No title

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.

S o m e History of t h e Hauptvermutung We offer a comment on the article

by Peter Hilton and and Ioan James on "The Whitehead Heritage" in the Winter 1997 issue of The Mathematical InteUigencer. The authors remarked that Whitehead was attracted to the "Hauptvermutung," but they only stated the conjecture for manifolds. In view of Whitehead's deeper involvement in complexes, it should be observed that the original conjecture was in this wider context, asserting that homeomorphic simplicial complexes are combinatorially equivalent. In this generality the fkrst counterexamples were obtained by John Milnor in 1961, using Whitehead's own simple homotopy theory. The first positive results on the more difficult manifold case were obtained by Andrew Casson and Dennis Sullivan in 1967. Their articles were not published at the time, but may now be found in The Hauptvermutung Book (K-Monographs in Mathematics 1, Kluwer, 1996, A. Ranicki ed.). A counterexample to the full manifold statement was found by Larry Siebenmann in 1969. Whitehead's work was not a direct primary ingredient of the manifold work. However, by then the "handlebody" theory introduced by Smale a decade earlier had become the principal tool used in manifolds. Handlebody theory is a translation of Whitehead's cell complexes to the manifold context. FRANK QUINN Virginia Tech Blacksburg, VA 24061-0123 USA e-mail: [email protected] ANDREW RANICKI University of Edinburgh Edinburgh Scotland, UK e-mail: [email protected]

I n t u i t i o n i s m and C e l e s t i a l Mechanics In "The solution to the n-body problem" (InteUigencer, vol. 18 (1996),

no. 3, 66-70), Florin Diacu refers to Brouwer's intuitionism in connection with a 1907 paper by Sundman. As he was trying to undo misleading folkmathematics, it is unfortunate that Prof. Diacu relied on R. L. Goodstein's Essays in the Philosophy of Mathematics (p. 5) as a source of information on the Annalen affair. Thus he relayed the folk-tale that Brouwer had "rejected all submitted papers that used reductio ad absurdum as a method of proof," as well as the equally erroneous story of the resignation and self-reelection (minus Brouwer) of the board of the Annalen. He could have got the story right by referring to the well-documented article by D. van Dalen (InteUigencer, vol. 12 (1990), no. 4, 17-31). None of the editors of the Annalen had accused Brouwer of imposing his philosophical position on authors. In the circular letter sent to his felloweditors, Btumenthal stated, "Brouwer has been a very conscientious and capable editor" (W. P. van Stigt, Brouwer's Intuitionism, North-Holland, Amsterdam, 1990, p. 102). Both Carath~odory and Einstein refused to sign the dismissal notice. As to the re-election of the editorial board (minus Brouwer), here were the editors of 1928: 1. under "unter Mitwirkung von": L. Bieberbach, H. Bohr, L.E.J. Brouwer, R. Courant, W. v.Dyck, O. H61der, Th. v.K~m~m, A. Sommeffeld; 2. under "gegenw~rtig herausgegeben von": D. Hilbert, A. Einstein, O. Blumenthal, C. Carath~odory; and here are the editors of 1929: 1: under "unter Mitwirkung von": O. Blumenthal, E. Hecke; 2. under "gegenw~rtig herausgegeben yon": D. Hilbert.

9 1997 SPRINGER-VERLAG NEWYORK, VOLUME 19, NUMBER 4, 1997

5

Prof. Diacu also conjectures that Brouwer might n o t have developed his intuitionism "had he k n o w n and understood Sundman's work." This seems far-fetched to me, considering Brouwer's own p r o n o u n c e m e n t s o n what led him in that direction (as for example in L.E.J. Brouwer, Collected Works, NorthHolland, Amsterdam, 1975, 472-476), a s w e l l a s his general opposition to the applications of mathematics. Of m a n y passages expressing this opposition, I quote this c o m m e n t o n the "games" of mathematics a n d logic: De par leur nature, ils ne devraient p a s s'immiscer darts la vie sociale. Celle-ci les ayant n~anmoins r~clam~s, ils subissent rinfluence des sciences pragmatiques tout en coop~rant, contre leur nature, aux transformations de la vie sociale qu'on appeUe le progr~s. Heureuse-ment, leurs plus beaux d~veloppements n'auront probablement jamais a u c u n rapport avec les questions techniques, ~conomiques ou politiques." (L.E.J. Brouwer,

Collected Works,Vol. 1, North-Holland, Am-sterdam, 1975, p. 503). t There may be constructivists who reject non-constructive proof because they think it "useless from a practical point of view." Brouwer was n o t one of them. His motivation for intuitionism lies elsewhere, and can be discerned only by a careful reading of his o w n words. VICTOR PAMBUCCIAN Department of Integrative Studies Arizona State University West P.O. Box 37100 Phoenix, AZ 85069-7100 USA e-mail: [email protected]

Florin Diacu replies: The excellent InteUigencer article by v a n Dalen came to m y attention some time after I completed m y article. Prof. P a m b u c c i a n is quite right. This illustrates that misleading folk-tales c a n be transmitted in print (the Goodstein book) as well as orally. Prof. Pambuccian goes o n to advance pertinent arguments against my specu-

lation on Brouwer's thinking. Let m e state m y reasons to leave it standing~ Constructivism is a normegligible part of Brouwer's intuitionism. The first chapter of On the Foundations of Mathematics, entitled "The Construction of Mathematics," makes clear that Brouwer has f o u n d e d his theory o n constructivism, i n d e p e n d e n t l y of a n y other r e a s o n s he might have had. But a s GOdel's result shows the feebleness of formalism, S u n d m a n ' s proof reveals the w e a k n e s s of intuitionism, which w a s well hidden up to now. I d o u b t that Brouwer would have remained insensitive to this fact, had he k n o w n it. Finally, I would like to t h a n k all those w h o have sent me c o m m e n t s o n my article. The list of n a m e s is too long to append. FLORIN DIACU Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3P4 Canada [email protected]

1By their nature, they ought not to be involved in human affairs. However, human affairs suck them in, and they come under the influence of the practical sciences, cooperating contrary to their nature in the transformations known as progress. Fortunately, their most beautiful developments will probably never have anything to do with technical, economic, Or political questions.

i) New SHELDON AXLER, San Francisco State University

LINEAR ALGEBRA DONE RIGHT Second Edition

This text for a second course in linear algebra is aimed at math majors and graduate students. The novel approach taken here banishes determinants to the end of the book and focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents-without having defined determinants--a clean proof that every linear operator on a finite-dimensional complex vector space (or an odd-dimensional real vector space) has an eigenvalue. A variety of inter-esting exercises in each chapter helps students understand and manipulate the objects of linear algebra. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite-dimensional spectral theorem. This second edition includes a new section on orthogonal projections and minimization problems. The sections on self-adjoint operators, normal operators, and the spectral theorem have been rewritten. New examples and new exercises have been added, several proofs have been simplified, and hundreds of minor improvements have been made throughout the text. Contents: Vector Spaces 9Finite-Dimensional Vector Spaces 9Linear Maps 9Polynomials ~ Eigenvalues and Eigenvectors 9Inner-Product Spaces 9Operators on Inner-Product Spaces ~ Operators on Complex Vector Spaces 9Operators on Real Vector Spaces 9Trace and Determinant 1997/256 PP., 21 ILLUS/SOFTCOVER/$29.95/ISBN0-387-98258-2 9 ALSOIN HARDCOVER:$59.95/ISBN 0-387-98259-0 9 UNOERGRADUATETEXTSIN MATHEMATICS

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8/97

Reference#H208

WALTER KIRCHHERR, MING LI,* AND PAUL VITANYI 1

The Miraculous Universa Distribution

W

hat is it, exactly, that scientists do? How, exactly, do they do it? H o w is a scientific hypothesis formulated? H o w does one choose one hypothesis over another?

It m a y be surprising that questions such as t h e s e a r e still discussed. Even m o r e surprising, perhaps, is the fact that the d i s c u s s i o n is still moving forward, that n e w i d e a s are still being a d d e d to the debate. Certainly m o s t surprising o f all, over the last 30 y e a r s o r so, the n o r m a l l y c o n c r e t e field of c o m p u t e r s c i e n c e h a s p r o v i d e d f u n d a m e n t a l n e w insights. Scientists engage in w h a t is usually called i n d u c t i v e reasoning. Inductive r e a s o n i n g entails making p r e d i c t i o n s a b o u t future b e h a v i o r b a s e d on p a s t observations. However, defining the p r o p e r m e t h o d of formulating s u c h predictions has o c c u p i e d p h i l o s o p h e r s through t h e ages. In fact, the British p h i l o s o p h e r David Hume (1711-1776) h a s argued convincingly that, in s o m e sense, p r o p e r ind u c t i o n is impossible [3]. It is impossible b e c a u s e w e c a n only r e a c h conclusions by using k n o w n d a t a a n d methods; s u c h a conclusion is logically a l r e a d y c o n t a i n e d in the starting configuration; consequently, the only form of i n d u c t i o n p o s s i b l e is deduction. P h i l o s o p h e r s have tried to find a w a y out o f this conundrum. To s e e w h e r e the d i s c u s s i o n s t a n d s today, let's put ourselves in the position of a b u d d i n g y o u n g scientist with a specific p r e d i c t i o n to make. Let's follow the young M i c e as she tries to win a bet. *Supported in part by the NSERC Operating Grant OGP0046506, ITRC, a CGAT grant, and the Steacie Fellowship. tpartially supported by the European Union through NeuroCOLT ESPRIT Working Group Nr. 8556, and by NW9 through NFI Project ALADDIN under Contract number NF 62-376 and NSERC under International Scientific Exchange Award ISE0125663.

Alice Is Offered A Bet Mice, w a l k i n g d o w n the street, c o m e s a c r o s s Bob, who is tossing a coin. He is offering o d d s to all p a s s e r s b y on w h e t h e r t h e n e x t t o s s will be h e a d s o r tails. The pitch is this: he'll p a y y o u two dollars if the n e x t toss is heads; y o u p a y him one dollar if the next t o s s is tails. M i c e is intrigued. Should she t a k e the bet? Certainly, if Bob is tossing a fair coin, it's a g r e a t bet. P r o b a b l y she'll win m o n e y in the long run. After all, she w o u l d e x p e c t t h a t half Bob's t o s s e s w o u l d c o m e up h e a d s and half tails. Giving up only one dollar on e a c h h e a d ' s toss and getting t w o for each t a i l s - why, in a while s h e ' d be rich! Of course, to a s s u m e that a s t r e e t hustler is tossing a fair coin is a bit of a stretch, and M i c e is no dummy. So she w a t c h e s for a while, recording h o w the coin c o m e s up for o t h e r bettors, writing d o w n a 1 for h e a d s and a 0 for tails. After a while, she has written 01010101010101010101. P e r h a p s Bob m a n i p u l a t e s the o u t c o m e s . C o m m o n s e n s e tells M i c e that she can e x p e c t foul p l a y w h e n she plays with Bob. What's h e r n e x t move? Research M i c e is n o w e q u i p p e d with d a t a (her r e c o r d of the t o s s e s she has o b s e r v e d ) a n d n e e d s to f o r m u l a t e a hypothesis concerning the p r o c e s s producing h e r d a t a - - s o m e t h i n g like, "The coin h a s a p r o b a b i l i t y of 1/2 of coming up heads." Or "The coin a l t e r n a t e s b e t w e e n h e a d s and tails." Which should it be? H o w d o e s one formulate a hypothesis? As w e

9 1997SPRINGER-VERLAGNEWYORK,VOLUME19, NUMBER4, 1997 7

said, Alice is no dummy. She first checks out what the great thinkers of the past had to say about it.

Epicurus The Greek philosopher Epicurus (342? B.C.-270 B.C.) is mainly known to us through the writings of the Roman poet Titus Lucretius Carus (100? B.C.-55? B.C.), who, in his long poem On the Nature of the Universe, popularized Epicurus's stoic philosophy among the Roman aristocracy. (What we moderns usually mean by "epicurean" has little to do with Epicurus, by the way.) Lucretius offers explanations for many natural phenomena and human practices (for example, he says, plausibly, that fire was delivered to humans by lightning), but he also admits that There are some phenomena to which it is not enough to assign one cause9 We must enumerate several, though in fact there is only one. Just as if you were to see the lifeless corpse of a man lying far away, it would be fitting to state all the causes of death in order that the single cause of this death may be stated. For you would not be able to establish conclusively that he died by the sword or of cold or of illness or perhaps by poison, but we know that there is something of this kind that happened to him. [9] This multiple explanations approach is sometimes called the principle of indifference. Bertrand Russell summarizes it as follows: "When there are several possible naturalistic explanations.., there is no point in trying to decide between them" [10]. In other words:

PRINCIPLE OF INDIFFERENCE: Keep all hypotheses that are consistent with the facts. (To be fair, it should be pointed out that Epicurean philosophy is not concerned with scientific progress but rather with haman happiness.)

William of O c k h a m The English cleric William of Ockham (1285-1349) is credited with formulating a different principle commonly called "Occam's Razor." He wrote, "Entities are not to be multiplied without necessity" and "it is vain to do with more what can be done with fewer." Again according to Bertrand Russell [10], "That is to say, if everything in some science can be interpreted without assuming this or that hypothetical entity, there is no ground for assuming it." As popularly interpreted, we have: OCCAM'S RAZOR: Among all hypotheses consistent with the facts, choose the simplest.

(of a model) is solely and precisely that it is expected to work . . . . Furthermore, it must satisfy certain aesthetic criteria--that is, in relation to how much it describes, it must be simple. [12] Of course, there are problems with this. Why should a scientist be governed by "aesthetic" criteria? What is meant by "simple"? Isn't such a concept hopelessly subjective? We're wading in deep waters now. However, we are wading not alone but together with the greatest scientist of all time, "Fortunate Newton, happy childhood of science!" in Einstein's phrase. Isaac Newton formulated in his Principia [8]:

N e w t o n ' s Rule #1 for d o i n g natural philosophy: We are to admit no more causes of natural things than such as are both true and sufficient to explain the appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for. Nature is pleased with simplicity, and affects not the pomp of superfluous causes. Thomas Bayes The English mathematician and cleric (clerics keep popping up in all this) Rev. Thomas Bayes (1702-1761) offered what is in essence a modified principle of indifference. Rather than accepting all hypotheses consistent with the facts as equal, he gave a method of assigning probabilities to hypotheses. B a y e s ' s Rule: The probability that a hypothesis is true is proportional to the prior probability of the hypothesis multiplied by the probability that the observed data would have occurred assuming that the hypothesis is true. [2] Suppose we have a priori a distribution of the probabilities P(H) of the various possible hypotheses. We want the list of hypotheses to be exhaustive and mutually exclusive so that 3: P(H) = 1, the summation taken over every possible hypotheses H. Assume, furthermore, that for all such H we can compute the probability I Pr(DIH) that sample D arises if H is the case. Then, we can also compute the probability Pr(D) that sample D arises at all: Pr(D) = ~ Pr(DIH ) P(H), summed over all hypotheses. From the definition of conditional probability, it is now easy to derive the familiar mathematical form of Bayes's Rule: Pr(HiD ) = Pr(DIH) P(H) Pr(D)

This is taken as given by most scientists and sometimes even explicitly stated. The great mathematician John von Neumann wrote,

Despite the fact that Bayes's Rule essentially rewrites the definition of conditional probability, and nothing more,

9 the sciences do not try to explain, they hardly try to interpret, they mainly make models . . . . The justification

1We use notation Pr( ) to distinguish computed probabilities from prescribed probabilities like the a priori probability P( ),

8

THE MATHEMATICALINTELLIGENCER

its interpretation a n d a p p l i c a t i o n are p r o f o u n d a n d controversial. The different H ' s r e p r e s e n t the p o s s i b l e alternative h y p o t h e s e s c o n c e r n i n g the p h e n o m e n o n w e wish to discover. The term D r e p r e s e n t s the empirically or otherw i s e k n o w n data c o n c e r n i n g this p h e n o m e n o n . The f a c t o r Pr(D), the probability o f d a t a D, is c o n s i d e r e d a s a normalizing factor, so that ~ Pr(HID) = 1, the s u m t a k e n o v e r all hypotheses. The f a c t o r P(H) is called the a priori, initial, o r prior p r o b a b i l i t y o f hypothesis H. It r e p r e s e n t s the p r o b a b i l i t y o f H being true b e f o r e w e have obtained any data. The p r i o r probability P ( H ) is often c o n s i d e r e d as the experim e n t e r ' s initial degree of belief in hypothesis H. The f a c t o r Pr(/~D) is called the final, inferred, o r posterior probability, w h i c h r e p r e s e n t s the a d a p t e d p r o b a b i l ity of H after seeing the d a t a D. In essence, B a y e s ' s Rule is a m a p p i n g from p r i o r p r o b a b i l i t y P(H) to p o s t e r i o r p r o b ability Pr(HID) d e t e r m i n e d b y d a t a D. Continuing to obtain m o r e a n d m o r e d a t a a n d repeatedly applying Bayes's Rule using the previously o b t a i n e d inferred p r o b a b i l i t y as the c u r r e n t prior, eventually t h e inf e r r e d probability will c o n c e n t r a t e m o r e and m o r e on the "true" hypothesis. It is i m p o r t a n t to u n d e r s t a n d t h a t one can f m d the true h y p o t h e s i s also, using m a n y e x a m p l e s , b y the law of large numbers. In general, the p r o b l e m is n o t so m u c h that in the limit the inferred probability w o u l d not c o n c e n t r a t e on the true hypothesis, but that the inferred p r o b a b i l i t y should give as m u c h information as p o s s i b l e a b o u t the p o s s i b l e h y p o t h e s e s from only a limited number of data. Given the p r i o r p r o b a b i l i t y of the h y p o t h e s e s , it is e a s y to obtain the inferred probability and, therefore, to m a k e informed decisions. Thus, with Bayes's Rule, w e k e e p all h y p o t h e s e s t h a t are c o n s i s t e n t with the observations, b u t we c o n s i d e r s o m e m o r e likely than others. As t h e a m o u n t of d a t a grows, w e h o m e in on the m o s t likely ones. But t h e r e is a n a s t y little p h r a s e there: "the experim e n t e r ' s initial belief in t h e hypothesis." H o w c a n a neutral o b s e r v e r have such an initial belief?. H o w can the p r o c e s s of assigning p r o b a b i l i t i e s get started? This is k n o w n as the p r o b l e m of assigning a priori probabilities. As a historical note: The m e m o i r [2] was p o s t h u m o u s l y p u b l i s h e d in 1764 by Bayes's friend the Rev. Richard Price. P r o p e r l y speaking, B a y e s ' s Rule as given is n o t due to Bayes. Pierre-Simon, Marquis de Laplace, w h o m w e will m e e t again l a t e r in this narrative, stated Bayes's Rule in its p r o p e r form and a t t a c h e d B a y e s ' s n a m e to it in [5]. W h e r e Does This Leave Alice? N o w that Alice k n o w s the thoughts of the ancients, w h a t s h o u l d she do. Should she t a k e the bet? Her b a s i c question is, "What p r o c e s s (that is, w h a t kind of coin) c a u s e d the s e q u e n c e 01010101010101010101?" That's a t o u g h one; so like any g o o d scientist, she first tries to a n s w e r a s i m p l e r question: Is Bob tossing a fair coin (one w h e r e h e a d s and tails are equally likely to c o m e up) or not? Since a fair coin could c a u s e such a sequence, Epicurus says w e c a n ' t rej e c t that hypothesis. (But w e can't reject a lot o f o t h e r hy-

potheses, either.) O c c a m says a c c e p t the fair coin hypothesis if it is s i m p l e r t h a n any other. (But he offers no help in d e t e r m i n i n g if "the p r o b a b i l i t y of h e a d s is 1/2" is simpler than, say, "the probability of h e a d s is 1/3.") Bayes's rule has s o m e intuitive a p p e a l here. The s e q u e n c e d o e s n ' t s e e m likely to have resulted from tossing a fair coin. Why not? w h a t d o e s Alice e x p e c t a fair-coin s e q u e n c e to look like? Randomness w h a t b o t h e r s Alice is that the s e q u e n c e of coin t o s s e s doesn't l o o k random. She e x p e c t s t h a t a fair coin p r o d u c e s a r a n d o m s e q u e n c e o f heads and tails. But w h a t is "random"? She h a s intuition a b o u t the c o n c e p t to be s u r e - 00101110010010111110 looks more random than 01010101010101010101--but, precisely, w h a t is m e a n t b y "random"? Again, let's review the thoughts of t h e sages. Dr. Samuel J o h n s o n (1709-1784), the great e i g h t e e n t h - c e n t u r y m a s t e r of conversation, had something to say on j u s t a b o u t all topics. His biographer, J a m e s Boswell, wrote: J o h n s o n w a s quite proficient in m a t h e m a t i c s . . . . Dr. Beattie observed, as something r e m a r k a b l e which h a d h a p p e n e d to him, that he c h a n c e d to see b o t h the No. 1 and the No. 1000 o f the h a c k n e y - c o a c h e s , the first and the last. "Why sir," said Johnson, "there is an equal c h a n c e for o n e ' s seeing t h o s e two n u m b e r s as any o t h e r two." He w a s clearly right; y e t t h e seeing o f two extremes, e a c h o f w h i c h is in s o m e d e g r e e m o r e conspicuous t h a n the rest, could n o t b u t strike one in a stronger m a n n e r t h a n the sight of any o t h e r t w o numbers. [1] Many of us (including Alice) w o u l d agree with Boswell. (Most o f us a r e n o t Samuel Johnson, o f w h o m it was also said, "There's no arguing with Johnson; for w h e n his pistol m i s s e s lure, he k n o c k s you d o w n w i t h the butt end o f it.") But w h y are the t w o n u m b e r s Dr. Beattie o b s e r v e d m o r e "conspicuous" t h a n any o t h e r t w o ? What does that mean? A r e n ' t t h e s e t w o n u m b e r s j u s t a s likely as any o t h e r two n u m b e r s (all pairs with equal p r o b a b i l i t y 1/1,000,000)? The g r e a t F r e n c h m a t h e m a t i c i a n Pierre-Simon Laplace (1749-1827) a d d r e s s e d the question o f w h y o u r intuition tells us that a regular o u t c o m e o f a r a n d o m event is unlikely: We a r r a n g e in o u r thought all p o s s i b l e events in various classes; a n d w e r e g a r d as extraordinary t h o s e classes which include a v e r y small number. In the game of h e a d s and tails, if h e a d c o m e s up a h u n d r e d times in a row, then this a p p e a r s to us e x t r a o r d i n a r y , b e c a u s e the alm o s t infinite n u m b e r of c o m b i n a t i o n s that can arise in a h u n d r e d t h r o w s are divided into r e g u l a r sequences, o r those in w h i c h w e observe a rule t h a t is e a s y to grasp, and into irregular sequences, t h a t are i n c o m p a r a b l y m o r e numerous. [5] What is r e g u l a r and w h a t is irregular? If Alice could convince h e r s e l f that the p a r t i c u l a r s e q u e n c e she o b s e r v e d is

VOLUME 19, NUMBER 4, 1997

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random, she c o u l d r e a s o n a b l y assign a high probability to the h y p o t h e s i s that Bob is tossing a fair coin, and she should take the b e t he offered. (In fact, betting strategies were a b a s i s for early definitions of r a n d o m n e s s - - i n essence, a s e q u e n c e of n coin t o s s e s is r a n d o m if y o u can't p r e d i c t the n t h t o s s by looking at the first n - 1 tosses. But s u c h definitions of rand o m n e s s ran into difficulties when a t t e m p t s w e r e m a d e to m a k e them m a t h e m a t i c a l l y precise.) Yet, the classical calculus of p r o b a b i l i t i e s tells us that 100 heads a r e j u s t as p r o b a b l e as any o t h e r sequence of h e a d s and tails, even though our intuition tells us that it is less "random" t h a n s o m e others. Laplace distinguishes bet w e e n the o b j e c t itself and a cause of the object: The regular c o m b i n a t i o n s occur m o r e rarely only because t h e y a r e less numerous. If w e s e e k a c a u s e wherever w e p e r c e i v e symmetry, it is n o t that w e r e g a r d the s y m m e t r i c a l event as less p o s s i b l e t h a n the others, but, since this e v e n t ought to b e the effect o f a regular cause o r that of chance, the first of these s u p p o s i t i o n s is m o r e p r o b a b l e t h a n t h e second. On a t a b l e w e see letters arranged in this o r d e r C o n s t a n t i n o p 1 e, and we j u d g e that this a r r a n g e m e n t is not the result of chance, not b e c a u s e it is less p o s s i b l e t h a n others, for if this w o r d w e r e n o t e m p l o y e d in any language w e w o u l d not s u s p e c t it c a m e from any particular cause, b u t this w o r d being in use a m o n g us, it is i n c o m p a r a b l y m o r e p r o b a ble that s o m e p e r s o n has thus a r r a n g e d the aforesaid letters than t h a t this a r r a n g e m e n t is due to chance. Let us try to turn Laplace's a r g u m e n t into a formal one. S u p p o s e w e o b s e r v e a b i n a r y string s o f length n and w a n t to k n o w w h e t h e r w e m u s t attribute the o c c u r r e n c e of s to pure chance o r to a cause. "Chance" m e a n s that the literal s is p r o d u c e d b y fair coin tosses. "Cause" m e a n s that t h e r e is a causal e x p l a n a t i o n for s having h a p p e n e d - - a causal explanation t h a t t a k e s m bits to describe. The p u r e c h a n c e of generating s i t s e / f l i t e r a l l y is a b o u t 2 - n . But the p r o b a bility of generating a cause for s is at least 2 - m . In o t h e r words, if t h e r e is s o m e simple cause for s (s is regular), then m < < n, a n d it is a b o u t 2 n - m t i m e s m o r e likely that s a r o s e as the r e s u l t o f s o m e cause than literally b y a rand o m process. It n o w r e m a i n s to m a k e this intuition operational.

Computer Science to the Rescue In the mid-1960s, t h r e e m e n - - R a y Solomonoff, Andrei N. Kolmogorov, a n d G r e g o r y C h a l t i n - - i n d e p e n d e n t l y invented the field n o w generally k n o w n as K o l m o g o r o v complexity. (Actually, S o l o m o n o f f was the e a r l i e s t b y a couple of years, and Chaitin last, b u t Kolmogorov, in the middle, w a s already w o r l d f a m o u s and his m a t h e m a t i c s impeccable, and his n a m e got a t t a c h e d to t h e field. As Billie Holliday sang, "Them t h a t ' s got shall get. T h e m that's not shall lose. So the Bible says, and, Lord, still it's true.") Solomonoff w a s a d d r e s s i n g Alice's p r o b l e m with Bayes's formula; h o w do w e assign a p r i o r i p r o b a b i l i t i e s to hy-

10

THE MATHEMATICAL INTELLIGENCER

p o t h e s e s w h e n we begin an e x p e r i m e n t ? K o l m o g o r o v and Chaitin w e r e addressing Alice's p r o b l e m of defining precisely w h a t is m e a n t b y a r a n d o m sequence. All t h r e e s a w t h a t the notion of "computable" lay at the h e a r t of t h e i r questions. They arrived at equivalent notions, showing t h a t t h e s e t w o questions are f u n d a m e n t a l l y related, a n d m a d e m a j o r strides t o w a r d a n s w e r i n g the age-old questions des c r i b e d above. (An extensive history o f the field c a n b e f o u n d in [7].) F o r t h o s e of us living in the c o m p u t e r age, the n o t i o n o f "computable" is p r e t t y m u c h intuitive. Many o f us have w r i t t e n programs; m o s t o f us have run computers. We k n o w w h a t a c o m p u t e r p r o g r a m is: it's a finite list o f ins t r u c t i o n s telling the c o m p u t e r h o w to calculate a particular function. Once you k n o w that, t h e notion o f a c o m p u t a b l e b i n a r y sequence o r "string" is both n a t u r a l and straightforward. C o n s i d e r a particular c o m p u t e r language like FORTRAN o r C + +. Once w e fix the language, w e can look at the prog r a m s t h a t t a k e no i n p u t - - y o u j u s t start t h e m running, a n d s o m e o f t h e m print out a b i n a r y string and stop. There a r e infinitely m a n y such p r o g r a m s . Now, let s b e a p a r t i c u l a r b i n a r y string. Some of the p r o g r a m s print out s; in fact, infinitely m a n y o f them do. Let's list them: P1, P2, P3. 9 9 and s o on to infinity. Among t h e s e programs, there is a shortest one. ( R e m e m b e r that p r o g r a m s t h e m s e l v e s a r e b i n a r y s t r i n g s - - t h i n k of object c o d e - - - s o w e can talk a b o u t the length o f a program.) H e r e ' s the k e y defmition: The c o m p l e x i t y of a b i n a r y string, s, is the length of a s h o r t e s t p r o g r a m which, on no input, prints out s. We'll use C(s) to d e n o t e the c o m p l e x i t y of s ( b u t we'll s h o r t l y r e p l a c e it with a v a r i a n t d e n o t e d K(s) that h a s t h e t e c h n i c a l p r o p e r t i e s we n e e d - - j u s t be prepared). Once w e have the definition, w e get a few e a s y facts. S u p p o s e s is a string n bits long. 1. C(s) <- n (plus s o m e constant).

This says that t h e r e is a p r o g r a m not m u c h longer t h a n s which will p r i n t out s. The one-line p r o g r a m "print(s)" will do the job. 2. There is a stliaag s for w h i c h C(s) >- n. This is true since t h e r e a r e only 2 n - 1 p r o g r a m s of length less than n, b u t 2 n strings o f length n. So at least one string m u s t satisfy this inequality. A string satisfying the s e c o n d inequality a b o v e m a y b e called random. Why? Such a string is its own s h o r t e s t description. In o t h e r words, it c o n t a i n s no regular pattern. F o r example, with the p h r a s e "a string of 10,000 zeros," one c a n d e s c r i b e a 10,000-bit string w i t h j u s t a few letters. But (plausibly) the s h o r t e s t w a y to d e s c r i b e the string 00100011101011010010 is b y writing it out. So using this idea, Alice is justified in feeling that 01010101010101010101 is n o t random. She can d e s c r i b e it with a pattern: "ten alternating O's a n d l's." But h o w d o e s that help h e r formulate a h y p o t h e s i s a b o u t the n a t u r e o f the coin Bob is flipping? T h a t ' s w h e r e Solomonoff's i d e a s c o m e in. Remarkably,

his initial idea gave rise to a w a y of assigning p r o b a b i l i t i e s to b i n a r y strings. If w e defime K(s) pretty m u c h as C(s) exp l a i n e d a b o v e (there are s o m e technical details w e ' r e postp o n i n g until the n e x t section), w e can assign a p r o b a b i l i t y to s as follows:

The first thing w e w a n t to do is to justify o u r calling a string w h i c h is its o w n s h o r t e s t d e s c r i p t i o n "random." Why should this definition be p r e f e r a b l e to any o t h e r we might c o m e up with? The a n s w e r to that w a s p r o v i d e d by t h e Swedish m a t h e m a t i c i a n Per Martin-Lbf (who w a s a postdoc o f Kolmogorov). Roughly, he d e m o n s t r a t e d that t h e P(s) 2 -K(s), definition "an n-bit string, s, is random iff C(s) >- n" enthat m a y b e taken as the a priori probability of s. This assures that every such individual r a n d o m string p o s s e s s e s s i g n m e n t o f probabilities is called the semimeasure uniwith certainty all effectively t e s t a b l e p r o p e r t i e s of ranversal f o r the class of enumerable semimeasures. We'll d o m n e s s t h a t hold for strings p r o d u c e d b y r a n d o m s o u r c e s j u s t call it the u n i v e r s a l d i s t r i b u t i o n . on the average. To s e e w h e r e this goes, t h i n k about the So what? What e x a c t l y d o e s this mean? (And h o w d o e s pre-Kohnogorov-complexity traditional problems of it relate to Alice's p r o b l e m ? ) First recall the t h r e e ancient w h e t h e r o r n o t the infinite s e q u e n c e o f decimal digits in p r i n c i p l e s for formulating hypotheses: the principle of in7r = 3 . 1 4 1 5 . . . can be distinguished f r o m typical o u t c o m e s difference, Occam's razor, a n d Bayes's Rule. As p o i n t e d out of a r a n d o m source. To d e t e r m i n e this, the sequence h a s earlier, Bayes's Rule is in a s e n s e a refinement of the prinbeen s u b m i t t e d to several statistical t e s t s for r a n d o m n e s s ciple of indifference. The i m p o r t a n c e of a priori p r o b a b i l called, appropriately, " p s e u d o - r a n d o m n e s s tests" (for exity is that it neatly c o m b i n e s all t h r e e principles. ample, w h e t h e r e a c h digit o c c u r s with frequency 1/10 Look at it this way: a n y s e n t e n c e can b e c o d e d into a within certain fluctuations). If ~- h a d failed any of t h e s e series of O's and l's. H y p o t h e s e s are sentences; so t h e y can tests, t h e n w e w o u l d have said that the s e q u e n c e is not ranb e c o d e d as binary strings. Since a priori p r o b a b i l i t y asdom. Luckily, it satisfies all of them. signs a probability to every b i n a r y string, it assigns a probWhat Martin-LOf p r o v e d w a s this: S u p p o s e y o u c o m e up ability to every hypothesis. But t h e r e ' s more. "Simple" hywith y o u r o w n definition of s u c h a "statistical test exposp o t h e s e s - t h e ones you favor u n d e r O c c a m ' s r a z o r - - a r e ing non-randomness." If y o u r definition is at all reasonable, p r e c i s e l y those with small complexity. then any string w h i c h m e e t s Martin-L6f's definition also If the c o m p l e x i t y is small, the a priori p r o b a b i l i t y is big. meets yours. Now, w h a t is "reasonable"? Here, w e have to So with this m e t h o d o f assigning probabilities to hypothee x a m i n e o u r intuition. First o f all, w e feel that m o s t strings s e s - a s required b y B a y e s ' s R u l e - - w e m a k e the s i m p l e s t are random, so w e d e m a n d that o f y o u r defmition. o n e s m o s t p r o b a b l e - - a s William of O c k h a m said w e (Specifically, w e d e m a n d that, o f all t h e 2 n strings of length should. n, at least 2 n (1 - 1/n 2) of t h e m do n o t fail y o u r randomOur solution of the induction p r o b l e m is to use Bayes's ness test.) Second, w e d e m a n d that t h e r e b e a c o m p u t e r rule with the single a priori probability P(s) = 2-K(s) in each p r o g r a m to e x e c u t e y o u r statistical t e s t - - i t m u s t be effecand every problem! Let's look at an example. Suppose w e tive. Technically, the set of all strings that don't m e e t y o u r have two working hypotheses, H1 and H2. Occam's razor says definition s h o u l d b e w h a t m a t h e m a t i c i a n s call recursively we should favor the simpler one. In this approach, that m e a n s enumerable, w h i c h m e a n s that t h e r e is a c o m p u t e r prothat w e should favor the one with the lower complexity (the gram that e n u m e r a t e s every string t h a t fails the randomone with the shorter description). Bayes's formula (as with ness t e s t - - t h a t is, is n o t random. the principle of indifference) says w e should keep t h e m both, F o r instance, s u p p o s e you define r a n d o m as "passing b u t assign probabilities to each one. The universal distribustatistical t e s t A." Now if an n-bit string, s, m e e t s Martintion satisfies both Occam and Bayes. Epicurus, too! Suppose L6f's definition of randomness, w e w a n t to p r o v e that it the shortest description of H i is 100 bits long, and the shortwill p a s s statistical t e s t A. Well, s u p p o s e it doesn't; in o t h e r est description of H2 is 200 bits long. Then we conclude that words, s u p p o s e it's one of the at m o s t 2n/n 2 strings that fail the probability o f / / 1 being the correct explanation is 1/21~176 test A. Then, h e r e is a d e s c r i p t i o n o f s: or about 8 X 10 -31, and that the probability of H2 being the The m t h string o f length n w h i c h fails t e s t A. correct explanation is 1/22~176or about 6 X 10 -61. These n u m b e r s also d e t e r m i n e their relative probabiliWe k n o w t h a t m is a n u m b e r b e t w e e n 1 and 2n/n 2. We m a y ties, so that w e can c h o o s e the m o s t likely one: H1 is a b o u t not k n o w w h a t n u m b e r m is, b u t w e k n o w it's in that range. 1030 t i m e s m o r e likely t h a n / / 2 . The length o f that d e s c r i p t i o n (if w e c o d e it in binary) inWe k e e p both h y p o t h e s e s (satisfying Epicurus), assign volves coding b o t h n (in log n bits) a n d m (in n - 2 log n p r o b a b i l i t i e s to our "initial beliefs" in t h e m (as B a y e s sugbits). This c o m e s to at m o s t n - log n bits (plus s o m e negg e s t e d w e do), and favor the simpler one with a higher ligible t e r m s w h i c h we ignore here); hence, w e can conp r o b a b i l i t y (so William o f O c k h a m won't feel left out). clude that C ( s ) < - n - log n. But t h e n s d o e s not m e e t One simple t h e o r y ties up a couple of millennia o f phiMartin-L6f's definition. losophy! To s e e that Martin-LOf's definition actually is itself such a r a n d o m n e s s test: In the first place, w e can a p p r o x i m a t e S o m e I)et.~il$ C(s) b y running all p r o g r a m s o f length at m o s t s (plus s o m e This section is for those readers w h o w o u l d like a f e w m o r e constant) for as long as it takes, in r o u n d s of one step o f details. each program. As s o o n as a p r o g r a m halts, w e c h e c k =

VOLUME 19, NUMBER 4, 1997

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w h e t h e r its o u t p u t is s, and if so, w h e t h e r this p r o g r a m b e a t s the c u r r e n t c h a m p i o n that o u t p u t s s (by being shorter.) In this way, w e a p p r o x i m a t e t h e s h o r t e s t p r o g r a m b e t t e r and b e t t e r as time goes by. This p r o c e s s will for each s, eventually d e t e r m i n e if it is not random. ( F o r r a n d o m s, the p r o c e s s m a y go o n forever.) This s h o w s also the seco n d property, n a m e l y that fewer t h a n 2n/n 2 strings fail Martin-L6fs test. F o r instance, let A t e s t C(s) <- n. That's the j u s t i f i c a t i o n for calling a string w h i c h is its own shortest d e s c r i p t i o n "random." It gets a bit stickier when you go into the details o f the universal distribution. You have to be a bit careful w h e n you t a l k a b o u t "shortest description." Of course, w h e n we talk a b o u t the description o f a string, w e mean, a s m e n t i o n e d above, a p r o g r a m which on no i n p u t will print out that string. But if w e w a n t to get p r o b a b i l i t i e s out of all this, w e are s u b j e c t to a certain k e y restriction: the probabilities m u s t a d d up to (no m o r e than) 1. We are f a c e d with the t a s k o f establishing that the sum o f all our a p r i o r i probabilities a d d up to no m o r e than 1. This a l m o s t killed S o l o m o n o f f s original idea. It w a s s o o n s h o w n t h a t if w e use t h e simple definition of shortest description, w e get that, for every n, t h e r e is an n-bit string, s, w h e r e the value o f C(s) is at m o s t log n. This m e a n s that for e v e r y n, t h e r e is a string s with P ( s ) at least 2 - log n o r 1/n. And, o f course, the infinite sum 1/2 4- 1/3 + 1/4 4- ." d i v e r g e s - - i t ' s certainly not one o r less! It was a b o u t a d e c a d e before Solomonoff's i d e a w a s resc u e d - b y Leonid A. Levin, a n o t h e r s t u d e n t o f Kolmogorov. Chaitin h a d the s a m e idea, b u t again later. The device is, i n s t e a d of considering the length of c o m p u t e r p r o g r a m s i n general, to c o n s i d e r only certain c o m p u t e r programs. Specifically, w e r e s t r i c t o u r attention to p r e f i x - f r e e comp u t e r programs, that is, a set o f programs, no one o f which is a prefLx of a n y other. (This is n o t too h a r d to imagine. F o r instance, if y o u design a c o m p u t e r language in w h i c h every p r o g r a m e n d s with the w o r d "stop" ( a n d "stop" m a y n o t a p p e a r a n y w h e r e else), the p r o g r a m s w r i t t e n in y o u r language form a prefix-free set.) The r e a s o n this a p p r o a c h saved the d a y is a k e y theor e m p r o v e d in 1949 b y L.G. Kraft (in his m a s t e r ' s thesis at MIT [4]). It says in the p r e s e n t p r o b l e m t h a t if w e restrict our attention to prefLx-free sets, then the resulting a p r i o r / probabilities will s u m to no m o r e t h a n 1. F r o m n o w on we'll use this slightly different definition of C(s), which w e d e n o t e b y K(s). So K ( s ) is the length o f the s h o r t e s t p r o g r a m for s a m o n g all prefLx-free syntactically c o r r e c t p r o g r a m s in o u r fixed p r o g r a m m i n g language. Thus, the universal distribution P(s) = 2 -g(s) m e e t s the requirements of p r o b a b i l i t y theory. Now, w h a t is o u r justification for calling it "universal"? Briefly, it's this: S u p p o s e y o u have defined a p r o b a b i l i t y distribution on strings. As long as it m e e t s a r e a s o n a b l e criterion ( n a m e l y that it be enumerable, which is w e a k e r than requiring that t h e r e is a c o m p u t e r p r o g r a m which, given s as input, will print out the probability y o u assign to s), then the universal distribution d o m i n a t e s yours, in the s e n s e that t h e r e is s o m e cons t a n t k, w h i c h d e p e n d s on y o u r p r o b a b i l i t y b u t n o t on s,

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for w h i c h k Pr(s) is as l e a s t at large as the p r o b a b i l i t y y o u a s s i g n e d to s. This is called "multiplicative domination" a n d w a s p r o v e d by Levin in t h e early seventies. In a way, this is similar to the idea of a "universal" Turing m a c h i n e which is universal in the sense that it can simulate any o t h e r Turing machine w h e n p r o v i d e d with an appropriate description of that machine. It is universally a c c e p t e d that the Truing machine is a mathematically precise version of w h a t our intuition tells us is "computable," and therefore the universal Turing machine can compute all intuitively c o m p u t a b l e functions [11]. The latter statement is not a m a t h e m a t i c a l one, it c a n n o t be proved: it is k n o w n as Turing's Thesis; in related form, it is called Church's ~ e s i s . Just as t h e Kolmogorov c o m p l e x i t y is minimal (up to an additive constant) among all description lengths that can be a p p r o x i m a t e d from above by a computational process, so d o e s the universal distribution multiplicatively d o m i n a t e (and is in a particular sense close to) each and every enum e r a b l e d i s t r i b u t i o n - - d i s t r i b u t i o n s that can be approxim a t e d from b e l o w by a c o m p u t a t i o n a l process. Hence, there are a lot of universalities here, and the Turing Thesis spawns. K O L M O G O R O V ' S T H E S I S : The Kolmogorov c o m p l e x i t y gives the s h o r t e s t d e s c r i p t i o n length a m o n g all d e s c r i p t i o n lengths that can be effectively a p p r o x i m a t e d from a b o v e a c c o r d i n g to intuition.

