No title

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

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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.

Unintended Consequences As a parent and teacher I read with great interest the articles "The Case Against Computers in K-13 Math Education" by Neal Koblitz and "Some Kinds of Computers for Some Kinds of Learning" by Dubinsky and Noss in the Winter 1996 Intelligencer. They reminded me of a story I tell my students. When my son was in the seventh grade he took a computer course. One of the software programs they used was intended to develop their typing skills. The program was designed as a game: faster, correct typing earned a higher score. My son discovered that if you typed the letter "a" followed by the spacebar over and over as fast as possible you could outdo those who were typing words. He didn't develop any typing skills, but he was very proud of the fact that he had the highest all-time score. S. P. Peterson Bell Laboratories 480 Red Hill Road Middletown, NJ 07748 USA e-maih [email protected] 6

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

R e m a r k s on H i l b e r t ' s 23rd P r o b l e m Shiing-shen Chern

This is the last problem of his famous Paris address in 1900.* At the beginning he said, So far, I have generally mentioned problems as definite and special as possible, in the opinion that it is just such definite and special problems that attract us the most and from this the most lasting influence is often exerted upon science. Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture--which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due---I mean the calculus of variations.

culus of variations. In m o d e r n terminology, it can be described simply and naturally. In an n-dimensional space M with the coordinates x i, 1 <- i <- n , suppose the integral is s=

F xl,. o

. x n, " ' dt"

..

dt, "'

dt J

The calculus of variations had a vigorous development through the pioneering w o r k of the Bernoulli brothers, Euler, Lagrange, Legendre, and so forth in the 18th century. From the study of the extremals it soon became clear that the g e o m e t r y of the space arising from the integral plays an important role. Riemann's great address was delivered in 1854; Finsler's thesis, written u n d e r the direction of Caratheodory, was published in 1918. Riemann restricted his attention to the Riemannian case, with a side r e m a r k on the general case. Hilbert did not explicitly relate the calculus of variations to metric geometry. H o w e v e r , he p u t emphasis on A. Kneser's book, w h e r e g e o m e t r y dominates, and w e n t in some detail into his invariant integral. The invariant integral is a powerful notion in the cal*We are especially fortunate to have this essay from Professor Chern, in that the u p d a t e on the Hilbert problems in The Mathematical Intelligencer (vol. 18, no. 1) dealt only sketchily with N u m b e r 23.

--Editor's Note THE MATHEMATICALINTELLIGENCERVOL.18, NO. 4 9 1996 Springer-VerlagNew York 7

where the function F(x, y), x = (x1. . . . . xn), y = (yl ..... y n), is smooth for y :~ 0 and linearly homogeneous in y: F(x, Ay) =

]~lF(x, y),

A~

•.

Interpreting s as the arc length, this is now called Finsler geometry, although the general case was included by Riemann in his classic paper. Let PTM (projectivized tangent bundle) be the space of tangent directions of M. We can take x i and yi as local coordinates of PTM, with yi as homogeneous coordinates. Then the linear differential form to = ~i

V

dxi

is well defined in PTM. This is essentially Hilbert's invariant integral, and we will call it the Hilbert form. The presence of the Hilbert form to in PTM in Finsler geometry is of fundamental importance. The regularity of the Finsler metric is equivalent to the condition to A (dto) n-1 :~ O. Thus, ro defines the contact structure on PTM. Hilbert treated the plane case, but remarked that it can be extended to a simple integral in higher dimensions, as we have done above. He also remarked that the treatment can be extended to double integrals. This is the geometry in a space where the "area" is defined. This was carried out by E. Cartan in [1], but not much developed. A fundamental problem is the equivalence problem: given two Finsler metrics, to find the conditions that they differ by a change of coordinates. For the simple

Hilbert's invariant integral, that is, Finsler's metric, was introduced b y . . . R i e m a n n . case of Riemannian metrics, this is the form of the problem solved by Christoffel and Lipschitz in 1870, which leads to the development of tensor analysis by Ricci. It is remarkable that the Hilbert form is the key to the solution of the equivalence problem. This was carried out by the author in 1948, but the paper [2] remains little noticed (see also [3], [4]). There were great developments in global Riemannian geometry in the past decades, but Finsler geometry was hardly touched. This was undoubtedly a loss to differential geometry, for Riemannian geometry is basically quadratic in character; its generalization to higher degree is natural. Clearly, the generalization could involve richer and more profound structures. Moreover, modern developments in differential geometry have provided the notions and tools to handle the general case efficiently. Recent works on global Finsler geometry have shown that almost all the results in Riemannian 8

THE MATHEMATICAL INTELLIGENCER VOL 18, NO. 4, 1996

geometry, the comparison theorems, Hodge theory, the Gauss-Bonnet formula, and so forth, can be carried over to the Finsler setting. A fundamental tool in all these developments is the Hilbert form. In 1974 there was a conference on the Hilbert problems. The 23rd problem was reviewed by Guido Stampacchia [5]. This beautiful review is restricted to the analytical side of the question. Hopefully, the above lines could form a complement. In July 1995 there was a conference on Finsler geometry in Seattle, Washington, under the auspices of the AMS. A Proceedings of the conference is expected in 1996 [6]. I wish to refer to these Proceedings for a modest beginning of global Finsler geometry (see also [7]).

References 1. E. Cartan, Les espaces mdtriques fondds sur la notion d'aire, Paris: Hermann (1933). 2. S. Chern, Local equivalence and euclidean conditions in Finsler spaces, Science Reports Nat. Tsing Hua Univ. 5 (1948), 95-121; S. Chern, Selected Papers, vol. II (1989), 194-212. 3. D. Bao and S. Chern, On a notable connection in Finsler geometry, Houston J. Math. 19 (1993), 135-180. 4. S. Chern, Riemann geometry as a special case of Finsler geometry, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (Shiing-Shen Chern and Zhongmin Shen, editors). Providence, RI: American Mathematical Society (in press). 5. G. Stampacchia, Extension of calculus of variations, Proc. Syrup. Pure Math. 28 (1976), 611-628. 6. Finsler Geometry, Contemporary Mathematics, Vol. 196 (Shiing-Shen Chern and Zhongmin Shen, editors). Providence, Rh American Mathematical Society (in press). 7 S. Chern, Finsler Geometry is just the Riemannian Geometry without the Quadratic restriction, Notices AMS, Vol. 43 (1996) 959-963. Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, CA 94720 USA

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-in-chief, Chandler Davis.

The "Indexed" Theorem Alfredo Octavio

Most mathematicians don't know about it, but there is a system to evaluate a scientist's research without reading a single paper. It is widely used in the Third World and it is called "citation analysis." It is based on a commercial product called the "Science Citation Index" (SCI for short). The Institute for Scientific Information (ISI), which markets the SCI, claims it lists the papers published in all major journals and cross-links each of them with the articles it cites. All you have to do, after buying a CD-ROM at $11,000 per year, is look for the citations a scientist has gotten in the past few years, and Voil?z!--you have a numerical, objective measurement of the worth of the scientist's work. That is, at least, what ISI would like you to believe. A first point is obvious: not all citations are created equal. It is clear that if a researcher's work is used by others to advance knowledge, it should be cited, and these citations constitute evidence about the merits of the work. But what about citations like " . . . the present work is related with [1]."? Or " . . . there is an error in the proof of Lemma 3.3.4 of [3]."? Or the citations in the present article? Obviously, these should not all be worth the same. But further inquiry about the SCI reveals even deeper problems. The Index, as it is also called, only counts certain citations, specifically those appearing in the so-called "Source Journals." This is a list of about 3300 scientific journals, out of the 70,000 or so published worldwide (see [4]). From statistics of citations in the Source Journals, ISI builds a list of journals which I will call the Long List, organized by areas and ranked according to the "impact factor" of the journal, which is proportional to the average numbers of citations per paper. The journals in the Long List have been used in promotions and grant systems throughout the Third World.

In Venezuela this is done by a government grant program called Programa de Promoci6n del Investigador, PPI for short. The PPI has three levels, plus an entry level for graduate students. The higher the level, the more money one gets. The criteria for entering, remaining, and being promoted in the system are based almost exclusively on the number of papers published. For example, to remain in the first level one must have a paper published within the last two years. Not a lot; but here is the catch:

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

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THEOREM. The only good journals are the ones in the Long List. And the following COROLLARY. The only papers to be counted are those that appear in Long List journals. You may think this is just one more example of Third World bureaucracy. I should inform you that the committee evaluating mathematicians (or physicists, or chemists) applying for a PPI grant is composed of researchers in these areas all of whom have PhD degrees from American or European institutions. To see these people use the Theorem and its Corollary only makes it more infuriating. Similar systems have been used (and objected to!) in Mexico and elsewhere (see [2]). One of the noxious effects of this kind of evaluation is the total exclusion of local journals (see [5] and [2]). In the past, a lot of mathematical communities developed by creating a new journal and exchanging it for other publications. To deny this opportunity to contemporary mathematicians of the Third World is, simply, stupid and unfair. So, can the Theorem be proved? To answer this, I should look at the Long List of journals and ask, is it complete and consistent? In other words, are there good journals that are not on the list? Are there journals on the list that shouldn't be there? I want to look into this issue, but I will restrict myself to mathematical publications. In the 1992 Long List (the newest one I could find, though I searched in the Indiana University Library and all the science libraries in Caracas), there are 119 journals in the Mathematics section and 85 in the Applied Mathematics section. To appreciate the lack of significance (statistical or otherwise) of these numbers one should take into account that Mathematical Reviews provides reviews from cover to cover of about 516 serials and has reviewed articles from over 3000 serials. Certainly, some of these 516 serials not in the Long List deserve to be included. The converse question is harder. All the journals in the Long List that I know about are considered good. Still, there is an objection: should everything published in the American Mathematical Monthly or The Mathematical Intelligencer (both Long List and Source Journals) be counted as a research paper in mathematics? One may be pardoned some skepticism. The word "journal" is also a problem here. To be included in the Long List, a publication has to come out at regular intervals, so papers in book series (or books!) are out of the running. According to the Corollary, a paper published in Contemporary Mathematics is worth nothing, while any paper published in Matekon (number 83 in the Applied Math. section) is worth as much as a paper published in Annals of Mathematics (number 1 in the Mathematics section). Again, Mathematical 10 THEMATHEMATICALINTELLIGENCERVOL.18,NO.4,1996

Reviews proceeds quite differently. It calls both, journals and book series, "serials," and it covers anything published in mathematics that they can get their hands on. Very obscure references come to light and get worldwide distribution thanks to Mathematical Reviews. There are other conditions a journal has to satisfy to be on the Long List. It can't be a new journal, because new journals don't already have a lot of citations. Remember that only citations in Source Journals count; consequently, it is very easy for a Source Journal to get onto the Long List. Next we ask, how can one become a Source Journal? Several conditions have to be met. The journal has to be published on time and provide English abstracts for its non-English papers, the members of its editorial board must have been cited often enough (in the SCI database, of course), and . . . it has to buy a $10,000 subscription to the Index! (see [5]). This last condition shows that in the ISI we are dealing with an institution incapable of embarrassment. The promised neoliberal future, in which venality is a virtue instead of a vice, is upon us. Imagine if Mathematical Reviews only provided reviews of papers in journals that paid for the privilege! It would certainly be a lot less useful than it is. There are 56 Source Journals in Mathematics and 47 Source Journals in Applied Mathematics (in the 1991 Source Journal List, the newest one I could find, see above). Looking at the list of Source Journals it is hard to believe that all of them pay, though probably big publishers and some universities pay up automatically. There are suspicions that ISI may relax the rules in some cases (see [5]). Can anyone prove the Theorem? I don't think so. What is more, I don't think any competent mathematician will claim that any two papers (in any list!) ought to be weighted the same. Or that a paper can be considered worthless just because it is in a book series or in a journal outside some list. I have asked several non-Venezuelan mathematicians about the SCI. Most of them don't know what it is. Very few have ever used it. None defends it as a way to find out who is a better mathematician or which journal is better. A typical answer came from the late Lee Rubel. To the question "Do you know what the Science Citation Index is?" he answered, "Yes." To the question "Have you ever used it to evaluate mathematicians or journals?" he answered, "No, I have used it only a few times and that for vanity." Other people, Louis Nirenberg among them (see [2]), have said that counting papers as a criterion for evaluation is only done by second-rate departments. The feeling is not exclusive to mathematicians: "Citation analysis is a beancounter approach," says Nature editor John Maddox [4]. "When it comes to promoting somebody, it's almost as bad to be presented with a list of citations to his various papers, as to be presented with the weight of those same papers in grams."

As you can see, none of these comments refers to the Long List. None of the mathematicians I talked to knew about either of ISI's lists, even if they had some idea what SCI was. In [2], the Mexican mathematician Jorge Ize asked several mathematicians about the SCI and evaluation in mathematics. The answers he got bear out my personal experience. Will this kind of evaluation hurt Third World mathematics? I think the answer is yes. Young researchers, especially if trained within the country, will tend to accept the requirements of this system as the definition of research. I already mentioned a bad effect: they won't publish in local magazines (nor national, nor regional). There are other consequences, harmful to the researcher's career. This type of system sends two messages: quantity is more important than quality; and going to conferences is irrelevant. The former produces shorter, less meaningful papers (published in lesser, though Listed, journals); the latter produces isolation from international research centers (as if that weren't enough of a problem for us already!). I don't disapprove of the PPI itself. I actually think the PPI can be beneficial to Venezuelan science. I do object to using the SCI to evaluate mathematicians. I have neither the time nor the inclination to make a long fight against this farce. I can only look up to my elders (sorry!) and ask, "Any ideas?" Epilogue: I have tried to stay in the PPI, so that people can't say I am just criticizing it out of envy. It is my obligation to tell you how I stand at this moment. I have been in the PPI (first level) since 1992. I renewed (successfully) in 1994; I have to renew again in 1996 and will probably fail. In the two-year period (1994-1996) to be considered, I have, so far, only two papers: one in Contemporary Mathematics and the other in Operator Theory: Advances and Applications. Both are book series and thus not on the Long List. But that is where m y mathematics appears. For the injustice that those articles do not count, it is small consolation that the present article does.

References 1. Margaret Bledsoe and Paul Garabedian, "On the weak solution of Burger's equation" (Spanish), Acta Cientifica Venezolana 44 (1993), 337-340. 2. Jorge Ize, Articulos de investigaci6n en matem~ticas y evaluaci6n, Ciencia 45 (1992), 157-173. 3. Alfredo Octavio, Dual algebras generated by commuting contractions, PhD dissertation, The University of Michigan, 1991. 4. Gary Taubes, "Measure for measure in science," Science 260, May 1993, 884-886. 5. W. Wayt Gibbs, "Lost science in the third world," Scientific American, August 1995, 92-99.

IVIC Caracas, Venezuela e-maih [email protected] THE MATHEMATICAL INTELLIGENCER VOL. I8, NO. 4, 1996

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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-in-chief, Chandler Davis.

Making Sense of Experimental Mathematics J. Borwein, P. Borwein, R. Girgensohn, and S. Parnes

Introduction Discovery and Verification. Philosophers have frequently distinguished mathematics from the physical sciences. While the sciences were constrained to fit themselves via experimentation to the real world, mathematicians were allowed more or less free reign within

12

the abstract world of the mind. This picture has served mathematicians well for the past few millennia, but the computer has begun to change this. The computer has given us the ability to look at new mathematical worlds that would have remained inaccessible to the unaided human mind; but this access has come at a price. Many of these worlds, at present, can only be known experi-

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

mentally. The computer has allowed us to fly through the rarefied domains of hyperbolic spaces and examine more than a billion digits of ~r, but experiencing a world and understanding it are very different. Most of these explorations into the mathematical wilderness remain isolated illustrations. Heuristic conventions, pictures, and diagrams developing in one subfield often have little content for another. In each subfield, unproven results proliferate but remain conjectures, strongly held beliefs, or perhaps mere curiosities passed like folktales across the Internet. It is our hope that by focusing on experimental mathematics today, we can develop a unifying methodology tomorrow. Where We're Coming From. The genesis of this article was a simple question: "How can one use the computer in dealing with computationally approachable but otherwise intractable problems in mathematics?" We began our current exploration of experimental mathematics by examining a number of very long-standing conjectures and strongly held beliefs regarding decimal and continued fraction expansions of certain elementary constants. These questions are uniformly considered to be hopelessly intractable given present mathematical technology. Unified field theory or cancer's "magic bullet" seem accessible by comparison. But their statements are beguilingly simple. We think our methods cannot yield counterexamples or proofs of the statements; our objectives were systematization and communication. For our attempted systematization of experimental mathematics we were concerned with producing data that were "completely" reliable and insights that could be quantified and effectively communicated. We initially took as our model experimental physics. We were particularly interested in how physicists verified their results and the efforts they took to guarantee the reliability of their data. Here we believe we can draw a useful distinction between experimental physics and mathematics. While it is clearly impossible to extract perfect experimental data from nature, such is not the case with mathematics. Reliability of raw mathematical data is often attainable. The most vexing of issues, we found, is the communication of insights. Unlike most experimentalized fields, Mathematics does not have a vocabulary tailored to the transmission of condensed data and insight. As in most physics experiments, the amount of raw data obtained from mathematical experiment will, in general be too large for anyone to grasp. The collected data need to be compressed and compartmentalized. To make up for this lack of unifying vocabulary, we have borrowed heavily from statistics and data analysis to interpret our results. For now, we can only try to present results in an intuitive, friendly, and convincing manner. Eventually we hope for a multileveled hypertextual presentation of mathematics, allowing mathe-

maticians from diverse fields to scan and interpret the results of others--across the language barrier that separates subdisciplines of mathematics. All these topics are addressed at much greater length in the research report [4] as well as [3]. For the remainder of the article we will take up the various models of experimental mathematics and ask how they might be integrated into the body proper of mathematics.

Experimental Mathematics Journal of. A current focal point for experimental mathematics is the journal called Experimental Mathematics. But does it really seek to change the w a y we do mathematics, or to change the w a y we write mathematics? We begin by attempting to extract a definition of "experimental" from the journal's introductory article [6] "About this journal" by David Epstein, Silvio Levy, and Rafael de la Llave. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, wellrounded and rigorous results. ([6], p. 1) Do conventional authors mean to disparage experimental results as inelegant, lopsided, and lax? We do not view these as essential properties of experimental mathematics but rather as traps that must be avoided. What is the journal interested in publishing? The editors continue to value the traditional mathematical virtues and are not averse to publishing proven results discovered experimentally. However, they note, "We consider it anomalous that an important component of the process of mathematical creation is hidden from public discussion. It is to our loss that most of the mathematical community are almost always unaware of h o w new results have been discovered" ([6], p. 1). It appears that the editors advocate a change in writing style, emphasizing the creative or synthetic aspect of mathematics as opposed to the deductive or analytic aspect. They hope that, "The early sharing of insights increases the possibility that they will lead to theorems: an interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere" ([6], p. 1). So what does an article published in that journal look like? A recent example is "Experimental evaluation of Euler sums" by D.H. Bailey, J. Borwein, and R. Girgensohn [2]. The authors describe how their interest in Euler sums was roused by a surprising discovery: In April 1993, Enrico Au-Yeung, an undergraduate at the University of Waterloo, brought to the attention of one of THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

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us the curious fact that

k•--1 1 +

.

. . . 21 +

+

k-2

~

4.59987

.

.

.

--~ ~ ' ( 4 ) - 177r4 360

([2], p. 17) This type of serendipitous discovery must go on all the time, but it remains a curiosity unless given a broader context. The authors now proceeded to provide a context. They systematically applied an integer relation-detection algorithm to large classes of sums of the above type, trying to find evaluations of these sums in terms of zeta functions (see box for details). Some of the experimentally discovered evaluations were then proven rigorously, others remain conjectures. While Au-Yeung's insight may fill us with amazement, the experimenters' approach appears quite natural and systematic. The Deductivist Style. The editors of Experimental Mathematics are advocating a change in the way mathematics is written, placing more emphasis on the mathematical process. Imre Lakatos in his influential though controversial book Proofs and Refutations [9] advocated a similar change. Lakatos begins his talk on Euclidean methodology in much the same tone as the introduction to Experimental

Mathematics. Euclidean Methodology has developed a certain obligatory style of presentation. I shall refer to this as "deductivist style." This style starts with a painstakingly stated list of axioms, lemmas and/or definitions. The axioms and definitions frequently look artificial and mystifyinglycomplicated. One is never told how these complications arose. The list of axioms and definitions is followed by the carefully worded theorems. These are loaded with heavy-going conditions; it seems impossible that anyone should ever have guessed them. The theorem is followed by the proof. ([9], p. 142)

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This is the essence of what might be termed formal understanding. We know that the results are true because we have gone through the crucible of the mathematical process and what remains is the essence of truth. But the "deductivist style hides the struggle, hides the adventure. The whole story vanishes, the successive tentative formulations of the theorem in the course of the proof-procedure are doomed to oblivion while the end result is exalted into sacred infallibility" ([9], p. 142). Perhaps the most extreme examples of the deductivist style are the computer-generated proofs guaranteed by Wilf and Zeilberger's algorithmic proof theory. The theory really provides a meta-insight into a broad range of problems involving "hypergeometric" identities. Yet the proofs produced by the computer, while understandable by most mathematicians, are uninteresting. We will not discuss the theory in any detail here but refer the reader to Doron Zeilberger's excellent elementary introduction in [14]. What we ask is how such proofs can contribute to our intuitive image of mathematics.

Zeilberger and the Encapsulation of Identity Putting a

Price on Reliability. At present, knowing the Wilf-Zeilberger (WZ) proof of an identity amounts to little more than knowing that the identity is true. In fact, Doron Zeilberger [13] has advocated leaving only a QED at the end of the statement, the author's seal that he has had the computer perform the calculations needed to prove the identity. The advantage of this approach is that the result is completely encapsulated. Just as one would not worry about how the computer multiplied two huge integers together or inverted a matrix, one now has results whose proofs need not be examined. If nothing but certainty mattered, we would be done. Fortunately, mathematics requires a great deal more. In

this section we will discuss the implications of this theory and Zeilberger's philosophy of mathematics as contained in Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture [14]. The two voices most strongly advocating truly experimental mathematics are also at times the most hyperbolic in their language. We will concentrate on Zei|berger, but G. J. Chaitin should not and will not be ignored. We will begin with D. Zeilberger's "Abstract of the future": We show in a certain precise sense that the Goldbach conjecture is true with probability larger than 0.99999 and that its complete truth could be determined with a budget of 10 billion. ([14], p. 980) Once people get over the shock of seeing probabilities assigned to truth in mathematics, the usual complaint is that the 10 billion is ridiculous. Computers have been getting better and cheaper for years. What can it mean that "the complete truth could be determined with a budget of 10 billion"? What is clear from the article is that this is an additive measure of the difficulty of completely solving this problem. If we k n o w that the Riemann hypothesis will be proven if we prove lemmas costing 10 billion, 2 billion, and 2 trillion dollars, re-

spectively, we can tell at a glance not merely what it would "cost" to prove the hypothesis but also where n e w ideas will be essential in any proof. (This assumes that 2 trillion dollars is a lot of money.) The introduction of "cost" leads immediately to consideration of a preoccupation that has overtaken the business world and is n o w intruding rapidly on academia: productivity and efficiency. It is a waste of money to get absolute certainty, unless the conjectured identity in question is known to imply the Riemann Hypothesis. ([14], p. 980) This comment is indicative of the feelings of a small but growing group of mathematicians w h o are asking us to look at not just the benefits of reliability in mathematics but also the associated costs. See, for example, A. Jaffe and F. Quinn in [7] and G. Chaitin in [5]. Still, we have not dealt with the central question. W h y does Zeilberger need to introduce probabilistic "truths" and h o w might we from a "formalist" perspective not feel this to be a great sacrifice? It's All About Insight. W h y is Zeilberger so willing to give up on absolute truths? The most reasonable answer is that he is pursuing deeper truths. In Identities in Search of Identities, Zeilberger advocates an examination of THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

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identities for the sake of studying identities. Still, as Wilf and others have pointed out, it is possible to produce an unlimited n u m b e r of identities. It is the context, the ability to use and manipulate these identities, that make them interesting. W h y then might we think that studying identities for their o w n sake m a y lead us d o w n the golden path rather than the garden path? We are n o w looking for what might be called metamathematical structures. We remove the mathematics from its original context and isolate it, trying to detect new structures. It is impossible to collect only the relevant information that will lead to the n e w discovery. One collects objects (theorems, statistics, conjectures, etc.) that have a reasonable degree of similarity and famiharity and then attempts to eliminate the irrelevant or the untrue (counterexamples). We are preparing for some form of eliminative induction. In this context, it is not unreasonable to introduce objects without being sure of their truth, for all the objects, whether proved or not, will be subject to the same degree of scrutiny. Moreover, if these probably true objects fall into the class of desired objects (i.e., they fit the n e w conjecture), it m a y be possible to find a legitimate proof in the new context. It is possible to use the WZ algorithm in such a w a y that it does not yield an absolute proof of some identity, but instead a probability value (possibly very high) for that identity to be true. H o w should we interpret such a result? W h a t should we do if we cannot prove the identity rigorously? If we agree with Chaitin, we m a y w a n t to introduce it as an axiom. I believe that elementary number theory and the rest of mathematics should be pursued more in the spirit of experimental science, and that you should be willing to adopt new principles. I believe that Euclid's statement that an axiom is a self-evident truth is a big mistake. 1 The Schr6dinger equation certainly isn't a self-evident truth! And the Riemann hypothesis isn't self-evident either, but it's very useful. A physicist would say that there is ample experimental evidence for the Riemann hypothesis and would go ahead and take it as a working assumption. ([5], p. 24) In this case, we have ample experimental evidence for the truth of our identity and we m a y w a n t to take it as something more than just a working assumption. We m a y want to formally introduce it into our mathematical system. What we need to avoid is the haphazard introduction of n e w axioms.

