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Letters

to

the

Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

Undigested Computer Output Among mathematicians writing about

mathematics, the late Richard Hamming was surely one of the most provocative, as is s h o w n by the review (vol. 20, no. 3) of his b o o k [2]. More of his stimulating and iconoclastic opinions can be found in his recent article in the MAA Monthly [3]. While trying to find the earliest writings on attractors in dynamical systems (information gratefully received!), I ran across Hamming's delicious 1965 review [1] of Stein and U]am's obscure pioneering w o r k [4] on computer simulations of dynamical systems; I cannot resist quoting it in full: This is a paper in the field of experimental mathematics using arithmetic. Experimental mathematics has always been widely practiced, but essentially only the arithmetic branch, centered around number theory, has achieved any status worth mentioning. The modern electronic computer is a handy sorcerer's apprentice for those who would enter the field. The paper considers m a i n l y the three-dimensional iteration transf o r m a t i o n s xi = Pi(xl, x2, xa), i = 1, 2, 3 where the Pi are cubics in xl, x2, x3. Some related transformations are also discussed. Many photographs of cathode ray tube displays are given, a fondness f o r citing large numbers of iterations and machine time used is revealed, and a crude classification of the

limited results is offered, but there appear to be no f i r m new results of general mathematical interest. It is an attractive idea that computers can produce m a n y specific cases of mathematical situations f r o m which we can obtain insight, but usually (though not always) it is the insight, not the specific cases, that has been the criterion f o r public presentation. One can only wonder what will happen to mathematics i f we allow the undigested outputs of computers to fill our literature. The present paper shows only slight traces of any digestion of the computer output. REFERENCES [1] R. Hamming, Review of Stein & Ulam [4]. Math. Reviews 29 (1965), # 6666, page 1248. [2] - - , The Art of Doing Science and Engineering; Learning to Learn, Newark, NJ: Gordon & Breach, 1997. [3] - - , "Mathematics on a distant planet," Amer. Math. Monthly 105 (1998), pp. 640-650. [4] P. R. Stein & S. M. Ulam, Non-linear transformation studies on electronic computers, RozprawyMatematycznevol xxxix. Inst. Mat. Polsk. Akad. Nauk. Warszawa: Panstwowe Wydawnictwo Naukowe (1964).

Morris W. Hfrsch Department of Mathematics University of California Berkeley, CA 94720 USA e-mail: [email protected]

9 1999 SPRINGER-VERLAG NEWYORK, VOLUME 21, NUMBER 2, 1999

3

Opinior

Real Fish, Real Numbers, Real Jobs N. J. Wildberger

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

here's good times ahead for biologists. The electronics revolution will plateau in the early decades of the new millennium, and quantum computing, gene manipulation, and pet cloning will kick in. We'll see bumper crops of biology graduates pouring into expanded university departments, research institutes, and biotech companies. Some decades later, however, with hemoglobin-monitoring wrist watches and desktop chicken mutators as plentiful a nuisance as mobile phones and laptops are now, the spotlight of public attention and funds will inevitably shift to consumer applications. Where will that leave the ranks of theoretical research biologists amassed by universities? My advice is to think now about saving the job prospects of future generations of academics by reconsidering and expanding the entire subject. That's where, I humbly suggest, toaster fish come in, and where mathematicians can provide some friendly help. A toaster fish is of course a creature with the head of a toaster (the pop-up kind) and the body of a fish. Thus Figure 1 shows a typical toaster fish. Toaster fish come in (at least) two different varieties--the North American kind, which operate on 120 volts, and the European kind, which use 220 volts. Already I hear a lot of you asking the obvious question: Can North

T

Please send all submissions to Mathematical Tourist Editor, Oirk H u y l e b r o u c k , Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

4

Figure 1. A toaster fish.

THEMATHEMATICALINTELLIGENCER9 1999SPRINGER-VERLAGNEWyOFIK

American and European toaster fish mate? This is typical of the important new research problems in the brave new world of biology I'm proposing. Biology needs to embrace the study of life in all its possibilities. There is no need to restrict attention to those creatures actually living, or having lived, or with the potential of living on planet earth and its immediate environs. These creatures are but a happy cross section of all possible life forms! By widening our consideration to arbitrary creatures, we can formulate new theories, create intellectual challenges, unleash our wildest imagination; and best of all, create jobs. Let's call this emerging field Life Theory. Toaster fish, baby universes, and slimy galactic superoctupi will all be embraced by this far-sighted new intellectual endeavour. But it won't be easy. There'll be colleagues who ridicule Life Theory. Those who call it a "theory about nothing" or "quasi-religions speculation on things which don't exist." Already one of my friends protests that Figure 1 amounts to little more than a poetic idea, since I have not "specified the internal workings" of the toaster fish figured. Fortunately, mathematicians have already encountered and triumphed over similar obstacles in their historic struggle to place real numbers squarely in the centre of all modern mathemati-

cal discourse. The M o d e r n Biologist can gain from a careful s t u d y of this imp o r t a n t achievement, w h i c h w e turn to n o w with an eye to the m a i n principles.

A Short History of Numbers and the Importance of Terminology The Neanderthals gave us 1, 2, and 3. The ancients a d d e d on 4, 5, 6, and so on, d i s c o v e r e d zero and negative numbers, and e m p l o y e d rationals like 22. 3 T h e y d i s c o v e r e d that o t h e r useful quantities like ~/2 and ~r w e r e n o t of this form a n d could only b e specified by algorithms, which subsequently allowed writing t h e m in binary e x p a n s i o n s like V ~ = 1.011010100... 7r = 11.00100100... The E u r o p e a n s a d d e d s o m e m o r e useful numbers: c o m p l e x n u m b e r s like 5 + 2i, and infmitesimal n u m b e r s like 1 + 3dx. It wasn't clear e x a c t l y w h a t t h e s e n u m b e r s were, b u t their use in the calculus was undisputed, so they remained. Until the n i n e t e e n t h century, that is, w h e n n e w s t a n d a r d s o f rigour r e v e a l e d that "an infmitely small quantity" was only a g r a m m a t i c a l construction bereft of any real m a t h e m a t i c a l meaning. So infinitesimals w e r e abandoned. But c o m p l e x n u m b e r s w e r e justiffed and constructible irrationals abounded, so we still h a d n u m b e r s galore for every conceivable problem. Then c a m e the greatest, the boldest leap of all. We introduced n u m b e r s so a b s t r a c t that they couldn't p o s s i b l y be n e e d e d for any concrete problem. How? We considered binary e x p a n s i o n s that couldn't be c o n s i d e r e d - - b y defimtion! Although this m a y s o u n d contradictory, it is not. Just c o n s i d e r a slimy galactic s u p e r o c t u p u s with an infinitely large brain, an infmite n u m b e r of arms, a n d an inclination to w a v e t h e m all simultaneously and independently either up or down. Looking at the configuration of arms at s o m e r a n d o m time gives a generally unconstructible sequence. As an aside, I mention that such a creature would be a useful friend, being able to solve the Goldbach Conjecture or the Riemann Hypothesis simply b y computation. And in an infinitesimally small amount of time to boot. More mundanely, an u n c o m p u t a b l e b i n a r y e x p a n s i o n is s i m p l y a b i n a r y ex-

p a n s i o n t h a t is not computable. It d o e s s e e m c u r i o u s that such a simple i d e a had e l u d e d all the great t h i n k e r s until the l a t t e r half of the nineteenth century. To envision such a n u m b e r w e p r o c e e d a s follows. R a n d o m l y w r i t e d o w n a string o f zeroes and ones; a d o z e n usually suffice, but if t h e audience is s k e p t i c a l I like to include more; and t h e n a d d three dots. Thus a = 1.011010111101... shows a typical unconstructible number. These n e w objects, when a d d e d on to the collection of rationals and constructible irrationals, gave us a w h o l e n e w d o m a i n of mathematical discourse. N o w a key point. What did w e call this n e w n u m b e r system consisting of largely chimerical creations of o u r imagination? Did we call them Arbitrarials? or Way-out n u m b e r s ? No. In a flash o f brilliance any PR p e r s o n w o u l d b e p r o u d of, w e called t h e m real n u m bers, a n d d e s i g n a t e d t h e m by the solidlooldng R. This inspired terminology is w o r t h emulating. I suggest you call the n e w a r e a o f M o d e r n Biology c o n c e r n e d with t o a s t e r fish, refrigerator fish, and so on the field of real fish. Those real fish actually f o u n d in the solar system will then have a special designation, subtly reinforcing the i d e a that it is unnatural to c o n s i d e r t h e m separately; let's say observable real fish. H o w m u c h m o r e of a b o t h e r it is to consider the latter. "Let a b e an observable real fish," you begin y o u r lecture. Where's it from? Is it green? What does it eat? s o m e o n e immediately asks. The preferable "Let a be a real fish" protects you from such unnecessary questions, which you can't ans w e r anyway! In mathematics, restriction to computable real numbers is a d e c i d e d nuisance. First one must worry a b o u t w h a t "computable" means. Which machine? Which language? About w h e t h e r or n o t p r o g r a m s halt. Simple operations like adding o r multiplying two numbers give us headaches. Real numbers free us from t h e s e hassles, which we p r e f e r to leave to the c o m p u t e r scientists anyway.

Training the Young H o w do w e get future g e n e r a t i o n s to t a k e t h e validity of real n u m b e r s for

granted? We i n d o c t r i n a t e t h e m early in their c a r e e r s w h e n t h e y are eager b u t impressionable undergraduates. Here's h o w we do it. First w e soften them up with a "Constructing the Real Numbers" blurb in t h e i r first calculus course. Needless to s a y w e d o n ' t really c o n s t r u c t real n u m b e r s as t h e y are b y definition unconstructible. But the p h r a s e sticks in t h e i r minds long after the details a r e forgotten. S o m e t i m e l a t e r w e e x p o s e them to the definition of a 1:1 c o r r e s p o n d e n c e , Cantor's w o n d e r f u l p r o o f t h a t there is no list of all b i n a r y sequences, a n d voil&! With a bit o f hand-waving a b o u t w h a t a set actually is, a v a s t and glit.tering n e w universe o f ordinals, cardinals, h i e r a r c h i e s o f infinities, and continuum h y p o t h e s e s opens up before their eyes. The c o n f u s i o n a n d apprehension of millenia cast aside, our students are s o o n fearlessly manipulating arbitrary unions o f arbitrarily m a n y p o w e r s o f u n c o u n t a b l e ordinals. It's a h e a d y experience, this first grasping hold o f infinity. But be warned; t h e r e will be s o m e s t u d e n t s w h o w o n ' t b u y this. After all, young p e o p l e a r e a t a d critical at this age. We explain to t h e m t h a t ff t h e y w a n t to get into "foundational questions" then t h e y h a d first b e t t e r m a s t e r the "predicate calculus o f first-order logic." Having b a r e l y m a s t e r e d the ordinary calculus, t h e y s e e m to find this a convincing argument. A further point. We d o n ' t start with y o u n g e r s t u d e n t s since t h e y get t o o easily confused. They w a n t to k n o w specifics like "How does one add o r multiply real numbers?" In fact one can't a d d o r m u l t i p l y real numbers, b u t o f course this s h o u l d n ' t a n d d o e s n ' t p r e v e n t us from using these o p e r a t i o n s on a dally basis. I m m a t u r e learners a r e likely to get s t u c k b y t h e s e subtleties. The snag is the following. Suppose we want to add the unconstructible real numbers 1.010101101000110 . . . and 1.100010010111001 . . . . Clearly the fLrst digits of the s u m a r e 10.11, b u t the n e x t digit requires m o r e information. H o w m u c h m o r e information? Possibly j u s t one m o r e digit, p o s s i b l y a trillion m o r e digits, and h e r e ' s the r u b - - p o s s i b l y infinitely m a n y m o r e digits! The complem e n t a r y p a t t e r n o f z e r o e s and o n e s

VOLUME 21, NUMBER 2, 1999

5

might continue to infinity, but if so we'll never know it. This prevents us from ever specifying a procedure to add real numbers. One might hope that trying some different definition of the reals using continued fractions or Dedekind cuts or Cauchy sequences of rationals might solve the problem, but it doesn't. It just transforms it to an equivalent one. But here's where religion, or Modern Biology, comes to our aid. It may be impossible for us to know whether a given complementary pattern of zeroes and ones continues indefinitely, but it is clearly not impossible for God, or a slimy intergalactic superoctopus, to know. So we are blessedly free to go on talking about adding arbitrary real numbers.

ca] objects too. This liberal approach is captured by what we might call the A x i o m o f F r e e d o m - - a n object is constructed, or specified (same thing), by simply stating it. To give but one of a multitude of possible uses of this principle, suppose you've fmally constructed a quasi-barrelled inverted amorphous fibre scheme S and all that remains is to exhibit the space P of all coherent quadratic functors from S to its dual. In years gone by this would have required fretting about precise definitions and concrete realizations. These days, you need only say, "Let P be the space of all coherent quadratic functors from S to its dual." Voile, the construction is complete.

Convincing the Professoriate This is a question primarily of critical mass. Once enough of the Academy accept and profit by the new bounds of the discipline, a critical momentum will prevent opponents from stemming the tide. First of all a leading luminary must be taken on board. In our case, that was David Hilbert, whose oft quoted "No-one shall expel us from the paradise that Cantor has created for us" becomes our motto. 1 Furthermore, we must convince our colleagues that opportunities for research are rife. In mathematics, the kind of free thinking that allows the pleasant contemplation of real numbers (I like to think of them as basking in the sun) also revels in more complicated abstractions. Unlike earher more squeamish generations, we are quite comfortable with arbitrary functions from ~ to (not just those given by formulae or computer programs but all the others too), with arbitrary operators on spaces of such functions, with arbitrary functionals on algebras of such operators and so on. Non-measurable functions, Banach-Tarski paradoxes, various versions of the Axiom of Choice, inaccessible cardinals---none of these ruffles our calm in the brave new world of twentieth-century mathematics. More than just jobs are opened up here though. Our easy attitude to sets can be extended to other mathemati-

Setbacks and Recovery At the turn of the century, with mathematicians divided in controversy over the validity and meaning of Cantor's new theory and the role of real numbers, disaster struck. In one of the most embarrassing intellectual deflations of all time, the entire new theory came crashing down with the discovery of fundamental new paradoxes. These weren't just the niggling kind of inconsistencies acknowledged from the start even by Cantor, they were the full-blown ~r = V 2 type of contradictions that ruin one's entire week. A dark time indeed, but ultimately only a minor setback. After some hesitant and unconvincing attempts to shore up the subject, we now see that worrying about such issues is largely a waste of time. Foundations are not really all that necessary. If pressed by philosophical types, we have a fall-back position. Mathematics is based on Axioms, and the Axioms for Set Theory were spelt out by Zermelo and Fraenkel. This is the Formalist position, also pushed by Hilbert, which got rather badly bruised by G6del's results in the 1930s, As a resuit, most of us closet Platonists aren't so keen on this official explanation, but who really wants to get embroiled in philosophical hair-splittings? An intellectual discipline reflects the spirit and values of the times, and these are times of pragmatism and expediency.

If pressed by our own ldnd, by mathematicians who try to point out the imprecision of our cherished fundamental concepts, we respond politely. We allow them their say, listen patiently, and then point out that they're basically only repeating Brouwer's critique of 1918. And what about those who obstinately refuse to understand? Who keep on insisting that you explain "What exact/y is a real fish?" Sigh. These people are quibblers---philosophical quibblers at that--and they are best ignored. After all, you've got a job to do.

Looking to the Future As pragmatism and expediency don't look to be departing the modern world anytime soon, I'm confident that the lessons mathematicians have learnt can be fruitfully applied to biology. Don't expect to convince your contemporaries about the importance of real fish. Recruit a few top names,

1Naturally we don't advertise, for example, that Gauss wrote, "1 protest--against the use of infinite magnitude as if it were something finished; this use is not admissible in mathematics. The infinite is only a fagon de parler; one has in mind limits approached by certain ratios as closely as desired while other ratios may increase indefinitely."

6

THE MATHEMATICALINTELLIGENCER

work on the future generations, keep the employment prospects firmly in view, and Life Theory will take its rightful place in the scientific world. As for me, this little advice-giving has got me to thinking. Mathematicians are themselves a bit hard pressed these

days for employment. The huge amount of time and effort required to rewrite m o d e m mathematics using only computable numbers, computable functions, computable everythings would keep us busy for decades. What an unpleasant lot of hard work that would be.

But hard work means jobs, grants, and pensions! Maybe Aristotle, Archimedes, Newton, Euler, Gauss, Kronecker, Borel, Lebesgue, Poincar6, Weyl, Brouwer, and the others were right. There is something fishy about real numbers.

VOLUME 21, NUMBER 2, 1999

'~

work on the future generations, keep the employment prospects firmly in view, and Life Theory will take its rightful place in the scientific world. As for me, this little advice-giving has got me to thinking. Mathematicians are themselves a bit hard pressed these

days for employment. The huge amount of time and effort required to rewrite m o d e m mathematics using only computable numbers, computable functions, computable everythings would keep us busy for decades. What an unpleasant lot of hard work that would be.

But hard work means jobs, grants, and pensions! Maybe Aristotle, Archimedes, Newton, Euler, Gauss, Kronecker, Borel, Lebesgue, Poincar6, Weyl, Brouwer, and the others were right. There is something fishy about real numbers.

VOLUME 21, NUMBER 2, 1999

'~

GIAN-CARLO ROTA

Combinatorial Snapshots* W

hen I was in high school, m y English teacher gave me to read a story by James Thurber, called "The Secret Life of Walter Mitty. "After rereading this story every f e w years, I decided that everyone has a Walter Mitty complex.

One w a y to u n d e r s t a n d a p e r s o n might b e to d i s c o v e r that p e r s o n ' s Walter Mitty fantasies. Most of our m a t h e m a t i c a l thoughts in high school or in college were Walter Mitty fantasies. When w e l e a r n e d a n e w piece o f math, w e w o u l d fmd ourselves fantasizing on its possible generalizations. As soon as w e u n d e r s t o o d binomial coefficients, w e fantasized about their generalization to the case w h e n t h e d e n o m i n a t o r is negative; the m o m e n t w e learned a b o u t derivatives, w e l a u n c h e d into derivatives of fractional order. If w e w e r e ever e x p o s e d to the Riemann zeta function, w e w o u l d romanticize s o m e n e w interpretation of this function that would give a w a y its secret. This lecture s h o u l d have b e e n given a n o t h e r title. It should be called "The Later Life of W a l t e r Mitty." It will consist of a s e q u e n c e o f displays of chutzpah b y a Walter Mitty w h o h a s lost his shyness. E a c h s n a p s h o t will deal with s o m e youthful fantasy that has p a r t i a l l y c o m e true.

First Snapshot: An Example of Profinite Combinatorics Let us begin with a piece of history-fiction, and imagine how Riemann might have discovered the Riemann zeta function. P r o f e s s o r R i e m a n n was a w a r e that a r i t h m e t i c density is of fundamental i m p o r t a n c e in n u m b e r theory. I r A is a subset of the set of positive integers N, t h e n the arithmetic density of the set A is defined to be

dens(A) = l i m 1 IA rq {1, 2 , . . . , n'-*~176 n

w h e n e v e r the l i m i t exists. F o r example, dens(N) = 1. flAp is the s e t of multiples of t h e p r i m e p, t h e n dens(Ap) = 1/p; w h a t is m o r e appealing is if o n e easily c o m p u t e s t h a t dens(Ap (q Aq) = 1/pq for a n y t w o p r i m e s p and q. If density w e r e a (countably additive) p r o b a b i l i t y m e a s u r e , w e w o u l d infer that the events t h a t a r a n d o m l y c h o s e n numb e r is divisible by either o f t w o p r i m e s are i n d e p e n d e n t . Unfortunately, arithmetic d e n s i t y s h a r e s s o m e b u t n o t all p r o p e r t i e s of a probability m e a s u r e . It is m o s t e m p h a t i c a l l y n o t c o u n t a b l y additive. After a p e r i o d of soul-searching, P r o f e s s o r R i e m a n n w a s able to find a r e m e d y to s o m e deficiencies of a r i t h m e t i c d e n s i t y b y a brilliant leap o f imagination. He c h o s e a real n u m b e r s > 1 and d e f m e d t h e m e a s u r e of a positive integ e r n to equal 1/nS; in this way, the m e a s u r e of the set N t u r n e d out to equal

n=l

Therefore, he could define a (eountably additive) probability m e a s u r e P~ on the set N o f positive integers by setting

P~(A) =

1 ~

1

*The third of three Colloquium Lectures delivered at the Annual Meeting of the American Mathematical Society, Baltimore, January 9, 1998.

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THE MATHEMATICALINTELLIGENCER@ 1999 SPRINGER-VERLAGNEWYORK

nJll

R i e m a n n t h e n p r o c e e d e d to verify w h a t he h a d s e n s e d all along, n a m e l y the f u n d a m e n t a l p r o p e r t y 1 Ps(Ap A Aq) = Ps(Ap)Ps(Aq) = - - . Pq

In o t h e r words, the events Ap and Aq that a r a n d o m l y chosen integer n be divisible b y one o f the two p r i m e s p or q are i n d e p e n d e n t relative to t h e probability Ps. The Riemann z e t a function w a s good for something, after all. I will n o w use a r h e t o r i c a l device that w a s effectively e m p l o y e d b y one of m y u n d e r g r a d u a t e teachers, P r o f e s s o r Bochner. In the classroom, P r o f e s s o r B o c h n e r w o u l d prefix t h e s t a t e m e n t o f a t h e o r e m b y the words: "Subject to t e c h n i c a l assumptions, t h e following is true," without, of course, ever disclosing w h a t his technical a s s u m p t i o n s were. P r o f e s s o r Riemann t h e n p r o c e e d e d to s h o w that, subj e c t to technical a s s u m p t i o n s on the set A, lim Ps(A) = dens(A).

S--*]

Thus, even though a r i t h m e t i c density is not a probability, it is u n d e r suitable c o n d i t i o n s the limit of probabilities. Long after Riemann w a s gone, it was shown, again subj e c t to technical a s s u m p t i o n s , that the probabilities Ps are the oniy probabilities d e f m e d on the set N o f n a t u r a l integers for w h i c h the events of divisibility by different p r i m e s a r e independent. This fact s e e m s to lend s u p p o r t to the p r o g r a m of proving results o f n u m b e r t h e o r y b y p r o b a bilistic m e t h o d s b a s e d o n t h e Riemann zeta function. Why didn't P r o f e s s o r R i e m a n n ever publish this wonderful i d e a o f his? The a n s w e r is not hard to fred. True, s o m e t h e o r e m s of n u m b e r t h e o r y can be p r o v e d p r o b a bilistically b y this limiting p r o c e s s - - f o r e x a m p l e Dirichlet's t h e o r e m on p r i m e s in a r i t h m e t i c progression. However, d e e p e r n u m b e r - t h e o r e t i c results have, to this day, e l u d e d this approach; for example, n o one has s u c c e e d e d in proving the prime n u m b e r t h e o r e m b y this method. P r o f e s s o r Riemann, a w a r e of this deficiency, t h r e w his n o t e s into the w a s t e b a s k e t and p r o c e e d e d to link the R i e m a n n z e t a function to the distribution o f p r i m e s in an altogether different way, b y stating the h y p o t h e s i s that b e a r s his n a m e a n d that r e m a i n s u n p r o v e d to this day. Why a m I telling y o u this bit of history-fiction? B e c a u s e I w a n t to p r o p o s e a n o t h e r probabilistic i n t e r p r e t a t i o n of t h e R i e m a n n zeta function t h a t is quite different f r o m the i n t e r p r e t a t i o n j u s t outlined. Let us c o n s i d e r a p r o b l e m in c o m b i n a t o r i a l e n u m e r a tion. Let us take a cyclic group of o r d e r r, s a y Cr. E v e r y c h a r a c t e r X of the group Cr has a kernel w h i c h is a subgroup of Cr. More generally, every sequence X1, X2, 9 9 9 Xs o f c h a r a c t e r s o f Cr has a j o i n t kernel which is also a subgroup of Cr; the j o i n t k e r n e l o f a sequence of c h a r a c t e r s is simply the intersection o f their kernels. If a s e q u e n c e X1, X2, 9 9 Xs of s c h a r a c t e r s is c h o s e n i n d e p e n d e n t l y a n d at r a n d o m , w h a t is the p r o b a b i l i t y that the j o i n t k e r n e l of the s e q u e n c e equals a given s u b g r o u p Cn of Cr?

The p r o b a b i l i t y of the event that t h e kernel o f a randomly c h o s e n c h a r a c t e r will c o n t a i n the subgroup Cn equals 1/n, as there a r e r c h a r a c t e r s o f the group Cr and r/n such c h a r a c t e r s will vanish on Cn. Therefore, the probability t h a t t h e j o i n t kernel o f a r a n d o m l y c h o s e n sequence X1, X2, Xs o f S c h a r a c t e r s shall c o n t a i n the subgroup Cn equals ( l / n ) s. Let us denote b y PCnt h e p r o b a b i l i t y that the joint k e r n e l o f the c h a r a c t e r s X1, X2,. 9 9 Xs shall equal the subgroup Cn. Then, w e have the identity 9

9

9

Here, w e use t h e fact that the p a r t i a l l y o r d e r e d set of subgroups o f a cyclic group Cr is i s o m o r p h i c to the partially o r d e r e d set o f divisors of the integer r. We n o w u s e the MSbias inversion f o r m u l a o f n u m b e r theory, t h e r e b y obtaining

Here,/~(]) is t h e MObius function of n u m b e r theor~j. After the change of variable d -- nj, w e can r e c a s t t h e right-hand side as follows: Pcn

1

1

= ~ ~ t~(j) 7

The variable j on the right ranges o v e r s o m e s u b s e t of divisors o f the integer r, which w e n e e d n o t w o r r y about. Now, if t h e s u m on the right r a n g e d o v e r all positive integers j , t h e n the right-hand side w o u l d equal 1

1

n s

~(s)'

that is, it c o u l d be e x p r e s s e d in t e r m s o f the inverse of the Riemann z e t a function. If we c o u l d c h a n g e o u r combinatorial p r o b l e m to get an u n r e s t r i c t e d s u m on the right-hand side, then w e w o u l d have a p r o b a b i l i s t i c interpretation o f the R i e m a n n z e t a function. This is d o n e by replacing the finite cyclic group Cn b y a profinite cyclic group. C o n s i d e r t h e group C~ o f rational n u m b e r s m o d u l o 1. F o r every positive integer n, the group C~ has a unique finite subgroup Cn w i t h n elements. The c h a r a c t e r group C* of C~ is a c o m p a c t group; it h a s a Haar measure, w h i c h is a p r o b a b i l i t y m e a s u r e P. The group C* is the d e s i r e d profinite group on w h i c h w e can generalize the p r e c e d i n g computation. The s e t o f all c h a r a c t e r s o f t h e g r o u p Coo (i.e., the set o f all e l e m e n t s o f the group C*) w h i c h v a n i s h on a subgroup Cn of C~ h a s H a a r m e a s u r e equal to 1/n. Thus, if w e c h o o s e a sequence X1, X2, 9 9 9 Xs of s c h a r a c t e r s of Coo independently a n d at random, the p r o b a b i l i t y that their j o i n t kernel will c o n t a i n the group Cn equals (1/n) s. If w e again denote b y Pc~ t h e probability t h a t t h e j o i n t kernel of a sequence X~, X2, 9 9 9 Xs of s c h a r a c t e r s equals the group Cn, then w e have the identity

I

~ Pc~,

VOLUME 21, NUMBER 2, 1999

9

where the s u m on t h e right is now infinite. Again, b y the M6bius inversion formula, w e obtain

Pc,, =

t

1

_1

tx(d/n) g---~= nS ~(s)"

This is the p r o m i s e d probabilistic i n t e r p r e t a t i o n of the Riemalm zeta function. Some p r o p e r t i e s o f t h e Riemann zeta function can b e p r o v e d p r o b a b i l i s t i c a l l y using this i n t e r p r e t a t i o n - - f o r example, the p r o d u c t formula. It remains to be s e e n w h i c h o t h e r p r o p e r t i e s o f t h e Riemann zeta function c a n b e p r o v e d in this way. The p r e c e d i n g a r g u m e n t is an i n s t a n c e o f a generalization of an e n u m e r a t i o n p r o b l e m on a finite set to an enum e r a t i o n on a p r o f m i t e set. Such a r e p l a c e m e n t of a finite set by a p r o f m i t e "set" w o r k s in o t h e r c o m b I n a t o r i a l p r o b lems. Will w e e v e r have a profmite c o m b i n a t o r i c s on profinite sets side b y side with c o m b i n a t o r i c s on finite sets?

Second Snapshot: The Cyclic Derivative The ordinary derivative of a p o l y n o m i a l in one variable has b e e n generalized b y H a u s d o r f f to p o l y n o m i a l s and formal p o w e r series in n o n c o m m u t a t i v e v a r i a b l e s as follows. Consider the a s s o c i a t i v e algebra C(a, b , . . . , c, x) genera t e d b y a set of letters {a, b , . . . , c, x}. The l e t t e r x is called a variable: all o t h e r letters a r e called constants. A monomial in this a s s o c i a t i v e a l g e b r a is w h a t y o u t h i n k it should be; it is a w o r d like

m = axbaxSbcxd. A p o l y n o m i a l is a linear c o m b i n a t i o n of monomials, and a formal p o w e r s e r i e s is defined as an inf'mite s u m of monomials, with suitable restrictions on the g r o w t h of degrees of the s u m m a n d s . F o r m a l p o w e r series in n o n c o m m u t a t i v e variables form an a l g e b r a C((a, b, . . . , c, x}). We will denote b y f ( x ) s u c h a formal p o w e r series. The H a u s d o r f f derivative of the m o n o m i a l m is comp u t e d as follows:

H ( m ) = H(axbax3bcxd) = abax3bcxd § 3axbax2bcxd + axbax3bcd. This definition is e x t e n d e d b y linearity to p o l y n o m i a l s and to formal p o w e r series. If m ' is a n o t h e r monomial, w e have the e x p e c t e d nile for finding the H a u s d o r f f derivative of a product: H ( m m ' ) = H ( m ) m ' + m H ( m ' ) . The Hausdorff derivative suffers from a m a j o r w e a k n e s s . There s e e m s to be no analog o f the chain rule for the differentiation of a function o f a function. F o r example, the H a u s d o r f f derivative of the p o l y n o m i a l ( a x ) n, w h e n the letters a a n d x do not c o m m u t e , is n o t equal to n ( a x ) n - l a . It is a mess. There is a n o t h e r notion o f derivative that d o e s satisfy a simple chain rule u n d e r functional c o m p o s i t i o n . It is the cyclic derivative, d e n o t e d b y the letter D. The cyclic derivative is deemed a s follows. First, define the truncation o p e r a t o r T as follows: (a) ff the first l e t t e r of a m o n o m i a l m is n o t the variable x, set T(m) = O.

10

(b) If the first letter of a m o n o m i a l m is the v a r i a b l e x, s o that m = x m ' , s e t T ( m ) = m'. (c) E x t e n d b y linearity to C((a, b , . . . , c, x)}.

THE MATHEMATICALINTELLIGENCER

The cyclic derivative of a m o n o m i a l m is defined in t e r m s o f the t r u n c a t i o n o p e r a t o r as follows: (a) Let p be the p o l y n o m i a l o b t a i n e d b y adding all cyclic p e r m u t a t i o n s of the m o n o m i a l m. (b) Set D ( m ) = T(p). (c) E x t e n d by linearity to all formal p o w e r series. F o r example, the cyclic derivative of the above m o n o m i a l m is c o m p u t e d in the following steps: S t e p 1. Write d o w n all cyclic p e r m u t a t i o n s of t h e m o n o mial axbax3bcxd. These a r e

xbax3bcxda, baxSbcxdax, ax3bcxdaxb, x3bcxdaxba, x2bcxdaxbax, x b c x d a x b a x 2, bcxdaxbax 3, cxdaxbax3b, xdaxbax3bc, daxbax3bcx. S t e p 2. In the above list, p e r f o r m one o f the following operations: (a) If the first letter o f a m o n o m i a l is not x, r e m o v e t h e m o n o m i a l from the list. (b) If the fn-st letter of a m o n o m i a l is x, r e m o v e the first letter. When w e p e r f o r m operations (a) and (b) on each of the m o n o m i a l s in the above list, w e obtain a shorter list, n a m e l y

bax3bcxda, x2bcxdaxba, xbcxdaxbax, bcxdaxbax 2, daxbax3bc. S t e p 3. A d d the m o n o m i a l s t h u s o b t a i n e d to get the cyclic derivative:

D ( m ) = D(axbax3bcxd) = bax3bcxda + x2bcxdaxba + x b c x d a x b a x + bcxdaxbax 2 + daxbax3bc. A n o t h e r e x a m p l e is as follows: The cyclic derivative o f the m o n o m i a l a x b x c x d x equals

D ( a x b x c x d x ) = b x c x d x a § cxdxaxb + d x a x b x c x + axbxcxd. The cyclic derivative of the m o n o m i a l (ax) n is t h e r e f o r e n ( a x ) n - la. Similarly, one c o m p u t e s D ( x + a) n = n ( x + a) n - t and, for formal p o w e r series, D(e x+a) = e x+a and D(e ~z) = eaxa. Remember, the letters a a n d x d o n o t commute! In t h e s e examples, the corresponding Hausdorff derivative is a mess. The cyclic derivative enjoys all p r o p e r t i e s e x p e c t e d o f the o r d i n a r y derivative; in particular, it satisfies t h e chain rule for the c o m p o s i t i o n o f t w o formal p o w e r series. To state the rules for t a k i n g cyclic derivatives m o r e generally, w e n e e d one m o r e o p e r a t o r , called the w r a p p i n g operator. The wrapping o p e r a t o r is d e f m e d as follows. Let

c1, c2,

.

.

.

,

Cn be any letters. If g(x) is any formal p o w e r se-

ries, set

(6ClC2. . . cn!g(x)) = ClC2 "'" C n g ( X ) + C2 " " C n g ( X ) C 1 + C3 "'" C n g ( X ) C l C 2 A- "" 4- c n g ( X ) C l C 2 " " C n _ 1.

I f f ( x ) is a n y formal p o w e r series, t h e w r a p p i n g o p e r a t o r

(C f ( x ) lg(x)) is defined b y linearity. Define

of an algebraic formal p o w e r series in noncommutative letters is, again, an algebraic formal p o w e r series. Despite t h e evidence that the cyclic derivative is the natural n o t i o n o f derivative for n o n c o m m u t a t i v e algebras, the theory as it is at p r e s e n t is n o t satisfying. The cyclic derivative is an empirical discovery. It n e e d s to be e n s c o n c e d in s o m e b r o a d e r algebraic theory, m u c h as the Hausdorff derivative h a s b e e n e n s c o n c e d in the t h e o r y of H o p f algebras.

Third Snapshot: Logarithms and the Binomial Theorem The E u l e r - M a c L a u r i n

(D(f(x))lg(x)) = T(C f(x)lg(x));

s u m m a t i o n f o r m u l a is o n e o f t h e

most r e m a r k a b l e formulas of m a t h e m a t i c s . F o r a suitable function f ( x ) o f a real o r c o m p l e x variable, it is stated as follows:

for example,

(D(f(x))ll) = D(f(x)). The cyclic derivative of t h e p r o d u c t of two "functions" is given b y the following identity:

D(f(x)g(x)) = (D(f(x)){g(x)} + (D(g(x))~f(x)).

f(x) + f ( x + 1) + f ( x + 2) + ... + f ( x + n) +n+l

= Bo

ix

vx

f ( y ) d y + B l ( f ( x + n + 1) - f i x ) )

+ ~B2 D( f ( x + n + 1) - f i x ) )

F o r example, one o b t a i n s D((1 - a x ) - t ( 1 - bx) -t) = (1 - a x ) - l ( 1 - b x ) - l ( 1 - a x ) - l a + (1 - b x ) - ] ( 1 - a x ) - l ( 1 - bx)-lb. No s u c h identity h o l d s for t h e Hausdorff derivative. The cyclic derivative of t h e p r o d u c t of any s e q u e n c e of f o r m a l p o w e r series is similarly c o m p u t e d by the w r a p p i n g operator:

D ( f l (x)f2(x) "'" fn(X)) = (D(ft(x)){f2(x)"" fn(X)> + {D(f2(x))lf3(x) "" f~(x)fl(x))) + ... + (D(fn(X)){fl(x)f2(x) "'" fn-i(X)). We c o m e n o w to t h e m a i n p r o p e r t y o f t h e cyclic derivative: t h e chain rule. Given t w o formal p o w e r s e r i e s f ( x ) a n d g(x) in C((a, b, . . . c, x)), a s s u m e t h a t t h e f o r m a l p o w e r s e r i e s g(x) d o e s n o t h a v e a c o n s t a n t term. U n d e r t h e s e c i r c u m s t a n c e s , t h e c o m p o s i t i o n f(g(x)) is w e l l def i n e d b y r e p l a c i n g g(x) f o r e v e r y o c c u r r e n c e o f t h e varia b l e x in the f o r m a l p o w e r s e r i e s f(x). Let us w r i t e Dg(f(x)) to d e n o t e t h e f o r m a l p o w e r s e r i e s o b t a i n e d b y s u b s t i t u t i n g g(x) in p l a c e o f e v e r y o c c u r r e n c e o f x in the cyclic derivative D ( f ( x ) ) of t h e formal p o w e r s e r i e s f(x). Then, t h e chain rule for t h e cyclic derivative g o e s as follows:

D(f(g(x)))

=

(Dg(x)~Dg(f(x))).

F o r example, we have

D(e axbx) = bxeaxbxa + eaxbxaxb. A m o r e elegant e x a m p l e is the following: D(e(l_ax ) 1) = (1

-

ax)-le (1-ax)-I (1

-

-

ax)-ta.

