures, in Topology of Low-Oimensional Manifolds, Proceedings, Sussex 1977 (R. Fenn, ed.), Springer Lecture Notes in Math. vol. 722, New York: Springer-Verlag (1979), 99-133. M. Sakuma and J. Weeks, The generalized tilt formula, Geometriae Dedicata 55 (1995), 115-123. O. Schreier, 0ber die Gruppen AaBb = 1, Abh. Math. Sem. Univ. Hamburg 3 (1924), 167-169. C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacific J. Math. 182 no.2 (1998), 329-358. P.G. Tait, On knots I, II, Ill, Scientific Papers, Vol. I, Cambridge: Cambridge University Press (1898), 273-347.

[Thil]

lThi2] [Thi3] [Thi4] [-i-ro] [Wks] BNel] [VVil]

M.B. Thistlethwaite, Knot tabulations and related topics, Aspects of Topology in Memory of Hugh Dowker 1912-1982 (James and Kronheimer, eds.), Cambridge, England: Cambridge Press, London Math. Soc. Lecture Note Series 93, 1-76. M.B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26(3) (1987), 297-309. M.B. Thistlethwaite, Kauffman's polynomial and alternating links, Topology 27(3) (1988), 311-318. M.B. Thistlethwaite, On the structure and paucity of alternating tangles and links, preprint. H.F. Trotter, Noninvertible knots exist, Topology 2 (1964), 275-280. J. Weeks, Convex hulls and isometries of cusped hyperbolic 3-manifolds, TopoL AppL 52 (1993), 127-149. D.J.A. Welsh, On the number of knots and links, Colloq Math. Soc. J. Bolyai 60 (1991), 713-718. R.F. Williams, The braid index of an algebraic link, Braids (Santa Cruz, CA, 1986), Contemporary Mathematics Series Vol. 78, Providence, RI: American Mathematical Society (1988).

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Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

Editor

I

Vhen he g r a d u a t e d top o f his y e a r in 1842 at C a m b r i d g e University, Arthur Cayley b e c a m e the Senior Wrangler of his day. He w a s the " c h a m p i o n student," the first in the order of merit, a position gained b y a m a t h e m a t i c a l trial of s t r e n g t h cond u c t e d o v e r six days of c o n t i n u o u s examination. The achievement had less to do with mathematical talent than with a fast writing technique coupled with mental and physical stamina. Occasionally a mathematician of merit, such as Cayley, could triumph in the competition, which w a s likened in the popular imagination to a horse-race. By acquiring this h a l l o w e d title, A r t h u r Cayley w a s well p l a c e d for a Trinity College fellowship, the p r e p a r a tory s t e p t o w a r d s a d m i s s i o n to the Anglican priesthood. He w a s a loyal Anglican w h o s e formative y e a r s w e r e s p e n t within the "wails" of the Church of England. The secondary school he attended, King's College London, was founded b y the Anglican establishment and o p e n e d in 1831 as a counterweight to the "godless" University College o p e n e d five years before in G o w e r Street, not two miles away on the other side of the metropolis. Pupils at King's were e d u c a t e d in an environment in which the Proprietors saw their duty to "imbue the minds of youth with a knowledge of the doctrines of Christianity as inculcated by the United Church of England and Ireland." [1] This atmosphere continued at Cambridge University, where religious observances were strictly upheld. Daily attendance at chapel was m a n d a t o r y - - a n d twice on Sunday. An e a r n e s t young m a n in the 1840s, Cayley w a s from a n e w g e n e r a t i o n of m a t h e m a t i c i a n s which b e n e f i t t e d from the m a t h e m a t i c a l reforms b e g u n b y the Cambridge Analytical Society (Charles Babbage, J o h n F. Herschel, George P e a c o c k ) in the early p a r t of t h e century. M e m b e r s of that Society w e r e apt to r e m e m b e r long j o u r n e y s to L o n d o n

W

by stagecoach, w h e r e a s Cayley c o u l d t a k e an e a s y j o u r n e y by steam-train to the capital w h e n the Great Northern railway w a s c o n n e c t e d to Cambridge in 1845. Unlike t h e earlier generation, who a b s o r b e d continental ideas a n d t r a n s m i t t e d t h e m locally, Cayley reversed the flow and t o o k his ideas to the continent. He was the first in England (and a l m o s t alone) to publish regularly in the Journal fiir die reine

und angewandte Mathematik (Crelle) founded by A. L. Crelle (1826) and Liouville's Journal de Mathdmatiques Pures et Appliqudes (1836). Cayley w a s a young p r o f e s s i o n a l a n d was cont e m p t u o u s of t h o s e who t r e a t e d mathematics as an avocation. Cayley r e p r e s e n t e d a n o t h e r b r e a k from the earlier generation. He restricted his scientific view, quite unlike the generalists at Cambridge; William Whewell w a s p e r h a p s an e x t r e m e example, of w h o m it was said that "omniscience w a s his foible." Cayley's natr o w e r focus w a s m o r e akin to that o f George P e a c o c k , his first-year t u t o r at Trinity. P e a c o c k taught Astronomy, an obligation be felt as h o l d e r of t h e L o w n d e a n c h a i r of G e o m e t r y a n d Astronomy, b u t he was m o r e at h o m e with m a t h e m a t i c s . Although in t i m e Cayley g r e w to b e m o r e like Whewell, his "omniscience" w a s restricted to t h e a r e n a o f m a t h e m a t i c s and not to t h e p a n o r a m a of all science. A m o n g s t the illustrious group of scientists w h o g r a d u a t e d from Cambridge in the early 1840s (R. L. Ellis, G. G. Stokes, J. C. Adams), Cayley was to become the representative "pure" mathematician. To b e sure, "mixed mathematics" is c o n s p i c u o u s in his w o r k (for instance, r e f e r e n c e s to optics in his pap e r s on group t h e o r y a n d m o r e substantiaily in his Reports on Dynamics), b u t his p r e o c c u p a t i o n was invariably with a s u b j e c t ' s m a t h e m a t i c a l c o n t e n t and he d i s p l a y e d s c a n t regard for considerations of an empirical kind. In 1845 Cayley b e c a m e a Cambridge

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Master of Arts. He was thus entitled to remain a fellow of Trinity College for a further seven years before taking Holy Orders. Fellows were paid from the college Chest, and though they were offered college accommodation, they had complete freedom of movement. They could pursue any interest, and they enjoyed social prestige. No wonder that fellowships were actively sought by the young men who aimed for the highest positions in the Mathematical Tripos order of merit, the barometer of undergraduate success. There was no obligation for a college fellow to accept a college or university appointment. Cayley did become an assistant tutor, but his main attention was on research and he took few pupils. As there was no requirement for a fellow to live in Cambridge, one popular diversion was to read for the legal bar at one of the "inns of court" in London. These institutions, four in number, had all the trappings of the Cambridge collegiate system. College architecture was duplicated at Lincoln's Inn, Gray's Inn, and the Inner and Middle Temples, and college hierarchy was replicated within their walls. The Seniority, dons, and undergraduates of a Cambridge college bore a strong resemblance to the benchers, barristers, and pupils of the inns. To enter these medieval institutions of learning was to "finish" an Oxbridge education. Cayley entered Lincoln's Inn in 1846 and was called to the bar in 1849.

Two Roads Diverge In 1849 Cayley saw several of his friends enter the academic world. In April, Hugh Blackburn, with whom he discussed the intricacies of geometry around the inns of court, abandoned the law to become professor of mathematics at Glasgow. In August, George Boole heard he had been elected to the mathematics chair in Cork; Cayley was one of the first to congratulate and envy him the prospect of long vacations (almost equalled, it should be said, by the long vacations at the inns of court). One appointment which may have attracted Cayley's ambition was the vacancy for the Cambridge Lucasian

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THE MATHEMATICALINTELLIGENCER

Arthur Cayley in the 1870s. Reproduced by permission from vol. 6 of his Collected Works.

professorship. But evidently he did not apply, and George Stokes was elected unopposed in October, 1849. Cayley stayed with the Law. His seven years of grace at Cambridge had not expired, but though his Anglican religious beliefs were orthodox, he felt no vocation for the priesthood. Had he taken this path, it might have led to a life of College teaching, capped, if he wished to marry, by one of the seventy church livings in the gift of Trim'ty. From the ranks of the undergraduates the "reading man" was in line to progress serenely along the rails: "After three years, he comes out as a high classic, or a wrangler; takes pupils, obtains a fellowship, and dies ultimately at an advanced age in the possession of a college living, virtuous, ignorant, happy and beloved." [2] If a secular appointment could be found, it invariably meant a low income. In 1851 T. H. Huxley was elected to the professorship of Natural History at London's Royal School of Mines. The following year he grumbled to his sister: "Science in England does everything--but pay. You may earn praise but not pudding." [3] The life of a London barrister in the 1850s could be precarious, but compared with the academic life, the Law

did have the potential of financial reward. If there had been any doubt as to Cayley's choice, he would have been swayed by family responsibilities. His father died in 1850, leaving him as head of the family with a mother and two unmarried sisters at home. His younger brother Charles, who described himself in 1851 as a "patent design agent," would have been of little help. He soon gave up this occupation to concentrate on his literary translations of Dante and Homer, and even at this stage was moving towards the life of an impecunious scholar. Both brothers were studious by nature, but Arthur appeared more in tune with the world. Charles managed to lose his capital in a railway speculation, and Arthur and the family rallied in support. Cayley showed the same steadfast behaviour in the inevitable friction between mathematical research and the immediate demands of legal work. His friend J. J. Sylvester remembered just one occasion when he saw him out of sorts with the conflict between the two lives. On one day the appearance of yet another legal brief was too much: "He flung the papers on the table with remarks more forcible than complimentary concerning the person who had distracted his attention at such an inopportune moment." [4l Legal chores only interfered occasionally with mathematics, but nevertheless his skills of legal draughtsmanship were valued by the closeknit circle of London's Real Property Lawyers. The transfer of landed estates encouraged a body of expertise in this aspect of the law, and was a monopoly enjoyed by barristers. (This is no longer the case, and solicitors, the other branch of English lawyers, deal with property. Barristers usually restrict themselves to appearances in court.) As now, senior barristers employed juniors to "devil" for them. During his time in the law, Cayley was actually content with the lowly "stuff-gown" of the junior counsel and did not "take silk," the distinguishing mark of the Queen's Counsel. His overriding passion was mathematics. On 3 June 1852 Cayley was elected to the Royal Society of London. Sylvester was a member of long standing and acted as Cayley's principal proposer: he

Stone buildings of Lincoln's Inn (drawing 1925). Cayley's Chambers are the far end, extreme left. By permission.

r e f e r r e d to Cayley as d i s c o v e r e r of "Hyperdeterminants" and eminent as a "geometer and analyst." No fewer than t w e n t y - o n e m e m b e r s o f t h e Victorian scientific establishment signed Cayley's certificate. In the 1850s Cayley began intensive w o r k on quantics--invariant theory, as it had been rechristened b y Sylvester. On election to the Royal Society, he began a definitive series of "memoirs on quantics." Invariant theory was a rich field which suggested other research, including the theory of partitions and matrix algebra (which grew out of a concern for notation), and it was the subject which lay at the centre of his understanding of projective geometry and the solution of polynomial equations.