Experiment and "Theory" In Advice to a Young Scientist, P.B. M e d a w a r defines four different kinds of experiment: the Kantian, Baconian, Aristotelian, and the Galilean. Mathematics has always

1There is no evidence that Euclid ever made such a statement. However, it does have an undeniable emotional appeal. 16 T.E MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

participated deeply in the first three categories but has s o m e h o w managed to avoid employing the Galilean model. In developing our notion of experimental mathematics we will try to adhere to this Galilean m o d e as m u c h as possible. We will begin with the Kantian experiments. M e d a w a r gives as his example generating the classical non-Euclidean geometries (hyperbolic, elliptic) by replacing Euclid's axiom of parallels (or something equivalent to it) with alternative forms. ([11], pp. 73-74) It seems clear that mathematicians will have difficulty escaping from the Kantian fold. Even a Platonist m u s t concede that mathematics is only accessible through the h u m a n mind, and thus all mathematics might be considered a Kantian experiment. We can debate whether Euclidean geometry is but an idealization of the geometry of nature (where a point has no length or breadth and a line has length but no breadth), or nature an imperfect reflection of "pure" geometrical objects, but in either case the objects of interest lie within the m i n d ' s eye. Similarly, we cannot escape the Baconian experiment. In Medawar's words, this is a contrived as opposed to a natural happening--is the consequence of "trying things out" or even of merely messing about. ([11], p. 69) Most of the research described as experimental is Baconian in nature, but also one can argue that all of mathematics proceeds out of Baconian experiments. One tries out a transformation here, an identity there, examines what happens w h e n one weakens this condition or strengthens that one. Even the application of probabilistic arguments in n u m b e r theory can be seen as a Baconian experiment. The experiments m a y be well t h o u g h t out and very likely to succeed, but the criterion of inclusion of the result in the literature is success or failure. If the "messing about" works (e.g., the theorem is proved, the counterexample found), the material is kept; otherwise, it is relegated to the scrap heap. The Aristotelian experiments are described as demonstrations: apply electrodes to a frog's sciatic nerve, and lo, the leg kicks; always precede the presentation of the dog's dinner with the ringing of a bell, and lo, the bell alone will soon make the dog dribble. ([11], p. 71) The Aristotelian experiment is equivalent to the concrete examples we employ to help explain our definitions, theorems, or the illustrative problems we assign to students Last is the Galilean experiment: [the] Galilean Experiment is a critical experiment--one that discriminates between possibilities and, in doing so, either

gives us confidence in the view we are taking or makes us think it in need of correction. ([11], p. 71) Ideally, one devises an experiment to distinguish between two or more competing hypotheses. In subjects like medicine, the questions are in principle clearer, although the Will Rogers p h e n o m e n o n complicates matters. 2 Does this medicine w o r k (longevity, quality of life, cost-effectiveness, etc.)? Is this treatment better than that one? Unfortunately, these questions are extremely difficult to answer, and the model M e d a w a r presents here does not correspond with the current view of experimentation in physics. N e w t o n i a n physics w o r k e d beautifully but ultimately was supplanted [10]; so it is n o w widely held that no a m o u n t of experimental evidence can p r o v e a theorem about the world a r o u n d us, and it is widely k n o w n that in the real world, the models one tests are not true. M e d a w a r acknowledges the difficulty of proving a result but has more confidence than modern philosophers in disproving hypotheses. If experiment cannot distinguish between hypotheses or prove theorems, what can it do?

"Theoretical" Experimentation Although there is an ongoing crisis in mathematics, it is not as severe as the crisis in physics. The untestability of parts of theoretical physics (e.g., string theory) has led to a greater reliance on mathematics for "experimental verification." This m a y be in part what led A r t h u r Jaffe and Frank Quinn to advocate what they have n a m e d Theoretical Mathematics (note that m a n y mathematicians think they have been d o i n g theoretical mathematics for years). It is not surprising that "theoretical mathematics," having been m o d e l e d on theoretical physics, has the closest parallels to parts of the Galilean (Popperian) m o d e l of experimentation. Before one can conduct an experiment, one m u s t formulate hypotheses. It is at this stage that a nonrigorous approach could really shine. Arthur Jaffe and Frank Quinn's "Theoretical

Mathematics": Toward a Cultural Synthesis of Mathematics and Theoretical Physics appears to be mainly a call for a loosening of the bonds of rigor. They are concerned about the slow pace of mathematical d e v e l o p m e n t s w h e n all the w o r k must be rigorously d e v e l o p e d prior

2The Will Rogers phenomenon or Simpson's Paradox. Both terms refer to the problems of reaggregation. Will Rogers once remarked, on hearing an acquaintance had moved from Ohio to California, that the man had thus raised the average IQ of both states. The term has become current in medical discourse: reassign subjects in a previous cancer study from low-risk to high-risk groups and both groups may appear to have performed better. The recognition of such issues is often attributed to E. H. Simpson (1951). In the baseball season in which two halves were played, a player winning the batting title in each half would not necessarily win the batting title for the whole season. [4/9 > 10/23 and 2/7 > 3/11, but (4+2)/(9+7) < (10+3)/(23+11).]

to publication. A n d yet they fear that a h a p h a z a r d introduction of conjectural mathematics will result in chaos. Their solution comes in two parts. They suggest that theoretical work should be explicitly acknowledged as theoretical and incomplete; in particular, a major share of credit for the final result must be reserved for the rigorous work that validates it. ([7], p. 10) They w a n t to ensure that there are incentives for following u p and proving the conjectured results. T h e y w o u l d distinguish between theorems, which are proved, and the conjectures of "theoretical mathematics." But this is w h a t mathematicians already do, albeit at a low level; that is, motivation and conjecture have become an integral part of m a n y mathematicians' output, although kept as a d d e n d u m s to more rigorous works. If w e find any fault in their proposals, it is that they have attempted to graft the strengths of theoretical physics onto mathematics without addressing the important check that experimental physics brings to it. It m a y be that "theoretical mathematics" cannot truly stand on its own, but needs a check from experimental mathematics. Being computationally oriented, we envisage especially computational conjectures and verification, but certainly there are other possibilities.

Conclusion We conclude with a definition of experimental mathematics.

Experimental mathematics is that branch of mathematics that concerns itself ultimately with codification and transmission of insights within the mathematical community through the use of experimental exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit. Results discovered experimentally will, in general, lack some of the rigor associated with mathematics, but will p r o v i d e general insights into mathematical problems to guide further exploration, either experimental or traditional. Conjectures experimentally verified will give us more confidence in our direction, even w h e n strongly held beliefs elude proof. One can hope to produce an intuitive view of mathematics that can be transferred in concrete examples and analysis, as opposed to the current system where intuitions can be transmitted only from p e r s o n to person. If the mathematical c o m m u n i t y as a whole were less splintered, we w o u l d probably r e m o v e the w o r d "codification" from the definition. But there are real communication problems between fields. Experimental investigators must make e v e r y effort to organize their insights and present their data in a m a n n e r that will be as widely accessible as possible. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

17

References 1. M. Atiyah, A. Borel, G.J. Chaitin, et al., Responses to "Theoretical Mathematics: Towards a Cultural Synthesis of Mathematics and Theoretical Physics" by A. Jaffe and F. Quinn, Bull. Am. Math. Soc. (2)30 (1994), 178-207. 2. D. H. Bailey, J. M. Borwein, and R. Girgensohn, Experimental evaluation of Euler sums, Experimental Math. 3(1) (1994), 17-30. 3. J. M. Borwein, P. Borwein, R. Girgensohn, S. and S. Parnes, Experimental mathematical investigation of decimal and continued fraction expansions of select constants (unpublished). 4. J. M. Borwein, P. Borwein, R. Girgensohn, and S. Parnes, Making sense of experimental mathematics, CECM Preprint 95:032 (1995). 5. G. J. Chaitin, Randomness and complexity in pure mathematics, Int. J. Bifurcation Chaos 4 (1994), 3-15. 6. D. Epstein, S. Levy, and R. Llave, de la, About this journal, Experimental Math. 1(1) (1992), 1-3. 7. A. Jaffe and F. Quinn, Theoretical mathematics: Towards a cultural synthesis of mathematics and theoretical physics, Bull. Am. Math. Soc. (2) 29 (1993), 1-13. 8. A. Jaffe and F. Quinn, Response to comments on "Theoretical Mathematics," Bull Am. Math. Soc. (2)30 (1994), 208-211. 9. I. Lakatos, Proofs and Refutations, Cambridge: Cambridge University Press (1970). 10. I. Lakatos, The Methodology of Scientific Research Programmes: Philosophical Papers Volume 1, Cambridge: Cambridge University Press (1978). 11. P. B. Medawar, Advice to a Young Scientist, New York: Harper Colophon (1981). 12. W. P. Thurston, On proof and progress in mathematics, Bull. Am. Math. Soc. (2)30 (1994), 161-177. 13. D. Zeilberger, Identities in search of identities, preprint (1992). 14. D. Zeilberger, Theorems for a price: Tomorrow's semi-rigorous mathematical culture, Notices Am. Math. Soc. 40(8) (1993), 978-981. Reprinted in The Mathematical Intelligencer 16 (1994), no. 4, 11-14.

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Contents: H. de Nivelle: Resolution Games and Non-Liftable Resolution Orderings. - M. Kerber, M. Kohlhase: A Tableau Calculus for Partial Functions. G. Salzer: MUltlog: an Expert System f o r Multiple-valued Logics. - J. Krajiffek: A Fundamental Problem of Mathematical Logic. - P. Pudlfik: On the Lengths of Proofs of Consistency. - A. Carbone: The Craig Interpolation Theorem for Schematic Systems. - I.A. Stewart: The Role of Monotonicity in Descriptive Complexity Theory. - R. Freund, L. Staiger: Numbers Defmed by Turing Machines.

Volume 1 1 9 9 5 . 2 f i g u r e s . V I I , 122 p a g e s . I S B N 3 - 2 1 1 - 8 2 6 4 6 - 7 Soft c o v e r D M 6 4 , - , a p p r o x . U S $ 4 4 . 0 0

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Patchworking Algebraic Curves Disproves the Ragsdale Conjecture Ilia Itenberg and Oleg Viro

Real algebraic curves seem to be quite distant from combinatorial geometry. In this article we intend to demonstrate h o w to build algebraic curves in a combinatorial fashion: to patchwork them from pieces which essentially are lines. One can trace related constructions back to Newton's consideration of branches at a singular point of a curve. Nonetheless, an explicit formulation is not familiar to m a n y mathematicians. This technique was developed by the second author in the beginning of the eighties. Using it, the first author has recently found counterexamples to the oldest and most famous conjecture on the topology of real algebraic curves. The conjecture was formulated as early as 1906 by V. Ragsdale [14] on the basis of experimental material provided by A. Harnack's and D. Hilbert's constructions [5,6].

A Combinatorial Look at Patchworking Initial Data. Let m be a positive integer (it will be the degree of the curve under construction) and T be the triangle in ~2 with vertices (0, 0), (m, 0), (0, m). Let ~-be a triangulation of T with vertices having integer coordinates and equipped with signs. The sign (plus or minus) at the vertex with coordinates (i, j) is denoted by cri,j. Construction of a Piecewise Linear Curve. Take copies Tx = sx(T),

Ty = sy(T),

Txy = s(T)

of T, where s = sx o sy and sx, sy are reflections with respect to the coordinate axes. Denote by T, the square T U Tx U Ty U T~. Extend the triangulation ~-to a symmetric triangulation of T,, and the distribution of signs cri,/ to a distribution at the vertices of the extended triTHE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

19

pair (T,, L) is called the result ofaffine combinatorial patchworking. Glue the sides of T, b y s. The resulting space is h o m e o m o r p h i c to the real projective plane RP 2. Denote b y ~ the image of L in ~, and call the pair (~, ~) the result of projective combinatorial patchworking. (See, for example, Figures 1-3). Let us introduce an additional assumption: the triangulation ~-of T is convex. This means that there exists a convex piecewise-linear function T ~ R which is linear on each triangle of ~-and not linear on the union of a n y two triangles of ~-. Figure 1. Initial data and the result of combinatorial patchworking of it.

angulation by the following rule: ~,io,i,~jEi3 j = 1, where e, 3 = + 1. In other words, passing from a vertex to its mirror image with respect to an axis w e preserve its sign if the distance from the vertex to the axis is even, and change the sign if the distance is odd. If a triangle of the triangulation of T, has vertices of different signs, select a midline separating pluses from minuses. Denote b y L the union of the selected midlines. It is a collection of polygonal lines contained in T,. The

P A T C H W O R K T H E O R E M : Under the assumptions above

on the triangulation 9 of T, there exist a nonsingular real algebraic plane affine curve of degree m and a homeomorphism of the plane R 2 onto the interior of the square T, mapping the set of real points of this curve onto L. Furthermore, there exists a homeomorphism •p2 __, ~ mapping the set of real points of the corresponding projective curve onto ~. Real Plane Algebraic Curves

The w o r d curve is k n o w n to be one of the most ambiguous in mathematics. Thus, w e had better specify the

Figure 2. Patchwork of a counterexample to the Ragsdale Conjecture with degree 10 and p = 32. 20 THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996

Figure 3. Patchwork of a counterexample to the Ragsdale Conjecture with degree 10 and n = 32.

type of curves to be considered. They are real algebraic plane curves, i.e., plane curves defined by equations f = 0, where f is a polynomial over the field ~ of real numbers. The constructions below give us real polynomials fix, y) of a given degree such that the curves f(x, y) = 0 are positioned in a complicated w a y (for this degree) in the plane ~2. For m a n y reasons we prefer projective curves. To a reader w h o is not comfortable with the projective plane, we offer the following motivations and definitions. It was probably Isaac N e w t o n [10] who first observed that a curve fix, y) = 0 in the plane ~2 is a more complicated object (e.g., to classify) than the cone generated by it in ~3. If m is the degree of f, then the cone is defined by the equation zmf(x/z, y/z)= 0. N e w t o n [10] found 99 classes of curves of degree 3 on ~2, but at the end of his text he noted that curves of all 99 classes can be obtained as plane sections of only 5 cubic cones. In the 19th century this observation and similar ones led to the notion of the projective plane and the idea that it is simpler to study curves in the projective plane than in the affine plane. The real projective plane ~p2 can be defined as the set of lines in R 3 passing through the origin (0, 0, 0). The

line passing through (0, 0, 0) and (x0, xl, X2) is denoted by (Xo : Xl : x2); the numbers Xo, Xl, and x2 are called homogeneous coordinates of (Xo : xl : x2). A cone in ~3 with vertex (0, 0, 0) can be thought of as a collection of lines passing through (0,0,0). This is a curve in the projective plane provided that the collection is one-parameter. An equation F(x, y, z) = 0, where F is a homogeneous real polynomial, defines a cone in ~3 with vertex (0, 0, 0) and hence a curve in the projective plane RP 2. Take a curve on ~2 defined by an equation fix, y) = 0 of degree m, shift it with its plane to the plane z = 1 in ~3, and consider lines passing through it and the origin (0, 0, 0). These lines lie on the cone zmf(x/z, y/z) = 0 and fill it together with its intersection with the plane z = 0. The corresponding curve on ~p2 is called the projective completion of the affine curve f(x, y) = 0. The study of real algebraic curves in the affine plane R 2 splits naturally into study of their projective completions and investigation of the position of the completions with respect to the line at infinity ~p2\[~2. A curve (at least an algebraic curve) is something more than just the set of points which belong to it. It is only slightly less than its equation: equations differing THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

21

by a constant factor define the same curve. M o d e r n algebraic geometry provides a lot of ways to define algebraic curve. We a d o p t the following definition, which at first glance seems to be overly algebraic. By a real projective algebraic plane curve of degree m we mean a h o m o g e n e o u s real polynomial of degree m in three variables, considered u p to a constant factor. (Similarly, by a real affl'ne algebraic plane curve of degree m we mean a real polynomial of degree m in two variables, considered u p to a constant factor.) If F is such a polynomial, then the equation F(x0, xl, x2) = 0 defines the set of real points of the curve in the real projective plane [~p2. Let ~A denote the set of real points of the curve A. Following tradition, we also call this set a curve, avoiding this terminology only in cases w h e r e confusion could result. A point (Xo:Xl:X 2) ~ ~p2 is called a (real) singular point of the curve defined by a polynomial F if the first partial derivatives of F vanish at (xo, xp x 2) (vanishing of the derivatives implies vanishing of the h o m o g e n e o u s polynomial: by the Euler formula deg(F).F(xo, xl, x2) = ~ i xi(c~F/axi) (Xo, Xl, X2))" A curve is said to be (real) nonsingular if it has no real singular points. The set ~A of real points of a nonsingular real projective plane curve A is a smooth closed one-dimensional submanifold of the projective plane. Then ~A is a u n i o n of disjoint circles smoothly e m b e d d e d in ~p2. A circle can be positioned in ~p2 either one-sidedly, like a projective line, or two-sidedly, like a conic. A two-sided circle is called an oval. An oval divides ~p2 into two parts. The part h o m e o m o r p h i c to a disk is called the interior of the oval. Two ovals can be situated in two topologically distinct ways: each m a y lie outside the other one--i.e., each is in the outside c o m p o n e n t of the c o m p l e m e n t of the o t h e r - - o r else one of them is in the inside c o m p o n e n t of the c o m p l e m e n t of the o t h e r - - i n that case, we say that the first is the inner oval of the pair and the second is the outer oval. In the latter case we also say that the outer oval of the pair envelops the inner oval. The topological type of the pair (~p2, ~A) is defined by the scheme of disposition of the ovals of ~A. This scheme is called the real scheme of curve A. In 1900 D. Hilbert [7] included the following question in the 16th problem of his famous list: what real schemes can be realized by curves of a given degree? The complete answer is k n o w n n o w only for curves of degree not greater than 7.

T-Curves N o w let us come back to the Patchwork Theorem. It states that for any convex triangulation ~-of T with integer vertices and a distribution of signs at vertices of 9 there exists a nonsingular real algebraic plane projective curve A of degree m such that the pair (~p2, ~A) is homeomorphic to the pair (2, ~) constructed as in the first section, i.e., the result of projective combinatorial patchworking. 22 THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996

In fact, a polynomial defining the curve can be presented quite explicitly.

Construction of Polynomials.

Given initial data m, T, ~-, and o-id as in the first section, and a convex function v certifying that the triangulation 7 is convex, consider the one-parameter family of polynomials

bt(x, y) =

~

crijxiyJt~iJ(

(i, j) runs over vertices of 7 Denote b y B t the corresponding h o m o g e n e o u s polynomials:

Bt(xo, Xl, x2) = x~bt(xl/xo, X2/Xo).

Polynomials

b t and B t

are called the results of affine and

projective polynomial patchworking.

DETAILED PATCHWORK THEOREM: Let m, T, T, cri,j, and u be initial data as above. Denote by b t and B t the nonhomogeneous and homogeneous polynomials obtained by the polynomial patchworking of these initial data, and by L and Sd the piecewise-linear curves in the square T, and its quotient space qZ, respectively, obtained from the same initial data by the combinatorial patchworking. Then there exists to > 0 such that for any t ~ (0, to] (1) b t defines an affine curve at such that the pair (N2, Rat) is homeomorphic to the pair (T,, L);

(2)

B t defines a projective c u r v e A t such that the pair R a t) is homeomorphic to the pair (2, ~).

(~p2,

A curve obtained by this construction is called a T-

curve. All real schemes of curves of degree <6 and almost all real schemes of curves of degree 7 have been realized b y the patchwork construction described above. On the other hand, there exist real schemes realizable b y algebraic curves of some (high) degree, but not realizable by T-curves of the same degree. Probably such a scheme can be f o u n d even for degree 7 or 8. The construction of T-curves is a special case of more general patchwork construction; see [17] and [13]. In this generalization the patches are more complicated: they m a y be algebraic curves of any genus with arbitrary N e w t o n polygon. Therefore, the patches d e m a n d more care than above. This is w h y we restrict ourselves here to T-curves. However, even w h e n constructing T-curves, it is useful to think in terms of blocks more complicated than a single triangle (made of several triangles). The rest of the article is d e v o t e d to applications of the p a t c h w o r k construction.

The Ragsdale Conjecture The year 1876 is often considered as the beginning of the topological study of real algebraic curves. Prior to that, topological properties were not separated from other geo-

o

0

o~ o

0

Figure 4. Real schemes of Harnack's and Hilbert's curves of degree 6.

metric properties, which are more subtle and could keep geometers busy with curves of a few lower degrees. In 1876, A. Harnack published a p a p e r [5] where he found an exact u p p e r b o u n d for the n u m b e r of components for a curve of a given degree. Harnack p r o v e d that the n u m b e r of components of a real plane projective curve of degree m is at most (m - 1)(m - 2)/2 + 1. On the other hand, for any natural n u m b e r m he constructed a nonsingular real projective curve of degree m with (m - 1)(m - 2)/2 + 1 components, which shows that his estimate cannot be i m p r o v e d without introducing new ingredients. It was D. Hilbert w h o made the first attempt to s t u d y systematically the topology of nonsingular real plane algebraic curves. The first difficult special problems he met were related to curves of degree 6. Hilbert suggested that from a topological viewpoint the most interesting are the curves having the maximal n u m b e r (m - 1)(m - 2)/2 + 1 of c o m p o n e n t s a m o n g curves of a given degree m. Hilbert's guess was strongly confirmed by the whole subsequent d e v e l o p m e n t of the field. Now, following I. Petrovsky, these curves are called M-curves. Hilbert succeeded in constructing M-curves of degree ->6 with the m u t u a l position of c o m p o n e n t s different from the ones realized by Harnack. H o w e v e r , he realized only one n e w real scheme of degree 6. See Fig. 4, where the real schemes of Harnack's and Hilbert's curves of degree 6 are shown. Hilbert conjectured that these are the only real schemes realizable by M-curves of degree 6, and for a long time claimed that he had a (long) proof of this conjecture. Although false (it was disproved by D. A. G u d k o v in 1969, w h o constructed a curve with the real scheme s h o w n in Fig. 5), this conjecture captured the essence of what in the thirties and seventies became the core of the theory. In fact, Hilbert invented a m e t h o d which allows one to answer all questions on the topology of curves of de-

~

gree 6. It involves a detailed analysis of singular curves which could be obtained from a given nonsingular one. The m e t h o d required complicated fragments of singularity theory, which had not been d e v e l o p e d at the time of Hilbert. It was only in the sixties that this project was completely realized. A complete table of real schemes of curves of degree 6 was obtained by Gudkov. We have mentioned that Hilbert's sixteenth problem is on this topic. Curious that he placed this sixteenth on his list! The number sixteen plays a very special role in the topology of real algebraic varieties. It is difficult to believe that Hilbert was aware of that. It became clear only in the beginning of the seventies (see Rokhlin's paper [15]). In 1906 V. Ragsdale [14] m a d e a remarkable attempt to analyze Harnack's and Hilbert's constructions in search of new restrictions on topology of curves. To a great extent, the success of her analysis was due to the right choice of parameters of a real scheme. Ragsdale suggested considering separately the case of curves of even degree m = 2k. Each connected component of the set of real points of a curve of even degree is an oval (i.e., divides ~p2 into two parts). An oval of a curve is called even (resp. odd) if it lies inside of an even (resp. odd) n u m b e r of other ovals of this curve. The n u m b e r of even ovals of a curve is denoted b y p, the n u m b e r of odd ovals b y n. It was Ragsdale who suggested distinguishing even and o d d ovals. Ragsdale p r o v i d e d good reasons w h y one should pay special attention to p and n. A curve of an even degree divides the plane ~p2 into two pieces with a c o m m o n b o u n d a r y NA (these pieces are the subsets of RP 2 where a polynomial defining the curve takes positive and negative values, respectively). One of these pieces is nonorientable; it is d e n o t e d by Np2. The other one is denoted by Np2. The numbers p and n are the f u n d a m e n t a l topological characteristics of RP} and Np2; namely, p is the n u m b e r of connected c o m p o n e n t s of Np2, and n + 1 is the n u m b e r of connected c o m p o n e n t s of Np2 (exactly one c o m p o n e n t of Np2 is nonorientable, so n is the n u m b e r of orientable components of Np2). Ragsdale singled out also the difference p - n, motivating this b y the fact that it is the Euler characteristic of Np2. It is amazing that essentially these considerations were stated in a paper in 1906. RAGSDALE OBSERVATION. M-curves of even degree m = 2k,

3k(k

P-

0

0o

Figure 5. Real scheme of Gudkov's curve of degree 6.

-

2

1)

For any of Harnack's

( k - 1 ) ( k - 2)

+1,

n=

2

For any of Hilbert's M-curves of even degree m = 2k,

(k - 1)(k - 2) 3k(k - 1) +l_
23

This motivated the following conjecture.

RAGSDALE CONJECTURE. For any curve of even degree m = 2k,

p--

3k(k-

1)

2

+1,

n -<

3k(k-

1)

2

p~

Writing cautiously, Ragsdale also formulated weaker conjectures. About 30 years later, I. G. Petrovsky [11,12] proved one of these weaker conjectures.

PETROVSKY INEQUALITIES. For any curve of even degree m -- 2k,

p-n

<

3k(k2

1)

+1,

n-p

<

pleted by Rokhlin [15], Kharlamov [9], and Gudkov and Krakhnov [3], marked the beginning of the most recent stage in the topology of real algebraic curves. Which of Ragsdale's questions are still open now? The inequalities

3k(k2

1)

3k(k - 1)

2

+1,

n~

3k(k-

1)

2

+1

have been neither proved nor disproved for M-curves. Ragsdale gave interesting reformulations of the first of these inequalities for M-curves. Below we present her reformulations together with the corresponding reformulations of the inequality n <- 3k(k - 1)/2 + 1.

RAGSDALE CONJECTURE ON M-CURVES. For a n y M - c u r v e of degree 2k IP - nl <- k2,

It is clear from [11] and [12] that Petrovsky was not familiar with Ragsdale's paper. But his proof runs along the lines indicated by Ragsdale. He also reduced the problem to estimates of the Euler characteristic of the pencil curves, but he went further: he proved these estimates using the Euler-Jacobi formula. Petrovsky also formulated conjectures about the upper bounds for p and n. His conjecture about n was more cautious (by 1) than Ragsdale's. Both the Ragsdale Conjecture above and Petrovsky's [12] are wrong. However, they stood for a rather long time: the Ragsdale Conjecture for n was disproved by O. Viro [16] in 1979. Viro's disproof looked rather like an improvement of the conjecture, since in the counter-examples n = 3 k ( k - 1)/2 + 1. In 1993 the RagsdalePetrovsky bounds were disproven by a considerable margin by I. Itenberg [8]: in Itenberg's counterexamples the difference between p (or n) and 3k(k - 1)/2 + 1 is a quadratic function of k (see below). The numbers p and n introduced by Ragsdale occur in many of the prohibitions that were subsequently discovered. While giving full credit to Ragsdale for her insight, we must also say that if she had looked more carefully at the experimental data available to her, she should have been able to find some of these prohibitions. For example, it is not clear what stopped her from making the conjectures which were made by Gudkov [2] in the late 1960s. In particular, the experimental data could suggest the formulation of the Gudkov-Rokhlin congruence [15]: for any M - c u r v e of even degree m = 2k p - n ~ k2 mod8.