One can prove that the cyclic derivative of a rational formal p o w e r series in n o n c o m m u t a t i v e letters is, again, a rational n o n c o m m u t a t i v e p o w e r series, and that the cyclic derivative

+ ~ , D 2 ( f ( x + n + 1) - f i x ) ) +

4 @ wq

o , ' ,

The Bn are t h e Bernoulli n u m b e r s a n d D is the ordinary derivative o p e r a t o r . The E u l e r - M a c L a u r i n f o r m u l a h a s p r o v e d v e r y useful for o v e r 200 years. Nonetheless, the E u l e r - M a c L a u r i n form u l a suffers f r o m a serious deficiency. The series on the right-hand side is a l m o s t n e v e r convergent, unless it reduces to a finite sum. Our question is the following: Is t h e r e a vector space o f functions w h i c h contains as m a n y of the elementary functions as possible, and a topology on such a vector space, relative to w h i c h the right-hand side o f the Euler-MacLaurin formula is a convergent series? The a n s w e r to this question is u n e x p e c t e d l y related to the a n s w e r to a n o t h e r question. What is the "right" generalization o f t h e binomial coefficients (~) w h e n k is allowed to b e a negative integer? This question leads, in turn, to a third question: H o w shall w e k n o w w h e t h e r a generalization of the b i n o m i a l coefficients is "right"? The a n s w e r to this third question is easy: a generalization of the binomial coefficients is "right" if it leads to a s e n s i b l e generalization of t h e b i n o m i a l theorem:

(aWx)n=Z

(nk)

When I w a s young, I used to t h i n k o f the binomial theo r e m as trivial. I think I have l e a r n e d m y lesson. A wellk n o w n p h i l o s o p h e r , I can't r e m e m b e r his name, w r o t e that the w h o l e universe can be inferred from a grain of sand. He s h o u l d have a d d e d that a great deal o f m a t h e m a t i c s c a n be d e r i v e d b y meditating u p o n the b i n o m i a l theorem. Let us t a k e t h e bull b y the h o r n s a n d s t a t e t h e "right" g e n e r a l i z a t i o n o f t h e b i n o m i a l coefficients. We p r o c e e d in the m o s t p e d e s t r i a n way, b y first g e n e r a l i z i n g the definition o f t h e factorial. Thus, let n b e a n y integer, p o s i -

VOLUME 21, NUMBER 2, 1999

11

tive o r negative. We define the R o m a n f a c t o r i a l [n]! as follows:

[n]!=n[ [n]! -

i f n - > 0;

(--1) n+l (-n-

1)!

to b e t h e unique indefmite integral of the f u n c t i o n f ( x ) t h a t h a s c o n s t a n t t e r m equal to zero. Do n o t worry, this will m a k e s e n s e in a moment. Define

X~)(x) = [n]!D -n log x.

if n < 0.

F r o m w h e r e d o e s this definition c o m e ? I c o u l d simply s a y that it works, b u t that w o u l d not be the w h o l e truth. The value of the R o m a n factorial [n]! for n negative equals the residue of the g a m m a function at the integer n - 1. Using the R o m a n factorial, w e define the R o m a n coefficients as follows:

Here, n is any integer, positive o r negative. The functions A~l)(x) a r e called the h a r m o n i c logarithms of o r d e r 1. F o r n positive, w e have

(

9~ l ) ( x ) = x n l o g x - l - ~ - ~

11

....

1)

and [ k ] ! ~ ~ kl!"

A~n(x) = ~nn"

When n -> k - 0, t h e R o m a n coefficients c o i n c i d e with the binomial coefficients. F o r all integers n a n d k, the R o m a n coefficients s h a r e all e l e m e n t a r y p r o p e r t i e s o f binomial coefficients, such as P a s c a l ' s triangle a n d so on. However, there are s o m e s u r p r i s e s in store; for e x a m p l e , for k positive, w e find

Does this m a k e any sense? Well, yes, b e c a u s e w e can find a generalization o f the binomial t h e o r e m t h a t goes with this. It is the following. Recall the p o w e r series e x p a n s i o n o f the logarithm: l o g ( x + a) = log x +

~ (--1) k+l a k k= ~ k Xk"

Of course, w e also have A(~)(x) = log x. We a r e n o w in a p o s i t i o n to s t a t e the generalization o f t h e b i n o m i a l t h e o r e m a s s o c i a t e d with the h a r m o n i c logarithms. It goes as follows:

~)(x+a)=Z

The a b o v e three identities are special cases o f this identity, for n = 0, 1, and 2. The generalization of the binomial t h e o r e m to h a r m o n i c logarithms gives nothing n e w for negative exponents, w h e r e it r e d u c e s to the identity

(x+a)-n=~" ( ~ )

We can r e c a s t this p o w e r series e x p a n s i o n in t e r m s of the R o m a n coefficients as follows: l o g ( x + a) = log x +

k=l

xk.

This is beginning to l o o k like a generalization of the binomial theorem, with the logarithm playing the role o f zeroth power. Another p o w e r series expansion in w h i c h the Roman coefficients m a k e their a p p e a r a n c e is the following:

[k]

However, for positive e x p o n e n t s , w e obtain a genuine a n d baffling generalization of t h e binomial theorem. It s t a t e s that the functions A~l)(x), for n positive, satisfy the ordin a r y binomial theorem, m o d u l o negative p o w e r s o f x. In o t h e r words, w e have t h e following identity: (x+a)

n

(l o g @ + a ) - i

11 3

n')

2

(x + a ) ( l o g ( x + a) - 1) = k=0 Do w e see a p a t t e r n ? Well, let us try y e t a n o t h e r p o w e r series expansion:

(x + a)2 (log(x + a) _ 1 _ l )

+[:]a210gx+s

kx2-k.

Now, w e can leap to a generalization. F o r suitable functions f ( x ) , set

D-if(x)

12

THE MATHEMATICALINTELLIGENCER

2

3

n

)

k "

The i d e n t i t y is valid m o d u l o n e g a t i v e p o w e r s o f x. M i r a c l e s o f c a n c e l l a t i o n a r e o c c u r r i n g in this identity. I wish I knew a combinatorial or probabilistic interpretat i o n o f this l o g a r i t h m i c g e n e r a l i z a t i o n of the b i n o m i a l theorem. So far, w e have a s s u m e d t h a t all series converge in t h e t o p o l o g y o f the c o m p l e x n u m b e r s . We will n o w c h a n g e t h e topology, while retaining the convergence. The motivation for t h e logarithmic t o p o l o g y w e a r e a b o u t to define is the a l g e b r a o f formal Laurent series. This t o p o l o g i c a l a l g e b r a m a y b e d e f i n e d by defining a t o p o l o g y on the a l g e b r a of rational functions in the variable x, a n d t h e n c o m p l e t i n g this a l g e b r a relative to the topology. The t o p o l o g y is so c h o s e n as to have linln____>oo X-n = O. E v e r y el-

ement of the completed algebra turns out to be a formal Laurent series, that is, a series of the form

We can n o w return to the Euler-MacLaurin summation formula:

~. anxn.

T H E O R E M . For every element f ( x ) o f the logarithmic algebra, the right-hand side o f the E u l e r - M a c L a u r i n series converges i n the logarithmic topology. For example, the following infinite series is convergent in the logarithmic topology:

n.~d

We want to perform an analogous completion process on another algebra: the algebra generated by all functions of the form x~(log x) t, where n is any integer, positive or negative, and where t is a non-negative integer. In order to specify which elements of this algebra are to converge to zero, we need a better-behaved basis of this algebra. This basis is provided by the harmonic logarithms of arbitrary order t. They are defined as follows: )~)(x) = [n]!D-n(log x) t

log x + log(x + 1) + log(x + 2) + .-- + log(x + n) = Bo((x + n + l) log(x + n + l) - x log x - n -1)

(1

+ Bl(log(x + n + 1) - log x) + ~ . w2 x + n + l Another example is the following. As y o u know, the sum

for every non-negative integer t and for every integer n. For example, we have

x k + ( x + 1) k + (x + 2) k + "" + (x + n) a can be expressed in closed form by the Euler-Maclaurin formula. The preceding theorem leads to analogous closedform expressions for sums of the form

=

for every non-negative integer n, and

x k log x + (x + 1) k log(x + 1) + (x + 2) k log(x + 2) + ... + ( x + n) k log(x + n).

22)(x) = 0

for negative n. Explicit expressions are k n o w n for the harmonic logarithms. For the harmonic logarithms of order 2, we have )t(02)(x) = (log x) 2, and for n positive, ~ ) ( x ) = x n ( (log x) 2 - ( 2 + 2 + ' " + 2 )

l~

+ 2

+---

§

and

The harmonic logarithms have other applications; let me mention one in closing. Recall the defmition of the shift operator of the calculus of finite differences: E a f(x) = f ( x + a). For n a non-negative integer, define the operator E1 as follows: E l , ~ ) ( x ) = )~)(x).

(

1

h(-2)(x) = 2x -n log x - 1

2

n

1)1 "

In ordinary notation, this is the same as saying E l x n = x n (log x - 1 - 1/2 - 1/3 . . . . .

For every non-negative integer t, the harmonic logarithms of order t satisfy the same generalization of the binomial theorem that we have already seen for the harmonic logarithms of order 1: "k(tn)(x+a)=Z[k] The harmonic logarithms are a basis of the algebra generated by all functions xn(log x ) t. We defme a topology on this algebra by requiring that lim n--~

=

o

-oo

for every non-negative integer t. This topology is called the logarithmic topology. The completion of this algebra relative to the logarithmic topology is the algebra of formal p o w e r series of logarithmic type, or logarithmic algebra. Every element of the logarithmic algebra is a linear combination of convergent p o w e r series of the form f(x) = Z t,n~--d

ranging over a finite set of values of t.

i/n).

One can prove the following two propositions: P R O P O S I T I O N . The operators E a a n d E1 commute. P R O P O S I T I O N . The restriction o f the derivative operator D to the subalgebra o f the logarithmic algebra generated by the h a r m o n i c logarithms X(nt)(x)for p o s i t i v e t (i.e., e x c l u d i n g the non-negative p o w e r s o f x ) is invertible. These two propositions can be used to obtain "logarithmic extensions" of special functions. Let us conclude with the simplest example: let us compute the logarithmic extension of the sequence of lower factorials, namely the polynomials ( X ) n = X ( X - - 1)(x - 2) ... (x - n + 1). This sequence of polynomials satisfies the difference equation A(X)n = n(X)n- 1, where A is the difference operator: Af(x) = f ( x + 1) -- fix). This sequence can be extended to negative n by setting 1

(x)_~ = (x + 1)(x + 2) .-. (x + n ) '

VOLUME 21, NUMBER2, 1999

13

and we have

~(X)-n

=

-n~)-n-l.

For positive n, we may define the logarithmic extension of this sequence by setting

(x)(l_)

=

(X)_ n

=

1 (x + 1)(x + 2) ... (x + n)"

For example, (x)~)l = 1/(x + 1). The elements (x)~)n belong to the submodule of the logarithmic algebra spanned by h~ ) (x), as n ranges over all integers. On this submodule, the operator A is invertible, and we can, therefore, set

(x)#) = a - n - , __1_1 x+l for all non-negative integers n. It turns out that the element (x)(0]) is given by the following series, convergent in the logarithmic topology: (x)~ t) =

BI

log(x + 1) + - 1+ x

B2 2(1 + x) 2

+

B3 3(1 + x) 3

But this is a familiar object: it is the W-function, heuristically introduced by Gauss. Gauss motivated the W-function as the "right" solution of the difference equation AW(x + 1) -

1

x+l"

We have now rigorously verified Gauss's guess. Further computations show that the elements (x)(11) and (x)(21) also coincide with special functions introduced by Gauss, namely the digamma and trigamma functions, which are at last rigorously defmed by infmite series convergent in the logarithmic topology. In a similar vein, one defmes logarithmic extensions of

the Bernoulli polynomials, the Hermite polynomials, &rid so forth, and one fmds that the asymptotic expansions of these polynomials reappear naturally as members of the logarithmic extensions of these functions. As a matter of fact, the logarithmic topology allows us to replace asymptotic expansions by series which are convergent in the logarithmic topology. In closing, two open problems may be mentioned. First, no closed-form expression is known for the coefficients of the expansion of a product

into a logarithmic power series. Second, we do not know a combinatorial or probabilistic interpretation of the Roman coefficients [~] in general. Thank you for listening. BIBLIOGItAPIlf Alexander, Kenneth S., Kenneth Baclawski, and Gian-Carlo Rota, A stochastic interpretation of the Riemann zeta function, Proceedings of the National Academy of Sciences 90 (1993), 697-699. Kung, J.P.S. (ed.), Clan-Carlo Rota on Combinatorics, Boston: Birkh&user (1996). Kung, J.P.S., M. Ram Murty, and Gian-Carlo Rota, On the R6dei zeta function, Journal of Number Theory 12 (1980), 421-436. Loeb, Daniel E., and Clan-Carlo Rota, Formal power series of logarithmic type, Advances in Mathematics 75 (1989), 1-118. Rota, Gian-Carlo, Bruce Sagan, and Paul R. Stein, A cyclic derivative in noncommutative algebra, Joumal of Algebra 64 (1980), 54-75. Department of Mathematics MIT Cambridge, MA 02139-4307 USA e-mail: [email protected]

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14

THE MATHEMATICALINTELLIGENCER

i~'rA~|i[~li,~.lq[-,.]~-ai~,[~i~-au,ani[~,,l~"-]i

Alexander

Shen,

Editor

PentangramA New Puzzle Klaus K0hnle

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

The Classic T a n g r a m The w e l l - k n o w n Chinese tangram is a puzzle consisting of seven pieces that can be arranged into either one square of area 2 or two squares of area 1 each:

not need to be profound (but it is allowed to be); it may not be directed only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions from readers--either new notes, or replies to past columns.

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

Although there are loads of other more or less funny shapes that can be built from those seven pieces, these two reveal best the immanent essence of the puzzle from the mathematical point of view: The tangram is based on V ~ as some sort of magic number. The ratio between any two lengths occurring as side-lengths of the seven pieces is some power of ~/'2. The right-angled, isosceles triangle occurs in three different sizes as pieces of the puzzle. The ratio between the hypotenuse and the legs of such a triangle is X/2. Furthermore, the hypotenuse of

I

a small triangle is just as long as the legs of the next larger one; in other words, the next larger triangle is scaled by a factor of V~. As a consequence, its area is just doubled. The same relation holds for the arranged squares shown above. The side-length of the big square is equal to the diagonals of the small squares, which are just V ~ times their sidelengths. Despite the triviality of all that has been said, it can help the solver of the puzzle. Since X/2 is irrational, a length that is the sum of a non-zero integer and a non-zero integer multiple of V ~ can never be an integer or an integer multiple of X/2. This implies that it is predetermined how the pieces may be rotated in order to be useful. Thus in the above illustration, all pieces in the small squares are rotated by an odd multiple of 4 in comparison with their appearance in the big square. And the preceding argument implies that this is necessary. Variants of t h e S a m e Idea The classic tangram is based on V~, the ratio between the side-length and the diagonal of a square. The angles occurring are all multiples of the angle between a side and a diagonal, namely ~ . Equally well one could design a similar puzzle where the pieces have to be arranged to regular hexagons, and the magic number would be V~, the length of the chord in a hexagon of sidelength 1. (The other chord, the diagonal, has length 2; and an integer magic number would not make for a very interesting puzzle.) Such a puzzle, based on V3, could be arranged to one big hexagon or to three small ones, where the chord-lengths of the small hexagons would be equal to the side-length of the big one. Here, all angles occurring should be multiples of 6. The sides whose lengths are even/odd powers of

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 2, 1999

15

w o u l d have to b e p l a c e d in a r o t a t i o n w h i c h is an even/odd multiple o f 6" A n o t h e r v a r i a n t w o u l d b e the a r r a n g e m e n t of equilateral triangles with ~ , which is the ratio b e t w e e n the height and the side-length, as the magic number. In this case, three big triangles w o u l d have the s a m e a r e a as f o u r small ones if one stuck to t h e principle that they b e s c a l e d b y the magic number. The angles s h o u l d again be m u l t i p l e s o f the smallest angle b e t w e e n the t w o straight lines w h o s e ratios of gg lengths defme t h e magic number, n a m e l y g. I do not a t t e m p t to give a c o m p r e h e n s i v e survey; let m e j u s t m e n t i o n t h a t for regular polygons with (say) 7, 9, 11, 13, o r 19 sides, t h e ratios b e t w e e n c h o r d s a n d sides are t r a n s c e n d e n t a l n u m b e r s , so I do not s e e a n y w a y to design a puzzle in the s a m e fashion from them. The variant t h a t a t t r a c t e d m e m o s t is t h e regular pentagon, w h o s e ratio b e t w e e n chord-length a n d side-length veritably is a m a g i c number, n a m e l y t h e g o l d e n ratio.

The Golden RaUo a n d its S q u a r e The golden ratio is ubiquitous in all s o r t s o f mathematics; I will resist the t e m p t a t i o n o f expatiating a b o u t it. Let us x/5+1 j u s t fLX u p o n ~ a s a b r i e f n a m e for it: ~ 2 As in the o t h e r variants, the puzzle shall b e such that its p i e c e s can b e a r r a n g e d into a n u m b e r of small p e n t a g o n s as well as into a n u m b e r of big pentagons. But this time the square of o u r magic n u m b e r is irrational; hence, for all integers n a n d m, n p e n t a g o n s of a certain size will o c c u p y an a r e a different f r o m that o f m p e n t a g o n s w h o s e sizes are s c a l e d b y ~. hlstead, w e will have to t a k e p e n t a g o n s of three different sizes with a scaling f a c t o r o f ~ b e t w e e n them. The a r e a s o f t h e s e p e n t a g o n s axe t h e n 1, ~2, and ~4 respectively, a n d t h r e e medium-sized p e n t a g o n s have the s a m e a r e a as a small a n d a big one together: 3~2=1

7r

ple of ~. But since Vn ~ 77: ~n + ~n+l : ~n+2 and a n y p o w e r o f ~ a p a r t from r = 1 is a s u m o f a n o n z e r o integer multiple of 89and a n o n z e r o integer multiple o f 2 ' s u c h a s t a t e m e n t can in this c a s e n o t b e made. This, at least in m y opinion, a d d s to the spirit o f s y m m e t r y a c c o u n t i n g for the fascination o f such a puzzle. Another consequence of Vn E 7/: ~d~ + ~n§ 1 = q~ + 2 is that, theoretically, any power o f q~as a goal length can be r e a c h e d by cumulating smaller p o w e r s of ~ regardless of h o w (i.e., with w h a t powers of ~) one has started to approach this goal. Of course, this is not true in practice, because no arbitrarily small p o w e r s of ~ occur as side-lengths of pieces in the puzzle; but still, this principle will have its bearing.

The Details of the Pentangram T h e s e principal features, arising from p r o p e r t i e s o f the pent a g o n a n d the golden ratio, do n o t at all d e t e r m i n e t h e details. Of the m a n y possibilities o f cutting a small a n d a big p e n t a g o n into pieces t h a t c a n be r e a r r a n g e d into t h r e e m e d i u m - s i z e d pentagons, I r a t h e r u n p r e m e d i t a t e d l y hit on one consisting o f thirteen p i e c e s of four different s h a p e s in up to t h r e e different magnifications, where, o f course, t h e magnification ratio is ~p. A r r a n g e m e n t of a big a n d a small pentagon:

+ ~ 4.

H o w did I hit on this simple equality? All that c a n b e said a b o u t the golden ratio boils d o w n to the simple equation q~2- q~- 1 = 0, which is usually u s e d as a definition for q~. We are intere s t e d here in d e s c r i p t i o n s of ~ ; so, w e j u s t adjoin

~=A and eliminate ~ f r o m the s y s t e m of t w o equations. The resuit is

A r r a n g e m e n t o f three m e d i u m - s i z e d p e n t a g o n s (with the s a m e pieces):

A 2 - 3A + 1 = 0, which is the s i m p l e s t possible s t a t e m e n t a b o u t A. Consequently, t h e a b o v e requirement t h a t a small and a big p e n t a g o n t o g e t h e r o c c u p y the s a m e a r e a as t h r e e mediumsized ones is the s i m p l e s t I could have m a d e . The s m a l l e s t angle b e t w e e n a side a n d a c h o r d in the ~T regular p e n t a g o n is g, h e n c e all angles o c c u r r i n g shall be multiples o f -~. In analogy to o t h e r cases, one is t e m p t e d to c o n j e c t u r e t h a t a side having a length t h a t is an odd p o w e r of ~ h a s to o c c u r in a rotation t h a t is an o d d multi-

16

THE MATHEMATICALINTELLIGENCER

The t w o different s h a p e s o f triangles are the only p o s sible o n e s given the obligation t h a t all angles b e m u l t i p l e s

of 5; the pentagon is a reminiscence of the shape that has to be arranged; and the quadrilateral is just one of the many alternatives with angles multiples of F and side-lengths powers of ~. As in the classic tangram, this division is not the one with the least number of pieces necessary to obtain the two p~_ncipal shapes, because such a minimal division would probably ease the task of s o l i n g the puzzle; more-

over it might decrease the number of other shapes that can be built. A small collection of such other shapes is given in the appendix; the reader is invited to discover more of them. Mittenwalder Stra6e 2 D-81377 MQnchen Germany

Appendix: A Small Collection of Other Arrangements

?

VOLUME 21, NUMBER 2, 1999

17

BENOIT B. MANDELBROT A N D MICHAEL FRAME

Thc Canopy and Shortest Path in a Self-Contacting Fractal Treo

rticle concerns the fractal trees that are obtained recursively by s y m m e t r i c bibranching. A t r u n k o f length 1 divides into two branches o f length r, each o f m a k e s an angle 0 > 0 ~ w i t h the linear extension o f the trunk. Each branch divides by the s a m e rule. Some basic i n f o r m a t i o n on such trees is f o u n d i n ter 16 of [FGN], on w h i c h this article elaborates. It is well k n o w n t h a t the b r a n c h tips o f t h e s e t r e e s can t a k e any dimension satisfying 0 < D -< 2. Moreover, w h e n 1 < D < 2, it is p o s s i b l e for different b r a n c h e s to have tips, but no o t h e r points, in c o m m o n . These trees, to b e called "selfcontacting," i n c l u d e p o i n t s one c a n n o t a c c e s s from infinity, e x c e p t by crossing a composite curve called the "hull." In the interesting cases, the hull includes a fractal called the "canopy." F o r 0 < 90 ~ the c a n o p y can be c h a r a c t e r i z e d in a n o t h e r way: as the s h o r t e s t p a t h along the b r a n c h tips from the u p p e r left c o r n e r to the u p p e r right corner. The self-contacting b r a n c h tips s c r e e n from inf'mity s o m e o t h e r b r a n c h

18

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAG NEW YORK

tips, t h u s providing s h o r t c u t s b e t w e e n p a r t s of the tree, effectively j u m p i n g over the s c r e e n e d regions. F o r 0 > 90 ~ the c a n o p y is d i s c o n n e c t e d b e c a u s e o f additional screening by b r a n c h segments. The s h o r t e s t p a t h along b r a n c h tips r e m a i n s a v a r i a n t o f the Koch curve, so the s h o r t e s t p a t h and c a n o p y no longer coincide. The fractal d i m e n s i o n s of the canopy, s h o r t e s t path, and t h e set o f b r a n c h tips are c o m p a r e d in the range 0 ~ < 0 < 180 ~ F o r certain ranges of 0, t h e canopy, shortest path, and t h e s e t o f b r a n c h tips a r e Koch curves. Consequently, the c o n s t r u c t i o n s p r e s e n t e d h e r e p r o v i d e alternative w a y s to d r a w Koch curves.

In addition, the angles 0 = 90 ~ and 0 = 135 ~ m a r k prof o u n d topological discontinuities in the c a n o p y a n d shortest path. Consequently, w e think of these as topological critical points.

The structure o f self-avoiding and self-contacting trees (with their canopies a n d s h o r t e s t paths) is instructive and entertaining. It s e e m s to m a k e the subtle d i s t i n c t i o n bet w e e n d e n u m e r a b l e and n o n d e n u m e r a b l e infinity c o n c r e t e a n d near-palpable.

BR(x,y) =

( x r cos(0) - y r sin(0), x r sin(0) + y r cos(0)) + (0,1) BL(X,y) =

( x r c o s ( - 0) -- y r sin(-- 0) + y r cos(-- 0)) + (0,1). T h e tip set is the set of limit p o i n t s o f all fmite compositions Of BR a n d BL applied to (0,1).

To include the t r u n k (and all the branches), for 0 -< 135 ~ a d d a third function Tr(x,y) = (O,sy),

A Classification

of Binary

Trees

In t h e p r e c e d i n g construction, e a c h b r a n c h is d e t e r m i n e d b y a finite n u m b e r o f c h o i c e s o f t h e form "bear left" o r " b e a r right," so each b r a n c h defines, in obvious fashion, an "address" that is a finite s e q u e n c e of letters L a n d R. Therefore, the b r a n c h e s a r e denumerable. F o r r - > 1, the o u t c o m e of this c o n s t r u c t i o n is easily s e e n to b e unb o u n d e d . F o r example, L R L R L R . . . = (LR) ~ defines a seq u e n c e in which every R b r a n c h is vertical a n d e v e r y L b r a n c h m a k e s an angle 0 w i t h the vertical. Thus, t h e total vertical e x t e n t of this b r a n c h sequence is

where 1 -- r 2 s--

1 + r cos(0)"

Here, s is the r e c i p r o c a l of the height o f the tree, h e n c e the vertical scaling factor of the trunk. Note that all the IFS t r a n s f o r m a t i o n s m u s t be contractions, so the t r u n k can b e g e n e r a t e d w i t h Tr(x,y) only s o long a s s < 1; that is, for 0 < 135 ~ However, BR and BL will generate the set o f b r a n c h tips for all 0. F o r 0 > 135 ~ r e p l a c e Tr with tyro functions

1 + r cos(0) + r 2 + r 3 c o s ( 0 ) + r 4 + ' " , diverging for r >- 1. However, if r < 1, a limit tree is r e a c h e d after an infmite n u m b e r o f branchings; it d e p e n d s o n 0 and will be d e n o t e d by ft. E a c h b r a n c h tip d e f m e s an a d d r e s s t h a t is an inf'mite s e q u e n c e o f L a n d R. A tip's add r e s s is the same as a n infmite sequence o f 0 and 2, hence, in turn, the s a m e as a point in the classic t e r n a r y ~ "~J C a n t o r set. .~ "~ \ Many geometrical p r o p e r ~ \ ties of t h e s e trees can be de9 d u c e d from the p o s i t i o n s of b r a n c h tips. Denote b y A1A2A3 9 9 9 the add r e s s of a b r a n c h tip a n d b y dn the n u m b e r of \ R ' s m i n u s the n u m b e r o f L ' s in A1A2A3 9 9 9 An. Placing the b a s e o f the t r u n k at the origin, this \ b r a n c h tip is l o c a t e d at t h e p o i n t with c o o r d i n a t e s x = r sin(d10) + r ~-sin(d20) + r 3 sin(d30) + ...,

(1) y

=

1

+

r cos(d10)

+

r 2 cos(d20) + r 3 cos(d30) + "".

W h e n the a d d r e s s is eventually periodic, c l o s e d e x p r e s s i o n s for the c o o r d i n a t e s c a n be found b y s u m m i n g the a p p r o p r i a t e geometric series. F o r example, the b r a n c h tip with a d d r e s s ( L R ) ~, a p o i n t of m a x i m a l height o f the t r e e for 0 ~ < 0 -< 135 ~ h a s y c o o r d i n a t e 1 + r cos(0) 1 r2 To g e n e r a t e pictures o f t h e set o f branch tips, t h e stand a r d m e t h o d already u s e d in [FGN] is n o w [B] r e f e r r e d to as "iterated function s y s t e m s " (IFS). The t w o functions required are

Trl(x,y) = (0,y/2), Tr2(x,y) = (O,y/2) + (0,1/2). The wide availability o f IFS s o f t w a r e m a k e s this a r e a accessible to c o m p u t e r experiments.

~~

lf-Avoidance

When 5r has n o d o u b l e p o i n t (i.e., no loop), it is said to be selfa v o i d i n g . If so, the b r a n c h tips are distinct p o i n t s and, like the p o i n t s in a Cantor set, are nondenumerable. They f o r m a self-similar fractal o f dimension D = log(2)Aog(1/r). That t h e scaling of the b r a n c h tips is identical to that o f the b r a n c h e s is illustrated b y t h e IFS formulation. In addition, it can be d e r i v e d f r o m the a d d r e s s e s o f a p p r o p r i a t e b r a n c h tips, using the m e t h o d we describe in the self-contacting case. F o r r -- 1/2, ~" is alw a y s self-avoiding, r e g a r d l e s s o f the value of 0. However, for 1/2 < r < 1, the tree m a y o r m a y not be selfavoiding, d e p e n d i n g on 0. Self-Contact

When the tip of s o m e b r a n c h also b e l o n g s to s o m e o t h e r branch, t h e tree is said to self-contact. Self-contacts a r e of t w o kinds: a tip m a y lie on a b r a n c h or t w o tips m a y coincide; b o t h kinds can be f o u n d on the s a m e tree. Tipto-tip self-contact will be s e e n to involve a generalization o f the familiar fact that in b i n a r y r e p r e s e n t a t i o n s of the points o f t h e interval [0, 1], the p o i n t s c o r r e s p o n d i n g to 0 . 0 1 1 1 1 . . . a n d 0 . 1 0 0 0 0 . . . a r e identical. Here, too, t h e d i m e n s i o n o f the tip set is log(2)/log(1/r).

Figure 1. The self-contacting # = 20 ~ tree.

VOLUME 21, NUMBER2, 1999

19

Analysis of T i p - t o - T i p S e l f - C o n t a c t

Note t h a t t h e self-similarity a n d l e f t - r i g h t s y m m e t r y o f the tree imply t h a t s e l f - a v o i d a n c e is g u a r a n t e e d if n o n e o f the b r a n c h e s a t t a c h e d t o t h e first left b r a n c h i n t e r s e c t s t h e linear extension, ~E, o f the trunk. "First" tip-to-tip serf-cont a c t o c c u r s w h e r e t h e r i g h t m o s t b r a n c h tip o f the left half of t h e tree lies o n ~ . H o w c a n this h a p p e n ? Not surprisingly, the a n s w e r d e p e n d s on 0. F o r 0 ~ < 0 --< 90 ~ to find the a p p r o p r i a t e b r a n c h tip, n o t e t h a t t h e b r a n c h LR is a vertical line s e g m e n t o f length r 2 a n d is at a h o r i z o n t a l d i s t a n c e r sin(0) f r o m ~ . The r i g h t m o s t b r a n c h tip o f this side of the t r e e is realized as the limit o f a s e q u e n c e of b r a n c h e s , e a c h a t t a c h e d to its p r e d e c e s s o r in t h e sequence; this tip lies on ~ if t h e lengths o f t h e h o r i z o n t a l d i s p l a c e m e n t s o f t h e s e b r a n c h e s s u m to 0. S u p p o s e N is the s m a l l e s t i n t e g e r for w h i c h NO >--90 ~ T h e d e s i r e d sequence is d e s c r i b e d b y the a d d r e s s LRN+I(LR)~; t h a t is, after the initial LR, t h e b r a n c h e s t u r n right until t h e first h o r i z o n t a l o r n e g a t i v e s l o p e branch. A f t e r that, t h e y alt e r n a t e b e a r i n g left a n d right. C o m b i n i n g all like terms, w e s e e the b r a n c h tip lies on ~ if its x c o o r d i n a t e is 0; t h a t is, if r s i n ( - # ) + r 3 sin(0) + ... + r N sin((N - 2)0) rN+ t rN+ 2 + ~ s i n ( ( N - 1)0) + ~ sin(N0) = 0.

(2)

The c o r r e s p o n d i n g equations for 0 < 90 ~ do n o t a d m i t s u c h c l e a n solutions.

Self-Overlap When t w o b r a n c h e s of ff i n t e r s e c t b e y o n d tips c o n t a c t i n g tips, the tree is said to self-overlap. This case is n o t of conc e r n in this article, but it d e s e r v e s a digression, b e c a u s e it s u p p o r t s interesting c o m p u t e r experiments. F o r r = l/X/2, t h e formal dimension is D = 2, suggestive of the plane-filling p r o p e r t y of Peano curves. This p r o p e r t y is o b s e r v e d for t w o v a l u e s of 0, for w h i c h 9- serf-contacts tip to b r a n c h a n d fills a p o r t i o n o f the plane. One o f these values is O = 90 ~ w h e n it is well k n o w n t h a t t h e tree fills a rectangle, a s is easily c h e c k e d b y h a n d and n o t e d in t h e caption o f P l a t e 155 o f [FGN]. However, this is not all. The definitions in this article do not e x c l u d e t h e 0 satisfying 0 > 90 ~ for w h i c h b r a n c h e s droop d o w n t h e t r u n k instead o f rising. It is easily c h e c k e d by h a n d that for 0 = 90 ~ + 45 ~ = 135 ~ the t r e e ~ fills a right isosceles triangle. F o r other v a l u e s o f 0, t h e r e is a massive serf-overlap. F o r 1/V~ < r < 1, t h e f o r m a l e x p r e s s i o n for t h e dim e n s i o n D = log (2)/log(i/r) satisfies D > 2. This inequality e x p r e s s e s nicely the fact t h a t the tree c a n n o t a v o i d m a s sive serf-overlap and ( i r r e s p e c t i v e of the choice o f 0) c o v e r s a p o r t i o n of the plane.

We shall analyze several explicit c a s e s in a moment. Figures 1 t h r o u g h 8 s h o w e x a m p l e s at interesting angles throughout t h e range 0 ~ < 0 < 180 ~ Figure 9 shows, as a function of 0, t h e critical contraction ratio r that e n s u r e s self-contact. F o r 0 ~ < 0 < 90 ~ this graph is o b t a i n e d b y solving Eq. (2) for t h e a p p r o p r i a t e N. F o r 0 > 90 ~ different b r a n c h tips m u s t b e used. F o r 90 ~ < 0 --< 135 ~ the relevant b r a n c h tip is L3(RL) ~176 Figures 5 and 6 give examples; for tip-to-tip serf-contact in this 0 range, r a n d 0 are r e l a t e d by r s i n ( - 0) + ~

r2

sin(--20) + ~

r3

sin(--30) = 0.

This equation c a n b e solved explicitly for r. r =

- c o s ( 0 ) - X/2 - 3 cos2(0) 4 cos2(0) - 2

(3)

In this range, the m i n i m u m r value is X/-3~, occurring at 0 = a r c c o s ( - l / X / - 6 ) . Examining Figure 9 reveals that for given r < X/-3~, tip-to-tip self-contact o c c u r s at only two values of 0, w h e r e a s for given r > X/3-~, tip-to-tip self-contact o c c u r s at four values o f 0. F o r 135 ~ < 0 --< 180 ~ the relevant b r a n c h tip is L2(RL)% Figures 7 a n d 8 give examples; for tip-to-tip serf-contact in this 0 range, r a n d 0 a r e related by r

The canopy initiator is

the line from (LR) | to (RL)=; the generator consists of the six seg-

r 2

1 - r 2 s i n ( - 0) + ~

Figure 2. The self-contacting 8 = 40 ~ tree. Self-contact results from the coincidence of LR4(I-R) = and RL4(RL) |

s i n ( - 2 0 ) = 0.

ments connecting the branch tips (LR) | LR2(LR) =, LR3(LR) =, RL~(RL) | RL2(RL) =, and (RL) | The corre-

LR4(/_R)= = RL4(RL) |

Again, this equation can be solved for r: 1 r = 2 cos(0)"

20

THE MATHEMATICAL[NTELUGENCER

sponding

binary

fractions

are

(LR)~--~ 1/3,

LR2(LR) | - , 5/12,

LR3(LR) ~ --> 41/96, LR4(LR) | --~ 83/192, RL4(RL) | --> 25/48, FIL3(RL) = --~

13/24, RL2(RL) | --~ 7/12, and (RL) ~ --~ 2/3.

Figure 4. The self-contacting ~ = 90 ~ tree. In the infinitely branched limit, this tree fills the rectangle. The circle in the middle indicates the branch tips LR3(LR) ~

Figure 3. The self-contacting ~ = 80 ~ tree. This rendering presents only 13 branchings, so the self-contacting nature is not apparent. Here, the canopy generator consists of four segments connecting the branch tips (LR) | LR2(LR) | LR3(LR) | = RL 3(RL) ~, RL 2(RL)% and

Ld(RL) |

R4(LR) = =

RL3(RL)=; the bot-

tom circle indicates La(RL) = = R3(LR) =, A s 0 crosses the topological critical value 0 = 9 0 ~ the topology induced by the contacts between tips changes throughout. All the links present for 0 < 90 ~ are broken and new links established.

(RL)% For comparison with the 0 = 900 tree, the branch tips L3(RL) = and R3(LR) = are indicated.