LEVIN'S THESIS: The universal distribution gives the largest p r o b a b i l i t y among all distributions that c a n be effectively a p p r o x i m a t e d from b e l o w a c c o r d i n g to intuition.

Repaying the Source So t h e n o r m a l l y c o n c r e t e field of c o m p u t e r s c i e n c e has c o n t r i b u t e d to an a b s t r a c t p h i l o s o p h i c a l d e b a t e w h i c h h a s o c c u p i e d the ages. One can, in turn, use the p h i l o s o p h i c a l i d e a s p r e s e n t e d to contribute to t h e field o f c o m p u t e r science. Several a r e a s of c o m p u t e r s c i e n c e have b e n e f i t e d f r o m the b a s i c c o n c e p t of u n i v e r s a l distribution. We will l o o k at o n e o f t h e m h e r e - - n a m e l y t h e a r e a called algor i t h m analysis. Algorithm analysis is c o n c e r n e d with determining the a m o u n t of time it t a k e s to run a p a r t i c u l a r p r o g r a m . Of course, the a m o u n t o f time a p r o g r a m t a k e s d e p e n d s on the size of the i n p u t to the program. F o r example, the n u m b e r o f s t e p s r e q u i r e d to s o r t n n u m b e r s using the sorting technique c o m p u t e r scientists call quicks o r t 2 will, i n the w o r s t case, b e p r o p o r t i o n a l to n 2. The italics are needed. A n o t h e r w a y to a p p r o a c h algorithm analysis is to d e t e r m i n e h o w fast a p r o g r a m runs on the average. This needs to b e m a d e precise. Let's l o o k c l o s e r at the p r o b l e m of sorting n numbers. We m a y as well a s s u m e o u r input is the n n u m b e r s il, i2, 9 9 9 in m i x e d up s o m e h o w , a n d w e w a n t to o u t p u t t h e m in order. The t i m e 2Briefly, the sorting algorithm known as quicksort is this: 9 Rearrange the list of numbers into two smaller lists, the left half and the right half, in such a way that every member of the left half is less than every member of the right half. 9 Sort the left half, and sort the right half. 9 Merge the two sorted halves into one sorted list.

required b y quicksort is p r o p o r t i o n a l to n 2 w h e n the input is the n u m b e r s a l r e a d y sorted. But, interestingly, for m o s t inputs (where the n u m b e r s a r e not even close to being in order), the time required will be p r o p o r t i o n a l to n log n. In o t h e r words, t h e r e is an input which will force quicks o r t to u s e n 2 (or s o ) steps, b u t for m o s t inputs, q u i c k s o r t will actually run m u c h faster than this. Now, i f all inputs are equally likely, the a v e r a g e running time for q u l c k s o r t is m u c h less than its running time in the w o r s t case. Again, the italics a r e n e e d e d . Are all inputs equally likely in "real" life? Actually, it d o e s n ' t s e e m that way. In "real" life, it s e e m s that m u c h c o m p u t e r time is s p e n t sorting lists w h i c h are nearly s o r t e d to begin with. What i n t e r e s t s us h e r e is the r e m a r k a b l e relation b e t w e e n w o r s t - c a s e running t i m e s and average-case running times r e v e a l e d b y the universal distribution. To reiterate, the term average implies the uniform distribution. The term worst case is independent of any distribution. It is s o m e h o w natural to e x p e c t that in m a n y c a s e s the average is better (in this c a s e lower) than the w o r s t case. It c a m e as a surprise w h e n it w a s s h o w n by t w o of us (NIL a n d PV) [6] that if w e a s s u m e the universal distribution, that is, if we a s s u m e that inputs with l o w c o m p l e x i t y ( o n e s with "short descriptions") are m o r e likely t h a n inp u t s with high complexity, t h e n the rtmning time w e exp e c t u n d e r this distribution is (essentially) the w o r s t - c a s e running time of the algorithm. This is n o t t o o h a r d to s e e in the case o f quicksort. As w e said, t h e w o r s t c a s e for q u i c k s o r t is w h e n t h e i n p u t is a l r e a d y sorted. In this case, the input h a s a s h o r t d e s c r i p t i o n ( n a m e l y "the n u m b e r s 1 t h r o u g h n in order"), w h e r e a s if the input is all m i x e d up, it is "random" a n d is its o w n s h o r t e s t d e s c r i p t i o n . U n d e r the universal distribution, t h e a l r e a d y s o r t e d i n p u t is far m o r e likely t h a n t h e u n s o r t e d input, and a g e n e r a l i z a t i o n of this c a u s e s quicks o r t to require n 2 steps. Why is this true for a l g o r i t h m s in general? Again, it's n o t t o o h a r d to s e e why, b u t the e x p l a n a t i o n is a bit m o r e abstract. S u p p o s e algorithm A r u n s fast on s o m e inputs and s l o w l y on others. Then, the p a r t i c u l a r input which c a u s e s A to run s l o w e s t has a s h o r t description, n a m e l y "that inp u t of size n which c a u s e s algorithm A to run slowest." This is a d e s c r i p t i o n o f length log n (plus s o m e c o n s t a n t ) w h i c h d e s c r i b e s a string o f length n. So the length-n string d e s c r i b e d has low c o m p l e x i t y a n d is assigned a high p r o b ability u n d e r the universal distribution. This is intuitively the r e a s o n w h y the universal distribution assigns high e n o u g h probability to s u c h s i m p l e strings to s l o w the average running time o f A to its worst-case running time. The universal distribution is a great b e n e f a c t o r for learning a n d induction; but it is so b a d for average running time a n d all o t h e r r e a s o n a b l e c o m p u t i n g r e s o u r c e s like comp u t e r m e m o r y use, that such distributions are n o w called "malignant" by c o m p u t e r scientists. T h e I,,Iniver,~al B e t We left Alice b a c k there a ways. Have w e really h e l p e d her? After all, Bob is a b o u t to flip the coin again, and it's time

for Alice to p u t up o r shut up. Well, w e are f o r c e d into the t h e o r e t i c i a n ' s defense here. Yes, w e have h e l p e d Alice. We have p r o v i d e d h e r with a solid framing of the p r o b l e m s h e confronts. If she's clever, she can m a k e a safe b e t with Bob. In fact, Alice plays the stock m a r k e t because, j u s t like Bob's offer, the profits go up all the time. However, in the s t o c k market, investment c o m p a n i e s tell you that "past performance is no guarantee for future performance." Alice knows about covering h e r position with side bets called "puts" and "calls." So, let's see h o w Alice can cover h e r position with Bob. What she can p r o p o s e is this: B o b flips his coin 1000 times a n d one part o f the s c h e m e is his original offer o f two dollars for one dollar p a y o u t on "heads." (Alice is a scientist, r e m e m b e r . This is lab work, a n d long lab hours are no d e t e r r e n t to her.) The results o f t h e s e 1000 flips a r e recorded, yielding a s t r i n g - - l e t ' s call it s---of 1000 r s a n d O's r e p r e s e n t i n g h e a d s and tails. With the s e c o n d part o f the scheme, Alice covers her position: Alice p a y s Bob one dollar a n d B o b p a y s Alice 21000-K(s) dollars. Now, if Bob's on the square, like the s t o c k market, he has to t a k e this side bet, since his e x p e c t e d p a y o u t is less t h a n o n e dollar. This follows f r o m Kraft's w o r k mentioned above, w h i c h sets the e x p e c t e d p a y o u t at Z 2-1~176176 2t000-K(s) -< 1, w h e r e the s u m is t a k e n over all b i n a r y strings s of length 1000. (The e x p e c t e d p a y o u t is actually smaller than 1 because t h e r e are p r o g r a m s that have length r 1000 and there are o t h e r p r o g r a m s than s h o r t e s t p r o g r a m s . ) If Bob is as h o n e s t as Alice's s t o c k b r o k e r (who a c c e p t s Alice's buying and selling orders, including h e r side-bet orders for p u t s a n d calls), Bob s h o u l d b e h a p p y to a c c e p t Alice's p r o p o s a l . In fact, he can e x p e c t to earn a little on the side bet. But if Bob's crooked, a n d his flips do not result in a r a n d o m string, b u t s o m e t h i n g like 000000 . . . 00000000000000, t h e n he'll receive 1000 dollars from Alice on the m a i n b e t but he'll p a y o u t something like 21000-1og 1000 dollars on the side bet. That's a b o u t - - w e l l , who cares? Bob d o e s n ' t have that m u c h money. If Bob's honest, this is no w o r s e t h a n his original proposal. But if Bob has any brains, he'll p a c k up and m o v e to a n o t h e r c o r n e r w h e r e Alice can't b o t h e r him, b e c a u s e in this game, Alice wins big if Bob is h o n e s t - - a b o u t 500 b u c k s - - a n d even bigger if he cheats! Using the universal distribution, w e have c o n s t r u c t e d the p e r f e c t universal bet that p r o t e c t s against all fraud. But t h e r e ' s a catch: n o n e of t h e s e s c h e m e s can actually be carried out. The c o m p l e x i t y o f a string is noncomputable. Given a string s, there is no w a y to d e t e r m i n e w h a t K(s) is. Alice c a n ' t d e t e r m i n e which o f h e r h y p o t h e s e s have low c o m p l e x i t y a n d which do not. The p a y o f f s c h e m e s she p r o p o s e s to Bob can't be calculated. So it a p p e a r s she's on h e r own. Don't leave it at that. An i d e a as e l e g a n t as the universal distribution c a n n o t be j u s t t o s s e d out. To m a k e the uni-

VOLUME 19, NUMBER 4, I997

13

versal side bet feasible, Alice can pay Bob one dollar and Bob pays Alice 21000- l e n g t h

of p

dollars for any prefix-free program p that Alice can exhibit after the fact and that computes s. This involves a computable approximation "length of some program p to compute s" of K(s) = "length of the shortest program to compute s." Consequently, Alice may not win as much in case of fraud because length of p >- K(s) by definition (and Kolmogorov's thesis). In particular, the scheme is not foolp r o o f anymore, for there may be frauds that Alice doesn't detect. However, for the particular bet p r o p o s e d by Bob, Alice only cares about compressibility based on deviating frequency of l's, because she can just bet that each bit will be 1. Such a betting strategy and side bet, based on counting the n u m b e r of l's in s and compressing s by giving its index in the set of strings of the same length as s and containing equally m a n y l's as s, are both feasible and foolproof. Wonderful Universal Induction We started out by asking how learning and induction can take place at all; and we have followed Alice in her quest to the universal distribution. Now for the full problem: from universal gambling to universal induction. It is a miracle that this ages-old problem can be satisfactorily resolved by using the universal distribution as a "universal a priori probability" in Bayes's Rule. Ray Solomonoff invented a perfect theory o f induction. Under the relatively mild restriction that the true a priori distribution to be used in Bayes's Rule is computable, it turns out that one can mathematically prove that using the single fixed universal distribution instead of the actually valid distribution (which m a y be different for each problem we want to apply Bayes's Rule to) is almost as g o o d as using the true distribution itself! This is the case both when we want to determine the most likely hypothesis and when we want to determine the best p r e d i c t i o n - - w h i c h are two different things. They are two different things because the best single hypothesis does not necessarily give the best prediction. For example, consider a situation where we are given a coin of u n k n o w n bias p of coming up heads, which is either Pl = 1/3 o r P2 = 2/3. Suppose we have determined that there is probability 2/3 that p = Pl and probability 1/3 that p = P2. Then, the best hypothesis is the most likely one: p = Pl, which predicts a next outcome heads as having probability 1/3. Yet, the best prediction is that this probability is the expectation of throwing heads, which is

2

1

71 + 72 =

4

To take the prediction case: Solomonoff has shown that using the universal distribution, the total expected prediction error over infinitely many predictions is less than a

14

THE MATHEMATICALINTELLIGENCER

fixed constant (depending on the complexity of the true a priori distribution). This means that the expected error in the nth prediction goes d o w n faster than 1/n. This is good news, and in fact, it is better news than any other inference m e t h o d can offer us. It turns out that these ideas can be used to prove Occam's Razor itself. Traditional wisdom has it that the better a theory compresses the learning data concerning some p h e n o m e n o n under investigation, the better we are enabled to learn, generalize, and predict u n k n o w n data. This belief is vindicated in practice but apparently has not been rigorously proved in a general setting. Two of us [PV and ML] have recently shown that, indeed, optimal compression is almost always a best strategy in hypotheses identification. For the different prediction question, whereas the single best hypothesis does not necessarily give the best prediction, w e demonstrated that, nonetheless, compression is almost always the best strategy in prediction methods. Statisticians like J o r m a Rissanen and Chris Wallace k n o w about Alice's problem. They have translated Solomonoff's ideas into workable (that is, easily computable) forms called "minimum description length" algorithms [7]. Such algorithms help m a n y Alices get on with practical problems nowadays, such as video scene analysis, risk minimization, and even playing the stock market. 3,a Acknowledgments We thank Harry Buhrman, Richard Fortnow, and John Tromp for comments.

Cleve, Lance

REFERENCES 1. J. Boswell, The Life of Samuel Johnson, New York: Doubleday and Co. (1945). 2. T. Bayes, An essay towards solving a problem in the doctrine of chances, Philos. Trans. Roy. Soc. 53 (1764) 376-398; 54 (1764) 298-331. 3. D. Hume, Treatise of Human Nature, Book I, 1739. 4. L.G. Kraft, A device for quantizing, grouping, and coding amplitude modulated pulses, Master's thesis, Department of Electrical Engineering, M.I.T., Cambridge, MA, 1949. 5. P.S. Laplace, A philosophical essay on probabilities, 1819. English translation: New York: Dover (1951). 6. M. Li and P.M.B. Vit&nyi, Worst case complexity is equal to average case complexity under the universal distribution, Inform. Process. Lett. 42 (1992), 145-149. 7. M. Li and P.M.B. Vitb.nyi,An Introduction to Kolmogorov Complexity and Its Applications, 2nd ed., New York: Springer-Verlag (1997). 8. I. Newton, Philosophiae Naturalis Principia Mathematica also known as the Principia, 1687.

3The authors have been informed of a scientific stock investment company using software based on approximations of the universal distribution hitting the stock market in the near future. The authors decline a priori any responsibility for any happenings resulting from any methods disclosed in this article. 4All these things and the accomplishments of the heroes in this story, Ray Solomonoff, Andrei N. Kolmogorov, Gregory Chaitin, Per Martin-L6f, Claus Schnorr, Leonid A. Levin, Peter Gacs, Tom Cover (who introduced universal gambling and portfolio management), and many others, are explained in [7].

9. Titus Lucretius Carus, The Nature of the Universe (Ronald Latham, translator), New York: Penguin Books (1965). 10. B. Russell,A History of Western Philosophy, New York: Simon and Schuster (1945). 11. A.M. Turing, On computable numbers with an application to the

Entscheidungsproblem, Proc. London Math. Soc., Ser. 2 42 (1936), 230-265; Correction, 43 (1937), 544-546. 12. J. von Neumann, Collected Works, Volume V, New York: Pergamon Press (1963).

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15

KEITH HANNABUSS

Sound and Symmetry

~

hen asked whether the sort o f m a t h e m a t i c s I do has a n y practical use, I usually reply that it can be used to design hi-fi systems. It seems to be easier to motivate the applications of group representations i n this area than i n m y m a i n area o f q u a n t u m theory. I w a s f i r s t m a d e aware

of this use o f group t h e o r y b y a fellow student, Michael Gerzon, w h o s e d e a t h in May 1996 at the age o f 50 r o b b e d the audio w o r l d o f one of its m o s t creative minds, although his n a m e will b e unfamiliar to m o s t m a t h e m a t i c i a n s [1]. After taking a d e g r e e in m a t h e m a t i c s at Oxford, Gerzon s t a y e d on to d o g r a d u a t e w o r k in the m a t h e m a t i c a l foundations of q u a n t u m theory. He was i n t e r e s t e d in countably c o n v e x s t r u c t u r e s in state spaces, a n d his w o r k w a s sufficiently impressive t h a t in 1970 he was a p p o i n t e d to a Junior Lectureship. (His t w o i m m e d i a t e p r e d e c e s s o r s w e r e Tony J o s e p h and E. Brian Davies, n o w well k n o w n for their w o r k in a l g e b r a a n d analysis, respectively, b u t t h e n also working on m a t h e m a t i c a l a s p e c t s of q u a n t u m theory.) At this point, his health, n e v e r strong, d e t e r i o r a t e d rapidly. When he tried to w r i t e up his results, he e n d e d up in the hospital, and he s o o n realized that his constitution w a s t o o frail for any n o r m a l a c a d e m i c career. Instead, he d e c i d e d to w o r k as an i n d e p e n d e n t consultant in t h e field of audio, w h e r e he was a l r e a d y starting to m a k e a n a m e for himself. (Although G e r z o n did n o t publish his early m a t h e m a t i c a l results, David E d w a r d s ' s p a p e r [2] on one o f Gerzon's theo r e m s gives a flavour o f his work.) In the late 1960s, several electronics c o m p a n i e s w e r e developing "quadraphonic" s o u n d systems. Gerzon w a s quick to see the w e a k n e s s o f each o f t h e p r o p o s e d fours p e a k e r s y s t e m s a n d set a b o u t designing one w h i c h w o u l d really work. He w a s t h o r o u g h in his a p p r o a c h , looking at all a s p e c t s o f the w a y directional information is perceived, from the c h a r a c t e r i s t i c s o f s p e a k e r s to p s y c h o a c o u s t i c s and the physiology of the ear. He l o o k e d n o t only at the general principles involved, but also h o w t h e s e principles might, in practice, b e implemented, w h i c h i n v o l v e d - - i n

16

THE MATHEMATICAL INTELLIGENCER 9 1997 SPRINGER VERLAG NEW YORK

Michael Gerzon (Courtesy of Stephen Thornton)

s o m e c a s e s - - s o m e quite s o p h i s t i c a t e d n e w n u m e r i c a l algorithms. The group theory, on w h i c h w e c o l l a b o r a t e d during 1969-71, was j u s t one small a s p e c t of this general programme. F r o m the start, he s t r e s s e d that, c o n t r a r y to m u c h of the

prevailing design p h i l o s o p h y at that time, it is n o t only the horizontal disposition of s o u n d s which matters. Sounds r e a c h the h e a d from all directions, and the s o u n d field can be r e p r e s e n t e d by a c o m p l e x - v a l u e d f u n c t i o n f on t h e unit s p h e r e of possible directions, w h o s e m o d u l u s ]f(x)l enc o d e s the a m p l i t u d e of the s o u n d at that instant f r o m a dir e c t i o n x in the sphere, S 2, and w h o s e a r g u m e n t gives the phase. The energy is p r o p o r t i o n a l to the integral o f If(x)l 2, and so for finite energy, w e r e q u i r e f ~ L2($2). E a c h s o u n d is r e c o r d e d on N channels o f disc, tape, o r w h a t e v e r m e d i u m one uses, e a c h channel of which can likewise c a r r y p h a s e a n d a m p l i t u d e information. ( F o r old-fashioned m o n o p h o n i c recordings N = 1, for s t e r e o N = 2, a n d for q u a d r a p h o n y N = 4.) R e c o r d i n g thus m a p s e a c h s o u n d field to a v e c t o r R f E V = C N, whilst p l a y b a c k t a k e s v E V to P v E L2($2). We shall t a k e R and P to b e linear maps, t h o u g h Gerzon w a s well a w a r e o f the i m p o r t a n c e o f nonlinear effects. When a r e c o r d i n g is p l a y e d back, after filtering through the fmite-dimensional s p a c e V, m u c h inform a t i o n has b e e n lost, b u t one can still try to optimize the s y s t e m within this constraint. One i m p o r t a n t criterion is that if one turns a r o u n d whilst listening to a r e c o r d to see w h a t the cat is doing in the corner, the string quartet s h o u l d n o t s e e m to j u m p o n t o the m a n t e l p i e c e or dangle from the chandelier. Mathematically, the system m u s t b e h a v e well under the a c t i o n of the r o t a t i o n group, SO(3). E a c h rotation g s e n d s a s o u n d field f to (U(g).f)(x) = f ( g - l x ) , and U is a r e p r e s e n t a t i o n of SO(3) on L2($2), that is, a h o m o m o r p h i s m to the group o f invertible linear operators. Rotational invariance c a n b e ens u r e d if V also carries a representation, D, o f SO(3), a n d R a n d P are intertwining o p e r a t o r s : D(g)R = RU(g) and U(g)P = PD(g) for all g ~ SO(3). (There m a y have to b e m o r e than N s p e a k e r s to achieve this ideal p l a y b a c k . ) These r e p r e s e n t a t i o n s c a n all be e x p r e s s e d as d i r e c t s u m s o f irreducible representations, and it is k n o w n that the rotation group has an irreducible r e p r e s e n t a t i o n D j in e a c h o d d d i m e n s i o n 22" + 1, w h i c h is unique up to equivalence. Each D j occurs just once as a s u m m a n d of U, w h e r e it acts on the s u b s p a c e of spherical harmonic fimctions of o r d e r j . In t h e simplest c a s e o f m o n o p h o n y , with N = 1, t h e r e is only r o o m for the trivial r e p r e s e n t a t i o n D o w h i c h m a p s e v e r y rotation to 1. In this case, w e would e x p e c t to r e c o r d j u s t the average of the s o u n d field over all directions, and to p l a y this b a c k as a c o n s t a n t s o u n d field, i n d e p e n d e n t of direction. There is actually a m o r e general c o n s t r u c t i o n o f the p l a y b a c k and recording maps. It is easily s h o w n that within e a c h irreducible r e p r e s e n t a t i o n space, there is a "zeroweight" vector, unique up to multiples, which is fixed b y Di(h) for all h in the s u b g r o u p o f rotations a b o u t a c h o s e n axis e. In fact, even w h e n a representation, D, is reducible, the direct sum of such v e c t o r s in its irreducible constituents gives a v e c t o r p fixed b y this subgroup. Any axis x E S 2 is the image o f e u n d e r a suitable r o t a t i o n g, a n d the m a p x = g.e --~ D(g)p is n o w well defined. E v e r y finited i m e n s i o n a l r e p r e s e n t a t i o n o f SO(3) is unitary with r e s p e c t to s o m e inner product, and w e set

Pv(x) = (D(g)p, v). It is e a s y to c h e c k that P i n t e r t w i n e s D and U, and so c a n be u s e d as a p l a y b a c k operator. (When D = D ~ this gives b a c k the a b o v e constant map.) B e c a u s e L2(S 2) a n d V b o t h have invariant inner p r o d ucts, r e c o r d i n g maps, R, can b e c o n s t r u c t e d as the adjoints of p l a y b a c k maps. Thus, for dS(g), the e l e m e n t of surface a r e a on t h e s p h e r e at g.e, a n d r a f i x e d vector, one t a k e s

R f = fs 2 f ( g ) D ( g ) r dS(g). (This w a s i n s p i r e d b y a p r o o f of the F r o b e n i u s r e c i p r o c i t y theorem; similar c o n s t r u c t i o n s later f o r m e d the basis of t h e generalized c o h e r e n t state a p p r o a c h to quantum theory.) Q u a d r a p h o n i c systems a r e n o w easily described. The only four-dimensional r e p r e s e n t a t i o n s o f SO(3) are the direct s u m o f four copies of D o o r D O@ D 1. The f o r m e r w o u l d j u s t give f o u r - s p e a k e r m o n o p h o n i c s (there is little p o i n t in duplicating representations), so one m u s t c h o o s e the latter case. The construction o f the p r e v i o u s p a r a g r a p h t h e n p r o v i d e s the r e c o r d i n g and p l a y b a c k maps. The recording m a p p i c k s o u t all zeroth- and first-order harmonics. In fact, this is j u s t typical o f w h a t h a p p e n s with N = n 2 channels, w h e n C N h o s t s t h e direct s u m D~ 1~ . . . $ D n-1. Having e s t a b l i s h e d the principles, one is left with t h e p r a c t i c a l questions o f h o w P a n d R s h o u l d be implemented, such as w h e r e s p e a k e r s and m i c r o p h o n e s should be loc a t e d and the signals mixed. M i c r o p h o n e s w e r e m o r e directly u n d e r the control o f the engineers; with P e t e r Craven, G e r z o n designed a regular t e t r a h e d r a l a r r a y of f o u r m i c r o p h o n e s to s a m p l e the incoming s o u n d s (Fig. 1). This "Soundfield" m i c r o p h o n e w a s first u s e d to m a k e an exp e r i m e n t a l r e c o r d i n g in Merton College chapel on 8 May 1971 a n d is still in c o m m e r c i a l p r o d u c t i o n [3]. Its uses w e r e n o t limited to quadraphony. Michael told me later with s o m e a m u s e m e n t t h a t the m i c r o p h o n e w a s very p o p u l a r with s o u n d engineers r e c o r d i n g live concerts, due to its e a s e of use. Ordinary s t e r e o m i c r o p h o n e s had to be p o s i t i o n e d a n d aligned very a c c u r a t e l y to give a g o o d result, a n d this w a s n o t always e a s y w h e n they h a d to b e strung a c r o s s a c o n c e r t hall; b u t t h e Soundfield microp h o n e w a s m u c h m o r e tolerant, as t h e signal could always be r e c e n t r e d electronically using a r o t a t i o n in V. [Actually one could r e p o s i t i o n the m i c r o p h o n e electronically too, using the Lorentz group SO(3,1), w h i c h also acts on C 4. Lorentz t r a n s f o r m a t i o n s could be u s e d to give m o r e emp h a s i s to t h e front, back, o r any c h o s e n direction. Since such t r a n s f o r m a t i o n s are i n t e r p r e t e d relativistically as velocity changes, w e d i s c u s s e d w h e t h e r to label the k n o b controlling s u c h a d j u s t m e n t s with "warp factors."] F o r playback, the l o u d s p e a k e r s w e r e also to be l o c a t e d tetrahedrally at f o u r of the eight c o r n e r s o f a rectangular r o o m (Fig. 2). (When nonlinear effects such as energy localization w e r e t a k e n into account, this t u r n e d out n o t to be the b e s t arrangement, and it was l a t e r r e p l a c e d by a "birectangular" design.) Surprisingly, the case of t w o - c h a n n e l s y s t e m s is r a t h e r

VOLUME 19, NUMBER 4, 1997

17

to the r a y through D(g)p. E v e n n o n l i n e a r m a p s c o m p a t i b l e with this could be used.] Shortly after this g r o u p - t h e o r e t i c a l work, Michael G e r z o n p r e s e n t e d me with a sequence of c o n j e c t u r e d inequalities c o n c e r n e d with the optimization of the r e c o r d ing a n d p l a y b a c k maps. No s o o n e r h a d I p r o v e d one t h a n he w o u l d return with a generalization. Although the original v e r s i o n s l o o k e d very different (and much m o r e complicated), they have a v e r y s i m p l e interpretation a n d p r o o f in t h e p r e s e n t context. The k e y to this simplification lies in the realization that a p a r a m e t e r called the directivity factor, w h i c h m e a s u r e s h o w well a s y s t e m resolves s o u n d s f r o m different directions, c a n b e i n t e r p r e t e d as Itr(PR)] 2 tr((PR)*(PR)) "

(As P R factors through V, it h a s finite rank, and so the t r a c e m a k e s sense.) Let E be the (finite r a n k ) o r t h o g o n a l p r o j e c t i o n onto the range of P in L2($2). The C a u c h y - S c h w a r z - B u n y a k o v s k y inequality for the trace i n n e r p r o d u c t on o p e r a t o r s gives ]tr(PR)] 2 = ]tr(E*PR)I 2 <- tr(E*E)tr[(PR)*(PR)],

Figure 1. The original tetrahedral microphone arrangement.

w h i c h s h o w s that the directivity f a c t o r is b o u n d e d a b o v e b y t r ( E * E ) = t r ( E ) , the r a n k o f E. To optimize the system, t h e directivity factor s h o u l d b e as large as possible, a n d t h e b o u n d is achieved if a n d only if P R = ~E, for s o m e s c a l a r A, which can be a s s u m e d nonzero, for vanishing P R w o u l d clearly not p r o v i d e g o o d reproduction. But then, (RP) 2 = R ( A E ) P = ARP, s o t h a t R P = AF w h e r e F is also a projection. Moreover,

(Reproduced by permission of the Audio Engineering Society.)

A t r ( E ) = tr(PR) = tr(RP) = A t r ( F ) ,

m o r e subtle t h a n the p r e v i o u s discussion w o u l d suggest, and an o r d i n a r y s t e r e o uses a rather different s y s t e m from that d e s c r i b e d here. F o r two channels, one w o u l d like to use an irreducible two-dimensional r e p r e s e n t a t i o n of the rotation group, b u t t h e r e is none. However, although the e a r is sensitive to relative phases, the overall p h a s e of a s o u n d d o e s n o t s e e m to m a t t e r (or at l e a s t n o t to affect the p e r c e p t i o n of direction). N o w the r o t a t i o n group d o e s have an irreducible p r o j e c t i v e r e p r e s e n t a t i o n (a h o m o m o r p h i s m up to factors o f m o d u l u s 1) in every dimension, and, in particular, it has the spin representation, D u2, in t w o dimensions. The r o t a t i o n s D1/2(h) a b o u t e do n o t leave any vect o r in the s p a c e fLxed, b u t they have t w o i n d e p e n d e n t eigenvectors w h i c h are simply multiplied b y u n i m p o r t a n t p h a s e factors. We m a y c h o o s e p to b e one o f t h e s e and then p r o c e e d a s before. The p l a y b a c k m a p n o w gives s e c t i o n s o f a line bundle over the s p h e r e r a t h e r than functions. B e c a u s e the bundle is nontrivial, t h e r e m u s t be a p h a s e d i s c o n t i n u i t y somewhere, but this c a n be s w e p t u n d e r the carpet, or, m o r e precisely, p l a c e d b e l o w the listener, in a p l a c e w h e r e it is least obtrusive. [Gerzon i n t e r p r e t e d P as an e m b e d d i n g of the sphere into the projective s p a c e of V, w h i c h t o o k g.e

18

THE MATHEMATICALINTELLIGENCER

s o that E and F have the s a m e rank. But F is a p r o j e c t i o n on the finite-dimensional s p a c e V, a n d so its rank is at m o s t N, a n d this is achieved j u s t w h e n F is the identity, 1y. Figure 2. The four speakers could be placed in an approximately tetrahedral formation at four nonadjacent corners of a room.

We have therefore shown that when the directivity factor is maximal, R P is a multiple of the identity; that is, recording the playback just gives back the original (apart from changes of volume and phase). Reversing the argument, we see that RP = A1Vimplies that PR = AE for some projection E, with E P = P. To ensure that E is an orthogonal projection, we add the assumption that E ~ P = P . (This condition shows that E = A - 1PR = A - 1 E * P R = E e E is selfadjoint, and, conversely, when E is orthogonal, we have E * P = E P = P.) Thus, the directivity factor is maximal precisely when R P = A 1 V a n d E * P = P. We have seen that the directivity factor is bounded above by the number of channels, N. This is analogous to the way in which limiting the frequency bandwidth restricts the ability to resolve rapidly changing sounds, and is related to Heisenberg's uncertainty principle in quantum mechanics. (A general theorem encompassing all these varied examples was proved by Alan Carey a few years later [4].) For rotationally invariant systems, the single condition that RP be a multiple of the identity is both necessary and sufficient for optimality. This happens because the adjoint of the intertwining property for R shows that U ( g ) R * = R * D ( g ) for all rotations g, so that R * a n d P both intertwine D and U. Because each irreducible representation occurs just once as a direct summand in U, and R has maximal rank, Schur's Lemma, which characterizes intertwining operators between irreducible representations, shows that P = R*a for some operator, a, on V that intertwines D. If RP = A1v, we deduce that P~P = PeR*c~ = ( R P ) * a = Aa,

so

Soundfield Mark V (Courtesy of

that E,p

= ~ - 1 R ~ p ~ p = R*cx = P.

We can now show that the rotationally invariant systems for irreducible D automatically optimize the directivity factor. The intertwining properties of R and P combine to give RPD(g) = RU(g)P = D(g)RP

for any rotation g. When D is irreducible, Schur's Lemma then tells us that R P is a multiple of the identity. Although R P always intertwines D, in general one can only conclude

that it is a linear combination of the projections Q j onto the irreducible components of D. The projections are orthogonal with respect to the trace inner product, so we have RP = j~ tr(Q3~P) tr(Q3~) 03"This is a multiple of the identity just when all the coefficients t r ( Q j R P ) / t r ( Q j ) are the same. By direct calculation, we discover that tr(QjRP) = 4 1 r ( p , Q j r ) , so that optimality imposes a condition on the choice of vectors p and r. As emphasised above, this formed only one aspect of the overall design strategy, later superseded by better techniques, but it did play an important part in the original development [5]. As well as the Soundfield microphone, Michael Gerzon and his collaborators put these ideas to use in the Ambisonics surround sound system. Unfortunately, by the time this system became available, the concept of quadraphonic systems had been so tarnished by those weaknesses of other systems which Gerzon sought to rectify, that commercial interest had waned. Moreover, the development of the system had been sponsored by the National Research and Development Corporation (now the British Technology Group), which missed its best commercial opportunities. (There is, however, the possibility that Ambisonics may influence the eventual standards for high-quality audio on the forthcoming Digital Video/Versatile Disc [61.) Michael Gerzon took these setbacks philosophically. By Soundfield Research Ltd.) the time of his death he had published over 90 papers on audio systems, including recent work on lossless coding with Peter Craven. In 1991, he received the gold medal of the Audio Engineering Society. Shortly before his death, he told me that he had realized that category theory was a natural tool for signal processing, and that he was thinking of writing a monograph on the subject. This was quite typical of Michael Gerzon. He first isolated the engineering problems that needed to be tackled, and then used any branch of mathematics which could give a solution, usually going well beyond the normal range of applied mathematical techniques.

VOLUME 19, NUMBER 4, 1997

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As Newton said of Cotes, "If he had lived we might have known something." Acknowledgments I am grateful to Peter Craven for reading the manuscript, providing additional information, and making numerous useful suggestions. Many of these I have incorporated, though I did suppress some practical details in the interest of clarity. I would also like to express my gratitude to Stephen Thornton for allowing me to use his photograph of Michael Gerzon, to the Audio Engineering Society for permission to reproduce the picture of the original tetrahedral microphone array which originally appeared in [5], and to SoundField Research Ltd for supplying the photograph of the Soundfield Mark V microphone. REFERENCES 1. Michael Gerzon, Obituary notices in the Guardian, 13 May 1996; Gramophone (August 1996), 123, and J. Audio Eng. Soc. 44(7/8) (1996), 669-670. 2. D.A. Edwards, On the ideal centres of certain partially ordered Banach spaces, J. London Math. Soc. (2) 6 (1973), 656-658. 3. http://www.proaudio.co.uWsndfield.htm 4. A.L. Carey, Group representations in reproducing kernel Hilbert spaces, Rep. Math. Phys. 14 (1978), 247-259. 5. M.A. Gerzon, Periphony with height-sound reproduction, J. Audio Eng. Soc. 21(1) (1973), 2-10. 6. K. Howard, Beyond stereo, Gramophone (August 1996), 116-118.