Maybe mathematicians trying to conjecture restrictions on some integer should keep this case in mind as evidence that restrictions can have not only the shape of an inequality but also a congruence. Proof of these Gudkov conjectures, initiated by Arnold [1] and com24

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

or, equivalently, p >--

(k - 1)(k - 2) 2

and

n > -

(k - 1)(k - 2) 2

Patchworking Harnack Curves In each area of mathematics there are objects which appear much more frequently than others. Some of them (like Dynkin diagrams) appear in several domains quite distant from each other. In the topology of real plane algebraic curves, Harnack curves play this role. It was not an accident that they were constructed in the first paper devoted to this subject. Whenever one tries to construct an M-curve, the first success provides a Harnack curve. Patchwork construction is no exception to the rule. In this section we describe, using the Patchwork Theorem, the construction of some Harnack curves of an even degree m = 2k. In what follows, all the triangulations satisfy an additional assumption: they are primitive, which means that all triangles are of area 1/2 (or, equivalently, that all integer points of the triangulated area are vertices of the triangulation). A polynominal defining a T-curve contains the maximal collection of nonzero monomials if and only if the triangulation used in the construction of the T-curve is primitive. A primitive convex triangulation of T is said to be equipped with a Harnack distribution of signs if: vertex (i, j) has the sign ..... if i, j are both even, and has the sign "+" in the opposite case. A vertex (i, j) of a triangulation of T is called even if i and j are both even, and odd otherwise. PROPOSITION. Patchworking applied to an arbitrary primitive convex triangulation of T with the Harnack distri-

O0

...

0

\,~

.-,/

~

ak(k- ~) Figure 6. The real s c h e m e of the s i m p l e s t H a r n a c k curve of degree 2k.

bution of signs produces an M-curve with the real scheme shown in Fig. 6. A n e x a m p l e of the construction u n d e r consideration is s h o w n in Fig. 7. Proof of Proposition: First, note that the n u m b e r of interior (i.e., lying in the interior of the triangle T) integer points is equal to (m - 1)(m - 2)/2, the n u m b e r of even interior points is equal to (k - 1)(k - 2)/2, a n d the n u m ber of o d d interior points is equal to 3k(k - 1)/2. Take an arbitrary e v e n interior vertex of a triangulation of the triangle T. This vertex has the sign ..... . All

adjacent vertices (i.e., the vertices connected with the vertex b y edges of the triangulation) are o d d , and thus they all h a v e the sign " + . " This m e a n s that the star of an e v e n interior vertex contains an oval of the c u r v e L. The n u m b e r of such ovals is equal to (k - 1)(k - 2)/2. Take n o w an o d d interior vertex of the triangulation. It has the sign " + . " There are t w o vertices with . . . . . and one vertex with " + " a m o n g the images of the vertex u n d e r s = sx o sy and Sx a n d sy (recall that sx a n d sy are reflections w i t h respect to the coordinate axes). Consider the i m a g e with the sign " + . " It is easy to verify that all its adjacent vertices h a v e the sign " - . " A g a i n this m e a n s that the star of this vertex contains an oval of the c u r v e L. The n u m b e r of such ovals is equal to 3k(k - 1)/2. But (k - 1)(k - 2) +

3k(k

2

-

1)

(m - 1)(m - 2)

2

2

so the c u r v e can only h a v e one m o r e oval. This oval exists because, for example, the c u r v e L intersects the coordinate axes.

~r

rd

tar

rat

rd~

~

~

~

tar

r~r

rat

rat

~

~

~

~

~

~

~

~r

rat

~ ~r

~r

rd

Figure 7. P a t c h w o r k of the s i m p l e s t Harnack curve of degree 10. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

25

oo

k.

~r

.,d

o9

20 Figure 8. Schemes of counterexamples to the Ragsdale Conjecture of degree 10. To finish the proof, we need only note that the union of the segments {x - y = -m, - m -< x,y-< m} U {x<-O, y = O, - m ~- x,y<-m} tO {x = 0, y_< 0, -m<-x,y<<_m}

is not contractible in ~ and contains only minuses. This means that 3k(k - 1)/2 ovals corresponding to odd interior points and encircling pluses are situated outside of the nonempty oval. []

C o u n t e r e x a m p l e s to the Ragsdale Conjecture The following theorem gives counterexamples to the Ragsdale Conjecture (or the conjecture of Petrovsky) [8]. THEOREM: For each integer k ~ 1 (a) there exists a nonsingular real algebraic plane projective curve of degree 2k with 3 k ( k - 1 ) + 1 + [ ( k - 3)2 + 4]; 2 8

P=

(b) there exists a nonsingular real algebraic plane projective curve of degree 2k with n

-

3k(k-1)2 + [ ( k - 3 ) 2" + 48]

Figure 9. Hexagon S and the patchwork fragment produced by it. 26

THE M A T H E M A T I C A L INTELL1GENCER VOL. 18, NO. 4, 1996

Remarks: The term [ ( ( k - 3)2+ 4)]/8 is positive when k _> 5 (i.e., we get counterexamples to the Ragsdale Conjecture in degree 10). On the other hand, it is known that there is no counterexample to the Ragsdale Conjecture among curves of lower degree. These counterexamples of degree 10 have the real schemes shown in Fig. 8. One of these counterexamples can be improved to obtain a curve of degree 10 with n = 32. The corresponding patchworks (for p = 32 and n = 32) are shown " in Figs. 2 and 3. Recently, B. Haas [4] improved the construction presented below and obtained T-curves of degree 2k with

p _ 3k(k - 1) + 1 + I k a - 7k+ 16 ] 6 Neither the counterexamples provided by the above theorem nor the curves constructed by Haas are Mcurves. Moreover, as we mentioned above, it is not known if the conjecture of Petrovsky holds for Mcurves. The only known counterexamples to the Ragsdale Conjecture among M-curves are the curves constructed by Viro [16] (see also [18]). It is curious that we did not succeed in presenting those M-counterexamples as T-curves. Proof of Theorem: Let us show, first, how to construct a curve of degree m = 2k with p = 3k(k - 1)/2 + 2. Suppose that the hexagon S shown in Fig. 9 is placed inside of the triangle T in such a way that the center of S has both coordinates odd. Any convex primitive triangulation of a convex part of a convex polygon is extendable to a convex primitive triangulation of the polygon. Inside of the hexagon S, let us take the convex primitive triangulation shown in Fig. 9 and extend it to T.

Figure 10. Real scheme of a curve of degree 2k with p = 3k(k - 1)/2 + 2.

Figure 12. Real scheme of a curve of degree 2k with p =

3k(k

-

1)/2 + 1 + a.

Figure 11. Partition of T for part (a).

To apply the Patchwork Theorem we need to choose signs at the vertices. Inside of S put signs according to Fig. 9; outside, use the Harnack rule of distribution of signs. It is easy to calculate that the corresponding piecewise-linear curve ~ has exactly one even oval more than the Harnack curve constructed above [i.e., now p = 3k(k - 1)/2 + 2]. One can verify that the curve obtained has the real scheme shown in Fig. 10. Consider the partition of the triangle T shown in Fig. 11. Let us take in each shaded hexagon the triangulation and the signs of the hexagon S. The triangulation of the union of the shaded hexagons can be extended to the primitive convex triangulation of T. Let us fix such an extension. Outside of the union of the shaded hexagons, choose the signs at the vertices of the triangulation using the Harnack rule. Calculation shows that for the corresponding piecewise-linear curve 1)

3k(k-

P-

2

+l+a,

where a is the number of shaded hexagons, and a = [ ( k - 3 ) 2"+ 48] This curve has the real scheme shown in Fig. 12. To prove part (b) of the theorem, let us take, again, the partition of the triangle T shown in Fig. 11 with the triangulation and the signs of each shaded hexagon co-

Figure 13. Partition of T for part (b).

Figure 14. Real scheme of a curve of degree 2k with n =

3k(k

-

1)/2 + a.

inciding with the triangulation and the signs of S. Fix, in addition, a triangulation of a neighborhood of the axis OY and the signs at the vertices of the triangulation as shown in Fig. 13 [the case k ~- 1 (mod 4)]. The chosen triangulation of the union of the shaded hexagons and the neighborhood of the y-axis can be extended to a primitive convex triangulation of T. Outside of the union of the shaded hexagons and the neighborhood of the y-axis, let us again choose the signs at the vertices of the triangulation using the Harnack rule. The corresponding piecewise-linear curve ~ has the real scheme shown in Fig. 14. In this case n = 3k(k - 1)/2 + a. [] THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

27

References 1. V.I. Arnold, On the location of ovals of real algebraic plane curves, involutions on four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms, Funktsional. Anal. Prilozhen. 5 (1971), 1-9 (in Russian); English translation in Functional Anal. Appl. 5 (1971), 169-176. 2. D. A. Gudkov and G. A. Utkin, The topology of curves of degree 6 and surfaces of degree 4, Uchen. Zap. Gorkov. Univ. 87, (1969) (in Russian) 14-20 and 154-176; English transl, in Amer. Math. Soc. Transl. 112, Series 2. "Nine Papers on Hilbert's 16th Problem" (1978), 9-14 and 123140. 3. D. A. Gudkov and A. D. Krakhnov, On the periodicity of the Euler characteristic of real algebraic (M - 1)-manifolds, Funktsional. Anal. Prilozhen. 7 (1973), 15-19; (in Russian); English translation in Functional Anal. Appl. 7 (1973), 98-103. 4. B. Haas, Les multilucarnes: Nouveaux contre-exemples la conjecture.de Ragsdale, C. R. Acad. Sci. Paris (in press). 5. A. Harnack, Uber Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), 189-199. 6. D. Hilbert, Ober die reellen Ziige algebraischen Curven, Math. Ann. 38 (1891), 115-138. 7. D. Hilbert, Mathematische Probleme, Arch. Math. Phys. 3 (1901), 213-237. 8. I. Itenberg, Contre-exemples a la conjecture de Ragsdale, C. R. Acad. Sci. Paris (S6rie I) 317 (1993), 277-282. 9. V. M. Kharlamov, New congruences for the Euler characteristic of real algebraic varieties, Funktsional. Anal. Prilozhen. 7 (1973), 74-78 (in Russian); English translation in Functional Anal. Appl. 7 (1973), 147-150. 10. I. Newton, Enumeratio linearum tertii ordinis, London (1704). 11. I. Petrovsky, Sur la topologie des courbes r6elles et alg6briques, C. R. Acad. Sci. Paris 197 (1933), 1270-1272. 12. I. Petrovsky, On the topology of real plane algebraic curves, Ann. Math. 39 (1938), 187-209. 13. J.-J. Risler, Construction d'hypersurfaces r6elles (d'apr6s Viro), Sdminaire N. Bourbaki (1992-93), no. 763. 14. V. Ragsdale, On the arrangement of the real branches of plane algebraic curves, Am. J. Math. 28 (1906), 377-404. 15. V. A. Rokhlin, Congruences modulo 16 in Hilbert's sixteenth problem, Funktsional. Anal. Prilozhen. 6 (1972), 58-64 (in Russian); English translation in Functional Anal. Appl. 6 (1972), 301-306. 16. O. Ya. Viro, Curves of degree 7, curves of degree 8 and the Ragsdale conjecture, Dokl. Akad. Nauk SSSR 254 (1980), 1305-1310 (in Russian); English translation in Soviet Math. Dokl. 22 (1980), 566-570. 17. O. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Math. 1060, New York: Springer-Veflag, (1984); pp. 187-200. 18. O. Ya. Viro, Plane real algebraic curves: constructions with controlled topology, Algebra Analiz 1 (1989), 1-73 (in Russian); English translation in Leningrad Math. J. 1 (5) (1990), 1059-1134. Institut de Recherche Mathdmatique de Rennes Campus de Beaulieu 35042 Rennes Cedex France Box 480 Uppsala S-75106 Sweden 28

T H E M A T H E M A T I C A L INTELLIGENCER VOL. 18, N O . 4, 1996

SpringerNewsMathematics Hans Hahn Gesammelte Abhandlungen / Collected Works L. Schmelterer, K. Sigmund (Hrsg./eds.) Mit einem Geleitwort yon / With a Foreword by Karl P o p p e r Like Descartes a n d Pascal, H a n s H a h n (1879-1934) was both an eminent mathematician a n d a highly influential philosopher. He founded the Vienna Circle a n d was the teacher of both K u r t G6del a n d Karl Popper. His seminal contributions to functional analysis and general topology h a d a huge impact on the development of m o d e r n analysis. H a h n ' s passionate interest in the foundations of mathematics, vividly described in Sir Karl Popper's foreword (which became his last essay) h a d a decisive influence u p o n K u r t G6del. l a k e Freud, Musil or Schiinberg, H a h n became a pivotal figure in the feverish intellectual climate of Vienna between the two wars.

B d . 1 / Vol. 1: 1995. XlI, 511 pages. Cloth DM 198,-, approx.US $140.00.ISBN 3-211-82682-3 The first volume contains H a h n ' s path-breaking contributions to functional analysis, the theory of curves, and ordered groups. These papers arc commented by H a r r o Heuser, Hans Sagan, a n d Laszlo Fuchs. B d . 2 / V o l . 2: 1996. Approx. 560 pages. ClothDM 198,-, approx.US $140.00.ISBN3-211-82750-1 The second volume of H a h n ' s Collected Works deals with functional analysis, real analysis and hydrodynaInics. The commentaries are written by Wilhelm F r a n k , Davis Preiss, a n d Alfred Kluwick. B d . 3 / V o l . 3: Approx. 480 pages. ISBN 3-211-82781-1. Will be published in FaU 1996. In the third volume, H a h n ' s writings on harmonic analysis, measure a n d integration, complex analysis and philosophy arc collected and commented by Jean-Pierre Kahane, Heinz Bauer, Lutger K a u p , and Wolfgang Thiel. This volume also contains excerpts of letters of H a h n a n d accounts by students and colleagues. Subscription price (only valid when taking all three volumes): 20 % price reduction

SpringerWienNewYork P.O.Box 89, A-1201 Wien New York, NY 10010,175 Fifth Avenue Heidrlberger Platz 3, D-]4197 Berlin Tokyo 113, 3-13, Hongo 3-ehome, Bunkyo-ku

David Gale* The end of David Gale's regime in this column, announced an issue ago but occuring only now, leaves us with a large gap. David is a hard act to follow. But wait till you see the column to come! The new Editor is Alexander Shen of Moscow. As before, readers are invited not only to be enter-

tained but to join in. We welcome guest articles, solutions to problems, and best of all, new light on them. Contributions can be sent to the column editor at [email protected] or to the Editor-in-Chief.

April Fool! After writing that tearful farewell column in the last issue it seems I'm still here. I hope you readers will excuse me, but editor Davis pleaded with me so pathetically that I had to agree to do just one more column, thus rounding out a full 6 years on the job. Here goes with three final items, a Pattern Problem, a Probability Paradox, and a Pretty Proof.

(*) The ij-th entry aij = mex{Si), where Sij is the set of entries to the left of it in row i or above it in column j.

Problems and Patterns If 5 is a set of non-negative integers, we define mex{S}

to be the smallest non-negative integer which is not a member of S. (mex is an abbreviation for minimum-excluded (Berlekamp, Conway, and Guy, Winning Ways).) The two-dimensional array A below, assumed to extend out to infinity, satisfies the following condition:

PROBLEM: What is the entry in row 777, column 1001? [Of course we are really asking for an explicit expression for the ij-th entry.] It is clear that there is one and only one array satisfying (*), namelyaoo must be zero. Hence, entries aOI and al0 are 1, etc., and one sees that in building up the matrix, row by row (or any other way), the entries at each stage are "forced," and one can grind out terms at will without resorting to computational technology. Here as elsewhere we index rows and columns starting with O. What emerges is highly unexpected. Indeed, it is hard to see how one could have guessed the result without this preliminary experimentation. Table 1 is a section of the array.

Table 1

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 3 2 5 4 7 6 9 8 3 a 1 6 7 4 5 10 11 2 1 a 7 6 5 4 11 10 5 6 7 a 1 2 3 12 13 4 7 6 1 a 3 2 13 12 7 5 4 2 3 a 1 14 15 6 4 5 3 2 1 a 15 14 9 10 11 12 13 14 15 a 1 8 11 10 13 12 15 14 1 a 11 8 9 14 15 12 13 2 3 10 9 8 15 14 13 12 3 2 13 14 15 8 9 10 11 4 5 12 15 14 9 8 11 10 5 4 15 12 13 10 11 8 9 6 7 14 13 12 11 10 9 8 7 6 17 18 18

a

10 11 8 9 14 15 12 13 2 3

a

11 10 9 8 15 14 13 12 3 2 1

1 6 7 4 5

7 6 5 4

a

12 13 14 15 8 9 10 11 4 5 6 7

13 12 15 14 9 8 11 10 5 4 7 6 a 1 1 a 2 3 3 2

14 15 12 13 10 11 8 9 6 7 4 5 2 3

a 1

15 16 17 18 14 17 16 19 13 18 12 11 10 9 8 7 6 5 4 3 2 1

a

·Column editors address: Department of Mathematics, University of California, Berkeley, CA 94720 USA. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO.4 © 1996 Springer-Verlag New York

29

Table 2 011 2 1 10 3 2 3 0 3 2 1 4 5 6 5 4 7 6 7 5 764 8 9 10 9 8 11 011 8 1 10 9 2 13 14 3 12 15 4 15 12 ~5 14 13 ~6 17 18 ~7 18

3 2 1 0 7 6 4 5 11 10 9 8 15 14 13 12

4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11

5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10

6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9

7 6 5 4 3 2 1

8 9 10 1 12 13 14 o 15 15 0 14 1 13 2 12 3 11 4 10 5 9 6 8 7

What is going on here? The partitioning of A in Table 2 shows what is happening. As noted, aoo must be 0, aOl and alO are I, and all is again O. If we call this 2 x 2 matrix All, then we see that the adjacent 2 x 2 matrices A 12 and A 21 to the right of and below All are All with 2 added to each of the entries, and An is All translated along the diagonal. We now have a 4 x 4 matrix A 44• To get the 4 X 4 matrices below and to the right of A 44, one adds 4 to the entries of A 44, and A 44 is then repeated on the diagonal giving an 8 x 8 array. Continue in this way to get a 16 x 16 array, and so on. The general structure is now apparent. The remarkable feature of the array is the totally unexpected role played by the powers of 2. There is nothing in condition (*) that would lead one to anticipate this. Having noted it, however, one then naturally writes the integers in binary notation, at which point the secret is revealed. Table 3 is an initial section of the array. DEFINITION: If i and j are integers, we consider their binary expansions as vectors over Z2' Then their nim sum, k = i
1 0

11 100 101 10 101 100 10 11 1 110 111 0 11 10 1 0 111 110 100 101 110 111 1 0 30

10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5

11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4

12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3

13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2

14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1

15 16 17 18 14 17 16 19 13 18 12 11 10 9 8 7 6 5 4 3 2 1 0

THEOREM: The entry aij in A is i
An interesting consequence of the Theorem is that A is the multiplication table for an abelian group structure on the non-negative integers, for the nim sum treats an n-digit binary expansion of an integer as an element of the n-dimensional vector space over Z2' One can verify that the entries of A satisfy the associative law. Tables 1-3 supply very convincing "evidence" for the correctness of our theorem, but the proof below is perhaps not completely trivial. Proof: Let i rand i'r = 0, so i' is less than i, thus, m = ai'i is to the left of aij' What about the original question, to find the entry in row 777, column 1001? Well, we have 777 ~ 1 1 0 0 0 0 1 0 Oland 1001 ~ 1 1 1 1 1 0 1 0 0 1 777
Table 3

0 1

9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6

10 11

110 111 100 101 10

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO.4, 1996

111 110 101 100

1000 1001 1010 1011

=

0011100000

so =

224.

Remarks: The theorem survives intact for n-dimensional arrays: namely, (*) becomes the condition that the entry at (iI' ... , ik, ... , in) is mexlentries at (ill ... , i 'k' ... , in) for all k, where i 'k < ik}· Then entry (iI' ... , ik, ... , in) turns out to be i1
of Nim with n piles, where pile k contains ik elements. People familiar with Nim will know that a position is a second-player win if and only if the nim sum of the ik is O. In general, we may think of i 1 EB i EB 2 ... EB in as the value of the Nim position (i 1, ••. , ik, ••• , in), and our theorem asserts that a player can move to any lower value by withdrawing from only one of the piles. Twopile Nim is of course not interesting, because the 0's occur exactly on the diagonal; but in the three-pile case, for example, 1 EB 2 EB 3 = O. One can create variations of the original problem by changing the definition of the set Sij- One interesting variation is to include in S the backward diagonals; that is, augment Sij to include all ai'j' with i' < i, j' < j, and i' - j' = i - j. For this case no explicit expression for the aij is known, but there is a method for finding the 0's. This corresponds to a 2-pile Nim-like game in which a move consists in reducing one of the piles by an arbitrary amount, as in ordinary Nim, or reducing both piles by the same amount. A Nim variation which is still unsolved is ordinary Nim with the additional possibility that once during the game either player, but not both, may "pass"; except that a pass is not allowed after all the pieces have been removed. Again, the last player to move is the winner. Let's say $100 for a solution.

Another Probability Paradox A random variable X with a known distribution takes on integer values from a finite ordered set S. For example, S might be the integers from 1 to 10. Let Pi denote the probability that X equals i, and let 0i denote the probability that X is greater than or equal to i. The "game" (against nature) r 2 consists in making a sequence of independent observations of values Xl' X2, ••. of X, where the sequence terminates as soon as some Xi is the "second highest," meaning that there is exactly one previous term of the sequence which is greater than or equal to Xi' The observer then receives a payoff equal to this Xi' The game r k is the same except that second highest is replaced by the k-th highest. The game r k is the same as r k except that the k-th highest is replaced by the k-th lowest. Example: The payoff to r 4 of the sequence (3, 7, 2, 5, 1, 6, 1,5*,9,2, ... ) is 5 since exactly three previous observations, X2, X 4, and X6, were 5 or greater.

Note: In the games

r 1 and r 1, the payoff is simply Xl'

Question: Which of the games r 2, advantageous for the observer?

r 17, r z, etc., is most

Answer: They are all equivalent. That is,

THEOREM: The probability distribution of the paYoff of r k and r k is the same as the original distribution on S. This statement seems at first counterintuitive. One feels that it ought to be better to bet on the second best rather than the 17th best or second worst. This theorem is essentially the discrete case of a result in probability theory known as Ignatov's theorem. The result can be proved by direct computation, looking at the combinatorial possibilities among the set of all sequences of observations and observing some identities among binomial coefficients. Instead, I present a short indirect proof which requires essentially no computation. The informal argument that follows can be easily formalized. It depends on a tricky characterization of the payoff of r k• Notation: given a sequence Z = (Xli X2, . . .) of observations, let Zr be the subsequence consisting of all terms of Z which are greater than or equal to r. Terminology: We will call the payoff of

r k the k-payoff.

LEMMA. The k-payoff of the sequence Z = (Xl' Xz, ...) is the smallest number r in S such that the k-th term of Zr is r.

Proof: If the game terminates with payoff r, this means that exactly k - 1 previous observations were greater than or equal to r; but this is means that r is the k-th term of 2 r• Finally, r must be the smallest number with this property, for if there were a smaller one, the game would have terminated earlier. The theorem follows directly from the lemma; namely, the probability that the k-th term of 2 s is [is not] 5 is p'/0s [Ds+dOs]. Further, if the k-th term of 2 s is not s, then it is greater than 5, and this gives no information on the value of the k-th term of 2 s + 1' Thus, the probability of payoff r is 0I(OZ/OI) ... (Or/ 0 r-l)(Pr/ Or) = Pro The fact that the theorem also holds for the games r k follows by symmetry. But one can say more than this. Suppose, given a sequence 2, one knows that the 2-payoff is, say, 5. What can one infer from this about the 3-payoff? The 17-payoff? Answer: nothing. In other words, given the sequence Z, if j is unequal to k, then the j- and k-payoffs are independent. This follows again from the lemma, which shows that the j-lk]-payoff depends only on the j-th lk-th] terms of the sequences 2 1, Zz, ... ; but these terms are independent since the original observations were independent. More generally, knowing the payoffs of any finite set F of values gives no information about values not in F, which is what is meant by independence.

A Pretty Proof One of the themes of this column from time to time has been variational or max and min problems. In this column a year ago John Halton, in "The Shoelace Problem," THE MATHEMATICAL INTELLIGENCER VOL. 18, NO.4, 1996

31

found the answer to the natural "everyday problem" of the most economical way to lace a shoe. The following exposition written by Michal Misiurewicz gives a variation and extension of Halton's result.

AO

A

1

Lacing Irregular Shoes Michaf Misiurewicz In his recent article [1], Halton considers the problem of the most efficient (in terms of the length of the lace) lacing of a shoe. Let us recall the problem. We are given 2(n + 1) distinct points (the eyelets) A o, AlJ ... , Ani Bo, Bl , . . . , Bn in the plane. A lacing is a path Co ~ Cl ~ ... ~ C2n +l , where {Co, Cl , . . . , C2n +1} = {A o, AlJ ... , An' Bo, BlJ ... , Bn}, Co = A o, C2n +l = Bo, and the points of the sets {A o, AI, ... , An} and {B o, BlJ ... , Bn} alternate along the path. Its length is ICOCII + ICl C21 + ... + IC2nC2n+ll. It is the length of the lace used, not including the slack pieces necessary to tie the lace. The problem is to find the lacing of minimal length. Halton shows that the standard lacingl is the shortest one. The standard lacing is the lacing A o ~ Bl ~ A 2 ~ B3 ~ ••• ~ A n- l ~ Bn ~ An ~ Bn- l ~ ... ~ A 3 ~ B2 ~ Al ~ Bo if n is odd, and A o~ Bl ~ A 2~ B3 ~ ... ~ Bn- l ~ An ~ Bn ~ A n- l ~ ... ~ A 3 ~ B2 ~ Al ~ Bo if n is even (see Fig. 1). The purpose of this article is to give a short proof of a considerably more general result. Halton assumes that the eyelets are arranged in two parallel rows with equal distances between consecutive eyelets. More precisely, there are constants v, w > 0 such that, up to isometry, Ai = (iv, w) and Bi = (iv, 0) (see Fig. 2). This is a crude approximation of a real situation (see Fig. 3), where both "horizontal" and "vertical" distances between eyelets may vary. Even the symmetry with respect to a horizontalline can be absent, especially if the shoe is old. I want to propose a proof of minimality of the standard lacing under an assumption that seems to be much closer to the real situation:

(l)

For any k, 1, the line through A k and BI separates the sets {Ai: i < k} U {B j : j < l} and {Ai: i > k} U {B j : j > l}.

In other words, if we view the"A" eyelets from any "B" eyelet (or the "B" eyelets from any "A" eyelet), they come in the natural order: A o, AI, ... , An (or Bo, Bl , ... , Bn). In fact, even this assumption is too strong. We need only the following one: IThis lacing is called American style in Ref. 1. However, I do not recall anybody, be it in America or in Europe, having his or her shoes laced in a different way.

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THE MATHEMATICAL INTELLIGENCER VOL. 18, NO.4, 1996

Figure 1. Standard lacing.