Definition Fractal

of the Canopy

of a Self-Contacting

Tree

Following [FGN], p. 242, one defmes the tree's hull or outer boundary, as the set of points that can be reached from far away by following a curve that does not intersect the tree. The hull is an intrinsically interesting concept and has been investigated for many fractal sets. In a serf-avoiding tree, this notion is without interest, because the hull is identical to the whole tree with its tips. More interesting are the serf-contacting trees that are illustrated in Figures 1 through 8 of this article. (They are adapted from Plate 155 of [FGN].) These figures show that the boundary includes two very different components whose structure depends on the sign of 0 - 90 ~ One component is made of straight intervals that are reached by approaching the tree "from below." In Figures 1 through 3, for which 0 < 90 ~ these intervals join in two broken lines that start in the root, move up by fanning to the right and to the left, and end in spirals. Each straight portion in these broken lines is a full branch of the tree. Moreover, these broken lines are of fmite length, therefore of dimension 1. The other part of the boundary is a fractal curve ~ made of points that can be reached by approaching the tree "from above." The curve 9 was considered in [FGN], p. 153, and called the canopy of the tree. Loosely speaking, ~ is made of branch tips that have somehow "coalesced'; in a moment, this idea will be made precise and we shall then evaluate the fractal dimension, ~r of the canopy. Because (~ is a curve, we have ~r - 1, and since is a subset of ~, we have ~ -< D. However, Figures 5 through 8, for which 0 > 90 ~ exemplify a totally different situation. The straight intervals

are not full branches but portions of branches, and the canopy can, at best, be defined as a dust of points: much of what looks like a canopy is screened from infinity by other portions of branches. Extrapolation

The s i g n i f i c a n c e o f t h e sign o f ~ - 90 ~ is c o ~ e d

ff t h e

tree is suitably extrapolated. The idea is to imagine that what is drawn in the figures is not a free-standing tree but only a branch in an infinite tree. Alternatively, one can "zoom in" a small portion of a tree drawn in the figures, with the constraint that this portion touches the boundary. When 0 < 90 ~ the extrapolation yields a simplified boundary: the portion made of branches vanishes and one is left with a piece of fractal canopy. When 0 > 90 ~ to the

Figure 5. The self-contacting # = 105 ~ tree. Both addresses of the indicated double points are La(RL) = = Ra(LR) |

RLa(RL)|

R4(LR) |

(LR)La(RL) = = (LR)Ra(LR) |

LR3(LR) | = L4(RL) =, and (LR)L4(RL) = =

(LR)LR3(LR)|

VOLUME21, NUMBER2, 1999

21

Figure 6. The self-contacting # = 130 ~ tree. Left to right: the top la-

Figure 7. The self-contacting 6 = 137 ~ tree. The # = 135 ~ tree (not

bels indicate the branch tips (LR) |

drawn) fills a triangle9 As # increases and crosses the topological

LR2(LR) |

RL2(RL) =, and (RL)|

the label on the trunk indicates the double point L3(RL) = = Ra(LR)~;

critical value # = 135 ~ the topology induced by the contacts between

the label below and to the left indicates the double point LR3(LR) | =

tips changes throughout, just as when 0 crosses the other topolog-

Ld(RL)|

the lowest labels indicate L2(RL) = and R2(LR) =,

ical critical point, 0 = 90 ~ For this value 8 = 137 ~ the shortest path nearly coincides with the Ces&ro curve illustrated on Plate 65 of

contrary, the e x t r a p o l a t e d b o u n d a r y c o n t i n u e s to include parts o f the b r a n c h e s . The

Structure

of the

Canopy

When

8 < 90 ~

Each piece o f (r c o n t a i n s b o t h single a n d d o u b l e points. F o r example, Figure 3 s h o w s clearly that the tips ( L R ) ~ and (RL) ~ c a n n o t b e r e a c h e d in any o t h e r way; t h e y are single points. R e p r e s e n t i n g L b y 0 a n d R b y 1, w e can "parse" each s e q u e n c e o f 0 and 1 as a b i n a r y fraction. Then, these two single p o i n t tips are r e p r e s e n t e d b y 0.010101 9 = 1/3 and 0 . 1 0 1 0 1 0 . . . -- 2/3. By contrast, f o r 4 5 ~ -- 0 < 90 ~ the tips LR3(LR) ~ a n d RL3(RL) ~ c o i n c i d e and form a double point. T h e s e tips are r e p r e s e n t e d b y the b i n a r y fractions 0.0111010101... = 1/2 - 1/24 a n d 0.1000101010... = 1/2 + 1/24; t h e y d e f m e an e x c l u d e d s u b i n t e r v a l o f [0, 1] cent e r e d on the midpoint. Every point of t h a t e x c l u d e d subinterval c o r r e s p o n d s to a tip that falls within ~- and not on its boundary; therefore, it d o e s not b e l o n g to the c a n o p y %. The e x c l u d e d subintervals can be o r d e r e d b y decreasing size; therefore, t h e y are denumerable. The s a m e is true o f the c a n o p y d o u b l e p o i n t s that c o r r e s p o n d to t h o s e intervals. The

Koch-like

Fractal

Structure

Dimension

When

of the

Canopy

and

THE MATHEMATICAL INTELLIGENCER

t w e e n ( L R ) ~ a n d (RL) ~, s h o w s that the c a n o p y is a Koch curve. The serf-similarity o f t h e c a n o p y allows us to u s e this t o p to c o m p u t e the c a n o p y dimension. Although t h e n u m e r i c a l value of the initiator length, l, d o e s n o t m a t t e r (units c a n be c h o s e n to m a k e it 1), w e do need a n e x p r e s sion for it in t e r m s of r a n d 8 to derive the scalings o f t h e generators. The left and right e n d s of the initiator are the b r a n c h tips ( L R ) ~ and (RL)% Thus, l is the difference of t h e x c o o r d i n a t e s of t h e s e tips. F r o m Eqs. (1), w e s e e t h a t l=sin(8)(r+r

3+...)-sin(-8)(r+r

3+...)-

2 r sin(8) 1-

r2

"

As to the g e n e r a t o r o f %, it d e p e n d s d i s c o n t i n u o u s l y on 0. Let us e x a m i n e a few cases.

Its

0 < 90 ~

The question w a s p o s e d in [FGN] and h a d s e e m e d difficult, b u t after sufficiently large n u m b e r s of a c t u a l t r e e s h a d b e e n p l o t t e d and e x a m i n e d with sufficient attention, the a n s w e r s b e c a m e self-evident. They are e l e m e n t a r y b u t tedious, because they d e p e n d on the value of 8. Figure 9 s h o w s that depending on the value o f r, self-contact o c c u r s for either two, three, or four t h r e s h o l d values of 0. As a function o f 0, the dimension o f t h e c a n o p y turns out to have negative discontinuities for 8 = 135 ~ and for all angles of the f o r m 0 = 90~ k > 1, and to b e right-continu o u s in the intervals b e t w e e n t h e s e critical angles. See Figure 10, w h i c h also includes the d i m e n s i o n D of the tip set, and the d i m e n s i o n ~y of the s h o r t e s t path. Inspection of the top p a r t of the canopy, the p a r t be-

22

[FGN] as a boundary between white and black.

Figure 8. The self-contacting ~ = 145 ~ tree. Left to right: the top labels indicate the branch tips (LR) |

LR2(LR) | = La(RL) =, and (RL)=;

the other (lower) label indicates L2(RL) | = R2(LR)| is the classical triadic Koch curve.

The shortest path

self-contact

r

.75

.50

I

I

I

45

90

135

I angle 180

Figure 9. The ~ l a t i o n b e ~ e e n ~ in degrees and the critical contraction ~ t i o r that ensures seW-contact Note that f o r r less ~ a n ~ , contact o c c u ~ at ~ o

By inspection, the structure o f the c a n o p y is clearest to the eye w h e n 0 is c l o s e to 90 ~ Therefore, let us begin b y c o m p a r i n g a plane-filling tree (Fig. 4) a n d one t h a t is close to filling (Fig. 3). As observed, the t o p o f Figure 4 is simply an interval o f dimension ~ = 1. J u s t b e l o w 0 = 90 ~ the c a n o p y "opens up" discontinuously, as s e e n in Figure 3, but its s t r u c t u r e r e m a i n s clearly

T h e R a n g e 4 5 ~ -< ~ < 9 0 =

Here, the mininial self-contact sequence o c c u r s w h e n the b r a n c h tip LR3(LR) ~176 c o i n c i d e s with RLS(RL)~176 t h a t is, f r o m Eq. (2), w e s e e 0 a n d r a r e r e l a t e d b y r sin(-0) + ~

self-

~ values. For l a ~ e r r ~ x c e ~ r = ~/2), seN-contact o c c u ~ at f o u r ~ values.

r3

sin(0) + ~

r4

sin(20) = 0.

dimension

/\ A

2.0-

1.0

0.5

I

I

I

I

45

90

135

180

angle

Figure 10. The dimension D of the tip set (top, continuous curve); the dimension &~ of the canopy (bottom, broken curve); and the dimension ~s of the shortest path (small circles). Note ~ = ~

f o r t ) < 90 ~ and 80 = D f o r e > 135 ~

VOLUME21, NUMBER2, 1999 '~3

recognizable. It is a Koch curve whose generator is made up of four intervals. The endpoints of the left interval are the b r a n c h tips ( L R ) ~ and LR2(LR)~; those of the next right interval are L R 2 ( L R ) ~ and LR3(LR)% F r o m the coordinates of these points, a n d t h e n by bilateral s y m m e t r y of the tree, and using Eqs. (1), we see the generator intervals have lengths rl = lr 2,

r2 = lr 3,

ra = Ir 3,

and

r4 = lr 2.

Again, &r j u m p s d i s c o n t i n u o u s l y from the value it takes'for 0 = 45 ~ to a larger value it takes for 0 = 45 ~ - e. The General Range 90~

-< 0 < 9 0 ~

- 1)

Figure 1 shows a typical e x a m p l e in this range. The generalization is obvious. In this range, minimal self-contact occurs w h e n r and 0 are related by Eq. (2) and A is the solution of the equation ~2 + ~3 + ... + /~N+I = 1/2

The d i m e n s i o n ~r of the canopy is k n o w n to be the (unique) solution of the Moran generating equation, namely (taking 1 = 1) the equation ~ , r ~ = 1.

Each passage through a value of the form 0 = 90~ a c c o m p a n i e d b y a j u m p up in the value of &r The LimitN ~

is

andD ~1

The Moran generating equation b e c o m e s Here, the Moran equation takes the form 2(r2~+r3~+'")-

2 r 28 + 2r 3~ = 1. It is convenient, in this article, to write r 8 = A, with h the solution of the third-order equation h 2 + As = ~/2. By contrast, in the plane-filling Figure 4, one could have said that A was the solution of the equation 2A2 = 1, which yields A2 = r 2a = 1/2, hence, ~ = 1. (For 0 = 90 ~ minimal serf-contact requires r = l/X/-2.) Therefore, &~ j u m p s disc o n t i n u o u s l y from the value &~ = 1 that it takes w h e n D = 2, to a larger value it takes w h e n D = 2 - e. T h e R a n g e 3 0 ~ -<

O<

45 ~

As 0 continually decreases to 45 ~ the c a n o p y reaches a second discontinuity b e c a u s e not only do the b r a n c h tips LR3(LR) ~ a n d RL3(RL) ~176 coincide b u t also LR4(LR) ~ and RL4(RL) :~ coincide (and, of course, so do m a n y other b r a n c h tips b e t w e e n these). As 0 decreases below 45 ~ LR4(LR) ~ a n d R L 4 ( L R ) ~ coincide at a smaller r value than do LR3(LR) ~ a n d RL3(RL) ~, so minimal serf-contact occurs w h e n LR4(LR) ~ coincides with RL4(RL)% See Figure 2. Thus, from Eqs. (2), we see 0 and r are related by r s i n ( - 0) + r 3 sin(0) + ~

r4

sin(20) + ~

r5

sin(30) = 0.

Passing 45 ~ adds two intervals to the generator, each of length given by the distance b e t w e e n the b r a n c h tips LR3(LR) ~ a n d LR4(LR)~; that is, the generator is a b r o k e n line of six intervals of lengths rl = lr 2,

r2 = lr 3, r3 = lr 4, r5 = lr 3, and r 6 = lr 2.

r4 = Ir 4,

The d i m e n s i o n &~ is n o w the solution of the Moran generating equation r~ = 2 r 2~ + 2 r 38 + 2r 4~ = 1, j=l

and )t = r ~ is the solution of the fourth-order equation

A2 + A3 + A4 = 1~.

24

Its real positive solution is r ~ = 1/2. At the same time as D o 1, one has r---)1/2; hence, the preceding equation yields &r ---) 1, which is w h e r e we started for 0 = 90 ~ This result was to be expected. We had k n o w n all along that the passage to the limit N - - - ) ~ m a k e s the canopy b e c o m e smaller and smaller. Moreover, we n o t e d already that because 9 is a curve and a s u b s e t of i~, one has 1 -< &~ <- D. When 0---> 0, D --> 1 a n d the c a n o p y necessarily b e c o m e s s m o o t h e r and smoother. As D decreases from 2 to 1, 0 decreases from 90 ~ to 0 ~ a n d we see that, as a n n o u n c e d , &r has negative discontinuities for 0 of the form 0 = 90~ and is right-continuous a n d increasing in the intervals b e t w e e n these discontinuities. See Figure 10. Knowing r as a function of 0 in each of the ranges 90~ - 0 < 90~ - 1), we obtain &~ by solving the corresponding Moran equation. The Value 0

THE MATHEMATICALINTELLIGENCER

=

90*

Here m i n i m a l self-contact requires r = l / V 2 . See Figure 4. To see the gap b e t w e e n the u p p e r left a n d right sides of the tree close, note that the b r a n c h tip L R 2 ( L R ) ~ coi n c i d e s with R L 2 ( R L ) ~, L R 3 ( L R ) ~ coincides with R L 3 ( R L ) ~, a n d every b r a n c h tip b e t w e e n L R 2 ( L R ) ~ a n d L R 3 ( L R ) ~ with x c o o r d i n a t e 0 coincides with the corres p o n d i n g tip o n the right side of the tree. A similar argum e n t s h o w s the gap b e t w e e n the lower left a n d right sides of the tree closes. Here, L 3 ( R L ) ~ coincides with R 3 ( L R ) ~, and L 4 ( R L ) ~ coincides with R 4 ( L R ) % In addition, from the IFS formulation, it is easy to see that the set of b r a n c h tips is a filled-in rectangle. Consequently, the c a n o p y consists of the edges of this rectangle a n d thus has d i m e n s i o n &r = 1. The

6

2r 2~ ~1 - -r 1 "

Cantor-like

Its Fractal

Structure

Dimension

when

of the Canopy

and

# > 90 ~

Unlike the situation w h e n 0 < 90 ~ for minimal self-contact n o w there are only two relevant b r a n c h tips, instead of one for each interval (90~ 90~ - 1)). For 90 ~ < 0 < 135 ~ m i n i m a l serf-contact is o b t a i n e d by requiring that the

b r a n c h tips L3(RL) ~ and R3(LR) ~ coincide; for 135 ~ < 0 < 180 ~ we require that L2(RL) ~ and R2(LR) ~ coincide. Also, unlike the situation w h e n 0 < 90 ~ p o r t i o n s o f b r a n c h e s s c r e e n from infinity s o m e of the b r a n c h t i p s that might visually a p p e a r to b e l o n g to the canopy. F i g u r e s 5 t h r o u g h 8 illustrate this situation. The effect o f this s c r e e n ing is to totally d i s c o n n e c t t h e canopy. Instead o f a variant on t h e Koch curve, it is n o w a variant on the C a n t o r set. The Range 90 ~ < 0<

135 ~

AS s e e n in Figures 5 and 6, the c a n o p y g e n e r a t o r c o n s i s t s of f o u r segments. The t w o on the left have c o m e r s (LR) ~ and LR2(LR) ~, and LR2(LR ~ and LR3(LR) ~ = L4(RL)~. [The s e g m e n t with c o m e r s LR3(LR) ~ and L3(RL) ~ = R3(LR) ~ is s c r e e n e d from infinity b y b r a n c h segments.] The first of t h e s e g e n e r a t o r s e g m e n t s h a s length ro = lr2; t h e s e c o n d h a s length rt = lr 3. Again, t h e tree is symmetric; hence, the Moran g e n e r a t o r equation b e c o m e s 2 r 2~ + 2 r 3~ = 1. So, r ~ ~ 0.565198..., the real r o o t of)t 2 + )t 3 ---- 1/2. F i g u r e s 5 a n d 6 s h o w the 105 ~ a n d 130 ~ trees, respectively.

Canopy

Dimension

as a Function

of o

Figure 10, a summary of the calculations presented here, is a graph o f t h e c a n o p y d i m e n s i o n 5r and the tip set dim e n s i o n D, a s a function of the b r a n c h i n g angle 0. The dim e n s i o n 6y o f the shortest path, d i s c u s s e d in the next section, is also shown. The discontinuities of ~ at the critical p o i n t s 0 = 90 ~ and 0 = 135 ~ deserve special mention. Comparing Figures 3 and 5 s h o w s why ~ a p p r o a c h e s t h e s a m e value as 0--> 90 ~ a n d as 0---) 90 ~ (not the value o f ~ at 0 = 90~ F o r both 0---) 90 ~ and 0---) 90 ~ the t o p o f the c a n o p y is gene r a t e d b y t w o c o p i e s o f the p o r t i o n s b e t w e e n the b r a n c h tips (LR) ~ and LR2(LR) ~, a n d b e t w e e n LR2(LR) ~ and LR3(LR)% T h e distances b e t w e e n t h e s e pairs o f tips app r o a c h the s a m e limits as 0--> 90 ~ Figures 6 and 7 illustrate the discontinuity at 0 = 135 ~ For 0 < 135 ~ exemplified by Figure 6, the generator of the top of the c a n o p y consists of four pieces. F o r 0 > 135 ~ exemplified b y Figure 7, segments of b r a n c h e s shield two o f these four pieces. Thus, the discontinuity at 0 -- 135 ~ represents a change in the number of generators, as do the discontinuities at 0 = 90~ N > 1. Only the 0 = 90 ~ discontinuity d o e s n o t involve a change in the n u m b e r o f generators.

The Value O = 135 ~

As in the 0 = 90 ~ case, here serf-contact requires r = 1/V~. To see the gap b e t w e e n the l o w e r left and right sides o f the tree close, note t h a t the b r a n c h tip La(RL) ~ c o i n c i d e s w i t h R3(LR)% L2(RL) ~ c o i n c i d e s with R2(LR) ~, a n d every b r a n c h tip b e t w e e n L3(RL) ~ and L2(RL) ~ with x c o o r d i n a t e 0 c o i n c i d e s with the c o r r e s p o n d i n g tip on the right side of t h e tree. In addition, f r o m the IFS formulation, it is e a s y to s e e that t h e set of b r a n c h tips is a filled-in triangle. AS in t h e 0 = 90 ~ case, 6 = 1. The Range O > 135 ~

As ~ i n c r e a s e s f r o m 135 ~ t o 180 ~ t h e b r a n c h t i p s f o r m a f a m i l y o f Koch curves, w i t h t h e g e n e r a t o r c o n s i s t i n g o f f o u r s e g m e n t s , t h e t w o o n t h e left with c o r n e r s (LR) ~ and L R 2 ( L R ) ~ = L3(RL) ~, a n d L3(RL) ~ a n d L2(RL)% All f o u r s e g m e n t s h a v e l e n g t h ro = r21, w h e r e r 2 v a r i e s f r o m 1/2 w h e n 0 = 135 ~ to 1/4 w h e n 0 = 180 ~. H e n c e , t h e M o r a n e q u a t i o n for t h e d i m e n s i o n o f t h e b r a n c h t i p s is simply 4r 2~=1

or

r ~=1/2.

F i g u r e s 7 a n d 8 s h o w t r e e s in this range. Here, b r a n c h s e g m e n t s c o m p l e t e l y s c r e e n t h e Koch c u r v e g e n e r a t o r w i t h c o r n e r s LR2(LR) ~ a n d L2(RL) ~ = R2(LR) ~, a n d the g e n e r a t o r with c o r n e r s R2(LR) ~ a n d R 3 ( L R ) ~ = R L e ( R L ) % The r e m a i n i n g g e n e r a t o r s for the c a n o p y h a v e c o r n e r s (LR) ~ a n d L R 2 ( L R ) % a n d RL2(RL) ~ a n d (RL)% T h e s e s e g m e n t s h a v e l e n g t h ro = r21, so t h e M o r a n equat i o n is 2r 2~=1 The Critical Value

or

r ~=1/~/2.

0 = 180 ~

Here, the tree c o l l a p s e s to its trunk.

Shortest-Path

Dimension

as a Function

of 0

As m e n t i o n e d earlier, for 0 -----90 ~ the s h o r t e s t p a t h along the b r a n c h tips from (LR) ~ to (RL) ~ c o i n c i d e s with the portion o f the c a n o p y b e t w e e n t h e s e points. F o r 0 > 90 ~ the c a n o p y d i s c o n n e c t s into a Cantor set and, thus, cannot b e the s h o r t e s t path. F o r 0 > 135 ~ the tip set b e c o m e s a K o c h - C e s ~ o curve and is the s h o r t e s t p a t h from (LR) ~ to (RL)% [Note t h e r e is a s h o r t e r p a t h in the tree, following those b r a n c h s e g m e n t s that s c r e e n p o r t i o n s o f the tip s e t and that lie a b o v e the line t h r o u g h (LR) ~ and (RL)% See Figure 8.] At O = 135 ~ the s h o r t e s t p a t h is a line segment. The range 90 ~ < 0 < 135 ~ is m o r e interesting. The shortest p a t h m u s t p a s s through the d o u b l e p o i n t s L3(RL) ~ = R3(LR) ~,

L4(RL) ~ = LRa(LR) ~

[and the c o r r e s p o n d i n g p o i n t R4(LR) ~ = RL3(RL) ~ on the right half; for simplicity, these c o r r e s p o n d i n g p o i n t s on the right half will no longer be mentioned], (LR)L3(RL) ~ = (LR)R3(LR) ~, (LR)L 4(RL )~ = (LR)LR3(LR) ~, . . . , (LR)nL3(RL) ~ = (LR)nR3(LR) ~, (LR)nL4(RL) ~ = (LR)nLR3(LR) ~, . . . . Note t h a t a s n --->~, t h e s e p o i n t s a p p r o a c h (LR) ~, as expected. The line s e g m e n t s connecting t h e s e points in order, together w i t h t h o s e connecting the c o r r e s p o n d i n g points on the right side, form the g e n e r a t o r o f the s h o r t e s t path. F o r m u l a s (1) a n d (2) could be u s e d to find the lengths o f these segments, and so d e t e r m i n e their scalings, but a much s i m p l e r a p p r o a c h is to use t h e IFS construction o f the tip set. First, BL t a k e s the tip set to that p a r t d e t e r m i n e d b y (LR) ~, LR2(LR) ~, LR3(LR) ~, L3(RL) ~, and L2(RL)~; BR

VOLUME 21, NUMBER 2, 1999

25

t a k e s the tip set to the c o r r e s p o n d i n g p a r t on the right. Letting A1 d e n o t e the s e g m e n t from L 2 ( R L ) ~ to L 3 ( R L ) ~ and A2 the s e g m e n t from R 2 ( L R ) ~ to R 3 ( L R ) ~ = L 3 ( R L ) ~, we see that the s e g m e n t f r o m L 3 ( R L ) ~ to L 4 ( R L ) ~ is B L ( A t ) the s e g m e n t from L 4 ( R L ) ~ t o ( L R ) L 3 ( R L ) ~ is B L B R ( A 2 ) the s e g m e n t from ( L R ) L 3 ( R L ) ~ to ( L R ) L 4 ( R L ) ~ is BLBRBL(A1), 9 . .

That is, these g e n e r a t o r segments have lengths r (length of A1), r 2 (length o f A2), r 3 (length o r A l ) , . . . . The initiator has length 2 r sin(0)/(1 - r2), and from their endpoints, we c o m p u t e directly t h a t A 1 and A2 have length 2 r 3 sin(0)/(1 r2). Taldng units so the initiator length is 1, the g e n e r a t o r for the s h o r t e s t p a t h consists of two s e g m e n t s of length r 3, two of length r 4, t w o of length r 5, a n d so on. The Moran equation is 2@3) 4 + 2(r4) d + 2(r5) d + . . . .

1

or (rd) 3

i ~ _ T d -- 1/2.

Thus,

r d ~ 0.589755, so d ~ log (0.589755)/log(r), w h e r e r is e x p r e s s e d in t e r m s o f 0 b y Eq. (3).

The Mix Map As m e n t i o n e d earlier, in the range 135 ~ < 0 < 180 ~ the tip set b e c o m e s a K o c h - C e s ~ r o curve. In this range, the double points have a d d r e s s e s L2(RL) ~ = R2(LR) ~

if S contains an even n u m b e r of elements, and m(S)10(00) ~

and

m(S)01(ll) ~

if S c o n t a i n s an odd n u m b e r o f elements. Here, m ( S ) den o t e s t h e obvious r e s t r i c t i o n o f m to a finite sequence. R e g a r d l e s s of the n u m b e r of e l e m e n t s in their c o m m o n initial string, sequences c o r r e s p o n d i n g to double p o i n t s are sent b y m to equal real n u m b e r s . As a function on b i n a r y sequences, the m i x m a p is clearly self-inverse. Viewing it as a m a p on the unit unterval requires m o r e care: equivalent b i n a r y e x p a n s i o n s d o n o t get sent to the s a m e numbers. F o r e x a m p l e , m ( 0 ( l l ) ~) = 0(01)~---> 1/6, w h e r e a s m ( l ( 0 0 ) ~) = 1(10) ~ = 5/6. Consequently, to v i e w m as a m a p [0, 1] ---) [0, 1], w e m u s t a d o p t some c o n v e n t i o n - - n o terminal strings o f all 0, for e x a m p l e - - a b o u t s e q u e n c e s representing p o i n t s in the domain. Figure 11 is a g r a p h of the mix map. To e m p h a s i z e the sizes of the jumps, in Figure 11 w e have d r a w n a line s e g m e n t b e t w e e n points o f the f o r m ( x , m ( x ) ) a n d ( y , m ( y ) ) , w h e r e x = s l ( 0 0 ) ~ and y = s 0 ( l l ) ~, s any fmite string of 0 a n d 1; that is, the binary strings x and y c o r r e s p o n d to the s a m e real number. Even without adding t h e line segments, the g r a p h has d i m e n s i o n 1. This can b e s e e n b y an argument a n a l o g o u s to t h a t showing that the p r o d u c t of t w o Cantor sets, e a c h o f d i m e n s i o n 1/2, has d i m e n s i o n 1. The longest line s e g m e n t c o n n e c t s m ( 0 ( l l ) : r = 0(01) ~ ---) 1/6 a n d r e ( l ( 0 0 ) :r - o 5/6, so has length 2/3. The s e c o n d longest line s e g m e n t s c o n n e c t m ( l l ( 0 0 ) ~) ---) 5/12 to m(01(00) ~) --> 1/12, and m(11(00) ~) ~ 7/12 to m ( 1 0 ( l l ) ~ --~ 11/12; b o t h have length 1/3. Continuing in this fashion, w e see t h a t the first variation of m is 2+2.

and

1 1+...+2n ~+4"~

"

1

n3 -9 2~ +

"'"

S L 2 ( R L ) ~ = S R 2 ( L R ) ~,

w h e r e S is any finite string of L and R. Figure 8 m a k e s this clear: note the l o c a t i o n s ofL2(RL) ~ = R 2 ( L R ) ~, L 3 ( R L ) | = L R 2 ( L R ) ~, a n d RL2(RL) ~ = R 3 ( L R ) ~, for example. Parsing t h e s e strings in the usual way, w e have

11/125/6

L 2 ( R L ) ~ = 00(10) ~ a n d R 2 ( L R ) ~ = 11(01) ~

and so on. Noting the a d d r e s s e s of these d o u b l e points, we define the m i x m a p a s t h e function on b i n a r y sequences taking the a d d r e s s e s of double p o i n t s to s e q u e n c e s corresponding to the s a m e real number; that is, the mix m a p is defined by m(blb2b3b4...)

5/12

= blb2b3b4 9 . . ,

w h e r e bi = 1 - bi. To see the c l a i m e d identifications, first note that L 2 ( R L ) ~ c o i n c i d e s with R 2 ( L R ) ~, a n d m(OO(lff~ = 01(11) ~ = 1/4 + 1/8 + 1/16 + ..., m(11(01) ~~ = 10(00) ~1761/2,

and the b i n a r y s e q u e n c e s c o r r e s p o n d i n g to S R 2 ( L R ) ~ are s e n t b y m to m(S)Ol(ll) ~

26

and

THE MATHEMATICALINTELLIGENCER

7/12

1/6 1/12 I

SL2(RL)

m(S)lO(O0) ~

~

and

1/4

1/2

3/4

Figure 11. The graph of t h e mix map, with jumps connected with line segments

Acknowledgment

which diverges. The qth variation is q+2"

q+4"

q+'"+2

n" ~

+""

Because 2n 9(3 9 2n-l) -q = (2/3 q) 9 (2 9 2-q) n-l, the qth variation converges for q > 1. Finally, by using different bases or modifying different length substrings, we can produce many other functions with different patterns of changes. For example, ml(blb2b3 9 9 .) = blb2b3b4b5b6 9 . 9 is another self-inverse function, whereas m 2 ( b l b 2 b 3 9 9 . ) -= b 2 b 3 b l b s b 6 b 4 9 9 9 satisfies m 3 = i d e n t i t y . This general theme can lead in other directions. For instance, by flipping values at increasingly widely separated positions, we produce functions that are extraordinarily mixing in the large, but less mixing in the small. However, this has (apparently) digressed from our study of trees, and so we leave it for another time.

Animating the Changes A clear understanding of the morphology change of selfcontacting trees as O increases from 0~ to 180~ is easily communicated through animation. Several efforts in this direction can be found at http://www.union.edu/PUBLIC/MTHDEPT/research/fractaltrees/

We thank the referee, whose thoughtful comments led to several extensions of the original version.

Further Reading Besides [FGN], Chap. 16, trees are discussed in [PJS], [L], and many other sources. The construction of trees as Lsystems is described in [PL]. Iterated function systems are used in [FGN], Chap. 20, and developed in [B] (which introduced the currently accepted term); see also [H].

References [B] M. Barnsley, Fractals Everywhere, 2nd ed., Boston: Academic Press (1993). [H] J. Hutchinson, "Fractals and Self-Similarity," Indiana University Journal of Mathematics 30 (1981 ), 713-747. [L] H. Lauwerier, Fractals: Endlessly Repeated Geometrical Figures, Princeton, NJ: Princeton University Press (1991). [FGN] B. Mandelbrot, The Fractal Geometry of Nature, New York: W. H. Freeman 1982. [PJS] H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers in Science, New York: Springer-Verlag, 1992. [PL] P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants, New York: Springer-Verlag (1990).

VOLUME 21, NUMBER 2, 1999

27

i~'J~-Tii[~li=r-ii[,~.allO-z,l==ai=l,,,ai|[;-].-ll

History of the IH[S and its Foundation by L6on Motchane* Louis Michel

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 submissions to the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith Cotlege, Northampton, MA 01063, USA; e-mail: [email protected]

28

Marjorie Senechal,

Editor I

don Motchane w a s b o r n l0 June 1900 to a Russian-Swiss family in St. Petersburg, and g r e w up there. At the time of the 1917 Revolution, while continuing his studies o f m a t h e m a t i c s a n d physics, he was active in an organization of young s t u d e n t s d e d i c a t e d to easing the sufferings o f t h e population. The following y e a r he j o i n e d his m o t h e r and his o l d e r b r o t h e r in Switzerland, w h e r e he w e n t o n with his s t u d i e s and did part-time w o r k as a cabinet-maker. When his f a t h e r j o i n e d t h e family a y e a r later, L~on w a s able to s p e n d a y e a r as an a s s i s t a n t in p h y s i c s at the University of Lausanne. F r o m 1921 he had to w o r k to supp o r t his parents. He w o r k e d first in Berlin as an artist's impresario, then in insurance. He moved to F r a n c e to stay in 1924, becoming a citizen in 1938. Many varied activities o c c u p i e d him, s o m e t i m e s extending b e y o n d France; he w a s an executive in several firms. He is married, with t w o sons: Didier, n o w a conseiUer maitre at the C o u r des Comptes; a n d Jean-Loup, a p h y s i c i s t at Paris 7. He h a s n o t forgott e n his m a t h e m a t i c a l bent; he j o i n e d the Soci~t~ Math~matique de F r a n c e a n d has p u b l i s h e d in m a t h e m a t i c s . He is also a g o o d pianist a n d c h e s s player. At the start of World War II he volu n t e e r e d to serve. A s s i g n e d to the artillery, he b e c a m e an officer c a n d i d a t e at t h e s c h o o l of Fontainebleau. After being demobilized in the s u m m e r of 1940 after the fall of France, he immediately j o i n e d the Resistance, w h e r e his duties were p r i m a r i l y in intelligence; he was w o u n d e d in a c t i o n 13 August 1944. F o r his s e r v i c e s he was a w a r d e d the Croix de G u e r r e a n d the Medal of the Resistance with rosette. At the s a m e time, he w a s w o r k i n g on the f a m o u s I~ditions de Minuit, which b r o u g h t out two c l a n d e s t i n e w o r k s of sociological t h e o r y by him u n d e r the

L

p s e u d o n y m Thimerais: Elements of Doctrine ( F e b r u a r y 1944) a n d Patient Thought (July 1943). His friends find his n a t u r e e v o k e d in his w o r d s in t h e introduction:

Thought, under pressure of the passing days, tends to the immediate. The obstacles to post-war recovery seem far away; nevertheless they are already present and must be considered. Difficulties can not be overcome without patient thought. Day-to-day worries m u s t not be allowed to conceal f r o m us the continuity of life, which makes today's efforts responsible for tomorrow's outcome. His faith in the future r e s t e d on absolute i n s i s t e n c e on social j u s t i c e a n d solidarity. After the war, while e x p a n d i n g his own p r o f e s s i o n a l activities, he w a s m o r e a n d m o r e interested in science. E n c o u r a g e d b y Paul Montel, he published n o t e s in the Comptes Rendus de l'Acaddmie des Sciences, in m a t h e -

L6on M o t c h a n e . (Used by permission of the IHES,)

*Edited version of an address 8 October 1998 in celebration of the fortieth anniversary of the founding of the Institut des Hautes Etudes Scientifiques.

THE MATHEMATICALINTELLIGENCER9 1999SPRINGER-VERLAGNEWYORK

matics and later in theoretical physics. He defended his thesis for the doctorat d'Etat in December 1954; it was entitled Invariant properties in simple convergence, and the committee consisted of Paul Montel, Arnaud Denjoy, Jean Favard, and Gustave Choquet. L~on Motchane came to have many scientific contacts in France and abroad. When did he conceive the project of founding a scientific institute? Perhaps as early as 1949. While he was visiting his brother Mexandre, an engineer in New Jersey, Alexandre introduced him to Robert Oppenheimer, the director of the Institute for Advanced Study in Princeton. From this they developed deep and friendly relations. Motchane visited Oppenheimer regularly and would write him asking advice. In the first visit to the IAS, L~on Motchane met C~cile Morette (later DeWitt) who was then working there; for her too he had numerous questions about the operation of the Institute, and remarks about how France could use such an institution. Certainly the IAS served as a model for the II-IES, but we have to bring out differences in their organization. It was not a matter of some benefactor creating a scientific foundation with a large endowment. In the twentieth century France has had some institutes for basic research set up by gifts of philanthropists (mostly foreign), but in the same area of science the only one was the Institut Henri Poincar~, and that is run by the Paris tmiversities and the Comit~ National de Recherches Scientifiques. To find a case more comparable to the IHES, we have to go back over a hundred years, to the Institut Pasteur! Motchane's vision inevitably seemed utopian. It was to found an institute of basic research in "Mathematics, Theoretical Physics, and Methodology of the Humanities." He would make only one requirement of its professors, that they be in residence for six months. They would be permanent (that is, named for life) and equal (in particu-

The early days of the Grothendieck seminar. (Used by permission of the IHI~S.)

lar, given equal pay). The director would be named for eight years, renewable (at most twice) for four years. The Scientific Committee would be composed of the Director, the permanent professors, and a smaller number of scientists coopted by them for terms of at most six years. M1 scientific decisions (and under the budget that inchided the naming of professors) would be made by the Scientific Committee and could not be reversed by the Administrative Council. At the end of the term of a Director, the successor would be chosen by the Scientific Committee; the Admires"trative Council had only the power to endorse or refuse their choice. And the Institute would have to operate on donations from business under the terms of the Law of 1901. This impossible project Motchane realized. It took unshakeable optimism, based on profound deliberation and on a thorough knowledge of the scientific world and of business circles; and it took high skills of negotia-

tion. The association was incorporated 27 June 1958 by a meeting of all the future members of the Administrative Council. It took place in the office of its fwst President, Joseph P~r~s, Dean of the vast Faculty of Science of the University of Paris. 2 The draft by-laws, which had been painstaldngly drawn up by Ldon Motchane and his legal advisor, were unanimously approved. Motchane was chosen as Director of IHr a position he kept until his retirement in 1971. Throughout these 13 years, by dint of his talent for human relations, and aided by some members of the Administrative Council, Motchane added to the ranks of the supporters. 3 Not without difficulties. Incorporation as a public service organization was essential to the functioning of the IHES, and was obtained in 1961. On this occasion two executives of oil companies wanted to put representatives from the Administrative Council onto the Scientific Committee to rule on the activities of pro-

2Along with Leon Motchane and his legal advisor Jean Robert, the group consisted of representatives of seven corporations--Renault, Esso Standard, French Shell, Petrole BP, Gaz de France, Compagnie generale de TSF (where Maurice Ponte was an ardent supporter), and Trefileries du Havre--and two philanthropists: a representative of Edmond de Rothschild, and Gabrielle Reinach, who was later to leave all her estate to the IHr enabling it to add a wing to the scientific building. 3Government agencies like the CEA (whose head Francis Perrin was an effective ally), the EDF, the Banque de France, the Caisse des Depots et Consignations; major corporations like Pont-a-Mousson, whose honorary president Andre Grandpierre succeeded Peres in 1962 as president of the Administrative Council.

VOLUME21, NUMBER2, 1999

fessors and visitors. Motchane held firm to the principles he had established, but it cost him the support of the two oil companies. In some cases also an industry member of the Administrative Council would be replaced in his executive position by someone lacking his devotion to IH]~S, and the company's support would cease. The first financial crisis came in 19659 The threat was not merely that the remarkable progress of the first seven years would be halted, but that the project might disappear! Prime Minister Pompidou was alerted, and undertook to arrange regular state funding; this soon became the primary source. Motchane, and later Grandpierre, the second president of the Administrative Council, tried to get subsidies from abroad, 4 but these did not last. The oldest foreign subsidy still continuing is that of English SRC (since 1970); a number of others were initiated during Nicholaas Kulper's Directorship (1971-1985)9 At present foreign contributions cover about onesixth of the budget, and French donors one-sixteenth. All of IH]~S's resources are in funds committed for only a few years. Let me turn to the scientific history of the IH]~S. It begins in two offices rented at the Fondation Thiers (in the

16th arondissement of Paris), one for the Director and the secretary (Annie Rolland), the other for the first two permanent professors: Jean Dieudonn6 (French, but then a resident of the USA) and Alexander Grothendieck (then 32, a stateless person educated in France). A weekly seminar in Algebraic Geometry was started in 1959, meeting in a room of the Fondation Thiers and attracting 20 to 30 mathematicians, from young students at the ]~cole Normale Sup6rieure to established professors. To this were added in 1960-61 the seminar of Claude Chevalley (who visited for the year) and lectures (held at various sites) by shortterm visitors: M. Atiyah, S.S. Chern, H. Grauert, H. Hironaka, J. Tits, A~ Weil, 9 In 1960 the Publications Mathdmatiques de I'IHES first appeared, quickly becoming an outstanding mathematical journal. This brilliant beginning was not matched by theoretical physics. Armed with some very good advice, Motchane in December 1959 offered the first professorship in Physics to Murray GellMann (Nobel Prize 1969), who was on a visit of several months to the Coll~ge de France. He imposed several conditions; it took six months for the Director to meet them, and another year for Gell-Mann to decline the offer. Motchane also invited a fair number of

visitors, but physicists have their own habits of work, and as far as I know only three of them agreed to visit this Institute without a campus: E. Caianello, A. Wightman, and G. K~llen. All that changed in autumn 1962, when the IHES moved into the magnificent estate of Bois-Marie, a 10hectare park in Bures-sur-Yvette. Oppenheimer, Peierls, and Weisskopf were then the physicist members of the Scientific Committee. I became the first permanent Professor of Physics9 Harry Lehmann arrived then too and stayed three years. He had been offered a permanent professorship, but soon decided not to resign from his chair at Hamburg and therefore not to accept the title of Professor at IH]~S. (He was later the representative of a German foundation on the Administrative Council.) By the way, Alain Connes is counted among the Professors, even though he declined the offer of the title upon getting membership in the Collbge de France; his office and most of his activities are here; he is listed by the IHI~S as the "L6on Motchane Professor" and has the same responsibilities as the permanent Professors. The list of visitors in theoretical physics soon became very important in quantity and quality. In mathematics, Ren6 Thorn had been approached years earlier, but came only in 19639 The Grothendieck seminar retained its high prestige, and

the l~ldments de Gdomdtrie algdbrique

The library of the IHES at Bois-Marie, Bures-sur-Yvette. (Used by permission of the IHES,)

(published in the "blue journal" of the IHES) became a monument of mathematics. The first foreign permanent Professor was the physicist David Ruelle, who came in 1964. Since then, ten more professors have been recruited, and only two of them are French, Alain Connes and Thibault Damour. Notice that except for Dieudonn6, professors at the IHl~S have been recruited at age between 25 and 40 (most often between 30 and 33)9 Both in its permanent and its visiting membership, the IHI~S is an international institution; since 1980 it is formally a French foundation. L6on Motchane had a residence built in 1967, l'Ormaille in Bures, to put

4First Fiat and Montecatini; also Euratom (though such support was strictly forbidden by its charter!); and for some years an American foundation, the American Committee for the Institute for Advanced Study--Europe, created for the purpose in 1965.

