No title

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.

The Composition Identity I write to protest an appalling mathe­ matical scandal: the lack of a standard notation for the composition identity. The nearest thing to a standard no­ tation is the abbreviation "id," assum­ ing the domain is clear from the con­ text. Note that "id" is two letters­ which won't do in an introductory cal­ culus text where the need first becomes urgent. The notation x� x is even worse. One "solution" (in the context of single-variable calculus) is to make ex­ plicit the often tacit identification of the independent variable with the compo­ sition identity. Then the common gaffe f= f(x) is correct, since the variable x represents the identity function on the domain off Unfortunately, the inerad­ icable habit of always thinking x' = 1, which this convention engenders, naively but inexorably mutates to u' = 1, thereby subverting the chain rule to (f( u))' = .f'Cu). Students then view cor­ rections to this faux pas as exceptions to the rule, notably when u represents a constant, or when x represents a func­ tion of t in a related-rates context. Adoption of a universal symbol for the identity function would do away with much if not all of this type of con­ fusion. I have seen the pound sign, the dol­ lar sign, and other special characters used in various contexts. For example, if # represents the identity function, then you can write (e#)' = e# instead Of the CUmbersome (X� eX)' =(X� ex), or (e-")' = e'", which is not true un­ less x' = 1, i.e., only when x represents the identity function. Uppercase i or "I" for identity is another possibility for more general application. The identi­ cally equal symbol with three horizon­ tal dashes (not possible to write in this text editor) is another possibility. I propose that Tbe Mathematical In­ telligencer take nominations for two or three years and then hold a vote. Our foresight will be taken for granted in future generations as we now take for granted the use of "0" for zero, the ad-

ditive identity, though it, too, was courageously adopted only when long overdue. My choice would be�. the Greek let­ ter iota, if it were available on standard keyboards. Forest W. Simmons Portland Community College Portland, Oregon USA e-mail: [email protected]

Reply Does the identity function need a sym­ bol of its own? Fifty years ago, Karl Menger made the case for a variable­ free calculus ( Calculus: a modern ap­ proach, Ginn and Co. , 1 955), but there is still no consensus. In this issue, For­ rest Simmons reopens the discussion. We hope his letter will spark a debate. Please send us your thoughts-and your suggestions. (Two candidates­ Menger j, Simmons �-are already in the running). We'll invite you to vote in two years or so. -The Editors

The Road to Reality In the Summer 2006 issue you pub­ lished two reviews of Roger Penrose's book Tbe Road to Reality. They bring to mind the standard politics of two­ party Anglo-Saxon democratic systems as trivialized by journals such as Newsweek, or rather, the "good cop­ bad cop" approach to criminals. Did you do that by mistake, or on the con­ trary, as a matter of pride, to try to im­ plant that approach into science? In the less than fortunate latter case, one can wonder why only two opposing views were presented. Why not, indeed, three, or even more opposing views? After all, why not bring some sort of circus into rather arid realms like mathematics? And now back to the two reviews. The first, shorter and quite sparse in detail, finds the book highly meritori­ ous and readable. The second finds quite a number of outstanding features,

© 2007 Sprtnger Science+ Business Media, Inc., Volume 29, Number 3, 2007

5

but that is totally and hopelessly drowned in a manifestly vicious over­ all prejudiced attitude and judgement. One can only wonder how a third, or perhaps, fourth and so on, review might have looked, had The Mathe­ matical Jntelligencer gone one better than the trivial Newsweek approach. I myself have had some arguments with Penrose on certain strictly mathe­ matical issues; thus I cannot be counted as one of his unconditional admirers. But I would like to say here, first and above all, that the subject of the book is by far the most fundamental and con­ sequential of the last few centuries. Second, for more than half a century now, science has discouraged scholar­ ship, especially wide-ranging and deep scholarship, in favour of narrowly spe­ cialized research production. Penrose happens to be one of the very few scholars, if not in fact the only one nowadays, with truly impressive depth and breadth. Consequently, even if his latest book were rather poor, which clearly it is not, one should appreciate his scholarship and his willingness to make the considerable effort to bring it into the public domain. Penrose, in this book, has given us a grand and most fascinating view of a fundamental and all-important field of science. A view that, hopefully, will tempt many in future generations to try to complete. For others who care to look at it, or to browse it, or read parts of it, the book may help them connect to things beyond, and no less impor­ tant than, day-to-day concerns or events. Elemer E Rosinger Emeritus Professor Department of Mathematics and Applied Mathematics University of Pretoria

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6

THE MATHEMATICAL INTELLIGENCER

2007 Springer Science+Business Media, Inc.

UM'ii

A Mathematician Called Bourbaki

H

ieremia Drexel (1 '581-1638) was a jesuit and professor of "the clas­ sics" at Augsburg ( Germany). He wrote many books in Latin on history and theology. Among these books is Aurifodina Artium et Scientiarum om-

nium; Excerpendi Sollertia Omnibus lit­ teraru m amantibus monstrata [ Gold Mine of Arts and Sciences, judiciously Chosen Extracts to Be Shown to Cul­ tured Amateurs] (Figure 1). In this book the author describes the

FRANCOIS l.AUBIE

Figure I.

In the frontispiece author.

is probably the

the

writer working by lamplight, at the right,

© 2007 Springer Science+Business Media, Inc .. Volume 29. Number 3, 2007

7

Figure 2.

The passage in question.

state of the arts, literature, the sciences, religion . . . as if for a scholar's guide. On page 265 of the second edition (Antwerpen, 1641), he lists the eighteen mathematicians he considers to be the best in the world. In this list we find ARCHIMEDES, COPERNICUS, KEPLER, . . . and "Georgius BURBACHIUS"! We note that BURBACHIUS is the natural latinized version of BOURBAKI. Let us translate the quotation de­ picted in Figure 2: " . . . in my opin-

On Two Fellows Who Wanted to Mal<e Money on Fluctuations JACEK MI�KISZ

8

THE MATHEMATICAL INTELLIGENCER

ion, the most remarquable doctors in mathematics, amongst the most recent, are Johannes REGIMONTANUS and his professor Georgius BURBACHIUS; the Scot Alexander ANDERSONIUS, the Prussian Nicolaus COPERNICUS, . . . " The puzzle is easily solved: Johannes Regimontanus (1 436-1476)-his real name was Johan Muller-is quite well known; he was the student of Georg Von Purbach ( 1423-1461), Viennese mathematician and astronomer. Thus it is simply an author's error: "BUR­ BACHIUS" instead of "PURBACHUS". One of my colleagues (D. Roux) no­ ticed that the coincidence is furthered by the first name of Nicolaus Coperni­ cus, which, in the printed text, lies di­ rectly under the surname Burbachius. But is it really a coincidence? A founding member of the Bourbaki group, and lover of old books, could have read this one and remembered this quotation (perhaps subconsciously).

I

write this to warn readers about two swindlers who hang around this neighborhood. One of them intro­ duces himself as a Mathematician and the other as a Physicist. They want to involve people in enterprises that look extremely promising. I will not soon forget the day when I was approached by the Mathemati­ cian. Listen-he said-I'll throw a coin. If heads comes up you will pay me a dollar. If tails comes up I'll pay you a dollar. Fair enough, I said. The proba­ bility of winning is 1/2 for each of us. After a while both of us agreed that the game was boring. Here is another one of my games­ said the Mathematician-! wonder what you will think of it. In addition to the true coin, I have two weighted coins; the probability of heads for the second coin is 3/4, and for the third coin it is 1/10. Which of these two coins I throw depends on my "financial status" (which may be given by a negative number). If the figure describing my funds is di­ visible by 3, then I throw the third coin, otherwise the second coin. As before, heads means a dollar for me and tails a dollar for you. Well? said the Mathe­ matician, and looked at me expectantly. At first glance the proposal didn't look too good for me. I reasoned that,

Xlim, UMR 61 72 CNRS Universite de Limoges 87060 Limoges Cedex France e-mail: [email protected]

Comment Back in 1994, I reported the existence of the name Bourbaki in a book printed in Berlin in 19 18, and I bet 10,000 Pold­ evian crowns that this was the earliest occurrence of the name in any book of mathematics (see The Mathematical In­ telligencer 16 ( 1 994), no. 1 , 3-4). My challenge has been superbly met by Fran<;:ois Laubie's marvellous historical example. I certainly must pay. Just al­ low me the time it takes to get into con­ tact with the Poldevian authorities . . . . Osmo Pekonen Agora Centre Fl-400 1 4 University of Jyvaskyla Finland e-mail: [email protected]

on the average, for one in three throws the figure describing my opponent's funds would be divisible by 3, and then the thrown coin would be the third coin, the one that favored me; but in 2/3 of the throws the coin used would favor the Mathematician. Hence in a sin­ gle throw the probability of a win for me was 1/3. 9/10

+

2/3. 1/4

=

7/15,

that is, less than 1/2. The Mathemati­ cian insisted that the game was the essence of honesty. I excused myself for a moment and did some figuring on the side. Let 0, 1 , 2, the remainders o f the division of the Mathematician's funds by 3, represent the state of our system. The probability of ending up in a particular state de­ pends on the state of the system at the previous moment and not on the whole history of our game. Therefore the evo­ lution of our system exemplifies a Markov chain. One can show that, with the passage of time, the frequencies of the occurrence of particular states of the system tend to certain limiting values. My computations showed that the lim­ iting frequency of throwing the third coin is 5/13, and thus greater than 1/3 (see the course on Markov chains in

Note 1 ) . The probability of a win for me in a single throw was asymptotically 5/13. 9/10 + 8/13 . 1/4

=

1/2.