(2)

For any i < j and k < 1, IAiBkl + IAjBII < IAiBII + IAjBkl.

To see that (l) implies (2), take i < j and k < 1, and as-' sume that (l) holds. Then as in Figure 4, the eyelets Ai and BI lie on opposite sides of the line through A j and BkJ and the eyelets Aj and Bk lie on opposite sides of the line through Ai and BI. Therefore, the segment AiBI intersects the segment AjBk. If D is the point of intersection, then by the triangle inequality we get IAiBkl

+ IAjBII < IAPI + IDBkl + IAPI + IDBII = lAm

+ IAPkl,

so (2) holds. One can try to consider an even more realistic setup, where the eyelets are points of three-dimensional space. However, a lacing should closely follow the surface of the shoe. Therefore, we can assume the eyelets to be points of the surface of the shoe, with distances measured along this surface. The reader can try to make precise measurements to verify whether in this setup his or her shoes satisfy assumption (2). But usually the piece of the shoe surface involved in lacing has a very small curvature. This means that we do not make a big error if we view this surface as a "bent" piece of plane. Smoothing it out leads us to our initial model. Now a quick glance is enough to verify whether assumption (1) is satisfied.

AO

0

o

o

o

o

a

o

Figure 2. Halton's arrangement of eyelets.

Figure 5. An arbitrary lacing.

Figure 3. A real shoe.

It is time to prove the main result of this article.

THEOREM: Assume that the set of eyelets {A o, AI' ... , An' Bo, ... , Bnl satisfies (2). Then the standard lacing is shorter than any other lacing. The idea of the proof is rather simple. Referring again to Figure 4, let us define a "move" to consist of replacing the pair of segments AiBI and AjBk by AiBk and AjB/. The only problem is that what we get as the result of such a move may not be a lacing. For instance, if we start with the lacing shown in Figure 5 and make a move that replaces segments A oB2 and An-IB I by AoB I and AIl - I B2, then what we get (Fig. 6) is not a lacing. Therefore, we will introduce objects that are more general than lacings, namely systems of arrows. In this larger class, all moves will be legitimate, and after a finite number of moves we will arrive at the system corresponding to the standard lacing. These ideas lead to the following formal proof. Proof of Theorem: Let L = Co ~ CI ~ ••• ~ C2n + 1 be a lacing (see Fig. 5). There is some k such that Ck = All' We

A.~

Figure 4. (1) implies (2).

Figure 6. Result of a move.

draw arrows from Ci to Ci + l for i = 0/ 1, ... , k - 1 and from Ci + l to Cj for i = k, k + 1, .. . 2n (see Fig. 7). Note that this system of arrows satisfies the following properties. (3)

Each arrow begins at an "A" eyelet and ends at a "B" eyelet or vice versa; all eyelets except .40, Bo, and An have one arrow coming in and one going out; eyelets A o and Bo have one arrow going out and none coming in; eyelet An has two arrows coming in and none going out.

A, J

Figure 7. A system of arrows obtained from the lacing from Figure 5. THE MATHEMATICAL INTELLIGENCER VOL. 18. NO.4, 19%

33

We define the length of a system of arrows as the sum of their lengths. Although not every system of arrows comes from a lacing, if it does, then its length is equal to the length of the corresponding lacing. Let us call the system of arrows obtained from the standard lacing the standard system. We will show that it has smaller length than any other system of arrows satisfying (3). Indeed, if this is not the case, then there exists a system of arrows :A satisfying (3) that has minimallength and is not standard. If the arrows A k ~ Bk+1 and Bk ~ Ak+1 for k = 0, ... , n - 1 are in :A, then by (3), :A consists of these arrows and the arrow Bn ~ An' This means that the system :A is standard, a contradiction. Therefore, we can take the smallest k such that at least one of the arrows A k ~ B+ 1, Bk+1~ Ak+1 is not in:A. We may assume that this is Bk ~ Ak+1' Thus, we must have in :A arrows Bk ~ Ai and B1 ~ Ak+1 for some j k + 1

'*

1.

J. H. Halton, The shoelace problem, Math. Intelligencer 17: 4 (1995), no. 4, 36-40.

Department of Mathematical Sciences

IUPUI Indianapolis, IN 46202-3216, USA e-mail: [email protected]

Evariste Galois

(Endbetrag) von Ilona Bodden

(Sum Total) by Ilona Bodden translated by Kurt Bretterbauer

Die Republik-woher sollte sie?wird auch nicht mehr Verstand besitzen als ihre VorgangerSie baut sich aus ME SCHE auf. Ich sterbe mit zwanzig Jahren als Opfer der einzig unendlichen Grosse, der menschlichen Dummheit. (Denn selbst das Universum ist endlich und somit nicht un-berechenbar.) Ich sterbe.

My calculations were wrong. The republic also will not-how should it?possess more common sense than its predecessorsit is built of PEOPLE. I die aged twenty as a victim of the only infinite quantity, human stupidity. (For even the universe is finite, and hence not uncalculable.)

I die. That is an equation with two unknowns:

Das ist eine Gleichung mit zwei Unbekannten: x Tod und y Gott = das unendliche ICHTS ...

34

References

Evariste Galois*

Meine Berechnungen waren falsch.

~Reprinted by

'*

and 1 k. Since any arrow ending at Ai with j :s k begins at Bi - 1 and any arrow beginning at B1 with 1 < k ends at A 1+ 1J we have j > k + 1 and 1> k. If we now make the move replacing the arrows Bk ~ Ai and B1 ~ Ak+1 by the arrows Bk ~ Ak+1 and B1 ~ Ai' the new system will still satisfy (3). However, by (2) (with i = k + 1), its length will be smaller than the length of :A, a contradiction. This completes the proof. •

permission.

THE MATHEMATICAL lNTELLIGENCER VOL. 18, NO.4, 1996

x Death plus

yGod = the infinite OTHING. Kurt Bretterbauer Institut fiir Theoretische Geodtisie und Geophysik Gusshaussfrasse 25-29/128 1040 Vienna Austria

Lifelong Symmetry: A Conversation with H. S. M. Coxeter Istv n Hargittai

Harold Scott MacDonald Coxeter (b. 1907, London) is Professor Emeritus in the Department of Mathematics of the University of Toronto. When Buckminster Fuller published his magnum opus Synergetics, he dedicated his work to H. S. M. Coxeter, characterizing him in the following way: "By virtue of his extraordinary life's work in mathematics, Dr. Coxeter is the geometer of our bestirring twentieth century, the spontaneously acclaimed terrestrial curator of the historical inventory of the science of pattern analysis." My wife and I visited Professor Coxeter on August 1, 1995, and the following conversation was recorded on that occasion. Istv~n Hargittai (IH): You have three first names. Which is the one you like most? H. S. M. Coxeter (DC): I prefer to be known as Donald. The original intention of my parents was to call me MacDonald Scott Coxeter but some stupid godparent said that I should be named after my father and they added Harold at the beginning. That made Harold MacDonald Scott. The initials then would look like a ship, H. M. S., Her Majesty's Ship. This is why they switched the two names, and it became Harold Scott MacDonald. What I have done lately is to use H. S. MacDonald Coxeter.

IH: Grandchildren?

DC: I have five grandchildren and five great-grandchildren. IH: You wrote somewhere that your hobby was music and

travel. When you listen to music, do you relate it in any way to geometry? DC- Not directly, but the artistic feeling that one has is very much the same in both cases. Before I took up mathematics, I was very interested in music, to the extent that I tried to compose. Between the ages of 7 and 14 I did a lot of musical composition, under the guidance of Tony Galloway, an old friend of m y family who was a very expert violinist and a sadly neglected composer. He taught me about the theory. I wrote a lot of piano pieces, and songs that my father used to sing. I was even so ambitious as to write a string quartet. However, very few of them are worth preserving. Two

IH: You have a son and a daughter. Did they follow your footsteps?

DC: Not at all. My son got interested in the church and took a degree, Master of Theology. As a minister he did not fully enjoy anything except the parish visiting, looking after unfortunate people. Eventually he gave that up and got a second degree as Master of Social Work. He did something about rehabilitation of drug addicts, then got interested in geriatric hospitals and getting supplies for them and he is still in that position now in the State of New Jersey. My daughter married an accountant. She is a Registered Nurse and lives in a small place between Toronto and Hamilton. We can visit her more easily than our son who is 800 km away. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

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T h e C o x e t e r s w i t h t h e i r p r e s e n t f r o m E s c h e r . ( 9 1996 M.C. E s c h e r / C o r d o n A r t - B a a r n - H o l l a n d . A l l f i g h t s r e s e r v e d . ) 3 6 THE MATHEMATICALINTELL1GENCERVOL.18, NO. 4, 1996

samples can be seen after the biographical sketch at the beginning of my new book, Kaleidoscopes. It was edited by F. A. Sherk, one of my former students. He collected 26 of my papers that had to do with symmetry. IH: Who turned your attention to geometry? DC: It was pretty much by myself. I was always interested in the idea of symmetry. When I was 14, I was in a boarding school in England, and happened to have some trivial illness; in the school sanatorium I was put in a bed next to a boy called John Flinders Petrie and he became a firm friend. (He was the only son of Sir Flinders Petrie, the great Egyptologist.) He and I looked at a geometry textbook with an Appendix on the five Platonic solids. We thought how interesting they were and wondered why there were only five, and we tried to extend them. He said, if you can put three squares around a corner to make a cube, what about putting four squares around a comer? Of course, they'd fall flat, giving a pattern of squares filling the plane. He, being inventive in words, called it a "tesserohedron." He called the similar

arrangement of triangles a "trigonohedron." Later on he said, what about the limitation of putting four squares around the corner and why not more than four? Maybe you can put six squares around the corner if you don't mind going up and down in a zigzag formation. Thus he discovered a skew polyhedron with "holes," a kind of infinite regular sponge. He also noticed that the squares in this formation belong to the cubic lattice. He saw that it can be reciprocated so that instead of six squares at each vertex you have four hexagons. He noticed that this could be obtained from the uniform honeycomb of truncated octahedra fitting together to fill space. The hexagons of the truncated octahedra come together, four at each vertex, and continue to form a sponge filling all space; so this was a second skew polyhedron. Then I said if you can have six squares and you can have four hexagons, why not even more: why not have six hexagons at the vertex as in the space-filling of tetrahedra and truncated tetrahedra? Then we extended the Schl/ifli symbol by which the cube is called {4, 3}, and we called these new polyhedra {4, 614} and {6, 4]4}, and {6, 6[3}, the number after the stroke indicating the nature of the holes one sees in the sponge. Before we left school, we went on to consider what would happen in four or more dimensions, and other things which later we learned had been discovered before, by L. Schl/ifli in Switzerland. IH: Did your friend also continue in geometry? DC: He did, and became quite clever at it. Unfortunately, because his father belonged to University College London, and my teacher wanted me to go to Cambridge, we went to different universities. He did quite well at University College and then the War came, W.W.II; he enlisted as an officer and was taken prisoner by the Germans. He organized a choir there. After the War ended and he was released, he went to a well-known school in southwest England, Dartington Hall, and he had a rather trivial job there. He never seemed to fulfill his early promise. He just became a tutor who looked after children who were not doing well in school. But he still corresponded with me, and it was he who noticed that when you take a regular polyhedron and look at the edges, you see that there is a zigzag of edges that go round and close up; for instance, if you take those edges of a cube that do not involve one pair of opposite vertices, they form a skew hexagon. We call this the "Petrie polygon," and it is now a well known property of a regular polyhedron to have a Petrie polygon: a skew polygon in which every two consecutive edges, but no three, belong to a face. IH: Is he retired now?

With Magdolna Hargittai, contemplating the mathematical sculptures of George Odom.

DC: No, he died. A very sad story. He married a very lovely lady and had a daughter and all went well. Then THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

37

somehow his wife got a heart attack and died. He was so distraught, missed her so terribly that he didn't k n o w where he was going, and he walked into a m o t o r w a y in England where the cars were going at a huge speed and he just didn't k n o w what was happening and one of them killed him, just two weeks after his wife died. This was about 24 years ago. II-I: Buckminster Fuller called you "the geometer of the twentieth century." How did you get to know each other? DC." This is a terribly exaggerated statement, but he was given to that sort of writing and speaking. He was a dear old man, and I was quite fond of him, but he had overblown his stars as a mathematician. He was really a very good architect and a very good engineer. His geodesic domes are really a wonderful thing. But w h e n he got into mathematics he was a little bit amateurish.

DC" As in all branches of mathematics, there is a tremendous increase in productivity. Research goes on, and m u c h of it I have no inkling of. If you only look at the development of Mathematical Reviews, w h e n they first started about 1940, it was quite a thin volume, and each m o n t h they got more a n d more. Eventually there were h u n d r e d s of thousands of papers being written, and so the later volumes are ever so much thicker than the original ones. IH" Geometry is very important in chemistry. We have simple but very helpful models of molecular geometry, but teaching them in a freshman chemistry course in the U.S. is rather hindered by the students' lack of knowledge of basic geometry. DC: It's even worse in England, where in school they teach almost no geometry.

II-I" Did he claim that he was a mathematician?

IH" Your books are full of quotations. How do you collect them?

DC" I think so, yes. He liked to invent different names for things. For instance, the cuboctahedron he called "vector equilibrium" or something like that.

DC" Just by noticing. I m u s t have read a lot, and I just remember them.

IH" How much interaction did you have with him?

IH: Do you return to books that you'd read before or just keep moving on to other books?

DC." Very little. Once Hendrina and I visited him in his

home in Southern Illinois. I have a friend w h o is a Professor of Philosophy in Carbondale, and while we were there, we visited Bucky's polyhedral house. As people passed by, they were very curious, and he finally had to build a high fence around the house so that people shouldn't see it and he could have some peace. IH" How far back can we detect the regular polyhedra in human history? DC" Of course, Plato wrote about them, and this is w h y they are called Platonic solids. Obviously the Pythagoreans k n e w them before that. Sometimes the archeologists find dodecahedral dice. That sort of thing is what I mean w h e n I say that we don't k n o w how far back they go. IH: In some of your writings you distinguish between crystallographic solids and others such as is the icosahedron and dodecahedron. Nowadays, however, this distinction is quite blurred. DC: That's true. Just look at the writings of Professor

Marjorie Senechal. I'm just reading her lovely book about Quasicrystals which refers to some recent papers of mine. IH: So even geometry is changing and evolving. 38 THEMATHEMATICAL INTELLIGENCER VOL.18,NO.4, 1996

DC: I just move on to other books. When I was y o u n g

I was very interested in stories by H. G. Wells and w h e n I was a student I was very interested in the plays of G. Bernard Shaw. II-I: You have had some connections with M. C. Escher. DC: First, at one of the International Congresses of Mathematicians which took place in Amsterdam, there was an exhibition by M. C. Escher. M y wife, being Dutch, naturally talked to him w h e n he was exhibiting his art to the mathematicians. So she got to k n o w him and that was very helpful; we kept up correspondence. Later I wrote an article for the Royal Society of Canada: m y Presidential address for Section III, on symmetry. It included a Poincar6-style model of the tessellation of (30 ~, 45 ~, 90 ~) triangles filling the hyperbolic plane so as to form a black and white pattern. Escher saw this and t h o u g h t it was just what he wanted. In some of his work he had got tired of filling the plane with congruent figures, fitting together, and he thought h o w nice it w o u l d be if they were not congruent but just similar and changed size while keeping their shape. Escher liked these things because they fulfilled his wish to m a k e a pattern in which he had fishes, for instance, of a good size near the center but getting smaller and smaller as he w e n t towards the circumference. He m a d e Circle Limit I, and then Circle Limits II, III, and IV. Circle Limit III was particularly interesting because it had four col-

ors besides black and white. It was closely related to the hyperbolic reflection group that I'd described. IH: Did you inspire him to this work? DC: That's right. He was very pleased with this idea. After he had seen that paper of mine he did Circle Limits III and IV. He had done Circle Limits I and II before. IH: Did he construct his drawings with precision?

DC: Extraordinarily well, yes. There was a very interesting apparent exception because in Circle Limit III, if you look at the rows of fishes following one another, they have white stripes along their backs so that the circle is filled with a pattern of white arcs that cross one another. It is remarkable that the spaces between the white arcs appear to form a tessellation of hexagons and squares. Yet the white arcs cross one another, three going through each vertex; therefore they cross at angles of 60 degrees. In particular, you seem to have triangles all of whose angles are 60 degrees, and that, of course, is wrong because such a triangle would be Euclidean and not hyperbolic. Bruno Ernst, in his book about Escher, The Magic Mirror, page 109, was similarly disturbed, saying, "In addition to arcs placed at right angles to the circumference (as they ought to be), there are also some arcs that are not so placed." I was interested in this and looked at it for a long time, and at last I realized what had happened. By careful measurement, I saw that all those white arcs meet the circumference at an angle which is very close to 80 degrees instead of 90 degrees. In fact, each of the white arcs does not represent a straight line in the hyperbolic plane but one branch of an equidistant curve. When you put it that way, everything falls into place, and you see that Escher did those drawings with extraordinary accuracy: when I worked it out trigonometrically I found that the angle of 80 degrees is actually arc cos [(21/4 - 2-1/4)/2] -~ 79o58'. IH: Was he aware of this? DC: Absolutely unaware. In his own words: " . . . all these strings of fish shoot up like rockets from the infinite distance at right angles from the boundary and fall back again whence they came." IH: Was it intuition?

DC: True intuition. He came to hear me give a lecture once, and I tried to make it as simple as possible; he said he didn't understand a single word. IH: Mathematicians and crystallographers recognized Escher before anybody else. What was his main appeal? DC: It was the appeal of symmetry.

IH: You give a definition of symmetry in one of your books and that definition, very geometrical, is based on congruency. How far do you think such a rigorous definition can be relaxed? DC: With Escher we've relaxed it to considering shapes that are similar instead of congruent. Groups of similarities are more general than groups of isometries. More precisely, groups of isometries occur as normal subgroups in groups of similarities. Part of the fascination for me was to look at presentations of groups. The groups have generators which satisfy certain relations. There is actually something they call a "Coxeter group," which means you have a certain number of generators of period two and you specify the periods of their products in pairs. It's a very simple idea but apparently nobody had put it like that as defining a particular family of groups. Then it turned out that some of the Coxeter groups have a relationship with Lie groups which I don't understand at all. I am very pleased, though, to see that these ideas have an application. IH: You mentioned before your wife's role in the contact with Escher. What does she do? DC: She is very artistic and appreciates music very much. She's been a wonderful wife to me, looking after me very carefully, and bringing up our children. IH: Did you know D'Arcy Thompson ? DC: He visited us about 1940. He had a tour of Canada

and actually stayed at our house. He was a wonderful man. His book On Growth and Form was very influential, and he brought out a huge second edition when he was 70 years old. He was extraordinary in combining interest in so many different things: in geometry, biology, and classical literature, languages, everything. Very remarkable. IH: How about Kepler? DC: I've been an admirer of Kepler ever since I read that it was he who invented names for all the Archimedean solids, such as the cuboctahedron. Although the names of the Platonic solids are ancient, these less regular figures were only named later. IH: One of the Archimedean solids, the truncated icosahedron, has now become very conspicuous as buckminsterfullerene, the name of the C6o molecule. Unfortunately the chemists who discovered it were not familiar with Kepler's work. DC: They thought this shape was discovered by Buckminster Fuller. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

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IH: The story of the discovery shows how useful geometry

is, even for chemists. DC: It also illustrates the fact that people who don't know any mathematics, if they happen to play with hexagons and pentagons, inevitably make that figure. This fact was demonstrated very well by a present that I once received from Mrs. Alice Boole Stott: a lampshade made of 12 glass pentagons and 19 glass hexagons, joined together by strips of lead, as in a stained glass window. I may as well tell you a little more about her. About 150 years ago an Englishman, George Boole, started what is known today as Boolean Algebra. He wrote a famous book on finite differences. He had five daughters and they were all distinguished in various ways. The youngest daughter, Ethel, married a Pole called Wojnicz so she is known as Ethel Lillian Voynich. She wrote novels, and one of these novels was called The Gadfly. That novel somehow appealed very much to the Russians at the time of the Soviet Union, and they made a movie of this book. The music for it was composed by Shostakovich. Sometimes one hears excerpts from this music; it's quite fascinating. Another one of Boole's daughters, the middle one, was called Alice; she married an actuary, Walter Stott. I got to know her very well, as it happened, through her nephew, Geoffrey Taylor, who was a mathematician and a Fellow of the Royal Society of London. He was in Cambridge when I was a student there and he introduced me to his aunt, Mrs. Stott, because he realized that she was interested in Archimedean solids as I was. She visited me and my mother, and I visited her very often in London. She was quite elderly and I was a student, so I called her "Aunt Alice." She got to know Dutch mathematicians because her husband happened to notice some articles by a Dutchman called Pieter Hendrik Schoute. Schoute was an expert concerning regular and semiregular polytopes in any number of dimensions, following in the footsteps of Schl/ifli. She was helpful to him and he was helpful to her. Between them they made a complete classification of uniform polytopes in four dimensions. He invited her to Holland and she was given an honorary degree by the University of Groningen. She didn't have a formal education. She was self-taught until she was taught by Schoute. Quite amazing. She had such a feeling for four-dimensional geometry. It was almost as if she could work in that world and see what was happening. She was always very excited when I had things to tell her, and she helped me in what I did. Through her I was introduced to some of the Dutch mathematicians. IH: Did you keep up your interest in Archimedean solids? DC: Yes. In 1950 I was one of the three authors of a paper on uniform polyhedra: a generalization of 40

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

Archimedean solids, the idea being that you have regular faces of two or more kinds and the same arrangement at every vertex. This is characteristic of the prisms and antiprisms as well as the Archimedean solids. If you allow the faces to cross one another, as Kepler did, then you get many more: 53 of these non-convex uniform polyhedra. I wrote a joint paper on these things with Jeffrey Miller (who died long ago) and Michael LonguetHiggins. There are two brothers Longuet-Higgins: Hugh-Christopher is a psychologist and Michael is an oceanographer. It is a unique case. The two brothers are not only Fellows of the Royal Society of London but for five years they were Royal Society Professors, both of them at the same time. So we wrote this paper in 1953, enumerating the uniform polyhedra, allowing them to be non-convex. S. P. Sopov in 1968, and J. Skilling in 1975, using electronic computers, verified that our list is, in fact, complete. IH: When did you leave England? DC: In 1936, when Hendrina and I moved to Canada. Before that I was a fellow of Trinity College. At one point the Princeton topologist Solomon Lefschetz came to visit Cambridge and talked to Professor M. H. A. Newman, who knew me. He happened to mention to Lefschetz that I showed promise in geometry. Lefschetz said that he would arrange for me to get a Rockefeller Foundation Fellowship to spend a year in Princeton. So I went there and was influenced a lot by his colleague Oswald Veblen, who had written a wonderful book on Projective Geometry. While at Princeton I thought about kaleidoscopes, groups generated by reflections, and what sort of fundamental region such a group would have. During a second fellowship in Princeton I was invited to Toronto by Gilbert Robinson, who had earlier been with me in Cambridge. He was a Canadian and had a job at Toronto. So I gave a lecture, and Samuel Beatty, Chairman of the Mathematics Department, must have liked my talk. For quite unexpectedly in 1936, back in England, I received a telegram from him, asking if I'd like to come to Toronto as an Assistant Professor. That was quite startling because usually one starts as a Lecturer and not as an Assistant Professor, so it was very flattering to be asked. I consulted Professor G. H. Hardy and my father; they both said that this was an offer one shouldn't turn down: you never know what's going to happen. In 1936, people already thought that war was possibly coming; so they said, take your newlywed wife and go to Canada, which we did. I met m y wife in 1935 in an English village called Much Hadham, where she was visiting from Holland a certain Mrs. Lewis. My mother introduced me to her neighbor, Mrs. Lewis. A beautiful young Dutch lady was there: Hendrina Brouwer. We liked each other, and

I invited her to come to Cambridge to see my rooms. Later she got a job in Cambridge, and we became engaged, and finally married in 1936. We thought that we would be going back to England in a few years, but then the War came and we remained in Canada.

poem and sent it to the magazine, Nature, where it was actually published. Somehow I got to know about this, and became fascinated by it, and generalized it in an article called "Loxodromic sequences of tangent spheres."

IH: What was your father's profession ?

DC: I've had 17 graduate students who went on to get their Ph.D. and most of them have done quite well. Thirteen of them are professors.

DC: He was a businessman, and at heart an artist. He belonged to the firm of Coxeter and Son, founded by his father and grandfather. They were manufacturers of surgical instruments and compressed gases, especially anesthetics. Nitrous oxide, N20, was their specialty. My father and his partner, Leslie Hall, invented a machine that had a controlled mixture of nitrous oxide and oxygen to give to a person undergoing an operation. The anesthetist would watch the patient and gave him more oxygen if he seemed to be failing and more nitrous oxide if he seemed to be coming awake. That has been used ever since by some hospitals. I wish it were used more; it is a wonderfully safe anesthetic. IH: Have you ever met John Bernal?

DC: I visited his laboratory in London. He worked with little balls of plastic clay; rolled them up, dusted them, put them together in large numbers and squeezed them to see what shapes they formed. I visited him because of my interest in sphere packing. He was a very fascinating person. Another man in the same direction was Frederick Soddy. He was the man who invented the name "isotope." I knew him because of his interest in the Descartes circle theorem. IH: Soddy was a chemist. DC" Yes, but he was also interested in geometry, just like you. I met Soddy around 1933. I visited him in his house on the south coast of England, and had a wonderful walk with him along the beach. He wrote an article for Nature about the problem of putting circles in contact with one another. The particular problem that started it was about four circles in an ordinary plane all having contact with one another. It's very easy to make three circles have contact; the fourth one will go in between in the middle or outside. So you have four circles in mutual contact. Soddy noticed that if you don't consider the radii themselves but their reciprocals, the curvatures of the circles, then the four curvatures satisfy a nice quadratic relationship: the sum of the squares of the curvatures is half the square of their sum. He didn't know that this was already discovered hundreds of years before by Descartes. Soddy wrote the theorem in the form of a

IH: How about your pupils?

IH: You have been retired for some time now, but stayed very active. DC: I have been retired for 23 years, but the University is kind enough to let me have this little office and so I g o on.

IH: Do you need any support for your work? DC: No, just this office. Of course, I have a pension which is an annuity. Then sometimes I get a hundred dollars for a lecture, and recently I was awarded a prize for research by the Fields Institute in Toronto and the Centre de Recherches Math6matiques in Montreal. IH: You don't use a computer. DC: No, I never used a computer. I'm too busy writing with pencil and paper. Fortunately, they have a very good secretary here who does word-processing. Then she says ! mustn't mind that I am fourth in line and she may have the paper typed by next Monday, but that's all right. IH: What's your next paper about?