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THE MATHEMATICAL INTELLIGENCER

As Newton said of Cotes, "If he had lived we might have known something." Acknowledgments I am grateful to Peter Craven for reading the manuscript, providing additional information, and making numerous useful suggestions. Many of these I have incorporated, though I did suppress some practical details in the interest of clarity. I would also like to express my gratitude to Stephen Thornton for allowing me to use his photograph of Michael Gerzon, to the Audio Engineering Society for permission to reproduce the picture of the original tetrahedral microphone array which originally appeared in [5], and to SoundField Research Ltd for supplying the photograph of the Soundfield Mark V microphone. REFERENCES 1. Michael Gerzon, Obituary notices in the Guardian, 13 May 1996; Gramophone (August 1996), 123, and J. Audio Eng. Soc. 44(7/8) (1996), 669-670. 2. D.A. Edwards, On the ideal centres of certain partially ordered Banach spaces, J. London Math. Soc. (2) 6 (1973), 656-658. 3. http://www.proaudio.co.uWsndfield.htm 4. A.L. Carey, Group representations in reproducing kernel Hilbert spaces, Rep. Math. Phys. 14 (1978), 247-259. 5. M.A. Gerzon, Periphony with height-sound reproduction, J. Audio Eng. Soc. 21(1) (1973), 2-10. 6. K. Howard, Beyond stereo, Gramophone (August 1996), 116-118.

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THE MATHEMATICAL INTELLIGENCER

B.B. M A N D E L B R O T

AND ST#PHANE

JAFFARD

Peano-POlya Motions, When Time Is Intrinsic or Binomial

(Uniform or Multifractal)

Peano Motions and Their Intrinsic Time Parameter "Peano curve" is the i m p r o p e r traditional t e r m for continu o u s m a p s of the interval to a d o m a i n o f n o n e m p t y interior in the plane. E a c h P e a n o c o n s t r u c t i o n is the limit of a s e q u e n c e o f "pre-Peano a p p r o x i m a n t s . " In the v a s t majority o f cases, these a p p r o x i m a n t s can be d e f o r m e d slightly to b e c o m e continuous one-to-one m a p s of the interval into t h e plane. The resulting graphs are serf-avoiding, that is, n e i t h e r c r o s s over (self-intersect) n o r touch (serf-contact). They can safely b e called "curves," b u t the limit o f the app r o x i m a n t s is (by design) a d o m a i n of the plane, t h e r e f o r e n o t a curve. Nevertheless, a serf-avoiding p r e - P e a n o app r o x i m a n t is o r d e r e d and c a n be p a r a m e t r i z e d in continuous (one-to-one) fashion b y a real parameter, w h i c h one can call time. Moreover, t h e s e constructions' limits P(t) c o n t i n u e to be p a r a m e t r i z e d in c o n t i n u o u s (many-to-one) fashion b y a time. Hence, the following suggestion: the imp r o p e r t e r m "Peano curve" ought to be r e p l a c e d in the lite r a t u r e b y the term "Peano motion," which is logically corr e c t a n d s o u n d s well. The echo of the familiar t e r m "Brownian motion" is intentional a n d eminently appropriate. Indeed, the c o r e of the a r g u m e n t in this article is easily e x p r e s s e d in t e r m s o f familiar physical heuristics for the r a n d o m p r o c e s s called Wiener's Brownian function, B(t). There is a m a j o r difference b e t w e e n the P e a n o and B r o w n cases. In the P e a n o case, the only intrinsic e l e m e n t is a n ordering of the p o i n t s in the image. F r o m t h e viewp o i n t o f the construction, the time t i t s e l f can b e r e p l a c e d freely b y any e v e r y w h e r e increasing function ~(t) having an e v e r y w h e r e increasing inverse t(r). In the B r o w n i a n case, to the contrary, all the usual c o n s t r u c t i o n m e t h o d s imply the s a m e time p a r a m e t e r . Could one also d e f m e an intrinsic time for the P e a n o m o t i o n ? This is easily d o n e in c o n c r e t e physical fashion, by requiring the m o t i o n to c o v e r equal a r e a s in equal times.

F r o m a t h e o r e t i c a l viewpoint, it is b e s t to define the intrinsic time on the graph of a function, without having to k n o w its generating algorithm. F o r t h e r e a d e r familiar with the H a u s d o r f f measure, let us o b s e r v e that it suffices to identify t i m e with the Hausdorff m e a s u r e t a k e n for a suitable gauge function such that the m e a s u r e is positive a n d finite a n d "intrinsic." The H a u s d o r f f m e a s u r e o f a d o m a i n of the plane, t a k e n in the p l a n e ' s d i m e n s i o n D = 2 and t h e gauge function h(p) = p2, is nothing b u t the domaln's area. As to the p o r t i o n from 0 to t of a B r o w n i a n graph, it is k n o w n that w h e n t h e gauge function is c h o s e n suitably, t h e H a u s d o r f f m e a s u r e is positive and finite, and p r o p o r t i o n a l to t. Without w o r r y i n g a b o u t t h e m e a n i n g of the symbol --, let us n o w follow the b e h a v i o r of B a n d P in intrinsic time. It is a s t a n d a r d p r o p e r t y of B r o w n i a n m o t i o n that dB(t) (dt)l/2; for example, in the s e n s e that limdt--. 0 log dB(t)/ 1 log dt = 5" Similarly, it will be s h o w n b e l o w that dP(t) ~ (dt) u2. It follows that b o t h B and P are nondifferentiable if f o l l o w e d in intrinsic time. But t a k e an alternative time, T, that is different from t a n d s u c h that dt ~ (d'r) ~, w h e r e a, the local H61der exponent, d e p e n d s on t. In that n e w time, dB(~') (dT") ay2 and dP(~') (d'r) aj2. This s h o w s that P i s nondifferentiable w h e n a(T) < 2, but differentiable at t h o s e p o i n t s w h e r e a(7) > 2. The goal of this article is to t r a n s f o r m this physical heuristic into a rigorous argum e n t a n d to relate the distribution o f the value of a(7) to the value o f the angle ~hwhich is s e e n in Figure 1 and defined below. The classification of the P e a n o - P 6 1 y a m o t i o n s according to their local regularity p r o p e r t i e s is a j o i n t w o r k b y the co-authors. The a p p r o a c h to this classification via t h e binomial multifractal m e a s u r e is due to Benoit Mandelbrot. An alternative a p p r o a c h relying on a S c h a u d e r basis exp a n s i o n is due to Jaffard and is p r e s e n t e d in Jaffard and Mandelbrot (1996). ~

~

9 1997 SPRINGER VERLAG NEWYORK, VOLUME 19, NUMBER 4, 1997

21

P61ya's Construction and a Heuristic Argument Peano's original example dates back to 1890, and the art of constructing variants flourished until 1925, then c e a s e d to b e active, only to a w a k e n with fractal geometry, since 1975. One o f t h e b e s t k n o w n variants is found in PSlya (1913), and a d v a n c e d a p p r o x i m a t i o n s a r e illustrated here in several ways. Figure 1 is an a b s t r a c t diagram to b e d i s c u s s e d in a moment. In Figure 2, to m a k e the c o n s t r u c t i o n legible, it is modified in several ways: (a) It is slightly distorted, so that the limit m o t i o n is n o t quite plane-filling; Co) r e c u r s i v e interp o l a t i o n is i n t e r r u p t e d after the intervals it g e n e r a t e s first fall b e l o w a p r e s c r i b e d l o w e r threshold; (c) the approxim a t i o n is n o t d r a w n as a b r o k e n line b u t as a b o u n d a r y bet w e e n b l a c k a n d white. In Figure 3, the p r o c e s s is m a d e legible b y a different graphic device: e a c h interval is r e p l a c e d b y a "boomerangshaped" icon, m a d e of t w o a t t a c h e d triangles. To return to Figure 1, the initiator o f the c o n s t r u c t i o n is the oriented unit interval [0, 1]. The generator of the con-

s t r u c t i o n is m a d e up o f the t w o s h o r t sides o f a right triangle with h y p o t e n u s e identical to [0, 1], o r i e n t e d to s t a r t at 0 a n d end at 1. This g e n e r a t o r is therefore fully specified b y a triangle; that is, b y the generating angle ~ n e a r t h e p o i n t 1, h e n c e by the quantities c = cos ~ o r s = sin ~. One can c h o o s e $ E ]0, ~-/4], h e n c e s -< c. In Figure 2, t h e right triangle is m a d e slightly o b t u s e for the s a k e o f legibility. The first c o n s t r u c t i o n stage starts with a single o r i e n t e d interval of length 1, and it e n d s with an oriented p i e c e w i s e linear curve (broken line) m a d e of t w o intervals o f lengths s a n d c, in that order. The s e c o n d stage ends with four intervals o f length s 2, cs, cs, and c2, in that order, a n d so forth. To e n s u r e that a triangle is filled without overlap, it sufrices t h a t its construction s h o u l d follow two rules. The side of length s always c o m e s first. O d d - n u m b e r e d c o n s t r u c t i o n s t a g e s p l a c e the g e n e r a t o r to the left of the o r i e n t e d outc o m e o f the p r e c e d i n g stage, a n d e v e n - n u m b e r e d s t a g e s p l a c e it to the right. Given a pre-Peano r e c u r s i v e a p p r o x i m a n t motion, t h e r e a r e m a n y alternative w a y s o f assigning a time p a r a m e t e r . W h e n the a p p r o x i m a n t s a r e recursive, it is also n a t u r a l for t i m e to be defined recursively. One can p r o c e e d as follows. Assign t = 0 and t = 1 to the e n d p o i n t s of the initiator, a n d a t t a c h to the right-angle c o r n e r of the g e n e r a t o r a t i m e equal to a p r e s c r i b e d real g ~ ]0, 1[. This g will b e called t h e generating fraction. Thus, the time span 0 < t < g is t h e t i m e t a k e n b y the m o t i o n to s w e e p a right-angle triangle of h y p o t e n u s e s and relative a r e a s2; during the time s p a n g < t < 1, the m o t i o n s w e e p s a right-angle triangle of

Figure 1. (top) Diagram explaining the construction of the Peano-Pblya motion. The points whose binary expansions start with 0 . 0 1 1 0 . . . are mapped into the shaded triangle. Figure 2. (bottom) [reproduced from Plate 57 of Mandelbrot (1982)]. This graph illustrates the path of a close variant of the Pblya plane-filling motion. This approximation is piecewise linear ("a broken line"), the interpolation being interrupted when it reaches intervals smaller than a prescribed minimum size.

22

THE MATHEMATICAL INTELLIGENCER

i i~_~, ~'f _~! ~ j ~ , , . _ :~-.~ j(~'j~'-~ f ~) ~ ~ j ~ _ _ . ~ f ~ r ~ ~

proximant is drawn as a triangle similar to the originator. To avoid illegibility, Figure,3 represents each interval by a smaller set, a boomerang whose area is proportional to that of the triangle. This procedure illustrates the progress of the Peano motion and implies that each b o o m e r a n g s \ j area is proportioned to the time taken to cover the triangle's hypotenuse. Iffhis figure is looked at out of focus, the j~ ~ k ~'~~ 1 ~ i average "blackness gives the impression of being more or ~ ] f ~k, , - - - . . . ~ lk I ~, | less uniform overall. , ~'f \ ~ ~ ~ ~ ~ ~ ~ ~ When 0 = v/4 [see Mandelbrot (1982), p. 65], j,(~ ~ ~ I ~ J ~ k 1~ ~ ~ ~ ~ ~ the two natural choices of g b e c o m e ~(~j f~_~ (~j j !f( ~ f ~~p~--.~R~=~_ identical, since g = s 2 = c 2 = 8 9 ,~,},~} (~__2~f*~ ~'! ~"'~ '~"'~ W h e n 0 r ~r/4, specifyingg =s2 de fines (by successive interpolations illustrated in Figs. 1, 2, Figure 3. [reproduced from Plate 64 of Mandelbrot (1982)]. This and 3) a useful one-to-one m a p t(T) from [0, 1] onto [0, 1], graph illustrates a broken strip that approximates the P61ya planewhich relates the two times and which is discussed in the filling motion, To make the motion perspicuous, each interval in the next section. Followed in the time T, the motion is given by approximation is first replaced by a broken interval that avoids selfH(~) = P[t(~)], hence II combines the properties of the funccontacts. Next, the broken interval is thickened into a boomerangtions P(t) and t(r). (In the context of stochastic motions, this shaped strip whose area equals the intrinsic time the motion takes combination is sometimes called "subordination.") For this to go along an interval in the approximation described in Figure 1. H(T), it will be shown in the next section that the heuristic core of the argument remains the same as sketched above for B(7). One can write dt ~ (d~)"(7); therefore, dH relative area c 2. Both of these triangles are geometrically (d~-)"(7)/2. Much depends, therefore, on the sign of a0-) - 2. similar to the original triangle. Once a time has been asWhen the time z is such that a0-) > 2, the function H(r) is signed to a node in an approximant, it will not change in nondifferentiable. To the contrary, when the time 9 is such later refinements. This recursive form of time is readily exthat a ( T ) < 2 , the derivative H'(T) exists and is zero. tended to the limit motion by a limit process. Therefore, given 0, the issue raised by Lax has been refoAs already stated, the intrinsic choice of g is the choice cused: Do any instants satisfy a(z) < 2? If so, how many? that m a k e s the motion cover equal areas in equal times. We shall see in the section after next that such instants are Hence, the intrinsic choice of time, t, corresponds to g = encountered for small values of 0. This explains the result s 2. The section after next shows that d P ( t ) - ( d t ) 1/2 holds obtained by Lax, and the motion H(~) ceases altogether to for all t in this time. As a result, the i n t r i n s i c behavior of be counterintuitive. the Peano-P61ya motion P(t) is as expected: n o w h e r e difBinomial Multifractal Maps and Measures ferentiable. The chord that joins the points P(t) and P(t § The function t(T) is one-to-one, and its graph, shown in dt) behaves in m a n y ways, depending on t; for s o m e t's, it Figure 4, deserves to be called the Besicovitch devil stairoscillates without end as dt --->O, for other t's, it winds endcase. Given 0 < r < 1, we defme the r-adic reals on [0, 1]. lessly. 1 These are, first, the real r that divides the interval [0, 1] in A second simple choice is g = 2" It was suggested by the ratios r and 1 - r, next the reals that divide the interP61ya and defines a time we shall denote by T. The resultvals [0, r] and [r, 1] in the same ratios, and so on. In these ing motion was investigated in Lax (1973), whose findings terms, the graph of t(r) m a t c h e s the g-adic points of the will be described below. Using this choice for g, the reguaxis of T with the s2-adic points of the axis of t. The first larity of the Peano-P61ya motion depends greatly on ~. If construction stage assigns to the "first-level intervals," > 30 ~ the motion is nowhere differentiable. If 0 < 30 ~ namely ]0, g] and ]g, 1], the increments s 2 and c2 oft. Next, there exists a noncountable set of points where the motion one assigns to the "second-level intervals" ]0, g2], ]g2, g], has a derivative, and if 0 < 15 ~ this happens almost every]g, 1 - (1 - g)2], and ]1 - (1 - g)2, 1] the increments s 4, where! s2c 2, s2c 2, and c 4, respectively. And so on. By uniform conImagine that a point m o v e s at uniform velocity along an vergence, this procedure defines a continuous function advanced approximant of the Peano-P61ya motion, that it t(~). But we shall see momentarily that t(r) is everywhere "seeds" matter continuously as it proceeds, and that this without a finite nonzero derivative of any order. In m a s s settles in a thin ribbon of uniform width. Such a ribLebesgue's terminology, t(~) is a s i n g u l a r function. bon is, of course, what the process of printing would use Let us n o w review the local properties of the function to represent our curve. It corresponds to the time gener1 t(7) in the multifractal binomial case when g = 2 and S 2 < ated by g = s/(s + c). (As the approximation b e c o m e s in2' as illustrated in Figure 4. It is obvious that for the dyadic creasingly refined, the ribbon width must d e c r e a s e points ~-= 0 and ~-= 0.5, the function t(~-) has a right derapidly.) rivative equal to 0, and for 9 = 0.5 and ~-= 1, it has a left The easiest value to illustrate is g = s 2, but the reprederivative equal to oo. sentation fills the triangle. Each interval of the Peano ap-

ti~-~j

J

~ ~ ~fl

~

VOLUME19, NUMBER4, 1997 23

If so, the H6lder e x p o n e n t a+(7) is the s u p r e m u m of all a such that t is C a at r +. A dyadic real is of the form 7 = p 2 -k for integers p a n d k. As is well known, each dyadic 7 has two binary develo p m e n t s - - o n e ending with O's a n d characterized by ~ = 1, a n d the other ending with l ' s a n d characterized by ~ = 0. The left and right H61der e x p o n e n t s of t(7) at these dyadic i n s t a n t s are amin = --log2 c2, which satisfies 0 < amin < 1, a n d amax = - l o g 2 s 2, which satisfies 1 < amax < ~. Dyadic i n s t a n t s are denumerable; hence, their H a u s d o r f f Besicovitch dimension is 0. Let us recall the definition of the Hausdorff-Besicovitch d i m e n s i o n of a set S on the line. For p > 0, one covers S with intervals of length Pm < P and one forms inf ~ p~ where the infimum is t a k e n o n all such coverings. The expression

/ /

j

J

lim {infp ~ . pdm}=

7

Figure 4. A binomial devil staircase of Besicovitch. This is the graph of the singular function t(~-) that relates the intrinsic time t corresponding to a value of g = s 2 < ~ to the time 9 corresponding to g - - ~ .1

To make strict mathematical sense of the heuristic relation dt ~ (dr) "(r) in the preceding section, we need to identify a(7) to a local form of the classical n o t i o n of the H S l d e r e x p o n e n t . The H61der e x p o n e n t is a s t a n d a r d measure of the local behavior n e a r 7 of the f u n c t i o n t(7); that is, of local behavior of the differential dt(7). The function t(7) has no nontrivial differential, a n d a properly defined dt('r) is a measure. When the H61der e x p o n e n t of t(7) at 7+ is -< 1, it is defined as a+(7) =

lim d r > 0, d r ~

l~ 0

+ dr) - t(7)l log(d~')

The H61der e x p o n e n t a - ( 7 ) is defined by replacing d7 > 0 by d r < 0. The graph of t(~) shows that the a's cover a whole range of values. We begin by defining ~(m) as the relative number of O's a m o n g the m first binary decimals of 7. Real ~'s such that ~(m) converges to ~ E ]0, 1[ as m ~ o~ will be called "~-normal." We start with dyadic reals, c o n t i n u e with ~-normal 7's, a n d e n d with the set of T such that ~(m) has no limit for m ---) ~, b u t satisfies either lim sup ~p(m) = ~p, or m---) ~

lim inf ~(m) = ~. m ---, oo

More generally, a function t is C a at T+ if there exists a polynomial p, w h o s e degree is at most the integer part of a, that satisfies It(7 + d.r) - t(7) - p(d~)l <- CId71 ~

24

Hd(S)

p~0

THE MATHEMATICALINTELLIGENCER

for

0
exists for all sets and all values of d, and is called the H a u s d o r f f (outer) m e a s u r e of S in the dimension d. D is t h e H a u s d o r f f - B e s i c o v i t c h d i m e n s i o n of S if for d < D, one has H d ( S ) = ~, and for d > D, one has H d ( S ) = O. This definition sets a n upper b o u n d to the sizes of the intervals (boxes) used in the covering, b u t allows them to be of different sizes. This feature is n o t essential for e l e m e n t a r y self-similar fractals, but is essential here because the sets entering in the theory of multifractals are everywhere d e n s e in [0, 1]. Boxes of identical sizes would attribute to every everywhere dense set a d i m e n s i o n equal to 1. Next consider a ~ n o r m a l 7 with ~ r 0 and ~ r 1. After k c o n s t r u c t i o n steps, the i n c r e m e n t s satisfy It(7 + d r ) t(7)[ - CId~h; h is called a coarse H61der exponent. As k ---) 0% it c a n be s h o w n that h--~ - ~ log2c2 - (1 - q~) log2s2. Observe that, w h e n 0 (hence s) is very small, there exist 4s with very large H61der exponents. A well-known t h e o r e m due to Eggleston (1949) tells us that the q>normal i n s t a n t s form a set whose Hausdorff-Besicovitch d i m e n s i o n is the "entropy" 6(~) = - ~ l o g ~ - (1 - ~)log(1 - ~#)[see Billingsley (1967).] This leaves us with reals for which ~ is not the lim, b u t either the lim sup or the lim inf of ~(m). These reals are covered by a little-known t h e o r e m of Volkmann (1958), which yields precisely the same e n t r o p y expression for ~. The H61der exponent a a n d the dimension 6 are expressed as functions of the parameter q~. By eliminating ~, we can express the dimension as a function of a. The graph of the resulting function f ( a ) is a concave curve. Figure 5 shows the graph o f f ( a ) for three values of 0 that will be seen below to exemplify three distinct behaviors of the motion II(7). In particular, 7 is almost surely a "normal n u m b e r " (in the sense of Emile Borel); that is, the asymptotic proportions of the decimals 0 a n d 1 are almost surely equal 1 to 3. Hence, the almost sure value of a is aa~ = 1

2

1

1

- 3 log2c - 3 l~ = -log2(cs) = 3 (amin + amax). (This aas is often denoted by ao.) The graph o f f ( a ) is symmetrical with respect to the vertical line having equation a = aas, and f ( a ) takes its m a x i m u m value t~ = 1 for a = aas.

ha)

45"

2.0 30~

37,5~

22.5~

15"

7.5~

Figure 5. Graphs of six representative examples of the binomial H61der spectrum f(a) corresponding to 0 which are multiples of 7.5 ~ These examples include the boundaries 15~, 30 ~, and 45 ~ of the three domains of variation of #, and additional O's half-way between those boundaries and # = 0.

g-adic, PoP~ (resp., POP[) is at l e a s t equal to the s m a l l e s t side of a g-lattice triangle, a n d At is at m o s t the a r e a o f a g-lattice triangle of level higher b y unity, h e n c e (AP)2/At > 2s4/cs. This c o m p l e t e s the p r o o f o f the s e c o n d p a r t of P r o p o s i t i o n B. To c o m p l e t e the p r o o f of the first part, let us e s t i m a t e P'P"2/At for t w o a r b i t r a r y times. We define Po as the l o w e s t - o r d e r g-adic p o i n t b e t w e e n p o i n t s P ' and P". N o w using the first part of P r o p o s i t i o n B in the g-adic case, PoP'2/(to - t') <- 4/s 2 and PoP"2/(to - t") <- 4/s 2. The angle P'PoP" being acute, one has p,p,,2 <_ p,po 2 + poP.2 while (t" - t') = (t" - to) + (to - t'). Thus, P'P"2/At < 4/8 2. When

Followed

Motions

Behave

in t h e T i m e

7, T h e P e a n o - P 6 1 y a

Paradoxically

THEOREM (Lax 1973). C o n s i d e r the P e a n o - P S l y a m o t i o n

There are two critical values of 6: (a) O = 30~ w h i c h is defined b y the condition that O/max ---- 2 , and (b) 0 = 15~ which is defined by the c o n d i t i o n that O/0 = 2. Observe that w h e n O (hence s) is very small, t h e r e exist p o i n t s with v e r y large H61der exponents. To m a k e the p i c t u r e m o r e complete, f(o/) -< O/, a n d the unique fixpoint w h e r e f(o/) = O/ can be s h o w n to corres p o n d to ~ = c 2, w h i c h yields f(o/) = O/= - c 2 log2c 2 - s 2 log2s2 < 1. This value o f O/andf(o/) is often d e n o t e d b y al.

H(T) w i t h the generating angle 0 a n d the g e n e r a t i n g f r a c tion g = ~. t For ~ > 30 ~ this II0-) i s n o w h e r e differentiable. F o r 15 ~ < O < 30~ H(~-) is n o n d i f f e r e n t i a b l e f o r alm o s t all .r a n d has a derivative H ' 0 - ) f o r a noncountable set o f values o f "~. F o r 0 < 15 ~ one has II'(T) = O f o r alm o s t all 7.

When

O/max ) .

Followed

Peano-P61ya Differentiable,

in I t s I n t r i n s i c T i m e t, t h e

Motion

Is A l m o s t

Nowhere

as Expected

PROPOSITION A. For all q,, the Peano-P61ya m o t i o n P(t) h a v i n g the g e n e r a t i n g angle ~ is n o w h e r e differentiable.

P r o p o s i t i o n A is a c o n s e q u e n c e of the following. PROPOSITION B. Denote by At the length o f a t i m e interval i n c l u d i n g t a n d by A P the corresponding change i n P. For all values o f t a n d o f At, (~a') 2 < 4

At

-- s 2'

a n d f o r every t, there e x i s t values o f At s u c h that (~)2

At

~

[2]S4.

Lcs j

P r o o f o f P r o p o s i t i o n B. Given a real to in ]0, 1[, cons i d e r g-adic s e q u e n c e s t~ and t~ with t{ -< t~ -< ... -< t~ --) to, t~ -> t"2 -~ " ' " -~ tk" ----> t 0. A S a result, the point Po = P(to) is a p p r o x i m a t e d by t w o p o i n t sequences P~', P ~ , . . . , P L - . . a n d P1," P"2, 9 9 9 Pk" 9 9 9 When to is itself g-adic of level h, then, for all k > h, PoP[~ (resp., POP[) is the h y p o t e n u s e of a g-lattice triangle, a n d ]to - t~[ (resp., Ito - t~[) is the a r e a o f this triangle. Therefore, (AP)2/At = 2/cs for t h e s e special a p p r o x i m a t i n g sequences. Every o t h e r a p p r o x i m a t i n g p o i n t P ' (resp., P") o f Po is b r a c k e t e d b y t w o s u c c e s s i v e distinct points in the s e q u e n c e [ P 0 - P'I <- ~ 0 - P~[ and It0 - t'l --> It0 - t~+ll, h e n c e (AP)2/At --< 2/cs. This p r o v e s P r o p o s i t i o n B for the g-adic values o f t. When t is n o t

P r o o f o f a strengthened f o r m o f L a x ' s theorem. We s e e that the b e h a v i o r o f dH(t) d e p e n d s on the position o f t h e critical value acfit = 2 relative to O/rain, O/max, a n d ~1 ( o / m i n + The fact that it is not p o s s i b l e for H(t) to have zero derivative e v e r y w h e r e is explained by observing that H ' 0 - ) = 0 w o u l d d e m a n d O/mi#2 > 1, a n d w e k n o w t h a t the binomial m e a s u r e satisfies O/rain< 1. The case ~b > 30 ~ The c o n d i t i o n for H(r) to be everyw h e r e nondifferentiable is O/max/2 < 1, from which follows O/max < 2, f r o m w h i c h follows 1/82 < 4, from which follows s > 1, from w h i c h follows 0 > 30~ This multifractal a r g u ment e x p l a i n s intuitively as well as rigorously the special role that the critical angle 0 = 30 ~ p l a y s in Lax's theorem. The case 0 < 30 ~ It involves a finer distinction. In ord e r for O/as to coincide with the critical value 2, it is nece s s a r y that cs = 88 from w h i c h 2cs = s i n ( 2 O ) - 3' 1 from which 0 = 15 ~ Again, the multifractal a r g u m e n t intuitively explains the critical role that the special angle 0 = 15~ p l a y s in Lax's theorem. The s u b c a s e 15 ~ < 0 < 30 ~ We s e e that O/~/2 > 2, a n d H(v) is a l m o s t surely not differentiable. This p r o v e s the Lax t h e o r e m in a s t r o n g e r form. Indeed, the "atypical" p o i n t s 7 w h e r e II'(T) exists and vanishes a r e the points w h e r e O/(~) < 2. We s e e that their H a u s d o r f f d i m e n s i o n is f ( 2 ) > 0; this c o n c l u s i o n is s t r o n g e r t h a n Lax's conclusion that they are n o n d e n u m e r a b l e . The subcase 0 < 15 ~ We see t h a t O/as/2 < 2, and H ' ( r ) exists and v a n i s h e s for a l m o s t all z. Again, w e see that the "atypical" p o i n t s w h e r e II'(7) fails to exist have the H a u s d o r f f d i m e n s i o n f ( 2 ) > 0, a result s t r o n g e r than Lax's c o n c l u s i o n that they are n o n d e n u m e r a b l e . The critical cases ~b= 30 ~ a n d ~ = 15 ~ They remain open. F o r 0 = 30 ~ P(t) almost everywhere satisfies a Lipschitz condition ]6P(t)/6t] < constant. What m o r e could be said?

VOLUME 19, NUMBER 4, 1997

25

Skew Ces~ro Motions, and Other Motions of Dimension D r 2. One can also r e p l a c e the PSlya g e n e r a t o r b y a c o n n e c t e d g e n e r a t o r m a d e o f t w o intervals o f lengths c < 1 and s < 1. F o r all c a n d s, the Moran equation CO + s D = 1 has a unique real solution that satisfies D > 1. Ordinarily, 1 < D -----2, and t h e e x p o n e n t D is the H a u s d o r f f - B e s i c o v i t c h d i m e n s i o n of t h e resulting one-to-one c o n t i n u o u s m a p of [0, 1]. The s a m e c o n s t r u c t i o n also e x t e n d s to c a n d s that yield D > 2. The resulting one-to-one c o n t i n u o u s m a p of [0, 1] is n o l o n g e r a curve b u t c o n t i n u e s to define a motion. The e x p o n e n t D is no l o n g e r a H a u s d o r f f Besicovitch dimension (because the HausdorffBesicovitch d i m e n s i o n of a p l a n a r s e t is b o u n d e d a b o v e b y 2); however, it is useful to call it a "latent fractal dimension." The g e n e r a l i z e d fractal m o t i o n s for c r s a n d D > 2 will b e c a l l e d s k e w Ces~ro m o t i o n s . Lovers o f the h i s t o r y of a s t r o n o m y m a y w a n t to call this m o t i o n Keplerian. The p r e c e d i n g a r g u m e n t s are basically unchanged. The intrinsic time is a c h i e v e d by setting the generating fraction equal to g = c ~ The m o t i o n P(t) in this intrinsic time satisfies dP(t) ~ (dt) up. When g r cO, one has dlI(~-) (d'r) ~(~)/D. The significant value of a b e c o m e s a = D, and Lax's t h e o r e m generalizes according to the p o s i t i o n of = D with r e s p e c t to aas and amax.

26

THE MATHEMATICALINTELLIGENCER

Acknowledgments A v e r y early version of this w o r k was p r e s e n t e d orally b y one of us (Mandelbrot) at the Mathematical C e n t e r Oberwolfach, Germany, in J a n u a r y 1987, during a w e e k long c o n f e r e n c e on fractals. Useful c o m m e n t s w e r e m a d e (there o r later) by J. Peyri~re, T. Bedford, and D. Gatsouras. REFERENCES Bedford, T. 1989. H61derexponents and Box dimension for self-affine fractal functions, J. Construct. Approx. 5, 33-48. Billingsley, P. 1967. Ergodic Theory and Information. New York: Wiley. Jaffard, S. and Mandelbrot, B. B. 1996. Local regularity of nonsmooth wavelet expansions and application to the POlyafunction, Adv. Math. 120, 265-282. Lax, P.D. 1973. The differentiability of Polya's function, Adv. Math. 10, 456-464. Mandelbrot, B. B. 1982. The Fractal Geometry of Nature. New York: W. H. Freeman and Company. Mandelbrot, B. B. 1995. Negative dimensions and H61ders, multifrac'~ tals and their HOlder spectra, and the role of lateral preasymptotics in science. J. P. Kahane meeting (Paris, 1993), J. of Fourier Anal. Appl. 409-432, special issue edited by J. Peyri0re and A. Bonami P61ya, G. 1913.0ber eine Peanosche Kurve. Bull. Acad. Sci. Cracovie, Serie A, 305-313. Volkmann, B. 1958. 0ber Hausdorffsche Dimensionenvon Mengen, die durch Zifferneigenschaftendefiniert sind, Vl. Math. Zeitschr. 68, 439-449.

ISTVAN HARGITTAI

A Great Communicator of Mathematics and Other Games: A Conversation with Martin Gardner

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n

artin Gardner (b. 1914) has had a unique career. He majored in philosophy at the University of Chicago and became a writer of international fame on mathematics and science, including some bestsellers. He created and edited a unique column, "Mathematical

Games", at Scientific American between 1957 and 1982. He has probably done more than anybody else to bring mathematics in human proximity for many non-mathematicians. His most recent book is The Night Is Large: Collected Essays, 1938-1995, St. Martin's Press, New York, 1996. For almost two decades he and his wife have lived in HendersonviUe, North Carolina. The following conversation was recorded in their home on September 21, 1996. I s t v ~ a H a r g i t t a i ( I H ) : Your 1977 p a p e r a b o u t the P e n r o s e tiling in Scientific American is often cited in r e s e a r c h rep o r t s in the quasicrystals field. This is r a t h e r unusual for a p o p u l a r magazine article. H o w did it c o m e about? M a r t i n G a r d n e r ( M G ) : S o m e h o w I h a d s t a r t e d a corres p o n d e n c e with R o g e r P e n r o s e and t h e n he visited me. To P e u r o s e it s e e m e d to be a trivial discovery, b u t to m e it app e a r e d very important, and I d e c i d e d that it d e s e r v e d m o r e e x p o s u r e than his original note p u b l i s h e d in a v e r y o b s c u r e periodical. Incidentally, t h e cover of that issue o f Scientific American w a s d e s i g n e d by John Conway. He did a lot of w o r k on the P e n r o s e tiling and h a p p e n e d to b e visiting m e at the time I w r o t e t h a t article. He sat down, t o o k a ruler,

THE MATHEMATICAL INTELLIGENCER 9 1997 SPRrNGER VERLAG NEW YORK

c o m p a s s , and a pencil, a n d s k e t c h e d out the cover, t h e n an artist at Scientific American m a d e a dramatic c o v e r for t h a t issue. I a m still following the quasicrystal story, and every time I c o m e a c r o s s something I a d d it to m y large collection o f clippings. P e n r o s e ' s n o n p e r i o d i c tiling is an outstanding rec e n t e x a m p l e o f something d o n e j u s t for the fun o f it that t u r n e d out to have significant p r a c t i c a l applications. This w a s a great surprise to everybody. It w a s really startling. III: H o w much formal training did you have in mathematics? MG: I h a d n o n e at all. I'm strictly a journalist, an amateur. I l o v e d m a t h e m a t i c s in high s c h o o l and m a d e g o o d g r a d e s in m a t h c o u r s e s but did n o t t a k e any m a t h in college. In 1932 I w e n t to the University of Chicago. It was an excit-

ing period, with R o b e r t wife c a m e to the class to Hutchins as President. He adt a p e - r e c o r d them, and I v o c a t e d a b r o a d liberal eduedited t h e m into a b o o k [Rudolf Carnap, An Introcation that you c o u l d do at y o u r o w n pace. If you p a s s e d duction to the Philosophy of the a p p r o p r i a t e test, y o u Science. E d i t e d by Martin Gardner. Latest edition: Dover c o u l d skip that class. A lot of Publications, New York, 1995]. p e o p l e g r a d u a t e d in j u s t a It's t h e only b o o k C a r n a p c o u p l e o f years. I did the ever d i d that w a s not highly usual four years though. Mortimer Adler w a s there. technical. E v e r y b o d y disliked him b u t IIt: Let's get b a c k a little in he s t i r r e d up a lot of excitetime. ment. S o m e o n e has called it MG: I w a s b o r n in Tulsa, the m a d m a n t h e o r y of educaOklahoma, in 1914. My f a t h e r tion. That is, if you have one was a geologist. He got his p e r s o n on the F a c u l t y w h o Ph.D. at the University of h a s strong, crazy views, it Washington, a n d w o r k e d for k i c k s up a lot of e x c i t e m e n t the Smithsonian for a while j u s t to o p p o s e the guy. A d l e r as a fossil hunter. The oil w a s s o r t o f like that. He c a m e b u s i n e s s was j u s t starting, so from a Jewish b a c k g r o u n d he w e n t to Tulsa w h e r e he and almost joined the s t a r t e d a small oil c o m p a n y Catholic Church but w a s a of his own. He w a s w h a t t h e y d e v o t e d Thomist. called a " w i l d c a t t e r " - - o n e He w a s the c e n t e r of conw h o g o e s a r o u n d singletroversy. He was a friend o f h a n d e d l y looking for oil. Hutchins, and Hutchins h a d S o m e t i m e s t h e y find it. More b r o u g h t him there a n d p u t often t h e y don't. him in the Philosophy When I was in high s c h o o l D e p a r t m e n t w i t h o u t checkm y a m b i t i o n w a s to b e c o m e ing with the H e a d o f the a physicist. I w a n t e d to go to Caltech, b u t Caitech at t h a t Department. The philosotime w a s taking you only p h e r s w e r e so i n c e n s e d b y after t w o y e a r s of general Hutchins doing that, that ale d u c a t i o n at s o m e other unimost all members of the Martin Gardner in his garden (photo by Istvdn Hargittai). D e p a r t m e n t resigned, left the versity. So I w e n t to the University o f Chicago and got h o o k e d on philosophy. University, and t o o k o t h e r jobs. So Hutchins h a d to t a k e After I h a d g r a d u a t e d a n d s p e n t a n o t h e r y e a r at graduA d l e r out o f the P h i l o s o p h y D e p a r t m e n t a n d c r e a t e a speate work, I d e c i d e d I didn't w a n t to teach. I w a n t e d to write. cial class for him called P h i l o s o p h y of Law, so he w a s in After Chicago I w e n t b a c k to Tulsa a n d got a j o b on the lothe Law School. cal n e w s p a p e r , The Tulsa Tribune, as a reporter. F r o m A d l e r and Hutchins t a u g h t the Great B o o k s c o u r s e s together. One of Hutchins's m a i n ideas was t h a t for a liberal there I w e n t to w o r k in the P r e s s Relations Office of the e d u c a t i o n you had to get a c q u a i n t e d with the classics, startUniversity o f Chicago. When the W a r came, I enlisted in the Navy, a n d s e r v e d for four y e a r s a s a yeoman. ing with Plato and Aristotle, and so on. He h a d a list of a b o u t a h u n d r e d books. But in the Great B o o k s c o u r s e t h e y When I got out o f the service I w e n t b a c k to Chicago. I w o u l d s p e n d a w h o l e s e m e s t e r on Plato a n d Aristotle. I could have gone b a c k to m y old job, but I didn't do t h a t t o o k this course. I also t o o k the survey c o u r s e s in the physb e c a u s e I s o l d a s h o r t story to Esquire magazine. That w a s the first t i m e I h a d b e e n p a i d for anything that I wrote. It ical s c i e n c e s and in the biological s c i e n c e s that y o u h a d to t a k e u n d e r Hutchins's plan, b u t I m a j o r e d in philosophy. w a s a s t o r y called "The Horse on the Escalator." So for C o m p t o n and F e r m i b o t h l e c t u r e d in the s u r v e y courses. a b o u t a c o u p l e o f years I lived from sales of fiction to After graduation I s t a y e d on for a y e a r of g r a d u a t e work, Esquire. I s o l d t h e m a b o u t a d o z e n stories. The first s t o r y was a h u m o r o u s one, and the second, called "The No-Sided lived n e a r the campus, a n d u s e d to see F e r m i bicycling to Professor," w a s b a s e d on topology. If you t a k e a strip o f his l a b o r a t o r y u n d e r the Stadium. p a p e r a n d give it a haft twist, it b e c o m e s one-sided. My plot My m o s t exciting c o u r s e w a s under Rudolf Carnap. He was a b o u t a p r o f e s s o r of t o p o l o g y w h o d i s c o v e r e d a w a y w a s a logical positivist from the Vienna School. I w a s so i m p r e s s e d by his c o u r s e that later, w h e n C a r n a p w a s in to fold a surface so it lost both sides, a n d j u s t disappeared. California, I p e r s u a d e d him to t a p e - r e c o r d his lectures. His After a while I d e c i d e d to m o v e to New York b e c a u s e