The Mathematician was right; the sec­ ond game was indeed honest. The Mathematician continued. Now that you know that the second game is hon­ est you'll he glad to play either one of the two games if all three coins are weighted so that the probabilities p;, i = 1 , 2 , 3, are less than before , say, Pl

=

1/2 - E,

P3

=

1/10 - E,

f>2 = 3/4- E,

where E is a fixed, small positive num­ ber. For then, as is easy to verify (please, kind Reader, do this!), the probability of your winning, in either game, is greater than 1/2. Of course­ r said-I'll be glad to play. Then came the suggestion I won't for­ get for a long time. The Mathematician suggested that, for the sake of variety, we should switch from one game to the other randomly. To avoid boredom­ he said-we'll play the first game with probability 1/2 and the second game with the same probability. After a few hundred runs, I realized with horror that my balance was deep in the red. The Mathematician left in a burry, and I em­ barked on a post mortem examination. The second game, like the first one, favors me. But if the figure of the Math­ ematician's funds is divisible by 3, which happens more or less in half the number of cases, he would not throw the third coin-which favored me-but the first coin. But then I could win or lose a dollar with equal probabilities. This may have been the source of my problem. I began to compute. I ana­ lyzed the Markov chain corresponding to the random combination of the two games with E 0. I concluded that the limiting frequencies of being in states 0, 1, or 2 were 245/709, 180/709, and 284/709, respectively. In the end, the probability of my winning in a move is

This means that if we play long enough, my funds will decrease in pro­ portion to the elapsed time with pro­ portionality coefficient - 18/709. Now the frequencies of visits of states of our Markov chain, and hence the expected values, depend continuously onE it fol­ lows that, for an appropriately small E, though each of the two games favors me, their random combination spells fi­ nancial disaster for me. Thus we are dealing here with an example of two random dynamics for each of which the expected value of a certain random variable goes up in time, whereas for the random combination of the two dy­ namics the expected value of this ran­ dom variable goes down in time. I urge you to check these results (ei­ ther analytically, on a piece of paper, or by simulating coin throws on a com­ puter). This may build up your resistance to other tricks of probabilistic swindlers. () ()()

I barely managed to come to after the encounter with the Mathematician when the Physicist knocked on my door. He wasted no time on preliminaries and showed me a sketch of his new device (see Figure 1 ) . It consisted of an axle with a fan at one end, a ratchet wheel with a pawl at the other end, and a spool with a thread in the middle. There was a little weight at the end of the thread. All components of the device were tiny. Due to random fluctuations-said the Physicist-there are moments when more particles hit the fan blades on one

side rather than on the other side. The situation is similar to Brownian motions of a particle in a suspension accidentally hit by particles of the surrounding fluid. But for the pawl, the fan blades would move now clockwise and now coun­ terclockwise. My device-the Physicist summed up--replaces variable fluctua­ tions with a single selected direction of rotation. The fluctuations supply work, and so we have a free source of energy. The device can be yours, hut, of course, not free. This time I was cautious. The en­ counter with the Mathematician taught me that the composition of two random dynamics, for which the state of the sys­ tem remains on the average unchanged, can result in unidirectional motion. But the pawl, like the little fan, is subject to impacts of surrounding air particles, per­ forms analogous Brownian motions and, as a result, every now and again goes accidentally up and lets the weight drop. In effect, the average shift of the weight is zero: the asymmetry of the pawl won't work miracles. Had the temperature around the pawl been lower than the temperature around the little fan, then the number of fluctuations of the pawl would have been smaller than that of the fan, and the Physicist's device would have really done work-at the expense of the energy drawn from the warmer environment and transmitted in part to the colder environment. 2 This is an ex­ ample of the Brown engine. Alas, there is one more difficulty: we would have had to maintain steady air temperatures

=

245/709( 1/2 . 1/2 + 1/2 . 9/10) + 180/709(1/2 . 1/2 + 1/2 . 1/4) + 284/709(1/2 . 1/2 + 1/2 . 1/4) = 691/1418, which is less than 1/2. Then the expected value of my profit for one move is 691/1418 - 727/1418

=

- 18/709.

Figure I

© 2007 Springer Science+Bus1ness Media, Inc., Volume 29, Number 3, 2007

9

around the fan and the pawl, and this, of course, would have involved an ad­ ditional cost. Yet another example of constructing perpetuum mobile (of the second kind) went down the drain. In the evening I talked with a Biol­ ogist friend. I told him about my math­ ematical and physical "adventures" ear­ lier in the day. The Biologist made an interesting comment. Maybe-he said­ Nature had found a way of exploiting microscopic fluctuations of particles in cells, and of using the energy liberated in biochemical reactions to transport useful cell components. Maybe the mol­ ecular cell motors function like Brown engines. I was glad the Biologist made no attempt to sell me molecular cell mo­ tors. We focussed instead on consum­ ing tasty food items, that is, transform­ ing the offerings of the head chef into simple organic compounds-an activity I recommend to all of my readers. 3

With my very best wishes, ]acek Mi�kisz, a physical-biological mathemati­ cian. NOTES

1 . A minicourse on Markov chains In our game we are dealing with a system that can be in states 0, 1, or 2. These states

�Springer

the language of science

correspond to the remainders in dividing the Mathematician's funds by 3. In the second game, the transition probabilities between these states, pij; i, j 0, 1, 2, are, respectively, 0, Po1 0.1, P o2 0.9, Poo P11 P22 . 75, P21 0.25. P12 0. 75, P10 0.25, P2o =

=

=

=

=

=

=

=

=

=

Of course, this system has infinitely many so­ lutions, but only one of them, namely, 7TQ

=

5/13,

=

6/13,

11"0 + 71"J +

11"2

=

1'

and this gives the asymptotic frequencies of vis­ iting the states of our system. In particular, the

memory loss, which can be briefly character­

figure of the Mathematician's funds is divisible

ized as follows: if we know the present, then the future does not depend on the past. Let 1r1j

by 3, on the average, in 5/13 of all throws. 2. The idea of a device utilizing Brownian mo­ tions to do useful work was first discussed in

denote the probability of the system being in state i at time t. Then the formula for total prob­ ability implies that

The above evolution of our system is an ex­ ample of a Markov chain. Note that we can go from one state to another in a finite number of steps. For such chains, with the passage of

1912 by Marian Smoluchowski, and was sub­ sequently developed by Richard Feynman (see Feynman, Lectures on Physics, vol. 1 , part 2, ch. 46). In 1996 Juan Parrondo wrote an (unpub­ lished) article titled "How to cheat a bad math­ ematician," in which he proposed certain para­ doxical gambling games. 3. The Polish original of this note appeared in

time, the frequencies of visiting particular states

Delta, a publication of the University of War­

of the system tend to values that are indepen­ dent of the initial state. In our case, the limiting

saw, and is used by permission. Translation by A. Shenitzer.

probabilities satisfy the following system of lin­ ear equations:

Institute of Mathematics, Polish Academy

1ro

=

1r2

=

11"1

=

0.2571"1 + 0. 7571"2, . 1 7ro + 0.25 71"2 , 0.9 7ro + 0. 7571"1.

of Sciences 00956 Warsaw 10 Poland e-mail: [email protected]

springerlink.com

.,. Journals, eBooks and eReference Works integrated on a single user interface .,. New powerful search engine .,. Extensive Online Archives Collection

THE MATHEMATICAL INTELUGENCER

7T2

2/13,

The probability of being in state i at time t + 1 depends only on the state we were in at time t. This is the so-called Markov property of

The world's most comprehensive online collection of scientific, technological and medical journals, books and reference orks

10

=

satisfies the condition

ringerlink

.,. 0

11"1

in 13 subject Collections

Viewpoint

Poetic Metaphor and Mathematical Demonstration: A Shallow Analogy M IRIAM LIPSCHUTZ-YEVICK

The Viewpoint 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. Viewpoint

J

an Zwicky in "Mathematical Anal­ ogy and Metaphoric Insight" [Zw] says that understanding a poetic metaphor feels like understanding cer­ tain mathematical demonstrations. She investigates the correspondences he­ tween the notion of metaphor primar­ ily as it is used in poetry and that of analogy in the development of mathe­ matical demonstrations. Although she clearly states that metaphors and math­ ematical analogies are not the same thing, she maintains that there are such fundamental similarities that they should both be considered as species of "analogical reasoning. " She posits that the sense of understanding, the "flash of insight" (the "I get itl" , the "Eu­ reka moment" ) on grasping a metaphor or a demonstration, is closely related in the two domains. Analogy, the drawing on associa­ tions, is all-pervasive in our thinking, in our language, and in our creative en­ deavors, be they artistic, scholarly, or everyday: concocting a new Italian recipe; racial stereotyping; formulating a legal opinion; making a medical di­ agnosis; the "coup de foudre" estab­ lishing a romantic link; and so on. As­ soCiation appears as an essential concept in the Fourier logic represen­ tation of brain function proposed by Karl Prihram [Prl, [LYll. Similes are imbedded in our language, carrying much of our meaning ( as proverbs, be­ fore they were so displaced by techni-

cal jargon, used to do). Zwicky's article abounds in them: "the field of reso­ nance", "lift off the page", "has no pur­ chase on", "cede pride of place" . . . . Hannah Arendt [A] wrote that "all con­ ceptual or metaphysical language is ac­ tually and strictly metaphorical. " Zwicky argues that metaphor and mathematical demonstration have spe­ cial kinship, in that in both, the new in­ sights derive from discovery of unsus­ pected analogies between facts long known but wrongly believed to be strangers to each other. But this kinship extends to all creative thinking! I maintain that analogical reasoning, being a generally present feature of thought, can not prove mathematical reasoning any closer to poetry than say­ ing that both are thought. Going beyond this commonality, we follow the divergent aspects in the further use of analogy in the two domains-"Points of Non-Correspon­ dence", as Zwicky calls them-and find two different "Languages of the Brain". To me, they look complemen­ tary (as the word is used in physics). Mathematical thinking analyzes; it is modelled, perhaps, by digital logic of networks [vN] . Poetical thinking embellishes; it more resembles holo­ graphic pattern recognition. Let us look at the dichotomy. Though the two use analogy differently, their symbiosis may suggest a more insightful mode of thought.

should be submitted to the editor-in­ chief, Chandler Davis.

Metaphorically Valid? While my husband and I were graduate students at M.I.T. during WWII, the young Walter Pitts. a brilliant protege of the great mathematician Nor­ bert Wiener, offered to deliver a lecture on "Sinkiewicz's Theorem" to an eager audience of graduate students. Pitts gave, as usual, a dazzling per­ formance. He proved the theorem moving seamlessly through a maze of lemmas and analogies, with frequent hand-waving to bypass the "obvious. " His ( almost poetic) presentation was received with applause and admiring comments. The lemmas were profound, the theorem still more so. Even though the lecture had the form of-and felt like-a proof, unbeknownst to us it was fiction. ( Sinkiewicz was in fact a Polish novelist.)