DC: At the moment I'm writing a paper on the trigonometry of hyperbolic tesellations. Escher may have known the solution intuitively, by trial and error, and I suppose he might have been interested in seeing precisely how to find the centers and radii of all his circular arcs. H.S.M. Coxeter Department of Mathematics University of Toronto Toronto M5S 3G3 Canada Istvdn Hargittai Department of Chemistry University of North Carolina at Wilmington 601 South College Road Wilmington, NC 28403, USA [email protected]

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

41

The Trigonometry of Escher's Woodcut "Circle Limit III" H. S. M. Coxeter

In M. C. Escher's circular woodcuts, replicas of a fish (or cross, or angel, or devil), diminishing in size as they recede f r o m the centre, fit together so as to fill and cover a disc. Circle Limits I, II, and I V are b a s e d on Poincar6's circular m o d e l of the hyperbolic plane, w h o s e lines appear as arcs of circles orthogonal to the circular b o u n d ary (representing the points at infinity). Suitable sets of such arcs d e c o m p o s e the disc into a theoretically infinite n u m b e r of similar "triangles," r e p r e s e n t i n g congruent triangles filling the hyperbolic plane. Escher replaced these triangles b y recognizable shapes. Circle Limit III is likewise b a s e d on circular arcs, but in this case, instead of b e i n g orthogonal to the b o u n d a r y circle, they m e e t it at equal angles of almost precisely 80 ~. (Instead of a straight line of the hyperbolic plane, each arc represents one of the two branches of an "equidistant curve.") Consequently, his construction required an e v e n m o r e i m p r e s s i v e display of his intuitive feeling for geometric perfection. The present article analyzes the structure, using the elements of t r i g o n o m e t r y and the arithmetic of the biquadratic field QX/2 + X/-3): subjects of which he steadfastly claimed to be entirely ignorant.

comings of Circle Limit I are largely eliminated. We now have none but "through traffic" series, and all the fish belonging to one series have the same colour and swim after each other head to tail along a circular route from edge to edge . . . . Four colours are needed so that each row can be in complete contrast to its surroundings. As all these strings of fish shoot up like rockets.., from the boundary and fall back again whence they came, not a single component reaches the edge. For beyond that there is "absolute nothingness." And yet this round world cannot exist without the emptiness around i t . . . because it is out there in the "nothingness" that the centre points of the arcs that go to build up the framework are fixed with such geometric exactitude. ([2], p. 109) The p u r p o s e of the present article is to d e m o n s t r a t e this " g e o m e t r i c exactitude" (see Fig. 2) b y finding the radii a n d centres of the first three sets of four c o n g r u ent circles that trace the backs of the "strings of fish." I n a t u r a l l y a s s u m e that the relevant arcs of these circles cross one another at equal angles of 60 ~, d e c o m p o s e the interior of the " b o u n d a r y " into alternate triangular a n d q u a d r a n g u l a r regions, a n d all cut the b o u n d a r y at the s a m e pair of s u p p l e m e n t a r y angles 09,

"~T-- 09.

Introduction Concerning his four Circle Limit w o o d c u t s , M. C. Escher wrote: Circle Limit I, being a first attempt, displays all sorts of shortcomings ... and leaves much to be desired . . . . There is no continuity, no "traffic flow" nor unity of colour in each row . . . . In the coloured woodcut Circle Limit III, the short42

The acute angle w a p p e a r s on the side of each arc w h e r e the regions are q u a d r a n g u l a r . An earlier article ([1], p. 24) used hyperbolic t r i g o n o m e t r y to p r o v e that

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

cos w = sinh(14 log 2) sinh 0.1732868 ~ 0.1741553.

Figure 1. Escher's Circle Limit III. 9 1996 M.C. Escher/Cordon Art-Baarn-Holland. All rights reserved.

Since cos(79~ ') ~ 0.17424, co scarcely differs f r o m the value 80 ~ which can easily be m e a s u r e d in Escher's woodcut. H e r e I obtain this expression for co b y a m o r e e l e m e n t a r y procedure.

T h e A n g l e o~ at t h e B o u n d a r y Figure 2 is a sketch of the m i d d l e p a r t of Escher's " f r a m e w o r k , " s h o w i n g the centres O~, at distances d~ = A O ,

f r o m the centre A of the b o u n d i n g circle, of radius 1, a n d s h o w i n g the radii r~ = O ~ X ~ .

F r o m the triangle XIAO1, w h o s e angle co at X1 is opposite to the side A 0 1 d 1, as in Figure 3, w e h a v e =

d 2 = 1 + r 2 - xrl,

(1)

x = 2 cos co.

(2)

where

THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996 43

X-

G

Figure 2. Escher's "framework."

Similarly, the triangle X 2 A 0 2 , whose angle ~- - o~ at )(2 is opposite to d2, yields

Because the angle between two intersecting circles equals the angle between their radii to a c o m m o n point, the triangle 0 1 A C has angles 2r rr/4, and ~-/12 opposite to sides

X 1

-

CO 1 = rI

C A =- d 2 -

that is, dl

rl

(4)

r2,

o,

v

A

~

C

v

0 2

Figure 3. Triangles w i t h angles w at X a and X3, 1r - w at X2. 44

dl _ rl _ d2 - r 2 sin(21r/3) sin(1r/4) sin(~-/12)'

(3)

d 2 = 1 + r 2 + x r 2.

AO 1 = dl,

respectively, as in Figure 4. Hence, we have

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

A

C

Figure 4. T h e s i m i l a r triangles

02 01AC and

02AB.

The similar triangle 0 2 A B , with angles 27r/3 and rr/4 opposite to sides AO 2 = d2

and

V2 : 1(0_

0-1),

~

- 1 ( 0 q- 0-1),

: 1(02 _ 5).

In this field, 0 is called an integer because it satisfies a monic equation, namely

B 0 2 = r2,

respectively, yields

04-1002+1 d2

G

=0.

r2

vT

When w e assert that "factorization is unique," w e disregard, as factors, the units, which are divisors of 1; for if st = 1, any number

Thus, d 2 = 3 2r 2 v ( v = l o r 2 )

and expressions (1) and (3) for dayyield quadratic equations for G: r 2 + 2xrl - 2 = O,

r 2-

n = nst

(5)

has the trivial factorization ns x t. In our approach to (7) w e replaced (1 -[- ~ -

2xr2 - 2 = O.

Solving these equations for the positive numbers G, w e find

~V/3)2

by

2(V3-

1 ) ( V 3 - V2).

This " f a c t o r i z a t i o n " loses its element of surprise w h e n w e face the obvious fact that 1/3 - ~ is a unit: ( V 3 + X / 2 ) ( V 3 - h / 2 ) = 1.

r I = - x + V ~ + 2,

r 2 = X + ~ X 2 + 2.

(6)

The First Two Circles From (4) we have Since V ~ x2 + 2

=

V21/2 +

2 -1/2

2-1/4V3, (6) yields

=

(Nf3 - 1)rl = 2(d2 - r 2) = (V6 - 2)r2. r I = 2-'/4(1 - V 2 + V 3 ) ~ 1.1081646, r 2 = 2-1/4(V2 - 1 + V 3 ) --~ 1.8047860,

In terms of x, this equation becomes, in turn, (X/3 - 1 ) ( - x + V~x2+ 2) = ( V 6 - 2)(x + X/-~x2 + 2), ( 3 - X / 3 - V6)x = - ( 1 + V 3 -

and, from (5), d 1 = 2 - 3 / 4 ( V 3 -- G

V6)X/~x2 + 2,

d2 = 2-3/4(G

{(3 - X / 3 - X/-6)2 - (1 + X / 3 - V6)2}x 2 = 2(1 + V 3 -

X/-6)2,

(2 - 2X/-3)(4 - 2V6)x 2 = 2{(V2 - 1)(1 + V 2 - V3)} 2,

q- 3) ~ 1.3572189,

- V 3 + 3) ~ 2.2104024.

From (4), A C = d 2 - r 2 = 21-(V3- 1)r 1 = 2

3/4(1 + V 2 -

V3)

0.4056164. (g3-

1)(G-

V2X 2 • (V2-

2)x 2 = ( V 2 -

1 ) 2 ( V ~ - 1 ) ( V 3 - Xf2), Finally, the triangles C A 0 1 and B A 0 2 are similar,

1)2,

x = 2-1/4(21/2 - 1) = 21/4 - 2 -1/4 = 2 sinh(88 log 2).

(7)

r2 V3 - 1 F1-G 2-

The Biquadratic field Q (X/2 + V3)

and

The numbers (a + b X / 2 + c V 3 + d X / 6 ) / q , where a, b, c, and d are integers and q is a positive integer, are easily seen to constitute a field ([3], p. 230). This field is called Q ( V 2 + X/3) because it can be expressed as the set of all rational functions of the special n u m b e r 0 = X/2 + X/3, in terms of which

(1 +

v5)(3-

=

v5 + G ) = V 2 ( - 1 + 2~/2 + ~ -

"V6);

hence A B = 2-3/4( - 1 + 2 V 2 + V 3 - V6) ~ 0.6605975. THEMATHEMATICALINTELLIGENCERVOL.18,NO. 4, 1996 45

22.7, 37.0, 6.92,

The Third Circle Looking again at Figure 3, we see that

27.8, 4.53, 20.46.

These distances agree perfectly with actual measurements in the w o o d c u t itself.

d 2 = 1 + r 2 - xr3

and, since the third circle passes t h r o u g h B, Acknowledgments

d 3 - r3 = AB.

I am grateful to J. Chris Fisher for the numerical computations and to Catherine Crockett for Figures 2, 3, and 4.

Thus,

d3 + r3 -

(1/AB) Y3 =

=

2r 3 _ 1 -

1 - xr 3 AB '

xr 3

AB,

AB - AB

_

2 + x/AB

References

1 - AB 2 2AB + x

1. H. S. M. Coxeter, The non-Euclidean symmetry of Escher's picture "Circle Limit III," Leonardo 12 (1979), 19-25, 32. 2. B. Ernst, The M a g i c M i r r o r of M . C. Escher, New York: Random House (1976). 3. G. H. Hardy and E. M. Wright, A n Introduction to the T h e o r y of N u m b e r s , 4th ed., Oxford: Clarendon Press, 1960. D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of Toronto Toronto M 5 S 3 G 3 Canada

2-1/4

3 /

= 2-1/4 (

3 /

= 2-1/4(

-3)

= 2-1/4 (

MOVING? We need y o u r new address so that y o u do not miss any issues of

= 2-1/4 (

THE MATHEMATICAL INTELLIGENCER.

--~ 0.3375915

Please fill out the form below and send it to: Springer-Verlag N e w York, Inc. Journal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485

and d 3 = r3 + AB

= 2 -3/4

Name Old Address (or label)

0.998189.

Address City/State/Zip

Since Escher's b o u n d i n g circle has diameter 41 cm, our results

Name r I ~ 1.10816,

dl ~ 1.3572,

r 2 ~ 1.8048,

d 2 ~ 2.2104,

r 3 -~ 0.3376,

d 3 ~ 0.9982

should be multiplied b y 20.5 to obtain the distances in centimetres: 46

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

New

Address

Address City/State/Zip Please give us six weeks notice.

Nestor of Mathematicians: Leopold Vietoris Turns 105 Gilbert Helmberg and Karl Sigmund

His latest research paper--a short note on Euclidean geometry--is currently in print. Some of his other recent publications deal with statistics. Leopold Vietoris, who was born on the fourth of June 1891, can see the statistics of age distributions from the vantage point of an outlier. He looks a bit frail today, and wears a hearing aid, but otherwise is in good shape. The family name dates back to the time when German intellectuals took to latinizing their names (cf. Mercator, Regiomontanus). The father of Leopold Vietoris was a railway engineer who expected his son to follow in his steps, but after reading a book on geometry, young Leopold switched to mathematics. The First World War broke out before he could complete his studies at the University in Vienna. He was immediately called to arms, and severely wounded in 1915 on the Russian front. After his recovery, his regiment entrained for Bosnia, but then was shunted to South Tyrol and the Italian front. This was a decisive turning point in the life of Vietoris. Stationed in a snow-bound Alpine hamlet, he started working on topology and learned to ski--this at a time when both the science and the sport were in their thrilling pioneer stage. The war, of course, soon caught up with him. As a front officer and a military mountain guide, Vietoris had his full share of it right up to the end, and was detained after the Austrian collapse in an Italian camp till August 1919. The treatment there was so decent that he was able to write his Ph.D. thesis on "continuous sets" (i.e., connected sets). After his return to Austria, Vietoris became

assistant, first in Graz to Weitzenb6ck (whose ideas are experiencing a remarkable revival lately) and then in Vienna. With his very next paper he obtained the coveted title of Dozent. The 1920s were a heady decade for topologists, and Vienna was as good a place to be as any, with Hahn, Menger, Reidemeister, and later Hurewicz and N6beling

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

47

ogy was later extended to general topological spaces. When Vietoris returned to Vienna and lectured there on topology, this led to the method of the Mayer-Vietoris sequence, which allows one to compute the homology group of X U Y in terms of those of X, Y and X n Y. (Walter Mayer, 1889-1953, was at the time the owner of a small coffee-house. Later he became Einstein's assistant and collaborator in Berlin and Princeton.) The theorem of Vietoris and Begle which relates the homology groups of a compact metric space and its image is also a classic. Furthermore, Vietoris introduced (at about the same time as Lefschetz, Alexander, and Pontrjagin) the notion of cocycles. And in 1931, he wrote jointly with Tietze (who had left Vienna in 1910) an encyclopedia article on the relation between the different branches of topology which is still well worth reading. By then, the other passion of Vietoris proved decisive: the High Alps. He preferred their eerie silence to t h e opinionated debates of the Vienna Circle and to the fervid atmosphere of the Austrian capital, which Karl Kraus had described as "laboratory for the apocalypse." In 1927, Vietoris eagerly accepted a position as associate professor in Innsbruck. In 1928 he returned to Vienna to become full professor at the Technical University (and to marry), but in 1930 he was offered a

Five years ago, Leopold Vietoris celebrated his hundreth birthday. He still l o o k s the same.

around. In the general commotion many ideas emerged independently and almost simultaneously in several places. Vietoris, who always was an extremely modest person, never engaged in priority debates (quite in contrast, for instance, to his young and fiery colleague Karl Menger). But Vietoris was the first to introduce filters (which he called "wreaths") and one of the first to define compact spaces (which he called "liickenlos"), using the condition that every filter has a cluster point. He also introduced the notion of regularity, and proved that (in modern parlance) compact spaces are normal. During a stay in Amsterdam with Brouwer (which was financed by a scholarship from the Rockefeller Foundation shared with Karl Menger), Vietoris became one of the founding fathers of algebraic topology. His contribution in this field is much better recognised: Saunders MacLane, for instance, wrote some 60 years later a paper entitled "Topology becomes algebraic with Vietoris and Noether." Indeed, Vietoris adapted Brouwer's technique of simplicial approximation and used it to define the concept of homology group for compact metric spaces. This notion of Vietoris h o m o l 48

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

When topology became algebraic, Vietoris (here in a portrait from 1930) had a hand in it.

Glory to the pioneers. In the good old days, skiers had to make do with one stick only (Vietoris has the longest). Notice the fair proportion of intrepid ladies.

chair in Innsbruck and did not hesitate one minute to accept. The m o u n t a i n - b o u n d capital of Tyrol was the ideal place for him. Quickly spotted as a committed m o u n t a i n fan in Innsbruck's faculty, Vietoris became involved with the

local school of glaciologists led b y the distinguished Finsterwalder. In the role of a Gletscherknecht, he carried the h e a v y instruments for geological measurements and set u p experiments in countless scientific alpine excursions. In due time, Vietoris started publishing himself

On the rocks. Kept fresh by a life among glaciers, Vietoris (here at age 80) feels truly at home at high altitudes.

THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996 49

on the blockstream of the Hochebenkar, a glacier-like formation of rock d6bris pasted together by ice, which he had come to k n o w like no one else. He also wrote on how to use the compass as an alpinist (rather than a sailor), on "geometry in the service of the mountaineer," and on the physics of skiing, and he held patent no. 100832 for a m e t h o d of using air photographs in cartography. At the start of the Second World War, Vietoris was enrolled again, and w o u n d e d in Poland. On reaching 50, he was allowed to resume his post at the university in Innsbruck. Vietoris rarely returned to topology, but started working on mechanical integration, on probability and statistics, and on real analysis (thus he developed a method to introduce the sine by a functional equation which has found its w a y into some textbooks). Much of this was continued during five decades of peace following the war, which were filled with scientific work, increasing

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academic recognition, and a rich family life (he has six daughters and altogether 51 descendants, at current count). His interests have returned to geometry; his first paper on the subject, which was almost finished before the First World War, is still his favorite. A n d the lure of the mountains did not abate. At the university skiing championships he routinely w o n the gold medal in his age class, ultimately by being the only one at the start. Ten years ago, doctors told him that he really ought to stop skiing. Vietoris said that he would think it over. Institut far Mathematik und Geometrie Universit/tt Innsbruck 6020 Innsbruck Austria Institut far Mathematik Universitft't Wien 1090 Vienna Austria

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9 Numerical Computation (Arithmetical and Numerical Analysis) 9 Equation and Inequalities (Algebra) 9 Geometry and Trigonometry in the Plane 9 Solid Geometry 9 Functions 9 Vector Calculus 9 Coordinate Systems 9 Analytic Geometry 9 Matrices, Detemainants, and Systems of Linear Equations 9 Boolean Algebra - Application in Switching Algebra 9 Graphs and Algorithms 9 Differential Calculus

9 Differential Geometry 9 Infinite Series 9 Integral Calculus 9 Vector Analysis 9 Complex Variables and Functions 9 Differential and Partial Differential Equations 9 Fourier Transformation 9 Laplace and z-transformation 9 Probability Theory and Mathematical Statistics 9 Fuzzy Logic 9 Neural Networks 9 Computers (Introduction to Pascal, C, C++, Fortran, and Computer Algebra) 9 Tables of Integrals

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Jeremy J. Gray* Enriques and the Popularisation of Mathematics In 1906 Federigo Enriques published a successful popular work, later issued in English in 1914 as The Problems of Science. The book, and its reception, tell us a lot about the popular understanding of science at the time. The years before 1914 were an exciting period for scientists and mathematicians. After Paris in 1900 and Heidelberg in 1904, the International Congress of Mathematicians came to Rome in 1908, and Cambridge (England) in 1912, while the peripatetic philosophers went to Paris, Geneva, Heidelberg, and in 1912 to Bologna, where Enriques presided. The Congresses offered images of the Italians in various ways: at Paris, Italians appeared as logicians. Naturally, as an eloquent spokesman for this community, Enriques benefited from the depth and range of his countrymen's work, and he was indeed regarded as drawing on that collective wisdom. W h a t D o e s the P u b l i c A s k from M a t h e m a t i c i a n s ?

Between 1900 and 1914 the American public mostly supported the progressives, who want money spent wisely on social reform. In this spirit Woodrow Wilson, on becoming President of Princeton in 1902, said, "Science and scholarship carry the truth forward from generation to generation and give the certain touch of knowledge to the processes of life" (quoted in Kevles, p. 70). James McKeen Cattell, who was one of the early American psychologists and a former student of Wilhelm Wundt in Leipzig, edited the Popular Science Monthly, from 1900. He used his influence to extol science and its uses, but the enterprise was not a lasting financial success. In 1915 the owner sold the journal, and Cattell brought out a new one, Scientific Monthly, which retained the popular science but dropped the social reform (Kevles, p. 96); reformers, led by Dewey, were no longer looking in such numbers to science. We get a sense of what was wanted from scientists by looking at what other writers asked of them. Paul Carus built up the Open Court Publishing House in Chicago *Column editor's address: Faculty of Mathematics, The Open University,MiltonKeynes,MK7 6AA, England.

and edited The Monist, where he published essays by Ernst Mach, Poincar6, and many others. Open Court published the English translation of Poincar6's books of essays, and Hilbert's Foundations of Geometry. Carus was himself a hugely prolific author, concentrating on philosophy and religion, and in these books his aim was to make a science of philosophy. His book on mathematics (The Foundations of Mathematics, 1909) opens with a lengthy, intelligent, and up-to-date account of nonEuclidean geometry. His sources are the American G.B. Halsted (translator of Bolyai and Lobachevsky into English) and the excellent text by Engel and St/ickel [1895]. So Carus had done his homework. From the existence of hyperbolic and spherical geometry he was led to ask what this can mean for our understanding of space. He wanted to retain a primitive concept of straightness (we need not discuss his attempt here), and so he had to confront the question of the true geometry of space. The authority he turned to at this point was C.S. Peirce, the wild man of American thought, who set him right by describing astronomical tests that discriminate between the three geometries. But Carus had read, indeed published, Poincar6, and knew that Poincar6 believed that the question was purely a matter of convention. Carus agreed with him too, saying that the straightness of the Euclidean line is the most convenient, allowing one to keep with Euclidean geometry, which in any case is a good approximation to the truth. This is an example of what I mean by an issue in mathematics. There is a real question: what is the nature of space? The question is mathematical: no one is out there anxiously measuring parallaxes; any pragmatic person would accept that the true geometry is Euclidean to a sufficient degree of accuracy. But apparently mathematicians have come up with other geometries, so the audience wants to know what that means, what these other geometries are, and how we can tell which is true, even though that question is entirely abstract. There is typically another issue involved, which has overtones of psychology. The late 19th century saw the start of serious professional psychology as an academic discipline. Mindful of the error into which Kant had

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO, 4 9 1996 Springer-Verlag N e w York

51

Federigo Enriques fallen, psychologists took care to make sure that their theories allowed human beings to discover nonEuclidean geometry. The pioneer in this regard was the energetic if unsystematic Wilhelm Wundt. Gradually psychologists turned from the investigation of the prerequisites for any kind of thought to the study of how people think. What are concepts, what is it to know something? What sort of an intellectual activity is the study of geometry, regarded as the elementary appreciation of space? A third issue also presented itself: logic. There was a psychological side to logic, and an abstract side. At the resolutely hard-edged, anti-psychology end stood Gottlob Frege. Dedekind, Russell, and Peano wished to reduce mathematics to logic or to derive mathematics from logic. Hilbert may have wished to reverse the process, at least on occasion, and derive logic from axiomatic set theory, but he was equally austere. At the other extreme one might place Poincar6, who roundly denigrated attempts to derive the integers from abstract sets. There was a widespread feeling, in logic as in geometry, that much had happened since the Greeks, and much of that recently. These are significant issues, worthy of attention. H o w the mind works, how we come to know the simplest but most essential features of the world, what logical thinking is--are real questions. And, as the public could easily discover, the experts found them thorny too.

Against the Priority of Deductive Thinking Enriques regarded his philosophical position as critical and positive. Part of his programme was an investi52

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

gation of logic, and an integral part of that was an investigation of psychology, which he hoped would ultimately be reducible to physiology. This reduction, which was of course very sketchy, was intended to explain how the laws of thought related to the world of phenomena. Enriques argued that we acquire a knowledge of the deductive logic of the mathematician through evolution as a species, and each of us is more or less born with it. It differs from the creative process called inductive logic, which brings with it the problem of explaining "the real meaning and the means of acquiring the more general and abstract concepts of geometry" (p. 100). But Enriques did not have in mind anything as simple as a logic of discovery and a logic of proof. Enriques pointed out that psychological conviction comes with the experience of finding that a hypothesis is not refuted. Indeed, he observed, this is exactly what the creators of non-Euclidean geometry had had to g o on; the rigorous proofs came a generation later. So Enriques rejected Kant's distinction between absolute knowledge and absolute ignorance in favour of a spectrum of degrees of knowledge. Within this spectrum, knowledge is acquired, he argued, by a mixture of nativist and empiricist methods. Association of ideas and abstraction, combined with logic, give rise to the explanation of the postulates of elementary geometry and the feeling of self-evidence that accompanies them. One part of a postulate will appear as a fact, which is a condition needful for uniting infinitely many examples, the other part as an axiom of a logical kind. Drawing on the so-called associative theory of Taine and Delboeuf as well as the work of Mach, he produced an analysis of how the different spaces of sensation are united into the general concept of space. There are tactile, visual, and muscular spaces: and of these the tactile-muscular is constructed with the notion of distance, and so has circles and spheres, whereas the line belongs to visual space, which is formalized in terms of projective geometry because it is not metrical. The anomalous position of the parallel postulate arises because it combines touch and sight (p. 229). In support of this position he offers us the historical observation that Saccheri made progress by departing from the then current definition of parallels as equidistant lines on the grounds that this was a complex definition. It is complex, according to Enriques, in that it derives genetically from several other conceptions. But matters are not so simple, even for Enriques. Earlier on, he had argued that Legendre's definition of a straight line as the shortest curve which can connect two points is flawed, because from it one cannot deduce that two distinct straight lines meet in at most one point. Yet the definition is not fatally flawed. Enriques said that it does teach something, and pupils find that it gives them a precise idea. So definitions do two things. Some, usually the more abstract, really define. Others, usually the initial ones, gesture. It is because these two kinds of definition

Albert Einstein and Enriques.

are inescapable that Enriques did not regard geometry as a purely formal, axiomatic science, and rejected the idea of a logic of discovery as something that can and must be kicked away by more formal methods before one can speak of knowledge. To do so, for him, would be to reject not only personal experience but history. In some respects, Enriques's psychologising makes his position close to Poincar6's, but Enriques rejected Poincar6's conventionalism, arguing that the cooledplate universe, for example, makes temperature a geometrical feature of the world, and for that reason this geometry is truly different from ours. None of this means, of course, that Enriques thought geometry was 3-dimensional and Euclidean. Other geometries, higher dimensions, non-Euclidean and even non-Archimedean spaces enter our thinking precisely through the method of abstraction, but they cannot transcend their origins. For Enriques, mathematics was conceptual, the knowledge it provided open to change. The initial phase of research into the principles of geometry sought merely to simplify Euclid's Elements. Searching criticism began with Riemann, "the deepest philosopher of geometry. Here, through the direct influence exerted by Herbart on his disciple, a psychological criterion also enters the field as a guiding interest." From this psychological origin stem the two main fields of geometry, the metrical and the projective, as already discussed.