VOLUME 19, NUMBER 4, 1997

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puzzle columns, problem columns, and there was a considthat was w h e r e all the publishers were. I could not exist erable literature of b o o k s on recreational mathematics. So I as a free-lancer, b u t I landed a j o b with Humpty-Dumpty j u s t started pulling material out of the b o o k s and magazines. magazine, a c h i l d r e n ' s publication. This w a s m y s o u r c e of After the column was going for a few years I began to h e a r i n c o m e for t h e n e x t eight years. I did all its activity feafrom creative mathematicians w h o were doing w o r k in the tures, a s h o r t s t o r y e a c h month, and a p o e m of m o r a l advice. During t h a t p e r i o d I m a r r i e d a New York girl, field, like Sol Golomb, the m a n who invented and studied polyominos. I began to get n e w material that wasn't in the Charlotte Greenwald, and w e have b e e n t o g e t h e r ever since. When w e got m a r r i e d w e were s o p o o r that w e were books. II-I: H o w a b o u t your children? m a r r i e d by a j u d g e friend of ours for free. My h o b b y is conMG: We have t w o boys. One of them, Jim, is on the F a c u l t y juring, and this j u d g e was an a m a t e u r magician. We also got o u r b l o o d tests free from a n o t h e r m a g i c i a n friend. o f the D e p a r t m e n t of Psychology, University of Oklahoma. His Ph.D. Dissertation at the University of Michigan w a s IH: Had you b e e n m o r e well off, w o u l d y o u have gone to a c h u r c h for y o u r marriage? on using c o m p u t e r s for treating the retarded. He h a s given MG: I don't know. My wife is Jewish a n d I w a s brought up us t h r e e grandchildren. The y o u n g e r son, Tom, living in Greenville, South Carolina, is a struggling artist. He gradas a Methodist. Neither of us goes to church. We are philosophical theists. We believe in God. I have e x p l a i n e d m y u a t e d at the Rhode Island School of Design a n d is trying u n s u c c e s s f u l l y to m a k e a living as a painter, so he s u p p o r t s d i s e n c h a n t m e n t with Christianity in a novel [Martin h i m s e l f b y various o d d jobs. He is unmarried. Gardner, The TTight of Peter Fromm, first p u b l i s h e d in 1973, n o w a P r o m e t h e u s paperback]. I I t : You m u s t have met a lot o f f a m o u s scientists a n d mathNumerous p r o m i n e n t A m e r i c a n s c i e n t i s t s have b e e n ematicians. quite frank a b o u t being atheists. Carl Sagan is a g o o d exMG: One is J o h n Conway, w h o is t r e m e n d o u s l y i n t e r e s t e d ample. S t e p h e n Weinberg is another. S o m e scientists are in r e c r e a t i o n a l mathematics. On a visit to see me he s h o w e d c a n d i d a b o u t it in public. To be P r e s i d e n t o f the United m e his n o w f a m o u s game of Life, a n d I wrote a c o l u m n States you n o t only have to p r e t e n d to believe in God, but a b o u t it. A n o t h e r n o t e d m a t h e m a t i c i a n I k n e w p e r s o n a l l y you have to p r e t e n d to be a Christian. I think that scienw a s Stanist~aw Ulam. tists are a b o u t equally divided b e t w e e n s e c u l a r humanism I b e c a m e a g o o d friend of Ron G r a h a m of Bell Labs, a or atheists a n d t h o s e w h o have s o m e s o r t o f religious faith, w e l l - k n o w n combinatorial m a t h e m a t i c i a n . He i n t r o d u c e d a b o u t the s a m e division as in the general public. m e to the late Paul Erd6s. With no h o m e of his own, E r d d s During m y eighth y e a r with Humpty Dumpty I s o l d m y w o u l d travel here and t h e r e to r o o m with o t h e r m a t h e first article to Scientific American. It w a s on m e c h a n i c a l maticians. It w a s E r d d s ' s m o t h e r in B u d a p e s t w h o k e p t his h u n d r e d s of p u b l i s h e d p a p e r s . When she died Ron t o o k logic machines. Later I did an entire b o o k a b o u t such machines. The s e c o n d article I sold t h e m w a s on hexaover. There is n o w an E r d 6 s r o o m at Bell Labs, a n d Ron flexagons. These are one-sided p a p e r s t r u c t u r e s that k e e p is k n o w n at Bell Labs as " E r d S s ' s mother." E r d 6 s was very c o n c e r n e d about getting old. w h e n w e changing their f a c e s w h e n y o u "flex" them. The flexagon met, his first question w a s "When did you arrive?" I l o o k e d a r o u s e d a lot of interest. One of its early co-inventors was Richard F e y n m a n . at m y watch, but G r a h a m Gerry Piel, the Publisher of Martin Gardner demonstrating a puzzle to Istvdn Hargittai (photo whispered to me that it was Scientific American, called by Magdolna Hargittai). Erd6s's way of asking, "When me into his office after the were you born?" He s p e a k s an hexaflexagon article appeared, Erd6sian language. and he asked m e if there was By the way, I have an enough similar material to E r d 6 s n u m b e r 2. That m e a n s make a regular department. I I have published with s o m e said there was. In January 1957 one w h o p u b l i s h e d with my first colunm appeared, Erd6s. I a m p r o u d to s a y I called Mathematical Games. I have m y E r d 6 s n u m b e r in quickly rushed around Mant h r e e different ways. hattan and bought all the I had some corresponbooks I could find on recred e n c e with Arthur Koestler. ational mathematics because I He e n j o y e d reading m y coldidn't own any at the time. umn, and once w r o t e t h a t it The column then continued w a s like playing over a c h e s s for 25 years, and it got steadily g a m e b y a top chess player. I more and more sophisticated. t h o u g h t that was nice o f him This was b e c a u s e I w a s learnto say. But w e d i s a g r e e d ing more and m o r e mathemata b o u t parapsychology. He ics. I began subscribing to w a s a fascinating writer. about six magazines that had Koestler had a lot to do

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with m y b e c o m i n g a w a r e of the evils o f Stalinism, especially by his novel Darkness at Noon. When I w a s quite young in college I flirted a bit with Marxism. This and m y disenchantment with Marxism are covered in m y novel The Flight of Peter Fromm. When I became aware of the evils o f Stalinism, that is w h e n I began to sympathize with E d w a r d Teller. Hitler's Nazism and Stalin's Communism w e r e equally wicked. I h a d s o m e c o r r e s p o n d e n c e with Richard F e y n m a n . When I wrote The Ambidextrous Universe he a g r e e d to l o o k over the m a n u s c r i p t to m a k e corrections. We exc h a n g e d several letters. I did get acquainted with Benoit M a n d e l b r o t w h e n he w a s living not far from us in Westchester, New York. I int r o d u c e d Conway to M a n d e l b r o t in our home. C o n w a y exp l a i n e d h o w P e n r o s e tiling h a d a fractal quality. We also t a l k e d a b o u t Robert A m m a n n , a young c o m p u t e r chap w h o h a d written to m e a b o u t P e n r o s e tiling. M a n d e l b r o t got so i n t e r e s t e d in A m m a n n ' s discoveries that he w e n t to M a s s a c h u s e t t s to l o o k him up in person. All I k n o w a b o u t A m m a n n was from Mandelbrot. He said A m m a n n w a s young and very shy. I l a t e r p u b l i s h e d s o m e t h i n g a b o u t w h a t are called A m m a n n bars, a v e r y i m p o r t a n t a s p e c t of P e n r o s e tiling. I n e v e r m e t Ammann. He s e e m s to have vanished into obscurity. Mandelbrot liked to travel a r o u n d to m e e t p e o p l e w h o did w o r k related to his field. I told him a b o u t Bill Gosper, one o f the great c o m p u t e r h a c k e r s of all time. He w a s the one w h o had d i s c o v e r e d the glider gun, an i m p o r t a n t structure in Conway's Life. G o s p e r p r o b a b l y k n o w s m o r e a b o u t c o n t i n u e d fractions than a n y b o d y in the world. He h a d disc o v e r e d an analog o f the snowflake curve that he called the "flowsnake". It w a s a w e i r d fractal. I s h o w e d it to M a n d e l b r o t who w a s so intrigued that he flew to California to m e e t Gosper. They b e c a m e g o o d friends. I w r o t e a c o l u m n on f r a c t a l c u r v e s t h a t d e s c r i b e d t h e s n o w f l a k e . I said it h a d an a n a l o g in t h r e e d i m e n s i o n s . If y o u t a k e a r e g u l a r t e t r a h e d r o n and k e e p s t i c k i n g little t e t r a h e d r o n s in t h e c e n t e r s o f its sides, it g r o w s like t h e s n o w f l a k e curve. The final result, I said, is a v e r y intric a t e c o r r u g a t e d surface, infinite in a r e a b u t e n c l o s i n g a finite volume. G o s p e r w a s s u s p i c i o u s of this r e m a r k . He w r o t e a c o m p u t e r p r o g r a m to find the a c t u a l s h a p e o f t h e s t r u c t u r e and it t u r n e d o u t to b e a cube! I w r o t e to M a n d e l b r o t a b o u t this a n d he said he k n e w a b o u t it all t h e time b u t t h o u g h t it w a s t o o trivial to m e n t i o n in his b o o k on fractals. This also h a p p e n s often w i t h Conway. E v e r y n o w and t h e n s o m e b o d y will s e n d m e s o m e t h i n g a n d I'll be e x c i t e d a b o u t it a n d m e n t i o n it to Conway. He'll s a y he h a d t h o u g h t o f t h a t y e a r s ago, b u t d i d n ' t b o t h e r to p u b l i s h it. C o n w a y is an a u t h e n t i c genius. My wife a n d I a r e v e r y fond o f him. Recently I gave a r a d i o interview a b o u t Coxeter. I have m e t him only once w h e n he l e c t u r e d in New York and w e s p e n t a short time together. I've had o c c a s i o n a l corres p o n d e n c e with him. He r e a d m y column. His Introduction to Geometry was a beautiful book, the first to be organized a r o u n d the c o n c e p t of symmetry. He sent m e a c o p y of

p a g e p r o o f s b e f o r e the b o o k c a m e out, and it had a g r e a t influence on me. Are y o u i n t e r e s t e d in s u p e r s t r i n g theory? E d w a r d Witten of Princeton, the leading s u p e r s t r i n g expert, lectured on the topic a few y e a r s ago at the University o f North Carolina at Chapel Hill. P e n r o s e w a s also lecturing t h e r e at the s a m e time. I d r o v e t h e r e to h e a r the t w o lectures. The confere n c e w a s to h o n o r H e r m a n n Weyl, a u t h o r o f a beautiful little book, Symmetry. I u n d e r s t o o d everything Penrose said and u n d e r s t o o d nothing Witten said. I also h a d lunch one d a y with Eugene Wigner. We h a d c o r r e s p o n d e d a b o u t his f a m o u s p a p e r on t h e i m p r o b a b l e effectiveness of mathematics. N o w that I think of it, yes, I've m e t quite a few famous scientists and mathematicians. C h a r l o t t e G a r d n e r : We should h a v e k e p t a log b o o k of t h e s e meetings. MG: Let m e c o m e b a c k to Wigner for a moment. He m a d e s o m e brilliant contributions to physics, b u t I never liked his a p p r o a c h to quantum mechanics. He flirted with the notion that t h e outside world w a s s o m e h o w a p r o j e c t i o n of c o n s c i o u s n e s s . In this sense he w a s n o t a realist. He w r o t e quite a bit a b o u t the m e a s u r e m e n t p r o b l e m in quantum mechanics, h o w b e s t to explain s u c h p a r a d o x e s as the one k n o w n as "Schr6dinger's cat." The cat is s u p p o s e d to b e n e i t h e r alive n o r d e a d till s o m e o n e o b s e r v e s it. However, the p e r s o n observing it is a n o t h e r q u a n t u m system. So t h e question is, "When d o e s the o b s e r v e r ' s o b s e r v a t i o n b e c o m e definite?" Wigner looks into the b o x and sees, say, a live cat. But Wigner is also a quantum system, so is Wigner real b e f o r e s o m e o n e o b s e r v e s Wigner looking into the b o x a n d seeing the cat? This is the " p a r a d o x o f Wigner's friend." But is the friend "real" until s o m e o n e o b s e r v e s him, and so on into an infinite regress? The question is, when d o e s the w a v e function of the cat finally c o l l a p s e so you k n o w for c e r t a i n w h e t h e r the cat is alive o r d e a d ? Wigner argued that it c o l l a p s e s when a hum a n c o n s c i o u s n e s s o b s e r v e s the cat. You don't have to m o v e on to Wigner's friend. When a h u m a n being with a c o n s c i o u s m i n d m a k e s a q u a n t u m m e a s u r e m e n t , then the wave function collapses (or, in a n o t h e r terminology, the state v e c t o r rotates) and the p h e n o m e n o n b e c o m e s real. Most p h y s i c i s t s think t h a t q u a n t u m effects are real w h e t h e r o b s e r v e d b y a c o n s c i o u s being or not. That the cat is either alive o r d e a d r e g a r d l e s s o f w h e t h e r a n y b o d y l o o k s at it. Wigner u s e d the h u m a n m i n d to cut off the infinite regress a n d m a k e the p h e n o m e n o n real. F r o m Wigner's point of v i e w t h e r e is a s e n s e in w h i c h even the entire Universe is n o t "out there" unless t h e r e are h u m a n s to observe it. This is a sort of flirtation with solipsism. Wigner was one o f t h o s e physicists w h o a r g u e d that you h a d to have h u m a n c o n s c i o u s n e s s to m a k e q u a n t u m effects real. To m e this is total nonsense. J o h n W h e e l e r is a n o t h e r p h y s i c i s t w h o s e a p p r o a c h is similar to Wigner's. Wheeler has also t a l k e d a b o u t the Universe n o t being "real" w i t h o u t h u m a n observers, m a k ing m a n the m e a s u r e of all things. W h e e l e r defends the notion that w i t h o u t h u m a n intelligence the Universe w o u l d have, as he p u t s it, only a "pale sort o f reality," a kind o f

VOLUME 19, NUMBER 4, 1997

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undifferentiated fog. Until human beings evolved you can't say the Universe is really "out there" as a well defined structure. Like almost all physicists, I am a realist. A m a j o r difficulty with the Wigner-Wheeler view is this. S u p p o s e life exists only on Earth. Did dinosaurs have e n o u g h consciousness to m a k e the Universe real? Evolution p o s e s a h a r d p r o b l e m for anti-realism. Einstein was v e r y m u c h a realist. He liked to p u t it this way, "Does the Moon exist only w h e n something like a m o u s e looks at it?" That w a s a w o n d e r ful choice of an animal. It focuses the p r o b l e m . If you n e e d a m o u s e to m a k e t h e Moon real, h o w a b o u t a butterfly or a bee? I a m also a m a t h e m a t i c a l realist. I a g r e e with P e n r o s e and a l m o s t all m a t h e m a t i c i a n s , that m a t h e m a t i c a l truth is discovered, n o t created. Sociologists w h o argue that mathematics is a p u r e l y cultural p h e n o m e n o n , like fashions in dress, drive m e up the wall. I think a huge n u m b e r is either a prime n u m b e r o r not before a n y b o d y p r o v e s it one w a y o r the other. In his b o o k The Emperor's New Mind, P e n r o s e t a k e s the Mandelbrot set, a p e c u l i a r kind o f fractal curve, as an e x a m p l e o f h o w m a t h e m a t i c i a n s d i s c o v e r things r a t h e r than invent them. You c a n n o t guess w h a t the M a n d e l b r o t set is going to l o o k like each time you go to a higher magnification. It's like exploring a jungle. You a r e n o t creating the Mandelbrot set, you are exploring it. III: H o w w o u l d y o u a s s e s s the i m p a c t o f y o u r popularizing m a t h e m a t i c s o v e r the years? MG: My writings have p r o b a b l y had an influence on persuading s o m e y o u n g p e o p l e to b e c o m e m a t h e m a t i c i a n s . Every n o w a n d t h e n I get a letter f r o m a P r o f e s s o r of Mathematics w h o s a y s he s t a r t e d reading m y c o l u m n w h e n he w a s in high s c h o o l a n d it got him i n t e r e s t e d in mathematics. There has b e e n a steadily increasing i n t e r e s t in teaching recreational m a t h e m a t i c s . I have b e e n subscribing to The Mathematics Teacher for a long time. T h e y are publishing m o r e a n d m o r e recreational articles. Especially on the j u n i o r high a n d high school level m a t h e m a t i c s is often taught in such a dull fashion that kids go to s l e e p in class. But if you give t h e m a game, a puzzle, a m a t h e m a t i c a l magic trick, or s o m e t h i n g that h o o k s their interest, everything changes. When I w a s in high school I had a g e o m e t r y t e a c h e r to w h o m I d e d i c a t e d m y first book. I loved geometry, and she was a great teacher. One day, w h e n I w a s doing well in class and had finished my lessons, I w a s sitting t h e r e trying to analyze the g a m e o f tic-tac-toe. I didn't k n o w at that time w h e t h e r t h e first o r s e c o n d p l a y e r c o u l d a l w a y s win, o r w h e t h e r the g a m e w a s a d r a w w h e n b o t h sides p l a y e d their best. The t e a c h e r c a m e a r o u n d a n d s n a t c h e d the pap e r on which I h a d b e e n doodling. She said, "When y o u are in m y class, Mr. Gardner, I w a n t you to w o r k on mathem a t i c s and not on anything else." She did n o t realize that analyzing tic-tac-toe c a n lead into d o z e n s o f nontrivial m a t h e m a t i c a l questions. IIt: H o w a b o u t n u m b e r mysticism?

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MG: I've always b e e n intrigued b y it but, of course, I d o n ' t t a k e it seriously. I invented this character, Dr. Matrix. I w o u l d do an interview with him once a y e a r for the Scientific American, and that w a s m y w a y of getting into a m u s i n g numerology. It's amazing that s o m e r e a d e r s t o o k t h e s e c o l u m n s seriously. I got lots of letters from r e a d e r s w h o w a n t e d to get in t o u c h with Dr. Matrix, h o w c o u l d t h e y r e a c h him, and so on. T h e y didn't realize t h a t I w a s s p o o f m g numerology. IH: Symmetry has played an important role in your writings. MG: I have always thought of s y m m e t r y in terms of group theory. If you perform an operation on something and it remains the same, it has that kind of symmetry. Nowadays broken s y m m e t r y is a hot topic. It's essential in the theories of cosmic evolution. If the Universe started with the Big Bang, then o u r Universe is a series of b r o k e n symmetries. I have invented a model of b r o k e n symmetry. You take a d e c k of cards, stand the cards on end and arrange t h e m in a circle. The cards stand side b y side, balanced, and form a perfectly symmetrical circle. Then you bang the table and the" cards collapse into an asymmetric pattern, a beautiful rosette, and it's impossible to predict w h e t h e r it will be left o r right. So it's a nice m o d e l of w h a t h a p p e n e d after the Big Bang w h e n the original symmetry broke. It's a neat little trick. The t h e o r y is that before the Big Bang there w a s perfect symmetry, that is, in t e r m s of quantum fields. Immediately after the Big Bang t h e t e m p e r a t u r e was t r e m e n d o u s l y high a n d there w a s a v e r y high degree of symmetry. As t h e Universe c o o l e d t h e r e w a s a s u c c e s s i o n o f b r o k e n s y m m e t r i e s . The Higgs field is r e s p o n s i b l e for the last broken symmetry. N o b o d y k n o w s if it exists o r not. If w e h a d built t h e super-collider w e c o u l d have found out w h e t h e r the Higgs field exists or not. R o g e r P e n r o s e is on r e c o r d saying r e c e n t l y that had the super-collider b e e n built, he w o u l d have b e e n very d i s a p p o i n t e d if they did find the Higgs particle. He h o p e s it d o e s n ' t exist! By t h e way, Conway in the p a s t few y e a r s has b e e n d e e p into s y m m e t r y a n d has an e l a b o r a t e classification o f symmetries. He loves to invent n e w vocabularies. Speaking a b o u t symmetry, l o o k out o u r window, a n d y o u m a y s e e a cardinal a t t a c k i n g its reflection in the window. The cardinal is the only b i r d w e have w h o often d o e s this. If it h a s a n e s t nearby, t h e c a r d i n a l thinks t h e r e is ano t h e r c a r d i n a l trying to invade its territory. It n e v e r realizes it is attacking its o w n reflection. Cardinals d o n ' t k n o w m u c h a b o u t m i r r o r symmetry! Martin Gardner 3001 Chestnut Road Hendersonville, NC 28792 USA Istv~n Hargittai Budapest Technical University H-1521 Budapest, Hungary and University of North Carolina at Wilmington Wilmington, NC 28403 USA e-mail: [email protected], [email protected]

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Structural Topology, or The Fine Art of Rediscovery Hen~ Crapo

Marjorie

Senechal,

Editor

The Structural Topology Research Group was an interdisciplinary group of mathematicians and engineers that took shape in the period 1974-1977, flourished as a close-knit group in Montreal until 1980, and then gradually came apart, becoming, in the words of its founder, the "loose mafia" it is today. The aim of this group, which I helped to form, was to bring mathematics, particularly projective geometry and invariant theory, to bear on theoretical problems in architecture and structural engineering. In telling its story, I will emphasize the period of close collaboration during which we began publishing our bilingual journal

Topologie Structurale/Structural Topology. This includes the decade during 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 all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissionsto the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA; e-mail: senechal@mink~

which I was most active in the affairs of the group, and can best appreciate the forces at play. But to find the beginning, we first go back another decade, to the preparations for Montreal's Expo '67. Germination Janos Baracs, now Vice-dean of Research in the Faculty of Architecture and Design at the University of Montreal, fled Hungary in 1956 and launched a career as structural engineer in Montreal. Janos soon achieved a certain notoriety for his participation in the conception of the theme buildings, the Canadian pavilion, and the "Man and His Environment" pavilion at Expo '67. These visually impressive forms (Figure 1) shook free of that simplistic architectural habit of our century, the cutting up of space into rectangular regions. Expo '67 opened his way to a professorship in the School of Architecture. There, Janos developed a popular course on the creation of form and structure, along the lines of Klein's Erlangen Programme: he taught that architectural forms should be syn-

I

thesized f'ffst as topological objects, which can then be successively enriched by the addition of projective geometric, then aff'me, and fmally metric properties. While recognizing that a client will expect a metrically defined concept from his architect, Janos showed that the architect must make his key decisions at the projective geometric stage. He called his course Structural Topology. Let's take a brief look at the projective geometric layer of this course, to show what Janos was able to do without the help of a research group, and how lucky we were to be able later to benefit from his years of preparation. Janos found he was able to use a single 2-dimensional view both to describe synthetic construction in 3-dimensional space, and to determine induced mechanical properties of plane frameworks. In Figures 2 and 3 you s e e two typical early problems from Janos's course: how to draw the complete section of the four faces of a tetrahedron with a plane Q determined by three points said to lie (a) on three edges, or (b) on three faces, of the tetrahedon. These figures are projectively determined, in the sense that if three points of the figure are required to lie in the drawing plane, there remains a single parameter of choice for the "height" of the tetrahedon; all other e l e m e n t s of the figure in space are then determined. The problem (b) is harder. Trial and error, aimed at completing a triangle with vertices on the top edges of the tetrahedron and edges through the given points, quickly shows there is a unique answer. But that's cheating: what is the construction? Hint: the line spanned by two of those points m e e t s the third face plane in a point that is well-determined. Once you have this, the rest is easy.

9 1997 SPRINGER-VERLAG NEWYORK, VOLUME 19, NUMBER 4, 1997

27

Figure 1. Man and his Environment, Expo '67, Montreal.

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THEMATHEMATICALINTELLIGENCER

C a n a d a could make a sigProjective determinificant contribution to n a c y w a s the key to the n e w sciences. A f t e r J a n o s ' s course, and his all, the country h a d students learned to s h o w n s o m e indepen"think" in t h r e e - s p a c e d e n c e with r e s p e c t to t h e through grasping this U.S., harboring o b j e c t o r s concept. But w h a t hapto and d e s e r t e r s from t h e p e n s if a figure is n o t prow a r in Vietnam, h a d jectively determined? d e m o n s t r a t e d its creative Two n e w p h e n o m e n a abilities in the design a n d arise: a drawing m a y have o p e r a t i o n o f E x p o '67, choices, or conditions (or a n d w o u l d s o o n be b u s y both). J a n o s s h o w e d t h a t Figure 2. To find the complete section by a plane, in a projectively determanufacturing snowmofor a general p o s i t i o n mined figure. biles for the Inult and rubd r a w i n g of a s p h e r i c a l ber-tired s u b w a y cars for p o l y h e d r o n with s verParis. The energies of t h e s e "Trudeau J a n o s furnished a rich catalogue of tices to be projectively determined, it Years" w o u l d c o n t r i b u t e much to t h e m u s t have no m o r e t h a n 2s - 2 edges. critical forms, explaining that polyheearly d e v e l o p m e n t of the Structural If t h e r e are c edges more, t h e r e are c dral liftings s e e m e d to explain the apTopology group. d e g r e e s of indeterminacy, thus choices p e a r a n c e of e x t r a degrees of f r e e d o m During the winter of 1973-4, for the spatial p o l y h e d r o n . If t h e r e are in p l a n e frameworks. His d r e a m w a s to Anatole Joffe, t h e n d i r e c t o r of t h e c edges fewer, t h e r e are c "negative explain critical forms of b a r a n d j o i n t Centre de R e c h e r c h e s Math6matiques, choices," or projective conditions. No s t r u c t u r e s in three-space in the s a m e met J a n o s B a r a c s on an airplane flight lifting at all is p o s s i b l e unless all t h e s e way, b y showing they satisfy projective from Montreal to New York. Seated toc o n d i t i o n s are satisfied. (Here I'm leav- c o n d i t i o n s related to their liftability to gether, one s u p p o s e s p u r e l y by chance, ing out lots of details that c o m e up higher-dimensional space. This is m o r e they d i s c o v e r e d t h e y s p e n t their d a y s w h e n one looks m o r e closely at t h e s e o f a r e s e a r c h p r o g r a m t h a n a conjecworking in the s a m e building, the conture, b u t w e chose to call it s i m p l y the questions.) verted m m n e r y that h o u s e d b o t h Baracs Conjecture. J a n o s w o u l d t h e n p o i n t out that the Architecture a n d Mathematics at t h e The late sixties and early s e v e n t i e s m e c h a n i c a l b e h a v i o r o f b a r and joint University of Montreal. J a n o s s e i z e d in C a n a d a w a s a p e r i o d b o t h o f optis t r u c t u r e s in the p l a n e d e p e n d s on the o c c a s i o n to enlist his neighbor's w h e t h e r or not t h e y can b e i n t e r p r e t e d m i s m a n d of intellectual ferment. That aid in solving s o m e of the m a n y mathas p r o j e c t i o n s of polyhedra. His fa- it w a s also the end of a long e c o n o m i c ematical p r o b l e m s he h a d b e e n formuvorite e x a m p l e s w e r e c o m b i n a t o r i a l b o o m w a s not yet clear. Two o p p o s i n g lating a b o u t f r a m e w o r k s and polyhef r a m e w o r k s with the t o p o l o g y of a trends, e a c h seeking to hide b e h i n d the dral forms. A n a t o l e w a s delighted: this s p h e r i c a l polyhedron, with 2 s - 3 other, u n i t e d to p r o m o t e interdisciplis e e m e d a s p l e n d i d w a y to d e m o n s t r a t e edges, and with no m o r e than 2s' - 3 n a r y research. Certain s e g m e n t s o f the that his r e s e a r c h c e n t e r could be useedges on any s u b s e t of s ' vertices, for a c a d e m i c left w e r e critical o f ivoryful a n d "relevant" to Quebec society. s'--2. General d r a w i n g s of such t o w e r i d e a l i s m and c o m p a r t m e n t a l i z e d They a g r e e d to set aside one entire d a y f r a m e w o r k s are j u s t rigid, o r isostatic, studies. At the s a m e time, g o v e r n m e n t on which J a n o s w o u l d p r e s e n t his proas the engineers say, and a r e incorrect a g e n c i e s w e r e b u s y developing program to the p e r s o n n e l o f the Centre. as d r a w i n g s of polyhedra. One projec- g r a m s a i m e d at pushing university retive condition is n e c e s s a r y for a good s e a r c h c a p a c i t i e s t o w a r d the service of Formation p o l y h e d r a l projection, as in Figure 4 industry. Both p r o c l a i m e d the desirThe Structural Topology R e s e a r c h (left), a n d this s a m e c o n d i t i o n guaran- ability o f b r e a k i n g d o w n the b a r r i e r s Group crystallized from that s e s s i o n t e e s an infinitesimal m o t i o n of the cor- b e t w e e n disciplines. There w e r e signs organized by Anatole. In the long rtm r e s p o n d i n g plane framework. Plane that even a "small" c o m m u n i t y like we caused more figures satisfying headaches than this projective j o y a n d funding condition are the for the Centre, critical forms of but Anatole reframework. m a i n e d a loyal Analogous techfan. niques apply even I was living in for f r a m e w o r k s Ontario at the with non-planar time, working at graphs, Figure 4 the University of Figure 3. To find the complete section by a plane. (right).

VOLUME 19, NUMBER 4, 1997

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(b) to characterize the critical forms of such structures by projections from higher-dimensional space, i.e., the Baracs Conjecatre,

Figure 4. Critical forms of plane frameworks 9

Waterloo. English Canada was generating so m u c h virulent propaganda aimed at undoing the Quebec "liberation" movement, that m y wife and I felt there had to be another side to the issue, and opted to take our family to Montreal for a sabbatical year. We would learn some French and see what was happening at first hand. Anatole provided me with an office at the Centre, so I was on hand for Janos's presentation. As I recall, there was a lively exchange of views. His many drawings and models both fascinated and confused us mathematicians, accustomed as we were to much less "figurative" notions of algebra and analysis. Of course we struggled and complained about a lack of precision in the definitions. Our response was to propose counterexamples. We realized only slowly that our task was to fred correct defmitions and assertions that would sharpen and confirm Janos's correct intuitions concerning the subject matter. The mathematicians who were willing to work in this fashion always formed a minority, both at that first meeting and in our subsequent work. Of the 10 or 15 mathematicians assembled to hear the presentation, three of us eagerly took up the challenge: Anton Kotzig, Ivo Rosenberg, and myserf. Anton wanted to apply what he knew about zonotopes to mechanical linkages with equal-length bars. Ivo wanted to study the combinatorial structure of Laplace expansions of minors of matrices controlling critical forms of bar and joint structures. At the start, I was fascinated by two problems: the flexibility of rings of hinged panels, and the cross-bracing of one-story buildings. Our small group enlarged gradually and serendipitously. An A.M.S. session

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in Vancouver attracted Ethan Bolker from Boston and Ben Roth from Wyoming, who collaborated to produce the first important theoretical advance in the subject: the description o f critical forms for frameworks forming bipartite graphs [4]. In September 1974 I organized two weeks of meetings in Waterloo, bringing together for the first time the handful of architects and engineers dealing seriously with geometric questions, and the increasingly active family of mathematicians working in combinatorial geometry and invariant theory. That meeting brought in Walter Whiteley and Neil White, who would later collaborate on a series of influential papers on statics and mechanics [5,7]. Walter plunged headlong into the work, and hasn't surfaced yet, m o r e than two decades later. Walter and I became traveling salesmen for the study of "structural rigidity." During the Waterloo meeting, after a particularly heated discussion period, we were surprised to find a brand-new polyester necktie with bold orange, brown, and white stripes hanging on a chair in the lecture room. I fear we didn't wait a decent interval, but w h e n no one claimed it, we decided to offer it as a prize for a positive solut i o n - r e p h r a s i n g and p r o o f - - t o the Baracs Conjecture. I was the self-declared custodian of the prize, and it's still hanging in my closet in France, waiting for the winner to claim it. The meetings p r o d u c e d not only new recruits, but also our formulation of the two basic problems of structural rigidity in 3-space: (a) to find a combinatorial characterization of generically isostatic frameworks, and

problems which remain unsolved to this day9 The cover of this issue of the InteUigencer has three revelant drawings. The large figure at the top is Janos's double-four configuration, one step in an attempt to explain critical forms in 3-space by section-projection from 5-space; it is a 4-dimensional generalization of the Desargues configuration. At the lower left, you find the most general critical form of an octahedral skeleton in 3-space. At the lower right (and also in Figure 5, below), we see h o w a framework in 3space, combinatorially equivalent to the 1-skeleton of a 4-polytope, can be a critical form without being a correct projection of such a polytope. (In a correct projection, the three red points would coincide.) Reflections on Rediscovery Rediscovery is particularly c o m m o n in interdisciplinary work, where people in the different disciplines are likely to have encountered your problem before, to have published in journals of which you have never heard, and to have stated their conclusions in a technical language you may not understand. Rediscovery also has a bad reputation, partly because it is poorly defined. Behind the term lurk two distinct and complementary activities, which I will call RD1 and RD2. RD1 could simply be called "mathematics," it's what you do in the heat of discovFigure 59 Critical form

of a

spatial framework.

ery b e f o r e you c h e c k to s e e if it has alr e a d y b e e n published. RD2 c o m e s after the often thankless t a s k of consulting all p o s s i b l e sources, w h e n y o u identify (and cite) the a n t e c e d e n t s of your work, that is, w h e n y o u d i s c o v e r the t r a c e s of y o u r i d e a in o u r cultural history. No act of r e d i s c o v e r y is c o m p l e t e (or established) until RD2 is accomplished, b u t this h a r d w o r k is too often left to a referee o r competitor. I have the i m p r e s s i o n that t h e r e is s o m e t h i n g quite n e w a b o u t rediscovery as a concept. My 1959 (2kg) edition of the Shorter Oxford English Dictiona r y has no definition o f it. Definitions I find in m o r e r e c e n t r e f e r e n c e w o r k s underline only the m e a n i n g RD1, but that in the F r e n c h d i c t i o n a r y Robert I a d d s a helpful example, t a k e n from Paulhan, which I translate: "The thoughts o f o t h e r p e o p l e d o n o t s e e m decisive to us until that i n s t a n t when, rediscovering t h e m for ourselves, w e feel t h e m to be very m u c h o u r own." This is c l o s e r to the s e n s e in w h i c h redisc o v e r y is a c o r n e r s t o n e o f science. Richard F e y n m a n strongly recomm e n d s discovering k n o w n results, "It's m u c h e a s i e r to u n d e r s t a n d things after you've fiddled with them, b e f o r e you r e a d the solution." I p a r t i c u l a r l y app r o v e J a n o s ' s attitude: "Rediscovery is an enriching process. It m a k e s you feel justified and supported." The s u m m e r o f 1975 I w a s in Nova Scotia, and s p e n t s o m e afternoons s e a r c h i n g the literature on statics and m e c h a n i c s in the Dalhousie University Library. Their collection is strong in this area, b e c a u s e an e s t a b l i s h e d engineering school h a d b e e n m e r g e d into the university at its founding, and its lib r a r y transferred. One l u c k y day m y eyes lit on a t e x t b y Luigi Cremona,

Graphical Statics: Two treatises on the Cn'aphical Calculus and Reciprocal Figures in Graphical Statics [3]. It h a d b e e n p l a c e d on the shelves on July 15, 1925, and had n e v e r b e e n b o r r o w e d b y a reader. The original library card was still in place, n e v e r s t a m p e d . This small v o l u m e w a s t h e R o s e t t a Stone for s u b s e q u e n t r e s e a r c h on rigidity of structures. It c o n t a i n s the refere n c e s to all the k e y literature of the y e a r s 1864-1888, from the invention of the m e t h o d of r e c i p r o c a l figures by the

Scottish physicist James Clerk Maxwell [1,2] a n d his colleague P r o f e s s o r Rankine, to the establishment o f a science of "graphical statics" b y t h e Germ a n e n g i n e e r Culmann. As C r e m o n a states in his preface to the English translation, "A great m a n y of t h e propositions, w h i c h form the Graphical Calculus of the p r e s e n t day, have b e e n k n o w n for a long time; b u t t h e y were d i s p e r s e d in various g e o m e t r i c a l works." I fear t h a t if w e had not d i s c o v e r e d this p a r t i c u l a r book, we w o u l d never have m a d e c o n t a c t with the literature of graphical statics. This s u b j e c t had long since b e e n d r o p p e d from t h e engineering curriculum in m o s t countries. W h a t few traces r e m a i n go usually b y the n a m e of "the funicular p o l y g o n method," an i s o l a t e d technique. References to the early literature die o u t a r o u n d the t u r n o f t h e century, a n d d i s a p p e a r totally by 1920. Walter a n d I followed C r e m o n a ' s b i b l i o g r a p h y b a c k to Maxwell, locating the early v o l u m e s of the Philosophical Transactions and the Royal Society Proceedings in the engineering library o f t h e University of Toronto. We f o u n d to o u r a s t o n i s h m e n t that reading Maxwell's first (1864) article on framew o r k s w a s like opening a p a g e from J a n o s ' s course. Maxwell u s e d t h e s a m e e x a m p l e s for the s a m e reasons, and s u g g e s t e d the same conjectures. The m a i n difference w a s that Maxwell conc e n t r a t e d on resolution of f o r c e s (thus on stresses), while J a n o s ' s a t t e n t i o n w a s f o c u s s e d on infinitesimal motions.