© 2007 Spnnger Science+ Business Media, I n c . , Volume 2 9 , Number 3, 2007

11

Proof vs. Gestalt: Two Modes of Creative Endeavor Poincare (P] defined Discovery as "ap­ pearances of sudden illuminations, ob­ vious indications of a long course of pre­ vious unconscious work. All that one can hope for from these inspirations which are the fruit of unconscious work, is the point of departure for such calculations. They must be done in the second pe­ riod of conscious work: results must be verified and consequences deduced." This is the dichotomy: on the one hand Zwicky's visual intuition, the "see­ ing as"; on the other, a rigorous proof, which requires analytic validation, a de­ rivation from axioms, or an algebraic computation. Creating a mathematical proof

Professor Norbert Wiener's lectures dur­ ing my graduate studies at M.I.T. ( 1943-1947) were revelations of the re­ searcher at work. The classroom had blackboards on three sides. Wiener, starting on the left wall, would write the theorem he intended to prove. He pro­ ceeded to accomplish this by assem­ bling a chain of valid deductions from various lemmas he had previously given in class. Talking to himself as much as to us-though with an occasional "just a minute, just a minute"-he would pro­ ceed from blackboard to blackboard, never losing his thread, though perhaps leaving us well behind. After much trial and error he might exclaim, "Let us do a Cesaro job on this" or the like. At the end (often close to the last space on the right wall blackboard, near the door) the proof would be complete. To the students he appeared to be pulling the proper tools from his toolkit,

as an experimental scientist would reach for instruments. If he thought of his tools in an analogic way, like the "Cesaro job", this was not the defining feature of his work. It is hardly necessary to insist on the magnitude of Wiener's discoveries [M]. No doubt he had to perceive un­ suspected analogies between facts long known but wrongly believed to be strangers to each other, as Zwicky says; but then came the checking; his lectures both displayed the checks and displayed his conviction of their importance. Creating a poem

Allow me to give some impressions of the experience of writing a poem. This poem was inspired by what I had learned in 1948 about the fate of my Jew­ ish classmates at a genteel girls' school in Antwerp. The school had a hateful dis­ ciplinarian Principal with a strict rule for the wearing of gloves. I had not thought of her for years. Only on recently en­ countering an acquaintance with similar background, and seeing her gloves and how she removed them to shake hands, as we were taught, did a chain of asso­ ciations start up which led to the poem [LY2]. Some lines from the end: "This one and that one," They picked out my former class­ mates. One by one they gathered them. Even the blonde, blue-eyed Berthe Perelman Whose name betrayed her. "A Jewess!" The Principal stood by her post at the head of the passageway, As the girls walked by to the trucks waiting outside.

MIRIAM LIPSCHUTZ-YEVICK was bom in Scheveningen, Holland,

and arrived in the U.S. in 1940 after a three-months-long fiight from the Nazis. She eamed her doctorate at M.I.T. in 1947 (one of the few

women in mathematics up to then). She was at Rutgers (University Col­ lege) from 1964 until retirement She has published

on

probability, on

her invention holographic logic, and on other areas, including a text

Mathematics

(or the Billions

for her remedial students. She is a deeply

devoted grandmother.

Miriam Lipschutz-Yevick

22 Pelham Street Princeton, NY 08540 USA

e-mail: [email protected]

12

THE MATHEMATICAL INTELLIGENCER

Did she check to see if the girls wore their gloves? The writing of the poem seemed to arise from a consciousness distinct in character from mathematics. After read­ ing Jan Zwicky's article I subjected my poem to critical scrutiny and noted nu­ merous metaphors. I might have started with a problem: my feelings of guilt for having escaped the fate of my friends; my buried wish for revenge on my Principal; the contrast between the en­ forcement of good manners and the brutality of the Nazis . . . but then-no poem. Yet the poem I did write holds all of this "compact in one." A poem, even a long one, may some­ times be grasped as an emotional whole; a lengthy proof can be recon­ structed only step by step, even if one has first grasped the general thrust. The thought processes in the writing of a poem do not take the conscious form "this reminds me of this reminds me of this . . . ", though they may do so in an attempt to implement an intu­ itive mathematical perception. Rather the search for "the right word" for use as poetic metaphor is often a tip-of-the­ tongue phenomenon, and in all cases feels almost the reverse of mathemati­ cal puzzling.

Conclusion Associations, "atoms" of analogies, guide our discoveries, be they poetic, mathe­ matical, judicial, culinary, amorous, or whatever. Sometimes the "right" analogy is discovered by a chain of steps. On other occasions it pops up sponta­ neously as though by "ghosting" the ob­ ject with another stored jointly in an as­ sociative hologram, or by the emergence

of a cluster of unconnected associations which together recreate the object. Comparing the domains on which analogy acts, I find another contrast: po­ etic analogy casts a wide net (the search for the telling metaphor runs through a wide web of relevant associations); mathematical analogy deals with con­ cepts appropriate to a particular theory, and the validation is deductive and se­ quential in character. When mathemat­ ical research does widen its scope, it is by generalization, which "by condens­ ing compresses into one concept of wide scope several ideas which seemed widely scattered before" (P6lya [Pol, p. 30) . Alas, this sometimes relies on for­ malism which obscures the ideas re­ lated, thereby impoverishing the mean­ ing and insight. "Mathematics does notfit all. " Cloth­ ing humanistic and social sciences in

mathematical garb is a technique fre­ quently used by scholars in the non­ quantitative fields ( The Phillips Curve, The Bell Curve, and so on) to over-awe a quantitatively uneducated population. Perhaps we agree that the technique can be pernicious. I have tried to argue that, likewise, poetry has nothing to gain from overstating its resemblance to mathematics-and vice versa. Rather our aim should be to teach the general public to appreciate the insights of the two domains. And beyond that, to un­ derstand and act upon the problems of our world with rational thought and em­ pathic feeling.

Jan Zwicl
tantly, its relationship to truth--only with the former. I suggest that if we adopt Hardy's notion of proof as a rhetorical flourish designed to get other people to see what we ourselves see, then there is a surprising parallel between mathemat­ ical proofs and the analogies they con­ firm, on the one hand, and poems and the metaphors to which they give ex­ pression, on the other. Poems, too, of­ ten serve as rhetorical flourishes which position an auditor or reader to grasp the insight the poem's composer wishes us to see. This argument-whether or not it is sound-is conceptually impos­ sible to mount if one fails to distinguish between in sight and demonstration. It is also impossible to mount if one moves, as Lipschutz-Yevick does, seamlessly between "metaphorical" and "poetical", the paired distinction on the literary side of the equivalence. "Poet­ ical thinking'' (p. 1 1 )-whether we mean by this effusive Victorian rhetoric or, simply, poetry-is not co-extensive with metaphor. Poetry (the reading of "poetical thinking" that interests me) can contain metaphors, and often does, but needn't (witness the excerpt from her own poem that Lipschutz-Yevick quotes). Metaphors can, and often do, live inside poems; but they also inhabit well-written prose, as well as garden-variety oral con­ versation. The distinction is as crucial to

I

n her helpful reply to my essay, Miriam Lipschutz-Yevick argues that it is not surprising that we should ex­ perience some kinship between mathe­ matical analogies and metaphors since "both are thought" (p. 1 1 ); but there the resemblance ends. "Mathematical think­ ing analyzes," she asserts, but "poetical thinking embellishes" (p. 1 1 ). A proof in mathematics is the result of "a deriva­ tion from axioms, or an algebraic com­ putation" (p. 12) while, for Lipschutz­ Yevick, "the search for 'the right word' " when writing a poem "feels almost the reverse of mathematical puzzling" (p. 12). Except for her claim that 'poetical thinking' embellishes, I couldn't agree more. Lipschutz-Yevick claims that I ar­ gue that "metaphor and mathematical demonstration have special kinship" (p. 1 1), but in fact, I do not. I am at pains to distinguish between mathematical in­ sight (or 'invention') and mathematical demonstration or proof, and wish to ar­ gue that metaphorical thought shares structural features-and, most impor-

REFERENCES

Fourier Logic," Pattern Recognition 7 (1 975), 1 72-2 1 3 .

[LY2] Miriam Lipschutz-Yevick, "Gloves," The Kelsey Review, 2003.

[M] Michael Marcus, review of "Dark Hero of the Information Age," Notices of the Amer. Math. Soc. 53 (2006), 574-579.

[P] Henri Poincare, The Foundations of Sci­ ence, The Science Press, New York, 1 929.

Chap. Ill, "Science and Hypothesis." [Po] George P61ya, Mathematics and Plausible Reasoning, Vol. 1: Introduction and Analogy in Mathematics, Princeton University Press,

Princeton, 1 954. [Pr] Karl Pribram, Languages of the Brain, Pren­ tice-Hall, Englewood Cliffs NJ, 1 971 . [vN] John von Neumann, The Computer and

[A] Hannah Arendt, quoted in Dwight Bolinger, Language the Loaded Weapon, Longman,

the Brain, Yale, 1 958.

[Zw] Jan Zwicky, "Mathematical Analogy and Metaphorical Insight," Mathematical lntel/i­

London, New York, 1 980; p. 1 43 . [LY1 ] Miriam Lipschutz-Yevick, "Holographic or

gencer 28 (2006), no. 2, 4-9.

my argument as the distinction between insight and demonstration. Why, then, does Lipschutz-Yevick re­ spond as though I had conflated both? I believe two things may be contribut­ ing to her misreading. The first is that it seems utterly obvious to her that both metaphorical insight and mathematical insight are, as she says, "thought" (p. 1 1) . This is actually all-or nearly all­ ! wish to establish! (I do also wish to draw out a few consequences of so con­ ceiving metaphor.) I am delighted that my central claim is something she thinks we should take for granted, but I sup­ pose I have spent too long among the skeptics to rest easy. To many of my colleagues in the humanities and to sev­ eral in the sciences ( vide P6lya, quoted in my essay), it is anything but obvious that metaphorical contemplation consti­ tutes a way of thinking. (Even some po­ ets-as Shakespeare's and Wordsworth's characterizations of metaphors under­ line--can, on occasion, claim not to take metaphorical discernment seriously as a form of thought. ) It is, many suggest, mere 'play', a gesture without signifi­ cant meaning, or, worst of all, a rhetor­ ical embellishment of ideas that might be rendered more clearly, if less attrac­ tively, in plain prose. (I will return to Lipschutz-Yevick's characterization of 'poetical thinking' as embellishment in

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

13

a moment.) It is because of this wide­ spread denial that metaphorical thought is genuine thinking aimed at truth that I felt it useful to develop the corre­ spondences with mathematical insight. Which brings us to the second rea­ son Lipschutz-Yevick may have con­ flared the distinctions at the heart of my argument: she is angry at specious at­ tempts by the ill-informed to garner prestige for their endeavours by link­ ing them to the hard sciences. I am grateful to her for voicing her concerns and believe I should have been more alert to the possibility of this reading, especially among an audience of math­ ematicians. Let me take a moment to clarify my intentions. I readily admit to being ill-informed about mathematics! But I must stress that I am not interested in mathematics' prestige, in and of itself. I am interested in using my colleagues' impression of that prestige to get them to reflect seri-

�Springer

the language of science

ously about why they underrate and ghettoize metaphorical insight. But if any among them is willing to bite the bullet and dismiss mathematical insight as thinking, then I am content to let them be consistent and dismiss meta­ phorical insight, too. The nihilisms en­ gendered by post-structuralism are per­ nicious, and, when skepticism about meaning goes very deep, there is little one can accomplish by way of argu­ ment. The argument, then, was directed at scholars who won't take metaphor seriously as thought, but who do think mathematical epiphanies count. As I say, I am grateful that Lipschutz-Yevick is not among their number. In closing, I would like to touch briefly on Lipschutz-Yevick's claim that "poetical thinking embellishes" (p. 1 1) . As I have indicated, i t i s precisely this attitude-assuming, this time, that she means metaphorical thought-that my essay hopes to dislodge, by arguing that

metaphor is as much a mode of epis­ temic insight as mathematical analogy. Since Lipschutz-Yevick grants this, why does she also appear to suggest that metaphor merely decorates? I do not know. She continues, in the same sen­ tence, with the suggestion that "poeti­ cal thinking . . . more resembles holo­ graphic pattern recognition [than it does the digital logic of networks]" (p. 1 1) . Again, assuming we are talking about metaphor, I couldn't agree more. But such pattern recognition, as I under­ stand it, has nothing to do with em­ bellishment, and registers instead a de­ gree of structural or shape-dependent correspondence.-Which, as I believe Lipschutz-Yevick would also agree, is what Kepler, and P6lya, and Guldin, and Bernoulli, and Leibniz are saying is involved in the flash of insight that is distinct from the hard work of demon­ stration, but which nonetheless sustains us in our search.