Enriques and Royce The Problems of Science was well received abroad. It was translated into German, and in two parts into

French. The publication of the first part, on logic, was the occasion for H.M. Sheffer at Harvard to call for an English translation (Philosophical Review 19 (1910), 462-3). As Sheffer may well have known, the English translation was by then under way. It was made by Katherine Royce, the wife of Josiah Royce, who was the senior professor of philosophy at Harvard and had taught Sheffer. Royce had met Enriques at the International Congress of Philosophers in Heidelberg in 1908 and been impressed by him. He wrote to Cattell in 1908 that Enriques's book "has the advantage over Poincar6's of going deeper into modern logical problems," and that "as the book of a modern geometer and a notable representative of the great Italian school of logic, it would occupy a novel place in the literature." Paul Carus agreed to publish the translation, but various administrative difficulties prevented Royce from finishing his wife's translation (as he had agreed to do) before the end of 1913. By then, however, Royce hoped in his Introductory Note to the book, it might be useful in combatting the recent rise of anti-intellectualism, which Royce feared would prefer easy, dramatic answers to patient, critical thought. Royce argued that although Enriques's reputation was founded on his treatise on projective geometry and his essay on the foundations of geometry (no mention of the work with Castelnuovo on classifying algebraic surfaces), as a philosopher he said much that pragmatists could accept. This was all the more surprising because the Italian book had been published in 1906, while the vogue for pragmatism had not begun until Heidelberg in 1908. Indeed, Enriques's form of pragTHEMATHEMATICALINTELLIGENCERVOL.18,NO.4,1996 53

counts. Enriques's interests in logic have nothing to do with the severe formalism of Peano, and everything to do with psychology. But these both enter in the popular perception of mathematics, and Enriques was taken to speak for both. Likewise Royce saw no problems in harmonising Enriques's ideas with any others he (Royce) happened to support. T h e V i e w from the 1990s

Josiah and Katherine Royce. Probably in their early Cambridge years. (From Josiah Royce, by Robert V. Hine. Photo courtesy of Nancy Hacker.)

matism was largely original, many-sided, and judicious. Royce went on to comment on many aspects of Enriques's diverse yet synthesising approach, before concluding by welcoming the book above all as a treatise on methodology. Royce himself is a major figure: many-sided and brilliant, in the view of Norbert Wiener. He graduated with distinction from the fledgeling university of his native California in 1875, and went to Leipzig where he studied logic and anthropology under Wundt. Then he went to G6ttingen to study under Lotze (he also took some mathematics courses). He married in 1880 and became a professor at Harvard in 1882. There he was an intimate friend of William James. Some time after 1900 he turned from philosophy and religion to the study of logic. In 1905 he published a lengthy paper, inspired by work of Kempe, on the close connection he saw between logic and geometry, citing work in the axiomatic style by Huntingdon at Harvard and Veblen at Princeton. So we see in Royce a breadth reminiscent of Wundt, who had impressed him as a young man; a turn towards mathematics; a definition of philosophy that embraced psychology as well as logic; and a sense of pragmatism. All these naturally pre-disposed him to the writings of Enriques. Insofar as they were typical--and they increasingly were---they indicate that in American intellectual circles Enriques had a ready-made audience. When a scholar is received as a spokesman, he is given credit for a point of view that is broader than his own. This happens in two ways. He is taken to represent a school to which he may barely belong, and he is assimilated to traditions that he may not subscribe to. He may be falsely praised and also falsely criticised on these ac54

THE MATHEMATICAL [NTELL[GENCER VOL. 18, NO. 4, 1996

The mathematical community has evolved sophisticated ways of reading Enriques's work in algebraic geometry, and we see most of it as either correct or easy to put right. It is harder for us today to accommodate his writing as a philosopher or populariser. He held a subtle position, according to which knowledge is inseparable from the means of knowing, logic from psychology. This has long been unfashionable in the sciences. It may be that cognitive psychology will reopen the avenues Enriques explored; there are signs that it has reached at least the philosophy of mathematics. But that field is not what it was; even its practitioners detect a widespread feeling that philosophy of mathematics has drifted out of the mainstream and is becalmed. Moreover, the English-speaking world is losing its sense that a philosopher is the right person to ask. France is different, of course, but there the work of Derrida and the iconoclastic, deconstructionist spirit of Bruno Latour, which has made him a guru on the practise and policy of science, offer no platform for popularisation. What these intellectuals propose is a social critique; their expertise is a skill, they merely claim to be quicker at unmasking presuppositions and seeing through mystiques than the rest of us. They do not offer a theory, but claim to be the enemy of all theories. They are market leaders in a nervous, disenchanted, cynical age. Enriques offered a position on the nature of knowledge that was original and sophisticated. His readers found a rare grasp of modern science, traditional philosophy, and contemporary psychology. If only someone would make a similar contribution to the present debates about the nature, meaning, and value of mathematics and science! References

Carus, P., 1909: The Foundations of Mathematics, Open Court, Chicago. Enriques, F., 1906: Problemi delle scienze, Zanichelli, Bologna. Enriques, F., 1914: The Problems of Science, translated by K. Royce with an introduction by J. Royce, Open Court, Chicago. Kevles, D. J., 1987: The Physicists; the History of a Scientific Community in Modern America, Harvard University Press, Cambridge, Mass. Engel, F., and St/ickel, P., 1895: Die Theorie der ParalMlinien yon Euklid bis auf Gauss, Leipzig, Teubner.

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Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the

famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? 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 follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.

The Bone that Began the Space Odyssey D. Huylebrouck

The Ishango Bone, a Mesolithic Mathematical Artifact

Classical Greek mathematics had its roots in Egypt and Mesopotamia; going further back in time, one has to turn to non-written proto-mathematics. From bones, strings, and standing stones, early mathematical reasoning can sometimes be read off quite clearly (see [Jos]). In the Lebombo mountains between South Africa and Swaziland, a 37,000-year-old baboon fibula was found, marked with 29 notches, while a 32,000-year-old bone from ex-Czechoslovakia has 57 notches (see [Bog]). These engravings are most probably simple tally marks to ease counting; a more interesting find is the so-called Ishango bone. It has carvings according to an unknown but intriguing pattern. Ishango is a little village on the shores of Lake Edward, one of the farthest sources of the River Nile, on the border of Uganda and Za~re (see Figure 1). In the surrounding mountains, Prof. Jean de Heinzelin did archeological excavations at a Mesolithic settlement about 20,000 years old. Some archaeologists believe the early Ishango-man was a pre-sapiens species. The later inhabitants of that region, who gave it the name Ishango, have no immediate ties with the primary settlement, which was buried in a volcanic eruption. Together with many stone tools and human remains, de Heinzelin excavated a little carved bone of about the 56

Fig. 1. The earliest mathematical artifact was discovered at Ishango, at the sources of the Nile.

THE MATHEMATICAL 1NTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

Fig. 2. The Ishango Bone from de Heinzelin's original representation. size of a pencil. It even looks like a writing device: it has a firmly fixed piece of quartz at one end (see Figure 2). It is generally supposed to be about 11,000 years old, but this is unsure, because the nearby volcanoes have upset the usual ratios of isotopes used by the carbon 14 method. There are three separate columns, each consisting of sets of notches arranged in distinct patterns. The upper photograph shows two rows of notches on the bone. Above, there are four groups composed of 11,

13, 17, and 19 notches (from right to left); below it, there are 11, 21, 19, and 9 notches, in that order. The third column has the indentations arranged in eight groups: 3, 6, 4, 8, 10, 5, 5, 7. The following observations suggest the Ishango man might have carved the bone according to some kind of pattern: (a) the markings on the second row obey the rule "10 + 1, 20 + 1, 20 - 1, 10 - 1";

Fig. 3. Marshack's interpretation of the bone as a lunar calendar. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

57

(b) the pairs of the third row are related b y duplication or halving, except for the final 5 and 7; (c) counting the carvings on the first r o w yields the prime n u m b e r s between 10 and 20; (d) the markings on the first two rows each add u p to 60. The interpretation of these observations is more speculative. Fact (a) could indicate the use of a n u m b e r system with base ten, while the next two observations seem to indicate some arithmetic was done. The final fact suggests the bone could have been a lunar calendar, since 60 corresponds to two lunar months; 48, the sum of the n u m b e r of carvings on the last row, accounts for about a m o n t h and a half. The arithmetical g a m e interpretation was favored by de Heinzelin. H e based his option on archaeological evidence, and c o m p a r e d the Ishango h a r p o o n heads to those found in n o r t h e r n Sudan and ancient Egypt. The professor emeritus of the Ghent University is an authority in the field of African archaeology, and suggested a link b e t w e e n the Ishango arithmetic and the c o m m e n c e m e n t of mathematics in ancient Egypt: It is possible to trace the influence of the Ishango technique on other African peoples by examining harpoon points at other sites. From central Africa the style seems to have spread northward. At Khartoum near the upper Nile there is a site that was occupied considerably later than Ishango. The harpoon points found there show a diversity of styles. Some have the notches that seem to have been invented first at Ishango. Near Khartoum, at Es Shanheinab, is a Neolithic site that contains harpoon points bearing the imprint of Ishango ancestry. From there the Ishango technique moved westward, but a secondary branch went northward from Khartoum along the Nile Valley to Nagada in Egypt.[...] The first example of a well worked out mathematical table dates from the dynastic period in Egypt. There are some clues, however, that suggest the existence of cruder systems in predynastic times. Because the Egyptian number system was a basis and a prerequisite of classical Greece, and thus for many of the developments in science that followed, it is even possible that the modern world owes one of its greatest debts to the people who lived at Ishango. Whether or not this is the case, it is remarkable that the oldest clue to the use of a number system by man dates back to central Africa of the Mesolithic period. No excavations in Europe have turned up such a hint. (After de Heinzelin [Hei], p. 109 and 116). The explanation of the patterns carved on the Ishango bone as a lunar calendar was s u p p o r t e d b y A. Marshack, w h o carried out a detailed microscopic examination of the bone (see [Mar]). H e found markings of different shapes and sizes, and m a d e a diagram w h e r e the phases of the m o o n c o r r e s p o n d e d to the Ishango notches (see Figure 3). Circumstantial evidence based on what we k n o w of the religious rituals of present-day peoples w h o still follow the hunter-gatherer life style of the Ishango, might favor his explanation. Whatever the interpretation, the patterns surely show the bone was more than 58

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

Fig. 4. At the Brussels Natural Science Institute, the mathematical tourist has to confront a dinosaur on the way to the bone.

a simple tally stick. But to credit the computational and astronomical reading simultaneously w o u l d be farfetched; as Joseph stated it, "a single bone may well col-

lapse under the heavy weight of conjectures piled upon it." From Africa to Belgium

and into Space

The bone was brought to Belgium b y its discoverer, Prof. J. de Heinzelin, and stored in the treasure r o o m of the Royal Belgian Institute for Natural Science in Brussels (Belgium). A mould, and several copies, were m a d e from the petrified bone, ensuring the survival of the cryptic information contained on the small and fragile artifact. It is not exposed a n y longer to the public, but the mathematical tourist in Brussels can, b y a simple written request, ask for the m u s e u m ' s authorization to see it (see Figures 4 and 5). Prof. William A. Hawkins,

Fig. 5. The Ishango bone (center), together with some harpoon heads, in a drawer of the Institute's treasure room.

Fig. 6. Kubrick used a metaphor to express man's odyssey from the first Mesolithic reasoning to the space technology of the future. (This sketch, based conceptually on an image from the motion picture "2001: A Space Odyssey," is printed with the permission of Turner Entertainment Co.) THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

59

pyramids. From the Nilotic epoch on, a mere 4000 years w e r e e n o u g h for science to reach space. M a y b e one day an inspired astronaut will allow the Ishango bone to continue its Kubrick-like voyage b y carrying the bone (or a c o p y of it) with him on a Space Shuttle mission. What w o u l d be fitting words to pron o u n c e w h e n the bone reaches its destination, space?

References [Bog] J. Bogoshi, K. Naidoo and J. Webb, The oldest mathematical artifact, Mathematical Gazette, 71 (458), 294 (1987). [Hei] J. de Heinzelin, Ishango, Scientific American, 206, June, Director of S U M M A (Strengthening Under-represented Minority Mathematics Achievements), is currently editing a poster on African and African American Pioneers in Mathematics. It includes images of the Ishango Bone and is to be distributed t h r o u g h colleges and universities in the U.S. A. Marshack, at the time of de Heinzelin's discovery, was in charge of supplying some historical and scientific b a c k g r o u n d for the NASA lunar program. H e questioned m a n y authorities in the administration and in the scientific c o m m u n i t y about the reasons for going into space. Finally, these discussions led him to the Ishango bone: My interest in space and science had been kindled, in part, because these were human activities, culturally specialized products of the human brain, and to me therefore, not much different from politics, religion, art or war. I could not in my thoughts, while writing and searching for the meaning of man and the space program, separate Dr. Jerome Wiesner, Special Assistant to the President for Science and Technology, or Yuri Gagarin and John Glenn, the first Soviet and American astronauts, or Dr. Lloyd V. Berkner, the scientist and science administrator who had suggested the International Geophysical Year to the world's scientists, from the extremely primitive natives I had met in New Guinea and Australia, or the starving farmers I had seen in India, or from the men who, thousands of years before, had hunted mammoth, reindeer, and bison and had painted the caves of Ice Age Europe, or from the later men who first farmed or built cities around the Mediterranean (after A. Marshack [Mar], p. 9-11). Incidentally, the Ishango site w h e r e de Heinzelin found the famous bone, is a w o r t h y location for scenes of movies such as 2001: A Space Odyssey. Kubrick showed our ancestor in a prehistoric landscape throwing a bone tool into the sky. By special effects, the bone gradually transformed to a space ship. This movie sequence can be seen as a metaphor to illustrate h u m a n progress from discoveries that look v e r y simple nowadays, to the technology of the space age. The picture in Figure 6 was compiled with Stanley Kubrick's images in mind. The front layer shows the Ishango excavation site, the rectangular rock being evidence of the archaeological work. The water of Lake E d w a r d behind it extends to Egypt, a t h o u g h t that led to the addition of the 60

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105-16 (1962). [Jos] G. G. Joseph, The Crest of the Peacock:Non-European Roots of Mathematics, Penguin Books, London (1992). [Mar] A. Marshack, Roots of Civilization, The Cognitive beginnings of Man's First Art, Symbol and Notation, McGraw-Hill, New York (1972).

Aartshertogstraat 42 8400 Ostends Belgium

The English Hammer-Beam Roof David Horowitz

In touring the English countryside, if one is willing to stray boldly from the larger motorways, then a drive down almost any country road will pass through small hamlets reminiscent of bygone eras. Though each has its own distinctive character, most share a general communal theme: main street, pub, common area or park, and church. The church often stands out amid the rest. Many of those in East Anglia are large edifices built during the fourteenth to eighteenth centuries by patrons who became wealthy as the wool trade flourished throughout the region. When the industry foundered the population relocated, leaving the church to dwarf the town and serve only a handful of remaining parishioners. A familiar architectural feature that appears in many of these large churches has become known as the English hammer-beam roof. It is an example of a triangulation network, or truss, which is characterized by a beam (the hammer-beam) protruding from each wall and supported by a wall post and a lower curved brace (Figure la). A vertical strut (the hammer-post) joined to the end of each hammer-beam enables the weight of the principal rafter to be carried down to the wall, which is often buttressed at the point from which the hammerbeam protrudes. There were several motivations for this form of Gothic roofing design. First of all, it permitted the spanning of wider halls where single beams of sufficient length were difficult to find. (In general, hammer-beams have a length of one-sixth to one-fifth of the entire spanning width.) When it became necessary to span larger halls, a "double hammer-beam" roof was sometimes used (Figure 2). Secondly, it shifted the point of application of the roof's outward thrust further down the wall (via

the wall post). Thirdly, it allowed the unencumbered vision of stained glass windows or other artwork at the end of the hall. And finally, it offered the aesthetic suggestion that the entire roof was "suspended" without the utilization of intervening columns and supports. Architectural historian John Hooper Harvey claims [2] that the hammer-beam roof is "one of the greatest of all triumphs of mind over matter." Although this form of roofing was executed as early as the fourteenth century (Pilgrims' Hall at Winchester, England, circa 1325), the use of the term "hammerbeam" does not seem to appear until the early 1800's, and its origin is in doubt. Some suggest that it is derived from the French heraldic term hamade, signifying three pieces cast into a triangular shape [1]. Others perceive the form of a hammer framed by the hammer-beam (as the head) and the wall post (as the handle). The wooden structural elements which form the hammer-beam roof are often ornately carved or decorated. One of the most masterful examples is in Saint Mary's Church at Bury St. Edmunds (Figure 3). It is described [6] in the words of the British architect and polemicist A. Welby Pugin (1812-1852): "At every pair of principals are two angels as large as the human figure, bearing the sacred vessels and ornaments used in the celebration of the holy sacrifice; these angels are vested in chasubles and dalmaticks, tunicles and copes, of ancient and beautiful forms; the candlesticks, thurible, chalice, books, cruets, &c., which they bear are most valuable authorities for the form and design of those used in our ancient churches." The Church of Saint Peter and Saint Paul in the village of Knapton in Norfolk offers a beautifully carved and painted double hammer-beam roof (Figure 4). (QUERY: Does there exist an example of a

THE MATHEMATICAL INTELL1GENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

61

Figure 1. Simple Hammer-Beam Roof. (a) illustrative sketch; (b) principal structural elements; (c) equivalent truss form

triple hammer-beam roof? If not, why not?) English hammer-beam roof construction was not limited to churches; a richly polished specimen can be found in the main hall of Trinity College, Cambridge (Figure 5). The renowned example in Westminster Hall, London is not a "pure" hammer-beam roof, as it incorporates great arched ribs to supplement its support (Figure 6). The stress analysis for the members of the hammerbeam roof is of interest in the field of statics [4]. The process can be simplified by replacing each curved brace in the roof (Figure lb) by an equivalent straight member (Figure lc) having a slightly lower stiffness ratio (Young's modulus). In Figure lc arrows pointing toward each other indicate members under tension (extension); arrows pointing away from each other indicate members under compression (contraction). For a deFigure 2. Double Hammer-Beam Roof

Figure 3. Saint Mary's Church, Bury St. Edmunds (c. 1424-1446)

62 THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996

Figure 4. Church of Saint Peter and Saint Paul, Knapton (1503-1504)

tailed analysis of the stresses within a h a m m e r - b e a m roof see [3] and [5l. Maxwell's Rule is a principle of architectural design used to insure that a truss will remain stable. The rule states that for a simply-stiff truss to be non-collapsible u n d e r loading the minimal n u m b e r of m e m b e r s n required to connect j joints is given b y the formula n = 2j - 3. (This formula is not foolproof w h e n the truss is attached to the environment; absolute verification of stability requires careful vector analysis. See [7].) Figure lc shows t h a t j = 10 and n = 17 for a simple h a m m e r - b e a m roof. In a double h a m m e r - b e a m roof j = 14 and n = 25. (EXERCISE: What are the values of j and n for an mh a m m e r - b e a m roof?) Exploring and analyzing the architectural motifs f o u n d in rural English villages is one of the pleasures of touring in Britain. Each adventure is almost certain to produce m a n y delights and surprises.

References 1. Architectural Publ. Soc., The dictionary of architecture, volume 4. London, Thomas Richards, 1863. 2. Harvey, J.H., Mediaeval architect. New York: St. Martin's, 1972. 3. Heyman, J., Westminster Hall roof. Proc. Instn. Civ. Engrs. 37 (1967), 137-162. 4. Pippard, A.J.S. and Baker, J.F., The analysis of engineering structures. London: Edward Arnold, 1968. 5. Pippard A.J.S. and Glanville, W.H., Primary stresses in timber roofs with special reference to curved bracing members. Building Research Technical Paper No. 2. London: H.M. Stationery Office, 1926. 6. Pugin, A.W.N., The true principles of pointed or Christian ar-

Figure 5. Hall at Trinity College, Cambridge (1604-1605) THEMATHEMATICALINTELLIGENCER VOL.18,NO.4, 1996 63

Figure 6. Westminster Hall, London (1394-1400) chitecture; set forth in two lectures delivered at St. Marie's Oscott. N e w York: St. M a r t i n ' s , 1973. 7. S c h o d e k , D.L., Structures. E n g l e w o o d P r e n t i c e - H a l l , 1980.

Cliffs, New

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Department of Mathematics Golden West College Huntington Beach, CA 92647-2748 USA

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Mathematical Encounters of the Second Kind P.J. Davis, Brown University, Providence, RI

Phil Davis was led to a career in mathematics by his failure as a young boy to Philip J. I)avis '; solve a certain problem in geometry and by his persistence in tackling it. Here is the problem, here is how the problem itself developed, and here are the human associations that surround it.

The Mathematical Experience Study Edition P.J. Davis, Brown University; R. Hersh, University of New Mexico & E.A. Marchisotto, California State University, Northridge 1995 487pp.

An Indispensable Tool for Teachers - -

The Companion Guide to the Mathematical Experience Study Edition 1996

In the course of his career, Davis learned that mathematics, while serving the many needs of science, technology, economics, etc., can also serve as a social connection among people of diverse origins, abilities, and interests. Here also is an introduction to a number of his mathematical encounters with individuals that range over centuries, over oceans, and vary in virtuality. The reader will meet Napoleon, Queen Hortense of Holland, Lord Rothschild, John Dee the mathematician and crystal gazer, the actress Elizabeth Bergner, and many others. 1996 Approx. 250 pp., 15 illus. Hardcover $24.95 (tent.) ISBN 0-8176-3939-X

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Penrose Tiling in Helsinki and Tokyo K.H. Kuo

After the sensational discovery of quasicrystals displaying fivefold rotational symmetry but no translational periodicity, Penrose tiling became a fashion not only in the quasicrystal community but also in architectural decoration, especially in universities and science museums. The following are two cases to add to the Penrose Tiling in Northfield, Minnesota, published recently in this journal (vol. 17, no. 2, 1995). In front of the new Science Museum of Helsinki, opened Spring 1989, the pavement leading to the main gate is decorated with an aperiodic tiling of kites (dark colour) and darts (light colour) first suggested by Roger Penrose. It is noteworthy that the matching marks to guarantee an aperiodic tiling are clearly shown. For all visitors coming to this museum aiming at learning some science, the first thing they experience is the aperiodic tiling displaying fivefold symmetry. In the Munich Museum, a Penrose tiling has also been used to decorate one of the inside walls. Penrose tiling can also be used to decorate a building. On the front face of the Student Hall of the Tokyo Metropolis University there is a Penrose tiling two stories high. Two female students are admiring this beautiful pattern showing the contour of metropolitan

Figure 2. Penrose tiling in Tokyo (photographed in spring 1991).

Tokyo. The ten-spoke cartwheel (part of it is enlarged and shown as an inset) symbolises the eternal progress of Tokyo.

Figure 1. Kites and darts in Helsinki.

K.H. Kuo P.O. Box 2724 100080 Beijing, China THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York 6 5

Penrose Tiling at Miami University David E. Kullman

After the appearance of Martin Gardner's Scientific American column on Penrose tilings, many architects saw their potential for decoration, especially of scientific buildings. 1 Are we alone in having such a decoration conceived before the Gardner column came out? Milton Cox of our Department, inspired by visiting speaker John H. Conway in September 1976, made the design. Cox unfortunately found, on returning from a trip, that about a quarter of the tiles had been laid with

ISee MathematicalIntelligencervo1.17(1995),no. 2, 54. 66

a 90-degree rotation from the original position, so that the axis of symmetry no longer matched that of the building. It was deemed preferable to rotate the tiling rather than the building to bring them into agreement. Bachelor Hall was dedicated in September 1979, it is still the home of our Department, and the tiling is still a focal point. Department of Mathematics and Statistics Miami University Oxford, OH 45056-1541 USA e-mail: [email protected]

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York

Jet Wimp* 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 be-

ing assigned a book to review, please write us, telling us your expertise and your predilections. Address the column editor, Jet Wimp.

Calculus from Graphical, Numerical, and Symbolic Points of View by Arnold Ostebee and Paul Zorn

Calculus teachers (and textbook authors) today might do well to have some answers, or some sort of identifiable position, on these issues. You can bet that the authors of this two-volume series, just like the authors of most any calculus series, are far more motivated by the appreciation of one of "our species' deepest, richest, farthest-reaching, and most beautiful intellectual achievements" than by the utility of the calculus in building bridges and skyscrapers. However, you can also bet that most of their students are motivated by other factors, the two most presentable of which are the utility of calculus as a tool in other disciplines and the belief that educated people have always studied this subject so it must be important for them. The instructor's appreciation of the calculus is in general unscathed by modern technology, both student motivations are sorely challenged by its advent. After all, we do not prepare the managers and users of a communications network to understand how satellites function, nor do we prepare auto mechanics to understand the sophisticated electronics of the computer chips that control aspects of the modern automobile. We do not prepare doctors to understand the workings of instruments for laser surgery. Common sense and societal consensus have led us to prepare the experts to understand the client, customer, or patient, to understand what tools are useful in meeting customer needs, and to understand how to use those tools. Yet there is no consensus at all as to whether these goals can best be achieved by teaching mathematics which is directly relevant to our lives and work or by teaching a mathematics driven by its own inner logic and aesthetics. Even if our goal is a strictly utilitarian one, neither history nor current research data give us reliable guidelines. Mathematics education, to be effective and satisfying, needs to be guided by real needs, both the intellectual needs of the individual student and the professional needs of the future profession for which the student is preparing. For students who somehow manage, despite all the modern obstacles, to get in touch with their intellectual needs, the rest usually takes care of itself. Perhaps fortunately, most students are not "intellectu-

Philadelphia: Saunders College Publishing, 1995. v. I: paperback, vi + 516 pp., US $15.50, ISBN 0-03-010602-8 v. II: paperback, vi + 413 pp., US $15.50, ISBN 0-03010603-6

Reviewed by Herb Clemens As Bill Davis of Ohio State University is fond of asking, "Maple or Mathematica (especially with front ends like Joy of Mathematica) enable even math/computer-phobic undergraduate students to do any calculation we ever dreamed of teaching infinitely faster and more accurately than we ever dreamed of requiring. So what should we teach and how should we teach it?" It's the calculator controversy of elementary and secondary school revisited "in spades" in higher education. On the other hand, most students do not do proofs and they remain unconvinced that proofs are good for anything except persecution (their own). "Teacher knows best," unlike a couple of generations ago, no longer works, even for the serious student. So when I was asked to review the new calculus series by Arnold Ostebee and Paul Zorn of St. Olaf College, I picked it up, worked through some random sections, and thoroughly enjoyed its lively, informal yet very coherent narrative style. I even learned a few things about Simpson's rule and cubic splines that I didn't know before. However, I found myself getting stuck again and again on an issue that presumably is not relevant to the task at hand, namely, the fundamental issue lurking in the background: what of calculus is worth teaching and to whom? Is the traditional calculus curriculum any longer important for our students? If so, should it be studied for its profundity, for its intellectual power and beauty, or for its utility? If it is useful, what makes it so? *Column Editor's address: D e p a r t m e n t of M a t h e m a t i c s , D r e x e l Univ e r s i t y , P h i l a d e l p h i a , P A 19104 USA.