Formation and Functioning of the Group Any fruitful c o l l a b o r a t i o n requires a b o d y o f s h a r e d concepts, a well-def m e d v o c a b u l a r y and notation, a n d energy d i r e c t e d t o w a r d c o m m o n goals. F o r t h r e e years, Janos, Walter, and I m a d e frequent contact, and w o r k e d with s t u d e n t s o n different a s p e c t s o f the program. In 1978, J a n o s and Anatole c o n c e i v e d an a d m i n i s t r a t i v e p l a n for a r e s e a r c h group u n d e r the aegis o f t h e School of A r c h i t e c t u r e a n d the M a t h e m a t i c s Research Centre. It w a s this p l a n which b r o u g h t t o g e t h e r the critical m a s s n e c e s s a r y to a d v a n c e our work, a n d was the birth o f the

Structural T o p o l o g y group, in institutional form. Once Janos, Walter, a n d I w e r e together, in the w i n t e r of 1978, the w o r k did i n d e e d p r o g r e s s rapidly. We set aside e a c h Thursday, all day, plus o n e other morning e a c h week, for o u r working sessions. F o u r to six participants w a s the norm. Besides Janos, Walter, and me, w h o were a l w a y s there, the m o s t frequent p a r t i c i p a n t s were o u r students, Nabil Macarios, architect, R a c h a d Antonius and Tiong Seng Tay, mathematicians. Walter a n d I had h o p e d t h a t o u r c o l l a b o r a t i o n with Janos w o u l d p u t us in working c o n t a c t with a larger c o m m u n i t y of engineers and architects. This wish w a s fulfilled on t w o h a p p y occasions: a visit b y Israeli architect Michael Butt, w h o l a u n c h e d a p r o j e c t on the a r c h i t e c t u r a l uses o f periodic s u r f a c e s o f negative curvature, and an exciting p e r i o d o f w o r k with Marc Pelletier, a young colleague of architect Steve Baer, d e s i g n e r of zonohedral building s y s t e m s for the American southwest. But in retrospect, it s e e m s clear that each o f us c o n t i n u e d to a c t as s p o k e s m a n for a well-defined s c h o o l of thought, as along s e p a r a t e b r a n c h e s of a tree. There w a s active e x c h a n g e at the root, in our seminar, b u t e x c h a n g e s o u t s i d e the s e m i n a r w e r e within a single discipline, usually m a t h e m a t i c s . Our b e s t c o n v e r s a t i o n s were often over a b r e a k for lunch at a small Hungarian r e s t a u r a n t n e a r the university. The table m a t s w e r e p r i n t e d only on one side; o u r first gesture, after ordering the c h i c k e n paprikash, w a s to turn the m a t s a n d begin drawing. I have many o f t h e s e p a p e r s still in m y files, with their s c a l l o p p e d b o r d e r s and images of w a i t e r s f r o m s o m e bygone era. It w a s m y j o b to p r e p a r e s u m m a r i e s of our sessions, filling in the p r o o f s when they b e c a m e available. The resuiting collection o f s o m e 250 p a g e s o f t e x t for t h a t first y e a r s e r v e s even tod a y as an i n d i s p e n s i b i e i n t r o d u c t i o n to t h e m e s t h a t w e r e n o t subsequently developed in p u b l i s h e d form. Meanwhile, Walter and I filled in missing p a r t s o f proofs of Maxwell's t h e o r e m (relating s t r e s s e d f r a m e w o r k s to p o l y h e d r a l projections), a n d confh'med that pro-

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j e c t i o n s of o r i e n t e d 4-polytopes are s t r e s s e d in 3-space (the easy half of the Baracs Conjecture). Our first attempt, as a group, to int e r a c t with a larger circle of mathematicians w a s in a special session on rigidity theory, at the regional meeting of the A. M. S. in Syracuse, New York, O c t o b e r 1978. To that meeting, Branko Griinbaum b r o u g h t an u p d a t e of his c o u r s e with G e o f f r e y Shephard, entitled "Lectures o n Lost Mathematics." It s e e m s that t h e y h a d earlier p l a n n e d an e x t e n d e d w o r k on questions of rigidity, b u t the o t h e r half of their p r o p o s e d project, that on tilings, had t a k e n on entirely u n e x p e c t e d p r o p o r t i o n s , b o t h in time and energy. B r a n k o s h a r e d with us their thinking on s t r u c t u r e s f o r m e d by cables in tension, struts in compression, b e t t e r k n o w n as "tensegrity" frameworks. The Journal

t r e m e l y talented group led by the Vietnamese student T. T. Luong. Their task: "to study the s y n t h e s i s of polyh e d r a l forms for a r c h i t e c t u r a l purposes, on scales ranging f r o m that of an individual dwelling to that of a small city, and with due r e g a r d for the interp l a y of sculptural a n d a r c h i t e c t u r a l aesthetics." (Figure 6.) ST#3 d e s c r i b e d a s u b s e q u e n t c o l l a b o r a t i o n with t h e Montreal s c u l p t o r Pierre G r a n c h e on the j u x t a p o s i t i o n of z o n o t o p a l forms. F o r t h o s e first t h r e e issues, w e relied on articles p r o d u c e d in our o w n "stable," and it soon b e c a m e doubtful t h a t the j o u r n a l w o u l d be able to stimulate a flow of articles o f truly interdisciplinary inspiration. In a sense, two c a m p s developed: t h o s e i n t e r e s t e d in w o r k i n g on the basic r e s e a r c h problems, and t h o s e i n t e r e s t e d in the cultural and architectural a s p e c t s of the subject, w h o n e e d e d S T as an outlet. T h o s e in the former c a m p r e f u s e d to c o m p r o m i s e the a b s t r a c t n e s s essential to g o o d m a t h e m a t i c s for the s a k e of clarity and simplicity, a n d p r e f e r r e d to p u b l i s h in s t a n d a r d m a t h e m a t i c a l journals with larger circulation. Those in the latter c a m p w e r e unwilling or unable to a p p l y that s a m e rigor a n d precision to their creative p r o c e s s . The b l a m e can b e s h a r e d equally, but the d r e a m should not be a b a n d o n e d . R e a d e r s h i p (and a u t h o r s h i p ) w a s one problem; language w a s another. In m y n o t e s of o u r m e e t i n g o f 21/09/78 I wrote, "The publication o f o u r p o s t e r

Also in 1978, Janos, Walter, and I l a u n c h e d a n e w journal, which w e called Structural Topology, o r ST for short. We d r e a m e d that the j o u r n a l w o u l d help us to focus our research, providing an outlet for clearly written s u m m a r i e s o f advances, and exchanges of o p e n p r o b l e m s with an ext e n d e d clan o f activists. This was not to be the case. ST#1 opened with an introduction to geometry in architecture by Michael Rubin, with lovely pen-and-ink sketches by Katia Montillet. Janos, Walter, a n d I introduced the three main t h e m e s o f our p r o p o s e d collaboration: polyhedra, rigidity, and s p a c e filling. I'm still quite fond of that first issue. However, J a n o s urged us to r e d i r e c t o u r sights in subsequent numbers, in ord e r to attract the s o r t of r e a d e r s a n d aut h o r s he k n e w w e needed. Accordingly, S T #2 was b a s e d on the p r o j e c t "Polyhedral Habitat," realized, u n d e r J a n o s ' s direction, by an ex- Figure 6, Polyhedral Habitat, project in 1977.

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advertising the Bulletin, p r i n t e d m o s t l y in English, has given o f f e n s e ' t o at least one individual in this university. We will supply a F r e n c h translation to be p a s t e d over the right h a n d page, to b e mailed to certain F r e n c h language establishments." It d i d n ' t t a k e long to realize t h a t Quebec g o v e r n m e n t funding o f a n English-language p u b l i c a t i o n w a s unthinkable, and so w e d e c i d e d on a bilingual f o r m a t - - e a c h article to app e a r b o t h in English and in French. This m e a n t that w e h a d to t r a n s l a t e all articles in one direction or the other, and to invent a mise-en-page that w o u l d s m o o t h l y i n c o r p o r a t e m a n y illustrations. But it w a s not a b a d decision. At heart, we w e r e in favor o f the language policy, and felt w e w e r e do= ing s o m e t h i n g positive. F u r t h e r m o r e , the bilingual f o r m a t a s s u r e d us m a n y r e a d e r s in France. However, it discouraged a good number of potential publishers. Stresses and Strains

ST #4 w a s a h a p p y e x p r e s s i o n o f o u r ability to solicit articles from a larger interdisciplinary group: six a r c h i t e c t s and engineers, t h r e e m a t h e m a t i c i a n s . The l e a d article w a s b y the d e s i g n e r Koji Miyazaki, w h o deserves s p e c i a l m e n t i o n for his s h e e r love for all that is Geometry. He n o w p u b l i s h e s his own j o u r n a l HyperSpaee, a l r e a d y in its fifth year, at the University of Kyoto. This w a s a high p o i n t in o u r optimism, and signs o f s e v e r e t r o u b l e s to come were already visible. At first, logistical p r o b l e m s s e e m e d the m o s t formidable. The j o u r n a l w a s entirely p r o d u c e d "in house," in a time before the first Macintosh, and, thus before desktop publishing, TEX, and laser printers. F o r younger readers I will briefly r e c o u n t o u r n o w unimaginable travail. With the help of a neigh-

b o r w h o h a n d l e d the p u b l i c a t i o n of a chain o f Quebec w e e k l y n e w s p a p e r s , w e gained access to an o p t i c a l character-detection device t h a t w o u l d read o u r specially c o d e d t y p e s c r i p t s into p u n c h e d tape for a p h o t o t y p e s e t t e r , w h i c h w o u l d in turn p r o d u c e c o p y for pasteup. This m e a n t in p r a c t i c e that I w o u l d fly from Montreal 450 k m to the old mining t o w n o f Val d'Or in northern Quebec, b o o k into a h o t e l for three days, run our t y p e d p a g e s t h r o u g h the scanner, c o r r e c t all its scanning errors b y hand, feed the t a p e s into t h e typesetter. Then, b o r r o w i n g t o o l s from the n e w s p a p e r p r o d u c t i o n team, I w o u l d do m y page layouts. At t h e end o f the third day, leaving b e h i n d m y seventy p a g e s o f layout for printing, I w o u l d fly b a c k to Montreal and w a l t a c o u p l e of w e e k s for the p r i n t e d c o p i e s to arrive b y bus. That w a s always o u r big day: time to b o x the i s s u e s a n d get t h e m in the mall. P r o b l e m s like t h e s e w o u l d eventually be solved, b u t o t h e r s w e r e intrinsic. Although w e s e n t o u t m a n y hund r e d s of free issues to individuals, to a c a d e m i c d e p a r t m e n t s a n d university libraries, it t o o k s o m e time for o u r circulation to b r e a k the 200 barrier. It gradually settled in at 280 faithful subscribers, m o s t of t h e m in the U.S.A., Canada, and France, b u t also in the r e s t of Europe, the Soviet bloc, Israel a n d Japan. We a p p r o a c h e d various publishers, s o m e of w h o m m a d e s o m e helpful suggestions, b u t w e n e v e r f o u n d one willing to t a k e on the journal. A l e t t e r from Gian-Carlo Rota, w r i t t e n at t h a t time, s h e d s s o m e light on o u r situation: T h a n k you for keeping m e up to d a t e on p r o g r e s s in y o u r s e a r c h for a p u b l i s h e r for Structural Topology. I have b e e n doing w h a t e v e r I can to p u t y o u in t o u c h with v a r i o u s publishers, and to advise y o u on the highways and b y w a y s o f reaching a n agreement. I s e e you have failed to h e e d m y m o s t i m p o r t a n t bit o f advice. You are m a k i n g a serious err o r b y using the t e r m "topology" in y o u r title. It is misleading; with a title like this, t h e j o u r n a l will be inv a r i a b l y sent to the w r o n g referees, a n d in the end no p u b l i s h e r will

t o u c h it . . . I h o p e you w o n ' t make further t r o u b l e for yourself for the s a k e of a word. You have argued that it's t o o late to change. I say r a t h e r that if you do n o t t a k e this m a t t e r seriously, y o u will do great harm to your project, y o u r publication, a n d even to y o u r work. Gian-Carlo was right: n a m e s m a t t e r in the w o r l d o f publishers a n d market- Figure 7. Walter and ing. But in one of t h o s e funny twists o f fate, it is exactly the a s p e c t s t h a t he c o n s i d e r e d Topology that m u c h later b e c a m e t h e focus of o u r w o r k o n polyh e d r a l p r o j e c t i o n s and on f r a m e w o r k s [6,9,10]. In r e t r o s p e c t , it is clear t h a t the difficulties o f p r o d u c i n g a n d selling the j o u r n a l d i v e r t e d us from h e a d - o n collision with the o t h e r questions w e s h o u l d have b e e n asking ourselves. The p r e s s u r e and frustrations o f producing the j o u r n a l w e r e draining all o u r e n e r g y a w a y from research. Then, too, w e c o u l d not count on p e o p l e outside o u r o w n circle to be fully supp o r t i v e o f the d e c l a r e d aims o f o u r research; t h e y h a d their o w n h o r s e s to drive. As a result, our o w n goals for the j o u r n a l c o n t r a c t e d over time. We bec a m e increasingly reluctant to e x p e n d a s u b s t a n t i a l p o r t i o n o f o u r energies p r o m o t i n g research, s a y on certain specialized classes of p o l y h e d r a o r on a r c h i t e c t u r a l s y s t e m s having only marginal (or unanalyzed) g e o m e t r i c content. If the j o u r n a l was n o t advancing (a m o t o r for) o u r own r e s e a r c h aims, w h y w e r e w e publishing it? In fact, I can't recall a single c a s e in w h i c h a p r o b l e m was solved b e c a u s e a question w a s p o s e d in an issue of ST. It s e e m s a fact of life, at least in o u r field, t h a t p r o b l e m s have to b e exc h a n g e d b y p e r s o n a l mall o r viva voce,

Janos, Sophia Antipolis, 1987.

and t h e n not even via the lecture hall. Good g e o m e t r y p r o b l e m s are p e r h a p s too p e r s o n a l for any o t h e r medium. Based on o u r s t u d y of space-filling synthesis I h a d c o m p o s e d a list o f "20 questions" for t h e b a c k p a g e s of ST #2. It is surely indicative of t h e influence of the j o u r n a l t h a t n o n e of our r e a d e r s ever t o o k up a n y o f t h e s e questions. In a sense, 1980 w a s t o o early for s e r i o u s w o r k on s u c h p r o b l e m s . Our a d v a n c e d "tools" w e r e c a r d b o a r d and plastic shapes, and J a n o s ' s drawing b o a r d , with its crinkly-sounding tracing paper. The d a y o f the Silicon G r a p h i c s machine and the G e o m e t r y Center h a d n o t arrived. As early as 1980 it also b e c a m e c l e a r that w e h a d n o t found a h a p p y b a l a n c e b e t w e e n credit for individual and t e a m work, b e t w e e n sole a n d joint authorship. We s e t t l e d on a routine acrossthe-board a c k n o w l e d g e m e n t of j o i n t w o r k in o u r p a p e r s , but this often left the i m p r e s s i o n that the author deserved, wanted, o r took, credit for the main ideas. Our c a r e f r e e e x c h a n g e o f ideas b e c a m e cautious, prudent. As the fabric o f o u r w o r k i n g relationship eroded, o u r i n t e r e s t s b e g a n first to specialize, t h e n to diverge. It would perhaps equally b e true to say that frequent c o l l a b o r a t i o n was no longer a positive influence on our work.

VOLUME 19, NUMBER 4, 1997

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Rediscovering the Classics Our adventures in rediscovery wdre not over when we had traced our subject back to 1864. I recently found that one of my favorite creations, a bracket polynomial condition

lad] [be] [cf] + [af] [bc] [de] = 0 for six directed lengths obtained from a complete quadrangle as in Figure 9, had been around for over 1600 years: Pappus stated this condition in the form BF IrpS~ r~v AE, ogro~ vb trn-b AB, FZ 7rpb~ vb brrb AA, EZ (Collection vii, 198, see [Loeb, pp 610-613]). Figure 8. Janos with Giacometti's cat, Fondation Maeght, St. Paul de Vence, 1987.

Walter became the expert on generic rigidity, recursive procedures, weavings, sheet structures, and tensegrity, collaborating with Neil White, then, in a way which has been crucial for the further development of the subject, with Bob Connelly. I drifted between zonohedral space-filling and the search for critical forms of structures, then focussed on the homological aspects of polyhedral drawings and rigidity, working with Tiong Seng Tay. Janos concentrated on his conjecture, upping the ante to five dimensions and combined section/projection operators in his search for the characterization of critical forms of spatial frameworks. Janos addressed these questions in pamphlets entitled "New Grazing Grounds" (1980). In retrospect, I feel it was our inability to pursue these deeper questions that spelled

the end of three-way collaboration of the founding members of the group. The end of my three-year contract at the university coincided with the retirement of our supportive Vice-rector, Maurice l'Abb~, and his replacement by a physicist who had other research groups to encourage. Support for ours was not renewed. Janos found himself again without a team of mathematicians. You could say this was the death of the research group. But we had created a much larger, loosely linked, international group of mathematicians with a shared passion, which manages to meet here and there (Hungary, Japan, the United States, Canada, France, S w e d e n . . . ) . If the Montreal group and the journal accomplished something of value, it did so in laying the foundations for this lasting comradeship.

Figure 9. A new-old relation on intercepts of sides of a complete quadrangle.

THEMATHEMATICALINTELLIGENCER

And Pappus's condition is equivalent to a theorem of Desargues, that "The three pairs of opposite sides of a complete quadrilateral are cut by a transversal in three pairs of conjugate points of an involution." Desargues's result was later used extensively by Cremona, which brings us back full circle. But such is the world of mathematics and mathematicians. Today much of the work of the Structural Topology group is being rediscovered by the computer graphics community, the only difference being that we're still around to complain. Plus ~a change,

plus c'est la mOme chose. BIBLIOGRAPHY

The items are given chronologically. For complete references to work in structural topology, some 160 titles in all, see the bibliographies of [8] and [10], below. The latter work gives a unique overall view of structural rigidity and its relatives today. [1] James Clerk Maxwell, On Reciprocal Figures and Diagrams of Forces, Phil. Mag. 4, 27 (1864), 250-261. [2] James Clerk Maxwell, On Reciprocal Figures, Frames, and Diagrams of Forces, Trans. Royal Soc. Edinburgh 26 (186972), 1-40. [3] Luigi Cremona, Graphical Statics: Two Treatises on the Graphical Calculus, and Reciprocal Figures in Graphical Statics, Oxford Univ. Press, 1890, translation by T. H. Beare of Le figure reciproche nelle statica grafica, 1872.

[4] Ethan Bolker and Ben Roth, When is a Bipartite Graph a Rigid Framework?, Pacific J. of Math. 90 (1980), 27-44. [5] Neil White and Walter Whiteley, The Algebraic Geometry of Stress in Frameworks, SIAM J. Algebraic Discrete Methods 4 (1983), 481-511. [6] Henry Crapo and Juliette Ryan, Spatial Realizations of Linear Scenes, Structural Topology 13 (1986), 33-68. [7] Neil White and Walter Whiteley, The Algebraic Geometry of Motions of Bar and Body Frameworks, SlAM J. Algebraic Discrete Methods 8 (1987), 1-32.

[8] Jack Graver, Brigitte Servatius, Herman Servatius, Combinatorial Rigidity, Amer. Math. Soc., Providence R. I., 1993. [9] Tiong Seng Tay, Neil White, and Walter Whiteley, Skeletal Rigidity of Simplicial Complexes, Europ. J. of Combin. 16 (1995), I: 381-403, II: 503-523. 10. J. E. Bonin, J. G. Oxley, B. Servatius, ed., Matroid Theory, Amer. Math. Soc., Providence R. I., 1996. In particular: Walter Whiteley, Some Matroids from Applied Discrete Geometry, 171-311, and Henry Crapo, Problems on Bar Frameworks, 413-415.

VOLUME 19, NUMBER 4, 1997

35

i ~ ' J ~ | , I : J , , ~ - | | [~r -

.

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-

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This column is devoted to mathematics for fun. What better purpose is there 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.

Alexander

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48

Editor

An Unfair Game In this game, our p r o b a b i l i s t i c intuition (at least mine) fails drastically. I l e a r n e d it from A. L. Brudno. Alice and Bob t o s s a fair coin. Before t h e y start, each o f t h e m selects a three-bit string, like 010 o r 111. A p l a y e r wins w h e n t h e s e bits a p p e a r (as c o n s e c u t i v e bits), a s s u m i n g that the o p p o n e n t has not w o n earlier. Bob politely suggests Alice c h o o s e h e r string first. As he explains, it gives h e r m o r e freedom: she m a y c h o o s e any string she wishes, and he has to r e s t r i c t hims e l f to the remaining s e v e n strings (it is n o t a l l o w e d to have the s a m e string for b o t h players for evident reasons). Should Alice believe him? Stop reading here and think a m i n u t e a b o u t this game. Unless y o u do the computations, y o u m a y be s u r p r i s e d to k n o w h o w b i a s e d the game is. In fact, Bob's c h a n c e s are (at least) t w i c e as g o o d if he is clever enough. F o r example, i f A c h o o s e s 010, then B can c h o o s e 001 a n d win with p r o b a b i l i t y 2/3. Let us c h e c k that. P0o, P01, Pl0 and P l l d e n o t e t h e p r o b a b i l i t i e s that A wins, assuming that the last two bits a r e given and n o b o d y h a s w o n earlier. The equations are P00 P01 Pie Pu

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]

Shen,

= = = =

P00/2, 1/2 + P11/2, P00/2 + Pol/2, P10/2 + Pll/2.

F o r example, the s e c o n d equation says that a f t e r . . . 01, the t w o e v e n t s 0 (then A wins for sure) and 1 (then A wins with p r o b a b i l i t y P l l ) are equally probable. The first equation implies that P00 = 0, and it is e a s y to s e e w h y it is. Indeed, after 00, the only c h a n c e for Alice is never get 1, a n d this event has zero probability. The only solution for this s y s t e m of linear equations is P00 -- 0, P01 = 2/3, Plo = P l l = 1/3. The a v e r a g e o f t h e s e four probabilities (that is, the probability for A to win the g a m e from the beginning) is 1/3.

THE MATHEMATICAL INTELLIGENCER 9 1997 SPRINGER-VERLAG NEW YORK

I

S o m e o t h e r strings are even w o r s e for her. F o r example, string 000 wins against 100 only with p r o b a b i l i t y 1/7. (If Alice h a s no luck in the first t h r e e rounds, she cannot win at all!)

Colorings and Coverings We c o n t i n u e o u r collection o f p r o b lems with u n e x p e c t e d solutions. Consider a convex polyhedron w h o s e f a c e s are all triangles. We w a n t to c o l o r its vertices in t h r e e c o l o r s such that e a c h triangle has v e r t i c e s o f t h r e e different colors. An e v i d e n t n e c e s s a r y c o n d i t i o n is that e a c h v e r t e x is incident to an even n u m b e r o f faces: the colors a r o u n d a v e r t e x alternate, a n d if the n u m b e r o f faces is odd, t h e r e is no c o l o r for t h e last v e r t e x (Fig. 1). C

? Problem: Prove that this c o n d i t i o n is also sufficient. It is a w e l l - k n o w n p r o b l e m b u t I do not k n o w its origin. The following solution w a s suggested b y M a x i m Kontsevieh. C o n s i d e r six c o p i e s of each triangle face o f t h e polyhedron; e a c h c o p y carries one of the six p o s s i b l e colorings of t h r e e its vertices. Now w e glue the neighboring triangles together, taking into a c c o u n t the coloring: t w o c o p i e s are glued only if b o t h c o m m o n v e r t i c e s have the s a m e color. (This p r o c e d u r e m a y e n c o u n t e r difficulties ff d o n e in t h r e e - d i m e n s i o n a l space, since different layers i n t e r s e c t each other.) Let us l o o k w h a t h a p p e n s a r o u n d any vertex. If the n u m b e r of a d j a c e n t triangles is odd, we get three t w o - s h e e t coverings ( e a c h similar to the R i e m a n n surface for V~z n e a r the p o i n t z = 0). However, if the n u m b e r is even, w e get a regular covering with six sheets. But

the surface of the p o l y h e d r o n is a topological sphere, so its f u n d a m e n t a l group is trivial, a n d this covering is split into six copies of the polyhedron, e a c h p r o p e r l y colored. That's all! F o r the torus, the situation is different: s o m e triangulations allow 3-colorings w h e r e a s others don't. Indeed, if w e c o l o r vertices along a f u n d a m e n t a l cycle, at the end w e m a y get a coloring which is inconsistent with the initial one. In this c a s e the triangulation d o e s n o t allow 3-coloring. And if this d o e s not h a p p e n for either fundamental cycle, a 3-coloring exists.

Tilings and Polyhedra Revisited Since the first 1997 issue of The Mathematical Intelligencer a p p e a r e d , I've r e c e i v e d m a n y c o m m e n t s a b o u t the E n t e r t a i n m e n t s column, a n d s o m e n e e d answering. First, a b o u t the cube a n d tetrahed r o n of equal volume that c a n n o t be split into equal parts. Several readers, including C. Kenneth Fan, P e t e r Freud, J o h n Stillwell, and Wim Veldman, m a d e i m p o r t a n t c o m m e n t s : I did not m e n t i o n that it is a f a m o u s Hilbert p r o b l e m [included in his list of m o s t important mathematical problems (1900)] and was s o l v e d b y M. Dehn (1900). The solution I e x p l a i n e d is due to H a d w i g e r (1948). This information c a n b e found in the survey w r i t t e n by V. G. Boltianskii, w h o also w r o t e a popular b o o k on the s u b j e c t (Hilbert's Third Problem, Washington, 1978). A n o t h e r i m p o r t a n t r e m a r k is a b o u t t h e use of t h e a x i o m o f choice. Let m e quote a letter from Pierre Deligne: . . . the use of Dehn's invariant to s h o w that one c a n n o t go from a cube to a regular t e t r a h e d r o n by cutting and pasting d o e s n o t in fact require the a x i o m o f choice. The arg u m e n t gives a c u t a n d p a s t e invariant in D := ~ | which is 0 for the cube and of the form ~ | a for the tetrahedron, where a is the dihedral angle and ~ r 0. If y o u w a n t e d to s h o w the e x i s t e n c e o f a m o r p h i s m m from D to Q - - o r R - - with m(t? | a) r 0, I e x p e c t you w o u l d indeed n e e d the a x i o m of

choice. However, you only n e e d to p r o v e that g | a r 0. F o r this, it suffices to c h e c k that a ~ 2 ~ / 2 7 r ; 2 . Indeed, ~ | is, essentially b y definition, the inductive limit of t h e X | Y f o r X C ~ and Y C ~U27r;2 finitely g e n e r a t e d subgroups. It h e n c e suffices to c h e c k that for all finitely g e n e r a t e d X and Y such that e E X a n d a E Y, e | a i n X | n o t O. This d o e s not require t h e axiom o f choice! Otherwise said: if a cutting and pasting t r a n s f o r m s the cube into the tetrahedron, it will involve only finitely m a n y lengths a n d angles. Let X and Y be the s u b g r o u p of ~ (resp., R/;2) g e n e r a t e d b y t h o s e lengths (resp., angles). Then Dehn's p r o o f gives ~ | a = 0 in X | Y. 9 The a x i o m of choice is n o t used, b u t a s I p r e s e n t e d the a r g u m e n t , tertium non datur is: to c l a i m t h a t a finitely g e n e r a t e d s u b g r o u p X of is i s o m o r p h i c to ;2n for s o m e n, o r f o r t h e resulting c l a i m that, if eEX, t h e r e is m: X---)7/ with m ( ~ ) r 0. This m a k e s t h e p r o o f n o n c o n s t r u c t i v e : as often, it is a use o f tertium non datur, r a t h e r t h a n a u s e o f the a x i o m o f choice, w h i c h t h r e a t e n s constructivity. Here, t h e r e m e d y is simple: d e f i n e X n o t as a s u b g r o u p o f ~ b u t a s abs t r a c t l y g e n e r a t e d b y the s e g m e n t s o c c u r r i n g in a cutting a n d pasting, a n d r e l a t e d b y the r e l a t i o n s a m o n g length o f s e g m e n t s n e e d e d . Otherw i s e said: t h e invariant lives in the i n d u c t i v e limit o f t h e X | Y, f o r X (resp., Y) in t h e filtering c a t e g o r y o f ;2-modules effectively finitely p r e s e n t e d ( c o k e r n e l of s o m e m a p 7/p ---) 7/q), given with a m a p to (resp., ~/7/). N o w a b o u t the tilings p r o b l e m (a rectangle that can be tiled b y rectangles e a c h having at least one integer side, has an integer side). It is a s h a m e that I didn't k n o w a b o u t the e x c e l l e n t article b y Stan Wagon " F o u r t e e n p r o o f s o f a result a b o u t tiling a rectangle" [American Mathematical Monthly 94 (1987), 601-617]. Several r e a d e r s t o l d m e of this article, a n d I strongly r e c o m m e n d it for the nice p r o o f s e x p l a i n e d in a very c l e a r way.

(Some o f t h e m w e r e i n d e p e n d e n t l y found by c o l u m n readers!) The a l r e a d y cited letter from P. Deligne gives one more: . . . A similar r e m a r k applies to the semi-integral r e c t a n g l e s you consider in the first p a r t of the article. Here, to a r e c t a n g l e A with sides parallel to the axes: A = [x0, xl] • [Y0, Yl], one a t t a c h e s

c(A) = (~(xl) - 6(x0)) | (8(yl) - ~(Y0)) in 77(~/~) | 77(n/~). This invariant h a s the virtues t h a t (i) it vanishes if x l - x0 o r Yl - Y0 is integral, i.e., if A is semi-integral, (ii) if a rectangle A is d i s s e c t e d into rectangles Ai, then c(A) = Z c(A~). In addition, the c o n v e r s e of (i) is true. As before, it suffices to c h e c k this for D~/?/ rep l a c e d b y a finite subset, in w h i c h case it is trivial. Nothing m o r e is required for y o u r proof. By the way, this v e r s i o n o f the argument w o r k s as well with 7/ rep l a c e d b y any s u b g r o u p o f E (and a different s u b g r o u p for e a c h axis is allowed). Let m e r e p e a t this nice p r o o f without mentioning t e n s o r p r o d u c t s explicitly. F o r any rectangle R c o n s i d e r its "imprint" i(R), defined as a formal linear c o m b i n a t i o n o f its four vertices: two o p p o s i t e c o r n e r s ( n o r t h e a s t a n d s o u t h w e s t ) are t a k e n with plus signs, the two o t h e r s with m i n u s signs. If a rectangle R is tiled b y s m a l l e r rectangles R1, 9 9 9 Rn, then

i(R) = i(R1) + --- + i(Rn). This can be s e e n b y looking at all possible j u n c t i o n types:

_l§

+

+

-

+

+

-

-

Now we identify p o i n t s w h o s e coordin a t e s differ by integers

(x, y) -- (x + m, y + n) for any integers m a n d n. After that, the imprint of any semi-integer rectan-

VOLUME 19, NUMBER 4, 1997

49

gle b e c o m e s zero (.plus vertices are c a n c e l e d o u t b y minus vertices). Therefore, if all R i are semi-integer rectangles, their imprints vanish a n d the imprint o f the rectangle R should vanish also. But this m e a n s that it is a semi-integer rectangle, too. (End of proof.) Let me m e n t i o n t w o o t h e r p r o b l e m s c o n n e c t e d with tilings. The first is well known: if a unit square is tiled b y squares, their sides are rationals. (The solution I k n o w involves linear algebra and even a bit of m a t h e m a t i c a l logic!) The o t h e r w a s s u b m i t t e d b y David Gale: Another Euler-type equation. In a t i l i n g of a r e c t a n g l e by rectangles, the horizontal (vertical) edges of the tiles are p a r t i t i o n e d into disjoint horizontal (vertical) s e g m e n t s , as s h o w n below. A c r o s s is a v e r t e x which is c o m m o n to four tiles:

5 Horizontal Segments

10 Tiles

() ()

2 Crosses

6 Vertical S e g m e n t s Prove that in a n y s u c h tiling #Segments - #Tiles + #Crosses = 3.

50

THE MATHEMATICAL INTELLIGENCER

H.S.M. COXETER

Numerical Distances Among the Spheres in a Loxodromic Sequence

C o n s i d e r t w o s p h e r e s with radii a and b, and a d i s t a n c e c b e t w e e n their centres. The n u m b e r

On the o t h e r hand, the n u m e r i c a l d i s t a n c e s D n ( b e t w e e n s p h e r e s 0 a n d n, o r m and m + n for a n y m ) not only could be b u t will turn out in every c a s e to b e

D = (c 2 - a 2 - b2)/2ab m a y conveniently be called t h e i r n u m e r i c a l distance, f f t h e s p h e r e s intersect, D is obviously the cosine of one of their t w o s u p p l e m e n t a r y angles of intersection; therefore, IDI is an inversive invariant, even if the spheres do n o t intersect. If t h e y coincide, D = - 1 ; if t h e y are orthogonal, D = 0; if t h e y are externally tangent, D = 1. If five s p h e r e s are mutually tangent, S o d d y ' s e x t e n s i o n of the Descartes Circle T h e o r e m states that the r e c i p r o c a l s en of their radii satisfy the equation

If the s p h e r e s are n u m b e r e d 1, 2, 3, 4, and 5 in a n y order, t h e r e is a No. 6, t a n g e n t to the last four, a No. 7, t a n g e n t to 3, 4, 5, 6, and so on. Similarly, there is a No. 0, t a n g e n t to 1, 2, 3, and 4. In fact, t h e r e is a so-called loxodromic sequence o f s p h e r e s in w h i c h every consecutive five are mutually tangent. The a b o v e equation s h o w s that every consecutive s i x satisfy an equation such as e0 + ~5 = ~1 + e2 + ~ + e4. F o r instance, the s e q u e n c e o f "bends" ~ c o u l d b e . . . , 6, 3, 1, 1, 1, 0, 0, 1, 1, 1, 3, 6 , . . 9 or

9 . . , 7, 4, 3, 0, 1, 1, 1, 0, 3, 4, 7 , . . . .

9 .., 49, 25, 13, 7, 5, 1, 1, 1, 1, - 1 , 1, t, 1, 1, 5, 7, 13, 25, 4 9 , . . . The astonishing fact is that t h e s e n u m b e r s satisfy the s a m e infinite s e q u e n c e o f equations such as Do + D5 = Dt + D2 + D3 + D4.

Soddy's Bowl of Integers Sommerville [7], p. 125, s h o w e d that the c o n t e n t V of a 4s i m p l e x with edges a]2, a13, 9 9 9 a45 is given b y

-24(4!)2V 2 =

0

1

1

1

1

1

1

0

a22

a23

a24

a25

1 a21

0

a23

a224 a 225

1 a2

a2

1 a2t

a242 a23

2 1 a51

2 a52

31

32

0

2 a53

2

2

a34

a35

0

a25

a2

0

It follows t h a t this d e t e r m i n a n t is equal to 0 w h e n a12, a 1 3 , . . . , a45 a r e t h e mutual d i s t a n c e s a m o n g five given points in E u c l i d e a n 3-space. In the special case w h e n the five p o i n t s a r e the c e n t r e s of five mutually tangent s p h e r e s having radii rl, r2, r3, r4, and r5, w e have a 1 2 = ?'1 +

r2,

9

.

9

9 1997 SPRINGER-VERLAG NEWYORK, VOLUME 19, NUMBER 4, 1997

41

with the u n d e r s t a n d i n g that r n is negative if the n t h circle encloses the others. Hence, 0

1

1

1

0

( r I + r2) 2

1 (r2 + rl) 2

1

1

1

(rl + rs) 2 (r~ + rt) 2 (q + rs) 2

0

(r2 + r3) 2

( r 2 + r4) 2

(r2 + r5) 2

0= 1 (r3 + rl) 2 (r3 + r2)2

(r3 + q)2 (r3 + rs) 2

o

1 @4 + r l ) 2 (r4 + r2) 2 (r4 + r3) 2

0

@4 + rs) 2

1 (r5 + rl) 2 (r5 + r2) 2 ( r 5 ~- r3) 2 (r5 + r4)2

= 16

0

1

1

1

1

1

1

-r21

rlr2

fir3

rtr4

rlr5

1

r2rl

-~2

r2r3

r2r4

r2r5

1

r3rl

r3r2

-r~

r3r4

r3r5

1

r4rl

r4r2

r4r3

--r 2

r4r5

1

r5rl

r5r2

r5r3

r5r4

-r25

0

=

0

16(rlr2r3r4r5) 2

51

52

53

54

55

51 - 1

1

1

1

1

52

1

-1

1

1

1

53

1

1

-1

1

1

54

1

1

1

-1

1

55

1

1

1

1

-1

Figure 1. Soddy's Bowl of Integers.

is said to be loxodromic if every consecutive five of them are mutually tangent. Then their bends en (the reciprocals of their radii) satisfy not only Eq. (1) but also

where 51 = l / r 1 , . . . , 55 = 1/r5. As Pedoe r e m a r k e d ([5], p. 630), in c o n n e c t i o n with the analogous result for circles in a plane, this determinant, being quadratic a n d symmetric in the ds, m u s t be equal to A ( ~ 4- "'" 4- 4 ) 4- B(51 4- "'" 4- 55)2

53 = 54 = 55 = 1,

SO that 3A + 9B = 0, and we have proved Frederick Soddy's three-dimensional counterpart of the Desargues Circle Theorem: I f f i v e spheres are m u t u a l l y tangent, the reciprocals 51, 99 9 55 o f their r a d i i s a t i s f y the equation 3 ( 4 4- d 4- 4 4- 4 4- 4 ) ---- (51 4- 52 4- 53 4- 54 4- 55)2.

(1)

Soddy ([6], p. 78) illustrated this result by placing a large n u m b e r of metal balls (of radii ~, ~, 1 1 etc.) in a h e m i 5 ' 11' spherical bowl, as in Figure 1, which shows the equators of the balls at the top of the bowl. The n u m b e r s are the reciprocals of the radii. The dotted line shows the sphere of radius 89that lies below the plane of the paper.

Loxodromic Sequences of Tangent Spheres A doubly infinite s e q u e n c e of spheres ...,

-2,

- 1 , 0, 1, 2, . . .

42 THEMATHEMATICALINTELLIGENCER

whence, by subtraction, (50 - 55)(350 + 355 - 50 - 251 - 252 - 253 - 254 - 55) = 0, so that

for certain c o n s t a n t s A and B. In the case w h e n the spheres reduce to a pair of parallel planes with three unit spheres sandwiched b e t w e e n them, we have 51 ---- 52 = 0,

3 ( 4 + 4 + 4 + ~ + 4 ) = (5o + O + 52 + 53 + 54)2

50+ 55 ---- 51 -4- 5 2 4 - 5 3 + 54

a n d f'mally 5.m

4- 5 m + 5 = 5 m + 1 4- 5 m + 2 4- e r a + 3 4- 5 m + 4

(2)

for all integers m. F r o m the balls in Soddy's Bowl we can select a s u b s e t forming a loxodromic s e q u e n c e with b e n d s 50 = - 1 ,

eI ---- e - 1 ---- 2, 52 = 5-2 = 53 = 5-3 = 3,

54 = 5-4 = 5, . . . .

In fact, Eq. (2) makes it easy to c o n t i n u e the sequence of b e n d s thus: . . . , 155, 81, 42, 23, 14, 5, 3, 3, 2, - 1 , 2, 3, 3, 5, 14, 23, 42, 81, 1 5 5 , . . . To obtain an explicit formula for en, we m u s t solve the linear difference equation (2) with the initial conditions 50 = - 1 ,

51 =

2,

52 = 3,

e- n =

5~.

(3)

Setting en = x - n in the linear equation 50-- 51-- 52--

53--

544- 55=

0,

w e obtain the quintic equation

x5-x 4-x 3-x 2-x+l=0

(4)

hyperbolic cosines. Accordingly, it s e e m s desirable to define

or

(x + 1){x 2 - (~/2 + 1)x + 1}{x2 + (~/2 - 1)x + 1} = O,

the n u m e r i c a l d i s t a n c e b e t w e e n any two spheres to be D = (c 2 - a 2 - b2)/2ab,

w h o s e r o o t s are (say) xt = -1,

x2 = 1 ( k / 2 + 1 + % / 2 V 2 - 1), x 3 = 1 ( ~ + 1 - ~ -

x4 = +(-~r

+ i + i ~ +

1),

1),

Xs=-

--2(-~/2+!-

i 2~/2"~+1).