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14

THE MATHEMATICAL INTELUGENCER

Juggling Braids and Links SATYAN l. DEVADOSS AND JOHN MUGNO

he art of juggling has been around for thousands of years. Over the past quarter of a century, the inter­ play between juggling and mathematics has been well studied. There has even been a hook [6] devoted to this re­ lationship, dealing with several combinatorial ideas. Nu­ merous juggling software is also available; in particular, Lip­ son and Wright's elegant and wonderful juggleKrazy [4] program helped motivate much of this paper. Most of the information useful to juggling can be accessed via the jug­ gling Infonnation Service webpage [3] . The goal of this pa­ per is to construct and study a map from juggling sequences to topological braids. An early form of this idea providing motivation can he found in the work of Tawney [7], where he looks at some classic juggling patterns.

T

J4 . The pattern in which the balls are thrown is periodic, with no start and no end to this pattern. )';. Throws are made with one hand on odd-numbered beats and the other hand on even-numbered beats. A throw of a ball which takes k beats from being thrown to being caught is called a k-throw. Condition )5 implies that when k is even (or odd), a k-throw is caught with the same (or opposite) hand from which it was thrown. Thus, a 5-throw starting in the left hand would end in the right hand 5 beats later, while a 4-throw starting in the left hand would end hack in the left hand after 4 heats. In this no­ tation, a 0-throw is a placeholder where no ball gets caught or thrown on that heat. The following definition from [2] is used to embody some of the rules above.

Juggling Sequences

DEFINITION 1. Let n EN. A juggling pattern is a bijection

Imagine a dark room in which glowing balls are being jug­ gled. Our idea is to envision the juggler walking forward while juggling, with the balls tracing out glowing lines. By joining the ends of these glow-lines in an appropriate man­ ner, a solid torus braid is obtained. We show that every solid torus braid (and thus every link) can be obtained this way from an appropriate juggling pattern. This is done by first constructing the trivial solid torus braid, and then mod­ ifying this to obtain all generators. The technical terms here will he explained as we go along. In our discussion, we remove everything that is not math­ ematically relevant. Thus, assume the juggler in question is throwing identical objects, referred to as halls. By conven­ tion, there are some basic rules we adhere to in juggling. Jl . The balls are thrown to a constant beat, occurring at certain equally-spaced discrete moments in time. )2. At a given heat, at most one hall gets caught and then thrown instantly. )3. The hands do not move while juggling.

f: 7l.--'> 7l.

:

t--'> t + df(t)

where df( t + n) = df(t) 2:: 0. A juggling sequence (or a site swap) is the sequence (dJrO), df( l), , df( n - 1)) aris­ ing from a juggling pattern. The number df( t) is the throw value at time t, and the number of terms in a juggling pattern is its period. Thus, a juggling sequence is simply used to keep track of succes­ sive throw values. Under this terminology, the sequence ( k) yields the k-ball cascade for k odd, and the k-ball foun­ tain for k even; the k-ball shower is given by the (2k- 1 , 1) pattern. A natural question to ask i s which sequences of numbers provide valid juggling sequences. Buhler, Eisen­ bud, Graham, and Wright provide an elegant criterion. ·

·

·

THEOREM 2 [2] A sequence (h(O), h(l) , . . . , h( n- 1)) of nonnegative integers is a valid juggling sequence if and only n -1}. In i(lh( i) + i mod n} is a pennutation q({O, 1, ·

·

·,

© 2007 Springer Science +Busrness Media, Inc .. Volume 29, Number 3, 2007

15

, ""

\ '

Figure I. Ladder

Figure 2. Braid

diagram of the (5, 1) sequence.

diagram of the (5, 1) sequence.

this case, the average of lh(i)) is some integer b, and the se­ quence describes a valid h-ball juggling sequence.

Ladder Diagrams A j uggling sequence can be represented graphically in sev­ eral ways. With the goal of constructing braids in mind, we display the sequences in a ladder diagram, with the dis­ tance between the hands shown in the vertical direction, drawn with respect to time. Figure 1 shows the ladder di­ agram of the (5, 1) sequence, where one hand is the bot­ tom line and the other is the top; we distinguish the two hands by using solid and open circles. The viewer is look­ ing at the juggler from above, and the straight lines corre­ spond to the juggled balls' paths traced out over time. The juggling pattern associated to the (5, 1 ) sequence is j(t)

=

{ tt ++

5

1

if t = 0

if t = 1

mod 2 mod 2 .

I t i s straightforward t o realize that the number o f balls used in a juggling pattern f is the number of orbits determined by f From basic physics, the height of a k-throw is propor­ tional to k 2 . Thus, for a crossing appearing in a ladder di­ agram, the line representing ball x will cross over the line

SATYAN DEVADOSS is an assistant professor

JOHN MUGNO started worl
at Williams. His research interests lie in the inter­

wrt:h Satyan Devadoss while an undergraduate at

play between topology, geometry, and combina­

Williams College. He is cunrently a doctoral stu­

torics-or more generally, in anything visual.

dent at the Universrt:y of Maryland. In his spare

When he is not at worl< his greatest pleasure is

time he enjoys soccer and frisbee, but desprt:e his

his wife and three children. Department of Mathematics Williams College

Williamstown, MA 0 1 26 7 USA

e-mail: [email protected]

16

representing ball y when x has a higher throw value than Figure 2 shows the braid diagram of the (5, 1) sequence, which encodes crossing information into the ladder dia­ gram. Remark. The physics of juggling is quite interesting. We re­ fer the reader to the pioneering work of Magnusson and Tiemann [5) for details, where dwell times, error margins and angle variations are discussed. Before proceeding, two problematic situations need to be addressed, both involving collisions of balls under our simplistic model. First, interactions between odd throws of the same value need to be discussed. As mentioned above, basic physics guarantees that odd throws of differing val­ ues will not collide in our model. However, physics also implies that two odd throws of the same value which cross in the ladder diagram will collide in reality. This is because both will have the same throw height, proportional to the square of their throw value (and also from symmetry) . Thus, under a simplistic juggling model, the classic juggling se­ quence (3) (the 3-ball cascade) cannot be juggled without the balls colliding in midair. The second problem involves interactions between two even throws. An even throw is tossed vertically in the air, to be returned to the same hand in an even number of beats. Thus any other even throw from the same hand will y.

THE MATHEMATICAL INTELLIGENCER

persistent efforts he has not developed practical j uggling expertise to go wrt:h the theoretical. Department of Mathematics University of Maryland

College Park, MD 20742-40 1 5 USA

e-mail: [email protected]

cause a collision if it is not thrown far enough apart in time. For example, under our juggling model, even a classic jug­ gling sequence such as (4) (the 4-ball fountain) cannot be juggled without the balls colliding in midair. Both of these problems can be resolved by adding a bit of realism to our juggling model. One lesson learned from actual juggling is that the hands move slightly during catches and throws. Indeed, a catch is made by moving the corre­ sponding hand slightly out of standard position, away from the other hand. After being caught, the ball is then carried temporarily as it is moved slightly in (closer towards the other hand) before release. 1 Thus, the following changes are made to the juggling rules, resulting in a modified model. ]2' . At a given beat, at most one ball gets caught (slightly before the beat) and then thrown (slightly after the beat). ]3 ' . The hands move slightly such that a ball is caught on the outside and thrown from the inside. It is not too hard to show that this minor modification is enough to resolve the collision issues present. In particular, when two throws with the same throw value cross in the ladder diagram, the earlier throw will cross over the later.

Braids and Links Braids have a rich history in mathematics , appearing in nu­ merous areas. We refer the reader to Adams [ 1 , Chapter 51 for an elementary introduction to this subject. Roughly, an n-braid is a set of n disjoint arcs running between two hor­ izontal bars, where every vertical plane between the two bars must intersect each arc exactly once. The endpoints of the arcs are given an ordered labeling. Starting with a trivial n-braid ( no crossings) , let u ; de­ note the braid where the strand at the i-th position crosses over the strand at the ( i + 1 )-st position. Similarly, let u j 1 represent the braid with the crossing of the strand at the i-th position under the ( i + 1 ) -st position. Since any braid can be obtained by repeatedly crossing adjacent strands, every braid can be expressed as a word, an ordered col­ lection of u elements. The left side of Figure 3 shows an example of a 4-braid; the right side shows how this braid can be expressed as the word u2 u ] 1 u-;, 1 u 1 . There is a composition map of two n-braids w1 U?., ob­ tained simply by attaching the n ordered endpoints of the arcs of w1 to the n starting points of the arcs of w2. Elements of the braid group 713 n are equivalence classes of n-braids un­ der certain relations. The following theorem by Artin 0926) gives the relations needed to identify isomorphic braids: •

•

•

•

ARTIN's THEOREM The braid group 713 11 is generated by {u1 , · ·, U11-1 l with the relations:

Figure 3.

An example of a 4-braid.

each meridional disk intersects each strand of the braid ex­ actly once. Treating the solid torus as a subset of [R3 makes the closed braid into a link in IR3 . That is to say, it is an embedding of a finite collection of circles in [R3 ; two such links are called equivalent if we can deform the first to the second without passing it through itself. Markov 0935) in­ troduced two new equivalence relations on words which algebraically represent these correspondences. Let w he a word in 713 11• R3. ( Conjugation) R4. (Stabilization)

w = u; · w · u ;- 1 . w = w · U n where U 11

E 713 n + l ·

MARKOV'S THEOREM The braid group up to the conjuga­ tion relation describes the solid torus braids. Tbe braid group up to both conjugation and stabilization relations describes links. Thus, the set of all solid torus braids (or of all links) can be generated by u elements and their inverses. The smallest part of the ladder diagram which can be used to tile it is called the fundamental chamber. More pre­ cisely, a juggling sequence has exact period n if it does not have another period m for any divisor m of n. Thus, a fun­ damental chamber for a juggling sequence with exact pe­ riod n has length 2n when n is odd and n when it is even. This asymmetry between odd and even length patterns arises from throwing from right and left hands. After an odd number of throws, the next throw will be from the oppo­ site hand; thus, doubling the pattern will be the first re­ peatable set of throws that ends on the correct hand.