Two Roads Converge

In the mid-1850s, Cayley began to apply for academic positions. Financial considerations may have suggested he s u p p l e m e n t his barrister's i n c o m e by taking other work. It was quite c o m m o n for mid-century "men of science" to hold several appointments simultane-

ously. Stokes s e e m e d particularly able in this regard, holding in the 1850s, the Lucasian professorship at Cambridge and the professorship of Physics at the Government School of Mines in London, t o g e t h e r with other m i n o r appointments. With his interest in obtaining an a c a d e m i c appointment, Cayley's rate of m a t h e m a t i c a l p r o d u c t i o n r o s e sharply. F o r the p e r i o d 1853-1856 he p u b l i s h e d (on average) 10 p a p e r s p e r year, b u t in the p e r i o d 1857-1860 this rose to 30. Cayley wrote e x p o s i t o r y papers, s u r v e y papers, "calculation" pap e r s as well as traditional r e s e a r c h papers. He w a s a central figure in m a t h e m a t i c s at h o m e and abroad. In 1855 he filled the fiftieth v o l u m e o f CreUe with eight p a p e r s p r i n t e d one after the other. In 1857 he s u b m i t t e d his p a p e r s on m a t r i x algebra ( u n n o t i c e d at the time b u t famous in retrospect), and a y e a r later he outlined a projective theory of distance to a small group assembled at the annual British Association Meeting in Leeds. He p e r f o r m e d valuable t a s k s of translating p a p e r s from German (for Ludwig Sch~ffli and

Gottold Eisenstein) and the activity of "translating" results in pure geometry to theorems in analytical geometry. Cayley a p p l i e d to the Board of Commissioners for the Affairs of India for m e m b e r s h i p of the b o a r d of e x a m i n e r s which w o u l d a d m i n i s t e r the newly ann o u n c e d c o m p e t i t i v e e x a m i n a t i o n s for the civil service. Evidently George Boole a p p l i e d too, but Cayley a n d Stokes w e r e a p p o i n t e d for m a t h e m a t ics, and t o g e t h e r t h e y p r e p a r e d the examination p a p e r s in July, 1855. [5] Cayley also c o n s i d e r e d taking private pupils. U n d o u b t e d l y a m a n of his attainments, h a d he b e e n a citizen o f F r a n c e or Germany, w o u l d have b e e n safely e n s c o n c e d in a university, b u t e m p l o y m e n t p r o s p e c t s for scientists in m i d < e n t u r y Britain w e r e meagre. [6] A chance for a m o r e substantial position came in early 1856 with a vacancy in the chair of Natural Philosophy at Aberdeen's Marischal College. The daily duties a m o u n t e d to lecturing for two hours from the beginning of N o v e m b e r to the beginning o f April on such subj e c t s as Statics, Dynamics, Hydrostat-

VOLUME20, NUMBER4, 1998 51

ics, Optics, and Astronomy. Cayley was hardly inclined to the idea of professorial lectures, which, in the Scottish university system, would be attended by large classes. Understandably, for it was at variance with the intimate Cambridge tutorial system. In fact, Cayley was a poor lecturer, and the hurly-burly of a Scottish lecture room would have presented a stiff challenge. The great inducement for him was the seven-month vacation. [7] Sylvester registered surprise that Cayley would be willing to quit London: We (I mean your friends) shall all be sorry to lose you by your going to such a distance as Aberdeen but if you think such a change would conduce to your advantage or happiness I can only wish that you may succeed in obtaining your wishes. I thought you used to express an objection to the Scotch system of lecturing and to taking anything but a pure Mathematical Professorship. [8] The successful applicant was the young Scot, James Clerk Maxwell, then in his mid-twenties and a fellow at Trinity College, Cambridge. Cayley next turned his attention nearer h o m e - - t o South Wales. In September 1857, his name became linked with the ill-fated Western University of Great Britain (an episode in British education that is largely unknown). [9] The position had its attractions, not least a salary at s which compared favourably with a professorship at Oxford or Cambridge. The planned institution was to be modelled on the privately organised Ecole

Centrale des Arts et Manufactures founded in Paris in 1829. The "university" was to be adapted to the "Wants of the Age" in which young men were to be taught the "accurate knowledge of needful things." Prospective students would have to be drawn from the middle classes since the enterprise was fundamentally a money-making venture. The university prospectus was signed by the Resident Council of the College, driven by its mainstay, William Bullock Webster, an entrepreneur with a ready eye for speculation. In the end, the ambitious plans came to nought; Webster fled the country, 52

THE MATHEMATGALINTELLIGENCER

King's College London, about 1830. By permission.

and several years later found his niche in one of Queen Victoria's prisons. Towards the end of 1858 the Lowndean chair of Geometry and Astronomy at Cambridge became vacant through the death of Peacock. Cayley thought he had a chance and wrote to Sir John Herschel for a testimonial: "I take the liberty of acquainting you that I am desirous of offering myself as a candidate for the Lowndes' Professorship at Cambridge and of requesting that if you should think fit so to do, would have the kindness to give me a testimonial in support of my application." [10] De Morgan, who was familiar with Cayley's mathematical work and was a fellow member of the Royal Society Catalogue committee, commended him to Herschel: "Cayley is a capital man for it." [11] The competition was strong. John Couch Adams had been Senior Wrangler in the year following Cayley, and was a renowned astronomer, despite the Neptune debacle (when national pride was dented by the discovery of the planet Neptune as a result of Leverrier's calculations rather than Adams's predictions). When Adams was appointed and Cayley heard of his own failure, Sylvester was quick in support: "I had not heard before your note of your d i s a p p o i n t m e n t - - i t is most vexatious and it annoyed me as much as a personal disappointment could do. I shall try and call upon you

very soon. You know how much pleasure and profit it always is to me so to do." [12] Another opporttmity for Cayley arose when the combination of the chair of Astronomy at Glasgow and Directorship of the Observatory fell vacant in the autumn of 1859. The salary was only s but there was an official house in the grounds. This time Cayley's reputation as a pure mathematician worked against him. William Thomson, an old Cambridge friend, entertained doubts and confided in Stokes: Cayley is thinking on being a candidate, but it will probably be considered and I believe justly that he is not physical enough. I say this to you in confidence, knowing that you will not misunderstand what I say of so good a friend of us both. I think Cayley ought to be provided for by the country--mathematician laureate would be his right p o s t - but there is no doubt but that popular or physical lines of science are not in his way, and that in a situation where either may be required, he might not be well placed. [13] The long road to Academe reached an end with Cayley's appointment to the newly established Sadlerian chair at Cambridge, which replaced seventeen college algebra lectureships, one in each of the Cambridge colleges. A

Journal of Pure and Applied Mathematics which itself faced extinction in the mid-1850s. Through residence in Blackheath, a p r o s p e r o u s suburb just south of the City, and within easy reach of London by suburban railway, he was able to build and maintain an extensive a c a d e m i c network, a range of activity which would have b e e n difficult from a fenland base. Cayley's London connections carried over to his early years at Cambridge. During the 1860s the university h a d yet to c a t c h the research habit, a n d the capital offered the greater o p p o r t u A breakthrough in invariant theory. Reproduced by permission nity for mixing with from the Sylvester papers, St. John's College. scientists. Though f o r m e r legal colleague w i s h e d him his d e c i s i o n to practice Law w a s not well: "one of the first o f living mathe- the life he ultimately wanted, his time maticians, whose rare c a p a c i t y and at- in London and his administrative int a i n m e n t s have n o w h a p p i l y b e e n v o l v e m e n t h a d p l a c e d him at the cent r a n s f e r r e d to a m o r e congenial em- tre o f British scientific activity. It m a d e ployment." [14] A u t u m n 1863 was the all the difference. only time w h e n Cayley i n t e r r u p t e d his a b s o r p t i o n with studying a n d writing REFERENCES m a t h e m a t i c s . In S e p t e m b e r he m a r r i e d 1. F. J. C. Hearnshaw, The centenary history Susan Moline, and, settling b a c k in the of King's College, London 1829-1929, m a r k e t t o w n w h e r e he h a d b e e n a stuLondon: Harrap (1929). p. 40. dent, he busied himself with the prepa2. John Smith, [pseudonyn for J. Delaware ration of his first c o u r s e o f l e c t u r e s - Lewis]. Sketches of Cantabs, 2rid Edn., on Analytical Geometry. London: George Earle (1849). p. 7. The years spent in London in the 3. T. H. Huxley to Elizabeth Huxley: 3 May 1850s had b e e n a p e r i o d w h e n he de1852: in Life and letters of Thomas Henry v e l o p e d a b r o a d e r p e r s p e c t i v e and enHuxley. 2 vols. Leonard Huxley, London: g a g e d his c o n s i d e r a b l e administrative Macmillan, (1900) vol. 1, p. 100. ability. He t o o k p a r t in c o m m i t t e e 4. Simon Newcomb, The reminiscences of an w o r k of the Royal Society a n d u s e d his astronomer, London and New York: skills as a barrister to r e o r g a n i s e the Harper (1903). p. 281 constitution of the Royal A s t r o n o m i c a l 5. George Boole to De Morgan: 3 January Society. He began his interest in the ac1855: in The Boo/e-De Morgan Correspontivities of the British A s s o c i a t i o n for dence 1842-1864, G. C. Smith, Clarendon the A d v a n c e m e n t of Science, a conPress: Oxford (1982). pp. 68-69. nection he m a i n t a i n e d all his life. He 6. Charles Babbage, The Exposition of 1851, w a s a keen s u p p o r t e r of the limping London: John Murray (1851).

J

4.

Cambridge and Dublin Mathematical Journal and its successor the Quarterly

8. J. J. Sylvester to Cayley: 22 February 1856: St. J. Coll. Camb. 9. Iolo Wyn Williams, The Western University of Great Britain, Collegiate Faculty of Education Journal (1966), Univ. College, Swansea, 32-40. 10. Cayley to Sir J. W. F. Herschel: 13 Nov 1858: Royal Society of London, HS.5.226. 11. Augustus De Morgan to Sir J. W. F. Herschel: 15 November 1858. in Memoir of Augustus De Morgan, Sophia De Morgan, London: Longmans Green (1882). p. 297. 12. J. J. Sylvester to Cayley: 22 December 1858: St.J.ColI.Camb. 13. William Thomson to G. G. Stokes: 6 October 1859: in The correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs. 2 vols. David B. Wilson. (ed.) Cambridge: Cambridge University Press (1990). vol. 1, p. 250. 14. Charles Davidson, Jacob Waley, Thomas Key, (eds.) Davidson's precedents and forms, 3rd Edn. London: William Maxwell (1873). vol. 3, Pt. 2, p. 1067 f.n.

7. Cayley to William Thomson: 14 February 1856: CUL Manuscripts: Add 7342.C59C. VOLUME 20, NUMBER 4, 1998

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SUSAN LANDAU

,/2 + l-our Different Views

Introduction H o w m u c h time d o e s it t a k e to factor p o l y n o m i a l s ? H o w can you efficiently tell if a p o l y n o m i a l has r o o t s e x p r e s s ible in t e r m s of radicals? Is t h e r e a fast m e t h o d to d e c o m p o s e a polynomial into l o w e r - d e g r e e c o m p o n e n t s ? S u p p o s e it is claimed that ~/~/2

-

1 = ~

-

~79

+

~ ;

h o w you can c h e c k if this is true? You have to study the underlying algebraic structure, but often the theorems are not conducive to efficient computation, and n e w u n d e r s t a n d i n g - - a n d new results---are needed. In this article I p r e s e n t s o m e t h e o r e m s that resulted from the effort to find fast m e t h o d s for algebraic simplification. It should be no surprise that, in a computational area, conjecture and e x a m p l e s go h a n d - i n - h a n d - - b u t only after the fact did I realize h o w closely. Long after I had experimented, conjectured, again experimented, and then proved did I discover that a simple exanlple--*~/2 + ~ / 3 - - s h e d s m u c h light on four seemingly unrelated results. In several cases, the theoretical ideas leap from the simple radical. And that caused me to think m o r e about the role of example. It is a fact little r e m a r k e d u p o n that Euler c o m p u t e d his w a y to the law of quadratic reciprocity. Gauss's calculations led him to the prime n u m b e r theorem. Similarly, Dedekind

and F r o b e n i u s c o m p u t e d their w a y to conjecture and prove a n u m b e r of results concerning group representations. Despite t h e s e d e m o n s t r a t i o n s of the p o w e r of computation, s u c h calculating fell out favor in the early part o f this century. By introducing a b s t r a c t m e t h o d s to algebra, Hilbert p r o v e d the basis theorem, the syzygy theorem, and the Nullstellensatz. Not long afterward, Noether employed similar a b s t r a c t m e t h o d s in h e r w o r k on ascending chain conditions. Computation w e n t out of vogue, eschewed in favor of abstraction. It was not unusual to see group theory taught without reference to a single concrete group, to find the fundamental theorem of Galois t h e o r y proved without the calculation of a single example. There are good reasons to rely on the abstract approach: it is powerful, and for m a n y areas of mathematics, even small e x a m p l e s can be remarkably difficult to compute (commutative algebra is one such). Yet, e x a m p l e s have m u c h to t e a c h us. E x a m p l e s can point to a flaw in reasoning, and e x a m p l e s can give students s o m e t h i n g to hold onto as t h e y a t t e m p t to grasp elusive theory. E x a m p l e s can d e m o n s t r a t e p a t t e r n s and l e a d to conjectures. But to those w h o w e r e m a t h e m a t i c a l l y raised in the a b s t r a c t school, it m a y be surprising to discover h o w m u c h e x a m p l e s can guide research. In this article, I p r e s e n t four results a b o u t c o m p u t a t i o n a l algebra s e e n from the p e r s p e c t i v e of ~/2 + X/-3. My m a i n

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purpose is illustrating four results in computational algebra, but along the way I hope to demonstrate the p o w e r of computation.