More conveniently, w e m a y write X1 =

--1,

X2 = r,

x 3 = r -1,

x4 = c o s p + i sin p,

x5 = c o s p - i sin p,

where r = 1(~/-~ + 1 + ~/2~v/2 - 1)

and cos

= 89

-

(5)

We c o n s i d e r the tentative solution en = Ax*{ + B ( x ~ + x~) + c(xn4 + x~) = A ( - 1) n + B ( r n + r -n) + 2C c o s np,

w h e r e the u n k n o w n c o n s t a n t s A, B, and C are to b e det e r m i n e d by the initial c o n d i t i o n s (3). Thus, - 1 = e0 = A + 2B + 2C, 2 = e l = - A + B ( r + r -1) + 2C cos p = - A + (~v/-2 + 1 ) B - ( ~ / 2 - 1)C, 3=e2 = A+B(r 2+r -2)+2Ccos2p = A + ( 2 V 2 + 1 ) B - ( 2 % / 2 - 1)C. Eliminating A, w e find (~f12 + 3)B - (~r (2~r - 1)B - ( 2 ~

- 3)C = 1, + 1)C = 4,

whence 13

7B- 2V~

I,

7C-

13

2V~

i

w h e r e a a n d b are their radii a n d c is t h e distance b e t w e e n their centres. When the s p h e r e s intersect, D is the cosine of one of t h e i r angles of intersection; therefore, IDr is invariant for the group g e n e r a t e d b y all inversions. When the two s p h e r e s coincide, we have a = b a n d c = 0, so D = - 1. F o r t a n g e n t spheres, c = la -+ b I and, therefore, D = _ 1. F o r disjoint spheres, D is the h y p e r b o l i c cosine of their inversive d i s t a n c e ([3], pp. 90, 429, [4], pp. 123-125).

A Self-similar Sequence of Spheres The quintic equation (4) suggests the construction of a special l o x o d r o m i c sequence of spheres w h o s e radii are in geometric progression. The root r is the ratio of this progression. Notice that Eq. (4) is a "reciprocal" equation w h o s e coefficients are palindromic, so instead of radii we could just as easily say that the "bends" are in geometric progression. The s p h e r e s 0, 1, 2 , . . . are t r a n s f o r m e d into 1, 2, 3 , . . . by a similarity, or, m o r e precisely, a d i l a t i v e rotation. One d a y in 1995, I d e s c r i b e d such a sequence of balls to the s c u l p t o r J o h n Robinson ([1], p. 63), as a p o s s i b l e s h a p e for a p i e c e o f a b s t r a c t art. During t h e n e x t year, he pursued this n o t i o n a n d m a d e the s c u l p t u r e F i r m a m e n t consisting of five w o o d e n balls, all t o u c h i n g one a n o t h e r a n d having radii p r o p o r t i o n a l to r - 2 , r -1, 1, r , r 2,

w h e r e r = 89 + 1 + ~r - 1) ~ 1.88 . . . . On February 9, 1997, as r e q u e s t e d by Robinson, his friend, P r o f e s s o r Ronnie B r o w n b r o u g h t this w o r k of art to m e at the Fields Institute as a p r e s e n t for m y 90th birthday. E x a m i n i n g F i r m a m e n t , one m a y o b s e r v e that the centres o f baUs 1, 2, 3, 4, and 5 lie alternately on o n e side and

7A=-3.

a n d finally We have thus found that 7en = ( - 1 ) n + 1 3 +

( _2 V13 _ ~

_ 1)(r n+r

-n)

-2[~2~v/~13 + 1 ) c o s np. This r a t h e r c o m p l i c a t e d result can be c h e c k e d b y observing that it implies e3 = - A + (4 + 2 \ / 2 ) B + ~ J - 2kf2)C = - A + 4(B + C) + 2 V 2 ( B - C) _3 7

8+26=3 7 7

.

Numerical Distances The notion of i n v e r s i v e d i s t a n c e ([2], p. 118), can b e ext e n d e d immediately from circles in the plane to s p h e r e s in three (or more) dimensions. Again, as in the two-dimensional case, the inversive distances b e t w e e n disjoint s p h e r e s in a loxodromic sequence are found to have integers for their

Firmament, b y

John

Robinson.

VOLUME 19, NUMBER 4, 1997 4 3

which apply to every loxodromic sequence because numerical distance is an inversive invariant. The results are collected in Table 1. Thus, the sequence of numerical distances includes 49, 25, 13, 7, 5, 1, 1, 1, 1, - 1 , 1, 1, 1, 1, 5, 7, 13, 25, 49. T h e Astonishing A n a l o g y With sufficient hard labour, we could continue this geometric investigation indefinitely; but there is no need to do so because we possess already 14 instances of the recursion formula

Oa

o_1

O-a

Dm + Dm+5 = Dm+l 4- Dm+2 4- Dm+3 4- Dm+4

(6)

and it is inconceivable that the 15th instance (D10 = 89) could fail. Clearly, this is the same linear difference equation as Eq. (2), but the initial values (3) have been changed to Do = - 1 , 04

o

o 2

-2

A L o x o d r o m i c S e q u e n c e Including T w o Parallel P l a n e s Suppose we invert a loxodromic sequence with respect to a sphere whose centre is the point of contact of two neighbouring spheres, say numbers 0 and 1. In the resulting new sequence, "spheres" 0 and 1 are a pair of parallel planes, whereas 2, 3, 4, - 1 , - 2 , and - 3 are congruent spheres sandwiched between them, as in Figure 2, where planes 1 and 0 are above and below, respectively. The radius o f the inverting sphere can be adjusted so that the six congruent spheres all have radius 1 and the sequence of ds is, by Eq. (2),

= 02

=

1,

D-

(7)

n = D n.

Still defining r and p as in Eq. (5), we consider the tentative solution Dn = A(1)

Figure 2 (top). The sandwich. Figure 3 (bottom). The centres.

the other of a certain plane C. Accordingly, the dilative rotation, whose axis is perpendicular to C, is an "opposite" (i.e., sense-reversing) similarity: the ratio of the dilatation is not simply r but - r .

D1

n + B(r n + r -n) +

2C cos np,

where the three linear equations to be solved for the three u n k n o w n s A, B, and C are n o w A + 2B 4- 2 C = - 1 - A + ( V ~ + 1)B - (X/~ - 1)C = t, A + ( 2 V 2 + 1)B - (2X/-2 - 1)C = 1. Eliminating A, we obtain (3 + V ~ ) B + (a - x / ~ ) c = 0, ( 2 ~ / 2 - 1)B - ( 2 ~ / 2 + t ) C = 2,

whence FT

=

1,

and finally

7 c = -3

-1

7A = - 3 .

We have thus found that 7Dn = (-1)n+13 + ( 3 / ~ - 1)(r n + r -n) - 2(3./1 + 1) cos up.

. . . , 6, 3, 1, 1, 1, 0, 0, 1, 1, 1, 3, 6 , . . . . The smaller sphere 5, resting on the triad 234, has radius 1/3, because e5 = 3. Similarly, - 4 stands on the lower plane O, below - 1 , - 2 , and - 3 . Figure 3 relates the centres of all these spheres. The six centres 02, 03, 04, O-1, 0-2, and 0 - 3 lie in the midplane between the two parallel planes 0 and 1, while 05 is at height 2/3 above that midplane and 0 - 4 is at the same distance below. Thus, o

and

o4 = 42 +

050 24=32+

+

+

=

=11 .

Although inversion may have changed the gs, this Euclidean investigation will enable us to calculate the D's,

44

THE MATHEMATICALINTELLIGENCER

Table

1. T h e C a l c u l a t i o n of

0 2

(8)

- b2)/2ob

Dn

= (c 2 -

n

Pair of Spheres

a

b

c2

2oh

Dn

0

2&2

1

1

0

--2

2

--1

1

2&3

1

1

4

2

2

1

2

2&4

1

1

4

2

2

1 1

c 2 - a= - b 2

3

--1 & 2

1

1

4

2

2

4

--1&3

1

1

4

2

2

1

5

--1 & 4

1

1

12

10

2

5

6

--2&4

1

1

I6

14

2

7

7

--3 & 4

1

1

28

26

2

13

8

-4 & 4

1/3

1

17 7/9

16 2/3

2/3

25

9

-4&5

1/3

1/3

111/9

108/9

2/9

49

Coordinates for the Centres in the Self-Similar Sequence

yield t h e u s e f u l e q u a t i o n s

A l t h o u g h it is " i n c o n c e i v a b l e " that Eq. (6) s h o u l d fail for s o m e m > 4, n o c o n s c i e n t i o u s m a t h e m a t i c i a n c o u l d feel e n t i r e l y satisfied b e f o r e s e e i n g a p r o o f t h a t is u n i v e r s a l l y valid. Since Eq. (8) i m p l i e s Eq. (7), w e s e e k a g e o m e t r i c p r o o f for Eq. (8). Analogy with the p l a n a r investigation of circles suggests a fresh a p p r o a c h using C a r t e s i a n c o o r d i n a t e s for t h e c e n t r e s On. The t w o - d i m e n s i o n a l a n a l o g u e of Eq. (8) is t h e solution

nn

(72 - X/5 ( x n - x -n) =

2X/5

72 cos n/3 -

3 + 2~f2 = (O10_1) 2 = A2(r - r - l ) 2 cos2p + A2(r + r - l ) 2 sin2p 4- /z2(r -- r - l ) 2 = A2(r2 + r -2) - 2A2 cos 2p + / ~ 2 ( r 2 + r -2 - 2) A2(2V2 + 1 - 1 + 2 V ~ ) +/z2(2%/2 + 1 - 2) = 4X/'2 h 2 + (2%/'2 - 1)/~2, 9 + 4~f2 = ( 0 2 0 - 2 ) 2 = A2(r 2 - r - 2 ) 2 cos22p + A2(r 2 4- r - 2 ) 2 sin22p + / ~ 2 ( r 2 - r - 2 ) 2 = A2(r 4 + r -4) -- 2A2 cos 4p 4- /~2(r4 + r -4 -- 2) = A2(4V2 + 7 - 7 + 4Xf12) +/x2(4X/-2 + 7 - 2) = 8 V ~ h 2 + ( 4 V 2 + 5)/x2

X/5

a n d their s o l u t i o n

([2], p. 117), of the linear difference equation h 2 = (3X/-2 + 2)/7,

Dm + Din+4 = 2(Dm+l + Din+2 + Dm+3),

/x2 = 3/7.

(11)

(9) We c o n c l u d e that t h e c e n t r e On of t h e n t h s p h e r e is g i v e n b y Eqs. (10) a n d (11), a n d s i n c e

w h i c h r e s e m b l e s the e q u a t i o n ~m 4- ~m+4 = 2 ( e m + l 4- Era+2 4- em+3)

A2//x2 = X/2 + 2,

r e l a t i n g t h e b e n d s of five c o n s e c u t i v e m e m b e r s of a loxod r o m i c s e q u e n c e of circles ([2], p. 112). This a s t o n i s h i n g a n a l o g y w a s cleverly n o t i c e d b y E.J. B a r b e a u w h e n h e s a w t h e list

the c e n t r e s of all t h e infinitely m a n y s p h e r e s circular cone

lie

o n the right

x 2 + y2 = ( V 2 + ~.2)z2.

1, 1, 1, 7, 17, 49, 145, 415 ([2], pp. 116-118), o f v a l u e s of D n = c o s h (tin), d u r i n g a l e c t u r e o n I n v e r s i v e D i s t a n c e a n d the Fibonacci N u m b e r s i n o u r G e o m e t r y S e m i n a r . It was, in fact, this o b s e r v a t i o n t h a t i n s p i r e d the p r e s e n t article. T h e "fresh a p p r o a c h " b e g i n s w i t h the dilative r o t a t i o n b y w h i c h t h e s p h e r e s 0, 1, 2 , . . . are t r a n s f o r m e d i n t o 1, 2, 3 , . . . . E x a m i n a t i o n of R o b i n s o n ' s s c u l p t u r e s u g g e s t s t h a t t h e a n g l e of r o t a t i o n is ~r + p, a n d the ratio of d i l a t a t i o n is - r . Thus, for all v a l u e s o f n, t h e c e n t r e O n of s p h e r e n (of r a d i u s r n ) m u s t b e of t h e f o r m

( A ( - r ) n cos n(~r + p), A ( - r ) n sin n(Tr 4- p), i z ( - r ) n)

(10)

for s u i t a b l e v a l u e s o f t h e coefficients h a n d / x . Since the five s p h e r e s - 2 , - 1 , 0, 1, a n d 2 are m u t u a l l y tangent, the central distances include O 1 0 - 1 ---- r + r -1 = ~ 4- 1, O20-2=r 2+r -2=2V2+1. Since also r - r -1 = ~

K n o w i n g t h e radii a n d c e n t r e s of all t h e spheres, w e c a n fmd the numerical distance

Dn = (c 2 - a 2 - b2)/2ab b e t w e e n s p h e r e s 0 a n d n, w h o s e radii a a n d b are 1 a n d r n, respectively, a n d w h o s e c e n t r e s a r e (h, 0,/x) a n d

()t(--r) n c o s n(~r + p), ~ ( - r ) n S i n n(~r + p), t~(-r)n), respectively. Thus, c 2 = h 2 ( ( - r ) n cos n(Tr + p) - 1) 2 4- ~ 2 ( ( - - r ) n Sin n(~- + p))2 + ~ t 2 ( ( _ r ) n _ 1)2 = ) ~ 2 ( r 2 n - - 2 ( - r ) n cos n ( I r + p) + 1) 4- /.t2(r 2n -- 2 ( - r ) n 4- 1) = (A 2 4- /~2)(r2n 4- 1) - 2A2(--r) n cos n(~" 4- p) -- 2 / x 2 ( - r ) n, c 2 -- a 2 - b 2 = (A2 + / x 2 - 1 ) ( r en + 1) - 2(A 2 c o s n(~r + p) + / x 2 ) ( - r ) n

-- 1,

r 2 -- r -2 = (~f2 + 1 ) ~ / 2 ~ / 2 -- 1, r 4 + r - 4 = 4Xf12 + 7, 2 cos p = 1 -- 2 V 2

The Astonishing Analogy Explained

and

2 cos 4p = 7 - 4X/-2,

and, s i n c e ab

= r n,

2 0 n = (~2 4- /.62 __ 1 ) ( r n 4_

3Nil2 - 2 7

r-n)

_

2A2 c o s n p -

(r n + r_n) _ 2 3~/2 +2 7

cos n p -

2/x2( - 1) n 6 ( - 1 ) n -~;

the coordinates that is, 7Dn = O1 = O-1 = 02 = 0-2 =

( h r cos p, - A r sin p, - / x r ) , (2tr -1 cos p, h r -1 sin p, - / z r - 1 ) , (Ar 2 cos 2p, - A r 2 sin 2p,/~r2), (Ar -2 c o s 2p, Ar -2 sin 2 p , / x r -2)

( - - 1 ) n + 1 3 4- ( 3 / ~ -- 1 ) ( r n 4- r - n ) -- ( 3 f ~ 4- 1) c o s up, in p e r f e c t a g r e e m e n t with Eq. (8) for all v a l u e s of n.

VOLUME 19, NUMBER 4, 1997

45

The Ten Points of Contact of Five Mutually Tangent Spheres Although we have used Cartesian coordinates, which belong to Euclidean geometry, to establish Eqs. (7) and (8), it is important to understand that these are theorems of the wider inversive geometry: the loxodromic sequence of spheres is not necessarily self-similar. In the same spirit, it is justifiable to employ coordinates to obtain new inversively-invariant properties of points of contact. The point of contact of spheres 0 and I must divide the straight segment O001 in the ratio r: 1. Similarly, the point of contact of spheres 0 and 3 divides the segment 0003 in the ratio r 3 : 1. Referring to the coordinates (10), including O0 = (& 0,/~)

and 03 = (At3 cos 3p, - h r 3 sin 3p, -/zr3),

we conclude that, for every m, the points of contact of the pairs of "adjacent" spheres, m and m 4- 1, and likewise the points of contact of the pairs m and m + 3, all lie on the horizontal plane Z----0.

It follows that for any loxodromic sequence of spheres (not necessarily serf-similar), these two infmite sequences of points of contact lie on a sphere (possibly reducing to a plane). Asia Weiss ([8], p. 249), called this "the fixed sphere C."

Recalling that any pentad of mutually tangent spheres can be extended (forward and backward) to an infinite loxodromic sequence, we see that for such a pentad 12345, the six points of contact

1 .2,

2.3,

3"4,

4"5,

1 "4,

and

2.5

all lie on a sphere. Moreover, the numbering in the pentad being artibrary, we have incidentally proved the following theorem of inversive geometry:

For any two of five mutually tangent spheres, their six points of contact with the remaining three spheres all lie on a sphere. Although it is practically impossible to visualize this result in Robinson's Firmament or Soddy's Bowl, we can easily do so in the "sandwich" (Fig. 2), and still more easily when the first four of the five spheres are congruent (so that the tetrahedron O 1 0 2 0 3 0 4 is regular) while sphere 5 envelops them all. In this case, if the chosen two are 1 and 2, their points of contact with 3, 4, and 5 form the two isosceles bases of an oblique triangular prism, one of whose side-faces is the square having for its vertices the four points

1 .3,

2.3,

2.4,

1 "4.

This oblique prism clearly has a circumsphere. If, on the other hand, the two chosen spheres are 4 and 5, their points of contact with 1, 2, and 3 form two equilateral triangles of different sizes, the top and bottom of a pyramidal frustum.

4,(}

THE MATHEMATICAL INTELLIGENCER

Since the tetrahedron 0 1 0 2 0 3 0 4 is regular, there is .an 1 lth sphere passing through the 6 points of mutual contact of the 4 congruent spheres. These six points are clearly the vertices of a regular octahedron, which obviously has a circumsphere. This yields another inversive theorem:

For any one of five mutually tangent spheres, the six points of mutual contact of the remaining four all lie on a sphere. Applying this to the "sandwich" (Fig. 2), we observe that the six points of mutual contact of 0, 1, 2, and 3 consist of 0-2,

0-3,

3.1,

1 .2,

forming a square, its centre 2.3, and the point at infmity 0.1, all lying in a plane. On the other hand, the six points of mutual contact of 1, 2, 3, and 4 form two equilateral triangles

1 .2,

1 .3,

1 "4

and

3.4,

4"2,

2.3

of different sizes, which are the top and bottom of an irregular octahedron. Combining these two inversive theorems, we obtain our final result:

For any 5 mutually tangent spheres, there are 15 spheres each passing through 6 of the 10 points of contact.

Postscript For the numerical distance Dn between two spheres at sequential distance n in a loxodromic sequence, the expression (8) can be simplified by referring to E.W. Hobson's P/ane Trigonometry (6th ed., Cambridge University Press, 1925, p. 105) for the expansion of 2cos nO in powers of 2cos 0, applying it to 2cos np, where, by (5), 2cos p = 1 - V2: 2 c o s n p = Z v=0 ( - 1 ) v

n

n - vV

(1 -- V 2 ) n - 2v.

The analogous series with e p/replaced by r is

rn + r - n = ~'v=O( - - 1 ) v

n

n--Vv

(1 4-

~)n-

2v.

When these expressions are substituted into (8), we observe that the number ( 3 / ~ - 1)(1 + V ~ ) m - ( 3 / ~ + 1)(1 - ~vz2)m is an even integer, so let us denote it by 2Um. Then, for all positive values of n,

7On = (-1)n+13 4- 2 ~, ( - 1 ) v

n(n v)

v=O

V

Un-2v,

(12)

where un-2v is determined by the familiar difference equation

u m = 2Um 1 4"- Um_ 2

w i t h t h e u n f a m i l i a r i n i t i a l v a l u e s u 0 = - 1, Ul = 2, y i e l d i n g u2=3,

u3=8,

u4=19,...

SpringerNewsMathematics

For instance,

Hans Hahn Gesammelte Abhandlungen / Collected Works

7D4=-3+2(u4-4u2+2u0)=-3+2(19-12-2)=7.

REFERENCES

1. Ronald Brown, Sculptures by John Robinson at the University of Wales, Bangor, Math. Intell. 16(3) (1994), 62-64. 2. H.S.M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae 51 (1996), 104-121. 3. H.S.M. Coxeter, Introduction to Geometry, 2nd ed. New York: Wiley

L. S c h m e t t e r e r ,

Like D e s c a r t e s and Pascal, Hans Hahn (1879-1934) w a s b o t h an e m i n e n t m a t h e m a t i c i a n and a highly influential philosopher. He founded the Vienna Circle and w a s the t e a c h e r of b o t h Kurt Gi3del and Karl Popper. His seminal c o n t r i b u t i o n s to functional analysis and general topology had a huge i m p a c t on the d e v e l o p m e n t of m o d e m analysis. Halm's pass i o n a t e i n t e r e s t in the foundations of mathematics, vividly d e s c r i b e d in Sir Karl P o p p e r ' s f o r e w o r d ( w h i c h b e c a m e his last essay) had a decisive influence upon Kurt GOdel. Like Freud, Musil or SchOnberg, Hahn b e c a m e a pivotal figure in the feverish intellectual climate of Vienna b e t w e e n the t w o wars.

(1969). 4. H.S.M. Coxeter and S. L. Greitzer, Geometry Revisited, Providence, RI: Mathematical Association of America (1975). 5. Daniel Pedoe, On a theorem in geometry, Am. Math. Monthly 74 (1967), 627-640. 6. Frederick Soddy, The bowl of integers and the hexlet, Nature 139 (1937), 77-79. 7. D.M.Y. Sommerville, An Introduction

K. S i g m u n d ( H r s g J e d s )

to the Geometry of n

Dimensions, London: Methuen (1929). 8. Asia Ivi~ Weiss, On Coxeter's Ioxodromic sequences of tangent spheres, in The Geometric Vein (C. Davis, B. GrOnbaum, and F. A. Sherk, eds.), New York: Springer-Verlag (1982).

N o w complete: Bd. 3 / Vol. 3: 1997. XIII, 581 pages. Cloth, US $142.O0.ISBN 3-211-82781-1 In the third volume, Hahn's writings on harmonic analysis, measure and integration, complex analysis and philosophy are collected and commented on by Jean-Pierre Kahane, Heinz Bauer, Ludger Kanp, and Christian Thiel. This volume also contains excerpts of Hahn's letters and accounts by students and colleagues.

Bd. 2 / Vol. 2: 1996. XIII, 545 pages. Cloth, US $130.00.ISBN 3-211-82750-1 The second volume of Hahn's Collected Works deals with functional analysis, real analysis and hydrodynamics. The commentaries are written by Wilhelm Frank, Davis Preiss, and Alfred Kluwick.

Bd. 1 ! Vol. 1: 1995. XII, 511 pages. Cloth, US $130.00.ISBN 3-211-82682-3 The first volume contains Hahn's ground-breaking contributions to functional analysis, the theory of curves, and ordered groups. Commentary by Harro Heuser, Hans Sagen, and Laszlo Fucks. To Order: Call 14~00-SPRINGER (8:30 AM - 5:30 PM ET) or Fax: 201-348-4505. By Mail: Springer-Verlag New York, Inc., P.O. Box 2485, Secaucus, NJ 07096-2485. Please include $3.00 for shipping one book, $1.00 for each additional. (Residents of CA, IL, MA, NJ, NY, PA, TX, VA, or VT, add sales tax. Canadian residents add 7% GST.) Payment may be made by check, purchase order, or major credit card.

~

SpringerWienNewYork Sachsenplatz4-6, P.O.Box 89, A-1201Wien New York,NY i0010, 175 Fifth Avenue HeidelbergerPlatz 3, D-14197Berlin Tokyo 113, 3-13, Hongo 3-chome, Bunkyo-ku

VOLUME19, NUMBER4, 1997 47

LEE SALLOWS

Thc Lost Theorem which I have collected, is extensive. As a l r e a d y noted, 3 • 3 magic squares are the s m a l l e s t and h e n c e simplest types, for w h i c h r e a s o n they are the earliest to a p p e a r in history, a s well as being the m o s t t h o r o u g h l y investigated squares o f all. I n n u m e r a b l e b o o k s and articles o n magic squares begin with a d i s c u s s i o n of 3 x 3 types, the p r o p e r t i e s of w h i c h have long b e e n r e g a r d e d as c o m p l e t e l y understood. Writing in the well k n o w n Mathematical Recreations published in 1930, for instance, Maurice Kraitchik begins b y saying t h a t " T h e t h e o r y o f the squares of the third o r d e r is simple a n d c o m p l e t e . . . " a n d then goes on to p r e s e n t that theory in just t w o p a g e s of text. Yet for all its e x t r e m e simplicity, t h e e l e m e n t a r y corr e s p o n d e n c e with p a r a l l e l o g r a m s t h a t I h a d s t u m b l e d u p o n while w o r k i n g on G a r d n e r ' s p r o b l e m has, to the b e s t of m y k n o w l e d g e , n e v e r p r e v i o u s l y b e e n identified. I feel sure that m a n y r e a d e r s will s h a r e m y incredulity on inspecting t h e t h e o r e m below. They m a y agree w i t h m e t h a t the c o r r e l a t i o n with p a r a l l e l o g r a m s it d e s c r i b e s is so basic t h a t it d e s e r v e s to be r e g a r d e d a s the f u n d a m e n t a l theo r e m o f order-3 m a g i c squares, a n d t h e very first thing t h a t any n e w c o m e r to t h e s u b j e c t s h o u l d learn. H o w then c o u l d s u c h a t h e o r e m have e s c a p e d the a t t e n t i o n o f every res e a r c h e r in t h e field from a n c i e n t t i m e s d o w n to the pres e n t day? The e x p l a n a t i o n lies in t h e o r t h o d o x focus on magic squares using natural numbers. Once o u r attention broadens to include squares that use complex numbers, the familiar integer t y p e s b e c o m e only a special case, preoccup a t i o n with w h i c h has o b s c u r e d the w i d e r picture. Moving b e y o n d this n a r r o w view, w e step into a r e a l m of g r e a t e r

"Almost t h e last w o r d h a s b e e n said on this subject" ---H.E. D u d e n e y on magic squares [1]. A magic square, a s all the w o r l d knows, is a square a r r a y o f n u m b e r s w h o s e s u m in any row, column, or m a i n diagonal is the same. So-called "normal" squares are o n e s in w h i c h the n u m b e r s u s e d are 1,2,3, and so on; b u t o t h e r n u m b e r s m a y be used. Squares using r e p e a t e d entries are d e e m e d trivial. We s a y that a square of size N x N is of ord e r N. Clearly, magic squares of o r d e r 1 l a c k glamor, while a m o m e n t ' s thought s h o w s that a square of o r d e r 2 c a n n o t b e realized using distinct entries. The s m a l l e s t magic squares o f any interest are thus of o r d e r 3. Writing in Quantum r e c e n t l y [2], Martin G a r d n e r off e r e d $100 to anyone able to p r o d u c e a 3 • 3 magic square c o m p o s e d o f any nine distinct square numbers. So far nob o d y has p r o d u c e d a solution, or p r o o f of its impossibility, although a n e a r miss I d i s c o v e r e d is the following: 1272

462

582

22

1132

942

742

822

972

a s p e c i m e n w h o s e rows, columns, and j u s t one o f the t w o m a i n diagonals s u m to the s a m e number, itself a square: 1472. ~ It w a s while tinkering in c o n n e c t i o n with this p r o b l e m that I was s t a r t l e d to d i s c o v e r an e l e m e n t a r y corres p o n d e n c e b e t w e e n 3 x 3 magic squares a n d parallelograms. The r e a s o n for m y surprise is w o r t h explaining. Magic squares have b e e n a special h o b b y o f m i n e for o v e r t w e n t y years; the literature on the topic, m u c h of

1The magic square nearest to satisfying Gardner's condition to date is due to Michael Schweitzer, and contains just six squares (*) 37629182553169* 132058870105969* 32021959864489

61629448152529 6723667084120g* 72843893529889

102451381817929* 2414471576449* 96844159129249*

By the way, there is a milder version of Gardner's challenge. In a "magic hourglass" like 5 2

1 4 7

6 3

the 2 rows, 2 diagonals, and single column all sum to the same number. It is amazing that had Gardner offered $100 for a magic hourglass using 7 distinct squares, his money would still be in his pocket.

9 1997 SPRINGER-VERLAGNEW YORK, VOLUME 19, NUMBER4, 1997 51

clarity and h a r m o n y . And at the very c e n t e r of that realm we shall find an u n e x p e c t e d prize, the a t o m i c magic square.

Standard Theory What m a k e s a 3 x 3 square magic? The w e l l - k n o w n algebraic f o r m u l a due to l~douard Lucas d e s c r i b e s the structure of every magic square of order 3: c-b

c+a+b

c-a+b

c

c+a

c-a-b

c-a

c+a-b c+b

Lucas's f o r m u l a c o n v e y s m u c h of the e s s e n t i a l information in a single swoop. In particular, we can s e e at a glance that the c o n s t a n t total, w h i c h is 3c, is equal to t h r e e times the c e n t e r number. A c l o s e r l o o k s h o w s t h a t w h a t e v e r the nine n u m b e r s u s e d in the square, they m u s t a l w a y s include eight 3-term arithmetic progressions, namely: 1: 2: 3: 4: 5: 6: 7: 8:

c + a , c, c§ c, c+a+b, c§ c+a-b, c-a+b, c+a+b, c+a-b,

c-a, c-b, c, c - a - b , c, c - a + b , c+a, c+a+b, c-a, c-a-b, c+b, c-a+b, c-b, c-a-b.

The identification of these arithmetic t r i a d s is a r e c u r r e n t feature in d i s c u s s i o n s of order-3 theory, a p o i n t we shall return to later, although it is rare to find an explicit list of all eight. Of course, j u s t as any magic square can b e r o t a t e d and reflected to r e s u l t in 8 trivially distinct squares that are d e e m e d equivalent, s o t h e r e are 8 trivially distinct rotations and reflections o f the formula, all o f t h e m i s o m o r p h i c to e a c h other, a n d again comprising one equivalence class. So m u c h for a bird's-eye view of the t h e o r y of order-3 magic squares a s it is m e t within the literature. Let us n o w turn our a t t e n t i o n elsewhere. Complex Squares C o n s i d e r F i g u r e 1, which d e p i c t s an a r b i t r a r y parallelogram, PQRS, c e n t e r e d at s o m e a r b i t r a r y point, M, on the Euclidean plane, with axes X and Y. Point O is the origin of the plane. The c o r n e r points, P, Q, R, and S, t o g e t h e r with T, U, V, a n d W, the m i d p o i n t s o f t h e s i d e s o f the parallelogram, as well as the center, M, c a n thus be identified with v e c t o r s o r c o m p l e x n u m b e r s of the form x + yi, in which x and y a r e the real n u m b e r c o o r d i n a t e s of each point, and i = ~ - - 1. Equally, the lines connecting t h e s e p o i n t s m a y themselves be i n t e r p r e t e d as vectors, three o f w h i c h are identified in the Figure: M T = a, M U = b, and O M = c. Note that given any t h r e e p a r t i c u l a r c o m p l e x v a l u e s for a, b, and c,

52

THE MATHEMATICALINTELLIGENCER

w e could i m m e d i a t e l y p r o c e e d to c o n s t r u c t the corres p o n d i n g parallelogram. N o w by the law for the a d d i t i o n of vectors, the p o i n t or c o m p l e x n u m b e r T (which is the v e c t o r O T ) is the result a n t o f the two vectors c a n d a, o r c + a. And likewise, it t a k e s b u t a glance to identify t h e remaining points on P Q R S in t e r m s of the vectors, a, b, a n d c, as i n d i c a t e d in the Figure: P = c + a - b, Q = c + a + b , R = c - a + b, S = c-a-b, U=c+b, V=c-a, W=c-b, a n d M = c. Looking n e x t at the 3 x 3 square s h o w n b e l o w left in Figure I, observe w h a t h a p p e n s w h e n its entries a r e rep l a c e d with their c o r r e s p o n d i n g e x p r e s s i o n s in t e r m s o f a, b, a n d c: c-b c-a+b c+a

c+a+b c c-a-b

c-a

c+a-b c+b

The o u t c o m e is nothing less t h a n a r e a p p e a r a n c e of Lucas's f o r m u l a for m a g i c squares o f o r d e r 3. The implication is as o b v i o u s as it is surprising; given any p a r t i c u l a r p a r a l l e l o g r a m on the Euclidean plane, a n d t h e n transcribing the c o m p l e x n u m b e r s c o r r e s p o n d i n g to its f o u r corners, four edge midpoints, a n d center, into a 3 • 3 matrix, in the s a m e w a y as above, the resulting square will a l w a y s be magic. Or alternatively, starting with any 3 • 3 magic square that u s e s c o m p l e x n u m b e r entries, w e will find that they define nine p o i n t s on the E u c l i d e a n p l a n e t h a t coincide with the four corners, four edge midpoints, a n d c e n t e r of a parallelogram. Figure 2 furnishes a conc r e t e example. Writing t h e s e values into a 3 • 3 m a t r i x in the s a m e p a t t e r n as before, t h e result is the following c o m p l e x magic square, w h o s e c o n s t a n t s u m is 3 + 6i:

IGURIE

-7+0i

8+10i

2-4i

10-2i

1+2i

-8+6i

0+8i

-6-6i

9+4i

In s u m m a r y , we have: THEOREM. To every parallelogram drawn on the plane

there corresponds a unique equivalence class of 8 complex 3 x 3 magic squares, and to every equivalence class of 8 complex 3 x 3 magic squares there corresponds a unique parallelogram on the plane. Or in a nutshell: r o t a t i o n s and reflections disregarded, e v e r y p a r a l l e l o g r a m d e f m e s a unique 3 x 3 magic square, a n d vice versa. In this light it is interesting to recall the eight a r i t h m e t i c p r o g r e s s i o n s previously identified in every 3 • 3 magic square. F o r just as a r i t h m e t i c p r o g r e s s i o n s of real n u m b e r s c o r r e s p o n d to equidistant p o i n t s along the real n u m b e r line, so arithmetic p r o g r e s s i o n s of c o m p l e x n u m b e r s corr e s p o n d to equidistant colinear points on the plane. See t h e n h o w the eight p r o g r e s s i o n s listed earlier p r e c i s e l y correlate with the eight sets o f 3 colinear p o i n t s lying along the four edges a n d four b i s e c t o r s of the p a r a l l e l o g r a m in Figure 1: 1: 2: 3: 4: 5: 6: 7: 8:

c§ c, c + b , c, c+a+b c§ c+a-b c-a+b c§ c§

c - a = T,M,V c - b = U,M,W c, c - a - b = Q,M,S c, c - a § = P,M,R c + a , c § 2 4 7 = P,T,Q c - a , c - a - b = R,V,S c+b, c-a§ = Q,U,R c - b , c - a - b = P,W,S

In the magic square literature to date, d i s c u s s i o n of theory n e v e r gets further t h a n identifying t h e s e progressions; n o w at last w e c a n see h o w the g e o m e t r y of the parallelog r a m explains their p r e s e n c e . F r o m o u r new p e r s p e c t i v e w e can see also h o w the correlation with p a r a l l e l o g r a m s h a s e s c a p e d p r e v i o u s notice. Traditionally magic squares have u s e d integers, w h i c h are entries without i m a g i n a r y c o m p o n e n t . The p a r a l l e l o g r a m s c o r r e s p o n d i n g to t h e s e s q u a r e s a r e thus c o l l a p s e d o r de-

FIGURE

generate s p e c i m e n s of zero area, m a k i n g their p r e s e n c e undetectable. In fact, a closer l o o k at o n e such p a r a l l e l o g r a m will p r o v e instructive, as well as p r e p a r i n g us for an une x p e c t e d development: the d i s c o v e r y of a lost archetype, the p r i m o r d i a l magic square. & Flaw in the Crystal I s u p p o s e that, until now, the m o s t obvious candidate for the title of a r c h e t y p a l magic square w o u l d have been t h e Chinese Lo shu, the simplest, oldest, a n d m o s t well-known square of all. Its m a g i c sum is 15:

2 7 6

9 5 1

4 3 8

Legend h a s it that Lo shu w a s first e s p i e d b y King Yti on t h e b a c k o f a s a c r e d turtle that e m e r g e d from the river Lo in the 23rd c e n t u r y BC. In fact, historical r e f e r e n c e s to the square d a t e from the 4th c e n t u r y BC; C a m m a n has argued that it p l a y e d a m a j o r p a r t in Chinese p h i l o s o p h i c a l and religious thought for c e n t u r i e s a f t e r w a r d [3]. In the West, the Lo shu has long b e e n held up as a paradigm, o r "one o f the m o s t elegant p a t t e r n s in the history of combinatorial n u m b e r theory", as Martin G a r d n e r has written. Nevertheless, taking a lens to this a n c i e n t gem, w e can disc o v e r an interesting irregularity in its crystal lattice. C o n s i d e r the Lo shu's flattened parallelogram, which is that s e g m e n t o f the real n u m b e r line b e t w e e n 1 and 9, along which lie its four corners, four edge midpoints, and center, occupying nine equidistant points. Recalling Figure 1, the relation b e t w e e n t h e s e p o i n t s and t h e i r p o s i t i o n s in t h e magic square can b e d i a g r a m m e d as follows: 3 6 9 2 5 8 1 4 7 Parallelogram

<=>

2 9 4 7 5 3 6 1 8 Magic Square

The d i s t a n c e b e t w e e n the p a r a l l e l o g r a m ' s corners at p o i n t s 1 a n d 3 (or 7 and 9) is thus 2 units, while the distance b e t w e e n t h o s e at p o i n t s 3 a n d 9 (or 1 and 7) is 6 units; a ratio in side lengths of 1:3. Thus the Lo shu parallelogram is n o t equilateral, w h i c h is a bit disappointing for a p a t t e r n w h o s e famous s y m m e t r y h a s w o n acclaim d o w n the ages, from the b a n k s o f the river Lo to the p a g e s o f Scientific American. It is beginning to l o o k as if that turtle w a s n o t quite as s a c r e d as King Yti had imagined. However, as a p r i s o n e r in Flatland, the Lo shu is d o o m e d to this imbalance: squash any equilateral parallelogram, and two p a i r s o f edge m i d p o i n t s a n d t w o c o r n e r s will coincide, forcing r e p e a t e d entries in the a s s o c i a t e d magic square. In o t h e r words, unless it is trivial, an a s y m m e t r i c a l p a r a l l e l o g r a m m u s t a c c o m p a n y every n o n c o m p l e x magic square, the one on the (mock?) turtle included. Where t h e n is the magic square with the symmetrical p a r a l l e l o g r a m that King Yti w a s denied?

VOLUME 19, NUMBER 4, 1997

53

The Atomic Square Of course the m o s t s y m m e t r i c a l case o f all is an equilateral p a r a l l e l o g r a m that is equiangular as well: the square. A magic square w h o s e a s s o c i a t e d p a r a l l e l o g r a m is again a square; the i d e a is at once compelling. But w h a t kind of a magic square w o u l d that be? To find out, all we have to do is d r a w a square on the plane, r e a d off the c o m p l e x values of its four corners, etc., and then write these into a 3 • 3 matrix in the usual way. Simplest of all is this canonical o r atomic case:

2"

I i

i -X-

-1

1

[

"X

-•

REFERENCES

I f

[1] H.E. Dudeney, Amusement in Mathematics, Dover (1917), New York, p. 119. [2] M. Gardner, The Magic of 3 x 3, Quantum, January/February 1996, pp. 24-6. [3] S. Camman, The Magic Square of Three in Old Chinese Philosophy and Religion, History of Religions 1(1961), pp. 37-80.