DEFINITION 3 Let :J he the map from juggling sequences to solid torus braids, defined by taking the closure of a fun­ damental chamber of the braid diagram.

•

Rl. R2.

U; . (J'i+ l . U ;

= (J'i+ l . U; . (J'i+ 1 ·

U; . (J'i = (J'j . U ;

if I i - .tl > 1 .

A natural operation to perform on a braid is closure, identifying the endpoints of the strands of the braid with the corresponding starting points. The result after closing a braid is a solid torus braid, a braid in a solid torus where

Note that there is no canonical map from juggling sequences to braids. The translation of the fundamental chamber pro­ duces alterations of the starting and ending points of the braid. However, taking the closure of the chamber, thereby pass­ ing to solid torus braids, removes all ambiguity. Figure 4 shows examples of :J evaluated on sequences (7, 1 , 1) and (5, 5, 5, 1 ), respectively. A fundamental cham-

1This is sometimes done in the opposite fashion, with a hand catching on the inside and throwing on the outside, called reverse juggling.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

17

Figure 4.

The juggling sequences (7 ,

1,

1) and (5, 5 , 5, 1).

ber in each braid diagram is shaded. By placing the solid torus braids in [R3 (that is, by allowing stabilization) , (7, 1 , 1 ) maps to the unknot and (5, 5 , 5 , 1 ) maps to the trefoiP The natural question is to ask which links arise from jug­ gling patterns. The following is the key result, the proof of which will consume most of the remainder of the paper. is

THEOREM 4 The map :f

sutjective.

This, along with Markov's theorem, results in the following:

COROLLARY 5 For every link, there exists a juggling se­ quence which maps to the link.

Constructing a Trivial Braid

Our method of attacking Theorem 4 is by constructing a jug­ gling pattern for every solid torus braid. We begin with the trivial solid torus n-braid, which will serve as a building block. Before providing the juggling pattern, we first describe it from the perspective of the halls (or strands of the braid). The first ball of the n balls is thrown at a constant heat of 3 units, being thrown at t = 0. The second ball is thrown at a constant beat of 33 units, being thrown at t = 1 . Let a1 + 32(k-Zl for k � 2. In general, 0 and let a k = 3° + Y + the k-th ball is thrown at a constant beat of 32k- I, being thrown at time t a�e- We denote this pattern as In, where n corresponds to the number of balls thrown. Figure 5 shows the braid diagrams for h , lz and !3 re­ spectively. We refer to the strand traced out by the k-th ball as the k-strand. Notice how a given strand always crosses over or always crosses under any other strand. The juggling pattern can then be defined as =

·

·

·

=

(1)

I.,(t)

Z'fhis could

18

=

{t + 0

Yk- 1 if t = (a k + m otherwise t+

be the right or the left trefoil,

THE MATHEMATICAL INTELLIGENCER

·

32k-1) mod 32 "- 1

depending on which hand releases

where k E { 1 , 2 , . . . , nl and m E 7L. Here, k keeps track of the balls and a k + m 32 k- l the beats in which the k-th ball is thrown. ·

THEOREM 6 Tbe function I.,(t)

is

a valid juggling pattern.

Proof. We need to show that Eq. (1) satisfies the criterion set forth in Theorem 2 . That is, we need to show l dln ( i) + i mod Y "- 1 1

(2)

=

l i mod

3 2n- I J ,

where i E 7L marks the position and dln(i) the throw value. This equality clearly holds at positions when no balls are thrown, for the throw value is 0 . Look at each ball sepa­ rately. The set of beats in which ball k is thrown (with throw length 3 2k- l ) is

(3) where m E 7L. This corresponds to the right-hand side of Eq. (2). The left-hand side of Eq. (2) for a given ball k is IYk- 1 +

(a k + m . 3 2 k-l) mod 3 2n-l) yk- l mod 32 " - 1 1 . = \a k + (m + 1) ·

Since m E 7L , this is a rephrasing of

0

(3).

THEOREM 7 The image ofI,, under:f n-braid.

is the trivial solid torus

Proof Since all throws of a given ball k are of the same height 32 k- J , the k-strand will cross over all }-strands when j < k, and cross under all /strands when j > k. This results in the trivial solid torus braid. D

Constructing the Generators

There is a well-known method of constructing new juggling patterns from old ones. Let (h(O) , h(l), h( n - 1)) be a ·

the " 1 " throw.

In other words,

the map J is defined only

·

· ,

up to reflections of

links.

0

wn at

•

I =

1 =

I, .

==

==

·

·

·,

•

paration of 34 uni

juggling sequence. Let a and h be integers such that 0 :S a < b :S n - 1 and b - a :S h(a) . We construct a new se­ quence which coincides with h(i) on all beats except at a and b. Namely, define g(i) h(i) for all i of. a, h, and let g(a) b( h) + ( h - a) and g(h) = h( a) - ( h - a). The fol­ lowing is immediate.

quence.

paration of I unit

10

The braid diagrams for I1 , h and

THEOREM 8 (g(O), g(l),

0

paration of 3 2 unit

I

I

•

throv n at

0

I

1=

thrown at

Figure 5.

0

0

g( n - l l) is a juggling se­

In other words, the sites b( a) and h(b) in which the balls land will swap positions. Indeed, the term site swap is used because of this property. In our context, when attempting to construct the gener­ ators of the braid group, the creation of a crossing is sim­ ply an extension of this idea of swapping sites. Starting with the identity braid In, the strands to be crossed are chosen and two of their throws are manipulated (swapped), cre­ ating a crossing in an otherwise trivial braid. Figure 6 shows a part of the braid diagram of 1,/ t), where only the k and k + 1 strands are depicted. The length of heats between Z two hands is scaled by a factor j k- 2

Over-crossing:

Let a);

==

h�

a k+ l and

_

a'k = (33

h� = _

a k + 32k+ l . Then

1) .

3z k- 2 < 3z( k+ l)- 1

=

dln(ak"J ,

satisfying the conditions of Theorem 8. We take a funda­ mental chamber of In( t) and swap the two sites d!,(a%) and d!,lb'k ) , resulting in

{

+ ( h); - a);) if t = a); mod 2 C/e,Jt) = J,(aJ:) - ( b'k - aJ:) if t = b'k mod 2 J,( t) otherwise. I,( b);)

Notice the length of its fundamental chamber is

·

2

·

32 "- 1 32 " - 1

·

32 "- 1 .

LEMMA 9 j(Ck,n(t))

is the solid torus n-braid with its k-strand crossing over its (k + 1)-strand.

Proof Theorems 7 and 8 guarantee that Ck, n( t) is a juggling pattern. We need to show the crossing information is as claimed. The throw value of ball k + 1 at a'k increases from 32( k+ ] )- l to y k- 1

+ Cb'k - aiD = (33 + 3 - 1 l .

3zk- z .

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3 , 2007

19

I I

Figure 6.

rati n

f 3 2k - 2 unit

..

•

Y.

O O O O

O O O

;�z

Braid diagram only of the k and k + 1 strands of I,z(t).

,.

:

(

__

Figure 7.

0 0

Braid diagram showing only the k and k + 1 strands of Ck,n(t) .

0 Figure 8.

Braid diagram only of the k and k + 1 strands of Ck,�(t).

Crossings will not interfere with other }strands for which j > k + 1 , because 32j- l

2:

32(k+ 2 )- 1

=

35 . 32k-2 > (33 + 3

.

- 1 ) 3 2 k- 2 .

Similarly, the throw value of ball k starting at position b'k decreases from 32k- l to

So crossings will not interfere with other }-strands for which j < k. Thus, the only crossings that can occur are between the k and k + 1 strands around the swap location. Figure 7 depicts the details: the k-strand at position b'k + y k- z has a throw value of 32k+ 1 , thus crossing over the (k + 1 )-strand thrown from b'k + Yk- l with a value of 32k- l . D

Under-crossing:

This is identical to the situation above, except for different choices of sites to swap. Let a� = a k+ 1 and b� = a k + 32k- l . Then b� - a� = 2 32k- z < 32C k+ l ) - l dln(a�), satisfying the conditions of Theorem 8 above. We take a fundamen­ tal chamber of In(t) and swap the two sites dlnCa�) and dln(b�), resulting in ·

20

THE MATHEMATICAL INTELLIGENCER

=

{

InCh�) + ( b� - a�) if t == a� mod 2 32 n- l � Ck, ( t) = In C a�) - ( b� - a�) i f t = h� mod 2 32"- 1 otherwise. In( t) � LEMMA 1 0 :J(Ck, (t)) is the solid torus n-hraid with its k-strand crossing under its (k+ 1)-strand. ·

·

Proof Again, Theorems 7 and 8 guarantee Ck, n(t) to be a juggling pattern. The throw value of ball k + 1 at a� de­ creases from 32Ck+ l) - l to 3zk- 1

+ ( h�

_

a�) =

y k- l + 2

.

3zk- z.

Thus crossings will not interfere with other j-strands where j < k. Similarly, the throw value of ball k starting at posi­ tion b� increases from 32k- l to 3zck+ D - l

_

( b�

_

a�)

=

3zk+ l

_

-.

2 . 3zk z

So crossings will not interfere with other }-strands, where j > k + 1 . Thus, the only crossings that can occur are be­ tween the k and k + 1 strands around the swap location; see Figure 8 for details. Notice the k-strand at position a� 3 2k- z with throw value of 32k- l crosses under the (k + 1)-

-

.·

.·

Figure 9.

:f 2(3) and :J3(3) mapping to the figure-eight knot and the Borromean rings.

strand at position a% having a throw value of 3 2k- J 2

.

3zk-z.

+

D

To finish the proof of Theorem 4, we construct a juggling pattern for every solid torus braid. Let w = aT, a;, · · a7, be a word describing a solid torus n-braid. Let ) I"(t n

=

{

111C

·

bp + < h71 - a:)

) � ) - ( b'11· - n' In ( n ""'�t '" "'1r. . . . 1,/t)

if t = a;, + Z( j - 1 ) · 32"- 1 if t = b'11· + 2( ,,. _ 1 ) . otherwise.