THE MATHEMATICALINTELLIGENCER

The lounge at the IHES, site of the famous daily tea. (Photograph by Marjorie Senechal.)

up visitors and their families. The IH]~S could not afford to own it, but got a 25year lease. It has obtained a loan to make possible purchase of the property, so as to hold on to an essential feature of its m e m b e r s ' way of life. Members' children are very well received by the schools, and their school friendships open family friendships with non-scientist French neighbors. The residences, lunch at the Institute, and tea every day at 4 are a pattern of life copied from the IAS in Princeton; they are well suited to the scientific life of the IHt~S and the intensity of the exchanges among its members. In his prophetic vision, Motchane used to speak of collaboration between mathematicians and theoretical physicists. During his Directorship, the most one could speak of was coexistence; but since then the IHt~S has bec o m e the one place in the world where the two disciplines have the most intense, the deepest, and the most productive interactions. To go farther into the rich scientific history of the IHES would take too long. At any given time there are six Professors, six long-term CNRS visitors, and thirty or forty other visitors, and their achievements and influence are great. An outline of the history can be found in the annual reports of the Directors: L. Motchane, N. Kuiper, M. Berger, J.-P. Bourguignon. There is already a Princeton thesis with the somewhat sarcastic title A cultural history

of catastrophes and chaos around the Institute des Hautes t~tudes Scientifiques, France, by David Aubin; it describes and analyses a part of the Institute's scientific activity in the decade beginning a bit before 1970. Let

me just give a statistic that m a y be impressive even to non-scientists: the IHI~S is unique in that two-thirds of its Professors of Mathematics (6 of the 9 it has n a m e d since its founding) have a Fields Medal. Of these, only Ren~ Thorn already had it before joining. Alexander Grothendieck, Pierre Deligne, Alain Connes, Jean Bourgaln, and Maxim Kontsevich got the Fields Medal while IHl~S Professors. Kontsevich and Mikha~l Gromov are now the permanent Professors of Mathematics, Dennis Sullivan having recently resigned. The Professors of Physics are David Ruelle, Thibault Damour, and Michael Douglas; former holders of the title are Jtirg FrShlich, Oscar Lanford, and myself. L~on Motchane retired in 1971. Annie Rolland, who had as general secretary put so much devotion, taste, and care into organizating the setting of Institute life, retired at the same time to b e c o m e Madame Motchane. The newlyweds lived several years near Aixen Provence, to let Motchane's successor have a go at running the IH]~S on his own. On their return to Paris, L~on Motchane gave further valuable help to IHES as Vice-President (later Honorary President) of the Administrative Council. He had the pleasure of seeing all his project realized as he had conceived it except the section on Methodology of the Humanities. Ren~ Thom took some steps in that direction, but the section was never created. The life of the Institute is the best memorial to L~on Motchane. Scientists of the whole world, and all of the French, m a y be grateful to him for what he did for the country and for science. In the park, near the library, there is a very handsome bust o f him, 5

to recall to those w h o knew him and to everyone that Ldon Motchane was the creator of the IHt~S.

5Made to mark the 40th anniversary of the Institute by Richard Rysanek, a sculptor living in Bures-sur-Yvette.

VOLUME21, NUMBER2, 1999 31

ROBERT J. MACG. DAWSON* AND WENDY A. FINBOW*

What Shape is a Loaded Everbody knows that the dice are loaded Everybody ro& with their fingers crossed --Leonard Cohen, Everybody Knows Loaded dice have been around for a long time. Indeed, no die is completely fair, and the earliest dice that have survived were either astragali-approximately polyhedral knucklebones of sheep-or fairly crude approximations to cubes. In either case, the probabilities of the various throws were far from equal, and games must have relied either on cheerful ignorance of probability or rules that gave neither player the benefit of the asymmetric dice. For instance, one would not want to play craps with obviously skewed dice, especially if they were provided by the house; but the game of liar’s dice might actually be improved by the innovation! The mechanics of loaded dice is quite complicated, and appears to involve not only the shape of the dice but their resilience. In this paper we will examine only the “statics”-considering dice that are so loaded that they can stand only on one facet. Imagine such a die, assumed to be a convex polyhedron, standing upon one facet, on a flat surface. If the center of gravity of the body is not above that facet, the body will fall over, landing upon another facet. The body may in turn be unstable on that second facet, and fall again. This process always lowers the center of gravity, and so cannot continue forever. However, the possibility of the body being stable on only one facet is not ruled out; it is interesting to determine under what conditions this can occur. Clearly, this problem may be generalized to polytopes of other dimensions. If the body is considered to be constructed of a material of uniform density, the position of the center of gravity is determined by the shape of the body. A polytope which is stable on only one facet when so constructed has been called monostatic. It is known that no polygon is mono*Supported by a grant from NSERC.

32

THE MATHEMATICAL INTELLIGENCER0 1999 SPRINGER-VERLAG NEW YORK

static, but there exists a monostatic polyhedron with nineteen facets [l]. This is an irregular 17-sided cylinder. Its cross-section is a bilaterally symmetric polygon whose halves each approximate an equiangular spiral. The ends are oriented obliquely, which renders the polyhedron unstable on those facets, and moves the center of gravity further towards the resting facet. Figure 1 shows two of these: one standing on an oblique end, the other in its stable position. The cylindrical rings that decorate the end facets are centered on a line through the center of gravity. Conway and Guy ask, in [l], whether it is possible for such a polyhedron to be stable on its smallest-diameter facet. They give a simple modification of their construction which achieves this. Let the final resting facet be pushed out slightly as in Figure 2. If the angle 13 is small

enough, the new facet in the middle may be made as narrow as desired, while the center of gravity will not change significantly and will not be above either of the other two

new facets. Repeating this construction yields a single resting facet, of diameter as small as desired. It was shown in [2] that a simplex can be monostatic in ten dimensions, but not six. The ten-dimensional example given there has all of its facets nearly perpendicular to a single plane; it is thus much larger in eight of its dimensions than it is in the other two. Subsequent work [3,4] has shown that there is no monostatic simplex in seven or eight dimensions. There is an ll-dimensional simplex [3] which falls sequentially onto all of its facets; this cannot occur in fewer dimensions [4]. Clearly, if the body is allowed to have nonuniform (but positive) density, the center of gravity may be anywhere in the interior of the body. Conway has constructed a tetrahedron which, with a suitably positioned center of gravity, is stable only on one facet. The sequence of falls (left to right) by which it rolls to its stable facet from its least stable facet is illustrated in Figure 3; the white dot indicates the nearest point on the surface to the center of gravity. Such a (polytope, center of gravity) pair will be called loaded, and the polytope will be called loadable. The question may be asked: which polytopes are loadable? This is clearly equivalent to asking which polytopes have internal points which are above only one facet. (When we say that

VOLUME

21,

NUMBER 2, 1999

33

a p o i n t is "above" a facet, we m e a n that the f o o t of the perp e n d i c u l a r to the f a c e t is within the facet.) It is easily s e e n that, no m a t t e r w h e r e the c e n t e r of gravity of a p o l y t o p e is, a p o l y t o p e cannot fall f r o m one facet to a n o t h e r unless the intervening dihedral angle is obtuse. We m a y formalize this as the F i r s t G o o d R e a s o n ( f o r s t a b i l i t y ) : I f a facet, or set of facets, is separated f r o m the r e m a i n i n g f a c e t s by a boundary of (d - 2)-faces w i t h nonobtuse angles, the polytope is not loadable. Such a p o l y t o p e c a n n o t fall f r o m a face inside the b o u n d a r y to one outside, or vice versa. An i m m e d i a t e r e s u l t of the 1GR is that, in the plane, no triangle can b e loadable, as a triangle m u s t have at least t w o acute angles, isolating the edge b e t w e e n them. Furthermore, no r e g u l a r simplex or m e a s u r e p o l y t o p e is loadable in any dimension. However, t h e r e are other obstacles to loading. C o n w a y has o b s e r v e d [1] that no regular polygon is loadable. F o r 2n-gons, this is easily d e d u c e d from w h a t w e m a y call the S e c o n d G o o d R e a s o n ( f o r s t a b i l i t y ) : I f a polytope and all its facets have central symmetry, it is not loadable. Every facet of such a polytope has an o p p o s i t e facet which is the other cap o f a right prism; and any p o i n t which is above one of these t w o facets is necessarily a b o v e the other. A zonotope is a p o l y t o p e that occurs as the Minkowski s u m o f a finite s e t o f line segments. Well-known e x a m p l e s are the square, the hexagon, and other regular 2n-gons in the plane; and the cube, the hexagonal prism, a n d the rhombic d o d e c a h e d r o n in R 3. Alternatively, it m a y b e s h o w n that a polytope is a z o n o t o p e if and only if every face of any dimension has central symmetry. Thus, it follows from the 2GR that no z o n o t o p e is loadable. As 1-faces (edges) are alw a y s centrally symmetric, a p o l y h e d r o n that satisfies the conditions of the 2GR is necessarily a zonotope. However, this d o e s not h o l d in higher dimensions; for instance, the 120-cell in R 4 has centrally-symmetric d o d e c a h e d r a l facets, but non-centrally-symmetric pentagonal 2-faces. Neither of the G o o d Reasons applies to the case of the regular (2n + 1)-gon, ( n > 1). With suitable loading, w e can m a k e such a p o l y g o n roll from any specified edge to any other. However, w e c a n n o t load it so as to m a k e it unstable on all edges b u t one. To explain this, w e m u s t look m o r e closely at the geometry. We first o b s e r v e that there is a p o i n t (the c e n t e r O) which is a b o v e all the edges, and that the set of p o i n t s above any facet is convex. F o r any interior p o i n t B, let B ' be the p o i n t at w h i c h the r a y OB m e e t s the boundary. If p e r c h a n c e B ' is a b o v e (or on) b o t h edge E a n d edge E ' , then so is the w h o l e ray, and in particular, B. Conversely, the set of points a b o v e any facet is closed, s o t h a t if there is a point B ' on an edge E that is above no o t h e r edge, there m u s t be interior p o i n t s n e a r b y on the s e g m e n t OB' that are above only E. This a r g u m e n t holds in any dimension, and m a y be formalized as the

THE MATHEMATICAL INTELLIGENCER

Boundary Principle: A polytope which has a p o i n t 0 that is above every facet is loadable i f and only i f there is a point on some facet that is above no other facet. It t h e r e f o r e suffices to s h o w that every b o u n d a r y p o i n t o f a r e g u l a r (2n + 1)-gon is a b o v e s o m e o t h e r edge. C o n s i d e r an edge AB that is o p p o s i t e a vertex M, with adj a c e n t vertices L and N. The s e t of p o i n t s of AB w h i c h a r e a b o v e L M has length ~1_s e c ( ~ / ( 2 n + 1)) and i n c l u d e s the m i d p o i n t and one end of AB; a n d the rest of AB is over MN. (Figure 4) What a b o u t higher d i m e n s i o n s ? We have s e e n that t h e 1GR p r e v e n t s the cube a n d t e t r a h e d r o n , and t h e i r analogues in higher dimensions, f r o m being loadable. (These a n a l o g u e s are the measure polytopes with 2d facets, e a c h a m e a s u r e p o l y t o p e o f d i m e n s i o n d - l; and t h e regular simplices with d + 1 facets, e a c h a regular s i m p l e x of dim e n s i o n d - 1.) The remaining regular p o l y t o p e s are: 9 the octahedron and cross polytopes in R d, with 2 d facets, e a c h a simplex of d i m e n s i o n d - 1; 9 the dodecahedron in R 3 with 12 p e n t a g o n a l facets; 9 the icosahedron in R 3 with 20 triangular facets; 9 the 24-ceU in R 4 with 24 o c t a h e d r a l facets; 9 the 120-cell in R 4 with 120 d o d e c a h e d r a l facets; and 9 the 600-ceU in R 4 with 600 t e t r a h e d r a l facets. It is n o t h a r d to s h o w that o f the regular polytopes, only the s i m p l e x e s and (2n + 1)-gons fail to have central symmetry. Thus, the 24-cell and 120-cell, w h i c h have centrallys y m m e t r i c facets, are n o n l o a d a b l e b y the 2GR ( t h o u g h w e n o t e d t h a t the 120-cell, w h o s e 2-faces are pentagons, is n o t a zonotope). We shall s h o w t h a t the remaining p o l y h e d r a - the c r o s s polytopes, d o d e c a h e d r o n , icosahedron, a n d 600c e l l - - a r e loadable.

FIGURE 4

M

L

IGURE

9 --Xd). The vertices of a n y facet are (SlXt, s2x2, . 9 9 , SdXd) (where si is either - 1 or 1), the points in it are {(Cl, 9. 9, Cd) : Z~=n+I SiCi = 1, SiCi > 0}, a n d the normal vector to such a facet is (Sl, s2, 99 9 Sd). We will consider the set of points that are in the facet h w h o s e vertices are (x~, x2, 9 Xd) a n d above some other f a c e t - - w i t h o u t loss of generality, a facet h ' with vertices (Xl . . . . , Xn, - X n + l , 999, --Xd). A p o i n t (Cl, 999, Cd) is in A if

d

= 1

(1)

i=l

and Cl, 99 9 Cd > 0. It is above h ' if adding some multiple of the n o r m a l vector ( 1 , . . . , 1 , - 1 , . . . , - 1) yields a p o i n t in h'; that is, if d

(ci + t ) i=1

~.

(ci-

(2)

t)= l

i=n+l

and Consider, first, the set of points on the b o u n d a r y of a d o d e c a h e d r o n which are above (but not on) a given facet9 We see (Figure 5) that these comprise a regular decagon inscribed in the opposite facet and five triangles on the facets neighboring it. If the dodecahedron's edge length is 1, the triangles have base equal to the edge of the decagon, 2%/5/5, and altitude (X/25 -10X/'5/10) sec(Tr - a), where is the dihedral angle of the dodecahedron; this is approximately 09 which is small enough that the triangles on a n y one facet are completely contained in the decagon 9 Thus, the points o n any facet which are outside the inscribed regular decagon are above n o other facet, a n d the d o d e c a h e d r o n is loadable. A similar a r g u m e n t w o r k s for the icosahedron. The facets of the 600-cell are regular tetrahedra. Like the icosahedron and dodecahedron, the 600-cell has central symmetry, but its facets do not; so opposite facets are oppositely oriented. The points on the b o u n d a r y that are above, b u t not on, a given facet comprise a regular octahedron inscribed in the opposite facet, and tetrahedra in the four facets that share a 2-face with the opposite facet9 These tetrahedra have bases, in the c o m m o n 2-face, that coincide with 2-faces of the octahedron; and their height is sec(~r T) times that of a regular tetrahedron with that base, where Tis the dihedral angle of the 600-cell. As sec(~- - T) ~ 19 a n d as the height of a regular octahedron m e a s u r e d perpendicular to a 2-face is the same as that of a regular tetrah e d r o n with the same face, the four such tetrahedra in any one facet are not wholly contained in the octahedron; but, as the illustration shows (Figure 6), the u n i o n of the octah e d r o n and the tetrahedra (whose apices are j u s t visible poking through the centers of the internal faces of the octahedron) is far from filling the tetrahedral facet of the 600cell. So, by the Boundary Principle, the 600-cell is loadable. We n o w t u r n to the most difficult case, that of the crosspolytopes. These exist in all dimensions, and m a y be repr e s e n t e d as the convex hull of all points with one coordinate _+1 and the rest 0. If (Xl, x2, 99 9 Xd) is a n o r t h o n o r m a l basis for R d, the vertices are (xt, x2, 999, Xd, -- xt, -- x2,

ct + t , . . . ,

Cn + t > 0 > Cn+ 1

--

t,...,

ca - t

(3)

for some t > 0. Subtracting (2) from (1) a n d isolating t d yields t = ~2 Zi=n+l ci. But substituting this o n the right of 2 d (3) gives, for n + l < - - j < - d , c j - ~ Z i = n + l c i < O and 2 • c i > g - ~ cj where the s u m m a t i o n is over {n + 1, . . . , j - 1 , j + 1 , . . . , d}. I f d > 2, this c a n n o t be satisfied w h e n cj is sufficiently greater than the other coordinates. Thus, for a n y d > 2, there exists 9> 0 s u c h that every p o i n t whose coordinates are a p e r m u t a t i o n of (1 - e, e - e2, 9 ed-1 - ed, ed) is in h but n o t over any A'; and thus the cross-polytope is loadable. (On the other hand, when d = 2 the cross-polytope is a square, which is n o t loadable.) Figure 7 s h o w s (on the left) the region within an oct a h e d r o n w h i c h is above m o r e t h a n o n e facet, a n d o n the right its complement 9 We c a n see that the i n t e r s e c t i o n of the c o m p l e m e n t with one facet of the o c t a h e d r o n consists of a h e x a g o n (above the opposite facet) and three t r i a n g u l a r spikes (above facets t o u c h i n g at exactly o n e vertex). In Figure 8 we see, for j u s t o n e facet of a 16-cell,

FIGURE 6

VOLUME 21, NUMBER 2, 1999

,35

36

THE MATHEMATICAL INTELLIGENCER

the region which is above no other f a c e t - - a n d its complement, which consists of one octahedron (above the opposite facet), four tetrahedral spikes, and six triangular laminae (actually o f zero thickness, but s h o w n slightly thickened). Should we be alarmed that the octahedron, icosahedron, and d o d e c a h e d r o n - - l o n g ignored as dice, but n o w enjoying considerable popularity as such in role-playing g a m e s - are vulnerable to loading? Is this, rather than Satanism and late nights, the true risk to the morals of garners? Well, not exactly; it is surprisingly difficult to move the center of gravity of a body very far towards the boundary by substituting a denser material for some of its volume. In the case of the octahedron, for instance, we can calculate exactly the regions which are above only one facet (the "pockets" in Figure 7). These comprise 48 small scalene tetrahedra, near the vertices; one of them has vertices {(0,0,1), (1/3,0,2/3),(2/9,1/9,6/9),(3/14,1/14,9/14)}, and the others are its images under the s y m m e t r y group of the octahedron. A s o m e w h a t protracted calculation shows that to move the center of gravity into one of these regions would require the die to be weighted with a substance more than 350 times as dense as the material from which the rest o f the die was made. This might just be done using (say) osmium (specific gravity 22.5) and a light plastic foam, but the deception would be rather obvious.

We attempted to construct a model using lead weights in one corner of a skeletal frame of light steel wire, with edges about 30 cm long. The first frame that we built was too heavy; the second was nearly light enough, but sagged under its o w n weight. We thus conclude that loaded dice in the shape of regular polyhedra, in the sense we have described, are more a mathematical than a practical curiosity. Acknowledgments

Most of the illustrations for this article were produced using POV-Ray version 3.01. This powerful piece of ray-tracing software is written by a team of dedicated programmers and made available to the public at no charge. Executable code for various operating systems, and even source code, are available from ftp.povray.org and various other sites. REFERENCES [1] J.H. Conway and R.K. Guy, Stability of polyhedra, SIAM Rev. 11 (1969) 78-82. [2] R.J.MacG. Dawson, Monostatic Simplexes, Amer. Math. Monthly 92 (1985) 541-546. [3] R.J.MacG. Dawson, W. Finbow, and P. Mak, Monostatic Simplexes II, Geometriae Dedicata 70 (1998) 209-219. [4] W. Finbow, On Stability of Polytopes, M.Sc. thesis, April, 1998, Dalhousie University.

VOLUME 21, NUMBER 2, 1999 3 7

BURKARD POLSTER

YEA WHY TRY I IER RAW WET HAT A Tour of the Smallest Projective Space

R e m e m b e r o u r d e a r colleague the Rev. T h o m a s P. Kirkman, b e s t k n o w n for the following classical p r o b l e m in c o m b i n a t o r i c s ?

Kirkman's Schoolgirls Problem Fifteen schoolgirls w a l k each day in five groups o f three. Arrange the girls' walks for a w e e k so that, in that time, e a c h p a i r o f girls walks t o g e t h e r in a group j u s t once. Recently, I r e c e i v e d a letter from T h o m a s which I feel obliged to s h a r e with you. It contains the r e a s o n s why T h o m a s d e c i d e d to flee our world, a p i c t o r i a l solution to his problem, a n d a lot of illustrations w h i c h c a n n o t be found in any t e x t b o o k . It also contains s o m e o f T h o m a s ' s m o s t r e c e n t insights into the nature o f his p r o b l e m , and s o m e sinister i m p l i c a t i o n s of his results o f w h i c h I think all of you should be aware.

The Nightmare Dear f r i e n d , . . . F o r m a n y y e a r s I had a s u s p i c i o n that there is something f l m d a m e n t a l l y w r o n g with o u r - - t h a t is, y o u r - - u n i v e r s e . In 1851 I finally figured o u t what! I w o k e up in the middle o f the night and the only thing I could rem e m b e r was this nightmare of me falling into s o m e kind

THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER-VERLAG NEW YORK

o f b o t t o m l e s s pit (Fig. 1). S o u n d s familiar? As usual, I h a d fallen a s l e e p thinking a b o u t geometry. P r o b a b l y it w a s bec a u s e o f this that I w o k e up in a m a t h e m a t i c a l frame of mind, thinking, "Let us a s s u m e t h a t I a m a flatlander a n d I w a k e up from the flat equivalent of m y falling nightmare. Then the last picture from m y d r e a m that I r e m e m b e r will l o o k like this." At this point, it d a w n e d o n m e that the fact t h a t parallel lines d o n o t m e e t is the r e a s o n for m y flat c o u n t e r p a r t having this terrible dream. Similarly, it is b e c a u s e t h e r e a r e p a r a l l e l p l a n e s that I k e e p w a k i n g up in the middle of the night. I h a d h e a r d r u m o u r s o f certain projective spaces in w h i c h t w o p l a n e s always i n t e r s e c t in a line, and I realized t h a t I h a d to travel to one of t h e s e distant worlds to e s c a p e t h e terrible nightmare. As y o u know, m y j o u r n e y w a s successful, and I have b e e n living a carefree life in the smallest p r o j e c t i v e space; a life d e d i c a t e d to r e s e a r c h in comb i n a t o r i a l mathematics. That is, carefree until 2 w e e k s ago, w h e n I w o k e up from a n i g h t m a r e again! Not the s a m e as before. It all has to do with m y r e s e a r c h c o n n e c t e d with the p r o b l e m which is n a m e d after me. Let me explain.

The Smallest Projective Plane R e m e m b e r that a projective p l a n e is a point-line g e o m e t r y that satisfies the following axioms.

Figure 3. A stereogram of the Fano plane.

Figure 1. The dream.

Two distinct p o i n t s are c o n t a i n e d in a unique line. Two distinct lines i n t e r s e c t in a unique point. 9 Every point is c o n t a i n e d in at least three lines a n d every line contains at least t h r e e points. 9 9

A s s o c i a t e d with every field K is a classical projective plane w h o s e points a n d lines can be identified w i t h the 1a n d 2-dimensional s u b s p a c e s of the 3-dimensional v e c t o r s p a c e over the field K. In t h e s e classical p r o j e c t i v e planes, the three a x i o m s are easily verified. F o r example, the first a x i o m c o r r e s p o n d s to the fact that two 1-dimensional subs p a c e s of a 3-dimensional v e c t o r s p a c e are c o n t a i n e d in exactly one 2-dimensional subspace. F o r c o m p l e t e n e s s ' s sake, I should r e m a r k that t h e r e are n o n c l a s s i c a l projective planes. The smallest p r o j e c t i v e plane is the Fano plane, that is, t h e projective plane a s s o c i a t e d with the field Z2. It has s e v e n p o i n t s and seven lines. Every line c o n t a i n s e x a c t l y t h r e e p o i n t s and every p o i n t is contained in e x a c t l y t h r e e lines. Figure 2 is a w e l l - k n o w n picture of this plane. In fact, it s e e m s to be the only p i c t u r e of this f u n d a m e n t a l geometry of w h i c h m o s t p e o p l e are aware. R e m e m b e r t h a t the "circle" counts as a line.

Figure 2. The traditional picture of the Fano plane.

I have t o admit that it is a nice picture, b u t is it really the only p i c t u r e w o r t h drawing, a n d is it even the b e s t m o d e l of this plane? Well, all the p l a n e s in the world t h a t I a m living in are F a n o planes, and I have to s h o w you at least t w o m o r e beautiful p i c t u r e s with which you are p r o b ably n o t familiar. The s t e r e o g r a m in Figure 3 s h o w s a spatial m o d e l of t h e F a n o plane. The s t e r e o g r a m can b e v i e w e d with either the parallel or t h e cross-eyed technique; that is, one of the techniques that y o u had to m a s t e r a c o u p l e of years ago to b e able to view random-dot s t e r e o g r a m s that w e r e in fashion in y o u r world. You can think o f this m o d e l as being inscribed in the t e t r a h e d r o n as follows. The p o i n t s are the centers o f the six edges of the t e t r a h e d r o n plus t h e center of the tetrahedron. The lines are the three line segments connecting the c e n t e r s o f opposite edges plus the circles inscribed in the four sides of the tetrahedron. E v e r y s y m m e t r y of t h e t e t r a h e d r o n translates into an a u t o m o r p h i s m of the geometry. The s y m m e t r y group o f the t e t r a h e d r o n has o r d e r 24. The p i c t u r e of the F a n o p l a n e in Figure 4 s h o w s t h a t n o n e o f the p o i n t s of the p l a n e is distinguished among t h e p o i n t s a n d t h a t no line is distinguished among the lines. In fact, the r o t a t i o n through 360/7 d e g r e e s a r o u n d the c e n t e r of the d i a g r a m c o r r e s p o n d s to an a u t o m o r p h i s m of o r d e r 7, which g e n e r a t e s a cyclic group o f a u t o m o r p h i s m s acting transitively on the point a n d line sets of the plane. T o g e t h e r with an a u t o m o r p h i s m o f o r d e r 7 like the o n e underlying Figure 4, the 24 a u t o m o r p h i s m s a p p a r e n t in the

Figure 4. The Fano plane: all points are equal!

VOLUME 21, NUMBER 2, 1999

39

Figure 5. The smallest projective space.

spatial model generate the full automorphism group of the Fano plane. It has order 24 • 7 = 168.

The Smallest Perfect Universe Associated with every field K is a (3-dimensional) projective space whose points, lines, and planes can be identified with the 1-, 2-, and 3-dimensional subspaces of the 4-dimensional vector space over the field K. Of course, there is also a set of axioms for projective spaces. I will not bother reminding you of these axioms, as, essentially, there are no examples of projective spaces apart from the classical ones associated with fields. The smallest projective space over the field Z2 has 15 points, 35 lines, and 15 planes. Each of the 15 planes contains 7 points and 7 lines; as geometries, they are isomorphic to the Fano plane. Every point is contained in 7 lines and every line contains three points. Furthermore, two dis-

Figure 6.

40

Hall's magical labelling.

THE MATHEMATICALINTELLIGENCER

tinct points are contained in exactly one line and two planes intersect in exactly one line. The diagram on the left in Figure 5 is a partial picture of this space. It shows all 15 points and 7 "generator lines." The other lines are the images of these generator lines under four successive rotations of the diagram through 360/5 degrees. Given a point p and a line 1 not through this point, form the union of all points of the lines connecting p with points of 1. This union is the point set of one of the planes of the space. All planes are generated in this way. The diagram on the right shows one such plane. Note that all lines connecting different points in such a plane are fully contained in the plane. As a point-line geometry, every such plane is really a Fano plane.

Hall's Magical Labelling Figure 6 is a construction of the smallest projective space due to m y friend Hall. Let SEVEN and EIGHT be the sets {1, 2 , . . . , 7} and {1, 2 , . . . , 8}, respectively. Label the points of the Fano plane with the numbers in SEVEN in all possible ways. Remember that the automorphism group of the Fano plane has order 168. This means that there are 7!/168 = 30 essentially different such labellings. On close inspection, it turns out that 2 among these 30 labellings have either 0, 1, or 3 lines (=triples of labels) in c o m m o n . There is a unique partition of the 30 labelled Fano planes into 2 sets X and Y of 15 each such that any 2 Fano planes in 1 of the sets have exactly 1 line in common. Now, the 15 points of the projective space can be identified with the 15 labelled Fano planes in either X or Y, and the lines with

Figure 7. Generalized quadrangle and spread.

the (7) = 35 triples of distinct numbers in SEVEN. A point (=labelled Fano plane) is contained in a line (=triple) if the triple is a line in the labelled Fano plane. Figure 6 is a labelling of the above model of the projective space with labelled Fano planes. The highlighted line corresponds to the triple 237.

this world, and I can handle the fact that there are disjoint lines in this space. Still, I discovered that there are 56 spreads of lines! Fix two of the elements of SEVEN. Then, there are five triples containing these two numbers. Every such set of five triples corresponds to a spread in the space. For example, the numbers 1 and 7 correspond to the spread in Figure 7. We can construct (7) = 21 of the 56 spreads contained in our space in this way. Here is a natural identification of the 56 spreads in our space with the (s) = 56 triples of numbers contained in the set EIGHT. Let xy8 be such a triple. Then, the spread associated with it is the spread associated with the two numbers x and y. Let xyz be a triple in SEVEN. Then, the spread associated with it consists of xyz itself plus the four triples in SEVEN which are disjoint from xyz.

For example, the generalized quadrangle which corresponds to the number 7 is the geometry depicted in Figure 7 on the left. Note that an ordinary quadrangle with its four vertices considered as the points and its four edges considered as the lines of a point-line geometry is a generalized quadrangle. Furthermore, just as in this prototype, the smallest n for which an n-gon can be drawn in a generalized quadrangle using only lines of the geometry is 4.

Packings--Solutions to Kirkman's Schoolgirls Problem Now consider any labelling of the Fano plane with elements of EIGHT. Then the seven spreads corresponding to the seven lines (=triples in EIGHT) of the Fano plane are pairwise disjoint. In fact, every line in the projective space is contained in exactly one of these seven spreads. Any set of seven spreads of our space which has this property is called a packing of the space and is, oh horror of horrors, just a "spread of spreads." Because every packing of the space corresponds to such a labelling of the Fano plane, there are 8!/168 = 240 packings of our space. Figure 8 shows one packing and the labelled Fano plane associated with it. Ironically, every packing corresponds to a solution of the problem which is named after me. Just identify the 15 girls with the 15 points, the "groups of 3" occurz~g during a week with the lines of the space, and the 7 walks with the 7 spreads of a packing. For a long time I thought that things cannot get any worse. I was mistaken.

The Nightmare Continues A spread of a geometry is a partition of its point set into

Hyperpackings--the One-Point Extension of the Fano Plane

disjoint lines or planes. Two parallel lines are quite scary, but the mere thought of a spread makes me want to hide somewhere. Fortunately, there are no spreads of planes in

Let us play the f o l l o ~ g game. Remove the 7 spreads corresponding to a packing from the 56 spreads of our space Try to find a packing among the remaining 49 spreads. If you find one, put it aside and try to find yet another one among the remaining 42 spreads, and so on until no more

Generalized Quadrangles Fix a number n in SEVEN. Then, there are (6) = 15 triples containing this number and 2 of the remaining 6 numbers in SEVEN. The 15 points of the projective space together with these 15 lines make a so-called generalized quadrangle; that is, a geometry satisfying the following axioms: 9 Two points are contained in at most one line. 9 Given a pointp and a line 1 that does not contain p, there is a unique line k through p which intersects 1.

~~

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Figure 8. Packing.

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245 467 356

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:ioOn Cfletharly5,6eSpIr~ad~pmetr~a8c~g::p::pkiong~stW~o: s a h O r ;

Figure 9.

One-point extension of the Fano plane.

VOLUME 21, NUMBER2, 1999 41

eight labelled Fano planes whose labels are contained in EIGHT and which are pairwise line-disjoint. Figure 9 is one way to construct such a set of labelled Fano planes. The one-point extension of the Fano plane has eight points; the seven points of the Fano plane plus one additional point. It has 14 lines containing 4 points each. These are the complements of the lines of the Fano plane in its points set, plus the lines of the Fano plane which have all been extended by the additional point. Note that any 3 distinct points of the geometry are contained in exactly 1 of the 14 lines. The points of the one-point extension can be identified with the eight vertices of the cube such that the lines turn into the following sets: 9 The vertex sets of the regular two tetrahedrons inscribed in the cube. 9 The vertex sets of the six faces of the cube. 9 The vertex sets of the six "diagonal rectangles." The points of the derived geometry at a point p of the one-point extension are the seven points different from p. The lines are the lines of the one-point extension containing p which have been punctured in p. Clearly, every such derived plane is a Fano plane. Label the one-point extension with the elements of EIGHT. This labelling induces a labelling of the eight derived Fano planes, and it is easy to see that any such set of eight labelled Fano planes derived like this has the "desired" property. Figure 10 shows the eight packings of a hyperpacking which corresponds to the labelling of the cube in the middle of the diagram. The de-

Figure 10.

42

Hyperpacking.

THE MATHEMATICALINTELLJGENCER

rived Fano planes at the points 1, 2 , . . . correspond to the packings in the upper left corner, in the middle above, and so on in the clockwise direction. Up to automorphisms, there are 30 different labellings of the one-point extension corresponding to the 30 essentially different labellings of the Fano plane. Unfortunately, not all hyperpackings can be constructed like this. In fact, there are 27,360 different hyperpackings!

Hyperhyperpackings Let us play another game. Remove the 8 packings corresponding to a hyperpacking from the 240 packings of our space. Try to find a hyperpacking among the remaining 232 packings. If you find one, put it aside and try to fmd yet another one among the remaining 224 packings, and so on until no more hyperpackings can be found. If this happens when no packing is left, you have constructed a hyperhyperpacking; that is, a partition of the 240 packings into 30 disjoint hyperpackings. Unfortunately, these hyperhyperpackings do exist. In fact, the 30 hyperpackings corresponding to the essentially different labellings of the onepoint extension form a hyperhyperpacking.

Hyperhyperhyperpackings? I do not know what other monsters are lurking in the shadows. Conceivably, it might be possible to construct hyper-

hyperhyperpackings, hyperhyperhyperhyperpackings, and so on ad infinitum. I do not know, and I do not dare to investigate any further. I think it is time to flee again. I just

0 0

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F i g u r e 11.

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The identification o f the different p a c k i n g s with the different labellings of the F a n o p l a n e w i t h e l e m e n t s of EIGHT can be f o u n d in [3] a n d [7]. The d i a g r a m of the generalized quadrangle in Figure 7 is called the doily a n d is due to Payne. Lots o f s t e r e o g r a m s a n d o t h e r p i c t u r e s of spatial and p l a n e m o d e l s o f the s m a l l e s t projective space a n d m a n y o t h e r finite a n d topological g e o m e t r i e s c a n be found in [61. Finally, I should a c k n o w l e d g e t h a t Kirkman is n o t k n o w n to h a v e fled o u r world in horror.

Inversive plane--the sign of the devil.

REFERENCES f o u n d out that other p r o j e c t i v e s p a c e s also c o n t a i n s p r e a d s a n d packings. So, I t h i n k I will try to turn m y s e l f into a flatl a n d e r and move to a p r o j e c t i v e plane. Wish m e luck that no o t h e r nightmares are waiting for m e there. Yours, apprehensively, Thomas P.S.: YEA WHY TRY HER RAW WET HAT! I j u s t discove r e d the sign of the devil right in the middle o f this univ e r s e while investigating the c o u n t e r p a r t o f the g e o m e t r y o f circles on the s p h e r e in this space. A sphere in this w o r l d is a set o f five p o i n t s s u c h that any line of the s p a c e inters e c t s the set in one o r t w o points. Every p o i n t of a s p h e r e is c o n t a i n e d in e x a c t l y one tangent plane. Hence t h e r e are five p l a n e s intersecting the s p h e r e only in one point. The remaining 10 planes i n t e r s e c t the sphere in 3 p o i n t s each. The sets of points o f the t h r e e n e s t e d regular p e n t a g o n s of p o i n t s visible in o u r m o d e l o f the space are t h r e e such spheres. The points o f t h e g e o m e t r y of circles a s s o c i a t e d with such a sphere are the p o i n t s o f the sphere. Its circles a r e the intersections with the s p h e r e of all t h o s e p l a n e s t h a t i n t e r s e c t the s p h e r e in t h r e e points. J u s t like t h e onep o i n t extension of t h e F a n o plane, this g e o m e t r y h a s the p r o p e r t y that three distinct p o i n t s are c o n t a i n e d in e x a c t l y one circle. If you d r a w t h e circles o f the g e o m e t r y associa t e d with the inner p e n t a g o n , you arrive at the following picture. (See Figure 11.) Ominous.

1. Beutelspacher, A. 21 - 6 = 15: a connection between two distinguished geometries, Am. Math. Monthly 93 (1986), 29-41. 2. Cameron, P. J., Combinatorics--Topics, Techniques, Algorithms, Cambridge: Cambridge University Press, (1994). 3. Hall, J. I., On identifying PG(3, 2) and the complete 3-design on seven points, Ann. Discrete Math. 7 (1980), 131-141. 4. Jeurissen, R. H., Special sets of lines in PG(3, 2), Linear AIg. Appl. 226/228 (1995), 617-638. 5. Lloyd, E. K., The reaction graph of the Fano plane, in Combinatorics and Graph Theory '95, Vol. 1, edited by Ku Tung-Hsin, Singapore: World Scientific (1995), pp. 260-274. 6. Polster, B., A Geometrical Picture Book, New York: Springer-Verlag (1998). 7. Van Dam, E., Classification of spreads of PG(3, 4)\PG(3, 2). Des Codes Cryptogr. 3 (1993), 193-198.