Factoring Polynomials How does one factor a polynomial over the rationals? One might wonder if the problem is decidable; an algorithm from an astronomer in 1793 shows it is. Let f(x) be a polynomial of degree n over 77. Compute the values f(0), f ( 1 ) , . . . , f(n), and then factor each of the f(i). Choose a set of factors d(0), d(1) . . . . , d(n), and interpolate to find a potential factor d(x) of f(x). Integer factorization is decidable, and because there are only finitely many sets of the d(i), factoring is decidable. However, I caution the reader not to implement this algorithm, as it takes exponential time even in the best case, namely when all the f ( i ) are prime. ( C o m p u t e r scientists define the size of a p r o b l e m to be the n u m b e r of bits used to represent the problem. Thus, the input size of "factor the integer n" is log n, as log n bits are n e e d e d to represent n. The s t a n d a r d viewpoint is asymptotic behavior, so I ignore constants, and, in particular, the base for the log function.) Since the 1970s, the standard method for factoring has been the Berlekamp-Hensel algorithm (see [1, 11]). This works by factoring the polynomial m o d p for some suitable choice of a prime p, and then "lifting" the factorization to one rood p2, then to m o d p4, and so on until the coefficients are sufficiently large that one has a factorization that "resembles" the factorization over the integers. For example, the polynomial xa - 8x3 + ar - 2 4 x - 6 factors into (x e + 2x + 3)(x 2 + 3)

(mod 5)

and into (x 2 -

Sx - 2)(x 2 + 3)

(mod 25)

and, finally, into (x 2 - 8x - 2)(x 2 + 3) over the integers. That's not so bad. The real issue is, does this algorithm always work? One can always factor m o d p, but will the lifting always be efficient? Are there polyomials for which the factoring blows up? Unfortunately, the answer is yes. Swinnerton-Dyer discovered certain irreducible polynomials that factor into linear or quadratic factors mod m for every integer m. Consider the polynomial x 8 - 40x 6 + 352x 4 - 960x 2 + 576. Over Q, this is irreducible. But it factors into (x 2 + 6x + 6)(x 2 + 6x + 3)(x 2 + x + 6)(x 2 + x + 3) (rood 7), and into four quadratic factors mod 49, and into four quadratic factors m o d 343, and so on. Indeed, this polynomial

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THE MATHEMATICALINTELLIGENCER

will factor into linear or quadratic polynomials (mod m) for every integer m. A polynomial of lower degree with the same property is x 4 - 10x ~ + 1. Its zero ~/2 § V ~ makes it one of a special class of polynomials discovered by Swinnerton-Dyer. These polynomials have zeros of the form

for a set of distinct primes P l , . . . , Pn. Galois theory explains w h y they split into so m a n y pieces mod p. Take an irreducible Swinnerton-Dyer polynomial, sayf(x) of degree 2n. Over Q, it has Galois group (Z/277)n. When p does not divide the discriminant off(x), the Galois group of f(x) over ~/pE is a subgroup of the Galois group off(x) over Q (ifp divides the discriminant, the Galois group off(x) over Z/p~ is a homomorphic image of a subgroup of the Galois group o f f ( x ) over Q). Finite extensions of finite fields are always normal (when one root is adjoined, all the roots are), and the Galois group is cyclic. Thus, the Galois group over Z/p~Y must be Z/277 or ~/~. The polynomial f(x) must give rise to an extension of degree at most 2 over E/p7/. Thus, f(x) splits into linear or quadratic factors mod p for every p. Suppose now one takes two Swinnerton-Dyer polynomials, say fl(x) with zeros ~/2 + ~f5 + ... + ~ and f2(x) with zero V 3 + ~ + ... + pV~2n2n.Then, f l ( x ) f 2 ( x ) is of degree 2n over Q but factors into 2 2n-2, 22n-1 or 2 2n irreducibles (mod p). (One must be careful to stay away from primes that divide the discrkminant of the polynomial, as factoring over such primes introduces repeated factors.) Recombining factors to find the factorization offl(x)f2(x ) over ~ involves at least 2 ~'' combinations.

When Does a Polynomial Have Solvable Zeros? Given an irreducible polynomial over the rationals, how can we tell if its zeros are expressible in terms of radicals? Galois theory gives a technique to discover the answer. That is, in principle. In practice, if f(x) is a polynomial of degree n, Galois's algorithm takes time greater than 2n! steps to determine solvability--even with today's computers, the technique is simply not viable for polynomials of degree higher than 5. There is another well-known method to solve this problem: f a c t o r f ( x ) over Q[x]/f(x), adjoin a zero of one of the remaining nonlinear irreducible factors, factorf(x) over the n e w field, adjoin another zero, and stop only when the polynomial splits completely. This is a faster technique than Galois's original method. Ignoring the size of the coefficients off(x), bootstrapping, as this method is called, takes 2 n steps to find generators for the splitting field o f f ( x ) over Q. Unfortunately, this is exponential in n. There is, however, a polynomial-time algorithm for the problem. The idea is quite simple: divide the the solvability question up into lots o f smaller solvability problems. Let a be a zero of the polynomialf(x). Suppose there is a field Q(fl) between Q and Q(a), Q C ~(/3) C Q(~). Then, a is expressible in radicals over Q if and only if a is expressible in radicals over (~(fl) and fl is expressible in terms of radicals over Q. There's no reason why one should stop with one intermediate field.

S u p p o s e I could find a m a x i m a l chain o f fields Q = Q(fl0) C ~ ( ~ 1 ) C "" C Q(fin) C ~(ot) = Q ( ~ n + l ) , where Q(fli) C F C Q(fl.i+l) implies F = ~(fli) o r F = Q(fli+l). Then, a is e x p r e s s i b l e in radicals over ~ if a n d only if a is e x p r e s s i b l e in radicals over Q(/~n) and fin is e x p r e s s i b l e in radicals o v e r Q(fln-1) a n d . . , and 81 is e x p r e s s i b l e in radicals over Q(fl0) = Q. In terms of group theory, I am looking at s u b g r o u p s of G~, the subgroup of the Galois group that fixes a. Let G act on a set ~ = {al, 99 9 an}. A C ~ is a block o f i m p r i m i t i v ity if for all (r ~ G, ~(A) n A = ~ o r A. The singleton sets and the full set ~ are always blocks; if these are the only b l o c k s of imprimitivity, then the group is acting primitively on t2. To say that there is no field between Q(fli) and Q(fii+ ~) is equivalent to saying the Galois group of the splitting field of Q(fli+l) over Q(fii) acts primitively on the set of zeros of the minimal polynomial of Q(fii+l) over Q(fi). Primitive solvable p e r m u t a t i o n groups are small. In 1982, P~lfy s h o w e d that a primitive solvable p e r m u t a t i o n group acting on n e l e m e n t s h a s no m o r e than n 3-25 e l e m e n t s [8]. So, if one could c o n s t r u c t t h o s e i n t e r m e d i a t e fields, the Galois group that is c o n s t r u c t e d w o u l d be acting primitively on the roots. If t h e e x t e n s i o n s were also solvable, by Phlfy's result they w o u l d b e small, and thus could b e comp u t e d quickly even b y b r u t e force. Gary Miller and I found a polynomial-time algorithm for finding m a x i m a l subfields b e t w e e n Q and Q ( a ) [6]; iterating this gives a m e t h o d for finding a maximal chain o f subfields. Here, I will p r e s e n t H e n d r i k Lenstra's i m p l e m e n t a tion of the Landau-Miller algorithm [7]. Let f ( x ) b e the irreducible p o l y n o m i a l of a o v e r Q. Suppose f ( x ) f a c t o r s into irreducible factors Ilhi(x) in L = Q[x]/f(x) = ~(a), w h e r e a is a zero o f f ( x ) . Then, for each irreducible factor h(x) o f f ( x ) in L, w e define the field Lh as follows: If h(x) = (x - 11) is a linear factor (i.e., if 1, can be written as a polynomial in a with coefficients in Q), let ~r b e the unique automorphism in the Galois group that t a k e s a to % and let Lh be the field of invariants of ~. Otherwise, if ~/is a zero of h(x), a nonlinear factor o f f ( x ) in Q[x]/f(x), then L h = (~(a) N ~('y). All the m a x i m a l subfields of L o c c u r a m o n g the Lh; they are those subfields of highest degree over Q ([7], p. 224). This follows from the simple observation that if G is a finite group with H C J C G subgroups with H r J, and no subgroup I of G such that H C I C J with H r I r J, then there exists ~ ~ G - H such that =J

< H,o'H(r -~ > = J

ifo-H~ ~=H, if o-Ho--~ r H.

One can r e p e a t this p r o c e d u r e [substituting Q(fli) for Q ( a ) ] to d e t e r m i n e a m a x i m a l chain of subfields b e t w e e n Q a n d Q ( a ) . Not only have w e d e t e r m i n e d solvability, b u t w e have also given a technique for d e t e r m i n i n g all subfields of a given field. Let us t a k e a simple Galois e x t e n s i o n but one with s o m e subfield structure. An obvious e x a m p l e to c h o o s e is Q(X/2,~/3) = ~ ( ~ / 2 + ~/3) --~ @[x 4 - 10x 2 + 1]/x; as w e

know, the p o l y n o m i a l x 4 - 10x 2 + 1 h a s zeros _+N/-2 _+ V3. Factoring t h a t p o l y n o m i a l over the field Q ( ~ / 2 + ~/3), the p o l y n o m i a l splits completely: x 4 - 10x 2 + 1

= (x - 10(~/~ + x/5) + (~/~ + ,JS)3)(x + 10(x/~ + ~/5) - ( ~ / ~ + *dS)3)(x + ~/~ + x/5)(x - (~/~ + x/-5)) = (x + ~/~ - ~/5)(x

- ~/~ + ~/5)(x

- ~/~ - ~/5).

(x + x/~ + ~/5) There are t h r e e 2-element b l o c k d e c o m p o s i t i o n s . The b l o c k d e c o m p o s i t i o n {(%/2+ N/-3, ~ v / 2 - ~ - 3 ) , ( - ~ f 2 + ~/3, - ~ / 2 - ~f13)} gives rise to the p o l y n o m i a l s x 2 - 2X/2x - 1 and x 2 + 2~,/--2x - 1 a n d c o r r e s p o n d s to the field Q(~/-2). The b l o c k d e c o m p o s i t i o n {(N/-2 + ~ / 3 , - ~/2 + ~/3), ( N / 2 - ~/3, - ~ / 2 - ~/3)} c o r r e s p o n d s to p o l y n o m i a l factors x 2 - 2 ~ / 3 x + 1 and x 2 + 2 ~ / 3 x + 1 and the field ~ ( ~ f 3 ) . A n d the b l o c k d e c o m p o s i t i o n {(~J2 + ~ f 3 , - ~/-3), ( - ~ / 2 + ~/3, ~ - ~/3)} c o r r e s p o n d s to polynomial factors x 2 - 5 - 2~/6 a n d x 2 - 5 + 2~/-6 a n d the int e r m e d i a t e field @(N/6). If one w a n t s a simple e x a m p l e of Galois theory, the field Q(~v/2 + ~/3) over Q is a nice one; it has a slightly complex subfield structure, with t h r e e nontrivial subfields. And the b l o c k d e c o m p o s i t i o n of the four zeros -+V~ _+ gives a simple but effective d e m o n s t r a t i o n of s o m e elem e n t a r y results in primitive p e r m u t a t i o n groups. Another a s p e c t of ~ / 2 + ~/3 has surfaced.

Polynomial Decomposition Multiplication is a fundamental m a t h e m a t i c a l operation; factoring, its reverse. But p o l y n o m i a l s are functions and have a n o t h e r o p e r a t i o n akin to multiplication, n a m e l y composition, f ( x ) = g ( x ) o h ( x ) or, equivalently, g(h(x)). C o m p o s i t i o n is interesting for a n u m b e r of reasons, including the fact that in composition, unlike polynomial multiplication, the degrees multiply. That c o m p l e x i t y m a d e p o l y n o m i a l c o m p o s i t i o n a potential c a n d i d a t e for an RSAtype c r y p t o s y s t e m . (RSA is a "public key" c r y p t o s y s t e m in which "easy" p a r t s of the c o m p u t a t i o n are public, and difficult-to-compute p o r t i o n s are private, thus providing security. See [9].) The p r o b l e m is also m a d e m o r e interesting b y Ltiroth's t h e o r e m [10], w h i c h tells us that if k is an arbitrary field, the fields b e t w e e n k ( f ( x ) ) and k(x) are in one-to-one c o r r e s p o n d e n c e with the d e c o m p o s i t i o n s o f f(x); e a c h field b e t w e e n k(f(x)) and k(x) can be written as k(h(x)) for s o m e (right) c o m p o s i t i o n f a c t o r o f f ( x ) . These w e r e a m o n g the motivations that D e x t e r Kozen and I h a d w h e n we l o o k e d at the issue of decomposition. Previous algorithms had relied on factorization; a t h e o r e m of Evyater a n d Scott, Dorey a n d Whaples, and F r i e d and MacRae s h o w e d that the univariate p o l y n o m i a l f ( x ) is dec o m p o s a b l e into g(h(x)) if and only if the multivariate polynomial h(y) - h(x) divides f ( y ) - f ( x ) . Barton and Zippel (and i n d e p e n d e n t l y Alagar and Thanh 1) used this to decompose: f a c t o r f ( y ) - f ( x ) , c o m p u t e potential d e c o m p o sition factors from divisors of f ( y ) - f ( x ) . If f ( y ) - f ( x )

11 am presenting the Barton and Zippel algorithm.