-u :IGURI

It is a square c e n t e r e d on the origin o f the plane, such that its four c o r n e r s a n d four edge m i d p o i n t s coincide with the 8 c o m p l e x integers i m m e d i a t e l y s u r r o u n d i n g the origin. The magic square c o r r e s p o n d i n g to this g e o m e t r i c square is consequently an a t o m i c p a r a d i g m o f its kind too: it is the smallest, m o s t p e r f e c t l y s y m m e t r i c a l m a g i c square, comp o s e d of the nine s m a l l e s t gaussian integers: -i

1+i

-1

-1+i

0

1-i

1

-1-i

i

The elegance o f this flawless p r i s m is b e y o n d compare. The two main diagonals and two central o r t h o g o n a l s are like four b a l a n c e d b e a m s pivoted on the c e n t e r number, the integer at t h e e n d of each b e a m offset b y its o p p o s i n g negative image, an equipoise reflected in the magic sum of zero. Rewriting the square in the form of v e c t o r s as o r d e r e d pairs, [a,b], its s t r u c t u r a l h a r m o n y r e a p p e a r s in the s h a p e of p a l i n d r o m i c r o w s a n d antipalindromic columns, a quality that is b e t t e r highlighted w h e n i r e p l a c e s - 1 , and the c o m m a s and b r a c k e t s are discarded: O1

11

TO

11

O0

11

10

11

01

THE MATHEMATICAL INTELLIGENCER

Analysing the square in t e r m s of Lucas's formula, w e find t h a t the variables a a n d b have here taken on the values of i and i, the real a n d imaginary forms of unity, while e is equal to zero. Could anything be m o r e natural, o r poetic? My interest in magic squares b e g a n a couple o f d e c a d e s ago w h e n I first e n c o u n t e r e d t h e Lo shu. I recall m y delight in exploring its s y m m e t r i e s , but I also recall m y disquiet in detecting a strange lopsidedness. In Lucas's formula, the variables a and b a p p e a r in t w o p a t t e r n s that a r e p e r f e c t m i r r o r images. In the Lo shu, however, a = 1, while b -- 3, a n u m e r i c a l i m b a l a n c e that c l a s h e d with the symm e t r y o f the patterns. A t t e m p t s to construct a square in w h i c h a = b, or a = - b , w o u l d n ' t w o r k either, b e c a u s e the result is then trivial. Yet a craving for s y m m e t r y is w h a t m a k e s the m a t h e m a g i c a l m i n d tick. D o w n the y e a r s this u n e a s e has continued to smoulder, until r e c e n t events b r o u g h t the p a r a l l e l o g r a m t h e o r e m to light, and with it the a t o m i c square, w h o s e s y m m e t r y is w i t h o u t flaw. It is a relief; I l o o k f o r w a r d to sleeping at nights once again.

mld,[.~]BL'~-"~q,[:-]|i/":||[--~-llll[o]|ldl~1Llm

Quetelet and Dandelin of Brussels lain Skinner

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have yoi~ encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

lan Stewart,

Editor

I

n Brussels, a city m o r e c e l e b r a t e d for its artistic treasures, t h e r e is a m o n u m e n t to a m a t h e m a t i c i a n - - o n e c e l e b r a t e d in his own time for contrib u t i o n s to r e s e a r c h but n o w s o m e w h a t n e g l e c t e d in o u r current fascination with "original thinkers" to the exclusion of t h o s e w h o s e organizational, political, a n d p r o m o t i o n a l skills m a d e res e a r c h possible. The m a t h e m a t i c i a n is L a m b e r t A d o l p h e J a c q u e s Quetelet. The m o n u m e n t is his statue in the gardens o f the Royal Academy. This s e e m s a p p r o p r i a t e for a city w h e r e E u r o p e - w i d e collaborative r e s e a r c h is n o w a d m i n i s t e r e d and the m o s t extensive o f statistical returns collated. Quetelet w a s b o r n in Ghent on 22 F e b r u a r y 1796, to a F r e n c h father a n d Walloon mother. He a t t e n d e d and, in turn, t a u g h t at local schools. Ghent's University h a d b e e n f o u n d e d in 1817, and Quetelet w a s a w a r d e d the n e w institution's first PhD for a thesis a b o u t caustics a n d conic sections. Appointm e n t in 1819 as m a t h e m a t i c s t e a c h e r at the restructured Ath~n~e de Bruxelles m e a n t moving to Brussels, w h e r e Quetelet, to his delight, e n t e r e d the society o f e d u c a t e d , cultured F r e n c h unw e l c o m e or u n h a p p y in their h o m e -

I

land. His original inclination had b e e n to train as a painter; he h a d p u b l i s h e d literary works. Born 12 April 1794, Germinal Pierre Dandelin, too, w a s of Franco-Walloon p a r e n t a g e and g r e w up in Ghent. After attending school with Quetelet, he w e n t to Paris to study at the Ecole Polytechnique, j u s t in time to fight with the s t u d e n t v o l u n t e e r s at Vincennes in 1814. Dandelin w a s exiled following Waterloo. In 1817 h e t o o k Dutch citizenship a n d so c o u l d w o r k as a milit a r y engineer. His first t a s k was to rep a i r the m a s s i v e fortress at Namur. In 1816, Dandelin and Quetelet comp l e t e d their first p u b l i c collaboration: writing and p r o d u c i n g an opera. Of this venture, Quetelet l a t e r w r o t e [1, m y translation], "It w o n s o m e praise, b u t t h a t didn't stop Dandelin declaring, aft e r the s e c o n d p e r f o r m a n c e , that he w o u l d be a m o n g t h e first to b o o t h e play if f o r e w a r n e d o f its r e t u r n to the theatre." Of m o r e i m p o r t a n c e to mathematics is that, f r o m 1819, they prod u c e d 17 m a t h e m a t i c a l papers. Use o f caustics e n a b l e d Quetelet to formalise s o m e of H u y g e n s ' s intuitive a r g u m e n t s in optics; his l a t e r p a p e r s included applications to astronomy. Dandelin's in-

The old Observatory building where the Quetelet family lived from 1832.

Please send all submissions to Mathematical Tourist Editor, lan Stewart, Mathematics Institute, University of Warwick, Coventry CV4 7AL England e-mail: [email protected]

9 1997 SPRINGER-VERLAG NEWYORK, VOLUME 19, NUMBER 4, 1997

55

terest extended to the practicalities of calculating solutions to polynomial equations. To determine their roots, he formulated (1825) a method now named for Gr~ffe, who published 12 years later. Primarily, though, these papers were about conics. Chasles observed [2, my translation], "Recently, Messrs Quetelet and Dandelin, in considering conic sections, obtained several very elegant new results." Notable was (under Dandelin's name) the following theorem: If a sphere is tangent to both a cone and a plane cutting that cone, then the point of contact between plane and sphere is a focus of the conic section formed by the plane and cone; the corresponding directrix is the intersection of the cutting plane and the plane defined by the circnmference of contact between cone and sphere.

Quetelet's statue in the gardens of the Royal Academy, of which he was the first Permanent Secretary.

Commissioned by an insurance company and considering births and deaths in Brussels, Quetelet's first statistical publication appeared in 1825. There followed analyses of regional variations in demography, tax, commerce, and crimes in the Netherlands. He showed, for the first time, how the ever-increasing amounts of data collected by post-enlightenment governments could be collated to give useful details about the lives of those governed, and enunciated a desire that, with such knowledge, the study of society might acquire a precision such as physics possessed. S u r l'homme et le ddveloppement de ses facudtds ou Essai de physique sociale appeared in 1835. In it Quetelet introduced his famous idea of the "the average man"; he perceived how human behaviour might be analysed in terms of variations from some average, and how statistics could be used to derive comparisons showing the impact of different influences on people. He noted that such human characteristics as the heights of soldiers showed a normal distribution. The analogy with Gauss's observation about errors posed the question: were human traits departures from an ideal? He insisted that an average is only meaningful for a continuous distribu-

56

THE MATHEMATICAL INTELLIGENCER

tion and is more reliable for larger populations. On a philosophical level, he pondered whether numbers could measure everything; explaining the need for a weight able to measure each publication's intrinsic merit (a factor often neglected by modem successors), he even showed how they could evaluate a writer's productivity. Despite doubts about the mathematical worth of his statistical work, there is no doubting the enormous contemporary impact of this challenge to broaden the use of quantitative analysis. Quetelet was elected to the Royal Academy in 1820; Dandelin in 1822. Quetelet was to serve as Director 1830-31; President 1832-34, while it was being reorganized as a Belgian national institution; and Permanent Secretary from 1834. Dandelin was to be appointed Director of the Science Section of the Academy in 1846. Of the five government education enquiries in which Quetelet participated, the most personally significant was when, on behalf of the Academy, in 1823 he reported the need for an observatory. In time the Government accepted this proposition and, in 1828, appointed Quetelet Director. Visits to other European observatories to pur-

chase the needed new instruments followed, so that he found himself in italy during the revolution of 1830-31. Upon his return to Brussels he saw the incomplete buildings of the Observatory damaged and occupied by the volunteers from Liege, but by 1832, the institution was functional, and its Director no longer involved in mathematical but astronomical, climatic, and geomagnetic research. Given that such investigations are global in scope, Quetelet quickly understood that international collaboration was essential for successful prosecution of research. Given the contacts made while visiting other observatories and as Secretary of the Academy, Quetelet was well placed to organize these. His enthusiasm and persistence ensured they were fruitful. In 1853, Brussels hosted two inaugural conferences: the International Climate Conference and the International Statistics Conference. He presided over both. Indeed, part of Quetelet's interest in statistics was prompted by a concern to standardise methods of comparing measurements made in different places by different people. His choice of friends and innate interest in social sciences meant Quetelet was always keen to educate an audience wider than mathematical specialists. Although declining a chair at the new Universit6 Libre in 1834, he frequently delivered public education courses and wrote popularizing texts. For specialized papers, Quetelet and his former PhD supervisor J.G. Gamier founded (1825) La correspondance mathdmatique et physique, and he also became founding editor of Bulletins de l'Ac ddmie royale de Belgique in 1832. In 1825 Dandelin was appointed Professor of Metallurgy at the new University of Li6ge, a position that freed him to pursue an interest in research. However, in 1830 he voluntarily resamed his military career by enlisting with the rebels, and thereafter supervised repairs to several fortresses. After an 1843 transfer to Brussels, Dandelin resumed research activities until his death on 15 February 1847. Quetelet married C6cile-Virginie Curtet in 1825; they had two children. Their son, Ernst Adolphe Franqois

Quetelet, succeeded his father as Director of the Observatory and established an independent reputation as a scientist. In 1855 Quetelet suffered a stroke that ended his research and he died 17 February 1874. Historical Context What is now Belgium had been annexed by France in 1794. Successive post-revolutionary governments in Paris reformed education, especially the secular tertiary level, and greatly increased the collection of data about every detail of the nation. Following the collapse of the Napoleonic Empire, Belgians were included in the United Netherlands (modern Benelux nations), and Brussels, again the intellectual and administrative centre for the southern provinces, was enlivened by immigrants for whom the reac-

tionary, restored Bourbon administration in France was unsatisfactory. Insurrection began in 1830. Within a year, there was an independent Belgium under Leopold I. He proved a vigorous patron of the sciences, as did his nephew Albert (later Consort of Queen Victoria), who spent 1837 in Brussels with Quetelet as his tutor. H o w to Find It Quetelet's statue is easily found. Starting at Rue Royal/Konings str., walk the full length of the street (Place de Palais) between the Royal Palace's front (on your right) and Brussels Park. The buildings housing the Royal Academy of Belgium are in front of you at the end of the street. The Old Observatory is on Place Quetelet, which is at a corner of the main ring road around Brussels' CBD,

about 300 meters east of porte de Schaerbeek/Schaarbeek poort. Acknowledgement The staff of the service d'optique et d'acoustique, Universit~ Libre de Bruxelles, are thanked for hosting a visit in 1995. REFERENCES 1. Ad. Quetelet, "G.-P. Dandelin", in Biographie Nationale (L'Academie Royale des Sciences, des Lettres & des Beaux-Arts de Belgique, Brussels, 1873), vol 4, col 663 2. M. Chasles, Apergu historique des methodes en gdem6trie (Gauthier-Villar, Paris, 1889), p 286 Department of Communications School of Electrical Engineering The University of New South Wales Sydney 2052, Australia e-mail: [email protected]

I) NEW GEORGE E. MARTIN, State University of NY at Albany

GEOMETRIC CONSTRUCTIONS Geometric constructions have been a popular part of mathematics throughout history. The ancient Greeks made the subject an art, which was enriched by the medieval Arabs, but which required the algebra of the Renaissance for a thorough understanding. Through coordinate geometry, various geometric construction tools can be associated with various fields of real numbers. This book is about these associations. As specified by Plato, the game is played with a ruler and compass. The first chapter is informal and starts from scratch, introducing all the geometric constructions from high school that have been forgotten or were never seen. The second chapter formalizes Plato's game and examines problems from antiquity such as the impossibility of trisecting an arbitrary angle. After that, variations on Plato's theme are explored: using only a ruler, using only a compass, using toothpicks, using a ruler and dividers, using a marked rule, using a tomahawk, and ending with a chapter on geometric constructions by paperfolding. The author writes in a charming style and nicely intersperses history and philosophy within the mathematics. He hopes that readers will learn a little geometry and a Tittle algebra while enjoying the effort. This is as much an algebra book as it is a geometry book. Since all the algebra and all the geometry that are needed is developed within the text, very little mathematical background is required to read this book. The text has been class tested for several semesters with a master's level class for secondary teachers.

Contents: Euclidean Constructions 9 The Ruler and Compass ~ The Compass and the Mohr-Mascheroni Theorem - The Ruler 9 The Ruler and Dividers ~ The Poncelet-Steiner Theorem and Double Rulers 9 The Ruler and Rusty Compass 9 Sticks ~ The Marked Ruler 9 Paperfolding - - The Back of the Book 1997/APP. 160 PP., 130 ILLUS./HARDCOVER/S35.00/ISBN 0-387 98276-O/ONDERGRADUATETEXTS IN MATHEMATICS

8/97 #H209

Re~nce

VOLUME 19, NUMBER 4, 1997

5 ~ t'

I'..=-~.~.,--lw:~4[.--

Jeremy

Gray

Riemann's Lecture Courses on Complex FunctionTheory

I

'hereas Weierstrass has become famous for lecturing to audiences of 200 or so in Berlin, his younger contemporary, Bernhard Riemann, is not remembered for his oratorical skills. His audiences at GSttingen were small, roughly 8 to 10 people, but select (Dedekind was among them), and copies of his lecture notes were treasured. 1 Fuchs reports transcribing a set of Riemann's notes for his own use in 1894 (see Riemann, Werke, p. 723). By the end of the century, his courses had inspired a number of books by others on complex function theory; for example, the first German book on the subject (Dur~ge, 1864), ran to four editions and was translated into English for the American market in 1896. In the absence of up-to-date treatments from the Berlin school, editions of Riemann's lectures began to be published. These were sometimes modemised, as Stahl did with the course on Abelian functions, sometimes much closer to the original, and accompanied by editorial notes, as in Stahl's edition of lectures on elliptic functions. Some copies of handwritten lecture notes survived in GSttingen, where they could be consulted, and even used by others, as Schering (Riemann's successor) did. The recent third edition of Riemann's Werke (Riemann, 1990) lists all the lecture courses Riemann gave (see pp. 712-713), and from them it is possible to build up a picture of what went on, and what a good generation of mathematicians thought it was worth trying to catch up with in the following 40 years.

W

T h e L e c t u r e s from 1 8 5 5 - 1 8 5 6 1861-1862

to

Riemann first lectured on complex function theory in 1855-1856 while working on his great memoir on Abelian

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

58

functions (Riemann, 1857b), and then on function theory and the hypergeometric series in 1856-1857 while working on his paper on the hypergeometric equation (Riemann, 1857a). Thereafter, one or another aspect of the topic was given in most years, alternating with mathematical physics, until his health gave way in 1864. There are various sources for this material. Extracts were published in the first edition of the Werke, and more in the Nachtrgige that accompanied the. second edition. Riemann's student Roch published his version of Riemann's course of 1861-1862, and it was one of the few published sources for Riemann's ideas for quite a number of years (Roch, 1863, 1865). In 1896 and 1899, H. Stahl published his own notes on the topic of the second half, the theory of Abelian functions and the theory of elliptic functions, respectively. Various other sets of notes survive, on the basis of which a detailed version has recently been published. 2 The G6ttingen archive (located at the Staats- und Universit~tsbibliothek, Handschriftenabteilung) contains Schering's 192 pages of notes covering the general theory (Cod.Ms.Riemann 37; see Box). The lecture course of 1861 begins with an account of the geometric meaning of complex quantities and their arithmetic. This was followed by a philosophical passage in which the extension of the number concept from the natural numbers to the integers, the rationals, and, fmally, the complex numbers was defended. The extension proceeds from an intuitive insight into the new quantities via the idea of inverting allowed operations--for example, inverting addition to obtain subtraction. When, asked Riemann, could one imagine the opposite of a quantity? He answered: when what is at stake is

~See Laugwitz (1996), pp. 149-151. Laugwitz's chapter on Riemann's work on complex function theory can be highly recommended. 2See Neuenschwander (1987). See also Neuenschwander (1990) for an account of the lecture notes and other surviving material.

THE MATHEMATICALINTELLIGENCER9 1997 SPRINGER-VERLAGNEW YORK

not a pure quantity b u t a relationship bet w e e n two quantities. This extension of the idea of quantity follows a modification in the meaning of the arithmetical operation. F o r example, c o m p l e x quantities are first thought o f as defining changes of position in the plane. One can then a s k for the conditions under which a meaning can be given to imaginary quantities. Riemann's lectures are obscure at j u s t this point (which may be w h y Roch omitted this passage). 3 He presented both an interpretation of % / ~ as a m e a n proportional between

+1 and - 1 , and an e n d o r s e m e n t of Gauss's views o f 1832, before offering the view that the geometric representation of c o m p l e x quantities, although it established their reality, was m o r e imp o r t a n t for making them intuitive. F r o m there, R i e m a n n t u r n e d to define a c o m p l e x function, w, as o n e which satisfied the equation 8w/Oy = i S w / O x . This e n s u r e d t h a t t h e function w w a s a f u n c t i o n o f the single v a r i a b l e z = x § y i , a n d t h a t its derivative d e p e n d e d only o n t h e p o s i tion o f z b u t n o t on the d i r e c t i o n o f

d z . Conversely, s u c h i n d e p e n d e n c e for a f u n c t i o n g u a r a n t e e d t h a t it w a s truly a c o m p l e x function. S u c h a funct i o n is c o n f o r m a l e x c e p t w h e r e t h e derivative v a n i s h e s , said Roch ( b u t n o t R i e m a n n in 1861), a n d it is a l s o h a r m o n i c , said Riemann, a l t h o u g h he d i d not u s e t h e w o r d . The integral of a c o m p l e x function along a p a t h w a s t h e n defined, and the question raised o f w h e n the integral a r o u n d a c l o s e d p a t h vanished. Riemann a n s w e r e d it by a Green's theo r e m approach, proving that

0x)

Ox

~

d~c g y =

The illustrations for this article are from Cod.Ms.Riemann 37. (Used by permission.)

f (X ~c + gay).

Because a Y / O x - OX/Oy = 0 w h e n X = u and Y = iv, Riemann concluded that the integral of a c o m p l e x function a r o u n d a c l o s e d c o n t o u r vanished, provided that the c o n d i t i o n s of his version o f Green's t h e o r e m a p p l i e d - - i n particular, that the function w w a s everyw h e r e finite and c o n t i n u o u s inside the contour. The e x a m p l e of the log function and the integral

d~/~ m o t i v a t e d an in-

vestigation o f w h e n the integral a r o u n d a closed c o n t o u r d o e s n o t vanish. Riemann s h o w e d that at points a j w h e r e l i m f ( z ) is in_f-mite a s z -+ aj, but lim(z aj)j~z) is finite a n d equal to cj, say, the integral enclosing the p o i n t z = a j once p i c k s up a contribution of 27ricj. F r o m this, h e d e d u c e d t h a t if the function is analytic e v e r y w h e r e in the interior, a n d t is an a r b i t r a r y interior point, then -

f(t) = ~

1 f zf(z) - t

dz,

with the familiar implications for the n t h derivatives: j%n)(t ) -

-

1-2-..n 2~ri -

s j

(z

J(z) _----+1 d z . t) ~

These are the Cauchy integral formulae, which Riemann could have read in s o m e of Cauchy's m e m o i r s and b o o k s he b o r r o w e d from the G6ttingen University Library as a s t u d e n t in 1847. 4 3Instead Roch gave a careful account of the number e (defined as lim(1 + 1/x) x for x--> oo) and the exponential function. 4See Neuenschwander (1981), p. 91.

VOLUME 19, NUMBER4, 1997 59

\

.'

*

0" 9 .

O

"

O -

~ / -

k'

f(t)

4"

,•a.,•r-•

~- o

,,,"

f f"'~( z- d) z I-%

IP. t ~."

The lectures t h e n c o n t i n u e d with an a c c o u n t of h o w a complex function can be developed inside a disc as a convergent p o w e r series, a n d inside an ann u l u s as a c o n v e r g e n t Laurent series. A function c a n n o t vanish along a curve without all the coefficients in a powerseries e x p a n s i o n vmtishing, so a nonzero complex f u n c t i o n c a n n o t vanish

60

f(z)(z - a) m 1 ~

THE MATHEMATICAL INTELLIGENCER

= ~

1 f zf(z) - t

dz

taken o n a circle centred at the origin a p p r o a c h e s a c o n s t a n t value as the radius t e n d s to infinity. The last result in this collection was the f u n d a m e n t a l theorem of algebra, but it required, said Riemann, some preliminary considerations of zeros and poles of t-mite order. These were s u m m e d u p in the t h e o r e m that the expression

:r, _%

5Plainly t h e c o n d i t i o n

given which have the same infiniti#s. In particular, if f ( z ) = ~ w h e n z = a but f ( z ) ( z - a) m does not, t h e n f h a s a pole of order m. 5 Similar definitions were given for a = ~ and for zeros of order m, called by R i e m a n n points where f is infinitely small of order m. So if a function has only finitely m a n y points, including z = ~, where it is inf'mite, t h e n it is a rational function, a n d if it is n e v e r infinite even at z = ~, t h e n it m u s t be a constant. R i e m a n n gave two proofs of Liouville's theorem: one by looking at the power-series expansion, a n d a n o t h e r direct p r o o f by observing that the integral

along a curve. Consequently, any identity b e t w e e n functions that holds for real values of their a r g u m e n t s holds also for complex values. T u r n i n g next to the power-series exp a n s i o n of a single-valued or monodromic function, R i e m a n n showed that such a function was k n o w n up to a c o n s t a n t as soon as functions are

oc a s z --, a is t a c i t l y s u p p o s e d .

= 1

f d l o g f ( z ) dz

2 qri

dz

(*)

evaluated o n the b o u n d a r y of a region is equal to the n u m b e r of zeros m i n u s the n u m b e r of poles inside the region, c o u n t e d according to multiplicity. Two proofs of the f u n d a m e n t a l t h e o r e m of algebra were then given. First, b y observing that a polynomial of order n is infinite to order n at z = % b u t expression (*) is zero for such functions; so the single nth-order pole m u s t be b a l a n c e d by a total of n zeros. The second is by evaluating (*) for the function f ( z ) / z n, a n d interpreting the answer, n. This, R i e m a n n remarked, is essentially Gauss's third proof of the f u n d a m e n t a l theorem. R i e m a n n t h e n gave a s u m m a r y of how definite integrals can be evaluated by c o n t o u r integration, before concluding the elementary part of his lecture course with a brief description of

fd

afro

. .

u

,-,,,0, ,,

in advance, and that it must change only by a constant value on crossing any cuts. He deduced correctly that an extremal for this integral is a minimum and that it is unique, but he gave no argument for its existence. It is not clear if he thought it obvious, established by his mentor, or too difficult to secure; there is not even an appeal to Dirichlet's principle. Riemann's avoidance of Dirichlet's principle here, when it generally plays such a central role in his thinking about complex functions, is intriguing. It had been present in the earlier lecture course, the one of 1855-1856 which his friend Dedekind attended, as the content of Cod.Ms.Riemann 37, mentioned earlier, attests. Here, there are many more references to original papers, and progress to the study of algebraic functions is altogether more rapid. Something like Roch's account of the Riemann mapping theorem can, however, be found, on pages 19-28. Did he repent of its proof?. Klein recorded Weierstrass telling him that "Riemann had never laid any particular value on finding his existence proofs with Dirichlet's principle," so it may be that he was happy enough to withdraw it, pending a better proof. 6 Perhaps he merely withdrew it from what he judged to be the truly elementary part of the subject suitable for a lecture course.

~,

p,

",blc/~ q

the branch points of a many-valued function. The comparison with the version published by Roch is instructive. As Roch himself commented: "The lectures of my revered teacher Riemann have been extensively used, as much for the manner of representation as for the detailed examples, and he has graciously given me permission to use their content completely." Faced with

the question of the existence of a minimizing function for the integral

L(,) = f f IVan/= + kkax/

\oy) j

at,

Roch first obtained some necessary conditions by looking at the variation of the integral. He found that the minimizing ftmction must satisfy the Canchy-Riemann equations, that one must specify its value on the boundary

Riemann's theory of Elliptic Functions Riemann's lectures on complex function theory were usually offered before a course on elliptic and algebraic functions. The course of 1861 was published by Stahl in 1899, with a chapter on the presentation of the theory of theta functions taken from the course of 1856. There he showed for the first time how geometric considerations of the Riemann surface defined by the function y2 = cubic or quartic in x illuminated the whole study of elliptic integrals and elliptic functions. In so doing, he solved a problem which had defeated both Jacobi and Cauchy, and made the first adaptation of Cauchy's approach to complex function theory

6Klein (1894), p. 492, n. 8.

VOLUME 19. NUMBER 4, 1997

61

to many-valued integrands. In the lectures he dealt only with the case w h e n the roots of the quartic either are real o r c o m e in c o m p l e x conjugate pairs, b u t there is no r e a s o n to s u p p o s e his analysis was r e s t r i c t e d to this case (as h a d b e e n Abel's a n d the first a t t e m p t b y Jacobi). Riemann t o o k the R i e m a n n surface, which he d e n o t e d T, cut it into a simply c o n n e c t e d region T' b y two cuts a and b, and s h o w e d t h a t the elliptic integral in its n o r m a l form m a p p e d T' onto a rectangle in the c o m p l e x plane. This illustrated t h e many-valued nature of the integral a n d the double periodicity o f its inverse. He s h o w e d that the p e r i o d s a r o s e a s integrals along the cuts; t h e y could n o w be s e e n to arise b y integrating along n o n c o n t r a c t i b l e c l o s e d curves on a t o m s . He defined a t h e t a function b y a p o w e r series and speedily o b t a i n e d the J a c o b i a n elliptic functions as quotients o f theta functions, by writing d o w n a quotient that had the right zeros and p o l e s and multiplying by an exponential factor to get the periods right. This only left a little trick to get the constant multiple r i g h t - a rather attractive use of Liouville's theorem. In the course of 1856 he gave an alternative account: he def'med the period parallelogram, P = {0, 1, ~-, 1 + ~-}

for the t h e t a function, pointing out that it w a s not, strictly, a p e r i o d i c function. Integration a r o u n d the b o u n d a r y s h o w e d that it had exactly one zero inside P, which could be found explicitly from the p o w e r series (v = r/2). Thence, following Jacobi, the o t h e r three theta functions, and a suitable quotient x := a 2 ~(v)/O2(v) was then d o u b l y periodic with a double pole at ~-/2. F r o m which it followed by counting p o l e s that d y / d x w a s the square root of a cubic, a n d so ~xx = sn(2Kv).

The Illustrations

1896 Elements of the theory of functions of a complex variable with especial reference to the methods of Riemann. transl. G.E. Fisher and I.J. Schwatt, Pennsylvania C.F. GAUSS

1832 Theoria residuorum biquadraticorum. Commentatio secunda, Comm. Soc. Reg. Gottingensis, 7(math.), 89-148; reprinted in Gauss Werke, 2 (1876), 93-148. C.F. KLEIN 1894 Riemann und seine Bedeutung for die Entwicklung der modernen mathematik, Amtlicher Bericht der Naturforscherversammlung zu Wien, in Gesammelte Mathematisehe Abhandlungen, 3, 482-497. D. LAUGWITZ 1996 Bernhard Riemann, Basel: Birkhtiuser.

The illustrations are t a k e n f r o m the d o c u m e n t Cod.Ms.Riemann, pages 172-173, 175, and 184. I c a n n o t determine w h o d r e w t h e m - - i t is surely unlikely that they were t a k e n d o w n like this u n d e r lecturing conditions. They illustrate very dramatically the Riemann surfaces o f certain hyperelliptic functions, and a question concerning periodicity and Abelian functions.

E, NEUENSCHWANDER

BIBLIOGRAPHY H. DUREGE 1864 Elemente der Theorie der Functionen einer complexen ver~nderlichenGrdsse mit besonderer Beff)cksichtigung der Schdpfungen Riemanns, Leipzig: Teubher.

B. RIEMANN

1981 Studies in the history of complex function theory, II, Bull. Am. Math. Soc. (new series) 5(2), 87-105. 1987 Riemanns Vorlesungen zur Funktionentheorie, allgemeiner Tell, Preprint nr. 1086, Technische Hochschule Darmstadt 1990 'A brief report . . . ' in Riemann, Werke, 855-8689 1857a Beitr&ge zur Theorie der durch die Gauss'sche Reihe F(a, fl, % X) darstellbaren Functionen, Abhandlungen der K(~niglichen Gesellschaft der Wissenschaften, Gdttingen, 7, in Werke, 67-83. 1857b Theorie der Abel'schen Functionen, J. reine Angew. Math. 54, in Werke, 88-144. 1990 Gesammelte Mathematische Werke, Wissenschaftliche Nachlass und Nachtr#ge, CollectedPapers (R. Narasimhan, ed.), New York: Springer-Verlag. G. ROCH 1863 Ueber functionen complexer Gr6ssen, Zeitschr. Math. Phys. 8, 12-26, 183203. 1865 Ueber functionen complexer Gr6ssen, Zeitschr. Math. Phys. 10, 169-194. H. SCHERING n.d. G6ttingen archive (Cod.Ms.Riemann 37) H. STAHL 1896 Theorie der Abel'schen Functionen, Leipzig: Teubner. 1899 Elliptische Functionen, Vorlesungen von Bernhard Riemann, Leipzig: Teubner.

62

THE MATHEMATICALINTELUGENCER

JEAN-MARC LI=VY-LEBLOND

If Fourier Had Known Argand ,, A Geometri cal Point of View on Fourier Transforms |

ourier's idea that (almost) any function could be represented as a s u m of harmonic functions certainly is one of the deepest and most fruitful in the whole history of mathematics. Beyond its immediate use in the hands of Fourier himself f o r solving differential equations of crucial interest in mathematical physics, this idea paved the way for many essential developments of modern mathematics, from abstract metric spaces to general eigenfunction methods, bridging analysis (series and integral expansions) with algebra (group theory), and developing into the whole field of harmonic analysis. Fourier's breakthrough continues to prove its fecundity, as in its recent (and belated) generalization to "wavelet analysis." Fourier's method is so important in the teaching of mathematics that it may be of some interest to reconsider it in a new perspective, all the more so since this revisiting is a very elementary one. Fourier transforms, whether discrete (Fourier series) or continuous (Fourier integral) are commonly introduced and presented in a purely analytical and algebraic setting. I wish to show here that they can be looked at geometrically. I hope that this visual approach, while it certainly does not pretend to replace the standard introduction, may be useful and pleasing. As a matter of fact, the basic idea is most simple, and the deepest question about it probably is one of understanding why it is not already familiar.

The D i s c r e t e Case: F o u r i e r S e r i e s The idea

Consider the general Fourier series

f(t) = ~ p

Cp exp(ipt),

(1)

where the summation on the index p runs over some subset of the integer numbers. I propose to take the elementary step of looking at the complex number f(/) through its Argand representation in the complex plane. It then appears as the vector sum of its components, that is, the successive complex numbers cp, each one being modified by a supplementary phase pt. This point of view is most helpful in the case of Fourier series with real coefficients, so I will, from now on, restrict myself to this case; f(t) is thus represented by the extremity of the polygonal contour formed by the successive components, that is, segments with respective lengths cp, each one making the same angle t with the preceding (and following) one--what could be called an isogonal polygonal line. Figure 1 exhibits the idea.

9 1997 SPRINGER-VERLAG NEW YORK, VOLUME 19, NUMBER 4, t997

63

The final result is the w e l l - k n o w n e x p r e s s i o n - - u s u a l l y c o m p u t e d as the sum o f the g e o m e t r i c series c o r r e s p o n d ing to ( 2 ) - -

,fit)

1

sin(nt/2) f(t) = a ~

e x p [ i ( n + 1)t/2].

N o w for the F o u r i e r - A r g a n d movie. As time g o e s on, the isogonal line folds in, so that the r a d i u s of the circums c r i b e d circle d e c r e a s e s for 0 < t < v and i n c r e a s e s b a c k for ~r < t < 2~-. The zeros o f f ( t ) have an obvious g e o m e t rical interpretation: they c o r r e s p o n d to those values o f t

t

Figure 3. Some Fourier portraits of the Fourier series with coefficients Cp = 1 (1 -< p -< 5), 0 otherwise.

Figure 1. A Fourier portrait.

4

It is most suggestive here to think of the variable t as the "time." Let us call the Argand diagram representing a Fourier series such as (1), the "Fourier-Argand portrait" of the funct i o n f at time t. The sequence of these portraits as the variable t runs over its range, showing the variation o f the function f, will be called the "Fourier-Argand movie" o f f (Fig. 2). The Simplest Example

)

Let us c o n s i d e r a F o u r i e r series with a f'mite n u m b e r of equal n o n z e r o positive coefficients. F o r definiteness, take cp=a cp=O

for l - - p - - < n, (2)

forpn.

Here, our g e o m e t r i c a l r e p r e s e n t a t i o n is so simple as to permit b y itself an explicit c o m p u t a t i o n of the F o u r i e r sum. ]ndeed, the F o u r i e r - A r g a n d p o r t r a i t of the function f ( t ) consists o f n equal s e g m e n t s with length a at successive angles t, all i n s c r i b e d in a circle with radius r = a12 sin(t/2)1-1. Seen from the c e n t e r of the circle, the isogonal line s u b t e n d s a total angle 0 = nt, a n d the value of the function is simply the c o m p l e x n u m b e r r e p r e s e n t e d b y the chord o f the c o r r e s p o n d i n g circular arc. E l e m e n t a r y g e o m e t r y suffices for computing its l e n g t h - - t h e m o d u l u s o f f ( t ) - - a n d d i r e c t i o n - - i t s p h a s e - - y i e l d i n g the results: If(t)l = 2 r sin(0/2)

f

and

,

,

,

i,

2

3

4

5

-i

-2

The sum of the series for t = 0 to 3

,

\

Arg[f(t)] = 0/2 + t/2.

Figure 2. Snapshots of a Fourier movie.

fit:) f ( t3) . ~ --"-~ ~ - ~

-~ ._. --. A

5

/:

($/(o)

-1

-2

Fourier portraits at t = O, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4

THE MATHEMATICAL INTELLIGENCER

such that the isogonal line closes onto itself; this occurs, of course, for t=2k~r/n ( k = 1, 2 , . . . , n - 1 ) . The Fourier-Argand portrait, for these special values, consists of regular n-gons, convex for k = I and k = n - 1, and otherwise steliated if n is odd, degenerate if n is even. The Fourier-Argand movie s h o w s the folding and unfolding of the Fourier-Argand portrait; see Figure 3 for the case n-5. Another possibility is to consider the real and imaginary parts of the function f separately, that is to say, to draw the Fourier-Argand portraits of symmetrical (C_p = c*) and antisymmetrical (C_p = - c ~ o ) Fourier series. In the present case, this leads to the two functionsfR andfx, with respective Fourier series

fR

i -- a 2

f(t)=

Cp = a

{% = 0 fr Cp sgn(p)a

f(t)

( - 1)E[t/~]

=

2 C2k+l-

1

i~- 2k +

1'

in,r~2

More Examples

1. A G e o m e t r i c F o u r i e r Series: Because of its simplicity, let us consider the Fourier series with coefficients

(3)

the sum of which is easily computed:

(5)

C2k

=

0

or cn - sin(n~-/2) ~

[Actually, because of the usual slight awkwardness associated with the correct choice of the coefficient Co, I have simplified by letting fR differ by the constant a from the real part of f as defined by (2).] Figures 4 and 5 sample the Fourier-Argand movies of fR andfz for the case n = 2.

iai < l,

(4)

[that is,f/t) odd, periodic with period 2~r, andf(t) = sgn(t), it] < Ir], with Fourier coefficients

for ~l > n and p = O for 0 < ~[-< n.

Cp = a~,

2'

that is, the classic Poisson kernel. It oscillates between a maximum value (1 + a)/(1 - a) at t = 0, 27r, etc. and a minimum value (1 - a)/(1 + a) at t = ~r, 3~-, etc. Its Fourier-Argand portraits give a direct visual feeling for this behavior (Fig. 6). 2. The S q u a r e Wave: More interesting is the typical discontinuous function

for ~l > n for ~]-< n,

[%=o

1-2acost+a

]"

(6)

Its Fourier-Argand portraits are s h o w n in Figure 7. The series has been truncated (Cp = 0 for p > 50); observe that the well-known Gibbs oscillations near the discontinuity are quite visible. Whereas in the preceding cases our pictures aptly illustrated what w e pretty well knew of the behavior of the Fourier sum, the contrary might be said here. The strange geometrical waggling of the isogonal line cries out for general formulation and proof. In any case, the visual interpretation underlines what was in Fourier's time the deepest mystery, that is, the ability of a sum of harmonically varying functions to represent a piecewise-constant function.

Figure 4. S o m e Fourier portraits of the series with coefficients Cp = 1 ( - 2 <- p -< 2), 0 otherwise. 1.5"

1

1

t= 1.2

~ t

~

~

~

_t1

L

1

-0.5

.5

-0.5-

t = 3.

0.5

0.5

0.5" iI

1.5

1.5 t = O.

I-

-1

-1-

-i,5

-1.5-

1.5!