·

32"- 1 ,

·

mod (Zr · Y"-1)

1 (_2 r · L221 mod J - 1 _)

air then bT = b� and a7 a�; similarly, if a7 = aj 1, then b� and a7 a�'. Theorem 7 along with the lemmas above guarantee I�((t) to he a juggling pattern; we leave it to the reader to provide details. We claim that :fWD maps to w. In Iif(t) , r copies of the fundamental chambers of I,l t) are used, one for each element in w; the length of its funda­ mental chamber is 2 r · 3 2 n - l . Each copy is altered by an ap­ propriate swapping of sites corresponding to the generating element in w. This alteration provides the appropriate cross­ ing needed. This completes the proof of Theorem 4. If a7

b;

=

=

=

=

Note that choosing the starting position of the fundamen­ tal chamber of /�\ t) yields an ordering of the position of the strands, determining w. This ordering is not necessar­ ily the ordering of the n balls, which are used to label the strands, since a site swap switches the strand formed by the k-th ball with the strand formed by the ( k + 1 )-st ball. However, since our elements are solid torus braids, we have the ability to slide our strands in the solid torus to the ap­ propriate order (due to conjugation in Markov's theorem).

Looking Forward Although our construction allows us to prove all links can he juggled, it is far from realistic or efficient. The throw values were chosen in powers of 3 (a prime) in order to

make transparent the construction of a valid juggling se­ quence, along with isolating crossings of two strands. This requires a juggling sequence with n balls to have throw values up to 3 2 rz- l . As juggling sequences with values of 9 are near impossible to perform, the method above is cer­ tainly not realistic. Let us now look at how realism can be introduced and measured. We begin with the map j from juggling sequences to solid torus braids. This map is based on taking the closure of the fundamental chamber. Thus, the classic (3) cascade sequence, having fundamental chamber of length two, is allowed to be "active" for only two beats, resulting in the unknot. But what if our juggler wishes to juggle longer, for more beats? Be­ cause we are going to dose the resulting braid, juggling mul­ tiple copies of the fundamental chamber can be allowed. De­ fine 3'"\./') to he the closure of k adjacent copies of the fundamental chamber associated to the juggling pattern] Fig­ ure 9 shows j2(3) and j3(3); the first maps to the figure-eight knot, the latter to the Borromean rings. Given a link, we ask to find the best juggling pattern which maps to it. To try to measure what "best" means, we need to look at a few factors. Given a link /, let j ( l ) he the set of juggling patternsfsuch that j k( j' ) maps3 to l for some k E N .

DEFINITION 1 1 The ball index of a link l is the minimum number of halls needed for juggling pattern/ for allfin j( /).4 The throw index of a link I is the minimum over all the max­ imum throw values of a juggling pattern f for all .f in j( l ). It is straightforward to show that for non-trivial links, the ball index must be greater than one and the throw index must be greater than two. Consider the trefoil as an ex­ ample. Two possible ways to construct it are by j 3(4 0) ,

3Strictly, :Jk maps a juggling sequence to a solid torus braid; we abuse terminology and sometimes refer to the composition of this map with stabilization. 4Ciearly, the braid index of a link is less than or equal to the ball index.

© 2007 Springer Science+Business Media, Inc., Volume 29. Number 3. 2007

21

and :}(5, 5, 5, 1), as shown in Figure 4. Since (4, 0) is a ball pattern, the ball index of the trefoil must be two.

2-

PROBLEM Study the properties of the ball index and throw index of links. An underlying issue to this problem is understanding :J . We have shown that :J i s surjective, but we d o not know much more about the map itself. Based on the figures above, up to conjugation, :J maps (7, 1 , 1)

� �

(5, 5 , 5 , 1) (3, 4, 5) � (4) �

u2u1 u:Z 1 <TI 1
simultaneous throws and varying throws (reverse juggling) can be encoded into more complicated models; Polster [6, Chapter 4] provides a nice introduction to these ideas. We end with the following:

PROBLEM Study braids and links obtained in more compli­ catedjuggling models. ACKNOWLEDGMENTS

We thank Colin Adams and Allen Knutson for helpful con­ versations, and Jake Tawney for motivating the problem. The first author is grateful to the National Science Foundation for partially supporting this project with DMS-0310354. REFERENCES

The first word is an element in the solid torus braid with three strands; the last three have four strands. Thus, an al­ gorithmic description of this map is needed, not based on crossing information in figures but on values of the jug­ gling sequences.

1 . C. Adams, The Knot Book, W.H. Freeman, New York, 1 994. 2. J. Buhler, D. Eisenbud, R. Graham, C. Wright, Juggling drops and descents, The American Mathematical Monthly 101 (1 994) 507-5 1 9. 3. Juggling Information Service (JIS), http: //www.juggling.org 4. A. Lipson and C. Wright, JuggleKrazy software. 5. B. Magnusson and B. Tiemann, The physics of juggling, Physics

PROBLEM For any juggling sequence, find an algebraic de­ scription of its solid torus braid.

Teacher 27 (1 989) 584-589.

6. B. Polster, The Mathematics ofJuggling, Springer-Verlag, New York, 2003.

The version of a juggling pattern used here is sometimes referred to as vanilla site swap. Allowing multiple jugglers,

� Springer

7. J. Tawney, Jugglinks, Master's Thesis, The Ohio State University, May 2001 .

of ldence

Q u i c k, what's the best way to keep up

Clat ur

with new devel opments

field?

answer i s Spri ngerAi ertsl As an alerts subscriber you'll get � Reliable •first-word• on upcoming books and journal content � Free e-book downloads �--�-

--=Ex�c=lustve tr al access to jou rnals

SpringerAlerts - it's fast, it's free, and best of all it's t1 xible sprlnger.com/springeralerts 01 1

22

THE MATHEMATICAL INTELLIGENCER

ip.iMMffliii§i.@hli ¥11 .Jih(ii

An Enig matic Pyramid FRANS A. CERULUS

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

D i r k H uylebro u c k , Editor

S

ome 25 kilometers northeast of Brussels (Belgium) lies the village of Wespelaar; a completely aver­ age place, were it not for the splendid park in which the Spoelbergh family has assembled a unique botanical collection of trees and shrubs. The park is not open to the public except on special occasions. In the park stands a small and in­ triguing pyramid with inner chambers having weird acoustic effects. Planting of the park began in 1797, when the prosperous Louvain brewer Leonardus Artois asked the Brussels ar­ chitect Ghislain Joseph Henry to plan a park around the mansion he had ac­ quired that would reflect his financial status. They were assisted in this by the advice of the Reverend Matthieu Verlat, erstwhile professor on the Arts faculty of the University of Lou vain, who coun­ seled the Artois family in legal and sci­ entific matters: an early example of syn­ ergy between business and academe. The park was conceived according to the latest fashion, set by the British landscape architects. All alleys were curved, and the strolling visitor would constantly confront a new view and a new mood . At strategic points little monuments, the so-called fahriques, would fix the visitor's attention and guide his imagination. He would meet a round Roman temple dedicated to the goddess Flora, a romantic lake leading

Please send a ll submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mai l: [email protected]

I

Figure I .

Locating Belgium in Europe

to a cave, a Chinese bridge and a pyra­ mid [Duquennel . These fabriques were conceived by Henry and built early on, certainly not later than 1818.

Why a Pyramid? How, in 1797, did an entrepreneur, a

professor, and an architect come to think of a pyramid as a garden orna­ ment? One should realize that historians had deceived themselves for two thou­ sand years over the mystery of Egyp­ tian civilization [Lamy & Bruhier]: myth prevailed over history until Champol­ lion found the key to Egyptian writing. All this time Egyptomania, a mental ail­ ment whose sufferers believe that the monuments of the old Egyptians come from a civilization much superior to their own whose secrets are now lost, had been endemic. It flared up in Rome during the first and second centuries when it was most fashionable to have an obelisk, and once more in the baroque period, when those obelisks were dug up and made into new mon­ uments to adorn the papal city. At that time (1653) the famous Jesuit Athanasius Kircher published his mo­ mentous work ( 1500 pages) on the Egyptian religion and on hieroglyphs, a mixture of ethnological learning and delirious imagination [Kircher, Eg.l. As the official papal obelisk expert, he conceived the plan for the fountain of the four rivers in Rome's Piazza Navona, executed by Bernini, which is crowned by an obelisk. He described the sym­ bolism of the monument in a special publication ([Kircher,Oh.] and [Rivosec­ chi] ) . We shall come back to this. A hundred years later intellectuals had been converted to rationalism, doubted the Church and the Bible, hut continued to believe in secret knowl­ edge and Egyptian myth. This belief found its way into the newly founded Masonic lodges, where Egyptian sym­ bolism was combined with the myth of the temple of Solomon in their cere­ monies. It is against this background and widespread awe for the symbolism

© 2007 Springer Science+Business Media, \nc., Volume 29, Number 3, 2007

23

is the clue to the basic figure that com­ mands the form of the pyramid: the ver­ tical section through a diagonal is an equilateral triangle (disregarding the truncation for the moment), of side 32 = 25 feet. The side of the base should then actually be 6.24 m and the slope of a face with the vertical 22° 208 = arctan

1

v6 ,

which agrees with my measurements. The orientation

Figure 2. The garden Pyramid, about 7 m high, in Wespelaar; underneath is an icehouse whose entrance is at the rear (from [Duquenne)) .

of mathematics that w e have to con­ sider Henry's project for a pyramid. He was probably a Mason and could not drop just any pyramid into the park; he would have been proud to report to his brethren the insights he cleverly hid in the dimensions and proportions of his symbol-laden building.

Decoding the Pyramid General form

The angle of the pyramid is rather sharp, hut it is truncated at the top by a small obtuse pyramid, as on an obelisk, and it stands over a cellar: the icehouse in which the ice from the lake was stored in summer. The apparent base of the pyramid measures about 6 . 1 0 m but the origi­ nal base is in fact hidden by the turf around it. Belgium had been absorbed by the French republic in 1794 and the metric system had been voted the same year, but in 1800 the old measures were still in use. In Wespelaar the unit of length was the Brussels foot, which measured 0.27575m [Vandewalle). It seems there­ fore one should look for dimensions of the pyramid that are multiples of that foot. The diagonals of the base are ori­ ented towards the cardinal directions, which suggests that the architect started his plans by drawing the diagonals. This

24

THE MATHEMATICAL INTELLIGENCER

The diagonals are not exactly oriented North-South or East-West-there is a deviation of 10 degrees to the east, which makes me think that it is not re­ ally the diagonals that have to face the cardinal directions. In the pre-Christian era the summer solstice was celebrated, a custom that lives on today in the "fires of St. John". In the Masonic symbolism the Orient and the rising sun as well as St John play a role. So I venture a hy­ pothesis: the pyramid is oriented so that the entrance to its inner chamber faces the rising sun of the summer solstice. The remaining discrepancy of 4 degrees could be due to a slower change of the magnetic declination in the years 1790-1800, which was not accounted for by the usual linear approximations in, e.g., the Encyclopedie, and which was probably missed by the architect when setting out the base of the pyra­ mid on the ground.