VOLUME20, NUMBER4, 1998 57

splits into m a n y factors of small degree, the algorithm t a k e s exponential t i m e to c o m p u t e a d e c o m p o s i t i o n . It is the old r e c o m b i n a t i o n of f a c t o r s p r o b l e m again. Kozen and I discovered a simple w a y to d e c o m p o s e polynomials f ( x ) w h e n the degree is not divisible b y the characteristic of the field [3]. We also found an elegant structure t h e o r e m that gives a m e t h o d for decomposition. The theorem gives an effective technique for d e c o m p o s i t i o n over finite fields; the t h e o r e m also applies in characteristic 0. We began b y generalizing the c o n c e p t of p o l y n o m i a l decomposition. Let k b e a field of arbitrary c h a r a c t e r i s t i c and let f ( x ) ~ k[x] b e of degree n = rs, n o t n e c e s s a r i l y irreducible or separable. Let/~ be the splitting field o f f ( x ) over k, and let ~ d e n o t e the Gaiois group of/~ over k. D e f i n i t i o n 1. A block d e c o m p o s i t i o n f o r f is a m u l t i s e t A o f m u l t i s e t s o f e l e m e n t s o f k such that, 9 f = [IAa~ I I ~ A ( X -- a), 9 ife~ ~ A ~ A, fi ~ B ~ A, and cr E cb s u c h that (~(c~) = ~, then B = a(A) = {a(y)iT E A}. A block d e c o m p o s i t i o n A is a n r • s block d e c o m p o s i t i o n i f IAI = r and ~1 = s f o r all A ~ A. This generalization of b l o c k d e c o m p o s i t i o n to multisets is useful in d e c o m p o s i t i o n , w h e r e p o l y n o m i a l s are n o t necessarily irreducible a n d m a y have r e p e a t e d zeros. Let c}n d e n o t e t h e j t h e l e m e n t a r y s y m m e t r i c function on m - e l e m e n t multisets:

B

What is the simplest p o l y n o m i a l that w e can u s e to illustrate T h e o r e m 2? Because d e g r e e s multiply w h e n polyn o m i a l s a r e composed, the l o w e s t - d e g r e e p o l y n o m i a l t h a t has a nontrivial d e c o m p o s i t i o n w o u l d be one of d e g r e e 4. The p o l y n o m i a l x 4 - 1 0 x 2 + 1 fits the r e q u i r e m e n t s o f T h e o r e m 2, and indeed, w e get a b l o c k d e c o m p o s i t i o n A

B

~/~ + ~/5 - ~ / ~ - ~/-5

~7~ - "75 -~/-~ + ~7~

We have A= c2(A) = 1 = c2(B), c21(A) = ~/~ + ~/5 + ( - ~ / ~ - ~/5) = 0 = - V ~ + V 5 + N/2 - ~ = c2(B), c.~(A ) = - 5 - 2 V ~ , c~(B) = - 5 + 2N/-g, h ( x ) = x ~ - Ox = x 2, g ( x ) = [x - ( - 1 ) 3 ( - 5 - 2~/6)][x - ( - 1 ) 3 ( - 5 + 2~/6)] = x 2 - 1 0 x + 1. Thus, w e have a d e c o m p o s i t i o n of x 4 - 10x 2 + 1 - - a dec o m p o s i t i o n that the o b s e r v a n t r e a d e r m a y have a l r e a d y noticed. 2 At this point, I might have realized that I should investigate N/2 + ~ for any algebraic investigation I might t r y - b u t I did not. Instead, I first e x p l o r e d a n u m b e r of radical e x p r e s s i o n s , and only t h e n realized that my familiar exa m p l e w a s a particularly e a s y one with which to illustrate the t h e o r e m .

Denesting Radicals

I-J rOB

R a m a n u j a n discovered that

We let c~n = 1. T h e o r e m 2 (Kozen a n d Landau [3]) Let f ( x ) E k[x] be m o n i c o f degree n = rs. The f o l l o w i n g two s t a t e m e n t s are equivalent: 9f = g o h f o r s o m e g, h ~ k [ x ] o f degree r a n d s, respectively. 9 There e x i s t s a n r x s block d e c o m p o s i t i o n A f o r f such that c~(A)=c~(B)~k

forallA, BEA,

~/~ N/~

- 1 = ~ i 7 9 - ~2-79 + ~ - 9 , -

~

~/7~/~

= 1/3C~/-2 + ~ -

19 =

~

-

~),

~/-3.

H o w can w e simplify n e s t e d radicals, going from c o m p l e x equations as displayed on the left-hand side to the simpler, denested version on the right-hand side? F o l l o w i n g [2], a f o r m u l a over a field k and its depth o f n e s t i n g are defined as follows:

O < - j < - - s - 1.

In the p r o o f of T h e o r e m 2, g and h are explicitly cons t r u c t e d from A, B, and A by 8--1

h = ~ . ( - 1 ) k c](A)x s-j, j-0

9 An e l e m e n t of k is a f o r m u l a of depth 0 over k. 9 An arithmetic c o m b i n a t i o n (A + B, A x B, A/B) of form u l a s A and B is a f o r m u l a w h o s e depth o v e r k is m a x ( d e p t h ( A ) , depth(B)). 9A root ~ of a formula A is a f o r m u l a w h o s e d e p t h o v e r k is 1 § depth(A).

with g d e t e r m i n e d either explicitly from g ( x ) = I I [x - ( - 1 ) s+1 c~(A)l A~Zl

or by the fact that f ( x ) =- g(h(x)).

Such a f o r m u l a is a n e s t e d radical. A nesting of a m e a n s any f o r m u l a A that can t a k e a as a value. Note that n t h r o o t s are multivalued, so a m b i g u i t y is an issue. See [5] or [4] for further details.

2Although in the previous section we had three different block decompositions [corresponding to the fields Q(x~2), Q(~/3) and Q(~f6)], under the more restrictive re quirements of Theorem 2 that c~(A) E k, we have only one block decomposition, corresponding to the single polynomial decomposition.

58

THE MATHEMATICALINTELLIGENCER

The formula A can be denested over the f i e l d k if there is a f o r m u l a B of l o w e r nesting depth than A such that A = B. Formula A can be denested i n the f i e l d L if there is a formula B = A of lower nesting depth than A with all of the terms (subexpressions) of B lying in L. Define the depth of a over k to be the depth of the m i n i m u m depth expression for ~. When given a formula A for a such that A can be denested, I will s o m e t i m e s say that a c a n be denested. And I will cheat a little by writing a primitive n t h r o o t of unity as a special symbol ~n rather than as a nested radical; this defines the depth of nesting to be 1 for a primitive root of unity that is not already in the field. Under what circumstances can a radical be expressed in terms of radicals with a lower depth of nesting? I discovered that each time I c o m p u t e d subfields of Q(a), where a was a nested radical, the subfields c o r r e s p o n d e d to a denesting. T h e o r e m 3. Suppose ct is a nested radical over k, w h e r e k is a field o f characteristic 0 containing all roots o f unity. Then, there is a m i n i m a l depth nesting o f c~ w i t h each o f its terms lying i n the splitting field o f the m i n i m a l p o l y n o m i a l of c~ over k. All roots of unity is a rather large extension over Q; in particular, it is an infinite extension. From a computational standpoint, such an extension is not viable. Roots of unity are needed to make the field extensions between k and L Galois. However, we can limit ourselves to adding only those roots of unity that are necessary, thus trading optimality of denesting for finiteness of the extension over Q. Let ~t denote the lth root of unity. T h e o r e m 4. Suppose c~ is a nested radical over k, w h e r e k is a f i e l d o f characteristic O. Let L be the splitting f i e l d o f k(cO over k, w i t h Galois group G. Let I be the least comm o n multiple o f the exponents o f the derived series o f G. I f there is a denesting o f c~ such that each o f the terms has depth no more than t, then there is a denesting o f a over k(~t) w i t h each o f the terms having depth no m o r e than t + 1 and lying in L(~t). We can restore optimality by allowing some additional roots of unity, those that arise from the original expression for or: C o r o l l a r y 5. Let k, ~, L, G, l, and t be as i n Theorem 4. Let m be the least c o m m o n multiple of the (mij), w h e r e the mij are the indices o f the roots i n the given nested exp r e s s i o n f o r o~. Let r be the least c o m m o n multiple o f (m, l). Then, there is a m i n i m a l depth nesting of ~ over k(~r) w i t h each of its terms lying i n L(~r). One of the simplest nested radicals is h / 5 + 2X/6; consider the field extension Q(V/-5- + 2~fl6) over Q. As we already know, the algebraic n u m b e r %/5 + 2X/6 satisfies the irreducible polynomial x 4 - 10xa + 1 over (~. The field

Q(~v/5 + 2~/-6) is of degree 4 over Q, and it has {1, ~v/5 + 2V~, 5 + 2X/-6, (%/5 + 2 V ~ ) 3} as a basis over Q. This basis is of a nice mathematical form: {1,o~,o~2,cr3}. But because

and 1, ~/2, h/3, and ~/6 are linearly indep_e_ndent over Q, {1, ~/2, ~r ~/6} is also a basis for Q(~v/5 + 2 - ~ ) over Q. Many people prefer the second basis; it seems more natural to them. Thus, ~ + h/3 provides a practical reason for investigating denesting, namely designating procedures for a symbolic computation system like Maple to simplify nested radicals, and _thus, for example, to transform the basis {1, Xfl5~+ 2X/-6, 5 + 2%/-6, ( ~ / 5 + 2N/6) 3} into {1, ~/2, V~, ~/6}. In computational algebra, the practical and the theoretical often go very much hand in hand. What Is t h e S i g n i f i c a n c e of All This? ~/2 + ~/3 is one of the simplest c o m b i n e d radicals that exists, yet it provides a wealth of information about algebraic structure. For example, studying it demonstrates the relationship between intermediate subfields and dec o m p o s i t i o n - - a relationship that led to the discovery of Theorem 2. In one sense, I have presented a curiosity: one simple equation that illustrates results about factoring polynomials over Q, finding subfields using minimal blocks of imprimitivity, determining decompositions of polynomials, and denesting. But I think there is a deeper issue here. For many of us, computation has gone the way of the slide rule. We use it occasionally to illustrate a theorem. Yet the tools of such symbolic computation packages as Maple, MacCauley, Grobner, and A X I O M make such algebraic computations far easier to perform than they have ever been. When, in the 1920s, the Hilbert and Noether school made the transition to abstract methods, it was greatly beneficial to mathematics. The multivariate computations in commutative algebra were too large to be done by hand, and the abstract methods achieved what computation could not. Unfortunately, the transition went much farther. Algebraists and mathematicians of many flavors pursued abstraction, and concrete examples rarely appeared. The result was a g a i n - - a n d a loss. We have a chance to recoup that now. The computational tools recently introduced by computer scientists and mathematicians enable us to solve much harder problems, in extensions of higher degree, with many variables. I am convinced that had I fully examined ~ + X/3, results in decomposition and denesting would have jumped out at m e - - o r o t h e r s - - y e a r s earlier. P r o o f is the backbone of mathematics. Examples can light the way. We should use them for teaching, exploring, and research.