Acknowledgements, Further Readings, and Some Remarks I w o u l d like to t h a n k G o r d o n Royle for c o n d u c t i n g an exhaustive c o m p u t e r s e a r c h to calculate the n u m b e r s of the different h y p e r p a c k i n g s a n d for suggesting the n a m e s for t h e s e n e w structures. By t h e way, G o r d o n and his colleague Rudi Mathon have classified a large n u m b e r of nonclassical projective planes. If y o u are interested in investing in real estate in one o f t h e s e planes, or if y o u w a n t to have one n a m e d after you, get in t o u c h with them. T h a n k s are due to Keith H a n n a b u s s for the title of this article. F o r an accessible i n t r o d u c t i o n to c o m b i n a t o r i c s r e l a t e d to Kirkman's schoolgirls p r o b l e m , see [2]. F o r m o r e inform a t i o n a b o u t the s m a l l e s t projective space, s e e [1-7]. In constructing Figure 6, I u s e d the different labenings in [5].

VOLUME21, NUMBER2, 1999

Ibd,[:-~JA~'i|,[ai,~.~,;l[c-~lb..]!l~.~---~;ll D i r k H u y l e b r o u c k ,

Anstice and Kirkman: Mathematical Clerics lan Anderson and Terry Griggs

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

Editor

ur first visit is to Madeley, in Shropshire, situated between the new town of Tefford and the Ironbridge Gorge, birthplace of the Industrial Revolution. The Madeley churchyard is of historical interest for several reasons. The octagonal church was designed by the famous Scottish engineer and architect Thomas Telford, and the churchyard boasts the tomb of the saintly Fletcher of Madeley, the Church of England methodist rector who was John Wesley's intended successor (but who failed to outlive his more illustrious leader). But there is also the Anstice connection. (Park in the Anstice Square car park, beside the new shopping centre.) The Anstice family owned a large share in the Madeley Wood ironworks, which flourished in the 18th and 19th centuries and had considerable influence in the neighbourhood. The Anstice family tomb can be found in the churchyard. Here lies Robert Richard Anstice, who did some extremely interesting mathematics which has ever since been largely ignored. Anstice was born in Madeley in 1813, the fourth son of William and Penelope Anstice. The eldest son, William, would

O

are scratched, birthplaces, houses, or memorials? Have you 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 foUow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

44

Close-up of Anstice's gravestone, Madeley

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

I

take over the family interests in the ironworks; their second son, Joseph, would become Professor of Classical Literature at King's College, London at the early age of 22, only to die six years later. Robert followed Joseph to Westminster School and then to Christchurch, Oxford, where he obtained a first in Mathematics and won a mathematical scholarship a year later. There is a mysterious gap of 10 years in our knowledge before he reappeared at Wigginton, near Tring, Hertfordshire, as the Church of England rector. He lived there from 1847 until his death in 1853. During his time at Wigginton, Anstice became interested in the mathematical work of another rector, Rev. T.P. Kirkman, who had written on the subject of Steiner triple systems (as they are now called). In one of his papers Kirkman gave an elegant construction of a resolvable Steiner triple system on 15 elements (the famous Kirkman 15 schoolgirls problem), malting use of what are now known as a Room square of order 8 and the Fano plane. Kirkman stated that a generalisation of this construction seemed very hard. Anstice then remarkably achieved the con-

The church at Croft, where Kirkman was rector, 1845-1892.

The Kirkman grave, recently restored by the British Combinatorial Committee

s t r u c t i o n of a cyclic r e s o l v a b l e Steiner triple s y s t e m on 2p + 1 e l e m e n t s for all p r i m e s p ~ 1 ( m o d 6). His w o r k conceals r e m a r k a b l e discoveries. He cons t r u c t e d infmitely m a n y R o o m squares a c e n t u r y before a n y o n e else, using the " m e t h o d o f differences" w h i c h in the 20th c e n t u r y has b e c o m e o n e of the basic techniques of design construction, a n d he c o n s t r u c t e d infinite families of cyclic Steiner triple s y s t e m s on 6k + 1 e l e m e n t s d e c a d e s b e f o r e s u c h families w e r e exhibited by Netto in 1893. He also s e e m s to have b e e n one o f the first

to use primitive r o o t s in the c o n s t r u c tion o f c o m b i n a t o r i a l designs; s e e [1] for f m ~ e r details. He is b u r i e d b e s i d e his parents, almost forgotten b y the m a t h e m a t i c a l community. He d e s e r v e s g r e a t e r recognition. His m o r e famous Church of England colleague, Thomas Pennington Kirkman, is b u r i e d less than an h o u r ' s drive north, in t h e c h u r c h y a r d at Croft n e a r Warrington, w h i c h is n o w in Cheshire though in Kirkman's time it w a s in Lancashire. The village is b e s t ap-

The Croft Rectory, in which Kirkman wrote his papers on Steiner triple systems

p r o a c h e d from the M6 motorway. Exit at junction 22 and follow the signposts down a country r o a d to Croft. The church is not e a s y to find. There is no tall spire to guide the traveller, and the church is n o t in the centre of the village. Even t o d a y it s t a n d s on the periphery, still in a rural setting very much as it m u s t have b e e n w h e n Kirkman was rector. The b e s t strategy is to a s k one o f the locals for e x a c t directions. Kirkman w a s b o r n in Bolton in 1806, one of several children b u t the only s o n o f John a n d Elizabeth Kirkman. He w a s e d u c a t e d at Bolton School, but left at the age of 14 to w o r k in the office o f his father's c o t t o n business. He continued his studies privately and at the age of 23 w a s able to finance himself to enter Trinity College, Dublin. He g r a d u a t e d in 1833, and the following y e a r r e t u r n e d to England a n d t o o k holy orders. He w a s c u r a t e first at Bury a n d then at Lymm. In 1839 he b e c a m e curate in charge of Croft, b e c o m i n g the first r e c t o r in 1845. On 8th June 1841 he m a r r i e d Eliza Ann Wright o f Runcorn, Cheshire, a n d their first child w a s b o r n on 23rd J u n e 1842. Kirkman w a s to r e m a i n at Croft for nearly 50 years, tending to the n e e d s of the p a r i s h which he did not neglect, a n d doing m a t h e m a t i c s . B e t w e e n 1847 a n d 1895, the y e a r o f his death, he p u b l i s h e d over 70 p a p e r s on a wide range o f m a t h e m a t i c a l top-

VOLUME 21, NUMBER 2, 1999 4'~

ics: designs, partitions, hypercomplex numbers, geometry, groups, polyedra (sic; Kirkman considered the spelling with an "h" as an affectation), and remarkably, 5 papers on knots, written when he was nearly 80, which appeared between 1884 and 1886. He was elected a Fellow of the Royal Society in 1857 and is nowadays considered to be the founder of combinatorial design theory. An excellent account of Kirkman's work is given in the paper by Biggs [2]. The Old Rectory where Kirkman lived and worked is adjacent to the church. It is now owned by a private family, but one can still imagine Kirkman living there, attending to his clerical and mathematical work. The church itself is likely to be locked, a protection against vandalism, but it is worth seeking the key from the present rector or one of the churchwar-

46

THE MATHEMATICAL INTELLIGENCER

dens. A photograph of Kirkman still hangs in the vestry, and the entire east window behind the altar is in his memory. It is a three-light window made by Mayhr and Co. of Munich; the central light being given by Kirkman's family and the other two by parishioners. It depicts the Good Shepherd with both St. Peter and St. Paul, and is a fitting tribute. But the most interesting memorial is the Kirkman grave. It is in the old part of the churchyard, and the first person to be buried there was Kirkman's son, Frederick, who died in 1879. It also contains the remains of a young child, Alicia Bury Wells--a granddaughter--as well as both Kirkman and his wife who died 12 days after him. Next to Kirkman's grave is the unusually shaped headstone of his sister, May, who died in 1876. The Kirkman grave has recently

been restored by members of the British combinatorial community. It is now a worthy memorial. REFERENCES

1. I. Anderson, "Cyclic designs in the 1850s; the work of Rev. R.R. Anstice." Bull. Inst. Combinatorics and its Applications 15(1995), 41-46. 2. N.L. Biggs, "T.P. Kirkman, mathematician." Bull. London Math. Soc. 13(1981), 97-120. Department of Mathematics University of Glasgow Glasgow, G12 8QW United Kingdom Department of Pure Mathematics The Open University Walton Hail Milton Keynes, MK7 6AA United Kingdom

Tourism to the Future Javier Bracho

ourist attractions are usually sights of the past. However, Mexico offers to the m a t h e m a t i c a l tourist t h e strange sensation of glimpsing the future. Spiced, of course, w i t h the t a s t y t r a d i t i o n s and joyful a t m o s p h e r e of a rich and unique culture. In p l a c e s like Mexico City, Guanajuato, Morelia, Cuernavaca, M6rida or O a x a c a - - w h i c h , on their own, are well w o r t h v i s i t i n g - - m a t h e m a t i c a l activity has f l o u r i s h e d at the h a n d s of m a n y n o t e w o r t h y mathematicians. Maybe so m a n y b e c a u s e t h e r e you b r e a t h e an intuitive s e n s e of movement-into-the-future w h i c h only growth gives a n d which c o u l d not fail to stimulate mathematics. Besides the various research c e n t e r s to visit, o r conferences to attend, t h e r e is a site in the south of Mexico City which r e p r e s e n t s what m a t h e m a t i c s can be in the future of science c e n t e r s or museums. In the last d e c a d e s the i d e a of "museum" has been t r a n s f o r m e d and has influenced the public view of science. A w a r e of this, the National University o f Mexico, w h o s e c a m p u s is an architectonic highlight o f the tourist Mexico, a d a p t e d an anachronic building as a science museum, n o w called "Universum." Its Mathematics Section blends art and m a t h in a unique a n d playful way. It has many of the usual artifacts one e x p e c t s in such spaces, b u t it also goes well beyond the normal. Various Mexican artists c o n t r i b u t e d w o r k s related to recent m a t h e m a t i c a l developments, from which the lay visitor gets an idea of mathematics m u c h closer to o u r own, and we mathematicians get a certain p r o p r i e t a r y pride. The m a i n hall, illuminated b y a t r a n s l u c e n t d o m e three stories up, h a s a P e n r o s e tiling floor c o n s t r u c t e d with h a n d m a d e s t o n e w a r e tiles. The artist, Juan Sandoval, t o o k the liberty to p l a y with contrasting tiles of various colors, textures and designs---of course, with only t w o shapes and the rules of Penrose---to highlight in various regions important patterns such as Batman and the cartwheel, the beginning o f t h e infinite s t a r a n d the infinite sun, a n d t h e principles of inflation and deflation. The i m p r e s s i o n it gives is o f a m a s t e r w o r k r e m i n i s c e n t of the A r a b i c tradition, a floor which invites to p r a y e r o r

T

contemplation. If y o u visit it, don't forg e t the v i e w f r o m the third f l o o r - - a n d also try to find t h e one little mistake w h e r e the rules w e r e broken. On one side of the P e n r o s e Mosaic there is a 3 • 4-meter M a n d e l b r o t mural made o f screen-size p h o t o g r a p h s , a n d on the o t h e r side s t a n d t h e kaleidoscopes. The Moebius i c o s a h e d r a l kaleidoscopes, k n o w n t h e r e as "Los Rolidoscopios," stand as a futuristic sculpture: a huge pair of interestingly shaped and placed shiny tetrahedra. The artist, Carlos Trejo, m a n a g e d to p u t three mirrors of m o r e t h a n h u m a n size on the p l a n e s w h e r e t h r e e reflections generate the s y m m e t r y group of the dodecahedron. E a c h k a l e i d o s c o p e is then a triangular c o n e w i t h the v e r t e x truncated, p r o d u c i n g a small (head-sized) opening, a n d a big (person-sized) one: the base o f the cone. Standing on t h e big opening, a k a l e i d o s c o p i c d o d e c a h e d r o n (or i c o s a h e d r o n ) is f o r m e d with w h a t e v e r v i e w c o m e s from t h e small triangular window. A face o r a simple c o l o r e d cloth unfolds and solidifies into a virtual p l a t o n i c solid, surr o u n d e d b y a n o t h e r one on which the viewer is s e e n 120 times. Well, t h a t ' s w h a t t h e o r y says, b u t reality, forcefully r e p r e s e n t e d b y the weight a n d width o f the mirrors, m a k e s the outside dodeca h e d r o n a little blurry. Mathematics have always influenced art. Here is yet a n o t h e r anecdote to a d d to the story. While Juan Sandoval was working on his ceramic P e n r o s e tiling, he was b e w i t c h e d b y the neighboring k a l e i d o s c o p i c images. He got the vice o f working with mirrors, and lucidly, he also h a d the s t a m i n a and creativity to p r o d u c e - - i n c o l l a b o r a t i o n with a m a t h e m a t i c i a n - - t h e n e x t generation of kaleidoscopes. Of which, after a couple of years, the first two mediumsized and m u s e u m w o r t h y s p e c i m e n s w e r e a d d e d to the hall. In a sense, t h e y a r e to 3-dimensional s p a c e w h a t the classic k a l e i d o s c o p e s are to the plane. They consist o f four mirrors placed on the planes w h o s e reflections generate a crystallographic group (of these groups there are essentially only three). Thus, the mirrors form an inward reflective tetrahedron, some of whose vertices are truncated to form triangular windows which serve both for looking

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 2, 1999 4 7

One of the Infinite Space Kaleidoscopes by Juan Sandoval surrounding the "Mosaico de Penrose" at Universurn.

and to provide an image for kaleido- But today, that slice of the future may scopic reproduction. The view inside is be seen only in Mexico. breathtaking. A perfectly coherent 3A9 dimensional elucidean space with triangulated solids floating harmoniously We thank Juan Sandoval for the use of his work, and Pedro Hiriart for phoaround---one's own face kaleidoscopized on some of them--and ex- tographing it. The kaleidoscopes are protected by U.S. patent #5,475,532, tending to infinity. The artist has mastered his medium. The precision of the mirror planes gets closer to the theoretical one on each new version. Thus, with high-quality mirrors, the image gets crisper and deeper. What photographs reveal is far from reality, where one is looking with both eyes and therefore depth is there--really or virtually, but there. Mathematics should play a more significant role in science centers and museums. Their relation with art must be exploited for that matter, serving to intrigue and lead young minds into their beauty. In the near future, new versions of the "Infinite Space Kaleidoscopes" will undoubtedly appear at other public sites throughout the world, or even at households in handy sizes. An inside view of an Infinite Space Kaleidoscope.

THE MATHEMATICAL INTELLIGENCER

held by Juan Sandoval and Javier Bracho. Instituto de Matematica, UNAM Ciudad Universitaria CP 04510 Mexico, DF Mexico e-mail: [email protected]

" M o s a i c o de Penrose" by Juan Sandoval, stoneware, 4 x 10m, 1992. A t Mathematics Section of the Universum-UNAM in Mexico City.

VOLUME21, NUMBER2, 1999 49

Felix Klein's Commemorative Plaque in Diisseldorf Nenad Trinajstid

he Dtisseldoff University is called the Heinrich Heine University, because the great German poet was born in that city in 1797 (he died in 1856 in Paris). However, it could have been named the Felix Klein University as well, since Dtisseldorf is also the birthplace of the famous mathematician. F~dhermore, the Dtisseldorf University is an excellent university for studies in mathematics and more generally in the natural sciences. The Medicine Department, founded almost two hundred years ago, had a Nobel Prize winner on its staff. Both Heinrich Heine and Felix Klein made beautiful abstract creations, and a full understanding of either Heine's poetry or Klein's geometry requires some advanced knowledge, though of course from different fields. The author has been visiting the Heinrich Heine University several

T

times per year since 1973. On several occasions, I went to the Goethe museum, housed in the Schloss J~igerhof, a castle built by Couven in 1763. I often walked along a street called the J~igerhofstrasse, unaware that I was passing a landmark. Indeed, like many tourists I went from the Goethe Museum to the Dtisseldorf Concert Hall or Tonhalle, located on the bank of the Rhine near the Oberkasseler Brticke. The Tonhalle was built by Kreis in 1926, and restored by Hentrich In 1978. There is a permanent exhibition of glassware. In the J~gerhofstrasse, the main Dtisseldorf park faces a row of modern, but not very attractive buildings on the other side of the street. One day, I noticed a modest black slab on a building in aluminum and glass, carrying the numbers 10 and 12. It happened to be a commemorative plaque in honor of Felix KleIn (1849-1925), one of the greatest German mathematicians, who was born at this exact location (see picture). The original building as well as the whole surrounding area was destroyed at the end of the Second World War. The present building houses the headquarters of Bankhaus Hermann Lampe, the Kommanditgesselschaft, and the Consulate of the Grand Duchy of Luxembourg, but architecturally it is not very Interesting. There is another, a more solemn slab on this buildIng, commemorating the ~ n n i s h painter Werner Holmberg (1830-1860). However familiar with mathematical lore, the visitor may need luck to discover Klein's birthplace. Tourist guides in Diisseldorf do not contain the slightest information about Felix Klein. The lady curator of the nearby Theatre Museum, the only building at the entrance of the park, was one of the very few people to know who Felix KleIn was, but she did not know he was born across the street.

The author showing his discovery: Klein's commemorative plaque in Di~sseldorf. It reads, In this house the mathematician Felix Klein was born, on April 25, 1849.

50

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

The Rugjer Bogkovi6 Institute P.O. Box 1016 HR-10001 Zagreb, Croatia

SUSAN HARRIS AND G R E G O R Y QUENELL

Knot Labelings and Knots Without Labelings

Introduction F r o m two recent introductory b o o k s on knot theory ([1], [7]), we learned about using labelings of knot diagrams to distinguish knots. For each prime p --> 3, a diagram o f a knot K has a valid mod-p labeling if and only if every diagram representing the knot type of K has a valid mod-p labeling. Writing down the first few examples that c o m e to mind shows quickly and easily that the trefoil and the figure-eight knots are distinct from one another, and that neither of them is equivalent to the trivial knot. (These facts are in close agreement with all of our experimental data.) We then wondered if there were any knots other than the trivial knot which fail to have mod-p labelings for any prime p. We learned s o m e w h a t later t h a t knot-theorists have long k n o w n that there are infinitely many such knots. In [4], it is shown that a diagram of a knot K has a mod-p labeling if and only if AK(-- 1) is divisible b y p , where AK(t) is the Alexander polynomial of K. When K is an odd torus knot, its Alexander polynomial AK(t) is readily c o m p u t e d (see [6], p. 265), and one finds that AK(-1) = 1 for every such knot. it can be s h o w n by other methods (see [10], p. 53, for example) that there are infmitely many distinct odd torus knots, so we get an infinite family of knots, none of which has a mod-p labeling for any prime p.

This a p p r o a c h provides a swift and decisive answer to our question, but it relies on rather advanced techniques. Because we are far from being specialists in the field of knot theory, we preferred to investigate mod-p labeUngs from a more elementary point of view. In doing so, we found that the study of knots provides some nice, hands-on applications of familiar techniques and results from group presentations, group representations, and number theory, as well as basic algebraic topology. Once we understood the algebraic significance of knot labelings (Lemma 5), we were able to p r o d u c e a very accessible demonstration that the odd torus knots have no mod-p labelings.

Definitions A knot is a mathematical model of a piece of rope which

has had a knot tied in it and then had its ends spliced together. Thus, a knot is basically an embedding of a circle in ~3. As a physical piece of rope has some positive thickness, a physical knot can't be pulled infinitely tight. This property is usually included in the definition of a mathematical knot by requiring that the embedded circle not come "arbitrarily close to itself." Embeddings of the circle in R3 which satisfy a condition of this kind are called t a m e knots. Knots that are not tame are called wild. Wild knots are not good models of everyday physical knots, and it

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 2, 1999

51

s e e m s they are n o t s t u d i e d very much. In this article, w h e n we say "knot," w e will m e a n "tame knot." We should c o n s i d e r two k n o t s as equivalent if the r o p e s t h e y m o d e l can be p u s h e d a r o u n d in R 3 so that they l o o k the same, at l e a s t up to scaling. Pushing a r o p e a r o u n d in s p a c e is m o d e l e d b y t h e m a t h e m a t i c a l n o t i o n o f an ambient isotopy, w h i c h is a family of h o m e o m o r p h i s m s f t : R 3 -+ ~3 w h e r e the "time" variable t runs t h r o u g h the interval [0, 1]. The initial h o m e o m o r p h i s m f 0 is the identity, and the familyft, c o n s i d e r e d as a function from ~3 X [0, 1] to ~3, m u s t be continuous. A n a m b i e n t i s o t o p y is like a movie: for each t E [0, 1], the m a p f i c o r r e s p o n d s to a single frame showing the p o s i t i o n o f o u r r o p e at time t. If K _ R3 is a k n o t a n d f i is a n a m b i e n t i s o t o p y such t h a t f i ( K ) is a k n o t for each t E [0, 1] (i.e., the r o p e doesn't p a s s t h r o u g h itself at any time during the movie), then the k n o t f l ( K ) is isotopy equivalent to K. It is intuitively clear ( a n d e a s y enough to c h e c k formally) t h a t i s o t o p y equivalence is i n d e e d an equivalence relation. When two knots are i s o t o p y equivalent, w e will s a y t h a t t h e y are o f the s a m e type. To study knots, w e d r a w pictures of them. A knot diag r a m of a k n o t K is a p r o j e c t i o n of K onto a plane, with gaps in the curve to indicate w h e r e p a r t s o f K c r o s s u n d e r o t h e r p a r t s of K. We also insist that e a c h p o i n t in the proj e c t i o n be the i m a g e o f no m o r e than t w o p o i n t s in K a n d that all intersections in the diagram be transverse. A knot i n v a r i a n t is s o m e quantity w e assign to a k n o t K that d e p e n d s only on the k n o t type of K, n o t o n the particular r e p r e s e n t a t i v e at which w e h a p p e n to b e looldng. Knot invariants t h a t w e can r e a d off a k n o t d i a g r a m are particularly useful, b e c a u s e t h e y can tell us i m m e d i a t e l y that two diagrams r e p r e s e n t different k n o t types.

Knot Labeling$ A strand in a k n o t d i a g r a m is an arc b e t w e e n one undercrossing and the next. If a knot diagram has n crossings, then it has n strands. A labeling of a k n o t d i a g r a m is a mapping from the s t r a n d s o f the diagram into s o m e set o f symbols. If we n u m b e r the s t r a n d s (in s o m e a r b i t r a r y w a y ) as 1, 2 , . . . , n, then w e c a n write d o w n any labeling as an ntuple (al, a2, 9 9 9 , a n ) , indicating that the s y m b o l ai is assigned to s t r a n d i.

Let p b e a prime. A mod-p labeling of a knot d i a g r a m is a labeling using the s y m b o l s {0, 1 , . . . , p - 1}. A m o d - p labeling (al, a2,. 9 9 an) is called valid if it satisfies t h e t w o conditions: MPL1

At least two different labels are used.

MPL2

At each crossing, the relation aj + ak -- 2ai -~ 0 ( m o d p ) holds, w h e r e i is the overcrossing s t r a n d a n d j and k are t h e s t r a n d s that form the undercrossing.

C L A I M 1. The existence o f a valid mod-p labeling is a knot invariant; that is, one d i a g r a m o f a knot K has a valid mod-p labeling i f and only i f every d i a g r a m o f every knot o f the s a m e type as K has a valid mod-p labeling. This is s h o w n in [7] by c o m b i n a t o r i a l methods. We will outline a slightly different p r o o f (also suggested in [7]) l a t e r on, b u t for now, let's a c c e p t Claim 1 and try it out on s o m e familiar knots. In Figure 1, it is e a s y to c h e c k that t h e m o d 3 labeling o f the trefoil a n d t h e mod-5 labeling o f the figure-eight knot are both valid. It is j u s t a shade m o r e difficult to c h e c k that the trefoil d i a g r a m can have no valid mod-5 labeling. [Hint: If (al, a2, a3) is a valid mod-p labeling, t h e n so is (al + c, a2 + c, a3 4- c), reading the entries m o d p, for any c o n s t a n t c.] By Claim 1, the two d i a g r a m s in Figure 1 actually r e p r e s e n t distinct knots. F u r t h e r m o r e , b e c a u s e the trivial knot (i.e., a n u n k n o t t e d loop o f r o p e ) h a s a d i a g r a m with only one strand, and no labeling o f s u c h a d i a g r a m can satisfy MPL1, t h e trivial knot has no m o d - p labeling for any p. Thus, n e i t h e r t h e trefoil n o r the figureeight k n o t is equivalent to the trivial knot. F o r o u r n e x t labeling, it will b e convenient to give o u r k n o t d i a g r a m an orientation, at l e a s t temporarily. This is e a s y to do. We orient a k n o t b y choosing a direction in w h i c h to traverse it. The k n o t orientation is inherited b y the d i a g r a m and m a y be i n d i c a t e d b y drawing a r r o w h e a d s along the strands. A crossing in an o r i e n t e d knot d i a g r a m is right-handed if an o b s e r v e r walking along the overc r o s s i n g s t r a n d in the p r e f e r r e d direction sees the undercrossing traffic a p p r o a c h i n g f r o m t h e right. Otherwise, t h e c r o s s i n g is left-handed. (See Fig. 2.) The h a n d e d n e s s o f a

0 0

Figure 1. A valid mod-3 labeling of a trefoil diagram (left) and a valid m o d - 5 labeling of a figure-eight diagram (right).

52

THE MATHEMATICALINTELLIGENCER

Figure 2. A right-handed crossing (left) and a left-handed crossing (right).

c r o s s i n g d o e s n o t d e p e n d on t h e orientation c h o s e n for the knot; reversing all the a r r o w h e a d s t a k e s a right-handed crossing to a n o t h e r right-handed crossing. Let H be a group a n d K be a knot. If we have an orie n t e d d i a g r a m of K with n s t r a n d s n u m b e r e d 1, 2 , . . . , n, w e c a n form an H-labeling o f the diagram by assigning a g r o u p e l e m e n t hi to e a c h s t r a n d i. An H-labeling (hi, h2, 9 9 9 , hn) is called v a l i d if the following two c o n d i t i o n s are satisfied. GL1

The set {hi, h2, 9 9 9 hn} is a generating set for H.

GL2

At e a c h right-handed crossing, t h e r e l a t i o n hihkhi -1 = hj holds, w h e r e i is the o v e r c r o s s i n g strand, j is the "incoming" u n d e r c r o s s i n g strand, and k is the "outgoing" u n d e r c r o s s i n g strand. At each left-handed crossing, with i, j , a n d k in the s a m e roles, the r e l a t i o n hihjhi -1 = hk holds.

B e c a u s e GL2 refers to "incoming" a n d "outgoing" strands, it a p p e a r s that the validity o f an H-labeling d e p e n d s on the o r i e n t a t i o n w e c h o o s e for o u r k n o t diagram. Although this is so, the e x i s t e n c e of a valid H-labeling is i n d e p e n d e n t o f o u r c h o i c e of orientation. This is b e c a u s e (hi, h 2 , 9 9 h n ) is a valid H-labeling o f a d i a g r a m M if and only if ( h i 1, h~- l, 9 h n 1) is a valid H-labeling of the diagram M' o b t a i n e d from M b y reversing its orientation. As w e did with mod-p labelings, w e n o w a s s e r t t h a t group labelings are useful, b u t w e defer giving the p r o o f until later. C L A I M 2. The e x i s t e n c e o f a v a l i d H-labeling is a k n o t invariant. As w e travel around a k n o t d i a g r a m with a valid H-labeling, t h e condition GL2 tells us that the label o n e a c h n e w s t r a n d w e e n c o u n t e r (beginning at s o m e u n d e r c r o s s ing) is conjugate to the label on the p r e c e d i n g strand. It follows that all the labels in a valid H-labeling of a k n o t dia g r a m m u s t belong to a single c o n j u g a c y class in H. As, b y GL1, t h e s e labels m u s t g e n e r a t e H, any group w e use to label a k n o t diagram m u s t b e g e n e r a t e d b y the e l e m e n t s o f a single conjugacy class. In particular, no nontrivial a b e l i a n group can be used in a valid group-labeling o f a knot. When w e turn our a t t e n t i o n to the dihedral groups, though, w e find j u s t w h a t w e ' r e looking for. Let p --> 3 be

a prime, a n d let Dp d e n o t e the d i h e d r a l group of o r d e r 2p, which w e p r e s e n t as Dp = (r,s: rp = s 2 = 1, rs = sr-1).

We list all the e l e m e n t s of Dp as {1,r,r 2, . . . ,

rp-1, s,sr, s r 2, . . . , srP-1},

and c o n s i d e r this group's conjugacy classes. E a c h element is conjugate only to its inverse, b u t t h e conjugacy class containing s also contains s r k for e a c h k. ( F o r even k, conjugate s b y r k / 2 ; for o d d k, conjugate s b y r ( P - k ) / 2 . ) Can e l e m e n t s o f the conjugacy class o f s generate D~? As each s r k h a s o r d e r 2, w e will n e e d at least t w o such elements to have any hope of generating Dp. So c o n s i d e r s r k and s r Z with k ~ 1 ( m o d p), and let H b e the subgroup of Dp g e n e r a t e d b y these two elements. Then H contains s r k s r I = r 1 - k . Because p is prime, I - k is relatively prime t o p , and e l e m e n t a r y n u m b e r theory tells us that s o m e p o w e r of r l-k is equal to r in Dp. So H contains r. But H also contains s r k, a n d it follows that H contains s. Thus, H = Dp. We s u m m a r i z e this discussion as a lemma. r k

L E M M A 3. L e t p >- 3 be a p r i m e . I f a set o f e l e m e n t s f r o m a s i n g l e c o n j u g a c y class o f Dp g e n e r a t e s Dp, the e l e m e n t s m u s t be o f the f o r m s r k. F u r t h e r m o r e , a n y two d i s t i n c t e l e m e n t s o f the f o r m s r k generate Dp. The c o n d i t i o n GL1 says that a valid Dp-labeling m u s t use only labels o f the form s r k, and it m u s t use at least two of them. This e c h o o f the condition MPL1 suggests that modp labelings a n d Dp-labelings are s o m e h o w related. A closer l o o k s h o w s t h a t mod-p labelings and D~-labelings are ac~ tuaUy identical. L E M M A 4. L e t p -> 3 be a p r i m e a n d M a k ~ o t diagran~. T h e n M h a s a v a l i d m o d - p labeling i f a n d only i f M h(1.~ a valid Dp-labeling. Proof. S u p p o s e M has n strands. By the a b o v e discussion, any valid Dp-labeling of M m u s t be of the form (sral, sra2, . . . . sran), w h e r e e a c h ai is in the s e t {0, 1 , . . . , p - 1}. We claim that the Dp-labeling (sr al, s r a2, . . . , s r an) is valid if and only if the mod-p labeling (al, a2, . 9 9 , a~) is valid. First, GL1 a n d MPL1 are equivalent in this situation, because each one simply requires the a p p e a r a n c e of two distinct ai.

VOLUME 21, NUMBER 2, 1999

53

To see that GL2 is equivalent to MPL2, consider a righthanded crossing with the usual cast of characters: i is the overcrossing strand and j and k are, respectively, the incoming and outgoing undercrossing strands. In the Dp-labeling, the condition GL2 at this crossing states that

1

(sraO(srak)(sra 0 l(sraj)-I .~ raj +ak-2ai;

X0

=

that is, aj + ak -- 2ai is a multiple of p. But this is exactly condition MPL2. Interchanging all the j's and k's in the previous paragraph shows that GL2 and MPL2 are equivalent at lefthanded crossings, as well. [] Figure 4. Representatives of the generators gl, g2, and g3 of the knot

We remark that although we still have not proved either Claim 1 or Claim 2, Lemma 4 at least shows that Claim 2 implies Claim 1.

Knot Groups The topology of a knot isn't useful for distinguishing one knot from another, for every knot is homeomorphic to a circle. Knot-theorists study, instead, the topology of the complement of a knot and, in particular, the fundamental group qVl(R3 - K), which is called the k n o t g r o u p of K. We will denote this group G(K). Since the topology of ~ 3 _ K is preserved by ambient isotopies, any quantity depending on G(K) is automatically a knot invariant. An element of G(K) is represented by an oriented closed path which begins at some fixed base point x0 ~ K, winds through the space around K, and then returns to x0. The composition operation in this group corresponds to concatenation of paths. The identity element is represented by a path that never leaves x0, or by a loop at x0 which never gets tangled up with any part of K, so that it can be shrunk back to x0 without getting caught anywhere. The inverse of the group element represented by a path ~is represented by the same path traced in the opposite direction. In Figure 3, for example, if path (rl represents a group element g, then path (r2 represents g-2. Path o,3 represents the identity, as it can be pulled clear of the knot. Given an oriented diagram of a knot K, there is a nice recipe for writing down a group presentation for G(K). We

group of a trefoil knot.

first establish a base point x0 somewhere off to the side of our knot diagram. Then, for each strand i in the diagram. , we write down a group element gi, represented by a closed path which begins at x0, crosses under strand i from, say, right to left, then crosses over strand i and returns to x0 without getting tangled up anywhere else in the knot. (See Fig. 4). It is not too surprising that the elements gi actually generate G(K). This is just saying that any x0-based loop through the space around K can be deformed into a sequence of loops, each of which leaves x0, circles one strand of K (in one direction or the other), and then returns to x0. For each crossing in the knot diagram, we have a relation among the generators gi. At a right-handed crossing where i is the overcrossing strand and j and k are the incoming and outgoing undercrossing strands, a path which passes underneath all three of these strands and circles the crossing once (see Fig. 5) represents the group element g i g k g ( l g f 1, because it can be deformed into four paths representing these generators. On the other hand, since this path can be pulled clear of K, it represents the identity in G(K), so we get the relation gigkgi~lgj -1 = 1. Hold Figure 5 up to a mirror to see that the corresponding relation at a left-handed crossing is gi-lgElgigj = 1. We now have a set of generators (one for each strand) and a set of relations (one for each crossing). The won-

"X---.Y Xo

Figure 5. At a right-handed crossing, the generators of a knot group Figure 3. Three paths in the complement of a figure-eight knot.

THE MATHEMATICALINTELLIGENCER

satisfy the relation g , g k g ~ - l g i I = 1.

derfully useful result is t h a t this set of relations is sufficient to define G(K). The r e c i p e we j u s t outlined is called t h e Wirtinger presentation o f G(K). See [10] for a p r o o f t h a t this p r e s e n t a t i o n actually does describe the k n o t group o f K.

Knot Groups and Knot Labelings Using the Wirtinger p r e s e n t a t i o n , w e can establish the imp o r t a n t c o n n e c t i o n b e t w e e n k n o t groups a n d group labelings o f knots. This c o n n e c t i o n will p u t the k n o t labelings, w h i c h we d e s c r i b e d in a combinatorial way, on a m o r e t o p o l o g i c a l footing. It will also allow us to use algebraic techniques to study the labelings t h e m s e l v e s and, in certain cases, to s h o w that valid labelings c a n n o t b e constructed.

1. I f any oriented diagram of K has a valid H-labeling, then there exists a surjective homomorphism f r o m G(K) to H. 2. I f there exists a surjective homomorphism f r o m G(K) to H, then every oriented diagram of K has a valid Hlabeling. We sketch the proof. F o r the first statement, s u p p o s e we have a diagram of K with a valid H-labeling. Then for each i, s t r a n d i in the diagram is associated with s o m e generator gi of G(K) and some label hi from H. The natural candidate for a h o m o m o r p h i s m from G(K) to H is a m a p taldng each gi to the corresponding hi. It turns out that each Wirtinger relation among the gi c o r r e s p o n d s to a relation (required by GL2) among the hi, so that the m a p taldng gi to hi can ind e e d be e x t e n d e d to a homomorphism. (To s h o w this forreally, one writes d o w n p r e s e n t a t i o n s for the two groups and applies Van Dyck's t h e o r e m (see [5]).) The condition GL1 guarantees that our h o m o m o r p h i s m is surjective. F o r the s e c o n d statement, w e are given a surjective hom o m o r p h i s m q~: G(K) --~ H a n d a diagram o f K. The natural w a y to label the d i a g r a m is to assign to each s t r a n d i the group element ~(gi), w h e r e gi is the s t r a n d ' s Wirtinger generator. The Wirtinger relations m a p to j u s t t h e r e l a t i o n s required b y GL2, and the fact that q~is surjective is e x a c t l y GL1. []

T3,s on

Torus Knots Let m and n b e relatively prime. An (m, n) torus knot, denoted T,n,n, is a simple closed curve t h a t w i n d s a r o u n d a s t a n d a r d t o r u s m times in the longitudinal direction and n times in the m e r i d i o n a l direction. More concretely, if w e let p: ~2 .....> ~3 be given by

p(x,y)

L E M M A 5. Let K be a knot and H a group.

Figure 6. The torus knot

Claim 2 n o w follows easily from L e m m a 5 and, in turn, establishes Claim 1. The i d e a o f labeling knot d i a g r a m s s e e m s to have originated with R. H. Fox. In [4], he e x p l a i n s h o w a knot group m a y b e s t u d i e d by considering its h o m o m o r p h i c images in metacyclic groups. Specializing to t h e dihedral groups leads directly to the mod-p labelings w h i c h w e r e our starting point.