The Inner Chambers Through the opening in the northeast face, one reaches in three steps a round inner chamber. It has a diameter of 1 2 feet; 4 niches, towards the corners of the pyramid, each originally contained Egyptian vases. The ceiling is a cap, formed by a hemispherical dome with an oculus of about 0.70 m diameter. This opens into a second chamber, sim­ ilar in form, but exactly one half of the dimensions of the lower chamber; in this there are no niches but four semicircu­ lar openings in the faces of the pyramid. The chambers are dimensioned and positioned such that two remarkable ra­ tios are realized.

Figure 3. Map of Belgium, locating B(russels), M(alines), L(ouvain), W(es­ pelaar).


=

=

i·

=

1 .618 . . .

'

commonly known as the golden ratio. The oculus as midpoint

The oculus divides the distance between the floor of the lower chamber and the top of the pyramid exactly in half. The truncation

If a cube were drawn with the same base as the pyramid, its upper plane would cut the pyramid exactly where it is truncated.

Acoustic Properties There are two remarkable phenomena, related to human speech, in the lower chamber. They can be explained by el-

N �-------.-+-.--�, 0 /

/

w

/

/

/

/

/

'

/

A'

A golden ratio

The distance from the oculus to the base of the pyramid ( 1 4 feet) divided by the side of the base is 0.618 . . . where

1 + Vs 2

0

Figure 4.

chamber.

'

'

/

'

/

'

/

'

/

'

'

X

6 m

Base of the pyramid and inner

ementary geometrical acoustics, i.e., acoustics on the model of geometrical optics, an approximation dependent on the wavelength of the sound compared to the size of the room. In the present case, for the pitch of normal speech, quite tolerable. Whispering dome

Under the spherical dome we experi­ ence the equivalent of the imaging of a light source by a concave mirror: words spoken near the wall are most distinctly heard near the wall opposite, from one niche, e.g., to the opposite niche. Was this planned by Henry' The phenomenon is reported in the first modern text on acoustics [Kircher, Phon.] by the very same Athanasius Kircher, mentioned earlier, who in this work shows himself a competent sci­ entist. In addition, the subject was cov­ ered in the course on sound that pro­ fessor Matthieu Verlat taught in the Arts faculty of the university of Lou vain ( [Go­ daertl, p.63).

•

•

•

Vanishing echo

Words spoken in the centre of the cham­ ber towards the oculus present no echo, while speech off the axis of the room, or directed sideways, reverberates for seconds. This is understood by the anal­ ogy with a system of a plane mirror (the floor) and a concave one (the domed ceiling) with a hole in the center. From a certain point on the axis, the sound waves are reflected by the dome and fo­ cussed on a lower point, but this point in turn is reflected by the floor through the oculus in the upper chamber from which the sound escapes to the outside, through the openings in the four sides.

The Symbols It is now not too difficult to look again at the pyramid through the eyes of an intellectual, perhaps a Mason, of the end of the eighteenth century. • The crowning of the pyramid by a small obtuse one and its position on a cave are very similar to the concept of the Bernini fountain: the pyramid or obelisk as a link between the lower and the higher world. Kircher ex­ plained that the apex stands for the

one God and the small pyramid whose sides are equilateral triangles that crowns the monument refers to the Trinity. The monument itself rep­ resents the angelic world: the link be­ tween heaven and earth. This is em­ phasized by the underlying cave, symbolic of the subterranean world. The equilateral triangle which com­ mands the form of the pyramid ap­ pears in Kircher as the symbol of the "immobile mover" ; the Mason would have called him the Architect of the Universe . The equilateral trian­ gle is also the basis for the hexagram or Solomon's seal. The golden ratio is fundamental in the construction of the pentagram, which should not fail in any magic circle [Goethe], and is conspicuously displayed in a masonic lodge. An ar­ chitect given to symbolism could not fail to smuggle the golden ratio into his creation, the mysterious number which many believe( d) confers beauty to works of art. The superposed chambers are inside the symbol of the angelic world, the road between this world and the Di­ vinity. They can be seen as succes­ sive stages of the human ascension.

I 2

That they are in the ratio of 1 to suggests the series 1 +

1

1

+ - +

-

2

4

1

-

8

+

2.

. . . =

Starting from the first chamber we end the infinite progression at the top of the pyramid, the Supreme Point.

Postscript Perhaps still other mathematical enig­ mas lie hidden in the elegant fab­ rique. As for their symbolic meanings, I have ventured a few , in the spirit of the late eighteenth century. Pyramids seem to inflame the imagination; they continued to do so in the nineteenth and the twentieth centuries and go on doing it, as a visit to a bookstore or the Internet will easily show. But those concern mostly the unmathe­ matical tourist. REFERENCES

[Lamy & Bruhier] Lamy, Florimond & Bruwier, Marie-Cecile, L'egyptologie avant Champol­ lion, Louvain-la-Neuve, 2005. [Duquenne] Duquenne, Xavier, Het Park van Wespelaar , Brussel 2001 . There exists a

French version of this work, Le Pare de Wespelaar.

[Kircher,Eg.]

Kircher,

Athanasius,

Oedipus

Egyptiacus, Roma, 1 653.

[Kircher,Ob.] Kircher, Athanasius, Obeliscus Pamphi/ius . . - - -

I

I I ,L I I I I I I

�M

I

I I _ _ jJ

_ _

I

I I I I I

2m .__...._..__.__..._.. 1 0 feet

Figure 5 .

Vertical section through cen­ tre, orthogonal to two sides. KLMN is a square, KIJN a harmonic rectangle. D = centre of dome; P = centre of circum­ scribed sphere; BE EF. =

. , Roma, 1 650.

[Rivosecchi] Rivosecchi, Valerio, Esotismo in Roma Barocca, Studi sui Padre Kircher,

Roma, Bulzoni editore, 1 982. [Vandewalle] Vandewalle, Paul, Oude maten, gewichten en muntstelse/s in Vlaanderen, Brabant en Limburg, Ghent, 1 984. [Goethe] Goethe, Johann Wolfgang, Faust, der Trag6die erster Teil; Studierzimmer, 1 808.

[Kircher,Phon.] Kircher, Athanasius, Phonurgia Nova, Campidonae (Kempten), 1 673.

[Godaert]

Godaert,

Paul

Matthieu

Verlat

(Pretre--Professeur a " Loven", chanoine . . . malheureux, conseiller et AMI de Ia famille ARTOIS), Beauvechain, 1 992 . lnstituut voor Theoretische Fysica K. U. Leuven Celestijnenlaan 2000 B-3001 Heverlee Belgium e-mail: [email protected]

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

25

Mathematic a l ly Bent

Col i n Adam s , Editor

A l
Opening a copy of The Mathematical Intelligencer you may ask yourself

uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am /?" Or even "Who am !?" This sense of disorienta­ tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Wil l i amstown, MA 01267 USA e-mai l : Colin.C .Adams@wi l l iams.edu

26

T

he name's Mangum. Mangum, P.I. That's right. I'm a principal in­ vestigator on a National Science Foundation grant, of the mathematical variety. You need a PI, you call me. I'm in the book. That's how this particular case came my way. It was a late Spring afternoon, and the constant California sun was try­ ing to weasel its way through the cracks in the mini-blinds hanging in my office windows at UCLA. I was busy filling my briefcase with a stack of papers that I wouldn't read that night, when the phone rang. Only this particular call was not of the usual variety. Most of­ ten, it's a co-author suspecting his part­ ner of co-authoring around. Wanting me to get proof. Or a chair who be­ lieves her department members are us­ ing office phones for personal business and wants the evidence so she can hang 'em out to dry. But this was different. It was Solomon Schmishmitt, a captain in the LAPD. The letters LAPD stand for the Los Angeles Police Department rather than some clever acronym for an outlandish mathematical organization. I know you were expecting otherwise, hut in this business you have to learn to expect the expected, even when you expect it the least. I knew Sol from the old days. We had been grad students together at Chicago. I majored in math. He majored in scotch. The last time I saw him, he was lying in the gutter outside the math building, in the pouring rain, too drunk to come inside and take his oral exams. Now that's dnrnk. When I went on to a post doc at UCLA, I heard he had finally dried out. Gave up the drinking, too. He realized that math wasn't his cup of tea, so he

THE MATHEMAnCAL INTELLIGENCER © 2007 Springer Science+Business Media, Inc.

got a degree in law enforcement from an on-line police academy. Printed his diploma on a color printer, and got a job in LA. Since then, he had stayed on the straight and narrow and built up a reputation as a good cop. Because of his grad school days, they gave him the math beat. After reminiscing about the various gutters he had spent time in, he got down to business. "Listen Dirk, I need your help. We have three dead algebraists. " "I know some departments that would be thrilled with that news, " I replied. "Most departments wouldn't want to lose one of these. Maclaunders, Hon­ eykey, and Nakanimji." The Holy Trinity. If algebraists marked fire hydrants like dogs, those three could soak down the Eiffel Tower, the Taj Mahal, and the Kremlin and still have plenty left over for actual fire hy­ drants. "That's quite a group of algebraists, " I said, "no pun intended." "None taken." "So let's hear the story. Were they at a conference?" "That's just it. They weren't together at the time. They were each at their home institutions when they died. About as far apart as any three points on the face of the earth. " Although it's true that Purdue and Tokyo are far apart, I didn't mention to Schmishmitt that you could do a lot bet­ ter for the choice of the third far-flung point than Notre Dame. He must have been lying in the gutter when we cov­ ered spherical geometry. "Think it's the work of a ring?" I asked. "Cut the puns, Mangum. Algebraists are dying. " "All right, all right. " I hated t o stop, but truth was I had run dry anyway. There are only two decent puns in all of mathematics, and I had already used them both. "How did they die?" "In each case, they wasted away.