Acknowledgments Warm thanks to John Cremona, Donald Goldberg, and Ann Trenk; their suggestions greatly improved this article. Supported by NSF grant CCR-9204630 and CDA-

VOLUME 20, NUMBER 4, 1998

59

9753055, a n d a g r a n t f r o m S u n M i c r o s y s t e m s . This w o r k w a s partially d o n e w h i l e t h e a u t h o r w a s visiting Cornell University. REFERENCES

[1] E. Berlekamp, Factoring polynomials over finite fields, Bell Syst. Tech. J. 46 (1967), 1853-1859. [2] A. Borodin, R. Fagin, J. Hopcroff, and M. Tompa, Decreasing the nesting depth of expressions involving square roots, J. Symbol. Comput. 1 (1985), 169-188. [3] D. Kozen and S. Landau, Polynomial decomposition algorithms, J. Symbol Comput. 7 (1989), 445-456. [4] S. Landau, How to tangle with a nested radical, Math. Intell. 16, no. 2 (1994), 49-55. [5] S. Landau, Simplification of nested radicals, SlAM J. Comput. 21 (1992), 85-110. [6] S. Landau and G. Miller, Solvability by radicals is in polynomial time, J. Comput. Syst. ScL 30(2) (1985), 179-208. [7] H.W. Lenstra, Jr., Algorithms in algebraic number theory, Bull. AMS 26(2) (1992), 211-244. [8] P. P&lfy, A polynomial bound for the orders of primitive solvable groups, J. Algebra 77 (1982), 127-137. [9] R. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public key cryptosystems, Communications of the ACM 21 (1978), 120-126. [10] B.L. van der Waerden, Algebra, Frederick Ungar Publishing Co. (1977). [11] H. Zassenhaus, On Hensel factorization I, J. Number Theory 1 (1969), 291-311.

Tracking the Automatic Ant And Other Mathematical Explorations For those fascinated by the abstract universe of mathematics, David Gale's columns in The Mathematical Intelligemer have been a prime source of entertainment. Here Gale's columns are collected for the first time in book form. Encouraged by the magazine's editor, Sheldon Axler, to write on whatever pleased him, Gale ranged far and wide across the field of m a t h e m a t i c s - frequently returning to his favorite themes: triangles, tilings, the mysterious properties of sequences given by simple recursions, games and paradoxes, and the particular automaton that gives this collection its title, the "automatic ant." T h e level is suitable for those with some familiarity with mathematical ideas, but great sophistication is not needed. Contents: Simple Sequences with Puzzling Properties 9 Probability" Paradoxes 9 Historic Conjectures: More Sequence Mysteries 9 Privacy Preserving Protocols 9 Surprising Shuffles 9 H u n d r e d s of N e w T h e o r e m s in a Two-Thousand-Year-Old Subject 9 P o p - M a t h and Protocols 9 Six Variations on the Variational Method 9 Tiling a Torus: Cutting a Cake 9 T h e Automatic Ant: Compassless Constructions 9 Games: Real, Complex, hnaginary ~ Coin Weighing: Square Squaring 9 T h e Return of the Ant and

the Jeep 9 G o 9 More Paradoxes, Knowledge G a m e s 9 Triangles and Computers 9 Packing Tripods 9 Further Travels with M y Ant 9 T h e Shoelace Problem 9 Triangles and Proofs 9 Polyominoes 9 In Praise of

Numberlessness 9A Pattern Problem, a ProbabilityParadoxand a Pretty Proof 9 Appendices:1. A CuriousNim-Type Game 9 2. The Jeep Once More and Jeeper by the Dozen 9 3. Nineteen Problemson Elementary, Geometry 9 4. The Truth and Nothing But the Truth 1998 I 256 PP. I HARDCOVER I $30.00 I ISBN 0-387-98272-8

Springer http:llwww.springer-ny.com

5/98

P*vmotio~z #14231

II:(:a,,[:a,,,l-1 Jet

Wimp,

Editor

I

Logical Dilemmas: The Life and Work of Kurt GGdel by John W. Dawson, Jr. WELLESLEY, MA: A. K. PETERS, 1997. xJv + 361pp. US $49.95, ISBN 1-56881-025-3

REVIEWED BY CRAIG N. B A C H

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biographer's job, simply stated, is to chronicle a life. That is not to say merely to compile a chronological list of facts and events; the biographer also is obliged to describe the significance of this information. Of historical writing, Hayden White states: "The events must be not only registered within the chronological framework of their original occurrence but narrated as well, that is to say, revealed as possessing a structure, an order of meaning, which they do not possess as mere sequence. "1 The remark is clearly applicable to biographies. However, not all facts and events are created equal. Some details need no e x p l a n a t i o n - either because they resonate with meaning for any reader or because their significance is well known to a specific audience. Other details need to have their significance explained, and still others require that they be related to additional facts and events before their import can be gleaned. Some details require different amounts of narrativization for different audiences. The biographer's job is to determine which details are worth mentioning and how much narrative explanation each fact and event requires. Consider the following three entries from the Annals of Saint GaU:2

A

709. Hard Winter. Duke Gottfried died. 710. Hard year and deficient in crops. 711. Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.

Why was the winter hard in 709? How important a figure was Duke Gottfried? To whom was he important? Were there no significant events in 711? A

contemporary r e a d e r would be hard pressed to determine the significance of these events without further elaboration or narrativization. There are, of course, biographies that overdo it. They fall under the heading of biographical fiction. These works link known biographical details with speculative narration. Bruce Duffy's The World as I Found It comes to mind. Here the author attempts to enliven the details of Wittgenstein's life and work with his own fictional explorations. At one point, Duffy goes as far as to pen a lurid account of Wittgenstein's inner thoughts regarding his sexual attraction to an officer during the First World War. Sometimes, too much narration can be a bad thing. John W. Dawson, Jr.'s recent book,

Logical Dilemmas: The Life and Work of Kurt G6del, is the first full-scale biography written about the life of Kurt GOdel. Incorporating details from several previously published accounts of G6del's life, 3 Dawson's own work with G6del's papers at the Institute for Advanced Study at Princeton, and his experience as co-editor of G6del's Collected Works, Dawson has collected an admirable amount of biographical information about the late logician. However, the inclusion of a few biographical details either without sufficient narrative or with highly conjecturai narrative impairs the early chapters of the book. A few examples will illustrate the point. Dawson claims that a school portrait of G6del (Figure 8) "confirms the image of a studious young man confident of his future." But many school bullies and underachievers contrive, on picture day, to look studious and polite for the camera. One can read too much into a photograph. Dawson may likewise read too much into a painting by Kolomon Moser,

Homage to the A m e r i c a n Dancer L6ie Fuller several pages later. The painting (Figure 9) depicts a woman wearing a

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 4, 1998

61

huge wing-like gown, her anus spread out in front of a dark background. In a footnote on the preceding page, Dawson states that the image perfectly captures "the underlying connotation" of the term 'moth' in German. Is the painting intended merely to represent a club where Adele (G6del's future wife) may have worked, or the feelings of G6del's family toward her? The fact that the image also graces the cover of the book leads one to believe that the painting is supposed to make some more encompassing point. Dawson occasionally resorts to empty conjecture. Concerning G6del's 1935 Atlantic crossing, Dawson remarks: "There is no record of his reaction to the crossing, which was presumably uneventful; but in view of his chronic digestive troubles (real, imagined, and self-inflicted) in later years, it would not be surprising if he suffered somewhat from seasickness" (page 97). The reader is left wondering whether it would not have been surprising, on the other hand, had G6del not suffered from seasickness. The failure to incorporate a few of the biographical details of G6del's early years can be forgiven, as the book is primarily concerned with the work and life of the adult logician. However, similar narrative problems can be found in chapter 3, where as preparation for discussion of G6del's work we are given a brief history of mathematical logic. The intended audience is not assumed to be familiar with the field: "I have not, however, presumed any acquaintance with modern mathematical logic, since even among mathematicians of the fwst rank such knowledge is often wanting" (Dawson, p. ix). The chapter is positioned to provide a background for G6del's main results: the Completeness Theorem for first-order logic, the Incompleteness Theorems, and the Independence of the Continuum Hypothesis.4 The digression into history covers the study of logic from Aristotelian syllogisms to Hilbert and Ackermann's 1928 text, Ga'undziige der theoretischen Logik. One desideratum of presenting such a large breadth of information in the short span of 16 pages is that all of the logical results outlined

6~

THE MATHEMATICAL~NTELLIGENCER

tie directly to G6del's work, and that these ties be clearly explicated. However, as in previous chapters, many details are left without appropriate explanation. For example, on page 39 Dawson states, "Of particular importance to G6del's own later work was Anselm's 9 ontological argument." The reader is left guessing what its import was. It is not until page 198 that we learn that Anselm's proof influenced an aborted attempt by G6del to defend his brand of Platonism, and on page 237 that we find out that G6del used an argument similar to Anseim's to "prove" his own ontological argument. In contrast, three paragraphs below the significance of Leibniz's work is neatly tied to G6del's proof of the First Incompleteness Theorem. Another example occurs on page 41. Dawson uses a quote from van Heijenoort's book, From Frege to

Gddel: A Source Book in Mathematical Logic, 1879-1931, to discuss Frege's analysis of propositions into functions and objects. Dawson states that this is Frege's most important contribution. He does not say why. Frege's work is well-known to logicians, but probably not to the intended audience. Cantor's notions of transfinite ordinals and cardinals are briefly mentioned on page 43. The reader would be much helped by further elaboration on the development of Cantor's new numbers as well as their contentious reception into mathematics. On page 46, Dawson informs the reader that, "Since Peano's axiomatization preceded the discovery of the logical and set-theoretic paradoxes, it did not arise in response to them." The reader wonders who might think that they were developed in reaction to the paradoxes and why. On pages 50-51, the method of expressing quantification using choice functions, the L6wenheim-Skolem theorem, the Entscheidungsproblem, and Post's work with truth tables are given in a single breath. 5 The section would be much helped by a few more, slower breaths. In general, all the relevant facts required to provide the proposed background are discussed, and the chapter

is intelligible, but not illuminating,, to those familiar with the material. However, Dawson covers too many results too quickly without linking them to G6del's work. At least he might have added brief summaries or section headings making the connections. In chapter 4, Dawson begins the discussion of G6del's main results. It is here that book hits its stride. Dawson provides a deft description of both G6del's Completeness Theorem and his Incompleteness Theorems. His discussion is pedagogically informative and well organized. Dawson's discussion of the impact of G6del's results on the mathematical community of the time is engaging. It has been said that any truly new discovery is met by disinterest (the result is nothing new), denial, and consternation before its significance is finally realized. The Incompleteness Theorems, with their shattering implications and potential to derail many a program, were not met with disinterest. However, as Dawson points out, the theorems were met with denial and consternation from some of the top mathematicians of the day. Dawson reports that Ernst Zermelo was quite hostile to G6del's First Incompleteness Theorem and endeavored to find fault with his proof. Zermelo's reaction to the theorem is surprising. As Dawson notes, Zermelo had fought bitterly for his Axiom of Choice. In defending the axiom, Zermelo argued that "principles must be judged from the point of view of the science, and not science from the point of view of principles fixed once and for all. ''6 His main argument was a pragmatic one--the choice axiom should be accepted because it gets the job done. By removing the philosophical underpinnings involved in the acceptance of his axiom system, his defense helped usher in the modern view of axiomatic mathematics 9 On the other hand, Zermelo's response to the First Incompleteness Theorem also shows him to be entrenched in an old view of logic. He was unable to appreciate the distinction, taken for granted today, between the syntax and the semantics of an axiom system, and therefore was unable to appreciate the relevant metatheo-