= ((2 + cos 2~ry) cos 21rx, (2 + cos 2qTy) sin 2~7x, sin 2~ry), then the i m a g e u n d e r p of the line s e g m e n t from the origin to the p o i n t (m,n) is an (m,n) t o r u s knot. Figure 6 s h o w s the t o m s k n o t T3,5 on the surface o f a s t a n d a r d toms. It is k n o w n ([9], P r o p o s i t i o n 7.5) t h a t the n u m b e r o f crossings in a n y diagram of an (m,n) t o r u s k n o t is at least the m i n i m u m of m ( n - 1) and n ( m - 1). F o r larger values of m a n d n, this m a k e s the Wirtinger p r e s e n t a t i o n o f G(T,~,n) unwieldy. However, it is e a s y to c o m p u t e the k n o t group o f a t o r u s knot using the Seifert-Van Kampen theorem. In fact, finding G(Tm,n)is literally a t e x t b o o k example (see [8], p. 136, o r [3], p. 92) of a Seifert-Van Kampen computation. Recall t h a t the Seifert-Van K a m p e n t h e o r e m d e s c r i b e s the f u n d a m e n t a l group of a p a t h - c o n n e c t e d union X] U X2 in t e r m s of t h e fundamental g r o u p s o f X1, X2, a n d X1 R X2. Roughly speaking, w e obtain a p r e s e n t a t i o n of ~rt(Xt U X2) by c o m b i n i n g the generators a n d r e l a t i o n s from ~'t(X1) a n d 7r1(X2), a n d t h e n throwing in an e x t r a relation for each generator of ~-t(X1 A X2). The e x t r a relations a c c o u n t for the fact t h a t a c l o s e d p a t h o- in Xt A X2 r e p r e s e n t s simultaneously s o m e e l e m e n t o f ~'1(X1) a n d s o m e e l e m e n t of ~h(X2). In ~rl(X1 U X2), the two group e l e m e n t s r e p r e s e n t e d b y (r m u s t b e equal, a n d so we include a r e l a t i o n that says so. To c o m p u t e the fundamental group of R 3 - Tm,~, w e place Tm,n o n a s t a n d a r d t o m s , as in Figure 7. The inter-

the surface of a torus, and a lO-strand diagram of

T&s.

VOLUME21, NUMBER2, 1999 55

I m a g e of y Image of x

rj

sr j

rk

I

II

sr k

III

IV

C a s e I Since ~(x) and ~(y) are b o t h p o w e r s of r, the entire image of ~ is contained in the cyclic subgroup of Dp g e n e r a t e d by r. Thus, ~ is n o t surjective. Case II Figure 7. The space R = X l rq X2 in the complement of T3,5.

Because ~ is a h o m o m o r p h i s m and x m = yn in

G(Tm,n), w e m u s t have ( ~ ( x ) ) m = ~ ( x m) = ~(y~) = (q~(y))n.

section of this t o r u s with the c o m p l e m e n t o f T i n , n is a ribb o n which w i n d s a r o u n d the t o r u s m t i m e s in the longitudinal direction and n t i m e s in the meridional direction. Let R denote this ribbon. The s p a c e X1 will b e the region inside the t o m s , along with R, and the s p a c e X2 will be the region outside the t o m s , along with R. We c h o o s e a p o i n t s o m e w h e r e in X1 A )(2 to serve as a b a s e p o i n t for all s p a c e s concerned. The space X1 is topologically a solid t o m s , so its fund a m e n t a l group is a free group on a single g e n e r a t o r x, corr e s p o n d i n g to a p a t h that m a k e s one trip a r o u n d the "hole" in the toms. It is n o t i m m e d i a t e l y obvious t h a t the s p a c e X2 is also topologically a solid t o m s , b u t it s h o u l d b e plausible that its f u n d a m e n t a l group is also a free group on a single g e n e r a t o r y, c o r r e s p o n d i n g to a p a t h t h a t p a s s e s once through the hole. The i n t e r s e c t i o n o f X1 and )(2 is the r i b b o n R. B e c a u s e R is topologically an annulus, its f u n d a m e n t a l group is also free on one generator, w h i c h is r e p r e s e n t e d b y a p a t h (r making one trip a r o u n d the annulus. F r o m the p o i n t of view o f Xb the p a t h (r m a k e s m trips a r o u n d the t o m s , so it repr e s e n t s x m. F r o m X2's p o i n t of view, (r p a s s e s n times t h r o u g h the hole in the t o m s , so it r e p r e s e n t s yn. The result of all this is that the knot group o f Tm,n, which is the f u n d a m e n t a l group o f X1 U X2, has the simple presentation G(Tm,n) = (x,y: x m = yn}.

Knots with No Mod-p Labelings A n ( m , n ) t o r u s k n o t is c a l l e d odd if b o t h m a n d n are

odd. To finish o u r p r o o f t h a t o d d t o m s k n o t s have no m o d - p labelings, w e n e e d to s h o w t h a t t h e r e c a n b e no s u r j e c t i v e h o m o m o r p h i s m f r o m the g r o u p G(Tra,n) to the g r o u p Dp w h e n m a n d n a r e b o t h o d d a n d p is an o d d prime. To begin, w e write d o w n the group p r e s e n t a t i o n s G(Vm,n) = {x,y: X m = yn) a n d D p = {r,s: v p = 1, s 2 = 1, s r = r - i s ) , and s u p p o s e that we have a h o m o m o r p h i s m ~. G(Tm,n) --> Dp. Next, w e apply C o m m o n e r ' s s e c o n d law of ecology (see [2]), "Everything must go s o m e w h e r e . " In the p r e s e n t instance, this says that ~(x) m u s t be either r k for s o m e k o r else s r k for s o m e k, b e c a u s e t h e s e a c c o u n t for all the e l e m e n t s of Dp. Similarly, ~(y) is either rJ or srJ for s o m e j. We c o n s i d e r the four p o s s i b l e cases:

5~

THE MATHEMATICALINTELLiGENCER

(1)

The o r d e r of srJ is 2 a n d n is odd, so (q~(y))n = (srJ)n = srJ. So srJ m u s t also be equal to ( ~ ( x ) ) m, which is r km in this case. However, r km is a p o w e r of r and srJ is not, s o this c a s e c a n n o t occur. C a s e I I I This is j u s t Case II with the roles of x a n d y interchanged. C a s e IV Because m and n a r e o d d and the e l e m e n t s srJ a n d s r k a r e of o r d e r 2, Eq. (1) implies that srJ = s r k. However, if x and y both m a p to t h e s a m e e l e m e n t and that e l e m e n t h a s o r d e r 2, then the i m a g e of ~ contains only t w o elements, so it can't be all of Dp. We have s h o w n that there c a n b e no surjective h o m o m o r p h i s m f r o m G(Tm,n) to Dp a n d have thus arrived b y fairly e l e m e n t a r y m e a n s at an a n s w e r to our question a b o u t labelings. T H E O R E M 6. A n odd t o r u s k n o t h a s no v a l i d m o d - p labeling f o r a n y p r i m e p >-- 3. REFERENCES

[1] Colin Adams, The Knot Book, New York: W. H. Freeman (1993). [2] Barry Commoner, The Closing Circle, New York: Alfred A Knopf (1972). [3] R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Graduate Texts in Mathematics Vol. 57, New York: SpringerVerlag (1977). [4] R. H. Fox, A quick trip through knot theory, in Topology of 3Manifolds and Related Topics (M. K. Fort, Jr., ed.), Engiewood. Cliffs, NJ: Prentice-Hall (1962), pp. 120-157. [5] Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics Vol. 73, New York: Springer-Verlag (1974) [6] Louis H. Kauffman, On Knots, Princeton, NJ: Princeton University Press (1987). [7] Charles Livingston, Knot Theory, Carus Mathematical Monographs Vol. 24, Washington, DC: Mathematical Association of America (1993). [8] William S. Massey, Algebraic Topology, An Introduction, Graduate Texts in Mathematics Vol. 56, New York: Springer-Verlag (1987). [9] K. Murasugi, On the braid index of alternating links, Transactions of the American Mathematical Society 326(1) (1991), 237-260. [10] Dale Rolfsen, Knots and Links, Wilmington, DE: Publish or Perish (1976).

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57

I',~,~-~_,~,.m

J e r e m y Gray, Editor I

The Poincar6 Mittag-Leffler Relationship Philippe Nabonnand

T h e Mittag-Leffler Swedishmathematician G5sta and the French mathematician Henri Poincar6 maintained a regular correspondence between 1881 and 1911. The Mittag-Leffier Institute in Djursholm (Sweden) keeps 259 of these letters (119 letters or rough drafts written by Mittag-Leffler and 140 by Poincar6)J By MittagLeffler's standards, this correspondence is not exceptionally big. Indeed, he was almost as prolific with Hermite, Weierstrass, Appell, or Palnlev& But, for Poincar~, his correspondence with Mittag-Leffler is by far his most important. It is not clear why Poincar6, who did not collaborate mathematically with Mittag-Leffler, maintained such an intense relation with him, and in what follows I shall try to explain that.

We can distinguish four periods in this correspondence. In the first one (10 letters between April and August 1881), the two mathematicians get to know each other. The creation of Acta Mathematica and Poincar6's papers about Fuchsian functions are the main subjects of the second period (about 50 letters between August 1881 and March 1887). The third period is devoted to Poincar6's participation in Oscar II's 60th Birthday Competition (about 50 letters between March 1887 and July 1890). In the last period, we can see two eminent mathematicians managing nominations and positions in universities and academies and nominations for Nobel Prizes (about 150 letters between May 1891 and September 1911).

i.e., complex functions which cannot be extended to the whole complex plane. Hermite had communicated his very high opinion of Poincard to MittagLeffier, and Mittag-Leffier wanted more information about the results in Poincar~'s thesis and his work on differential equations: When do you think you will publish your research on differential equations? I am waiting for it impatiently. I cannot see from M. Hermite's account ff your results are the same as M. Fuchs has published recently, or ff your research is yet more general. [Mittag-Leffler to Poincar6, 11 April 1881--IML]. Poincar~ was not yet well known at that time: he had published only one communication on the qualitative theory of differential equations and two about Fuchsian functions. On the other hand, Mittag-Leffler was already a

Making Contact In 1881, Mittag-Leffler wrote to Poincar~ about "lacunary functions," Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

~8

Mittag-Leffler (by courtesy of MittagLeffler Institute).

G.

1The entire Poincare correspondence will be edited by the Archives Poincare (Universite Nancy 2). The first volume will be the annotated correspondence, with a commentary, of the Poincar6-Mittag-Leffler correspondence, and it will be published early this year.

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEW YORK

Mittag-Leffler, who was an adept of the Berlin School of rigour, agreed with Weierstrass's point of view and often criticised Poincar6's manner of writing mathematics: What do you make of Poincar~'s second paper "Sur les fonctions fuchsiennes'? It is indeed regrettable that he is not a graduate of a German University. As full of new ideas as his papers are, they leave, it seems to me, far too much to be desired in their formal presentation. [MittagLeffler to Weierstrass, 11 May 1883---

IML] Henri Poincar6 (by courtesy of

Mittag-Leffler

Institute).

recognised mathematician. He had been one of the more brilliant of Weierstrass's students, and his theorem on the existence of meromorphic functions with prescribed poles and zeros [Mittag-Leffier 1879] is a free generalization of Weierstrass's result about the existence of holomorphic flmc-tions with prescribed zeros [Weierstrass 1876]. Therefore, it is not surprising that to Hermite, Mittag-Leffier adopts a patronising attitude. 2 Moreover, in the first version of his paper [Poincar6 1881a], Poincar6 did not quote Weierstrass's results concerning analytic functions, and in particular those about functions with "lacunary spaces"; Mittag-Leffier "frankly and loyally" tells him he must. [Mittag-Leffier to Poincar~, 22 May lSSl--~L] Mittag-Leffler always defended Weierstrass's work firmly. Over the years he became worried about Weierstrass's priority, and more generally the diffusion of his work, because Weierstrass did not publish all his results and a great part of them were only communicated in his lectures. Consequently, many were known only to his students, and French mathematicians were unaware of them. Mittag-Leffler wrote to Poincar6 on several occasions about questions of priority concerning Weierstrass.

Mittag-Leffier's opinion of Poincar6's work always remained ambiguous and ambivalent. On the one side, he admired the "genius" of Poincar~, on the other, he was also critical of his lack of rigour: But he [Poincar6] . . . writes with too little care and his memoirs are full of inexactitudes. That is something which can only be said between us! One must let the great geniuses follow their own paths and accept with gratitude what they give us, even if one might hope to receive it in more digestible form. [MittagLeffier to Hermite, 27 October 1887 IML]

This opportunity of publishing Poincar6's great papers helped MittagLeffier to decide to embark on his project of creating a new mathematical journal. During the 1870s, the level of French mathematical journals, such as the Journal de mathdmatiques pures et appliqudes and the Journal de I'Ecole Polytechnique, had declined [Gispert 1996]. During the same period, German mathematical journals, such as the Journal fi~r die reine und angewandte Mathematik (Crelle's Journal) or the newly-founded Mathematische Annalen, reflected the great activity and creativeness of the German mathematical community. However, these journals were not very widely read outside the natural domain of the German language, and particularly not in France. Mittag-Leffler took it into his head to create a new journal which would be a bridge between German and French mathematicians. In addition, thanks to his friendship with both Hermite and Weierstrass, he expected the collaboration of the two communities. But favourable circumstances would not be enough. Mittag-Leffler remembered that the success of Crelle's Journal was grounded on Abel's major contributions on elliptic functions. He now argued that the diffusion of the new theory of Fuchsian functions needed a new journal:

The C r e a t i o n of t h e A 9 Mathematica

Nevertheless, in 1881, impressed by the notes about Fuchsian functions and convinced by Hermite's arguments, Mittag-Leffler understood that Poincar6 was a real mathematical genius: I congratulate you heartily on the great success you have had in your research, and I find that our dear master M. Hermite was entirely right when he wrote to me that you are a veritable mathematical genius. I only wish you would publish a great work where you would bring together all your researchers into Fuchsian functions. [Mittag-Leffler to Poincar6, 22 June 1881--IML].

I assure you that your discoveries will compete with those of Abel and that your functions are the most remarkable since the discovery of elliptic f u n c t i o n s . [ . . . ] Now I have a proposition for you. We, the Scandinavian mathematicians, have a project of publishing a new mathematical journal modeled on Crelle's journal. [ . . . ] Now, we, Mr. Gyld6n and I, have thought that you, a Frenchman, will be so generous as to assure the success of our journal. Would you agree to submit your memoir "Sur les groupes fuchsiens" for publication in the first issue of the journal? [ . . . ] I ask you to say nothing about our project because the realization of this project de-

2"M. Poincare est un jeune homme encore, je suppose." [Letter from Mittag-Leffler to Hermite, 6th April 1881 --AS]

VOLUME21, NUMBER2, 1999 59

p e n d s on you. If you decline, m y opinion is t h a t w e shall have to w a i t t w o or t h r e e years. It is only the huge a d v a n t a g e o f publishing y o u r discoveries that c a n convince me. [Letter from Mittag-Leffier to Poincar~, 29 March 1882--IML] Poincar~ agreed to publish his five great p a p e r s on F u c h s i a n functions in the n e w journal, Acta Mathematica. The letters in which Poincar~ promised his help to Mittag-Leffier are lost, and w e can only conjecture w h y Poincar~ decided to publish in Acta Mathematica. First. we can p r e s u m e that Poincar~ felt flattered b y Mittag-Leffier's proposition. Moreover, Poincar~ w a s con~ n c e d that he h a d to move quickly to establish his p r i o r i t y a n d his reputation, and that he m u s t m a k e his t h e o r y k n o w n in Germany, t h e p r e e m i n e n t nation for m a t h e m a t i c s . In addition, w e m u s t not forget t h e quarrel b e t w e e n Klein and Poincar~ a b o u t the priority and the t e r m i n o l o g y for F u c h s i a n functions. Poincar~ t o o k c a r e to m a k e his results k n o w n to s o m e p r o m i n e n t mathematicians: Authors are, in general, impatient to get their reprints, not to pass t h e m around to their friends, but to send copies as s o o n as possible to ten or a dozen great n a m e s to w h o m they wish to m a k e their w o r k s known. [Letter from Poincar~ to EnestrSm, 3 June 1884 CHS] So Poincar~ n e e d e d to diffuse his w o r k in Germany. He had published a survey article on Fuchsian functions [Poincard 1882] in Mathematische Annalen, Klein's journal, but, on account of the G e r m a n o p h o b i a in France after the w a r of 1870, w e can conjecture that he could n o t publish his main papers in a G e r m a n journal, and MittagLeffier's offer allowed him to be read in Germany without offending the nationalist ideology o f his compatriots. 3 By publishing in Acta Mathematica, he could r e a c h b o t h a F r e n c h and a G e r m a n audience, g o o d for his reputation in b o t h countries. If Mittag-Leffler built the s u c c e s s ofActa Mathematica

on Poincard's participation, Poincar~ built his international f a m e on his m a n y publications in that journal. Poincar~ p u b l i s h e d 10 p a p e r s in the first ten issues of Acta Mathematica (676 of the 1594 p a g e s w r i t t e n by the F r e n c h authors). Not all w e r e on F u c h s i a n functions, for Poincar~ was especially creative a n d prolific during this period. F o r example, in 1883 he s h o w e d that a m e r o m o r p h i c function o f t w o variables is t h e quotient of two h o l o m o r p h i c functions: I have tried for a long t i m e to find if a m e r o m o r p h i c function F(x, y) c a n be always w r i t t e n in the f o r m G(x, y)/Gl(x, y) b u t w i t h o u t reaching a satisfying result. I k n o w that Mr. Weierstrass c o n s i d e r s this probl e m as one of the m o s t essential and one of the m o s t difficult in Analysis. A n d you a n n o u n c e to m e that you a r e on the w a y to the solution. I h o p e that you will let m e k n o w without delay when you have a definitive result. [Letter from MittagLeffler to Poincar~, 5 D e c e m b e r 1882--IML] Other i m p o r t a n t p a p e r s are "Sur l'~quilibre d'une m a s s e fluide" (1885a), a n d "Sur les int~grales irrdguli~res des ~quations lin~aires" (1885b); this last p a p e r was the o c c a s i o n o f a p o l e m i c with Thorn& These m a t h e m a t i c a l disp u t e s are almost the only circums t a n c e s in which Poincar~ b r o k e t h r o u g h his reserve. I h a v e a l r e a d y m e n t i o n e d his very s h a r p c o n t r o v e r s y with Klein. Mittag-Leffler r e f e r r e d to this p o l e m i c on several occasions, as on 18 July 1882, w h e n he told of his visit to Schwarz and S c h w a r z ' s fury: It w a s not the s a m e thing with Mr. Schwarz. I found him full of indignation with you. He thinks that he is the first to have given an e x a m p l e of t h o s e groups that you call Fuchsian which cannot be found in the theory of elliptic functions. [ . . . ] F r o m your p o i n t of view, you have to find in this squabble b e t w e e n G e r m a n mathematicians about y o u r n e w n a m e s a p r o o f of the i m p o r t a n c e o f y o u r dis-

3Moreover, Poincar6 was born in Nancy, in Lorraine, and half of Lorraine had been annexed by Germany.

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coveries. Mr. Schwarz did not h i d e the fact that especially the F u c h s i a n functions were the cause of his fury.

[IML] Before answering, Poincar~ quickly s t u d i e d S c h w a r z ' s p a p e r s and conc l u d e d t h a t he s a w no r e a s o n to c h a n g e his mind: I do n o t h o p e to mollify Mr. Schwarz. What are the r e a s o n s for his fury? First, he is in a rage bec a u s e he had in his h a n d s an imp o r t a n t result a n d did n o t k n o w h o w to t a k e advantage o f it. I cann o t d o anything a b o u t that. Next, h e is dissatisfied with t h e n a m e Fuchsian. He w o u l d p r e f e r Schwarzian. I have s t a t e d the reas o n s w h y I cannot d o anything a b o u t that either. [Letter f r o m P o i n c a r d to Mittag-Leffier, 27 J u l y 1882--IML] Poincar~ replied to Weierstrass's questions a b o u t his note "Sur l'int~gration des ~quations diff~rentielles" in the s a m e way. In this paper, Poincar~ h a d s h o w n that there is always a change of variables so that solutions of a differential equation are representable b y a convergent series for all values o f the n e w variable. Weierstrass's objection c o n c e r n e d collisions in the three-body problem. Poincar~ a n s w e r e d that in case o f collision, his n e w variable t e n d s to infinity, and so, after the collision, "the formulas do not give anything [...] and that is the b e s t they have to do." Poincar~'s a n s w e r to the old M a s t e r is abit airy... Later, after his s u c c e s s in O s c a r II's Prize competition, Poincar~ c a m e to Mittag-Leffler's help in a p o l e m i c w i t h the S w e d i s h a s t r o n o m e r Gyld~n. In his prize-winning paper, Poincar~ disc u s s e d t h e c o n v e r g e n c e of the n e w exp a n s i o n s (without secular t e r m s ) u s e d by the a s t r o n o m e r s . Present-day geometers have end e a v o r e d to replace these developm e n t s b y s o m e n e w ones, w h i c h c o n t a i n only t r i g o n o m e t r i c terms. Recently t h e y have succeeded, a n d

Mr. Gylddn's series or Mr. Lindstedt's contain only terms of the form A sin at

or

B cos at.

[ . . . ] . Nevertheless, that is not the end. We may ask if these series are convergent, and as the presence of "small divisors" results in making some terms very big, this convergence is dubious. This work will show that these doubts are grounded; all of these series are divergent. However, I have to exclude those proposed by Mr. Gyld6n in his last memoir. I do not have any way to know if they are convergent or divergent. [Poincar6 1888, p. 6] Without having read Poincar6's paper, Glyd6n maintained the convergence of his own expansions and claimed priority over Poincar6 about some results concerning asymptotic solutions. King Oscar required a response from Mittag-Leffler, so MittagLeffler called for Poincar6's help. In his answer, first, Poincar6 said that reading Gyld6n's papers [1887] and understanding his results was very hard. He explained also his own method of reading: Shall I confess to you that I think Mr. Gyld6n's style a bit tedious and it is very difficult for me to read it? I am used, when I read a memoir, to glance over first quickly so as to have a general impression, then come back to the points which seem to me obscure. I find it more convenient to do proofs over than to examine thoroughly those of the author. My proofs are generally far poorer, but they have for me the advantage that they are mine. [Letter from Poincar6 to Mittag-Leffier, 5 February 1889--IML] As Gyld6n's algorithm is not precisely defined, deciding its convergence is hard. Poincar6 showed that Gyld6n's argument is not rigorous, and concluded pitilessly: [Gyld6n's proof] amounts to accepting the following principle: Every series with variable less than 1 is convergent unless we have very

serious reason to doubt it. [Letter from Poincar6 to Mittag-Leffler, 1 March 1889--IML] Later in the same letter, Poincar6 admitted that Gylddn's convergence arguments are sufficient for calculations in astronomy but "not enough for the geometer." These mathematical polemics are the few occasions where Poincar6 is somewhat expansive. In his university correspondence he was very reserved and showed no personal feelings. He rarely referred to political and social events. Furthermore, although interested in academic and university life, unlike Mittag-Leffler he did not participate in the usual gossip and intrigue of the scientific community. T h e O s c a r II P r i z e Thanks to the book of J. Barrow-Green [1997], the story of Oscar II's Competition is now well known. For the 60th birthday of Oscar H, the King of Sweden, who was a friend of the sciences, Mittag-Leffier organized a mathematical competition and a commission whose members were Hermite, Weierstrass, and Mittag-Leffier. They asked four questions "which from different points of view equally engage the attention of analysts, and the solution of which would be of the greatest interest for the progress of science" [Announcement of the Oscar Competition]. The first one, posed by Weierstrass, concerned the n-Body Problem, and in particular the stability of the orbits. Poincar6 decided to compete:

I have not forgotten the King Oscar's prize, and I will even say that I have been thinking only about this prize for one or two months. [Letter from Poincar6 to Mittag-leftier, 16 July 1887--IML] Poincar6 tried to answer the first question. The main result of his work was a proof of stability in the case of the restricted 3-Body Problem. Poincar~ obtained this result quickly and hoped to obtain a more general one: Nevertheless, I have obtained some interesting results and I want to

quote one of them. It concerns the special case where the first and second bodies have a fmite mass and the third a zero mass. The first and second bodies trace circles around their mutual center of gravity and the third moves in the plane of these circles. In this special case, I have found a rigorous proof of stability and a way to determine precise bounds on some parameters of the third body. [ . . . ] I hope now to be able to attack the general case and by the 1st of June, if not to have completely solved the question (I do not hope that), at least to have found some results which will be sufficiently complete to be sent in for the competition. [Ibid.] Poincar6 arranged his memoir around this result. He expressed his disappointment in the introduction: The present memoir has been undertaken to answer the fn'st of the four questions of the competition; but the results I have obtained are so incomplete that I should hesitate to publish them if I did not know that the importance and the difficulty of this problem gives some interest to anything concerning it, and that we may expect a definitive solution only after a long succession of attempts. [ . . . ] I have had to restrict myself with a special case. I have only handled the equations of motion in the case [ . . . ] of two degrees of freedom. [Poincar6 1888, p.

5-81 During the summer of 1888, MittagLeffler and Weierstrass studied Poincar6's memoir. Mittag-Leffler was very enthusiastic, but thought that Poincar6's work was very difficult and that some proofs were incomplete. Poincar6 answered his questions with 100 pages of supplementary notes. On 21 January 1889, King Oscar awarded the Prize to Poincar6. But, at the beginning of July, MittagLeffier transmitted to Poincar6 some questions he had been asked by Phragin6n, who was his assistant for A c t a Mathematica. They concerned the convergence of expansions of the

VOLUME 21, NUMBER 2, 1999

61

asymptotic solutions introduced by Poincar& Poincar~'s answer formed the last supplementary note "Sur les solutions asymptotiques" [1888, p. 251-256]. It seems that these questions spurred Poincard to investigate the whole proof of the convergence of asymptotic solutions more precisely and unfortunately to fmd an irreparable error. On 1 December, Poincar~ announced in despair to Mittag-Leffier that the major part of his memoir was flawed. I will not conceal from you the distress this discovery has caused me. In the first place, I do not know if you still think that the results which remain [ . . . ] deserve the great reward you have given them. [Letter from Poincar~ to Mittag-Leffler, 1 December 1889--IML] Poincar~'s proof of stability was based on the fact that asymptotic surfaces, i.e., surfaces generated by asymptotic solutions, are closed and so other solutions are excluded. 4 This followed from a more general lemma, but this lemma was wrong. We may imagine that for a while Poincar~ hoped to save his stability result by proving the closure of the asymptotic surfaces directly from the convergence of the asymptotic solutions. Indeed, on 10 December, Hermite wrote to Mittag-Leffler that the error was not as important as Poincar~ had believed and that it is only a question of reshaping his "admirable work." But Poincar~ discovered that these expansions were not convergent but asymptotic. Unfortunately, these series are not convergent. [ . . . ] But although divergent, cannot we make use of them? [ . . . ] We can say that the series we have obtained in this paragraph stand to the asymptotic solutions with small value o f / z in the same manner as Stirling's formula stands to the Eulerian functions. [Poincar~ 1890, p. 384-386]

Although many results survived, in particular the divergence of Lindstedt's series and the recurrence theorem, it was nevertheless a disaster. Mittag-Leffier kept cool and answered that he did not regret having given the prize to Poincar6. He busied himself with his usual efficiency. By then, Poincar~'s memoir had been printed and partially delivered to the editors of Acta Mathematica, so recove~_ng the copies without giving away the reason was the most immediate problem. When that was done MittagLeffier proposed that Poincar~ write and publish a new memoir. And now here is what I propose you do and what will be, from my point of view, the most honorable for you and for us. You write a new memoir in which you put what remains from your original memoir, the developments which are in the notes, and whatever you think right to add. You write for this new memoir an introduction in which you say that it is a modification of the prize memoir, and that you have given some developments which were only sketched there and corrected an error. [...] I think that this story has to be kept between us until the publication of your memoir. [Letter from Mittag-Leffler to Poincar~, 5 December 1898--IML] Although Weierstrass complained that he had learned the news of Poincar~'s error through gossip, Mittag-Leffier succeeded rather well in keeping the whole story secret. On 20 December, after Poincar~ had accepted Mittag-Leffler's deal, 5 MittagLeffler considered that the crisis was settled from the institutional point of view. Secure at home, he encouraged Hermite to praise Poincard's work at the Acad~mie des Sciences. At the official meeting of the Institute, you must absolutely speak about his memoir. Otherwise, everyone will be astonished and will

begin to wonder why you do pot. Moreover, you can really praise the memoir without fear of exaggerating, because in any case, it will remain as one of the masterpieces of our day. [Letter from Mittag-Leffler to Hermite, 20 December 1889--AS] He added: The advice will be very useful to Mr. Poincar6, who maybe will in the future give up his regrettable habit of stating results whose proof he knows only imperfectly. [ . . . ] Of course, I share your opinion about Poincar6 and his genius, but he has imposed too much on everybody, and someone who will not condescend to be more rigorous should not hold such a prominent position as he has had. He is still young, he will reform, and mathematical sciences will gain. [Ibid.]

The Nobel Prizes in Physics One of the most interesting points of the Poincar6-Mittag-Leffler relationship is Mittag-Leffler's attempt to obtain the Nobel Prize in Physics for Poincar& His wider objective was to have the Nobel Prize in Physics awarded to theoreticians. It was a new round of his fight against the group at the Swedish University, supporters of "the scientific methods of Ostwald and Arrhenius. "6 In her book The Beginnings of the Nobel Institution [1984], Crawford shows that nothing less than the definition of the field of physics was at stake in the opposition between experimentalists and theoreticians over the Nobel Prize in Physics. Mittag-Leffler explained his strategy to Painlev6: Now I am doing my best to obtain the Nobel Prize for Lorentz. It is necessary to put Lorentz ahead of Poincar& First, Lorentz is more directly a physicist and then one must have a report signed by a sufficiently competent authority. Poin-

4Nowadays asymptotic surfaces are called "stable and unstable manifolds." 5In particular, Poincare agreed to pay the expenses of the new printing (3585 Swedish crowns). In comparison, the value of the prize was 2500 crowns and MittagLeffler's annual wage was 7000 crowns [Domar 1982]. 6Ostwald in Leipzig, who won the Nobel Prize in Chemistry in 1909, and Arrhenius in Stockholm, who won the Nobel Prize in Chemistry in 1903.

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car~ is writing the report. If I succeed, [...] ! will have succeeded in opening the door to theory, which was surely in the ideas of Nobel, and then first Poincar~, and you will come afterwards. [Letter from Mittag-Leffler to Painlev~, 18 July 1902--IML] However, Mittag-Leffler had to agree to compromise and to accept the division of the Prize between Lorentz and Zeeman, who was more of an experimentalist. Between 1904 and 1909, Poincar~ was nominated several times for the Nobel Prize in Physics on Darboux's initiative. In 1910, MittagLeffler thought that the circumstances were propitious. He wrote to Appell in order to coordinate the French and Swedish attempts: The time is come when we can hope to make Poincar~ winner of the Nobel Prize. I send enclosed with the next mail a proposal written by Fredholm that he subjects to your judgment and one by Mr. Darboux. He has made considerable use of the proposal made by Darboux this year. The most important thing is first to establish the prominent part played by pure theory in physics and then to conclude with the proposition to give the prize for discoveries defined by a sufficiently simple formula. After some discussion, we have found this formula in Poincar~'s discoveries concerning the differential equations of mathematical physics. I think that we will win with this program. [Letter from Mittag-Leffier to Appell, 28 Novem-ber 1909---IML] Mittag-Leffler added that the nominators had to avoid "mathematics" and refer to "pure theory" because "like those who are only experimentalists, members of the Nobel committee for Physics are scared silly by mathematics." Appell, Darboux, and Fredholm signed a report "sur les travaux d'ordre physique de Poincar&" In the introduction, they claimed that it was ira-

possible to ignore the very important part played by the progress of theory in the advancement of the sciences. They emphasized that physicists had to propose theories and to build mathematical tools: To the extent that physics wants to understand the hidden and inner mechanism of things, the role of hypothesis increases, and to the same extent the physicist's need for better analytical tools also grows. [Report on "les travaux d'ordre physique de M. Poincar~"--CHS] Poincar~'s nomination was justified by his contributions to the general and correct solution of problems of mathematical physics. Then Mittag-Leffler sent to the whole physics community a circular asking them to nominate Poincar~ for the Nobel Prize. 34 eminent physicists or mathematicians supported the proposition in favor of Poincar~. 7 Unfortunately, the experimentalists and Arrhenius's network were not convinced, and the Nobel Commission awarded the prize to an experimentalist, Van der Waals. We have again been beaten, this time for the Nobel Prize. This crowd of naturalists who do not understand anything about the fundamentals of things has voted against us. They fear mathematics because they don't have the slightest possibility of understanding anything about it. [Letter from Mittag-Leffler to Poincar~, 6 December 1910---IML] ARCHNES

AS--Archives of the Academie des Sciences-Paris (France). CHS--Center for the History of Sciences--The royal Swedish Academy of Sciences-Stockholm (Sweden). IML--Mittag-Leffier Institute--Djursholm (Sweden). BIBLIOGFL&PI.IPf

Andersson, Karl Gustav [1994] Poincar6's Discovery of Homoclinic

Points. Archive for History of Exact Sciences 48, 133-147. Barrow-Green, June [1994] Oscar II's Prize Competition and the Error in Poincar6's Memoir on the Three Body Problem, Archive for History of Exact Sciences 48, 107-131. [1997] Poincar6 and the Three Body Problem. American and London Mathematical Societies. Crawford, Elisabeth [1984] The beginnings of the Nobel Institution, Cambridge/Paris: University Press/Editions de la Maison des Sciences.

7[See Crawford 1987]. Nevertheless, some physicists like Rutherford did not agree with Poincare's nomination because they thought that Poincare was not a real physicist.

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[1987] The Nobel Population 1901-1937, Berkeley/Uppsala: Office for History of Sciences and Technology/Office for History of Science. Domar, Yngve [1982] On the foundation of Acta Mathematica, Acta Mathematica 148, 3-8. Gispert, Helene [1996] Une comparaison des journaux franqais et italiens dans les annees 1860I875, in [Goldstein-Gray-Ritter, 1996], 389~406. Goldstein-Gray-Ritter [1996] L'Europe math~matique, Paris: Editions de la Maison des Sciences de I'Homme. Gylden, Hugo [1887] Untersuchungen 0ber die Convergenz der Reihen, welche zur Darstellung der Coordinaten der Planeten angewendet werden, Acta Mathematica 9, 185-294. Mittag-Leffier, G6sta [1879] Extrait d'une lettre a M. Hermite par

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M. G. Mittag-Leffler, Bulletin des Sciences mathematiques et astronomiques 14, 269-278. Poincare, Henri [OEuvres] OEuvresde Henri Poincare, 1-11, Paris: Gauthier-Villars, 1916--1956. [1881a] Preliminaryversion of the Poincare's paper "Sur les fonctions & espace lacunaire" [1881b]--IML. [1881b] Sur les fonctions & espace lacunaire Acta Societatis scientarum Fennicae 12 (1883), 343-350; OEuvres, 4, 28-35. [1882a] Sur I'integration des equations differentielles,Comptes rendus 94, 577-578; OEuvres, 1, 162-163. [1882b] Sur les Fonctions uniformes qui se reproduisentpar des substitutions lineaires, Mathematische Annalen 19, 553-564; OEuvres, 2, 92-105. [1883] Sur les fonctions & 2 variables, Acta Mathematica 2, 97-113; OEuvres,4, 147161. [1885] Sur I'equilibre d'une masse fluide

animee d'un mouvement de rotation, Acta Mathematica 7, 259-380; OEuvres, 7, 40140. [1886] Sur les integrales irregulieres des equations lineaires, Acta Mathematica 8, 295-344; OEuvres, 1,290-332. [1888] Sur le probleme des trois corps et les equations de la dynamique avec des notes par I'auteur--memoire couronne du prix de S. M. le Roi Oscar II. Printed in I889 but not published--lML. [1890] Sur le probleme des trois corps et les equations de la dynamique, Acta Mathematica 13 (1890), 1-270; OEuvres, 7, 262490. [1901] Analyse des travaux scientifiques de Henri Poincare faite par lui-m~me, Acta Mathematica 38 (1921), 3-135. Weierstrass, Karl [1876] Zur Theorie der eindeutigen analytischen Functionen, Abhandlungen der Kdnigl. Akademie der Wissenschaften vom Jahre 1876, 11-60; Werke, 2, 77-124.