THE MATHEMATICAL INTELLIGENCERVOL. 18, NO. 4 9 1996 Springer-Verlag New York 67

als," so learning calculus is not primarily about satisfying some inner drive for deep quantitative understanding and rigor. Both types of needs, intellectual and practical, are complicated. My intellectual needs as a math student may or may not be the same as yours. I may learn ritualistically, beginning with pattern recognition through repetition, and be unable, or unwilling, to "conceptualize" until a pattern or ritual is in place. You may be unable or unwilling to learn a ritual or algorithm until you understand the concept behind the ritual, and satisfy yourself that the ritual is "important." On the "practical" level, professional and personal needs are almost impossible to predict, especially during early educational years. Even for the individual college student, professional needs are difficult to predict. In these times, no job/profession is secure, and we will all be called upon to change what we do, both in our work and personal lives, every few years. The authors of this series seem to me to be firmly in the "concept-first" camp, and to operate there with considerable verbal and pedagogical sophistication. The books seem very "user-friendly," at least for the bright, articulate, verbally sophisticated undergraduates at places like St. Olaf's. The authors are trying to be the students' friends, to appeal to their sense of humor, to embellish, be whimsical, be interesting, and create a sense of shared purpose. We are in this together, they implicitly say, this exploration of one of the most profound intellectual achievements of modern humankind. Let's make a (statistically outrageous) comparison of this calculus book [OZ] with the classic Thomas and Finney (7th edition) [TF] by choosing one chapter, that on the Chain Rule. TF is four and one-half pages, OZ a page longer. TF has nine example computations, OZ eight. TF offers a "proof" of the Chain Rule halfway through (relying on the concept of "increment &t" of dubious parentage) whereas OZ "settle for plausible evidence." The OZ chapter is built around a real narrative, a kind of conversation about the Chain Rule such as one might have about a new set of running shoes. TF's few paragraphs are much tighter, serving mostly to tie "rule boxes" to examples. OZ's exercises give more emphasis to graphing and numerical solutions; TF's contain more serious applications to physics and engineering. These chapters, if typical, raise some interesting questions. By drawing the reader's attention away from computational ritual in favor of "getting the idea of what a particular calculus concept is about," could OZ possibly frustrate the "ritual-first" learner? Are the authors in effect taking the position that nothing has been lost, since the "ritual-first" learner is really a "ritual-only" learner and since computational ability per se, is no longer very useful? On the other hand, is OZ's assertion that a "fully rigorous proof of the chain rule runs quickly into delicate technical problems" as useful to those few rigorously 68 THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996

logical students as TF's "proof," even though it may raise further questions? The authors state in a paragraph entitled "Concept vs. rigor" in the introduction to Volume I that they "emphasize only those [proofs] which we believe contribute significantly to understanding calculus concepts." This raises the serious question of what "concept-first" means if logical rigor is only exemplified occasionally, rather than being the framework on which the course structure is built. In fairness, there is certainly something to the approach of concept without too much rigor. An awfully lot of beautiful and interesting calculus, both conceptual and computational, was done in the century or more before there was anything like rigor. (The party was over, the riches were spent, and then around 1800 the accountants moved in, some would say.) Most of us probably worry that, by eschewing logical rigor, we frustrate the relatively rare but precious student whose need for intellectual precision is his or her entryway to mathematics and science. We tell ourselves that those precious few will, by and large, fill in the logic on their own and that, in any case, we must be guided by the needs of the larger group. After a while, most of us professional mathematicians take (are reduced to?) that position in our teaching, simply because the more rigorous approach has worked out so disastrously in practice. Suppressing our lament that strict logical reasoning is just not in the cards for most of our students, we make the necessary compromises with reality. But we often do this without putting forward a very clear concept of what "the needs of the larger group" really are. We've never come up with a really good answer, and the technological challenge mentioned above only makes matters worse. These books, it seems to me, try to avoid this unanswerable, or at least unanswered, question by focusing on the student as a person and on the individual's relationship with the material. As for the focus on the student, the accessible, spiraling narrative style of presentation is attractive, as is the regular interjection of whimsical, throw-away humor. It is as if the authors are saying, "Trust us, we're going on an intellectual journey together, you're our friend and we know what we're doing . . . . And by the way, look how this is related to t h a t ! . . . And by the way, isn't this subject impressive and interesting?" Such a style projects energy, enthusiasm, and caring, so it can hardly go wrong. I'm probably not alone in believing that, if the mathematics is correct and the style is intelligent, energetic, and caring, any style of presenting mathematics, from the most traditional rote learning to the most modern forms of constructivism, will produce good results much of the time. The authors can certainly be forgiven for their failure to answer the metaquestions. Others don't either. But this approach of "exploring together" and worrying about the rigor, appropriateness, or utility of the out-

comes later, only postpones the day of reckoning. A few years from now, will I have anything left from what I have learned, and will what I have be useful, or pleasing, or profound? All of which leaves me still stuck and unable to make a reasonable evaluation of this delightfully written new calculus series!

Department of Mathematics University of Utah Salt Lake City, UT 84112-1107 USA [email protected]

Number Words and Language O r i g i n s J. Lambekt When mathematicians meet in the lounge, they don't usually chat about mathematics. Instead, they discuss hockey games, personal computers and, not infrequently, the origin of words. Mathematicians with an amateur interest in philology included Newton and Hamilton, while others pursued a more professional interest: to mind come Eratosthenes and Wallis and, most notably, Grassmann, who has been regarded as the founder of Indo-European philology. More modestly, the present author too has carried out some linguistic investigations in syntax and morphology, although his love of etymology has been confined to reading the contributions of others. Number words, like kinship terms, tend to be rather conservative and bear witness to the genetic relationship between languages. If we look at English five and French cinq, we observe that these two words have not a single sound in common, even the written letter i is pronounced quite differently in the two cases. Yet five is related to German fanf and cinq is derived from Latin quinque, which even the layman will recognize as being related. But don't infer that German f corresponds to Latin q in both places; the story is a bit more complicated. Both Latin, from which all Romance languages (Portuguese, Spanish, Catalan, French, Italian, Romanian, and a few minor ones) are descended, and English, a member of the Germanic language group (which also includes Scandinavian, Dutch, and German, plus many dialects), are members of the vast Indo-European language family, which, even before the spread of European civilization in the last few centuries, stretched all the w a y from Iceland to Ceylon and includes many other language groups, such as Celtic, Baltic, Slavic, Albanian, Greek, and most of the languages spoken in Iran, Afghanistan, Pakistan, and India. For example, the six English words describing members of the nuclear tThe author wishes to thank Merritt Ruhlen, Chandler Davis, and David Sussman for helpful comments.

family have recognizable cognates in Sanskrit and go back more than 4,000 years (see, e.g., Bhargava and Lambek [1992]). A hypothetical Indo-European proto-language has been carefully constructed by philologists; in it the original word for 5 is supposed to be *penkwe. (Reconstructed forms are marked with an asterisk; I have often simplified the spelling of such forms to make them more accessible to non-specialists.) According to the rules of Germanic sound change, discovered by nineteenth-century philologists, German f is indeed descended from proto-Indo-European p. This explains the initial consonant of fanf, but not the final one, nor is there such an explanation for the initial consonant of quinque; presumably both arose by alliteration. However, we may conjecture that behind German fiinf there lies a pre-proto-Germanic root like *ring, which also gave rise to English finger (see Klein [1971], who ascribes this hypothesis to A. Meillet). Anthropologists are alleged to know of people who have no words for numbers other than 1, 2, and 3, alias 'many." This possibly apocryphal story would not imply that these people don't know how to count, only that they do so by gestures and not verbally. Now, IndoEuropean (and many other) languages originally possessed three forms of the noun, called 'numbers,' to wit: singular, dual, and plural. The dual form has mostly disappeared, but is represented by the last vowel of Latin duo and is said to survive in the middle letter of the English word both. A dual also occurs in Hebrew sh'nayim, the masculine word for 2, Hebrew being a member of a different language family that used to be called 'Hamito-Semitic.' This family includes ancient Babylonian, modern Arabic, ancient Egyptian, the Berber languages of North Africa, and many distinct languages spoken in Ethiopia. Most people, once they learn to count beyond 3, use their fingers to do so, hence the ubiquitous scale of 10, based on the biological fact that we have ten fingers. But do we? The question is whether the thumb counts as a finger. If not, a hand contains four fingers, not five! Indeed, etymologists have surmised that, at one time, the Indo-Europeans did count by fours. While some linguists are skeptical about this theory, there being no evidence for a scale of 4 in extant languages (see, e.g., Greenberg [1990] or Zaslavsky [1973]), we shall briefly consider the evidence in its favour. English eight is related to Latin octo, which still retains the dual ending o, and in almost all Indo-European languages the word for 9 is closely related to the word for 'new,' e.g., German neun/neu; in French the word neuf means both 9 and new. This suggests that 9 was conceived as a new number after the eight fingers had been used up. According to this theory, which you find in etymological dictionaries, Indo-European *okto originally meant something like 'a pair of fours.' Admittedly, this argument would have been more convincing, had *okto THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

69

retained a recognizable trace of Indo-European *qwetwor, meaning 4, or of some word meaning 'hand.' I don't know when the Indo-Europeans finally acknowledged the thumb as a finger and adopted our decimal terminology, but it must have been over 4000 years ago, before their dispersion from their original homeland. English ten, German zehn, French dix and Latin decem all go back to a reconstructed *dekm(t). Menninger [1969] speculates that this might mean something like 'two hands,' seeing two in *de and hand in *km(t). Plausible as this explanation may appear, mainstream philologists derive *dekm(t) from a word meaning 'finger,' which survives in our Latin loan-word digit and native English toe; in fact, German zehn closely resembles German Zehen, meaning 'toes.' The root of this word is quite ubiquitous and goes back to proto-IndoEuropean *deik, to show, which survives in German zeigen with the same meaning, in English teach, and in Latin dicere, to say. As we shall see later, this root may in fact be much much older. There is nothing of interest I can say about our word one. Concerning our word two, we note that it embodies the former dual ending o. Curiously, the word for 2 resembles that for 'tooth,' not only in English, but in most Indo-European languages. This is not to say that there is any obvious semantic connection between two and tooth, only that the initial consonants underwent the same changes from proto-Indo-European *duwo and *dvun respectively. By what seems a strange coincidence, Hebrew sh'nayim, the masculine word for 2, may also be read as meaning 'pair of teeth,' though it must be admitted that Hebrew sh here corresponds to two different sounds in Arabic. We shall return to the resemblance between 2 and tooth later. What about three? Menninger [1969] speculates that this originally meant something like 'many,' and he relates it to French tr~s, meaning 'very.' At least one scholar, Brunner [1969], suggests that three is related to the Semitic word for 3, as in Arabic talat, although this seems somewhat doubtful. Let us now look at six and seven, Latin sex and septem, Polish szedd and siedem, etc., going back to proto-IndoEuropean *seks and *septm. Many people have been intrigued by the analogy with Hebrew shesh and sheva, Arabic sittun and sab'un and ancient Egyptian sas and sefex. Does this mean that the Indo-European and Hamito-Semitic words for 6 and 7 go back to an earlier language from which both proto-Indo-European and proto-Hamito-Semitic are descended? According to Russian linguists, both Indo-European and Hamito-Semitic belong to a superfamily they call 'Nostratic.' However, the American linguist Greenberg [1969] disagrees with this view and incorporates IndoEuropean into a so-called Eurasiatic superfamily, which also includes Finnish, Hungarian, Turkish, Mongolian, Korean, Japanese, and even Eskimo! Although Eurasiatic has considerable overlap with Nostratic, in 70

THEMATHEMATICAL INTELLIGENCERVOL.18,NO.4, 1996

his opinion, Hamito-Semitic belongs to a quite distinct Afro-Asiatic superfamily, which also includes Hausa, a language spoken in Nigeria. He believes that any resemblance between Eurasiatic and Afro-Asiatic must go back to an earlier stage, though much later than the hypothetical origin of all languages, at least 50,000 years ago. Indeed, Ruhlen [1994a] presents evidence for a group consisting of Afro-Asiatic, Eurasiatic, and Amerind, between 20,000 and 30,000 years ago. Is the resemblance between the Indo-European and Hamito-Semitic words for 6 and 7 then a coincidence? It could also be that one group borrowed these words from the other. Menninger [1969] considers this idea, but concludes that, if so, it must have been the Semites who did the borrowing. On the other hand, according to a plausible hypothesis by Dolgopolsky [1988], it was the Indo-Europeans who borrowed a number of words from the Semites, among them the words surviving in English as six, seven, goat, star, and wine. His scenario places the homeland of the Indo-Europeans in Anatolia (modern Turkey) about five or six thousand years ago, where they were in close cultural and commercial contact with the Semites to the south. Curiously, the Bible makes a similar but wider claim in Genesis 10: that modern humans emerged from Anatolia about 5000 years ago. In particular, it asserts that the Jupiter-worshipping Indo-Europeans descended from Japhet, son of Noah, and the HamitoSemitic people from his brothers Ham and Shem. The biblical account may have been written about 1000 BC and is confined to a survey of all the people known to the Jews at the time of Solomon. It does not recognize any other language families, and it classifies Elamite as Semitic, although this extinct language is now believed to be more closely related to the Dravidian family (see Ruhlen [1987]). Dravidian is now represented by Tamil, Telugu, and other South Indian languages, but it also survives in a small pocket of Brahui speakers in Pakistan. Incidentally, the Bible in Genesis 2:2 also hints at the origin of the word for 7 when it says that God rested on the seventh day. The Hebrew words for 'rest' and 'seven' resemble one another, and the former survives in our word Sabbath. I don't know whether this was intended as a playful pun or as a serious etymology. Actually, the seven-day week goes back to the ancient Babylonians and presumably was meant to measure the intervals between the four phases of the moon, notwithstanding the theory proposed by Philo of Alexandria and later adopted by St. Augustine that God created the world in six days because 6 is a perfect number. Let us take another look at the intriguing resemblance between two and tooth, which pervades the IndoEuropean as well as the Semitic languages. We may reject as unlikely the hypothesis that the antecedent of two was borrowed like those of six and seven. Though different languages may independently represent the num-

ber 5 by a word for 'hand,' it is not plausible that they would separately come up with a word for 'tooth' representing the number 2. If not accidental, the resemblance between the words for 2 and 'tooth' must then be inherited. Greenberg and his followers believe that all languages have a common origin. In fact, Bengtson and Ruhlen produce a list of 28 words belonging to this hypothetical proto-language (see Ruhlen [1994a]). A surprising number of these words have recognizable descendants in m o d e m English; to mention just a few: hand, hound, who, queen, milk, man, mind (a Greek cognate of which is contained in the first part of mathematics), and toe, which we have met before and to which we shall return later. Looking at this reconstructed proto-language, we find the word *pal for 2. The list does not contain a word for 'tooth,' but the rival list of proto-Nostratic words contains *phal meaning 'tooth', and such a word is still preserved in Telugu palu. Is this a mere coincidence? Perhaps the best attested word on the BengtsonRuhlen list is the reconstructed word *tik, whose IndoEuropean descendants *dekm(t) and *deik we have already met. Other alleged descendants appear in over half of all the language families of the world, usually with the meaning of 'finger' and often standing for the number 1 or 10. The original Indo-Europeans also had a word for 100, usually rendered *kmto. At an early stage, they split into two groups, which are traditionally classified according to whether they preserved the initial k or transformed it into s; they are the so-called Kentum and Satem groups. Both English and French belong to the Kentum group, although English, like other Germanic languages, has transformed the k into h, while paradoxically, in French, the k (from Latin c) became s (still spelled c) after all. Both Klein [1971] and Schwartzman [1994] derive *kmto from *dekm-tom, supposedly meaning 'big ten.' While this appears to be the accepted explanation, Bengtson and Ruhlen seem to suspect that *kmto may be derived from an original form *kano, meaning 'arm' or 'hand' and surviving as English hand. As we saw, Menninger goes even further and wonders whether *dekm(t) should be explained as meaning 'two hands,' which is logical enough but would contradict the widely accepted relation to digit and the Bengtson-Ruhlen derivation from *tik. Our word thousand has cognates only in the Germanic and Balto-Slavic subfamilies of Indo-European; the latter may have borrowed their word from the former. It has been analyzed as meaning something like 'fat hundred' and its first component is said to be related to our word thumb, meaning the 'fat finger.' Bengtson [1987] derives the Indo-European words for 10, 100, and 1000 all from ancient forms meaning 'finger(s)' and 'hand(s).' In this survey, I have ignored compounds such as eleven and twenty and confined attention to traditional

number words, neglecting recent innovations with transparent etymologies such as million and billion. These were introduced during the French revolution; but there is no international agreement on the size of the billion. The cardinality of the empty set was first recognized as a number in India, though our word zero is derived from Arabic sift, meaning 'empty.' Cardinalities of infinite sets were called aleph by Cantor, after the first letter of the Phoenecian-Hebrew alphabet. This letter was so named after a Phoenecian word for 'ox,' and it resembled the head of an ox; but alef is also the Hebrew word for 1000. N e w number words are introduced whenever they are needed and may have strange etymologies. One of my then pre-school sons, after watching a rocket launching on TV, conjectured that the number below zero is called blastoff. The reader interested in finding out more about the origin of number words may wish to look at the fascinating account by Menninger [1969, 1982]. (She should be warned, though, that the English version of Menninger's book has not been proofread too carefully; for example, the letter p is frequently confused with a similar symbol standing for th.) For the origin of other words occurring in mathematics she is referred to Schwartzman [1994]; for the origin of languages, to the books by Ruhlen, in particular the popular account The Origin of Language [1994b]. While Greenberg's hypothesis that all languages have a common origin is still considered controversial in some professional circles, I for one am fairly convinced of its truth. In summary, we have looked at both recent and older speculations about the origin of number words and on how they reflect the origin of languages, as well as at the controversies still surrounding these questions. If the author dares make any suggestions for further research in this field, it is to observe that sometimes resemblances between two words of one language are more easily traced to other languages than are the individual words. This is particularly striking when the two bear no obvious semantic relationship to one another, as for the accepted pair hand and hound. More speculative is the pair tooth and two, where the resemblance may just be a coincidence.

References J.D. Bengtson, Notes on Indo-European '10', "100' and "1,000', Diachronica 4 (1987), 257-262. J.D. Bengtson and M. Ruhlen, Global etymologies, in: Ruhlen 1994, 277-336. M. Bhargava and J. Lambek, A production grammar for Sanskrit kinship terminology, Theoretical Linguistics 18 (1992), 45-60. L. Brunner, Die gemeinsamen Wurzeln des semitischen und indogermanischen Wortschatzes, Francke Verlag, Bern 1969. A. Dolgopolsky, The Indo-European homeland and lexical contacts of proto-lndo-European with other languages, Mediterranean Language Review 3 (1988), 7-31. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

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J. Greenberg, Indo-European and its closest relatives: the Eurasiatic language family, Stanford University Press 1996. , Generalizations about numeral systems, in: K. Denning and S. Kemmer (eds.), On language: Selected writings by Joseph H. Greenberg, Stanford University Press 1990. J.A. Hawkins and M. Gell-Mann, The evolution of human languages, Addison-Wesley Publishing Company, Reading, MA 1992. E. Klein, A comprehensive etymological dictionary of the English language, Elsevier, Amsterdam 1971. K. Menninger, Number words and number symbols, MIT Press, Cambridge 1969. M. Ruhlen, A guide to the world's languages L Stanford University Press 1987, 1991. , On the origin of languages, Stanford University Press 1994. , The origin of language, John Wiley and Sons, New York 1994. , Proto-Amerind numerals, Anthropological Science 103 (1995), 209-225. S. Schwartzman, The words of mathematics, The Mathematical Association of America, Washington, D.C. 1994. C. Zaslavsky, Africa counts, Prindle, Weber and Schmidt, Boston 1973. Department of Mathematics and Statistics McGill University Montreal, Quebec H3A 2K6 Canada

Eight Recent Mathematics Books Reviewed by Jet W i m p

Potential Theory in the Complex Plane by Thomas Ransford

readable for the novice, and Hille's treatment in the second volume of Analytic Function Theory (1959) is too brief and too much oriented toward function theory. The present book satisfies a palpable need. About the only real variable background required is a little knowledge of Borel measures, and what is necessary is smoothly developed in an appendix. As the author assures us, you don't need to romance Borel measures, only to have a flirting acquaintance with their properties. The author is careful to define terms that may be unfamiliar, for example, "compact exhaustion," and to reassure us when a concept is something we needn't usually worry about; for instance, open subsets of the complex plane have compact exhaustion, as (trivially) do all compact spaces. When one is trying to learn only enough about something to use it, it helps to know what to fret about and what to take for granted. The book begins with broad discussions of harmonic and subharmonic functions and then delves into potential theory. The treatment of polar sets is captivating. Next comes the Dirichlet problem, and then, in Chapter 5, capacity. This is, hands down, the most satisfying treatment of this highly intuitive yet devilishly subtle concept. The author talks about actually computing capacity and gives an amusing and useful table listing the capacity of some common sets: ellipses, intervals, triangles. The book closes with an application of potential theory to various problems in analysis. An excellent text; my compliments to the author.

Contests in Higher Mathematics: Mikl6s Schweitzer Competitions 1962-1991 edited by G~bor J. Sz6kely

Cambridge: Cambridge University Press, 1995. London Mathematical Society Student Texts 28, x + 232 pp. Hardcover, ISBN 0 5621 46120 0 $54.95; softcover, ISBN 0 5212 46654 7 $19.95.

N e w York: Springer-Verlag, 1996. vii + 569 pp. US $39.00, ISBN 0 387 94588 1

This book is a engaging addition to the estimable London Mathematical Student Text Series. These texts are generally short works, seldom over 250 pages, designed to make contemporary mathematics accessible to today's graduate students. The quality of exposition that prevails in this series is high, and this book is no exception. It is, I believe, the best treatment of potential theory available. Potential theory has proved to be of crucial importance in many areas of contemporary analysis: in special function theory, where it is used to analyze the root distribution of orthogonal polynomials and the behavior of the coefficients in their recursion formulas; in numerical analysis, for instance, estimating the error of Gaussian integration procedures; and in approximation theory, namely, the problem of uniform approximation. Unfortunately, current treatments do not recommend themselves to someone who wants to master the use of the theory as a tool. Tsuji's book, Potential Theory in Modern Function Theory (Chelsea, 1975), is all but un-

I am a sucker for two things: undefended cats and problem books in mathematics. I have a house full of the former and a library full of the latter. When I'm feeling selfcritical, I condemn my infatuation with problem books as a waste of time, something like Nintendo or surfing the web. On the other hand, I sometimes think that gratitude for such books--is this a positive addiction?--is more appropriate. Several of my published papers had their origins in intriguing mathematical problems I discovered in books I reviewed. In 1894, the Hungarian Mathematical and Physical Society initiated a high school competition for mathematical students. Among the winners of this event, at one time or another, were people with familiar names: Fej6r, Haar, K~rm~in, (Marcel) Riesz. The success of this competition led to the establishment in 1949 of a college-level contest, named after Mikl6s Schweitzer. The contest problems are suggested by prominent Hungarian mathematicians, so they tend to reflect mainstream mathematical thinking in Hungary. The problems in the current book

72

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

are a continuation of those given in a previous book, Contests in Higher Mathematics, 1949-1961, published by the Kiad6 Academy of Budapest in 1968. The present book is a member of the Springer series Problem Books in Mathematics, edited by Paul Halmos. All the books in this series are reasonably priced. The statement of the problems---usually with the original proposer identified--is predictably brief, about 50 pages. The solutions of the problems are predictably protracted, almost 500 pages. The problems are taken from all areas of mathematics and none is easy. Yet none is oppressively difficult, provided the solver knows a little about the problem area. There are many delicious morsels in this book. I'll give two of my favorites.

Let yl(x) be a continuous positive function on [0, A] and y,+](x) --

2fxoX/yn(t) dt,

n = 1, 2 , . . . .

Show limo~ yn(X) = x 2 uniformly. My first reaction, before I even attacked the problem, was predictably avid: "I can generalize that!" Alas, the problem-solver had already done it.

Let f(n) denote the maximum number of right triangles determined by n coplanar points. Show that lim fin) n2 -- ~,

n ---* ~

lim f(n) = 0. n 3

n ---> ~

The proposer of the first is not given. The proposer of the second is Paul Erd6s. Part of the success of any problem book lies in elegance of design. Are the problems well separated from each other? Is access to the solutions made easy? In short, browsing in a problem book should not be a problem. The publishers, Springer-Verlag, have not let us down with this volume: it's sleek, eumorphous, wellpaced, and scrumptiously typeset. Those sharing my disorder should either avoid this book at all costs or immediately order it. I'm not certain which piece of advice is the more responsible.

Catastrophe T h e o r y by V. I. Arnol'd Third revised and expanded edition, translated from the Russian by G. S. Wassermann, based on a translation by R. K. Thomas N e w York: Springer-Verlag, 1992. xiii + 150 pp. US $29.95, ISBN 3 540 54811 4 This unusual little paperback, translated from the Russian, has the qualities we tend to associate with Russian expository mathematical writing: a highly personal style, imaginatively presented examples, nontechnical language, sometimes puzzling asides, and de trop observations taken from the world at large.