retical results. As Dawson points out, There doesn't seem to be a similar "suspicious." Dawson also relates a reit was this that blinded him to the value concern when it comes to the political mark G6del made about the deleteriof G6del's theorem. Zermelo sat both views of mathematical logicians. I have ous effect that the 1946 Republican as proponent and detractor of the mod- never heard, for example, of logicians landslide had on the quality of movies e m conception of logic. being anxious that Frege's anti-Semitic shown at the time (p. 169). "In addiThroughout the remainder of the views may have infiltrated his logical tion, he believed it was 'too early' for book, Dawson's discussions of G6del's work and thereby tainted their own. another World War, since the first two mathematical results are neatly woven The conceptual distance between polhad been separated by 25 years and into accounts of his personal life, his itics and mathematical logic is too only 17 had elapsed since the second" philosophical beliefs, and his ongoing great. Having said that, it is still inter(Dawson, p. 209). battle with mental instability. The book esting to view how a first-rate logician Such political opinions seem not to also is filled with enjoyable anecdotes. responded to the rise of fascism in be much different from his opinions on The most famous involves GSdel's citi- Europe. other non-academic topics. For examzenship hearings. Dawson tells how For the most part, G0del seems to ple, "While at Notre Dame he had told Einstein and Morgenstern attempted to have ignored it. On page 140, Dawson Menger that he believed publication of keep G6del busy so that he would not mentions "the only known instance in some of Leibniz's works had been mention to the presiding judge the in- which he [GSdel] prefaced his signasuppressed by a hostile conspiracy" consistency he claimed to have found in ture with the words 'Heil Hitler.' " (Dawson, p. 166). Many of G6del's the U.S. constitution. The anecdote pro- Otherwise, GOdel had a staunch apoviews seem to reveal the inner turmoil vides more than just humor. It is an in- litical stance. His lack of political perand paranoia that eventually led to his sightfifl glimpse into the personalities of spective cost him the friendship of Karl downfall. The overarching aim of Einstein, Morgensteru, and GSdel away from their desks. 7 Dawson's book is to explain Zermelo sat both as proponent and G6del's mental instability in reOne area of G0del's life outdetractor of the modern conception side aca-demia that is reported lation to his work and personal of logic. with little comment is politics. life. He begins the book with the following: GSdel lived in Austria during the rise of Hitler's Germany and during Menger. Dawson states that Menger the Anschluss. How did he respond? considered G6del's concern for his "G6del's choice of profession, his Platonism, his mental troubles, and Early in the book, Dawson mentions own position, vis-h-vis his academic (without explaining any relevance to rights, to be misplaced in relation to much else about him may thus be attributed to a sort of arrested deGSdel!) Heidegger's appointment to the the greater wrongs being perpetrated velopment. He was a genius, but he rectorship of the University of Freiburg in Europe at the time. The degree to was also, in many respects, a man/ and his support for the Third Reich. which G6del could remain apolitical is Hans-Georg Gadamer 1~ relates the fop most strikingly revealed in a 1940 rechild" (Dawson, p. 2). lowing tale concerning Heidegger's res- mark he made to his friend Oskar The remark lays out the main theme of ignation from that same post: "Indeed, Morgensteru. Upon G6del's arrival in the book and makes the implicit after Heidegger resigned from the rec- Princeton, two years after the Austrian promise that these connections will be torate, one of his Freiburg friends, see- Anschluss, Morgenstern asked him for discussed and clarified. In the last ing him in the streetcar, greeted him: his views about the situation in Austria, chapter of the book, Dawson tries to 'Back from Syracuse?' " (page 429). whereupon G6del replied, "The coffee is make good on his promise. The reference is to Plato's ill-fated at- wretched" (Dawson, p. 153). Too often we hear glib truisms G6del's political indifference raises tempt to teach his philosophy to about the faint line between matheDionysius of Syracuse. It seems that interesting questions. Do intellectuals matical genius and insanity. The life of Dionysius wanted little to do with hold a greater responsibility for their Kurt G6del provides a chance to give Plato's teachings and opted for a life o f political actions? Do they have a them substance. Dawson takes up the pleasure and despotic rule. Heidegger, greater capability for intelligent politidaunting task of trying to thread like Plato, thought that his philosophy cal action? One need only think of G6del's rationalistic and causally dewas uniquely relevant to a current po- Einstein's and Russell's political interministic view of the world, his litical power--Heidegger's Dionysius volvement to conclude that GOdel's is Platonist belief in a mental reality diswas Adolf Hitler s. Noting the nature of not the only answer. tinct from the physical, with his paraG6del's later views on politics are Heidegger's work and his own claims noia and overly burdensome selfabout his philosophy's relation to noteworthy only for their absurdity. concern (hypochondria), into a single, National Socialism, many philosophers For example, Dawson states that cohesive narrative. While his remarks wondered whether there indeed might G0del thought that "secret powers" are highly speculative, the discussion be deep conceptual links between were at work hampering Roosevelt's and the questions it raises are comHeidegger's philosophy and that of the programs and that the circumstances National Socialists. 9 surrounding Roosevelt's death were pelling.

VOLUME 20, NUMBER 4, 1998 6 3

There are certain e x p e c t a t i o n s one brings to reading a b i o g r a p h y of an imp o r t a n t and influential thinker. One h o p e s to learn b i o g r a p h i c a l details t h a t illuminate the w o r k in n e w and interesting ways, to c o m e to a greater app r e c i a t i o n a n d k n o w l e d g e of the scholar's work, to find h u m o r o u s a n d thought-provoking anecdotes, and to c a t c h a glimpse of h o w an intellectual functions in t h e w o r l d outside the confines of academia. In the case of Kurt GSdel, there is the a d d e d e x p e c t a t i o n that his life-long b a t t l e with mental and emotional instability be clarified, a n d p e r h a p s explained, in the c o n t e x t of his life and work. After a slow start, P r o f e s s o r D a w s o n ' s b i o g r a p h y of Kurt GSdel m a n a g e s to m e e t most, if not all, of these e x p e c t a t i o n s . REFERENCES 1. White, Hayden, "The Value of Narrativity in

the Representation of Reality," pg. 5, On

2. 3.

4.

5.

6.

Narrative, ed. W.J.T. Mitchell, University of Chicago Press (1981). Ibid., pp. 7-10. The most notable discussions are given in Wang, Hao, Reflections on Kurt G6del. Cambridge, MA: MIT Press: (1987) and Kreisel, Georg. "Kurt G6deh 1906-1978." Biographical Memoirs of Fellows of the Royal Society 26 (1980): pp. 149-224. Dawson's stated intention is to provide a background for four seminal papers that influenced G6del's work. These papers in turn provide a backdrop for a discussion of G6del's main results. It may be illuminating to add that Ludwig Wittgenstein independently developed a truth-table method in his 1921 Tractatus Logico Philosophicas. Zermelo, Ernst, "A New Proof of the Possibility of a Well-Ordering," (1908) in Heijenoort, Jean van, From Frege to Gddel: A Source Book in Mathematical Logic 1879-1931, Cambridge, MA, Harvard University Press (1967).

(Continued from p. 3)

The Ancient

point, b(0), a n y w h e r e interior to the square, then fi(b(t)) has the required p r o p e r t y in s o m e n e i g h b o r h o o d of t = 0 w h e r e fl(b(t)) is well defined.

The authors of "The Q u e s t for Pi" (The Mathematical Intelligencer, vol. 19, no. 1, pp. 50-57) cite the Biblical passage in 1 Kings 7:23 as evidence that the ancient H e b r e w s u s e d 3 as the value for it. If they h a d r e a d a few v e r s e s farther they might have quest i o n e d this value. In 1 Kings 7:26 w e r e a d "its rim [or brim] w a s like the rim of a cup, like a lily blossom." The "sea" w a s not a simple cylinder; it flared outw a r d at the top. The p a s s a g e is not giving us the technical specifications of the "sea" but rather describing w h a t a visitor might see in the temple. The obvious w a y to m e a s u r e such an object w o u l d be to stretch a r o p e a c r o s s it from rim to ~ n and to p a s s a r o p e around it b e l o w the rim. The resulting diameter and circumference are for two different

REFERENCES

[1] J. L. Denny, A continuous real-valued function on En almost everywhere 1-1, Fund. Math. LV (1964), 95-99 [2] G. J. Foschini, Almost everywhere one-toone functions and an n-cube decomposition, Journal Math. Anal.and Appl., 31, No. 2, (1970), 314-317

Gerard J. Foschini Lucent Technologies Crawford Hill Laboratory 791 HolmdeI-Keyport Road Holmdel, New Jersey 07733-0400, USA

THE MATHEMATICAL INTELLIGENCER

Hebrews

a n d 7r

7. On the close friendship between G6del and Einstein, Hao Wang remarks, "It is hard to find in history comparable examples of intimacy between such outstanding philosopher-scientists." Op. cit., pg. 3. 8. Heidegger was not the only person to lay claim to the true philosophy of National Socialism. For an extended discussion, see Sluga, Hans, Heidegger's Crisis: Philosophy and Politics in Nazi Germany, Cambridge, MA, Harvard University Press (1993). 9. Karl LOwith, Hans Sluga, Jacques Derrida, Philippe Lacoue-Labarthe, and J0rgen Habermas, to name a few. 10. Gadamer, Hans-Georg, "Back from Syracuse?," Critical Inquiry 15, no. 2 (1989), 427-430. Department of Humanities and Communications Drexel University Philadelphia, PA 19003 USA

circles. We are not told h o w far the rim projected, so we cannot r e c o v e r the value for 7r. The m o r e thorough reading of the p a s s a g e does exonerate the H e b r e w s from the implication that they had a less accurate value for ~- t h a n s o m e of their neighbours. Others have advanced this interpretation of the passage, but their w o r k does not s e e m to be widely known in the m a t h e m a t i c a l community. A. Zuidhof (Biblical Archeologist, vol. 45, pp. 179-184) and R. C. G u p t a (Ganita Bharati, vol. 10, nos. 1-4, pp. 51-58) b o t h include this observation a m o n g o t h e r interesting points. George C. Bush 66 Meadowbrook Drive Bedford N.S., B4A 1R1 Canada

B--"]i~.~,,i,==~.]=,[:~-,-

Robin

Wilson,

Editor

Maccabees

Spinnino Tops Pinning Tops is the title of a

beautiful little book written by Mich~le Audin [1]. A heavy symmetrical top is a symmetrical body in a uniform gravitational field when one point on the synmletry axis is fixed (see [2], p. 213). As Audin explains in her introduction, "spinning top" is a recent abbreviation reserved for the cases where the differential equations in Hmniltonian form describing the motion of the top are completely integrable--the Euler top, the Lagrange top, and the Kowalewski top; the Euler and Lagrange cases are treated in many classical books. Much progress has been made by applying the techniques of symplectic geometry--in particular, reduction of the phase space using the momentum mapping that describes the symmetries of the system; an excellent recent reference is [3]. A method that also permits a study of the Kowalewski top is that of the Lax pairs with spectral parameter and the associated spectral curve, which is the one applied by Audin. A spinning top is the subject of a stamp issued in Israel on the occasion of Hanukka (the Festival of Lights), 1997. The stamp reproduces a traditional Dreidel or Hanukka top, a children's toy. It bears the four letters Nun, Gimmel, He and Shin. Such tops were used in Germany in the early Middle Ages for a game whose outcome could be Ganz ("all"--take all the money), Halb ("half"--take half the money), Stell ("put"--add more money), or Nichts ("nothmg"--take nothing), with initials

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics and Computing, The Open University, Milton Keynes, MK7 6AA, England

1-8),

and

Josephus

lighting of lights during this festival. The Babylonian Talmud (Shabbath 21b), citing the opinions of the two great rabbis of the mishnaic period, explains that lights were to be lit throughout the eight days, in arithmetic progression

Yvette Kosmann-Schwarzbach

S

x.

(Antiquities, XII. ii. 7) refers to the

Gimmel, He, Shin, and Nun in Hebrew script. In m o d e r n times, these four letters are taken to be the initials of the four words in the sentence, Nes Gadol Hayah Sham ("A great miracle occurred there"). This phrase refers to the victories of Judas Maccabeus over the Seleucidm which permitted the purification and rededication of the temple in Jerusalem in 164 B.C.E. In memory of that event, an eight-day festival was decreed (I Maccabees iv. 41-59; II

from 8 to 1 (according to Shamal) or 1 to 8 (according to Hillel). The Talmud also mentions that when the temple was liberated, only one flask of ritually suitable oil was found, but that it lasted for eight days. Thus, the stamp also depicts a stylized domestic Hanukka candelabra (hanukkia) with eight branches (and an additional branch for a candle to light the others). A total of 2 + 3 + . . . + 9 = 44 candles is needed for the eight days of the festival; were it to last n days, then n(n + 3)/2 candles would be needed! REFERENCES 1. M. Audin, Spinning Tops. A Course on Integrable Systems, Cambridge University Press, 1996. 2. H. Goldstein, Classical Mechanics, 2d. edn., Addison-Wesley, 1980. 3. R. H. Cushman and L. M. Bates, Global

Aspects of Classical Integrable Systems, Birkh~user, 1997. Centre de Mathematiques Ecole Polytechnique F-91128 Palaiseau, France