II-'-:a,J[:a,,,~--ll

Jet

Wimp,

Editor

I

The Man Who Loved Only Numbers: the Story of Paul ErdGsand the Search for Mathematical Truth by Paul Hoffman NEW YORK: HYPERIONPRESS, 1998, 289 PP. US $22.95, ISBN 0786863625 REVIEWED B Y MARION COHEN

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,

or most of his adult life, the mathematician Paul ErdSs was homeless. But never did he have to spend a single night on the street or in a shelter. In fact, some might say that he was the very opposite of homeless: he had many homes---those of his some 485 collaborators. However, there is at least one parallel that could be drawn between Erd6s and the stereotype of a homeless person: on the one hand, he was an interesting human being; on the other hand, he was not quite functional as a human being at all. Erdds spent just about every waking moment of his life chasing mathematics and mathematicians. It s e e m e d to be a m e r r y chase. At any rate, the chase was not difficult for him; indeed, for him it came as easily as breathing, eating, and walking. As most human beings brush their teeth once or twice a day, that is close to the n u m b e r of theorems which Erd6s and his co-authors produced in a day. With the speed and alacrity of Superman (a Superman who rarely took off his cloak to become Clark Kent), Erdds knocked off fundamental results, not only in number theory (as the title of Hoffman's book would suggest), but geometry, probability theory, approximation theory, interpolation theory, set theory, combinatorics, and real and complex analysis. He was not only a great problem poser and solver, but a theory creator as well. He virtually invented probabilistic number theory,

F

partition calculus for large cardinals, extremal graph theory, statistical group theory, and the theory of random graphs. He had written his first paper while just a freshman in c o l l e g e - a simpler p r o o f of Chebysh~v's theorem on primes between any number and its double. And by the end of his senior year, his thesis on primes in arithmetic progression had won him a doctorate as well as a bachelor's degree. (Throughout his life, p ~ n e s and arithmetic progressions remained his pet topics.) There are many "Erd6s stories" attesting to both his abilities and his disabilities. He didn't know functional analysis at the time but he gave, on the spot, a two-line p r o o f of a theorem that others had proudly shortened to 30 pages. Or he would attend a lecture, hear the speaker mention an m~solved problem, then raise his hand to offer the solution. Talk about simultaneous equations! Erdds would work on several problems, with several colleagues, at once. "He went around the room, like a grandmaster playing chess," recalls Bruce Rothschild, one of his collaborators. Sometimes, in addition, he would be actually playing chess! Indeed, Erd6s was the ultimate collaborator. Although, as I said, Erd6s had a socializing problem, he did, from the very start of his career, have a knack for math-socializing. "Erd6s's questions were always just right," says Richard Guy, another Erd6s collaborator. " . . . But Erd6s not only asked the right questions. He asked them of the right person. He knew better than you yourself knew what you were capable of." Hoffman concludes in that section of his book that Erd6s "set a record for coming up with good problems and seeing that somebody solved them." So important was the mathematical community to Erdffs that he went on record as criticizing Andrew Wiles for holing himself up for seven years and

9 1999SPRINGER-VERLAGNEW YORK,VOLUME21, NUMBER2, 1999 ~ 5

keeping the proof of Fermat's Last Theorem to himself. "The problem would have been solved a lot quicker if he had shared his work," Erd6"s said. (But I believe there are many ways to crack a nut.) "Erdds's forte," says Hoffman, "was coming up with short, clever solutions, and he devoted considerable energy to providing shorter and simpler proofs of theorems already proven by others." The Prime Number Theorem, about the distribution of primes, and the earlier mentioned Chebysh~v's Theorem are examples of these. "I'm always sayIng," Hoffman quotes Erdds, "that the SF [meaning Supreme Fascist, Erdds's term for the God he didn't believe in] has this transfinite B o o k . . . that contains the best proofs of all mathematical theorems . . . . " Euclid's proof of the infmity of primes is an example of this; Wiles's proof of Fermat's Last Theorem is not. Erd6s himself admitted that this Book business "should not be taken too seriously"; there is indeed "non-Book" math that is, if not pretty, at least necessary and, for the moment, sufficient. Wiles did at least convince us that Fermat's Last Theorem is true. Erd6s never owned nor rented a house or apartment. Instead, his sundry collaborators and their families put him up (and, as we will see, put up with him). He carried with him just two pieces of baggage containing one suit, many pairs of silk underwear, his mathematical notebooks, and (at least until a certain point in his life) a radio. He certainly never had a checkbook, credit card, or driver's license, and there's a story in Hofflnan's book about how once, detained by the police on a minor "loitex~ng" matter, he offered up one of his math papers (the police accepted it). His various hosts were happy to provide meals, laundry service, chauffeur service, room service. One suspects that Erd6s's messy and thoughtless habits involving things like showers and grapefruits must have caused his hosts to wish they could provide private kitchen and bath as well. (One of his hosts did build an addition to his house specifically for Erdds.) "Another roof, another proof" was Erd6s's motto, and that's what hap-

THE MATHEMATICAL INTELLIGENCER

pened (or sometimes it was "only" a conjecture). He would begin each stay by arriving, often unexpectedly, on a colleague's doorstep and declaring, "My brain is open." Then the two, or more, brains would work, most of the day and night, until the "mathematician in residence" was ready to move on to another residence. For only two short periods of his life did Erd6s have a regular job. The rest of his income came from prize money and lectures. Of course, his expenses were minimal--phone calls, snacks on the road. Erd6s could afford to, and did, give lots of money to charities, sometimes scholarships, and----one of his trademarks--prizes for the solutions to various problems he posed. Erd6s could not bear to spend too long a period of time not doing math. My favorite story, related in more detall on pages 242-3, concerns the mundane matter of Erd6s's eyes. He needed to have a cataract operation in one of them and he wanted, during the operation, to read a math book with the other eye! In compromise, doctors arranged for a local mathematician to stay with him and talk math. As Hoffman aptly puts it, "If it wasn't math, Erd6s wouldn't be bothered."

Romance and Reality I know the feeling. When I was in junior high school, I fell madly in loveat-first-sight with math. I especially loved ( a + b ) ( a - b ) = a 2 - b 2 a n d I began making up my own formulas and, soon, theorems and theories. I wrote about math In my diary, even wrote a couple of adolescent poems about math. In high school I much preferred doing a math problem to going bowling or to a school dance. I was struck by the sheer romance of the idea of spending 24 hours a day with any one thing, and a handful of years later, when I fell in love---not with math but with a p e r s o n - - I was still struck by the idea of spending 24 hours a day with that person. Over the years, although I have continued to be "bothered" with plenty of things besides math, I can still very well understand, with Erd6s, that if one "has" math, one might not need anything else. The Erd6s story is exciting and romantic, and so is Hoffman's book

(which wisely tells many exciting ~and romantic stories about other mathematicians besides Erd6s). However, I am now aware that, when one tries to carry this romantic idea to all levels, it gets inconsistent with not only reality, but with other people's romantic ideas; it can even get ugly and unromantic. The "magnificent obsession," as Hoffman calls it, can sometimes have the consequences of a non-magnificent obsession. This brings us to examples, which abound, of his disabilities, his social disabilities. Erd6s was far from autistic, but he was certainly what people sometimes call an "absent-minded professor." He was often as oblivious to the realities of the people he came in contact with as Archimedes was of the soldier who was soon to kill him. In fact, Erd6s's absent-mindedness very often took the form of just plain inconsiderateness (and I'm wondering whether absent-mindedness can ever fail to do that). We are told on page 7, "Erd6s would let nothing stand in the way of mathematical progress. When the name of a colleague in California came up at breakfast in New Jersey, Erd6s remembered a mathematical result he wanted to share with him. He headed toward the phone and started to dial. His host interrupted him, pointing out that it was 5:00 A.M. on the West Coast. 'Good,' Erd6s said, 'that means he'll be home.'" One could chuckle, and I admit I did, but Erd6s meant it. (When I told my 12year-old son the story, he did not chuckle, but murmured, "How extremely annoying.") I don't believe that "mathematical progress" is synonymous with satisfying one person's need to do whatever he pleases whenever he pleases. It could have waited 'til morning. Erd6s's obliviousness towards other human beings seems to me to border on passive--sometimes active-aggressiveness. Page 50, again quoting Bruce Rothschild: "Erd6s was allowed [meaning he allowed himself] to think about many problems at once, but he expected his collaborators to focus on the problem at hand. 'No illegal thinking,' he'd say when he sensed their minds wandering." Same page, quoting collaborator Michael Golomb: "ErdUs was playing

c h e s s with a 'local m a s t e r ' n a m e d Nat Fine, w h o m Erd6s c o u l d b e a t only r a r e l y . . . . I saw Nat with his h e a d bet w e e n his hands, d e e p in t h o u g h t . . . w h i l e E r d 6 s s e e m e d to b e e n g r o s s e d in studying a voluminous e n c y c l o p e d i a of m e d i c i n e . . . . I a s k e d him, 'What are y o u doing, Paul? A r e n ' t y o u playing against Nat?' His a n s w e r w a s 'Don't int e r r u p t m e . . . I a m proving a theor e m . ' " T h e r e s e e m e d to b e a kind o f s u b t l e unwritten a g r e e m e n t b e t w e e n E r d 6 s and his a s s o c i a t e s t h a t this kind o f thing w a s to be p e r m i t t e d . But not really an agreement: E r d 6 s d i s p l a y e d the s a m e sort of b e h a v i o r with p e o p l e h e ' d n e v e r met. Page 125, Louise Straus, wife of an Erd6s collaborator and a mathematician herself: "He'd j u s t s h o w up at o u r place, and we never k n e w h o w m a n y d a y s he was going to stay 9 I r e m e m b e r during the night hearing crashing sounds. The w i n d o w s h a d no sash cords 9 If you o p e n e d t h e lock, t h e y ' d c o m e crashing down. He . . . could n e v e r figure out h o w to gently l o w e r t h e w i n d o w s . . . (nor) h o w to m a n a g e t h e shower 9 He c o u l d n e v e r shut t h e f a u c e t s off. Water ran out on the floor 9 The linoleum buckled, a n d the d o o r w o u l d n ' t shut again. H e ' d go outside to t h e p a y p h o n e and d r o p coins in it all night, calling m a t h e m a t i c i a n s . . , and a s k i n g friends to c o m e o v e r to o u r p l a c e . . . . He never a s k e d us first if w e w a n t e d m o r e guests." A n d at parties, "he always had t r o u b l e tying his shoes, 9 . . ] r e m e m b e r him sticking his foot o u t at the party, asking p e o p l e to tie his shoe." Page 129, Anne Davenport, w i d o w o f an E r d 6 s collaborator: "He w a n t e d m e to call his m o t h e r in Budapest. I a s k e d him w h a t h e r n u m b e r was. He said t h a t she didn't have a phone. I a s k e d him h o w he e x p e c t e d m e to call. He said h e r n e i g h b o r h a d a phone. I a s k e d him for the n e i g h b o r ' s number. 'I d o n ' t know,' he said. 'That's for y o u to fred out.' " Hoffman is not in denial with res p e c t to ErdSs's t e n d e n c y t o - - i n the w o r d s of a friend of E r d 6 s "make his reality overtake yours," b u t he also s e e m s to s t e e r clear o f any flat criticism. Not once, for example, d o e s he u s e - - e i t h e r himself or as a q u o t e - -

w o r d s like "arrogant," "sponging," or "inconsiderate." Erd6s s e e m e d to have felt he w a s entitled to b e the w a y he was. Hoffman quotes him as saying, "Louis the F o u r t e e n t h said, 'I a m the state'; Trotsky could have said, '! a m society,' and I say 'I a m reality.' .... No one who k n e w him would disagree," adds Hoffman, b u t I, who didn't know him, would disagree. No one is the whole of reality. "He w a s n ' t an easy houseguest," Hoffman g o e s on, quoting a friend of Erd6s. "But w e all w a n t e d him a r o u n d - - f o r his mind." One could, of course, c o n j e c t u r e that, "his mind" aside, w o r k i n g with E r d 6 s b r o u g h t clout. But "mind" or clout, I w o n d e r w h y his h o s t s and other "Uncle Paul sitters" (as the m a t h e m a t i c a l c o m m u nity p u t it) w o u l d n ' t at least feel s o m e r e s e n t m e n t , let alone anger. P e r h a p s t h e y did, b u t did n o t voice it.

"1 am reality." E r d 6 s ' s attitude t o w a r d w o m e n w a s primitive. "To c o m m u n i c a t e with Erd6s," s a y s Hoffman, page 8, "you h a d to l e a r n his language," a n d t h o u g h s o m e o f his co-authors w e r e w o m e n , E r d 6 s ' s language did n o t reflect res p e c t for w o m e n . Some of it is c h a r m ing a n d harmless; "epsilon" m e a n t child, "epsilon-square" grandchild. However, "boss" m e a n t woman, "slave" m e a n t man, and "captured" m e a n t married. ("Stupid," m u t t e r e d m y 12-yearold son. "What a jerk!") A n d t h e stuff o f page 134 m a k e s m e wince: " 'In Hungary, m a n y m a t h e m a t i c i a n s drink strong coffee,' said Erd6s. 'At t h e m a t h e m a t i c a l institute t h e y m a k e particularly g o o d coffee 9 When P o s a w a s not quite fourteen, I offered him a little coffee, w h i c h h e d r a n k w i t h an infinite a m o u n t o f sugar. My m o t h e r w a s very angry t h a t I gave the little b o y strong coffee 9 I a n s w e r e d that P o s a could have said, "Madam, I do a m a t h e m a t i c i a n ' s w o r k a n d d r i n k a m a t h e m a t i c i a n ' s drink 9 I s a w a movie m a n y y e a r s ago w h e r e a l a d y sees a b o y of sixteen drink w h i s k e y with an o l d e r m a n and is shocked. The b o y says, "Madam, I do a m a n ' s w o r k and d r i n k a m a n ' s drink.' ....

To clinch it, on the s a m e p a g e Hoffman again quotes Erd6s: "When Lov~sz w a s still a n epsilon, in the first y e a r of high school, he a n d . . , a fell o w m a t h e m a t i c i a n c o u r t e d the s a m e boss-child, also a m a t h e m a t i c i a n a n d n o t a b a d one as b o s s e s go." I can't help w o n d e r i n g w h e t h e r E r d 6 s ' s female c o l l a b o r a t o r s bristled. Erd6s never h a d a wife o r a lover o f either gender. Recall, "if it w a s n ' t math, Erd6s w o u l d n ' t b e bothered." E r d 6 s a d m i t t e d that, with regard to sex, he h a d both p h y s i c a l and psychological problems, b u t he didn't view them a s important. He s e e m e d h a p p y being "bothered" only with math. One could speculate on his relationship with his mother, w h o m he loved and who was v e r y averse to his having a woman. Also, E r d 6 s had an unfortunate legacy; days after his birth his t w o sisters, aged three and five, died suddenly of scarlet fever 9 A y e a r and a half later, his father w a s taken to a prison camp and did n o t return until his s o n was seven 9 One could also speculate that being a child prodigy, and later an adult prodigy, can be a b u r d e n as well as a blessing. A n d when anyone has a p o w e r - - b e it political o r mathematic a l - i t can be h a r d to k n o w how to place that p o w e r in the s c h e m e of one's lee. Since Erd(bs s e e m s not to have ever been in therapy, one can only speculate, b u t I venture that Erd6s could have benefltted from therapy; so also would his public, and his math 9 As s o m e o n e w h o writes serious poetry (also non-serious) about the experience o f mathematics, I have been v e r y interested in h o w mathematicians o t h e r than myself feel about math. Mathematicians through the ages have described m a t h as "beautiful," "elegant," "exciting," "harmonious," "artistic," o r even m e r e l y "interesting" o r "cool9 They have talked in terms o f "passion," "the s e a r c h for lasting b e a u t y and ultimate truth," and "order that transcends the physical world." E d n a St. Vincent Millay wrote, "Euclid alone has looked on b e a u t y bare," and in his b o o k Hoffman often refers to t h e o r e m s and theories as "jewels" or "chestnuts." There have b e e n religious mathematicians, a n d t h e n religion enters into it. Ramanujan said, "An equation for

VOLUME 21, NUMBER 2, 1999

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m e has no m e a n i n g unless it e x p r e s s e s a thought o f God." (How, I can't help asking, a s s u m i n g t h e r e is a God, can an equation not e x p r e s s a thought of God?) And Kronecker, even if he believed that "God c r e a t e d the integers," struggled with all the exciting non-integers that k e p t occun~ng to him. P e r h a p s guilt, ambivalence, and suffering e n t e r into it. Ego definitely h a s e n t e r e d into it. There s e e m s to b e m u c h vying for leadership in the m a t h e m a t i c s community. Thus Erd6s, t h o u g h he thoroughly believed in this c o m m u n i t y and did m u c h to a d v a n c e it, c o u l d also proclaim, "I a m reality." (Here I think of the familiar joke: A p h i l a n t h r o p i s t or activist explains, "I w a n t to m a k e this a b e t t e r w o r l d for m e to live in!") So yes, s e l f - c e n t e r e d n e s s and o t h e r hang-ups e n t e r into it, and E r d 6 s is a case in point. H o f f m a n ' s b o o k a d d e d to a n d e n h a n c e d m y s e n s e of the emotions a s s o c i a t e d with math, and h a s confirmed t h a t t h e s e e m o t i o n s are n o t simple. "The search for b e a u t y a n d truth" is n o t t h e s a m e thing a s b e a u t y and truth themselves.

Genius and Personality Defects O n e can't help w o n d e r i n g w h y E r d 6 s ' s

colleagues a n d t h e i r families allowed him to m a n i p u l a t e t h e m so blatantly. I have tried to identify several o f E r d 6 s ' s m a n i p u l a t i v e g a m e s (games which everyone, c o n s c i o u s l y and unconsciously, p l a y s at o n e time o r ano t h e r b u t w h i c h in E r d 6 s ' s case got p l a y e d all o u t o f p r o p o r t i o n ) . E r d 6 s s e e m s to have gotten himself into the p o s i t i o n o f a guru, courting situations in w h i c h p e o p l e in his cult w a i t e d for h i m - - i n particular, w a i t e d to see w h a t he w o u l d do o r say next. On page 136 H o f f m a n d e s c r i b e s a p a r t y E r d 6 s attended; t h e r e w e r e m a n y people at that p a r t y w h o w a n t e d to m e e t a n d talk with Erd6s, a n d E r d 6 s k n e w that. B u t t h e h o s t h a d a n old sick father, in b e d upstairs, w h o could n o t att e n d the party, a n d E r d 6 s d e c i d e d to w a n d e r up a n d s p e n d the evening with the sick old man. Hoffman cites this as an e x a m p l e o f E r d 6 s ' s c o m p a s s i o n and generosity, a n d no d o u b t it w a s in part, but there w e r e v e r y p o s s i b l y m o r e subtle things going on as well. E r d 6 s

THE MATHEMATICAL INTELUGENCER

n e e d e d to be w a i t e d on, w o n d e r e d about, and in general singled out. This s e e m s r e m i n i s c e n t o f a n o t h e r m a n i p u l a t i v e game, withholding. Withholding m e a n s using as a t r u m p c a r d w h a t e v e r it is that p e o p l e have c o m e to e x p e c t and/or n e e d f r o m you. A m o t h e r withholds t h e e x p r e s s i o n of love from h e r t e e n a g e d d a u g h t e r until t h e daughter will, say, c u t h e r hair. A child refuses to be h e r usual h a p p y self until h e r p a r e n t s b u y h e r t h a t toy she s a w in that store window. E r d 6 s p l a y e d his t r u m p c a r d ( w h i c h was his

Erd6s's manipulative games " o p e n brain") slightly differently, a n d w i t h h e l d a n d brought it o u t m o r e unpredictably. Page 176, depicting a s c e n e involving a group o f mathematic i a n s s p r a w l e d a r o u n d a h o t e l room: "Erd6s didn't s e e m to b e p a y i n g attention. He was slumped over in a chair, his h e a d in his hands, like an invalid in a nursing home. But every few minutes he p e r k e d up and suggested a line o f attack to one o f his colleagues, w h o then s c r a m b l e d to implement the master's suggestion. The others w a i t e d patiently for ErdSs to have a flash of insight about their problem. Sometimes w h e n ErdSs raised his head, he fooled them. They leaned forward in anticipation of a tip. But instead of sharing a mathematical inspiration, he uttered an aphoristic s t a t e m e n t having to do with d e a t h - 'Soon I will be cured of the incurable disease o f life,' or 'This meeting, like life, will soon come to an end, b u t the meeting w a s much more p l e a s a n t ' - - a n d then b o w e d his head again . . . . " Possessing a trump card can be a burden and a mixed blessing; it's not necessarily easy to avoid using it in manipulative ways. But Erd6s didn't even try. As the epigraph to C h a p t e r 3, Hoffinan quotes a n o t h e r a c q u a i n t a n c e o f Erd6s: "My own g r e a t e s t d e b t to E r d 6 s arises from a c o n v e r s a t i o n 50 y e a r s ago in the Hotel P a r c o d e l Principi in Rome. He c a m e up and surp r i s e d me by saying, 'Guy, veel you have a coffee?' I don't d r i n k m u c h coffee, b u t I w a s intrigued as to w h y t h e g r e a t m a n h a d singled m e out. Coffees w e r e a dollar each, a b o u t s t a n d a r d to-

day, b u t t h e n it s e e m e d a small fortune. When w e got o u r coffee, Paul said, 'Guy, you a r e eenfeeneetely reech; l e n d m e $100.' I w a s amazed, n o t so m u c h at t h e request, but r a t h e r at m y ability to satisfy it. Once again, E r d 6 s k n e w me better than I know myself..." P e r h a p s I m i s s e d something; I fail to s e e t h e c h a r m in that. What I s e e is E r d 6 s ' s t e n d e n c y to m a n u f a c t u r e scenarios in w h i c h he called the s h o t s (as well as a t t e n t i o n to himself). In fact, it s e e m s as t h o u g h s o m e h o w the p o s i t i o n he o c c u p i e d in m a t h e m a t i c a l circles b e c a m e s u c h that w h a t he said w a s alm o s t defined as wisdom. The wors h i p p e r s s u r r o u n d i n g Erd6s a c c e p t e d his manipulations. Expecting, o r assuming, is a n o t h e r dodge. Your child says, "I c a n ' t w a i t 'til w e go to Disneyland," w h e n no s u c h trip h a d b e e n promised. A s a l e s p e r s o n says, "It's so difficult, isn't it, to d e c i d e w h i c h one," w h e n you h a d n ' t i n d i c a t e d you w e r e planning to buy a n y one. A n d Erd6s, w i d e - e y e d and open-brained, s h o w s u p on y o u r d o o r s t e p at 8:30 A.M., "Let k b e the greatest i n t e g e r . . . ." T h e b u r d e n o f breaking t h e bizarre a n d r i d i c u l o u s gambit is p u t on you. Along with this "expecting" might go being generally u n a w a r e o f reality and, again, expecting others to "let his reality overtake" theirs. Pages 59-60, d e s c r i b i n g the first meeting o f t h e 17year-old E r d 6 s and the 14-year-old future collaborator, Vaszonyi, at t h e latter's f a t h e r ' s shoe store: "Erd6s p r o v e d to b e a ' w e i r d o ' (in the w o r d s of Kathy, the s h o e s t o r e s a l e s p e r s o n ) f r o m t h e start. He k n o c k e d on the s t o r e door, w h i c h w a s no m o r e the c u s t o m in Hungary t h a n it w a s here. After t h a t unusual civility, he d i s p e n s e d with all int r o d u c t i o n s and conversational pleasantries a n d c h a r g e d to the back. " ' G i v e m e a four-digit n u m b e r , ' he demanded." Page 16: "Like all of Erd6s's friends, G r a h a m w a s c o n c e r n e d a b o u t his drugtaking. In 1979, Graham b e t E r d o s $500 that he couldn't stop taking amphetamines for a month. Erd6s a c c e p t e d the challenge, a n d went cold t u r k e y for 30 days. After Graham paid up---and w r o t e the $500 off as a business e x p e n s e - E r d 6 s said, 'You've s h o w e d m e I'm n o t an addict. But I didn't get any w o r k

done. I'd get up in the m o r n i n g and s t a r e at a b l a n k p i e c e of p a p e r . I'd have no ideas, j u s t like an o r d i n a r y person. You've s e t m a t h e m a t i c s b a c k a m o n t h . ' " ("Is that s u p p o s e d to b e funny?" a s k e d m y 12-year-old son.) Page 73: "Erd6s didn't hesitate to use his special language o u t s i d e mathematical circles. F o r instance, he once a s k e d B a r b a r a Piranian, p r e s i d e n t of the League of Women Voters in Ann Arbor, Michigan, 'When will you b o s s e s take the vote a w a y from the s l a v e s ? ' " These shortcomings are due, not to a n y mysterious genius, b u t to psychological problems and p e r s o n a l i t y defects. And my point is not to trash Erd6s, n o r to imply that a mathematician has to b e a perfect (or even a good) person, b u t to emphasize that it is imp o r t a n t for society to p l a c e things in perspective. What w a s great about Erd6s and w h a t wasn't n e e d to be recognized (even if there are connections b e t w e e n the two). H o f f m a n ' s b o o k could have done m o r e analyzing; that might have been m o r e socially responsible, both to the world and to the mathematical commtmity. It w o u l d also have b e e n m o r e in the interests o f truth.

the beginning or as an appendix, o f a s u m m a r y biography. I might also quibble with t h e b o o k ' s title; it m a k e s E r d 6 s s o u n d autistic. I w o n d e r w h e t h e r it w a s the e d i t o r w h o insisted u p o n t h a t tit i e - - p e r h a p s to t h e end o f getting a quote f r o m Oliver Sacks. F o r me, the s a m e title with t h e w o r d "only" elimin a t e d w o u l d have rung t r u e r (even c o n s i d e r i n g that m a t h is n o t only numbers, as t h e g e o m e t r y p r o b l e m des c r i b e d o n page 74 would indicate). There w a s a smattering o f m a t h e m a t i c a l m i s t a k e s (perhaps no m o r e than in t h e average m a t h t e x t b o o k , w h i c h is a d m i r a b l e considering t h a t Hoffman is admittedly n o t a m a t h e matician). At times Hoffman s e e m e d to confuse infinity with aleph-null. (But he did n o t fall into the trap o f over-explaining p r i m e numbers.) The m a t h e m a t i c a l m i s t a k e s are so c o m m e n d a b l y few in n u m b e r that I can c o m p l e t e l y list t h e m (or at least the o n e s I n o t i c e d ) in this review: Page 107, next-to-bottom line: Isn't the other diagonal the principal diagonal? Page 200: Is Euler's Conjecture a generalization of Fermat's Last Theorem? I think not; for w h i c h n d o e s that conj e c t u r e b e c o m e FLT? Page 217: Hoffman begins to summarize calculus: "The infmitely s m a l l c o m e s into play in 'integration' " - - a s t h o u g h it didn't c o m e into p l a y in differentiation as well. (On that s a m e page, b y t h e way, a p p e a r s the only typo, o t h e r than m i s p l a c e d q u o t a t i o n marks, t h a t I could locate---1/2n ins t e a d of the correct 1/2n. Given all t h e m a t h in t h e book, this is a c o m p l i m e n t to the c o p y - e d i t o r a n d printer.)

the n u m b e r of t e r m s a p p r o a c h e s infinity, the s u m is e x a c t l y 1." I would rep l a c e " a p p r o a c h e s " with "is." Hoffman s e e m s to have a misconception a b o u t irrational versus trans c e n d e n t a l numbers. Page 212: "repeating b u t n o n t e r m i n a t i n g decimals like the square r o o t of two." Huh? And on p a g e 229, t r a n s c e n d e n t a l n u m b e r s a r e d e f i n e d as "nonrepeating, nonterminating decimals"; again, w h a t a b o u t the square r o o t of two? I was also c o n f u s e d b y page 266. "What is t h e largest n u m b e r o f edges a graph can have?" Isn't any polygon, with as m a n y e d g e s as one pleases, a graph? Is t h e r e p e r h a p s a missing adjective b e f o r e "graph"? But m a t h e m a t i c a l m i s t a k e s do n o t p e r m e a t e t h e b o o k , and I'm sure they'll b e c o r r e c t e d in t h e n e x t edition. The b o o k m o s t definitely c o n v e y s the spirit of mathematics and mathematicians and, in doing so, is a p p r o p r i a t e l y gentle, non-invasive, respectful, humorous, and humble. In fact, it s e e m s c l e a r that, while n o t a m a t h e m a t i c i a n himserf, Hoffman h a s in s o m e w a y b e e n bitten by t h e bug. And h e r e ' s a n o t h e r suggestion, purely for fun: H o w a b o u t refining the defmition o f "Erd6s number": In the c a s e of E r d 6 s n u m b e r greater t h a n one, keep it the same; in the case o f E r d 6 s n u m b e r one, m a k e it the reciprocal of the n u m b e r o f p a p e r s the pers o n w r o t e with E r d 6 s ?

Can a Non-Mathematician Write About Math? P e r h a p s the greatest m e a s u r e of a b o o k ' s w o r t h is the e x t e n t to which it stimulates a reader's thinking, or puts into perspective its r e a d e r ' s sense of self. I came away with a s h a r p e n e d "A Little Bit Crazy"? s e n s e of m y own strengths a n d weakThe title o f t h e last c h a p t e r is "We n e s s e s as a math person, a few mathem a t h e m a t i c i a n s a r e all a little bit matical ideas for a p r o b l e m I'm workcrazy" (a quote f r o m Landau), and t h a t ing on, a n d a burst o f mathematical relatively s h o r t c h a p t e r goes on to reenergy different from w h e n I've j u s t flect on r e a s o n s w h y that might b e e n reading a m a t h b o o k o r article. I w a s also interested in the What was great about ErdSs and b e so. But I t h i n k that this "craziness" is no m o r e an attribute o f several passages which c o m p a r e what wasn't need to be recog- m a t h e m a t i c i a n s than, say, o f m a t h to other sciences, a n d I enphysicists, artists, writers, o r j o y e d adding my o w n specula- nized (even if there are conneceven "regular" people. Yes, E r d 6 s tions (such as disagreeing with tions between the two). h a d bizarre ways, likewise H a r d y Erd6s's "Math is the only infinite and GSdel, b u t so did Beethoven field"). and Van Gogh. Also, t h e r e are p l e n t y The w a y s in which the b o o k could Also, h e r e ' s a m i n o r suggestion for o f m a t h e m a t i c i a n s w h o are n o t have b e e n i m p r o v e d ( b e s i d e s probing the e n d o f p a r a g r a p h one on t h a t page: "crazy," even "a little bit." I do believe m o r e d e e p l y into E r d 6 s ' s social dys- "1/2 + 1/4 + 1/8 + . . . No m a t t e r h o w there are aspects o f m a t h e m a t i c s that functionalism) seem m i n o r in compari- m a n y finite t e r m s you c h o o s e to s u m could t e n d to m a k e a m a t h e m a t i c i a n "a son. First, the chronology s o m e t i m e s in this series, you'll never get the numlittle bit crazy"---or a t t r a c t to m a t h s e e m e d confusing; that situation might b e r 1, although you can get as c l o s e to p e o p l e w h o a r e already "a little bit have b e e n remedied by the addition, at 1 as y o u want. But in the limit, w h e r e

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crazy." But, again, that's true of every discipline. Also, there exists the myth of the Mad Doctor, the Suffering Artist, and now the Crazy Mathematician. In that last chapter, Hoffman should have balanced his portrayal of "a little bit crazy" with speculations like these (and then perhaps elaborated on those properties of math that might tend to make one "a little bit crazy"). As it is, I feel that Hoffman has unfortunately contributed to a less truthful public image of mathematicians, and this is no exaggeration. For example, already a review of the b o o k (Washington Post, Aug. 2) refers to mathematicians as "psychologically fragile". (The review was worded to include all mathematicians.) That review goes on to rather grossly exaggerate "a little bit crazy." To summarize: I was often moved by Hoffman's main character Erd6s but I found that I was also on my guard against such empathy, and antagonized by ErdSs's faults. However, I was not on guard against being moved by the book itself, by its depiction of its more universal character, mathematics. Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 USA

GOODBYE,DESCARTES The End of Logic and the Search for a New Cosmologyof the Mind by Keith Devlin NEW YORK, JOHN WILEY& SONS, INC., 1997, x + 301 pp. US $27.95, ISBN 0-471-14216-6

REVIEWED BY DON FALLIS

ince Aristotle, attempts to understand h o w h u m a n beings reason have been dominated by the study of logic. However, despite the great advances in logic over the last few millennia, the scientific study of human reasoning has not progressed nearly as far or as fast as have the natural sci-

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ences. In Goodbye, Descartes, Keith Devlin tries to explain why it has been so difficult to give a good scientific account of h o w haman beings reason and communicate. His diagnosis is "that the existing techniques of logic and mathematics--indeed of the traditional scientific method in general--are inadequate for understanding the human mind" (viii). (Aristotle also pointed out that formalization has its limits and that we should only "expect that amount of exactness in each kind which the nature of the subject admits" ([1], 8).) Devlin gives an extensive account of the development of logic from Artistotle to Frege and then describes the application of logic to m o d e m linguistics and artificial intelligence. This a c c o u n t (which takes up over half of the book) provides an entertaining, not overly technical, and fairly accurate picture of the history of logic. He does m a k e a few sweeping generalizations that might bother historians of philosophy, but they are forgivable in a popular presentation. (In fact, in order to be an effective teacher, one often has to be willing to say a few false things.) According to Devlin, the guiding principle in the development of logic has been the view that "rational ('logical') thought is a kind of mental calculation that follows certain prescribed rules, in many ways not unlike arithmetic" (4). In line with the title of the book, I will refer to this view as the Cartesian model of h u m a n reasoning. It is this model that Devlin thinks has just about reached its limits. Early this century, logicians were having great success capturing the rules of correct mathematical reasoning with the Cartesian model. (There is some dispute, however, over whether o r not these rules describe w h a t mathematicians actually do; see, e.g., [2].) Following up on this success, researchers used the Cartesian model to try to understand h o w h u m a n beings in general reason and use language. One way in which this research has been carried out is trying to program computers to reason and use language as h u m a n beings do. After all, if human reasoning is just "a kind of mental calculation," it should be possible to pro-

gram a c o m p u t e r to perform the same calculations. Unfortunately, after some great early successes, m o d e m linguistics and artificial intelligence have rtm into difficulties. Devlin presents a number of examples that seem to cause serious problems for the Cartesian model. For example, although human beings correctly interpret the following two sentences (from chapter 1) without a second thought, it is hard to imagine h o w to write down a set of formal rules which would allow a computer to do the same. Safety Goggles Must Always Be Worn Inside the Building. Dogs Must Always Be Carried on the Escalator. There is nothing about the syntactic structure or the meaning of the w o r d s involved which explains w h y you do not have to go find a dog to carry whenever you want to ride the escalator. Another example of this sort can be found in the recent gangster film, Donnie Brasco. When the title character is asked what the phrase "forget about it" means, we learn that the phrase can signal agreement, disagreement, enthusiasm, and that "sometimes it just means 'forget about it.' " I t is not easy for anyone who is not one of the gangsters, much less for a computer, to keep from getting confused. By giving an analysis of examples like these, Devlin presents a strong case that the Cartesian model will not ultimately be able to provide a successful scientific account of h o w human beings reason and communicate. It is important, however, to clarify exactly what goal(s) this model is falling to achieve. First, we might hope that our model of h a m a n reasoning would capture the actual reasoning processes of h u m a n beings. Unfortunately, a n u m b e r of experiments (see, e.g., [3]) indicate that even when people act in accordance with certain formal rules, they often do not do so consciously. Second, we might hope that people act in a c c o r d a n c e with the formal rules given by our model, even if only unconsciously. In this case, our model might allow us to simulate h u m a n reasoning with a computer. Unfortun-

ately, this p a r t i c u l a r goal has t u r n e d c o m m o n s e n s e reasoning [that] w e use o u t to be unattainable o u t s i d e of fairly in a n e v e r y d a y context" (267). We t e n d r e s t r i c t e d domains, s u c h a s c h e s s play- to ignore t h e relatively a b s t r a c t statising. (Researchers c o n t i n u e to have tical d a t a a b o u t the frequency of b l a c k s u c c e s s in s o m e of t h e s e r e s t r i c t e d do- c a b s a n d blue c a b s and to focus premains; see, e.g., [4].) Nevertheless, d o m i n a n t l y on the eyewitness report. even if w e do not k n o w e n o u g h of the F u r t h e r m o r e , by giving m u c h m o r e rules of h u m a n r e a s o n i n g in enough weight to t h e evidence of o u r o w n e y e s detail to simulate h u m a n reasoning and, b y extension, to the e y e w i t n e s s rewith a computer, w e might still have a p o r t s of others, w e are reasoning in a fairly successful scientific a c c o u n t of w a y "that evolution and p e r s o n a l hish u m a n reasoning. Unfortunately, as tory have d e v e l o p e d to a p o i n t w h e r e t h e following e x a m p l e ( f r o m c h a p t e r it is e x t r e m e l y reliable" (267). 11) indicates, even this m a y b e b e y o n d Although context is clearly an imt h e s c o p e o f the Cartesian model. portant notion, researchers have found A t a x i c a b is involved in a hit-and- it very difficult to generate context out run accident. The Black Cab C o m p a n y of the ']areclse, eternal, context-free o w n s 85% o f the c a b s in t o w n and the rules" (262) of the Cartesian model. Blue Cab C o m p a n y o w n s t h e rest. An What w e can do instead, however, is e y e w i t n e s s to the accident, w h o can adopt context as an incompletely anac o r r e c t l y distinguish b l a c k c a b s from lyzed primitive in our scientific study of b l u e c a b s 80% of the time, s a y s that the human reasoning. In the last few chapc a b involved in the a c c i d e n t w a s blue. ters, Devlin describes how a n u m b e r of Is it m o r e likely that the c a b involved c o n t e m p o r a r y research projects, such as in the a c c i d e n t was blue o r t h a t it w a s Jon Barwise's Situation theory, have inb l a c k ? Probability t h e o r y is n o w a ma- corporated the notion of context in this j o r c o m p o n e n t of t h e C a r t e s i a n m o d e l way. After reading Good-bye, Descartes, o f h u m a n reasoning; a s t r a i g h t f o r w a r d I am inclined to share Devlin's optimism a p p l i c a t i o n of a t h e o r e m o f p r o b a b i l i t y about the potential of these research t h e o r y s h o w s that it is m o r e likely t h a t projects to increase our understanding t h e c a b was black. Nevertheless, m o s t of h o w h u m a n beings reason and comp e o p l e (at least t h o s e w h o have never municate. However, in the final chapter, t a k e n a c o u r s e in p r o b a b i l i t y theory) he m a k e s a couple of claims (one of a r e inclined to say t h a t it is m o r e likely which a p p e a r s to be the thesis of his t h a t the cab was blue. In o t h e r words, book) that go beyond what is waITanted m o s t p e o p l e do n o t act in a c c o r d a n c e by the evidence he provides. with t h e formal rifles given by the First, in his analysis of t h e t a x i c a b C a r t e s i a n model. case a n d a n u m b e r of similar e x a m Of course, the failure o f the Car- ples, Devlin is not content j u s t to ext e s i a n m o d e l would not m e a n that w e plain w h y p e o p l e r e a s o n as t h e y do. He have to give up on o u r goal o f m~der- claims t h a t "it is n o t at all i r r a t i o n a l to standing h o w human beings reason. In r e a s o n in this way" and that "you c o u l d fact, Devlin indicates h o w t h e notion of only b e a c c u s e d of irrationality if, context can be used to clarify s o m e of faced with a clear explanation o f the t h o s e e x a m p l e s that c a u s e d p r o b l e m s a p p l i c a t i o n of Bayes' law in this case, for the Cartesian model. The gangsters y o u refuse to change y o u r original in Donnie Brasco are n o t c o n f u s e d by evaluation o f the e y e w i t n e s s ' s evithe multifarious meanings of "forget dence" (265). However, there is an ima b o u t it" b e c a u s e the c o n t e x t in which p o r t a n t distinction b e t w e e n explaining the p h r a s e is used gives it a fairly un- h o w p e o p l e r e a s o n and suggesting h o w a m b i g u o u s meaning. Similarly, w e are p e o p l e o u g h t to reason. Although t h e n o t misled into thinking t h a t w e have Cartesian m o d e l m a y have failed a s a to f'md a dog to carry if w e w a n t to ride d e s c r i p t i v e t h e o r y of h u m a n reasoning, the escalator b e c a u s e the aforemen- p r o b a b i l i t y t h e o r y is still a v e r y p o w t i o n e d sentence e v o k e s a particular erful n o r m a t i v e theory. ( D e s c a r t e s ' s context. Also, we d o n o t t e n d to o b e y Rules f o r the Direction of the Mind the rules of probability t h e o r y in the w e r e originally offered only a s guidet a x i c a b case b e c a u s e w e e m p l o y "the lines for h o w p e o p l e should r e a s o n . )