1,5'

14

1

t = 1.8

t= 0.2

t=3.6

0.5

0.5

l

-'1

_o.5~ _ _ ~

~

-1

-1

-I

-1.5

-1.5

-1.5'

1.5

1.5

1.5t= 0.6

1

t=6.5

t = 2.4

0.5

-'1 -0.5

-1 -1.5

t

-:i -0

J

0.5 84

/

I1

-:1 -0.5

-0.5

-1

-i

-1.5

J

- 1 . 5 84

VOLUME 19, NUMBER 4, 1997

t =0.i

=i.i

3

i

0:5

-2 -1.5 -i -0.5

t =2.1

i

-

'2

-

I'. 5

0.5

- '2

- 1'. 5

-:1

-0.5

-i

-i

t =1.6

3

-

i

-'2 -i .5 -'i -0.5

-i ~

t =0.6

3

'I - 0'.5

0 !.5

i

-'2

-

i'. 5

t =2.6

-

0 I

.5

'i -0'.5

-1

3

-:2 -i:.5 -:I - 0 ~

-1

> 0:5

i

--1'

Figure 5. Some Fourier portraits of the series with coefficients cp = sgn p ( - 2 -< p -< 2), 0 otherwise.

The Continuous

Case:

Fourier

Integrals

The Idea

Consider the general Fourier integral

f ( t ) = ] F(p) exp(ipt) dp,

(7)

where the integration runs over the real line; suitable existence conditions are supposed. Define the "incomplete Fourier integral"

f(t; q) =

~ F(p) exp(ipt) dp,

(8)

with asymptotic values

f(t;-~)=0

and ~(t; + ~ ) = f ( t ) .

For a given time t, the point representing the complex number~(t; q) follows, as q runs from - ~ to +~, a curve Ff(t),

66

THE MATHEMATICAL INTELUGENCER

extending from the origin to the point representingf(t); this curve is the Fourier-Argand portrait o f f at time t, and its evolution, as time goes on, is the Fourier-Argand movie off. Again, it will be more perspicuous to confine our attention to the real case: in (7) and (8), I will put only real functions F. The Simplest Example

As in the discrete case, the simplest example is that of a function constant over an interval and vanishing elsewhere, that is,

F(p) = a, 0 < p < P , F(p) O, p < O a n d p > P .

t=3.

0.5

t=O.

0.5 84

~

t=5.

i

t=l.

84

t=4.

0.5

?

v

e s

,

i

2

3

i

-0.5

0.5

2

-0.5

-0.5

-0.5 84

0.5

\

1

2

2

-0.5

t=2-

0.5

i

t=7.

i

2

3

- 0 . 5 84 Figure 6. (above) Some Fourier portraits of the series with coefficients cp = 3 -~~

Figure 7. (below) Some Fourier portraits of the series with coefficients

0.2

t=1.55

cp = sin{p~/2)/(p~/2)(-50 -< p -< 50), Co = O. O.'8 0.2

t=O.

0 . 2 84

1'

i ' .,

t=l 92

0.I

o.~2

o14 o'.6 o'.a

1

1.'2

-0.] -0.2

-0.2,

0.2

t=0.2

i-0.4"

0.1

i

~.~-~

1.~2

-0.1

-0.8

i i-0.6,

-1 0.2-

-0.2 0.2

i-0.8.

t=O .4

0.27

-0.

0

:

:

'2

1.

8

112

0.2

~

ol

o.~6

o:8

1'

, [2

-0.2- 1

-0.2-

-0.4-

-0.4-

-0.6-

-0.6-

-0.8-

t[2

2-

t=O.

o ,o

8

-0.80.2

-0.1 -0.3 -0.4

0.2

-0.1 ~

t=t 9

~176

016

1.2

-1 t=1.575

0-21

-0.2

-l.i

-0.4

-C

-0.6

-0 6

-0.8

-0 8

-0.2

-0.2

o.,,

t=O. 6

-0.2. O.

t=1.565

r2

(

-0.2

-~

t=l 9 4

/

012

04

06

018

i

12

H

-0.3 -0.4

-1

VOLUME19, NUMBER4, 1997 67

/

0.8

\

O. 6 ~ " , ~

0.1J ,

v

-0.2

]

I

I

I

0.2

0.4

0.6

0.8

1 !

- --

0.15 -0.i

-0.2-

I

I

0.05

0.I

I

0.1

Fourier portraits at times t = 9, 10, 11, 12

Fourier portraits at times t = 0, 1, 2, 3, 4, 5, 6, 7, 8 Figure 8. S o m e

-0.05

Fourier portraits o f t h e f u n c t i o n f(t) = 2 ( d t ) sin(Pt/2) exp(iPt/2).

Writing the differential of the incomplete Fourier integral, d J = a exp(iqt) dq

(0 < q < P),

it does not take long to recognize that the Fourier-Argand portrait o f f is but a circular arc with total length aP subtending an angle Pt (Fig. 8). Here again, geometry suffices to give, at once, the expression f ( t ) = 2 a sin(Pt/2) exp(iPt/2) t

for the complete Fourier integral. The Fourier-Argand movie exhibits an arc of a circle with constant length but varying radius ]a/t], decreasing for t > 0, so that it shrinks by curling up. It is quite intuitive to expect the modulus o f f to show an oscillatory behavior with decreasing amplitude. In particular, the endpoint of the Fourier-Argand portrait, representing the value of the functionf(t), clearly goes through the origin each time the circle closes onto itself, that is, for Pt = 2k~r,

k integer (nonzero).

We thus obtain a very direct interpretation (and calculation) of the zeroes off(t).

In the complex plane, this means that the element of arc of the Fourier-Argand portrait Ff(t) has length ds = F(q) dq,

while it makes an angle r = qt

with the real axis. It is immediate how to compute the radius of curvature of Ff(t) at the point parameterized by q (and for "time" t): R = ds _ F(q) dcp t

(10)

In other words, the Fourier-Argand movie of a function results from a global rescaling of its curvature by the time t. This geometrical interpretation of the Fourier integral is best understood by reversing the argument. Let s be a (sufficiently regular) curve in the plane. Consider the generic point Q on the curve; denote by s(Q) the curvilinear abscissa along F, some origin being chosen, and by ~(Q) the angle of the tangent with the horizontal axis. The curve F is given, in the complex plane, by s

z(Q) = fV exp[i~(s)] ds

(11)

A Geometrical Interpretation of Fourier-Argand Portraits: Percurvation

d q f = F(q) exp(iqt) dq.

68

Let R ( Q ) ds/dcPiv be the radius of curvature at Q. Its functional relationship with the slope angle r that is the function R(
The considerations of the preceding paragraph on the curvature of the Fourier-Argand portrait allow an immediate generalization. The infinitesimal increment of the incomplete Fourier integral is given by

THE MATHEMATICAL1NTELL1GENCER

(9)

F:

z ( V ) = fQ R ( ~ ) exp(icp) d~.

We now define a continuous family F(t) of curves obtained from F by scaling the curvature at each point Q of the curve by a c o m m o n factor t; we will call this transformation from F to F(t) percurvation 1, and say that F(t) is a percurved form of F. Of course, in the process, the functional relationship between the slope and the curvilinear abscissa changes as well; a little reflection shows that the curve F(t) is given by the formula

z(V) = yQ R(cp/t) exp(icp) dcp = t

F(t):

fr R(p) exp(/pt) dp,

(12)

where the change of variable r = pt yields the second expression. Comparison with the equation following (11) shows that the Fourier-Argand portrait of a hmetion f(t) at time t is the percurved form of the Fourier-Argand portrait off(t0) at some reference time to (with factor t/t0). It also proves that in the general Fourier integral (7), the integrand F (that is, the inverse Fourier transform o f f ) is, up to a factor t, the radius of curvature of the Fourier-Argand portrait offl More Examples

Here are a few more functional behaviors. 3. G a u s s i a n Function: Figure 9 show successive Fourier-Argand portraits of the Gaussian function

f(t) = ~

4. F o u r i e r T r a n s f o r m o f T r u n c a t e d Sine Function: Let us take for Fourier integrand

F(p) = sin(2p) F(p) 0

for ~l < ~r for ~l ~ ~r,

resulting in the function 4i

f(t) = t2 _--~ sinOrt). Its Fourier-Argand portraits (Fig. 10) consist of the cycloidal curves defmed by a circle with constant diameter of 1/4 rolling on a circle with varying diameter of I / I t - 21; the rolling is on the outside or inside of the circle, depending on the sign of (t - 2). Note the particular case of t = 2, where the Fourier-Argand portrait is a true cycloid with rectilinear base and the function reaches its maximum value. 5. T h e Sign F m a c t i o n : The sign function (suitably normalized) f(t) = ~ sgn(t) is the Fourier transform of

F(p)

(ip) -].

=

Due to the scaling property of F ( h o m o g e n e o u s o f degree - 1 ) , the Fourier-Argand portraits o f f ( t ) (Fig. 11) do not depend on the time t (up to a symmetry, depending on its sign). They s h o w a most interesting spiral, with the property of being invariant under percurvations.

e x p ( - t2/4),

corresponding to the Fourier transform F(p) = e x p ( - p 2 ) . These various percurved forms all have a radius of curvature varying as a Gaussian function of the slope angle.

1The word uses the intensive suffix per-, as in "permutation" or "perturbation." It is a pleasure to thank Barbara Cassin for her linguistic advice

Zeros

The graphical representation of Fourier series or integrals via their Fourier-Argand portraits (or movies), besides its interest for visualizing some of the main aspects of Fourier transforms, enables one to get an intuitive grasp of several general theorems. This is the case for s o m e results on the zeros of Fourier transforms. Since a zero of a functionf(t)

Figure 9. Some Fourier portraits of the Gaussian Function exp(-t2). t=2.

t= 0.

1i

-0.5.

-0.5

115

f<~ q ~

t= 3.

t= 1.

t= 1.5

fq

1:5 ~

1:5

0'.5

i

i;5

0:5

t=7. i

1:5

o',5

t= 1o. I' i.'5

-0.5

-0.5

-0.5

5

{

-0.5

-0.5.

.

t = 4.

0:5 -0.5

i

-0.5

0

)

t= 2.5

t= 0.5

-

~ i i .5

\ -0.5

t= 3.5 .5

~ ~79 -0.5

VOLUME19, NUMBER4, 1997 69

3

3l

t=l.6

t=l.

3

t=2.

t=3.

2

2

1

1

-'1

{

-'1

-1

-1

3} t=l.3

-1

-1

3

3

t:l.9

2

t=3.~

t=2.5

2

2

2

1

-

{

-i

-'1

{

-'1

-I

Conclusions The very elementary nature of the preceding considerations, and their promise, make one wonder why they have not been put forward earlier--in particular, by Fourier himsell The geometrical representation of complex numbers

70

THE MATHEMATICALINTELLIGENCER

-1

-i

Figure 10. Some Fourier portraits of the truncated sine Fourier integrand: sin(2p)

occurs whenever its Fourier-Argand portrait closes onto itself at the origin, it becomes almost obvious that a Fourier series with positive coefficients, or a Fourier integral with a positive integrand, has only discrete zeros; for the extremity of its Fourier-Argand portrait, curling up forever in the same sense, can pass through the origin only now and then (the simplest examples, Figures 3 and 8, are typical in that respect). More specific results can be obtained as well. For instance, it is left to the reader to see that a Fourier series with at most N nonzero positive coefficients has no zeros for t < ~r/(N - 1); indeed, for an isogonal line with N segments to close onto itself, it is necessary that the Nth segment have made at least half a turn with respect to the first one (and consideration of such a line where the first and last segments are equal and much larger than the other ones shows that the limit can be reached).

k'~'l

if ~o! < ~r, 0 if ~oI _> ~r.

via their Argand representation goes back to the beginning of the nineteenth century, just a few years before the breakthrough by Fourier. But this geometrical point of view gained acceptance rather slowly. Ivor Grattan-Guinness, in his C o n v o l u t i o n s

in

French

Mathematics

1800-1840

(Birkh~tuser, 1990), comments [The] most notable feature [of the French response] is silence; there was little interest at the time and Legendre's assessment of Argand's suggestion as "an object of pure curiosity" is typical. The main reason for the uninterest was that the algebraic theory of complex numbers had been exhaustively developed, so that there seemed to be no need for alternatives. (p. 256) The impression that complex numbers were not problematic is strengthened by other literature; for example, Lagrange's treatise on algebra (1808) often referred to complex roots of equations, but made no special issue of complex numbers themselves. Hence there was no reason why Argand's pamphlet (1806) should make an impact, quite apart from the obscurity of its author (he is said to have been a Paris bookseller).

2-

-1-'.5

j

/

-:1

- 0 . 5 ~'~

'.5

J

J

t=0 t
J i

1 5

|

1

--

0I

t>O

0~5

o5

-1 i

-1.5

-:1

-0.5

:5

-2. Figure 11. Fourier portraits of the Heaviside-type function f(t) = ~r sgn(t).

We may understand better in this context, why Fourier did not make use of complex numbers in the first place, let alone of their Argand's representation. The question remains of why Argand diagrams have stayed rather unused in Fourier theory up to the present time. [Note however, the use of what we have called here Fourier-Argand portraits by Michel Mend~s-France in his study of exponential sums (which are special cases of Fourier sums): Michel Mend~s-France, "The Planck constant of a curve," in J. B~lair and S. Dubuc, Fractal Geometry and Analysis, Kluwer, 1991), pp. 326-365 (see especially p. 346), and additional references therein. The geometrical representation of a Fourier complete sum in the complex plane (i.e., the trajectory of the extremity of the FourierArgand portrait--but not the line itself) has also been used in a very specific context by Main Robert, "Fourier Series of Polygons," Am. Math. Monthly 101 (1994), 420-428.] But, as the French saying goes, il n'estjamais trop tard

pour bien faire. ACKNOWLEDGMENT

It is a pleasure to acknowledge useful discussions and correspondences on this article with Marc Diener, Jean-Pierre Kahane, Main Laverne, Juliette Leblond, Michel MendbsFrance, Jean-Paul Marmorat, and Fr~ddric Pham. Most of this work was done at the University of Venezia (Dipartimento di Matematica), thanks to the kind invitation and interest of Michele Emmer.

VOLUME 19, NUMBER 4, 1997

71

II--r

Jet

Wimp,

Editor

I

An Introduction to the Mathematical Theory of Inverse Problems by Andreas Kirsch NEW YORK: SPRINGER-VERLAG, 1996. VIII + 282 PP. US $59.95, ISBN 0-387-94530-X

R E V I E W E D BY DAVID C O L T O N

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.

72

ome years ago, I was complaining to a friend that I felt my personal and professional life were disjoint and that I was essentially living two separate lives. My friend, who was not a mathematician, feebly asked (probably with little hope of understanding my answer!) what kind of mathematics I did. Realizing the difficulty of explaining mathematics to a nonmathematician, I tried to put it in as simple a manner as possible, explaining that I worked in the area of inverse problems, where typically a solution does not exist and, if it does, it may not be unique. Furthermore, small variations in the information available can cause wildly different answers. I still remember the broad smile that appeared on my friend's face when he said, "But that's exactly how life is! Why do you think your life is disjoint? It sounds completely connected to me!" I think this story explains why, until recent years, mathematicians totally ignored the area of inverse and improperly posed problems. After all, if a solution doesn't even exist, what can a mathematician possibly say? At the same time, physicists ignored such problems on the basis that if the solution, whatever it may be, does not depend continuously on the measured data, then it cannot possibly have any relevance to understanding the physical world. As do I, both mathematicians and scientists thought that such problems were totally removed from critical analysis and the scientific method.

S

THE MATHEMATICALINTELLIGENCER9 1997 SPRINGER-VERLAG NEW YORK

However, by 1960, cracks began to appear in the wall mathematicians and physicists had constructed between themselves and so-called "improperly posed" problems. Indeed, in Volume 2 of Methods of Mathematical Physics by Richard Courant and David Hilbert, the following paragraph appears on page 230: Existence, uniqueness, and stability of solutions dominate classical mathematical physics. They are deeply inherent in the ideal of a unique, complete and stable determination of physical events by appropriate conditions at the boundaries, at infmity, at time t = 0, or in the past. Laplace's vision of the possibility of calculating the whole future of the physical world from complete data of the present state is an extreme expression of this attitude. However, this rational ideal of causal-mathematical determination was gradually eroded by confrontation with physical reality. Nonlinear phenomena, quantum theory, and the advent of powerful numerical methods have shown that "properly posed" problems are by far not the only ones which appropriately reflect real phenomena. So far, unfortunately, little mathematical progress has been made in the important task of solving or even identifying and formulating such problems which are not "properly posed" but still are important and motivated by realistic situations. In a book of 800 pages this is essentially all that is said about improperly posed problems. However, an awareness that such problems may be important was beginning to creep into the consciousness of the mathematical community. This change of view was, of course, being actively encouraged by a small number of mathematicians at the time. Fritz John, Carlo Pucci, Larry Payne, and John Cannon, among others, were

systematically investigating improperly posed initial-value problems for partial differential equations. Such problems arise in the context of looking at the inverse problem to the more classical direct problem. A typical example considered by these mathematicians was the backward heat equation. Consider the one-dimensional heat equation

au(x, t) _ a2u(x, t)

o~

at

with boundary conditions u(O, t) = uOr, t) = O, t ~ O, and initial condition u ( x , O) = u o ( x ) ,

O <- x <- v.

The direct problem is to solve the classical initial-boundary-value problem, where the initial temperature distribution u0 and f'mai time T are given and it is desired to determine u(., T). In the inverse problem, on the other hand, one measures the final distribution u(., T) and tries to determine the initial temperature u0. By separation of variables, it is easily seen that the solution of the inverse problem can be f o u n d by solving the integral equation of the first kind

u(x, T) = --~

k(x, y) Uo(Y) dy, O<_x<_~r,

where

k(x, y): = ~

e -n2T sin(nx) sin(ny).

n=l

From this formulation it is clearly seen that the inverse problem is improperly posed (sometimes called ill-posed) in the sense that u0 does not depend continuously on the data u(., T) in any reasonable norm; i.e., by the RiemannLebesgue lemma, large perturbations in u0 of the form C sin(mx) can result in only small errors in u(., T) f f m is large. The work of John, Pucci, Payne, Cannon, and others served to accomplish one of the tasks p o s e d in Courant-Hilbert, i.e., a large class of physically relevant improperly posed problems arose from a study of inverse initial-boundary-value problems. At about the same time, many physicists

were involved in the study of inverse scattering problems for Schr0dinger's equation with a spherically symmetric potential (see [3] for a historical discussion), and this led to another class of inverse and improperly posed problems of interest to both physicists and mathematicians. Note that in contrast to inverse initial-boundary-value problems, the inverse scattering problem is nonlinear, for it is required to determine a coefficient in a differential equation from measured scattering or spectral data. In 1962-1963, the field of linear inverse and improperly posed problems underwent a dramatic transformation with the publication of papers by A. M. Tikhonov [14] and D. L. Phillips [11]. In particular, Tikhonov and his school introduced the key idea of an admissible regularization strategy for solving ill-posed operator equations of the form K x = y, where K is a linear compact operator between Hilbert spaces X and Y. According to Tikhonov, a regularization strategy is a family of linear and b o u n d e d operators R~ : Y--) X, > O, such that lira ~ R~Kx = x for all x 9 X. Now suppose we measure y to within an error 3, i.e., we do not k n o w y but rather y~ 9 Y with IlY - Y~H -< & Then, as 8 tends to zero, we would like to c h o o s e a = a(8) tending to zero such that R~y ~ tends to x. Such regularization strategies are called admissible. What Tikhonov did was to establish an admissible regularization strategy for operator equations of the form K x = y , In particular for Fredholm integral equations of the first kind such as the one appearing in the case of the backward heat equation. The m e t h o d proposed by Tikhonov for solving the ill-posed operator equation K x = y is to minimize the functional

Jo(x) := IIKx -yil 2 + 4 4 2 for an appropriate choice of the regularization parameter ~. At this point, it is worthwhile to remember the statement of Lanczos: A lack of informa-

tion cannot be remedied by any mathematical trickery! Hence, in order to choose the parameter a, we need some extra information. This extra information can be either a priori (before

starting the actual computation) or a

posteriori (after starting the computation). As Kirsch is careful to point out, Tikhonov's method is not the only one available to solve the ill-posed equation Kx = y. Indeed, in some ways, the Landweber iteration or conjugate-gradient method is preferable. Landweber's method is to use the steepest descent algorithm to minimize the quadratic functional I~r(x-y~ll 2, where the algorithm is stopped as soon as the mth iteration x m,~ satisfies IIKxm,~ - Y~II-< r8 for some r > 1. The conjugate-gradient method proceeds similarly, but uses the conjugate-gradient instead of the method of steepest descent to minimize ilKx - ySII2. These three methods all deal with linear ill-posed problems In infinitedimensional spaces, and are carefully discussed in Chapter 2 of Kirsch's b o o k (Chapter 1 is an introduction to basic concepts with many examples). However, in practical situations these equations are often first discretized, leading to badly conditioned finite linear systems. The question is then, How well does the solution of the finitedimensional problem approximate the solution of the infinite-dimensional problem? This leads to the use of projection methods (in particular, the Galerkin and collocation methods) to solve ill-posed problems, the subject matter of Chapter 3 of Kirsch's book. A particularly attractive part of Chapter 3 is a lucid and complete discussion of S y m m ' s equation 1 f0a r

lnlx -

Yl ds(y) =fix), x E al~,

where gI C R 2 is a bounded, simply connected region with analytic boundary a~, and f E C(MI) is a given function. Under these assumptions error estimates are obtained for the solution of this equation by both Gaierkin and collocation methods. The last section of Chapter 3 of Kirsch's b o o k is devoted to the Backus-Gilbert m e t h o d for "solving" the finite m o m e n t problem

f~ kj(s) x(s) ds = yj, j = 1 , . . . , n,

VOLUME 19, NUMBER 4, 1997

73

where yj are given real numbers and kj ~ L2(a, b) are given functions. This method does not primarily aim to solve the moment problem, but rather to determine how well all possible models x can be recovered. The presentation of this method by Kirsch is very well done--I know of none better in the literature. The method is deceptively simple: let t E [a, b] be a fLxed parameter, and determine numbers ~j = ~j(t) that minimize

j: [s - tl2 j=~l kj(s) ~j 2 ds subject to

fb n.~ kj(s) cpj ds = 1. a

J=l

The Backus-Gilbert solution Xn of the moment problem is then defined to be

x,~(t) = ~ yj ~j(t),

t E [a, b].

j=l

Now let x ~ Ht(a, b) be any solution of the moment problem, where H 1 is the usual Sobolev space. Then, under a few mild assumptions, Kirsch shows that there exists a positive constant en independent of x, such that

IIx - x4L2 -< nltX'llL2 and en tends to zero as n tends to infmity. Note that, in general, the Backns-Gilbert solution x~ is not a solution of the moment problem! Until this point in Kirsch's book, only linear inverse problems have been discussed. Many of the ideas for the regularization of linear inverse problems have been extended to the nonlinear case (cf.[5]). However, in general, for nonlinear problems, one needs a detailed knowledge of the direct problem for progress to be made: there is no "universal method" for nonlinear inverse problems. Furthermore, numerical methods for nonlinear inverse problems are typically more heuristic, with regularization parameters and stopping rules chosen by trial and error. The two most popular nonlinear inverse problems are the inverse spectral problem [9,12] and the inverse scattering problem [4,13], and these are the subject of the last two chapters. Both of these problems come under the class of parameter identification problems--in this case, the problem of determining a coefficient in a dif74

THE MATHEMATICALINTELUGENCER

ferential equation from measured spectral or scattering data. The inverse spectral or eigenvalue problem considered by Kirsch is to determine the coefficient q in the regular Sturm-Liouville problem

d2u(x) dx 2

+ q(x) u(x) = )tu(x), 0_<x_< 1,

u(0)=0

and

hu'(1)+Hu(1)=0

from a knowledge of the eigenvalues {An), where h and H are real numbers with h 2 + H 2 :> 0. A simple example shows that it is, in general, impossible to recover q from the set (An} unless more information is available; in particular, either the knowledge that q is an even function with respect to 89or the knowledge of a second spectrum. However, with such knowledge, it is shown in Chapter 4 that q is uniquely determined. The proof is based on a careful examination of the direct problem. The final section of Chapter 4 is devoted to the use of Newton-type methods to reconstruct q numerically from a knowledge of only a finite number of eigenvaines. The entire chapter is serf-contained and, together with Rundell's article in [2], provides the best introduction to the regular inverse spectral problem that I know. The last chapter of Kirsch's book is devoted to the inverse scattering problem of determining the index of refraction n from a knowledge of the farfield pattern corresponding to h3u + k2n(x)u = 0 in R 3 u(x) = e ikx'd + uS(x)

r~

r ( ausor - i k u S ) = O,

where d is a vector on the unit sphere S 2, k > 0 is the wave number, r = Ixl, and u s is the scattered field corresponding to the incident field e ikx'd. It is assumed that m := 1 - n has compact support. The far-field pattern u~ = uo:(:~, d), 9 = x/Ixl, is determined by the asymptotic behavior of the scattered field given by

uS(x) =

eikr

u~(:9, d) + O(1/r2). r As was to be expected, the analysis of the inverse problem is based on a careful investigation of the direct scattering problem using Rellich's lemma and the Lippmann-Schwinger integral equa-

tion. Included in this discussion is a scholarly presentation of the analytic properties of the far-field pattern. The last two sections of Chapter 5 use information about the direct problem to investigate the inverse problem. In particular, Kirsch gives a complete proof of the uniqueness of the solution to the inverse problem based on Hfihner's construction of the periodic Faddeev-Green's function [7], as well as a survey of three numerical algorithms for solving the inverse problem due to Gul]nan and Klibanov, Kleinman and van den Berg, and Colton and Monk. Although not completely selfcontained, this chapter provides an excellent brief introduction to the central themes of inverse scattering theory and is recommended to anyone wish-" ing to learn something about this important area of applied mathematics. Kirsch's book should become a classic in the field. To compare it to four of my favorite books on inverse problems, it is at a higher level than Groetsch's broad-based survey [6], but in many ways it is easier to digest than the scholarly books of Baumeister [1], Kress [8], and Louis [10]. What makes Kirsch's book so good is that he doesn't try to do too much, yet still manages to carefully treat a number of major topics in both linear and nonlinear inverse problems. In particular, his discussions of the inverse eigenvalue problem and the inverse scattering problem are excellent, mainly because he has taken the time to discuss systematically the corresponding direct problems. He has managed to write a true introduction to the field while maintaining a solid level of mathematical rigor. Of particular help to the student in this regard is an appendix giving the basic definitions and theorems from elementary functional analysis that are used in the book. If you want to find out what the field of inverse problems is all about, this is the book you should read! REFERENCES

1. J. Baumeister, Stable Solutions of Inverse Problems, Braunschweig: Vieweg-Verlag (1987). 2. K. Chadan, D. Cotton, L. P&iv&rinta,and W. Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, Philadelphia: SlAM Publications (1997). 3. K. Chadan and P.C. Sabatier, Inverse

5.

6.

7.

8. 9.

10. 11.

12.

13.

14.

and Electromagnetic Scattering Theory, New York: Springer-Verlag (1992). H. Engl, Regularization methods for the stable solution of inverse problems, Surveys Math. Ind. 3 (1993), 71-143. C. W. Groetsch, Inverse Problems in the Mathematical Sciences, Braunschweig: Vieweg-Verlag (1993). P. H&hner, A periodic Faddeev-type solution operator. J. Diff. Eq. 128 (1996), 300308. R. Kress, Linear Integral Equations, Berlin: Springer-Verlag (1989). B. M. Levitan, Inverse Sturm-Liouvilte Problems, Utrecht: VNU Science Press (1987). A. K. Louis, Inverse und schlecht gestellte Probleme, Stuttgart: Teubner-Verlag (1989). D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. CompuL Mach. 9 (1962), 84-97. J. POschel and E. Trubowitz, Inverse Spectral Theory, Boston: Academic Press (1987). A. G. Ramm, Multidimensional Inverse Scattering Problems, New York: LongmanWiley (1992). A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soy. Math. Dokl. 4 (1963), 10351038.

Department of Mathematical Sciences University of Delaware Newark, DE 19716 USA e-mail: [email protected]

Special Functions: An Introduction to the Classical Functions of Mathematical Physics by Nico Temme NEW YORK: JOHN WILEY AND SONS, 1996, 259 p. US $54.95 ISBN 0-47111-313-1

REVIEWED BY RODERICK WONG

athematics is like music; it goes with the trends of the day. The popular ones may not always

M

wnopm earely sell compareo to country music or rock-and-roll. About 50 years ago, special functions were considered important in the field of analysis. They occupy half of the classic books such as Whittaker and Watson [11] and Copson [4]. They were, and still are, frequently used in physics and engineering. However, in recent years, many mathematicians, both pure and applied, have held the view that with the invention of powerful computers, special functions are no longer needed or have become obsolete. If this is the prevalent view, then does it mean that we can avoid using the exponential function, the logarithmic function, and the trigonometric functions? Higher transcendental functions such as the gamma function, the Bessel functions, and the Legendre functions are only a level higher than the elementary functions just mentioned. To scientists, these functions play the same role as the elementary functions do to high school students. Although there are already many books on special functions and some of them have become classics, most of these books are not suitable as texts, and most of the textbooks being used are not written by specialists. The present book was written by Nico Temme, a specialist who has spent 30 years working on this subject. He clearly knows what to include and how to present the material. Properties of special functions should be simple to state, and the formulas should not be too long, although their proofs can be lengthy and complicated. Functions involving too many variables are best excluded from a textbook such as this. Temme starts Chapter 1 on Bernoulli, Euler, and Stirling numbers. The Bernoulli numbers are discussed in most books on special functions, but Euler and Stirling numbers are rarely mentioned in textbooks on this subject. A particularly attractive inclusion in this chapter is the Boole summation formula. While the Euler summation formula deals with sums of positive terms, the Boole summation formula treats sums of terms with alternating signs. In Chapter 2, Temme lists some usefnl theorems in analysis. These theo-

LAL~ U J . U ~ I U J[ ~LLLLII ILi::LLIUIL ~I,JLLULILLIC~Ia, LtUJ.L.

Also included in this chapter is a brief description of a powerful method in asymptotics, the saddle-point method. Chapter 3 deals with the gamma function. The material in this chapter is standard and can be found in most books on special flmctions. The only formula here which is usually not discussed in introductory books on special functions is an asymptotic expansion for the ratio of two gamma functions. Chapter 4 is on differential equations. It begins with the method of separation of variables, which reduces partial differential equations to ordinary differential equations. It then proceeds to discuss the power series solutions to ordinary differential equations near regular and singular points. It ends with short sections on Sturm's comparison theorem, integral representation for solutions of differential equations, and the Liouville transformation. It was a wise decision to include these topics. Hypergeometric functions are discussed in Chapter 5, including a short introduction to the q-functions, which is currently a hot topic. Orthogonal polynomials are dealt with in Chapter 6. Except for Legendre polynomials, the classical orthogonal polynomials are mentioned only briefly. It would have been nice to include some of the more recent results on these classical orthogonal polynomials. For instance, the inequality given on page 158, m

~/1 - x 2 IPn(x){ <- ~ n ' valid f o r - l < x < l , n= 1,2,..., has been improved by Antonov and Holgevnikov [2], who showed that the n on the right-hand side can be replaced by n + 1. Furthermore, Lorch [8] has generalized this result by showing that the ultraspherical polynomial P~ ) (x) satisfies (sin 0)~lP~)(cos 0) I < 21 -

,{r(,~)}-l(n

+ ,~)h-1

for 0 < A < 1 and - 1 - < x -< 1. Up to now, the most general result in this direction is probably that of Chow, Gatteschi, and Wong [3]. They proved VOLUME 19, NUMBER 4, 1997

75

mequaJ]ty 1

1

ices )

_F(q_ + 1) r(89

X

•

el

(n+q)

N-q-

1

n

'

whereN=n+• 2 1) a n d q = max(a, fl). These results are not too long or complicated to state, and they could have been mentioned (without proofs) at the end of the chapter for interested readers. In Chapter 7, Temme discusses confluent hypergeometric functions. Instead of the older notation (P and T used by Whittaker and Watson [11] and by Erd61yi, et al. [5], Temme prefers the M and U functions introduced by J.C.P. Miller (see [1, Chapter 13 and 19]) and recommended by F.W.J. O1ver [10]. As special cases of the confluent hypergeometric functions, he briefly mentions the Whittaker functions, the parabolic cylinder functions, error fimctions, exponential integrals, Fresnel integrals, and incomplete gamma functions. Chapter 8 comprises the usual material on Legendre functions. The presentation is both clear and to the point. Among all special functions, Bessel functions are the ones that are probably most frequently used. They are presented in Chapter 9. Even for such a well-studied function, new and interesting results are continually being discovered. For instance, it has recently been proved that the k-th positive zero J,,k of the Bessel fimction J. (x) has the upper and lower bounds at u --"~

2~

at V s < J v , k < u -- - -

~

2'~

3 V~ +

289 2

~0 ak--rw

for all v > 0 and all K = 1,2,... where a~ is the K-th negative zero of the Airy function. The surprising fact is that these are exactly the first few terms in the asymptotic expansion

3 2~ 2----~v~ + - ~ a~-~ + ......

ak

j,,t--v -

,

as v--, ~, k being fixed. (This result has not been published. For the cases k = 1 and 2, see [6] and [9]0 Chapters 10 and 11 are the most

76

THE MATHEMATICALINTELLIGENCER

mat taKe me warceslan coorolnates to other coordinate systems such as cylindrical coordinates, spherical coordinates, elliptic cylinder coordinates, parabolic cylinder coordinates, and oblate spheroidal coordinates. In addition, Temme gives formulas for operators such as the gradient, Laplacian, divergence, and curl in different coordinates. This is indeed a very useful chapter for people working in applied fields. In Chapter 11, Temme considers some special functions that are related to statistical distribution functions. These include the error functions, incomplete gamma functions, and incomplete beta functions. They are related to the normal (Gaussian) distribution, the hZ-distribution, the beta distribution, and the noncentral ~-distribution. For each of these special functions, uniform asymptotic expansions are provided. Most of this work was done by Temme himself. Chapter 12 concerns elliptic integrals and elliptic functions. The presentation here is very attractive: it includes a derivation of the period of a simple pendulum, a discussion of the connection between elliptic integrals, and the iteration by the arithmetic geometric mean (AGM). There is also some information on numerical computation of elliptic integrals. The final chapter is on numerical aspects of special functions. It begins with a short summary of the literature on software programs for special functions, and lists some packages that are available for symbolic computations. The method for numerical computation based on recurrence relations is discussed in some detail, including a section on Miller's algorithm. Each chapter includes a short section of remarks and comments for further reading. Some recent papers on special functions, especially those related to their asymptotic behavior, are mentioned here. All chapters end with long lists of exercises, which include many of the important properties of special functions. All in all, this is an excellent book and is highly recommended as a text for courses on special functions. Even researchers in mathematics, physics, and

ence. In my view, it is comparable to the two well-known texts by Lebedev [7] and by Olver [10]. The printing of this book and the layout of the material are also very pleasing. Important formulas on each page are highlighted in boxes for easy reference. REFERENCES

1. A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS Appl. Math. Series 55, Washington, D.C., 1964. 2. V.A. Antonov and K. V. Hol~evnikov, An estimate of the remainder in the expansion of the generating function for the Legendre polynomials (Generalization and improvement of Bemstein's inequal-. ity), Vestnik Leningrad Univ. Math. 13 (1981), 161-166. 3. Y. Chow, L Gatteschi, and R. Wong, Bemstein-type inequality for the Jacobi polynomial, Prec. A.M.S. 121 (1994), 703--709. 4. E. T. Copson, Theory of Functions of a Complex Variable, Oxford University Press, London, 1935. 5. A. Erd61yi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953. 6. T. Lang and R. Wong, Best possible upper bounds for the first two positive zeros of the Bessel function J~ (x): The infinite case, J. Comput. Appl. Math. 71 (1996), 311-329. 7. N. N. Lebedev, SpecialFunctions and Their Applications, Prentice-Hall,EnglewoodCliffs, NJ, 1965. 8. L. Lorch, Inequalities for ultraspherical polynomials and the gamma function, J. Approx. Theory 40 (1984), 115-120. 9. L Lorch and R. Uberti, "Best possible" upper bounds for the first positive zeros of the Bessel functions--the finite part, J. Comput. Appl. Math. 75 (1996), 249-258. 10. F. W. J. Olver, Introduction to Asymptotics and Special Functions, Academic Press, New York, 1994. 11. E.T. Whittaker and G. N. Watson, A Course of Modem Analysis, Cambridge University Press, London and New York, 1927. Department of Mathematics City University of Hong Kong Tat Chee Avenue, Kowloon Hong Kong e-mail: [email protected]

9 1997 SPRINGER-VERLAG NEW YORK, VOLUME 19, NUMBER 4, 1997

"~'~

I[...l,~.z,,z,zit.z.z-,,,r=-z;.-- R o b i n

Wilson

Mathematical Prizes June Barrow-G?een and Robin Wilson

I

n 1888, the Russian mathematician

I Sonya Kovalevskaya

(1850-91)

gained international recognition when she won the Prix Bordin of the French Academy for her memoir on the rotation of a rigid body. Largely due to the originality of her work, involving the recently developed theory of Abelian functions, the value of the prize was raised from 3000 to 5000 French Francs. Kovalevskaya had earlier come to prominence when in 1874, as a private student of Weierstrass in Berlin, she gained a Ph.D. from GOttingen University for her work on partial differential equations, Abelian integrals and Saturn's rings, and so became the first female mathematician to be awarded a doctorate in mathematics. Ten years later, encouraged by the Swedish mathematician G~sta Mittag-Leffier, she was appointed assistant professor in Stockholm. Promoted to fifll professor in 1889, she was the first woman to hold a chair of mathematics in modern Europe. These stamps featuring Kovalevskaya were issued in 1951 and 1996.

Oscar II, King of Sweden and N o r w a y ( 1 8 2 9 - 1 9 0 7 ) , studied mathematics at the University of Uppsala. He became an active patron of the subject, supporting various publishing enterprises, including the founding of the journal Acta Mathematica. He promoted mathematical lectures, made awards to individual mathematicians and--most famously--sponsored an international competition to celebrate his 60th birthday in 1889. This competition, with a prize of 2500 Swedish

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

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THE MATHEMATICAL INTELLIGENCER 9 1997 SPRINGER VERLAG NEW YORK

Crowns, invited mathematicians to submit a memoir on one of four given topics in mathematical analysis. It was judged by Mittag-Leffler, Hermite, and Weierstrass, and the winner was the French mathematician Henri Poincar6. This stamp featuring King Oscar II was issued in 1891.

Henri

Poincar6

(1854-1912)

won King Oscar's prize for a memoir on the 3-body problem. The published version of his memoir appeared in Acta Mathematica in 1890 and contains the fLrst description of mathematical chaos. The memoir also provided the foundation for Poincar6's celebrated threevolume Mdcanique Cdleste. Arguably the most brilliant mathematician of his generation, he contributed to many different areas of mathematics and mathematical physics. He pioneered work in automorphic functions, the qualitative theory of differential equations, algebraic topology, and celestial mechanics, while making notable contributions to probability theory, potential theory, electricity, and thermodynamics. He was a masterful lecturer and wrote several popular works on the theory of scientific reasoning. This stamp was issued in 1954 to commemorate the 100th anniversary of his birth. June Barrow-Green Faculty of Mathematics and Computing The Open University Milton Keynes MK7 6AA England