9 1998 SPRINGER-VERLAG NEW YORK, VQLUME 20, NUMBER 4, 1998

65

Wilson, Robin. Abel. (19.1) 80. Wilson, Robin. Stamps of Unusual Shape I. (19.2) 72. Wilson, Robin. Stamps of Unusual Shape II. (19.3) 76. Wilson, Robin. Isaac Newton. (20.3) 71. Wilson, Robin. See Barrow-Green, June, and Wilson, Robin. Wilson, Robin. See Flood, Raymond, and Wilson, Robin. Wilson, Robin. See Hannabuss, Keith, and Wilson, Robin. Wilson, Robin. See Pisanski, Tomaz, and Wilson, Robin. Wimp, Jet. Review of Selected Problems in Real Analysis Volume 107, AMS Series of Translations of Mathematical Monographs, by B.M. Makarov, M.G. Goluzina, A.A. Lodkin, and A.N. Podkorytov. (16.4) 68-72. Wimp, Jet. Review of Polynomials and Polynomial Inequalities, by Peter Borwein and Tmnas Erdelyi. (18.3) 76-79. Wimp, Jet. Review of Potential Theory in the Complex Plane, by Thomas Ransford. (18.4) 72. Wimp, Jet. Review of Contests in Higher Mathematics: Miklos Schweitzer Competitions 1962-1991, edited by Ghbor J. Szdkely. (18.4) 72-73. Wimp, Jet. Review of Catastrophe Theory, by V.I. Arnol'd. (18.4) 73-74. Wimp, Jet. Review of Real Computing Made Real: Preventing Errors in Engineering and Scientific Calculations, by Forman S. Acton. (18.4) 74-75. Wimp, Jet. Review of The Queen of Mathematics: An Introduction to Number Theory, by W.S. Anglin. (18.4) 75-76. Wimp, Jet. Review of A Primer on Nonlinear Analysis, by A. Ambrosetti and G. Prodi. (18.4) 76-77. Wimp, Jet. Review of Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists, by Dennis Shasha and Cathy Lazere. (18.4) 77-78. Wimp, Jet. Review of Five Hundred Mathematical Challenges, by

Edward Barbeau, William Moser, and Murray Klamkin. (18.4) 78-79. Wimp, Jet. Review of A Tour of the Calculus, by David Berlinski. (19.3) 70-75. Wimp, Jet. Review of The World According to Wavelets, by Barbara Burke Hubbard. (19.3) 70-75. Wimp, Jet. Review of Calculus Lite, by Frank Morgan. (19.3) 70-75. Wimp, Jet. Review of To Catch the Spirit: the Memoir of A.C. Aitken, by A.C. Aitken, with a biographical introduction by P.C. Fenton. (20.2) 62-79. Wimp, Jet. Review of Determinants and Matrices, by A.C. Aitken. (20.2) 62-79. Wimp, Jet. Review of The Case against Decimalisation, by A.C. Altken. (20.2) 62-79. Wimp, Jet. Review of Gallipoli to the Somme: Recollections of a New Zealand Infantryman, by A.C. Aitken. (20.2) 62-79. Wong, Roderick. Review of Special Functions, by Nico Temme. (19.4) 75-76. Woodin, W. Hugh. Large cardinal axioms and independence: the continuum problem revisited. (16.3) 31-35. Wo~niakowski, H. See Traub, J.F. and Woiniakowski, H. Wu, H. Review of Functions and Graphs, by I.M. Gelfand, E.G. Glagoleva, and E.E. Shnol. (17.1) 68-75. Wu, H. Review of The Method of Coordinates, by I.M. Gelfand, E.G. Glagoleva, and A.A. Kirillov. (17.1) 68-75. Wu, H. Review of Algebra, by I.M. Gelfand and A. Shen. (17.1) 68-75. Yarnall, Keith. Review of An Equation that Changed the World: Newton, Einstein & the Theory of Relativity by Harald Fritzsch. (19.2) 67-68. Yarnall, Keith. See Milligan, Lloyd and Yarnall, Kenneth. Yellin, Joel. A scholar's tale. (13.4) 27. Yevick, Miriam L. The happy (nonformalist) mathematician. (14.1) 4-6. Zalcman, Lawrence. Review of

Littlewood's Miscellany, edited by B~la Bollabhs. (ll.1) 63-65. Zalcman, Lawrence. Mathematicians sweep 1988 Wolf Prizes. (11.2)39-48. Zalcman, Lawrence. Review of DarsteUung und Begri~ndung einiger neuerer Ergebnisse der Funktionentheorie (dritte, erweiterte Aufiage) by Edmund Landau and Dieter Gaier. (11.4) 61-63. Zalcman, Lawrence. Review of Discrete Thoughts: Essays on Mathematics, Science and Philosophy, by Mark Kac, Gian-Carlo Rota, and Jacob T. Schwartz. (12.3) 81-83. Zalcman, Lawrence. Review of The Apprenticeship of a Mathematician, by Andr~ Weil. (15.4) 64-68. Zambrini, J.C. Schr6dinger's time reversal and quantum mechanics. (19.2) 5-6. Zdravkovska, Smilka. Listening to Igor Rostislavovich Shafarevich. (11.2) 16-28. Zdravkovska, Smilka. To my partner, to Allen Shields. (12.2) 4-7. Zeilberger, Doron. Theorems for a price: tomorrow's semi-rigorous mathematical culture. (16.4) 11-14. Zeilberger, Doron. How Joe Gillis discovered combinatorial special function theory. (17.2) 65-66. Zerger, Monte J. Student questions you love to hate. (16.4) 29-30. Zerger, Monte J. A quote a day educates. (20.2) 5-6. Zerner, Martin. Review of Mathdmatiques d Venir, by Karine Chemla and Ivar Ekeland. (13.2) 76-79. Zerner, Martin. Review of Mathematics Tomorrow, by Lynn Arthur Steen, (13.2) 76-79. Zhang, D.Z.C.N. Yang and contemporary mathematics. (15.4) 13-21. Zhang, S. See Schweiginan, C., and Zhang, S. Zweifel, P.F. See Nonnenmacher, Dirk J.F., Nonnenmacher, Theo F., and Zweifel, P.F. Zwicker, William S. See Brams, Steven J., Taylor, Alan D., and Zwicker, William S.

Books Reviewed

Aitken. Reviewed by Jet Wimp. (20.2) 62-79. Aitken, A.C. Determinants and Matrices. Reviewed by Jet Wimp. (20.2) 62-79. Aitken, A.C. The Case Against Decimalisation. Reviewed by Jet Wimp. (20.2) 62-79. Aitken, A.C. GaUipoli to the Somme: Recollections of a New Zealand Infantryman. Reviewed by Jet Wimp. (20.2) 62-79. Ambrosetti, A. and Prodi, G. A Primer on Nonlinear Analysis. Reviewed by Jet Wimp. (18.4) 76-77.

Anglin, W.S. The Queen of Mathematics: an Introduction to Number Theory. Reviewed by Jet Wimp. (18.4) 75-76. Arnol'd, V.I. Catastrophe Theory. Reviewed by Jet Wimp. (18.4) 73-74. Arbib, Michael and Ewert, J~rg-Peter (editors). Visual Structures and Integrated Functions. Reviewed by Shimon Edelman. (15.4) 68-70. Ascher, Marcia. Ethnomathematics: a Multicultural View of Mathematical Ideas. Reviewed by David Wheeler. (14.4) 64-66. Barbean, E.J. Polynomials (Problem

Acton, Forman S. Real Computing Made Real: Preventing Errors in Engineering and Scientific Calculations. Reviewed by Jet Wimp. (18.4) 74-75. Adams, Colin. The Knot Book; An Elementary Introduction to the Mathematical Theory of Knots. Reviewed by De Witt Sumners. (19.1) 74-75. Aitken, A.C., with a biographical introduction by P.C. Fenton. To Catch The Spirit: The Memoir of A.C.

VOLUME 28, NUMBER 4, 1998

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Books in Mathematics). Reviewed by George Szekeres. (14.1) 78-79.

Barbeau, Edward, Moser, William, and Klamkin, Murray. Five Hundred Mathematical ChaUenges. Reviewed by Jet Wimp. (18.4) 78-79.

Bednarek, AI, and Ulam, Fran~oise, eds. Analogies Between Analogies. The Mathematical Reports of S.M. Ulam and His Los Angeles CoUaborators. Reviewed by Reuben Hersh. (14.4) 71-73.

Berggren, Lennert, Borwein, Jonathan, and Borwein, Peter. Pi: A Source Book. Reviewed by Dan Sctmabel. (20.3) 64-65. Berlinski, David. A Tour of the Calculus. Reviewed by Jet Wimp. (19.3) 7O-75. Berndt, Bruce C. Ramanujan's Notebooks, Part III. Reviewed by George E. Andrews. (16.3) 69-71. BSlling, Reinhard (editor). A Photo Album for Weierstrass. Reviewed by R.B. Burckel. (18.1) 78-79. Borst, Arno. Das mittelalterliche Zahlenkampfspiel. Reviewed by Benno Artmann. (11.3) 77-79.

Borwein, Jonathan M. and Borwein, Peter B. Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity. Reviewed by John Todd. (11.3) 73-77.

Borwein, Peter and Erdelyi, Tamas. Polynomials and Polynomial Inequalities. Reviewed by Jet Wimp. (18.3) 76-79. Boskoff, W.G. Hyperbolic Geometry and Barbilian Space. Reviewed by Victor Pmnbuccian. (20,4) Brains, Steven J. Theory of Moves. Reviewed by Marc Kilgour. (19.3) 68-70. Bressoud, David. A Radical Approach to Real Analysis. Reviewed by lvor Grattan-Guinness. (17.4) 68-70. Brezinski, Claude. History of Continued Fractions and Padd approximants. Reviewed by William B. Jones. (15.3) 71-73.

Brieskorn, Egbert and Knorrer, Horst. Plane Algebraic Curves. Reviewed by Art Schwartz. (12.2) 74-78. Casti, John L. Searching for Uncertainty: What Scientists Can Know About the Future. Reviewed by Manfred Schroeder. (16.3) 72.

Chipman, Susan F., Brush, Lorelei R., and Wilson, Donna M. (editors). Women and Mathematics: Balancing the Equation. Reviewed by Neal Koblitz. (13.1) 77-78. Colton, D. and Kress, R. Inverse Acoustic and Electromagnetic Scattering 77teory. Reviewed by Pierre Sabatier. (16.2) 73-75. Connes, Alain. Noncommutative Geometry. Reviewed by Jonathan Block. (20.1) 73-75.

'78

THE MATHEMATICAL INTELLIGENCER

Coolidge, Julian Lowell. The Mathematics of Great Amateurs (2nd ed). Reviewed by David M, Burton. (14.3) 68-69. Corduneanu, Constantin. Integral Equations and Applications. Reviewed by Thomas S. Angell (16.1) 63-67.

Croft, H.T., Falconer, K.J., and Guy, R,K. Unsolved Problems in Geometry. Reviewed by Dennis DeTurck. (15.1) 71-72. Davies, Paul ed. The New Physics. Reviewed by Philip W. Anderson. (14.1) 70-71. Davis, Philip J. Spirals: from Theodorus to Chaos. Reviewed by Michele Emmer. (18.1) 75-78.

Davis, Philip J. and Hersh, Reuben. Descartes' Dream: the World According to Mathematics. Reviewed by Robert Osserman. (11.2) 66-70. Dawson, John W. Jr. Logical Dilemmas: The Life and Work of Kurt G6del. Reviewed by Craig N. Bach. (20.4) 61-64. Devlin, Keith. The Joy of Sets. Reviewed by J. Donald Monk. (17.2) 71-73. Dixmier, Jacques. L'Aurore des Dieux. Reviewed by David M. Bressoud (16.3) 72-74. Dudley, Underwood. A Budget of Trisections. Reviewed by Ian Stewart. (14.1) 73-77.

Duren, Peter, assisted by R.A. Askey, tt.M. Edwards, and U.C. Merzbach. A Century of Mathematics in America. 3 vols. Reviewed by W.H.J. Fuchs and Lee Lorch. (13.4) 74-78. Earman, John. Bayes or Bust?: a Critical Examination of Bayesian Confirmation Theory. Reviewed by Evelyn Mitchell. (16.4) 66-68. Educational Poster. Solving the Quintic. Reviewed by Eric Schechter. (17.3) 71-73.

Egorov, Yu. V. and Shukin, M.A. Partial Differential Equations I: Foundations of the Classical Theory, Encyclopedia of Mathematical Sciences, Vol. 30. Reviewed by David Colton. (15.3) 69-71. Ekeland, Ivar. Mathematics and the Unexpected. Reviewed by Cathleen S. Morawetz. (13.3) 81-83. Emmer, Michele. Special Issue of Leonardo, Visual Mathematics, Reviewed by Harold L. Dorwart. (16.1) 70-72. Ewing, John, ed. A Century of Mathematics. Reviewed by Underwood Dudley. (17.2) 73-74. Farin, Gerald. Curves and Surfaces for Computer Aided Geometric Design (2nd ed). Reviewed by Len Bos (14.3) 66-68. Faux, I.D. and Pratt, M.J. Computational Geometry ,fbr Design and

Manufacture. Reviewed by Art Schwartz. (12.2) 74-78. Ferguson, Claire. Helamon Ferguson: Mathematics in Stone and Bronze. Reviewed by J.W. Cannon. ((18.2) 73-75.