We ignore the guidelines of probability t h e o r y at o u r o w n peril. Although abstract statistical d a t a m a y not have b e e n an i m p o r t a n t factor in our evolutionary history, in the age o f information statistical d a t a is often crucial to making g o o d decisions. It w o u l d s e e m to be foolish to s t a n d p a t with the reasoning skills t h a t h a p p e n e d to be selected for on the savannah. Second, Devlin's ultimate conclusion is "that w e n e e d to l o o k for w a y s o f u n d e r s t a n d i n g t h a t go b e y o n d the limits of the t r a d i t i o n a l m e t h o d s of science and m a t h e m a t i c s and that challenge s o m e of the b a s i c a s s u m p t i o n s o f science going b a c k to Plato, Aristotle, Descartes, Galileo, and Bacon" (282). Admittedly, the s t o r y he tells clearly indicates that w e n e e d to try to unders t a n d h u m a n r e a s o n i n g a n d communication in a n e w way. Nevertheless, there is an i m p o r t a n t distinction bet w e e n (a) r e p l a c i n g one scientific m o d e l (in this case, the Cartesian model) with a n o t h e r and (b) drastically modifying t h e "traditional scientific method." In t h e early 19th century, the d o m i n a n t scientific m o d e l required a purely m e c h a n i c a l a c c o u n t of all physical p h e n o m e n a . Mechanism, like the Cartesian m o d e l of h u m a n reasoning, ran into difficulties (in particular, with e l e c t r o m a g n e t i c phenomena); scientists w e r e f o r c e d to a d d the notion of fields to their m o d e l of the physical universe. However, in a b a n d o n i n g mechanism, scientists did not t h e r e b y a b a n d o n the "traditional scientific method." E v e n if the shift a w a y from the Cartesian m o d e l t u r n s out to be a s i m p o r t a n t a n d far-reaching in its implications as the shift to Maxwell's Field theory, it is still j u s t g o o d old scientific p r o g r e s s in the tradition o f "Descartes, Galileo, and Bacon." Part o f the r e a s o n that Devlin considers the shift a w a y from t h e Cartesian m o d e l to b e such a radical d e p a r t u r e f r o m t h e "traditional scientific method" is that he s e e s an unp r e c e d e n t e d role for m a t h e m a t i c s in the scientific s t u d y of h u m a n reasoning. (Being a m a t h e m a t i c i a n and a logician, Devlin d o e s n o t w a n t to banish m a t h e m a t i c s a n d logic from the s t u d y o f h u m a n r e a s o n i n g altogether.) However, despite his claims to the contrary,

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the role that he sees for mathematics in the scientific study of human reasoning is not unlike the role that mathematics has always played in science. For instance, Devlin says that "the goal is to use mathematics together with other ldnds of reasoning to gain and increase our understanding of various complex human phenomena" and that, thus, the mathematics does not have to be "completely rigorous" (289). However, scientists have never been overly concerned with rigor when they apply some mathematics to a scientific problem (consider infinitesimals or, more recently, the Feynman path integral; see, e.g., [5]); they are satisfied as long as the mathematics increases their understanding of (and improves their ability to make predictions about) the phenomena under investigation. Despite these criticisms, the story that Devlin tells is important and he tells it well. I highly recommend Goodbye, Descartes as an introduction to this material--even though the implications of the story are not quite as dramatic as his rhetoric suggests. His title notwithstanding, not even Devlin would say that the scientific study of human reasoning and communication has seen "The End of Logic." REFERENCES

[1] Aristotle, Nicomachean Ethics, translated by H. Rackham, (Cambridge: Harvard University Press, 1934). [2] Don Fallis, "The Epistemic Status of Probabilistic Proof," Journal of Philosophy, April 1997, 165-186. [3] Antoine Bechara, Hanna Damasio, Daniel Tranel, and Antonio R. Damasio, "Deciding Advantageously Before Knowing the Advantageous Strategy," Science, 28 February 1997, 1293-1295. [4] Gina Kolata, "Computer Math Proof Shows Reasoning Power," Math Horizons, February 1997, 22-24. [5] Mark Steiner, "Mathematical Rigor in Physics," in Proof and Knowledge in Mathematics, edited by Michael Detlefsen (New York: Routledge, 1992), pp. 158-170. School of Information Resources University of Arizona Tucson, AZ 85719 USA

Indiscrete Thoughts by Gian-Carlo Rota (edited by Fabrizio Palombi) BOSTON, BASEL, BERLIN; BIRKH,~,USER, 1997, HARDCOVER US $36,50 XXII + 280PP. ISBN 0-8176-3866-0 REVIEWED BY LAWRENCE ZALCMAN

he title of this Rota omnibus inevitably suggests a sequel to Discrete Thoughts, that wonderful collection of essays, occasional pieces and obiter dicta by Mark Kac, Jack Schwartz, and Rota. The heady cocktail of Rota's brilliance, Schwartz's penetrating insight, and Kac's wisdom made that volume required reading for all lovers of mathematics. The current volume reveals that Rota's coruscating intellect has lost none of its sparkle, but inevitably one misses the balance provided by his former coauthors. In point of fact, Indiscrete Thoughts (as hinted broadly by Reuben Hersh in one of the two forewords to the book and made explicit by the publishers on the back cover, the "Indiscrete" of the title is really meant to be understood as "Indiscreet") is a discrete union of memoirs, philosophical pieces, collected jottings and book reviews, held together by the thread of a common authorship. Here and there, a fugitive piece from Discrete Thoughts (such as the eulogy "Stan Ulam" and a couple of very short book reviews 1) has also managed to creep in. Not unexpectedly, the quality varies rather widely. After all, only the mediocre are always at their best; and Rota is anything but mediocre. The first part of Indiscrete Thoughts, entitled "Persons and Places," is largely a memoir: of Princeton and Yale (and, to a lesser extent, MIT) in the 1950s, and of Stan Ulam. This material is every bit as gripping as when it first appeared in the Notices and The InteUigencer. Best of all are the sharpeyed portraits of mathematicians Rota has known. Rota hates hagiography and, going beyond "warts and all,"

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adopts an "all the warts" approach, especially to such Princeton luminaries as Church, Feller, Artin, and Lefschetz. Although he has confessed (p. 195) to "an occasional embellishment of reality," the treatment here is, so far as I can tell, scrupulously fair. I was particularly pleased to see my old teachers John Kemeny (p. 7) and A1 Tucker (p. 23) given their due. The piece on Yale is largely the story of Dunford and Schwartz's Linear Operators, with some necessary historical background and an occasional amusing digression thrown in. There is also an admiring treatment of Dunford and an extremely warm appreciation of Schwartz, an altogether fitting expression of Rota's filial piety. for his thesis advisor. Most fascinating of all is the discussion of Ulam, which comes in three installments: an imaginary dialogue, entitled "The Barrier of Meaning"; what appears to be a graveside eulogy; and a moving memoir, one of the very best of its kind, entitled "The Lost Caf&" If Schwartz was Rota's mentor, Ulam was his guru. The only picture in the book is a 25-year-old photo of Rota and Ulam standing (almost) arm-inarm and smiling (the former beatifically, the latter diabolically). "The Lost Cafd" is thick description at its fmest. Rarely, if ever, has a contemporary mathematician been given so rounded and perceptive a portrait by someone who knew him so well and saw so deeply into his soul. According to the editor of this volume (p. 269), Rota paid dearly for the publication of this piece. For the reader, at least, it was worth the price. Sandwiched between the memoirs is a piece entitled "Combinatorics, Representation Theory, and Invariant Theory," part revisionist history and part polemic, liberally seasoned throughout with mathematical gossip. My favorite plum from this Christmas pudding is Frobenlus's letter of recommendation for Hilbert (then a professor at KOnlgsberg), who was being considered for a position at GSttingen: "He is rather a good mathematician, but he will never be as good as Schottky."

1Compare the reviews of Quine, Ontological Relativity (p. 255) and Hofmann, Leibniz in Paris (p. 256) in Indiscrete Thoughts with p. 264 of Discrete Thoughts.

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All in all, mathematicians will want to read and reread all the essays in Part I of Indiscrete Thoughts. The second and longest section of the book, entitled "Philosophy: A Minority View," includes the frequently reprinted essay "The Pernicious Influence of Mathematics Upon Philosophy" (cf. Schwartz's 1962 essay "The Pernicious Influence of Mathematics on Science," reprinted in Discrete Thoughts); three essays entitled "The Phenomenology of Mathematical X," where X = Truth, Beauty, and Proof; and several pieces on phenomenology, one of Rota's principal (nonmathematical) interests. Rota is one of the very few research mathematicians working in core mathematics (not logic) who is involved with philosophy in a serious way, so his essays on the philosophy of mathematics command particular attention. "The Pernicious Influence..." seems to have stirred up a real hornets' nest within the community of analytic philosophers, but mathematicians are likely to find it largely uncontroversial. Still, it may be noted with some amusement that Rota's own account (p. 75) of the collaboration of Everett and Ulam on the theory of branching processes provides a refutation of his later claim (p. 99) that "no mathematician will ever dream of attacking a substantial mathematical problem without first becoming acquainted with the history of the problem," and that, pace Rota (p. 96), we have it on excellent authority 2 that mathematicians do indeed use the axiomatic method as a means of discovery and proof. Mathematicians are likely to fend "The Phenomenology of Mathematical X" more promising reading. I was surprised to discover myself disagreeing with Rota, time after time, on the aesthetic judgments expressed in the essay on Mathematical Beauty. Here are just a few points where we disagree.

Claim: "It is . . impossible to find beautiful proofs of theorems that are not beautiful." (p. 122) Comment: I offer the following counterexample. .

T h e o r e m : Let S be a countable set. Then S has uncountably many subsets all of whose pairwise intersections are finite (or empty). Proof: Let S be the rationals, and for each real number take a sequence of distinct rationals converging to it. Claim: "[T]he prime number theorem is a beautiful r e s u l t . . , but none of its proofs can be said to be particulariy beautiful." (pp. 122-3) Comment: Presumably, Rota hadn't seen Zagier's arrangement 3 of D.J. Newman's proof when he wrote this. (Though published only in 1997, it has been floating around for close to 20 years.) Claim: "Morley's theorem, stating that the adjacent trisectors of an arbitrary triangle meet in an equilateral triangle is unquestionably surprising, but neither the statement nor any of the proofs are [sic] beautiful." (p. 123) Comment: De gustibus non est disputandum, but this judgment seems idiosyncratic in the extreme. To my mind, Morley's theorem is one of the most beautiful theorems in plane geometry, surely the most beautiful discovered in modern times. Claim: "The basic notions of this field [scil., category theory], such as adjoint and representable functor, derived category, and topos, have carried the day with their beauty, and their beauty has been influential in steering the course of mathematics in the latter part of this c e n t u r y . . . " (p. 124) Comment: Carried the day? I am speechless. Claim: "The theory of the Lebesgue integral, viewed from the vantage point of one hundred years of functional analy-

sis, has received more elegant presentations than it deserves." (pp. 124-5) Comment: Rota really seems to have it in for Lebesgue (cf. p. 225). Here he breaks Rule Number Three (p. 205) of his "Ten Lessons for the Survival of a Mathematics Department" and comes close to contradicting the first of the claims listed above. Anyhow, hasn't Saul Bellow taught us that "Nobody deserves anything"? Claim: "[M]athematical research does not strive for beauty." (p. 128) Comment: Gauss and Jacobi must be spinning in their graves. You get the idea. These essays are nothing ff not engaging. The more purely philosophical essays are another story. I fear that most mathematical readers will fend them impenetrable. Hersh says as much in his foreword (p. xi) when he assures the reader that "You're within your fights if you decide to take phenomenology in small doses" and that "you can look forward to the easy delights of Indiscrete Thoughts coming next." Rota's editor, the philosopher Fab~zio Palombi, is even more explicit (p. 268), pronouncing the essay "The Primacy of Identity" "hard to read" and "doubt[ing] that any reader who is not a dyed-in-thewool Heideggerian can understand his chapter," "Three Senses of 'A is B' in Heidegger." If so, why include these pieces in the book? Ditto for the nasty little note entitled "The Barber of Seville, or the Useless Precaution," characterized by Palombi (p. 269) as "a rude letter sent to a well-meaning and dignified German philosopher of science." Including such essays in a volume whose readership seems guaranteed to consist almost exclusively of mathematicians can only be described as overt self-indulgence. Things come back down to earth

2Hermann Weyl, "A half-century of mathematics," American Mathematical Monthly 58 (1951), 523-553. "One very conspicuous aspect of twentieth-century mathematics is the enormously increased role which the axiomatic approach plays. Whereas the axiomatic method was formerly used merely for the purpose of elucidating the foundations on which we build, it has now become a tool for concrete mathematical research." (p. 523). Cf. Andr6 Weft, "The Future of Mathematics," American Mathematical Monthly 57 (1950), 295-306. "Only a few backward spirits still maintain the position that the mathematician must forever draw on his 'intuition' for new, alogical or 'prelogical' elements of reasoning. If certain branches of mathematics have not yet been axiomatized.., this is simply because there has not yet been the time to do it." (p. 297) 3D. Zagier, "Newman's short proof of the Prime Number Theorem," American Mathematical Monthly 104 (1997), 705-708.

VOLUME21, NUMBER2, 1999 73

with the final section of the book, listed as "Readings and Comments" in the table of contents but clearly labeled "Indiscrete Thoughts" on page 193 and at the top of subsequent evennumbered pages. The first two pieces, an after-dinner speech entitled "Ten Lessons I Wish I Had Been Taught" and "Ten Lessons for the Survival of a Mathematics Department," are solid gold. A chapter entitled "A Mathematician's Gossip" consists mostly of oneliners along the lines of the "Discrete Thoughts" of Discrete Thoughts, many of them extended (not always successfully) to five, ten, twenty, or even thirty lines. A typical example (the elaboration of which continues for some 28 lines) is "The history of mathematics is replete with injustice." (This can be generalized to "The history of X is replete with injustice.") One also finds a number of substantive comments on areas of contemporary mathematics here. There follow twenty-odd pages of book reviews, some long, many short. The short ones are mostly dispensable, though that judgment most defmitely does not apply to Rota's review of Husserl's Vorlesungen fiber Bedeutungslehre. ("Compared to Husserl's, the paradise that Cantor bequeathed to us is a limbo.") The longer reviews are all worth reading, though it is infuriating to have to turn to the end-notes to discover which book is being reviewed. Rota belongs to a generation that takes pride in having learned Latin and likes to let the world know it, so the reader should be prepared to field the occasional quotation from Cicero (pp. 212, 234) and Horace (p. 249), as well. On the whole, one has the feeling that Rota could have been better served by his editor. Palombi's epilogue is a curious mix of awed breathlessness and pointed criticism, in which he calls Rota a "phenomenologist manqu&" Does Palombi know the meaning of the word manqud? If so, "With friends like these . . . . " Palombi notes correctly Rota's penchant for appropriating ideas and for-

mulations from other authors (especially Croce and Ortega y Gasset). He does not, however, always follow up on his generally correct instincts. Thus, in connection with the chapter "The Barrier of Meaning," he writes that "Rota has assured me that the conversation with Stan Ulam described in this chapter really took place, but I have some doubts." For the record, the little piece of Ulam-Rota dialogue in lines -9 to -7 ofp. 58 appears at the very bottom of p. 2 of Discrete Thoughts, spoken by Frank Ramsey to Ludwig Wittgenstein. Similarly, commenting on the very last sentence of the book ("When pygmies cast such long shadows, it must be very late in the day," Rota's snapshot review of a tome entifled, not very promisingly, Recent Philosophers), Palombi writes, "I was at first convinced that this sentence had been lifted out of Chargaff..., and I was ready to add a footnote to that effect, but to my chagrin I could not locate the sentence in any of Chargaffs writings." In point of fact, it appears in the middle of the first column of page 641 of what Chargaff has called4 "the most widely read of my articles," "Preface to a Grammar of Biology," Science 172 (1971), 637-642. Amazingly, Rota actually seems to have improved on the original, which reads "That in our days such pygmies throw such giant shadows only shows how late in the day it has become." (Can I be the only one to have noticed that the shadows are just as long in the morning---or are we all late sleepers?) In my (overwhelmingly favorable) review5 in The InteUigencer of Discrete Thoughts (often quoted by the publishers in advertisements for that book, without proper attribution), I voiced my complaints about the numerous misspellings and other solecisms that plague the text, the lack of any indication as to where the individual pieces in the book had originally appeared, the repellent dust jacket design, and the grossly inflated price. The publishers seem to have taken the last two criticisms to heart: the cover design of Indiscrete Thoughts does not offend the

4In his autobiography, Heraclitean Fire, Rockefeller Univ. Press, New York, 1978, p. 109. 5The Mathematical Intelligencer 12 (1990), no. 3, 81-83.

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THE MATHEMATICALINTELLIGENCER

eye, nor does the price offend the pocketbook. (My copy cost me a mere 60% of what Discrete Thoughts was selling for a few years ago.) There are also far fewer misprints (though one has "distintuished" on p. xxi, "Bahgavad Gita" on p. 253, and "Truth" for "Proof' on p. 270, 1. 12), misspellings ("Marcus" for Markus p. 28, "Blavatzky" for Blavatsky p. 79, "Zaslawsky" for Zaslavsky p. 252), malapropisms ("salntlihood," p. 127, isn't a word; presumably saintliness is intended), and split infinitives (two occur on line -12 of page 237). But, unaccountably, there is still no listing of where those of the papers which axe reprinted here first appeared. Read Indiscrete Thoughts: for its account of the way we were and what we have become; for its sensible ~/dvice and its exuberant rhetoric; and, yes, for the phenomenology, too, at least as it relates to the nature and practice of mathematics. Department of Mathematics and Computer Science Bar-Ilan University 52900 Ramat-Gan Israel e-mail: [email protected]

Principles and Practice of Mathematics by COMAP (Consortium for Mathematics and its Applications) NEW YORK: SPRINGER-VERLAG, 1997, xi + 686 pp. ISBN 0-387-94612-8, US $64.95

REVIEWED BY MARTIN ERICKSON

~tor

II

me, the book Principles and Practice of Mathematics, written

under the direction of COMAP (Consortium for Mathematics and its Applications, a nonprofit organization dedicated to improving undergraduate mathematics education), with support from the National Science Foundation, accomplishes something similar to what the movie "N is a Number" (a tribute to Paul Erd6s) accomplishes. It

shows us that mathematics is an evolving art and science, practiced by living people from all nations and wales of life. The book is intended for use as a first-year undergraduate mathematics text. Reading the COMAP book I learned about: an undergraduate summer intern doing cryptology at the Center for Communications Research; the art of M. C. Escher and the 17 plane crystallographic groups; the "one-child" rule in China and the role of probability theory in public-policy decisions; and the design of the space shuttle engine and scheduling theory. At every turn there are interesting--and current--mathematical ideas and applications. There are sections on facial recognition (which may someday help automatic teller machines to identify customers), state lotteries (perhaps informed consumers will be wiser consumers?), and the art of Kenneth Snelson (an illustration of tension vectors). There are spotlights on probability and the law, the classification of finite simple groups, Buckeyballs, "Meals on Wheels," and robot arms. There are discussions of driver's license number check digits, X-ray tomography, genetic algorithms, computer animation, and computer chip design. There are accounts of a programmer analyst using the mathematics of bin-packing to improve efficiency in an automated furniture warehouse, the discovery of Huffman encoding, and the use of fuzzy logic in designing temperature control systems. Much of what is presented is twentieth-century mathematics. It is relevant, it is fascinating, and many teachers may want to share information such as this with every student--math major or not--who comes to class wondering "How do we use mathematics?" The coverage of math topics is broad rather than deep. There are chapters on change (covering difference equations and an introduction to differential and integral calculus), position (coordinate systems and vectors), linear algebra (matrices, eigenvalues and eigenvectors), counting (permutations and combinations), graph theory (cycles, game trees, and heuristic algorithms), analysis of algorithms (timecomplexity and big-oh notation), the

logic and design of "intelligent" devices (Boolean functions and logic circuits), chance (probability and expected value), and abstract algebra (symmetry, groups, and coding theory). The emphasis is on exposition of ideas rather than on rigorous deduction. I assume that the goals of the authors are to (1) give every student (math major and nonmajor) some notion of the spectacular range and applicability of mathematics in the twentieth century, (2) challenge students to think creatively about problems and not just do routine calculations, and (3) provide some math culture background for students going on to major in mathematics. The authors are Frank Giordano~ Chris Arney, Robert Bumcrot, Alan Tucker, Rochelle Wilson Meyer, Paul Campbell, Michael Olinick, and Joseph Gallian. Division of Mathematics and Computer Science Truman State University Kirksville, MO 63501 USA e-mail: [email protected]

teUigencer will be familiar with many of the authors as well as the editor. They will find the range of topics similar to many of those that appear in this jottrnal, too; although mainly taken up with accounts of the historical development of the principal areas of mathematical activity up to the beginning of this century, the volumes also include a variety of other topics such as mathematics education, ethnomathematics, recreational mathematics, and even mathematics on postage stamps. This cooperative effort of one hundred thirty-three scholars collects in one work an epitome of much current scholarship in the history of mathematics. On the whole, it probably gives more complete coverage of the history of mathematics up to the beginning of the twentieth century than any other single reference work (excluding works of many volumes like the Dictionary of Scientific Biography). It should therefore appeal to anyone who is interested in the history of mathematics. How will the mathematicians, students of mathematics, and instructors-the majority of the readers of The

Mathematical InteUigencer--use this

Companion Encyclopediaof the History and Philosophy of the Mathematical Sciences, 2 vols. edited by I. Grattan-Guinness. LONDON AND NEW YORK: ROUTLEDGE, 1994, xiii + 1806 PP., US $199.95, ISBN 0-415-03785-9 REVIEWED B Y PAUL WOLFSON

'hat is mathematics? Is it a sequence of theorems ordered by their proofs? Is it a collection of methods of computation? Is it a language of abstractions? The volumes under review, which survey the historical development of mathematics, encourage us to ask these questions ,and the editor's own answers obviously influenced this work. Ivor Grattan-Guinness has recruited a remarkable collection of historians, mathematicians, and scientists to write this encyclopedia. Readers of The Mathematical In-

W

work? The encyclopedia is divided into fourteen parts: (0) Introduction; (1) Ancient and non-Western traditions; (2) The Western Middle Ages and the Renaissance; (3) Calculus and mathematical analysis; (4) Functions, series and methods in analysis;: (5) Logics, set theories, and the foundations of mathematics; (6) Algebras and number theory; (7) Geometries and topology; (8) Mechanics and mechanical engineering; (9) Physics and mathematical physics, and electrical engineering; (10) Probability and statistics; (11) Higher education and institutions; (12) Mathematics and culture; (13) Reference and informarion. Thus, each of Parts 3 through 10 discusses one of the "major:branches of mathematics" [p. 5]. A reader of The Mathematical Intelligencer who is curious about a mathematical subject may get an overview from one o r more articles in the encyclopedia and then, using the bibliography, follow up the subject. The reader should not overlook the wealth of usefnl information in Part 13, Reference and information, especially Albert C.

VOLUME21, NUMBER2, 1999

75

Lewis's "Select bibliography of general sources." Occasionally one of the few recent textbook surveys of the history of mathematics (such as [2] and [4]) is stronger in some particular area, but this book is usually a very good place to begin. One particularly rich vein is applications. In the introduction [p. 5], the editor announces his intention to improve upon the existing general histories of mathematics with their "fight treatment given to applications, especially from the eighteenth century onwards; and the common ignoring of most of probability and statistics." The editor is rightly proud of this encyclopedia's coverage of probability and statistics and of such topics as cartography, meteorology, telecommunication theory, and capillarity, as well as more expected topics such as hydrodynamics or electricity and magnetism. Although [2], [4], and [5] all discuss some applications, and Kline [5] showed special interest in the influence of classical physics on mathematics, this encyclopedia is the first book that I would recommend to a student looking for an introduction to the history of a particular area of applied mathematics. Even sixteen hundred pages place narrow constraints on an account of a field of learning that has experienced three millennia of development. As a result, the articles occasionally so compress information that they read like an annotated chronology. On the other hand, some authors use their allotted space well by creating a useful sketch in broad strokes. I mention two out of the many excellently written articles. The text of Alexander Jones's "Greek mathematics to AD300," for example, covers less than eleven pages, but his annotated bibliography helps to compensate for the fact that in this small space he must cover the origins of Greek geometry, Euclid's "Elements," problem-solving and construction, the work of Archimedes and Apollonius, and "the problem of [the] decline" of pure mathematics in Greek antiquity. The next two articles, also by Jones, on "Greek applied mathematics" and "Later Greek and Byzantine mathematics," help to fill in the picture of the Greek tradition, as does Jurgen G. Schonbeck's article on "Euclidean and

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THE MATHEMATICALINTELLIGENCER

Archimedean traditions in the Middle Ages and the Renaissance." Still, less than 25 pages (about 1.5% of the encyclopedia) deals directly with ancient Greek mathematics. If you want a real discussion of the sources, content, and ancient influence of Euclid, for example, you will find more in [2], [4], or [5]. I would have preferred to see more space devoted to Greek mathematics in this encyclopedia, at the expense, if necessary, of such luxuries as an article on "Stamping mathematics." (What a waste, too, of the learning of its author, Hans Wussing.) Umberto Bottazzini supplies another example of space well used. His "Three traditions in complex analysis: Cauchy, Riemann and Weierstrass" traces the subject through the entire nineteenth century in thirteen pages and explains the motivatiotm--from integration, geometry, and rigor which inspired the three mathematicians named in the title and which led to the results of their followers. (Surprisingly, the author does not list [6] in his bibliography, an article that documents interrelations that Bottazzim has room only to suggest.) Many readers of The Mathematical InteUigencer will share my wish that the account of the development of mathematics extended farther into the twentieth century, providing more background to today's mathematics. The editor explicitly disavows this aim, saying quite rightly [p. 9] that "such a b o o k . . , would form a volume complementary to this one." Still, the encyclopedia might have reached further into the twentieth century without extensive additions. For example, Ian Stewart's article, "Lie Groups," is so tightly written that in just over four pages the author conveys a good sense of the origins and classification of Lie groups, comprehensible even to an advanced (American) undergraduate with no previous exposure to the subject. A few additional pages in the same style would have made it a proper background for an important part of today's mathematics. Again, Erhard Scholz covers a remarkable amount of material in his "Topology: geometric, algebraic," but in a slightly longer article, we might have read about the in-

fluence of m o d e m algebra upon algebraic topology and xrice versa. In his article, Scholz tell us that the concept of manifold entered differential geometry gradually, enabling geometers to take a global point of view. He thus approaches from the topological side the same material that Karin Reich approaches from the geometrical side in her "Differential geometry." How interesting it would have been if they had had room to continue! As I said, however, these wishes show the extent to which the encyclopedia has already succeeded. Readers may turn to Bourbaki's "Elements of the History of Mathematics" [1] for an introduction to contemporary mathematics. Using it together with this encyclopedia will give readers a broader view of the history of the subject and will prepare them to be more critical of some of the historical assertions found in [1]. Now that we have this resource, how shall we use it? For those who use it as their first source in the history of mathematics, I suggest being careful in two ways. Firstly, one should realize that ideas that were historically closely related may show little apparent connection today. The editor introduced his organization of topics (similar to the American Mathematical Society's current Mathematics Subject Classification scheme) principally, as he says [p. 6], "for the benefit of the reader, whose own understanding of mathematics is likely to follow divisions of this kind." This organization "begs historical questions concerning the existence or identification o f . . . [a certain] branch or topic, as such, in the fLrst place" [p. 5]. Therefore, beware of the fragmentation created by the organization of the encyclopedia into fourteen parts, each with several subdivisions. For example, Part 3 contains Kirsti Andersen's "Precalculus, 16351665" and Niccolo Guicciardini's "Three traditions in the calculus: Newton, Leibniz and Lagrange." Some 600 pages separate these articles from Craig Fraser's "Classical mechanics"--which includes a two-page discussion of Newton's Principia--and by 80 more pages from "Ballistics and projectiles," by A. R. Hall and I. Grattan-Guinness, and then "The pendulum: Theory, and

nation of y - - b u t the determination its use in clocks," by Paul Foulkes. Geometries and topology rather than of p. Moreover, Euler's programme Again, two hundred pages separate the in one of several other sections? True, of modular equations led nowhere, first brief mention of Newton's use of the article mentions the bridges of so that the breakthrough to PDE's series from its further discussion in Konigsberg and Euler's polyhedral forhad to walt another 10 years, until Lenore Feigenbaum's "Infinite series mula in its section on graph theory, but D'Alembert introduced them in his and solutions of ordinary differential it also discusses permutations and work on mechanics. [p. 455] equations, 1670-1770." Each of these combinations, partitions, and various articles is good in itself, but the reader problems of counting and designs. who does not know the connections Again, why do we fred Jeremy Gray's Again, in discussing Pappus's theorem and a few other theorems of anwill not be encouraged to find them article, "Finite-dimensional vector spaces," in this same part, rather than tiquity, Jeremy Gray remarks [p. 897]: here. "Modern commentators from Michel The creation of a separate part for in Algebras and number theory or in Chasles 1837 to B. L. van der Waerden higher education and institutions en- one of the parts devoted to analysis? A 1961 have indicated how profitably courages the same disjunction on a paragraph in the article suggests both these theorems can be seen as part of larger scale, by underwriting the divi- a reason and a missed opportunity for projective geometry; but it is unlikely sion of the history of mathematics making connections: along internal-external lines. Such a that they were so regarded in Classical times." division is especially regrettable be... [T]he study of vector spaces genMost mathematical textbooks, monocause many of the contributors to eralizes naturally to the infinitegraphs, and papers make little attempt these volumes are the very scholars dimensional case. But the infmite sums of vectors require a discussion to convey the value of their work. The who have brought the history of mathsubject is supposed to speak for itself. ematics to the same level of sophistiof their convergence if they are to This convention leaves the impression cation as the best history of science. make numerical sense, and so inOccasionally they overcome the strucvolve questions of topology. For of a cumulative subject in which every result is of equal importance (although tural limitations. In this part, for exthat reason they are not discussed here; see the articles on functional some results are even more imporample, Karen Hunger Parshall and tant!). Here the history of mathematics David E. Rowe's article, "The United analysis, integral equations and harcan offer a good antidote. By looking States of America, and Canada," intemonic analysis. [p. 951] back, we see that changing interests grates an account of the development One should beware of modern dominated mathematics at different of institutions with the concurrent deperiods, and that deep questions from velopment of research programs in translations of older mathematical ideas. As Grattan-Gninness wrote in the world of applications or from mathematics. Despite the editor's inmathematics itself usually motivated troductions to the parts and the cross- The Intelligencer in 1993 [3, p. 6]: "Of the best work. This is one of the ways references within the articles, we find course, modernizing old ideas is a perthat history can contribute to the other inconvenient (if less misleading) disjunctions. For exam- Confronting the student not with the education of mathematics students. Grattan-Guinness aptly ple, Part 12, Mathematics and culture, contains separate arti- problem but with the finished solution quotes Arthur Koestler in his introduction to the encyclopedia cles on "Art and architecture," means depriving him of all excitement. [p. 11]: The traditional method of "Symmetries in mathematics," confronting the student not with and "Tilings." The first of these the problem but with the finished sodiscusses frieze groups, wallpaper fectly legitimate process, even a good lution means depriving him of all exgroups, and tilings, while the second way to do research; but to identify this citement, to shut off creative impulse, discusses these topics again, as well as with the history of those ideas is a proto reduce the adventure of mankind to several other types of symmetries. found mistake, and the denigration of a dusty heap of theorems. History can These three articles are in a separate the latter a sign of cultural sterility." restore the sense of a growing field. part from the article "Crystallography," Fortunately, you can fmd articles that This point is clearly close to the ediand from the discussion of the origins sensitively avoid anachronistic catetor's heart, and he is quite eloquent and early history of the theory of gories. For example, in "Partial differgroups in "Fundamental concepts of ential equations" Jesper Ltitzen asks, about it. It suggests a further criterion by which to judge the organization and abstract algebra." (In-cidentally, the arDid Euler solve partial differential content of the encyclopedia. ticle on "Ethno-mathematics," also in As a test, let us consider two famous Part 12, does not mention frieze or equations here? Certainly the equaproblems of number theory: the tions he analysed were PDE's, and wallpaper patterns.) Riemann hypothesis and Fermat's Last A few points of organization may he had a uniform method of treatTheorem. Despite the role that it has simply puzzle the reader. Why, for exing them. However the problem he played in history, the Riemann hyample, does one find the article set himself was not the solution of pothesis makes its appearance in this "Combinatorics" in the section titled the equation--that is, the determi-

VOLUME 21, NUMBER 2, 1999

77

encyclopedia merely as a brief coda to a short discussion of Hadamard and de la Vall6e-Poussin's proof of the Prime Number Theorem (which did not use the Riemann hypothesis). In the same article, "Number Theory," Fermat's Last Theorem appears only as one of several conjectures of Fermat. Yet Hilbert, in his famous address on "The Future of Mathematics," delivered to the International Congress of Mathematicians in 1900, singled out the latter as a problem of great influence on the course of mathematics: Fermat has asserted, as is well known, that the diophantine equation (x, y, and z integers) is unsolvable--except in certain selfevident cases. The attempt to prove this impossibility offers a striking example of the inspiring effect that such a very special and apparently unimportant problem may have upon science. For Kummer, spurred on by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a cyclotomic field into prime factors--a law which today, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and the significance of which extends far beyond the boundaries of number theory and into the realm of algebra and the theory of functions. All this before the twentieth-century developments that have recently culminated in Andrew Wiles's proofi Wisely, the editor allowed Gunther Frei more pages than most for his article, "Number Theory," and Frei has certainly included a great deal of value. Wouldn't readers have seen the growth of mathematics more clearly, however, from a discussion of the history and influence of the conjectures of Fermat and Riemann? More generally, would-

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THE MATHEMATICAL INTELUGENCER

n't the editor's laudable pedagogic on an animal horn with a pointed aims have been better served by an or- stone, fending properties of special ganization of themes that emphasized functions in mathematical analysis, the genesis of important concepts and formulating the idea of recursion for methods? Similarly, clear discussions use in computing, forming predatorof the influences of philosophy and prey models for the mathematics of anpolitics upon mathematical practice imal behaviour, finding a complete would have contributed to a deeper un- axiom system for integral domains, exderstanding of the development of amining the distribution of prime nummathematics itself. Did the natural bers, developing a theory of capillary philosophical stances of Newton and flow in order to refme the accuracy of Leibniz influence their work on the cal- barometers, studying latent roots of culus? Why was the mechanics of con- matrices, searching for criteria to detinuous media so important in nine- tect singular solutions of differential teenth-century physical--and hence equations, interpreting the passage of mathematical--research? light in terms of non-Euclidean geomDespite these criticisms, the ency- etry, and many, many more things. [p. clopedia's richness extends beyond in- 5] dividual details to its cumulative effect. We may be grateful for the editor'.s Mathematics, said Hilbert, is what math- breadth of knowledge of the field and ematicians do. But who is a mathemati- of its experts (to whom we are also cian? In the sixteenth century, John Dee obliged), and for the energy that alcontributed to cartography, navigation, lowed him to complete it. the creation of mathematical instruments, and education. He argued that all REFERENCES these activities, as well as astronomy, 1. Bourbaki, Nicolas, Elements of the History geography, warfare, architecture, perof Mathematics, trans. John Meldrum, New spective, music, astrology, thaumamrgy, York, Berlin, Heidelberg: Springer-Verlag and more, were parts of mathematics. A (1994). century later, Christopher Wren bal- 2. Boyer, Carl B., and Merzbach, Uta C., anced activities in observational astronHistory of Mathematics, New York, omy, mechanics, geometry, the design Chichester, Brisbane, Toronto, Singapore: of mathematical instruments, and John Wiley & Sons (1989). more. When his time became more and 3. Grattan-Guinness, I., A Residual Category: more occupied with the architecture, Some Reflections on the History of he did not cease to be a mathematician. Mathematics and its Status, The MatheOn the contrary, Newton referred to him matical Intelligencer 15(1993), no. 4, 4-6. as one of the three "greatest geometers 4. Katz, Victor J., A History of Mathematics, of our times." Grattan-Guinness's hisNew York: Harper Collins College Publishers torical view of mathematics includes (1993). much more than the proving of a se- 5. Kline, Morris, Mathematical Thought from quence of theorems. Ancient to Modern Times, New York: O n e of the main lessons to learn Oxford University Press (1972). from the encyclopedia is the huge va- 6. Neuenschwander, E., Studies in the history riety of problems, theories, and techof complex function theory I1: interactions niques of which mathematics is comamong the French school, Riemann; and posed. It has encompassed, analysing Weierstrass, BAMS (2) 5 (1981), 88-105. the flow of water in a canal, calculating the path of a comet from observa- Department of Mathematics tions, introducing statistical tests of West Chester University of Pennsylvania significance in medicine, recording the West Chester, PA 19383-000I USA daily passages of the heavenly bodies [email protected]

II-'ll~4,,~.me'z.,a,t~-,-

Robin

International

Congresses of Mathematics

Chicago 1893

Wilson,

Editor

As part of the 400th anniversary celebrations of Columbns's voyage to America, a World Congress of Mathematicians took place at the World's Columbian Exposition in Chicago in 1893. Forty-five mathematicians attended and the opening address, on 'The present state of mathematics', was given by Felix Klein of GSttingen,

Moscow 1966

Helsinki 1978 Warsaw 1982/3

ZuHch 1994 Kyoto 1990

one of only four participants from abroad. Since that first meeting, twenty-three International Congresses have been held, the most recent taking place in Berlin last summer. In recent times, several ICMs have been commemorated by postage stamps. The fwst of these was the Moscow congress of 1966, attended by 4000 mathematicians from 49 countries. Twelve years later, the congress in Helsinki attracted delegates from 83 countries. In 1982 a set of four stamps was issued to commemorate the International Congress in Warsaw. This set featured the mathematicians Stefan Banach, Wac~aw Sierpifiski, Stanisfaw Zaremba and Zygmunt Janiszewski (see the Stamp Corner, Vol. 14, No. 2). In the event, the uncertain political situation necessitated the postponement of the congress to 1983. The first ICM to be held outside Europe or North America took place in 1990, in Kyoto. The design of the commemorative stamp shows an origami polyhedron. In 1994 the congress was held for the second time in ZtLrich. The commemorative stamp features Jakob Bernoulli and his law of large numbers; the likeness of Bernoulli was taken from a portrait painted by his brother Nicholas. The design for the 1998 congress stamp consists of a solution of the 'squaring-the-rectangle' problem of dividing a rectangle with integer sides into unequal squares with integer sides. The background consists of spirals made up from the decimal digits of ~-.

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Please send all submissions to the Stamp Corner Editor,

4~ 17

Robin Wilson,

Faculty of Mathematics and Computing, The Open University, Milton Keynes, MK7 6AA, England

80

Berlin 1998

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAG NEW YORK

Squaring the rectangle.

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