What is catastrophe theory? Well, many of us may remember that several years ago it was the regnant mathematical fad. Almost every physical law or phenomenon-from insect mating rituals to the curdling of milk--found its interpretation as a cusp, a fold, a bifurcation. Its proponents, swept to impossible heights of hype, discovered in it an afflatus denied them by the more mundane religious passions. These paeans today make pretty uncomfortable reading. "Catastrophe theory . . . . "Ren6 Thorn enthused in 1974, "favorizes a dialectical, Heraclitian view of the universe, of a world which is the continual theater of the battle between 'logoi'.... It is a fundamentally polytheistic outlook to which it leads us: in all things one must learn to recognize the hand of God . . . . Just as the hero of the Iliad could go against the will of a God, such as Poseidon, only by invoking the power of an opposed divinity, such as Athena, so shall we be able to restrain the action of an archetype only by opposing to it an antagonistic archetype..." Thorn goes on to hint at catastrophe theory's potential for resolving such eschatological issues as failure, success, illness, and death. But I will spare the reader any more of his fervor. Well, the new religion had feet of clay, as Mark Kac opined, when he discovered a simple and common chemical reaction which refused to submit to the predictions of catastrophe theory. "Is this just nature being unkind to mathematicians?" he sneered. With all fads, the hype eventually dissipates, and we are then left to contemplate in relative detachment whatever of true value it has bequeathed to the mathematical canon. This book, despite its occasional agitation, tends to view the discipline with coolheaded and occasionally very critical hindsight. " . . . we can try to use this information to study large numbers of diverse phenomena and processes in all areas of science ... in the majority of works on catastrophe theory, however," he cautions, "a much more controversial situation is considered, where not only are the details of the mapping to be studied not known, but its very existence is highly problematical." The theory, the author points out, goes back to the work of the American mathematician Hassler Whitney, who in 1955 proved that every singularity of a smooth mapping of a surface onto a plane, after an appropriate small perturbation, can be described by folds and cusps. The topic Whitney studied came to be called singularity theory, and that theory coupled with its applications was later termed catastrophe theory by the English mathematician E. C. Zeeman. The publication of a selection of Zeeman's papers, Catastrophe Theory: Selected Papers, 1972-1977 (Addison-Wesley, 1977), started an avalanche of articles of decreasing rigor and increasing hyperbole, popularity, and delirium. As an example of one of the speculative uses of catastrophe theory, the author quotes Zeeman's analysis of the "creative personality" in terms of three parameters, T H E M A T H E M A T I C A L INTELLIGENCER VOL. 18, N O . 4, 1996

73

technical proficiency = T, enthusiasm = E, and achievement = A. The theory, applied to the resulting surface in three-dimensional (T, E, A) space, allegedly furnishes us with truths about the profound issue of human creativity. We even have catastrophe theory models of drug addiction! ~ Here and in many other instances, catastrophe theory, if it does anything, only provides a dubitable verification of conclusions we can already reach because of our knowledge of human behavior. Arnol'd will have none of this moonraking. Brutally, he tallies the shortcomings of Zeeman's construct, declaring, "The deficiencies of this model and many similar speculations in catastrophe theory are too obvious to discuss in detail. I remark that articles on catastrophe theory are distinguished by a sharp and catastrophic 2 lowering of the demands of rigor . . . . " He slyly quotes a passage from the obscure Russian novelist V. M. Doroshevich (1864-1922): "I think, dear, that all this decadence is nothing more than a way of approaching tradesmen." Arnol'd's discussion is reality-oriented. One of the first concepts he introduces is that of a catastrophe machine, a simple device consisting of a board, a cardboard disk, pins, pencil, and rubber bands, which he invites the reader to build. The apparatus exemplifies beautifully the sophisticated ideas of the bifurcation of equilibrium states in elasticity. The resulting catastrophe curve has four cusps. Next, he talks about possible bifurcation states, and then, in short but highly readable sections, he discusses loss of stability of equilibrium, caustics and wave fronts. I found the chapter "The Large Scale Distribution of Matter in the Universe" tremendously exciting. As the reader undoubtedly knows, tile inhomogeneity of the universe has long puzzled cosmologists; in its early stages of development, the universe was homogeneous. What caused the transition from homogeneity to nonhomogeneity? Zel'dovich, in 1970, proposed an explanation that is mathematically equivalent to the formation of singularities, a topic covered in this book's previous chapters. The treatment takes only four pages, and it is gripping. The remaining chapters deal with singularities in optimization problems, smooth surfaces and their projections, and applications. In a welcome appendix, Arnol'd examines the origins of catastrophe theory and shows that it did not originate with Whitney nor with any other single person. Huygens, Cayley, Jacobi, Kronecker, Poincar6, all explored similar concepts. He can't resist a swipe at the deterioration of the mathematical idiom. "The unsophisticated texts of Poincar6 are difficult for mathematicians raised on set theory," he observes, ironically. 1S. J. Guastello, Cusp and butterfly catastrophe modelling of two opponent process models: Drug addiction and work performance, Behavioral Science 29, 258-262 (1984). 2I don't know whether this pun exists in the Russian, or is merely the whimsy of the translator.

74 THE MATHEMATICALINTELLIGENCERVOL.18, NO. 4, 1996

He points out that whereas Poincar6 would have said, "The line divides the plane into two half-planes," modern mathematicians write, "The set of equivalence classes of the complement R2/R 1 of the line R 1 in the plane R 2 defined by the following equivalence relation: two points A, B ~ R2/R 1 are considered to be equivalent if the line segment AB connecting them does not intersect the line R 1, consists of two elements." He claims he is quoting from an actual textbook. I believe him.

Real Computing Made Real: Preventing Errors in Engineering and Scientific Calculations by Forman S. Acton Princeton, N.J.: Princeton University Press vii + 259 pp. US $29.95, ISBN 0 691 03663 2 "Who," I asked myself, "is Forman S. Acton?" Actually, I knew: the author of the 1957 book, Analysis of Straight Line Data (John Wiley and Sons) and the 1970 b o o k Numerical Methods That Work, recently reprinted by the Mathematical Association of America. What I meant was, "Who is Forman S. Acton to write a book on error analysis of numerical methods?" Yes, professor emeritus of computer science at Princeton, but hardly a household name. A scant three papers on numerical analysis to his credit. I briefly riffled the pages. No mention of norms, no mention of Banach spaces. No matrix inequalities, no condition numbers, no closure, no completeness. We all know you have to have such things to discuss error analysis intelligently: Isaacson, Keller, and Davis have predicated their reputations on convincing us of this. Instead, this book gives us Latin quotations, frequently intercalated photographs of Thai temple statues, a beginning chapter labeled "An Exhortation," nursery rhymes ("As I was going up the stair, I met a man who wasn't there . . . . "), text printed upside down (to discourage peeking at a solution), and headings done in a script like the titles of a Charlie Chan movie. Acton, apparently, runs a reductionist numerical atelier. "When the volume and sophistication of your problems demand [new] weapons, you will know it." He quotes approvingly Hamming's famous comment, "The purpose of computing is insight, not numbers." He provides gentle admonitions and occasional homilies. If you can't understand how to solve quadratic equationy the example of which begins his discussion--and keep the error under control, he seems to be saying, you won't understand error in normed spaces, either. It may be that in this book he is trying to make the point that the appreciation of error is as much a humanistic enterprise as it is a mathematical one. If so, he is on firm philosophical ground. I have long puzzled over the reasons w h y students find the concept of error so elusive. What could be more intuitive than the closeness of one object to another? My

students happily reel off three figures from their desk calculators w h e n they should k n o w it's not enough, seld o m try to reconcile the n u m b e r they have calculated with the realities of the problem, and blissfully give me an occasional negative n u m b e r w h e n I ask for the cosh of something. A n d m y students are engineers. I think Acton has p u t his finger on the problem, and for doing so he deserves some sort of decoration. H e says, "[numerical methods] are also the vehicle for teaching something m o r e important: how to visualize the shape of a function and ultimately, h o w to use that shape to construct algorithms that process that function accurately and efficiently." H e continues with the trenchant observation, " G e o m e t r y is as important as algebra and calculus here, with constant switching of the viewpoint between them." Yes indeed. Check it out. I'll bet you'll find that students w h o can't c o m p r e h e n d numerical error are also those with the weakest geometric intuition. This beautifully written book doesn't have a lot of fancy mathematics in it, but it can help close the gulf between blind calculation and true understanding. There are many, m a n y exercises, and sedulously devised solutions. I sense that Acton is a veteran of long standing of the classroom wars. H e knows h o w students think and h o w they fail to think, and it's to this kind of voice that those w h o value the role of numerical analysis in building a tolerable technological society should be listening.

The Queen of Mathematics: An Introduction to Number Theory by W. S. Anglin Dordrecht: Kluwer Academic Publishers, 1995. x + 389 pp.; US $155.00, ISBN 0 7923 3287 3

Each of the writers of the new books on n u m b e r theory has staked out a claim for a different part of the territory, although there is, inevitably, some overlap. Anglin's relies heavily on simple continued fractions (continued fractions whose coefficients are integers). Thus, it invites comparison with the book Continued Fractions, b y A. M. Rockett and Peter Sz/isz (World Scientific, 1992). That book has some virtues, but after beginning their discussion equably, the authors become fastidious and nit-picky b e y o n d endurance, and reading the book is like watching a hyperactive spider at work on a web. Anglin is simply a fine writer (the philosophical background coming through, perhaps?) and the material never gets away from him. Everything is u n d e r control, and each topic receives its p r o p e r modicum of attention. Nevertheless, this is a v e r y curious book; to some extent, in its curiosity lie its virtues. The book is Volume 8 in Kluwer's Graduate-Level book series. Yet it requires almost no b a c k g r o u n d - - n o t more than a d o z e n pages even use calculus. One could use it to teach a highly motivated high school class. Anglin has respect for the cultural aspects of the subject, and often gives a historical or authorial context for the result at hand. I therefore f o u n d it pretty strange that he is so lackadaisical in providing references in other places. Let m e illustrate. From page 5, In 1875 Edouard Lucas, who had been a French artillery ofricer in the Franco-Prussian war, challenged the readers of the Nouvelles Annales de Mathdmatiques to prove the following: A square pyramid of cannon balls contains a square number of cannon-balls only when it has 24 cannon-balls along its base. In other words, the only nontrivial natural number solution of 12+22+

One of the pleasures of serving as book review editor of this journal is having d u m p e d in m y lap a continual profusion of books on n u m b e r theory. N o n e of the books I have examined so far is an unqualified success, but the subject brings out the best in potential writers. The intellectual b e a u t y of the subject seems to vivify authors and stoke their mental acquisitiveness to a white heat. It is axiomatic that w h e n an author (or teacher!) is interested in the subject, w e probably will be, too. The author of the present book is a tyro---as far as I can discern, Anglin's b a c k g r o u n d is in p h i l o s o p h y and in expository mathematical writing3--but that doesn't m e a n the book should be dismissed.

interdisciplinary Ph.D. in mathematics and philosophy from McGill in 1987, and prior to his present position at McGill he was a postdoc in philosophy of religion at Notre Dame. Also the author of Mathematics: a Concise History and Philosophy (Springer-Verlag, 1994). Anyone considering the use of that book as a text should be prepared to tolerate the occasional incursion of the author's religious affections, such as his attempt to buttress up Descartes's lame ontological arguments by an appeal to authority. 3An

9

+n 2=m 2

is n = 24, m = 90. Anglin points out that the problem waited for a solution until 1918, w h e n G. N. Watson solved it by using hyperelliptic functions. Later in the book, Anglin offers an elementary proof he himself obtained b y simplifying a proof due to D. G. Ma. Anglin should have p r o v i d e d this reference. 4 The Watson reference is also missing. 5 The bibliography contains only 10 entries, far too few for a textbook with such aspirations. The absence of calculus in the book m a y have w o r k e d to the author's advantage, as there is evidence he is uncomfortable with analytic arguments. For instance, he

4Sichuan Daxue Xuebao 4, 107-116 (1985). Z h e n Fu Cao claims, in MR91e:11026 that he a n d Z. Y. Xu g a v e a n e l e m e n t a r y proof of this fact in 1985 in Kexue Tongbao. These C h i n e s e journals are only intermittently abstracted in Math Reviews. This p r o b l e m h a s a connection w i t h the f a m o u s p r o b l e m of Mordell, a b o u t w h i c h so m u c h h a s been written.

5Messenger Math. 48, 1-22 (1918-1919). THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 4, 1996 75

wants to show that CV~Tc e-L2 ~0 et2 dt = 0(1), L --~ o% c > 0. His proof requires three lemmas. Note, however, that the order of the quantity on the left is unchanged by making the lower limit of the integral, say, 1. N o w integrate by parts with U = 1/2t, dV = 2tet2dt. Also, there is a great deal of flummery about integrals of the sort

fcc+i~ eft ~ dt, c > O, o~ < O. These are,

of

course, just

Gamma functions, despite the complex contour. Chapter I is titled "Propaedeutics" (< Gk., pro = before + paideO = instruct) and contains material about primes, Bernoullii numbers, Diophantine equations, Fermat's last theorem (now out of date), and the M6bius function. Chapter 2 treats continued fractions, Chapter 3 congruences, Chapter 4 the equation x 2 - Ry 2 = C. Chapter 5 talks about classical construction problems, not a typical subject in a number theory book. Even more unusual is the inclusion of the proof of Lindemann's famous theorem about the transcendence of It. The polygonal number theorem occurs in Chapter 6, and Chapter 7 is devoted to analytic number theory. In particular, the author derives Rademacher's spectacular series for the partition function--a topic usually confined to advanced books on analytic number theory. Rademacher's result is one of the great achievements of analysis in this century, and the author's devotion to it is clearly reflected in the considerable effort it must have cost him to present it in a way that is commensurate with the undemanding level of the book. There is enough uncommon material here to lend admirable new gloss to the subject, and few books on number theory are as well written as this one. However, the astronomical price of $155 makes me wonder whether Kluwer has misjudged its market.

A Primer on Nonlinear Analysis by A. Ambrosetti and G. Prodi Cambridge: The Cambridge University Press, first paperback edition (with corrections) 1995. vii + 171 pp. US $22.95, ISBN 0 521 48573 8 I get a creepy feeling when I open a book and see the first chapter labeled "0." In the past it has usually meant that I must expect a heinous snow job. Wasn't it Halmos who, in the introduction to his Measure Theory, remarked that the reader shouldn't be discouraged if he finds he doesn't have the prerequisites to read the prerequisites? But in this case, I shouldn't have worried. There's no more economical or lucid introduction to the subject than this great little book, a paperback member #34 of the fine paperback Cambridge Studies in Advanced 76

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

Mathematics. I sometimes think that this must be the golden age of expository writing, particularly with Springer-Verlag getting into the act, issuing one superb text after another. It's a great time to be a book review editor. What's noteworthy about this book is the seamless w a y the authors fuse mathematical statements with concrete and challenging physical applications--water waves, the heavy string, the restricted three-body problem. If you plan to give a rather abstract course in (primarily linear) functional analysis, you might consider following it with a segment based this material, which both demonstrates the power of functional analysis and serves as an introduction to nonlinear methods. The mathematics is robust. The zeroth chapter gives some of the background required: Banach spaces, the Fredholm alternative, function spaces, a statement of the general elliptic boundary value problem. I was amused " to find the first chapter called "Differential Calculus," the authors' little joke, perhaps. The "differential" signifies, of course, Fr6chet and Gateaux derivatives. Intoxicating stuff, indeed. Occasionally, the authors slip their leashes, threatening to menace the reader with derivatives in convex topological spaces and even pseudotopological spaces--but only as an afterthought; they soon wander back to more worldly matters, such as Nemitski operators. Chapter 2 states and proves an invaluable local inversion theorem, which guarantees that a map between Banach spaces will possess, at least locally, an inverse. This theorem places on a rigorous foundation what engineers know as the process of linearization, that is, gaining insight into a nonlinear problem locally through a study of its linear approximation. By now, the authors have developed enough tools to tackle serious issues, one of the first being a description of T-periodic solutions of the differential equation x" + g(x,x')= eh(t), where g is continuously differentiable and h is continuous and T-periodic. This would have been a good place to treat Newton's method in normed spaces, a la Kantorovich and Akilov, Functional Analysis in Normed Spaces (Pergamon, 1964). It is the cleanest application I know of the Fr6chet derivative. However, the authors seem interested only in physical applications, more specifically, physical applications modeled by differential equations. Chapter 2 closes with a discussion of the implicit function theorem, the stability of orbits, and autonomous systems. Chapter 3 treats global, as opposed to local, inversion theorems. The main theorem, sometimes called the monodromy theorem, goes back to Hadamard in the finite-dimensional case, and to Caccioppoli and Levy for general Banach spaces. In Chapter 4 we find semilinear Dirichlet problems and a wealth of their applications, to resonance problems and problems with asymmetric nonlinearities. Chapter 5 states general results on bifurcation, and Chapter 6 has many applications of the ma-

terial: to the rotating heavy string, to the B6nard problem (which involves convective motions in a heated fluid), to small oscillations for second-order dynamical systems. I would have liked to see some discussion of solitons here, via the Korteweg-deVries equation, and of nonlinear variational problems. However, incorporating all of my proposed additions would make the primer no longer a primer. I realize the authors had to draw the line somewhere. An abstract version of the Hopf bifurcation theorem starts Chapter 7, and the Lyapunov center theorem and the restricted three-body problem furnish the conclusion of this remarkable little book. Though it may be familiar to many, let me describe the restricted three-body problem. I make the point to my differential equations classes that, historically, Newtonian mechanics 6 and its formulation in terms of differential equations enabled humans for the first time to uncover truths about the physical world that were divorced from the often bogus conclusions provided by Aristotelean introspection. 7 The three-body problem provides an excellent paradigm of the power of mathematical modeling. The problem deals with three bodies, P1, P2 (called primaries), and Q with masses M1, M2 and M3, respectively, acting under Newtonian gravitation. It is assumed that M3 is insignificant compared to M1 and M2. There are several other rather minor restrictions imposed on the bodies, but the resulting motion is still of great practical interest. It can be shown that in this system there are five equilibrium points, Lj, j = 1, 2 , . . . , 5. Three are called Euler points; two are called Lagrange points. Among the several fascinating facts deduced in this chapter is that in a neighborhood of the Euler points, as well as the Lagrange points (these require several additional assumptions), the system has a family of periodic solutions. There are five pages of very challenging exercises. Overall, the English of the book is supple and unflawed--a consequence, I suspect, of diligent editing by the people at Cambridge. There are occasional indications that the editors were asleep at their desks, or unduly cowed or even hypnotized by the mathematics; witness the following passage: "Even if such a theorem is a classical result often understood in the current literature we think useful to have given here an elementary version, in the frame of Banach spaces." I have stressed to the desk editors of one of the journals of which I am an editor that they must not allow themselves to be intimidated by the mathematical expertise of the writers whose manuscripts they are scrutinizing.

6To be fair, I should also mentionthe name of Galileohere.

7From the beginning of his career, Newton was skeptical of Aristotelean science, despite his high regard for the Greek philosophers: "Amicus Plato amicus Aristoteles magis arnica veritas," he scribbled in a notebookin 1664.

Good English writing requires its own set of skills, and has its own indubitable authority.

Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists by Dennis Shasha and Cathy Lazere New York: Springer-Verlag, 1995. xi + 289 pp. US $23.00, ISBN 0 387 97992 1 Who could fail to be delighted by the title of this book? Friends who have long since tired of hearing me talk about the books I've been reading do a double take when I mention this one. Dennis Shasha is a young professor of computer science at NYU's Courant Institute, Cathy Lazere is a professional writer. The reader will conclude, correctly, that this is a book aimed at the popular market. It's issued under the colophon Copernicus, a new imprint of Springer-Verlag, and the SpringerVerlag name is tendered in such delicate lettering as to suggest the august publisher's discomfort at venturing onto the unseemly turf of bestsellerdom. The computer protogenists discussed here fall into four classes, which the authors describe as follows: 1. Linguists: How should I talk to the machine? 2. Algorithmists: What is a good method for solving a problem fast on my computer? 3. Architects: Can I build a better computer? 4. Sculptors of machine intelligence: Can I write a computer program that can find its own solutions? There are unwelcome lessons in this book for those

of us who teach. Reading a few of the biographies will force us to look at our unsuccessful students---our declared failures--in a new light. Several of these scientists were dropouts. Take the case of John Backus. Backus's father was chief chemist for Atlas Powder Company, but it seemed his scientific talent wasn't passed on to his son. Backus was born in 1924 in Philadelphia and went to the Hill School--a venerable intermediate institution--in Pottstown. "I flunked out every year," he says. "I hated studying. I was just goofing around." Next, Backus had a run at the University of Virginia, attempting to major in chemistry. He detested the labs, though. He spent most of his time partying, waiting to be drafted. He joined the Army in 1943, and because of his performance on an aptitude test, the Army consigned him to a pre-med program. Bizarrely, while attending, he was diagnosed with a tumor of the skull and had a metal plate patched into his head. Soon after, he enrolled in New York Medical College. "I hated it," he says. He quit and rented a small apartment in New York. "I really didn't know what the hell I wanted to do with my life. I decided that what I wanted was a good high-fi set . . . . " After a little experimentation with stereo systems and reading about their design, he decided that he might be THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

77

interested in math and enrolled in Columbia University's undergraduate program. He found he disliked calculus. By the spring of 1949, when he was 29 and a few months short of graduating with a B.S., he lost interest. One day he happened into the IBM computing center on Madison Avenue and was shown the SSEC, one of IBM's first electronic machines. He mentioned to his guide that he was looking for work and she urged him to talk to the director. "I said no, I couldn't. I looked sloppy and disheveled. But she insisted and so I did. I took the test and did OK." He was hired to work on the SSEC. To appreciate the ironies inherent in this drama, it helps to know that the state of computer programming in 1949 was Mesozoic. Programming was extremely costly because everything was done in assembly language, only one step removed from the machine language of binary digits. Backus, along with Harlan Herrick, created a program called Speedcoding, which supported floating-point calculations and helped to resolve some of the vexing problems caused by scaling of numerical quantities in computer programs. Working so extensively with assembly language programming convinced Backus that computers needed a high-level language, one which could translate mathematical problems into machine language. In 1953, only 4 years after being hired, he wrote a memo to that effect to his boss, Cuthbert Hurd. Interestingly, John von Neumann, who was a consultant to IBM at the time, advanced persuasive arguments against the project. He felt such a language would alienate programmers from the implications of their computations. And he didn't attach much importance to the costs of programming. Amazingly, Cuthbert sided with Backus. Von Neumann's objections were disregarded and Backus was allowed to hire an ill-assorted passle of both experienced programmers and novices straight from high school to help him. Their avowed intention was to create a language that would make programming easy. By 1957 the language that we know as FORTRAN was up and running. I find this story quite unsettling. N ow I understand w h y the medieval troubadours composed such fervent poems to Dame Fortune. The slightest breeze could have wafted Backus into a totally different profession: shoe salesman, greengrocer, copy editor. The professions in which we ultimately find ourselves, like all the events in our lives, are disturbingly contingent. Was Backus's genius translatable? Would he equally have been a genius at choosing shoes sizes for suburban matrons, judging the state of ripeness of a cantaloupe, prescribing the proper use of "that" and "which"? Who knows. And if he hadn't wandered into the office of IBM in 1949, what would be the state of computers today? There are 14 other equally mesmerizing hum a n dramas in this book. Every biography in this enthralling collection is a story, a narrative in the most profound 78

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996

sense of the word. Popularization or not, SpringerVerlag has published a book of which it should be proud.

Five Hundred Mathematical Challenges by Edward Barbeau, William Moser, and Murray Klamkin Washington, DC: The Mathematical Association of America, 1995. ix + 225 pp. US $29.50, ISBN 0 88385 519 4 I was browsing through this book on 40th Street in West Philadelphia the other day and a frenzied chorus of car honking broke out behind me. The light had changed. Well, after all, this is another problem book . . . . 8 Different, though, from the one discussed above. It's much more elementary; many of the items are accessible to high school students. The need for calculus is minimal, and the editors suggest the collection be described as "problems in pre-calculus mathematics." The problems first appeared in a series of five booklets published by the Canadian Mathematical Society. They were so popular that the present editors decided to issue an edited and revised version of all five. The editors are three of the leading problemists 9 of the day, so the problems are bound to be well chosen and the solutions lucid and well coiffed. The organization of the book makes it a superb pedagogical instrument. First come the problems, stated with no particular ordering (i.e., problems from combinatorics, arithmetic, algebra, inequalities, geometry and trigonometry) and the analyses are all jumbled up. There are 47 pages of them. There are 164 pages of solutions. Then comes a section the authors call a Toolchest, which is a 12-page list of useful facts drawn from the previous 6 branches of mathematics. This feature makes the book highly serviceable for the student. Throughout the book are interspersed fables concerning mathematicians and occasional bons roots. They are wonderful. When Leo Moser was playing in a chess tournament in Toronto in 1946, a bystander was heckling the players. "Chess is a complete waste of time," the heckler shouted. "It has no relation to any other branch of knowledge." "How about mathematics?" Moser asked him. "I have studied mathematics for many years," the man replied. "and know that chess has no relation to any of the four branches of

SThis,and countless other anecdoteswe have all traffickedin, furnish ample evidencethat the driving--and presumably other--privileges of mathematicians should be subjectto some restrictions.A great one, about A. N. Whitehead, is given in the present book. He was cautioning a student about a theory of logic. "This," he said, "should be taken with a grain of er... um... ah..." "Salt,Professor?"the student suggested. "Ah yes," Whitehead said brightly, "I knew it was some chemical." 9Maybe a new word is needed. Questilargitors?

mathematics." "What branches do you mean?" Moser asked. "You know," the heckler snapped disdainfully, "addition, subtraction, multiplication, division."

Show that for any positive integer n, [V~n + X/n + 1] = [~n-n + 2]. Find a closed-form expression for [V~] + [V2] + . . . +

The second, attributed to C. W. Trigg, editor of Mathematical Quickies (Dover, 1985), is probably as perceptive a definition as any of good mathematics. An elegant solution is generally considered to be one characterized by clarity, conciseness, logic and surprise. What are the problems like? I was d r a w n - - m a y b e m y m o o d - - t o some of the puzzlers involving the greatest integer function, [.]:

[~r

2 -- 1 ) 1 / 2 ] .

The book is a paperback, done in a large, elegantly printed format. I suggest you try it out on some of your talented undergraduate students.

Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 USA e-maih [email protected]

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996 7 9

1996 Anniversaries The following mathematicians all celebrated anniversaries last year. 400 years: Ren6 Descartes (born 1596) has been called the father of modern philosophy. His Discours de la mdthode, a treatise on universal science, appeared in 1637; it was wrongly titled on the stamp below, but later corrected. The third appendix, entitled La g6om6trie, contains Descartes's fundamental contributions to analytical geometry, as well as material on the classification of curves and the so-called Descartes' rule

of signs. 350 years: Gottfried Wilhelm Leibniz (born 1646) aimed to build the whole of knowledge from a few basic principles. This aim led to plans for a universal language for mathematical logic and the construction of a number of calculating machines that could add, subtract, multiply, divide and find square roots. Although Newton had priority for the invention of calculus, Leibniz published his results first and his notation proved to be more reliable than Newton's.

Robin Wilson*

250 years: Gaspard Monge (born 1746) developed a method for representing three-dimensional objects in the plane, thereby initiating the subject of descriptive geometry, and wrote the first book on differential geometry. He served on the committee that established the metric system in 1791, and was director of the ]~cole Polytechnique. He was Minister for the Navy in 1792-3 and accompanied his close friend Napoleon on an expedition to Egypt in 1798. 200 years: Lambert Adolphe Jacques Quetelet (born 1796) was a Belgian statistician who proposed the notion that naturally occurring distributions tend to follow a normal curve. He established a central statistical bureau that was imitated around the world. 150 years: Friedrich Wilhelm Bessel (died 1846) was a German mathematician and astronomer whose measurements on 50,000 stars first allowed the accurate determination of interstellar distances. In 1817, while investigating a problem of Kepler, he introduced the Bessel function Jn(x) of order n; this stamp depicts the Bessel functions Jo(x) and h(x).

Descartes

Leibniz

Monge

Quetelet

Bessel

*Column editor's address: Facultyof Mathematicsand Computing,The Open University,MiltonKeynes,MK7 6AA. England. 80

THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York