Fischer, Gerd, ttirzebruch, Friedrich, Scharlau, Winfried, and Tornig, Willi (editors). Ein Jahrhundert Mathematik, 1890-1990: Festschrift zum Jubildum der DMV. Reviewed by David E. Rowe. (13.4) 70-78.

Gelfand, I.M., Glagoleva, E.G., and Shnol, E,E. Functions and Graphs. Reviewed by H. Wu. (17.1) 68-75.

Gelfand, I.M., Glagoleva, E.G., and Kirillov, A.A. The Method of Coordinates. Reviewed by H. Wu, (17.1) 68-75. Gelfand, I.M., and Shen, A. Algebra. Reviewed by H. Wu. (17.1) 68-75. Gleick, James. Chaos: Making a New Science. Reviewed by John Franks. (11.1) 65-69.

Goodstein, David L., and Goodstein, Judith R. Feynman's Lost Lecture. Reviewed by Graham W. Griffiths. (20.3) 68-70.

Graham, R.L., Gr~tschel, M. and Lovfisz, L. (editors). Handbook of Combinatorics. Reviewed by Herbert S. Wils (19.2) 68-69.

Graham, Ronald L., Rothschild, Bruce L., and Spencer, Joel H. Ramsey Theory. Reviewed by Richard K. Guy. (15.1) 70-71. Gregory, J.A. (editor). The Mathematics of Surfaces. Reviewed by Art Schwartz. (12.2) 74-78. Hamming, Richard W. The Art of Doing Science and Engineering: Learning to Learn. Reviewed by Roger Pinkham, (20.3) 67-68. ttargittai, Istvfin, ed. Symmetry: Unifying Human Understanding. Reviewed by Michele Emmer. (12.4) 75-78.

Hargittai, Istv~n, and Hargittai, Magdolna. Symmetry Through the Eyes of a Chemist. Reviewed by Michele Emmer. (12.4) 75-78. Hubbard, Barbara Burke. The World According to Wavelets. Reviewed by Jet Wimp. (19.3) 70-75. Ibramigov, Nail H (editor). Handbook of Lie Ovvup Analysis of Differential Equations: Applications in Engineering and Physical Sciences. Reviewed by Robert Gihuore. (19.1) 71-74. Illmer, Detlef. Rhythmomachia. Reviewed by Benno Artmann (11.3) 77-79.

Jones, A., Morris, S.A., and Pearson, K.R. Abstract Algebra and Famous Impossibilities. Reviewed by Israel Kleiner. (15.3) 73-75. Joseph, George Gheverghese The Crest of the Peacock. Reviewed by D.J. Struik. (14.4) 66-68.

Kac, Mark, Rota, Gian-Carlo, and

Schwartz, Jacob T. Discrete Thoughts: Essays on Mathematics, Science and Philosophy. Reviewed by Lawrence Zalcman. (12.3) 81-83.

Katok, Anatole, and Strelcyn, JeanMarie. Invariant Manifolds; Entropy and Billiards; Smooth Maps with Singularities. Reviewed by Ya. B. Pesin. (18.3) 74-75. Katz, Victor J. A History of Mathematics: An Introduction. Reviewed by Judith Victor Grabiner. (16.4) 73-76. Kirsch, Andreas. An Introduction to the Mathematical Theory of Inverse Problems. Reviewed by David Colton. (19.4) 72-75. Klee, Victor and Wagon, Stall. Old and New Unsolved Problems in Plane Geometry and Number Theory. Reviewed by Kenneth Falconer. (15.1) 72-75. Koblitz, Neal. A Course in Number Theory and Cryptography. Reviewed by J.H. Loxton. (15.2) 63-67. Korneichuk, N. Exact Constants in Approximation Theory. Reviewed by T.M. Mills. (16.1) 68-70.

1894-1964. Reviewed by Adrian Riskin. (17.4) 75-77. Massey, William. A Basic Course in Algebraic Topology. Reviewed by Peter Hilton. (15.4) 62-64. Maz'ya, V.G., and Prossdorf, S. Linear and Boundary Integral Equations, Encyclopedia of Mathematical Sciences, Vol. 27. Reviewed by David Colton. (15.3) 69-71.

Melter, Robert, Rosenfeld, Azriel, and Bhattacharya, Prabir, eds. The Geometry of Vision. Reviewed by Shimon Edelman. (15.4) 68-70. Meyer, Y. Wavelets: Algorithms and Applications. Reviewed by Mary Beth Ruskai. (17.4) 70-73. Morgan, Frank. Calculus Lite. Reviewed by Jet Wimp. (19.3) 70-75. Moritz, Robert Edouard. Memorabilia Mathematica: the Philomath's Quotation Book. Reviewed by Donald M. Davis. (17.2) 69-70.

Naturforschende Gesellschaft, ed. The Bernoulli Edition: The Collected Scientific Papers of the Mathematicians and Physicists of the BernouUi Family. Reviewed by David Speiser. (14.4) 63-64.

Kozlov, Valerii V., and Treshchev, Dnfitrii. Billiards, a Genetic

Nikulin, V.V., and Shafarevich, I.R. (translated by Miles Reid).

Introduction to the Dynamics of Systems with Impacts. Reviewed by Ya. B. Pesin. (18.3) 74-75. Kress, Rainer. Linear Integral Equations. Reviewed by Thomas S. Angell. (16.1) 63-67.

Geometries and Groups. Reviewed by John Stillwell. (11.4) 63-67. Omnes, Roland. The Interpretation of Quantum Mechanics. Reviewed by Robert Gilmore. (18.1) 70-75.

Landau, Edmund, and Gaier, Dieter.

Calculus from Graphical, Numerical, and Symbolic Points of View. Reviewed by Herb Clemens. (18.4) 67-69. Parker, Marla (editor). She Does Math!: Real-Life Problems from Women on the Job. Reviewed by Marci Perlstadt. (19.2) 69-71. Penrose, Roger. The Emperor's New Mind: Concerning Computers, Minds and the Laws of Physics. Reviewed by Marjorie Senechal. (14.2) 72-77. Pipkin, Allen C. A Course on Integral Equations. Reviewed by Thomas S. Angell. (16.1) 63-67. The Poetry of Wallace Stevens. Comments by Jonathan Holden. (12.1) 77. P41ya, George. The Pdlya Picture Album: Encounters of A Mathematician, edited by G.L. Alexanderson. Reviewed by Lee Lorch (11.2) 70-71.

DarsteUung und Begri~ndung einiger neuerer Ergebnisse der Funktionentheorie (dritte, erweiterte Aufiage). Reviewed by Lawrence Zalcman. (11.4) 61-63. Lang, Serge E. Undergraduate Analysis: 2nd Edition. Reviewed by David M. Bressoud. (20.1) 76-77. Lord, E.A., and Wilson, C.B. The Mathematical Description of Shape and Form. Reviewed by Art Schwartz. (12.2) 74-78. Liitzen, Jesper. Joseph Liouville 1809-1882, Master of Pure and Applied Mathematics (Studies in the History of Mathematics and Physical Sciences 15). Reviewed by J. Dieudonn6. (14.1) 71-73.

Makarov, B.M., Goluzina, M.G., Lodkin, A.A., and Podkorytov, A.N. Selected Problems in Real Analysis Volume 107, AMS Series of Translations of Mathematical Monographs. Reviewed by Jet Wimp. (16.4) 68-72.

Marsland, T. Anthony, and Schaeffer, Jonathan eds. Computers, Chess, and Cognition. Reviewed by Robert Levinson. (15.2) 67-71. Masani, Pesi R. Norbert Wiener, 1894-1964. Reviewed by Philip J. Davis. (17.4) 73-75. Masani, Pesi R. Norbert Wiener,

Ostebee, Arnold, and Zorn, Paul.

Porter, David, and Stirling, David G. Integral Equations: a Practical Treatment from Spectral Theory to Applications. Reviewed by Thomas S. Angell. (16.1) 63-67. Porter, Roy (Consultant Editor). The Biographical Dictionary of Scientists, 2nd edition. Reviewed by Donald M. Davis. (17.3) 73-75. Ransford, Thomas. Potential Theory in the Complex Plane. Reviewed by Jet Wimp. (18.4) 72.

Reid, Constance. The Search for E. T. Bell Also Known as John Taine. Reviewed by David M. Bressoud. (16.3) 72-74. R4nyi, Alfred. A Diary on Information Theory. Reviewed by Gregory J. Chaitin. (14.4) 69-70. Resnikoff, Howard. The Illusion of Reality. Reviewed by Shimon Edelman. (15.4) 68-70. Richards, Joan. Mathematical Visions: The Pursuit of Geometry in Victorian England. Reviewed by Thomas Drucker. (14.2) 77-79. Roman, Steven. An Introduction to Coding and Information Theory. Reviewed by S.C. Coutinho. (20.3) 65-67. Rosenthal, Eric. Advanced Calculus of Murder. Reviewed by Mary W. Gray. (12.1) 77-79. Rothstein, Edward. Emblems of Mind: the Inner Life of Music and Mathematics. Reviewed by Leonard Gillman. (202) 60-64. Sagan, Hans. Space-FiUing Curves. Reviewed by John Holbrook. (19.1) 69-71. Schattschneider, Doris. Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M.C. Escher. Reviewed by Marjorie Senechal. (14.2) 72-77. Schmalz, Rosemary. Out of the Mouths of Mathematicians: A Quotation Book for Philomaths. Reviewed by Donald M. Davis (17.2) 69-70.

Shasha, Dennis, and Lazere, Cathy. Out of Their Minds: The Lives and Discove~es of 15 Great Computer Scientists. Reviewed by Jet Wimp. (18.4) 77-79. Steen, Lynn Arthur. Mathematics Tomorrow. Reviewed by Martin Zerner. (13.2) 76-79. Stenger, Frank. Numerical Methods Based on Sinc and Analytic Functions. Reviewed by Kenneth L. Bowers. (18.2) 71-73. Stewart, Ian. The Problems of Mathematics. Reviewed by Cathleen S. Morawetz. (13.3) 81-83. Stewart, Ian. The Problems of Mathematics, 2nd edition. Reviewed by David M. Bressoud. (15.4) 71-73. Stewart, Iau. Nature's Numbers. Reviewed by Freeman Dyson. (19.2) 65-67. Stillwell, John. Mathematics and its History. Reviewed by John Fauvel and Abe Shenitzer. (14.3) 69-73. Stoppard, Tom. Arcadia: A Play. Reviewed by Mary W. Gray. (17.2) 67-68. Straffin, Philip D. Game Theory and Strategy. Reviewed by Marc Kilgour. (19.3) 68-70. Stroock, Daniel W. Probability Theory: An Analytic View. Reviewed by Peter Whittle. (18.3) 71-74.

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Struik, Dirk J. A Source Book in Mathematics, 1200-1800. Reviewed by Craig G. Fraser. (11.4) 68-70. Sz~kely, G/tbor J. Contests in Higher Mathematics: Miklos Schweitzer Competitions 1962-1991. Reviewed by Jet Wimp. (18.4) 72-73. Temme, Nico. Special Functions. Reviewed by Roderick Wong. (19.4) 75-76. Ulam, Stanislaw. Sets, Numbers and Universes, and Science, Computers, and People. From The Tree of Mathematics. Reviewed by Reuben Hersh. (14.4) 71-73.

vos Savant, Marilyn. The World's Most Famous Math Problem. Reviewed by Lloyd Milligan and Keith Yarnall. (16.3) 66-69. Weil, Andr6. The Apprenticeship of a Mathematician. Reviewed by Lawrence Zalcman. (15.4) 64-68. Wiener, Norbert. Invention: The Care and Feeding of Ideas (introduction by S.J. Heims). Reviewed by Philip J. Davis. (17.4) 73-75. Wilf, Herbert. generatingfunctionology. Reviewed by E. Rodney Canfield. (15.2) 71-75.

Winfree, A r t h u r T. When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias. Reviewed by Leon Glass. (15.1) 67-70. Wolfram Media. The Mathematics Book, 3rd edition. Reviewed by Stan Wagon. (19.3) 59-67. Wolfram Research Inc. Mathematica 3.0. Software. Reviewed by Stan Wagon. (19.3) 59-67. Zeh, H.D. The Physical Basis of the Direction of Time. Reviewed by John C. Baez. (16.1) 72-75.

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