Over a period of a month. Just stopped eating. Nobody can explain why. All three passed away within a week of one another. " "Maybe they weren't hungry." "We figured that much out, Mangum. Now the question is why . " " S o how d o I fit in?" "I called you because I know you've done work in algebra. I figure it's got to he an inside job. I want you to go to AlgebraFest, taking place next week at the University of Texas. I already got you on the hill. " "You mean I ' m giving a talk?" "Yeah. The title is "On the peelabil­ ity of semi-subring modules. " I figured that was vague enough that you could throw something together over the weekend. " 'Thanks a lot," I replied. My plans to spend the weekend alone with a hot conjecture were evap­ orating quickly. "By the way," I asked, "what is peel­ ability'' "I don't know," said Schmishmitt. "Make something up." A week later, I found myself milling around with 100 other mathematicians in front of a registration table in the lobby of RLM, the tallest mathematics building in Texas. "Well, well, well. If it isn't Dirk Mangum, P.I. Now what would bring you to AlgebraFest? I doubt it's the muffins." I looked into the bulging eyes of Hal Balony, a module guy from Springfield State. "Hello, Balony. I haven't seen you since you announced a proof of the Re­

the face. Truth was I hadn't pinned the details of the talk down yet, and it was coming up at 3:00. "You'll have to come to my talk, Balony, if you want to hear about peel­ ability." "Sure, Mangum. I'll come to your talk if you come to mine at 5 :00. I'll be in­ troducing some amazing math, math like you've never seen before. It is truly addictive. " He smiled the kind of smile that makes your skin want to crawl up your neck and hide under your hair. "We'll see. Balony," I said. "I may need to wash my underwear. I only brought one pair.'' I reached deep down into the bas­ ket of muffins and pulled out a bran muffin the color of a UPS truck. Taking a huge bite, I said "Dee-ricinus," spray­ ing Balony with crumbs. Turning quickly, I walked away. As soon as I made the corner, I disgorged the ined­ ible mass into a fiscus tree planter. I spent the next hour in the library working on a definition for pee/ability. Then I wandered in on a few desultory talks, but you could find more excite­ ment at a grading session for the final exam of a large lecture remedial course. Truth was that nobody at the confer­ ence had the mathematical chops that the dead algebraists had had, and the future of algebra was looking pretty bleak. A few minutes before 3:00, I found my way to the lecture hall. It was sur­ prisingly full. As I walked down to the podium, I heard a buzz. Several audi­ ence members pointed to me. The chair of the session stood up. "Well, I know you are all excited to

home (pause), a conjecture, a big con­ jecture, tantalizingly close. So close he could feel its breath on his neck. He was on the verge of one of the great discoveries of mathematics. But there was just one concept he couldn't grasp. One concept he didn't understand. " "That young mathematician went on to become one of the most famous, if not the most famous mathematician of his generation, a name everyone in this room would recognize . . . . But did he figure out this concept? Did he ever solve the conjecture?" "No, he did not. (Long pause. ) Def­ inition. Let G be a group. Let x be an element of G other than the identity. Let N be a normal subgroup of G that con­ tains x, if such exists. Taking the quo­ tient of G by N, x is trivialized. Yes, x is dead. Killed in the quotient process. " I looked out over the audience, hop­ ing to see someone, anyone becoming flustered. But they all stared back at me, waiting expectantly. "We think of the cosets of G/N as onion layers making up the onion that is the entire quotient group. This is the concept that the famous mathematician whose name we need not utter missed. That groups are merely certain types of onions, Vidalia, red, yellow. . . . Iden­ tifying the type of onion determines the group-theoretic properties. And the x? We say the x is peelable." I went on for another 40 minutes, but most of the audience lost interest once I got into my theory of groups as relish and other condiments. After my talk, I slipped away to give Schmishmitt a call, and see if he had uncovered anything. He had.

traction Conjecture and then retracted it

hear about this new concept of peel­

"Listen, Magnum, do you know a Hal

a week later. " "Just like you, Mangum, to bring up the one mistake I ever made in my ca­ reer, a mistake I made 1 5 years ago, and to ignore all of the revolutionary results I've been responsible for since then." "If you mean the Balony subgroup, the Balony semi-simple ring, and the Balony semi-literate suhalgehra, then yes, I have ignored them. And it doesn't seem to have caused me any harm. What a hunch of, of . . . malarkey." Balony turned red in the face. "You should talk Mangum. Peelabil­ ity' What the hell is that?" Now it was my turn to turn red in

ability. I will turn the podium over to Professor Mangum, who will explain it to us all . " I stepped u p t o the podium. Sitting in the front row was Hal Balony, wear­ ing a dismissive sneer. An expectant si­ lence fell over the lecture hall. I looked down at the podium, grasp­ ing it with my hands, letting the tension rise, letting it go on much longer than they expected. Then, when they could hardly stand it another moment, I looked up and said, " 1 946. A cramped office on the Princeton campus. A non­ descript mathematician sits in his office day after day, night after night, disre­ garding his young wife and child at

Balony?" " I've had the pleasure," I said drily. "Well, get this. Balony was on leave at Purdue when Honeykey died. He was seen with Honeykey in his office right about the time Honeykey stopped eating. We think he might have poi­ soned Honeykey or somehow infected him with a virus. Maclaunders and Nakanimji received packages from Pur­ due shortly after this, probably sent by Balony. Keep an eye on him. " I had seen Balony slip into one of the seminar rooms after my talk, shut­ ting the door behind him. I strolled ca­ sually down the hall, and quietly opened the door.

© 2007 Spnnger Science+ Business Media, Inc., Volume 29, Number 3, 2007

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Balony was lecturing to the empty room, practicing his talk. He stopped when he heard me come in. "Well, Mangum," he said. "I enjoyed the melodramatic aspects of your talk, but the content was on the meager side. It would have been nice if you proved something. " "It was a preliminary report, Balony. But I'm not here to discuss my talk. I 'm here to talk about Honeykey. " "What's to talk about? He stopped eating, and now he's dead. That's what happens when you don't eat. End of story." I grabbed Balony by the collar. "Story's not over, Balony. You poi­ soned him, didn't you?" Balony snorted, as he knocked my hands away. "That is funny," he said. "The great Dirk Mangum. The hot shot from UCLA with his big fat NSF grant. You call your­ self a P . I . , and you don't have any idea what is going on. " "All right then, Balony. Why don't you tell me? I know you were at Pur­ due when Honeykey stopped eating. I know you mailed a package to Nakan­ imji and to Maclaunders, and they both stopped eating after getting the pack­ ages. Some kind of virus I suspect. " "A virus? Oh, that's good Magnum. It is a virus, but not the kind of virus you're thinking of. " "As confessions go, that's good enough for me," I said. "You can bore the cops with the details. Let's go, Balony." "Wait a second, Mangum. " He flipped on the projector. "Do you know Gauss's Last Lemma?" Who didn't? Biggest open problem in the world. "Here it is. And look here. We have all the pieces to the puzzle right in front of us. " I looked at the set of equations. Amazingly enough, it appeared he was right. There was the solution to the Kleinhold issue, which had dogged po­ tential proofs up to now. And there was an end-run around the uncountability of the necessary axioms. An axiom that assumed them all away. It was amaz­ ing. This might actually lead to a solu­ tion to the entire problem. I stared in disbelief. Balony laughed from what seemed a great distance. I sat down to ponder the implications. It

28

seemed like the solution was unbeliev­ ably close. I quickly began to make mental calculations. If the Toeplitz op­ erator was semi-sufficient, then conju­ gating the commutator by a self-adjoint orthonormal pseudo-canonical basis would do the trick. Could it be1 Could this be the solution of the greatest prob­ lem in the history of mathematics? Everything around me faded in impor­ tance. I felt an incredible need to fin­ ish it off. . . . 0 0 0

"And now, I would like to present Pro­ fessor Hal Balony, who will tell us about his latest work." Balony stood at the podium. "2005, " he said. "A small office at Springfield State. An earnest researcher working late into the night. He stum­ bles across something. Something very big. He finds himself inches away from a theorem like no other in the history of mathematics (pause). What theorem was that? How did he almost prove it? Feast your eyes on this!" As he flipped on the projector, I rushed into the room. "Don't look," I yelled. "Don't look at it. " Of course, this caused everyone in the room to look at Balony's slide. The result was almost instantaneous. Jaws slackened as minds turned to contem­ plating the tantalizingly close result. "Too late, Mangum," said Balony, a malevolent grin stretched across his face. "They've seen it." I sprinted down the aisle, leaped onto the stage, and ripped the overhead cord out of the wall socket. The screen darkened immediately. But as I looked out over the audi­ ence, I could see I was too late. Every­ one was lost in thought, sinking deeper into the abyss. Balony sneered at me. "You blew it, Mangum. Now it's only a matter of time before they starve to death, too wrapped up in the attempted solution to remember to eat. "But I have to admit, I'm a bit sur­ prised to see you here. Why aren't you trying to solve it? Are you immune? Is your mind too small to grasp the full implications? Too slow to realize how close it is? It only kills the better math­ ematicians, you know, leaving room for the rest of us. Lucky for you. Funny, isn't it? You're just like me. "

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Sc,ence+ Business Media, Inc.

"No, Balony, I 'm nothing like you. I 'm like the rest of them. I couldn't stop thinking about it." "Then how is it you're not paralyzed like everyone else? How is it you are here?" "Because, Balony, I solved it. " "No way, " said Balony, a look of hor­ rified disbelief on his face. "If Hon­ eykey, Nakanimji, and Maclaunders couldn't solve it, then how could you?" "Funny thing, Balony. Sometimes, just once in a while, luck plays a role in the math biz. In this case, turned out that there was one simple piece of the puzzle that those other mathematicians didn't have. One little idea. " "What's that?" asked Balony. "It's called pee/ability," I said as I plugged the projector back in. I put a clean overhead slide on the glass, and then I started writing. Slowly, the mem­ bers of the audience began to turn their attention to what I was doing. I set up the equations necessary and then I wrote down the punch line. There was an audible sigh. For some of these mathematicians, this was the closest they would ever get to orgasm. Balony collapsed onto a chair, stunned by what had just happened. "Okay, Balony, now let's hear the rest of your story. Where did you get the idea? I don't believe for a second it was yours. " Baloney cradled his head in his hands. "Honeykey must have found it," he said dully. "I went to see him at his of­ fice one day. He never did like talking math with me. Considered it a waste of his time. His door was open and there he was sitting at his desk in what ap­ peared to be almost a trance. He could hear me, but clearly his thoughts were far far away. "I looked down on the desk in front of him and saw what he was thinking about, an outline of a possible proof of Gauss's Last Lemma. I scooped it up since he wasn't about to notice. Went back to my office and read it, but it didn't do anything to me. I could see what it was, but it didn't look that close to me. "After a couple of days, I could see what was happening to Honeykey. He was fading away, losing contact with the real world. Couldn't eat. Couldn't sleep. It was obvious he wouldn't last

long. That's when I sent copies to Nakanimji and Maclaunders." "Why did you do that?" "Do you have any idea, Mangum, what it's like to be a second-rate math­ ematician? To never get grants? To prove theorems that everyone ignores' Do you? No, you don't. "I always wanted to he the best al­ gebraist in the world. I tried working hard to get there, spending every wak­ ing moment on mathematics. But after a while, it became apparent that wasn't going to do the trick. That left only one other option. Eliminate all the alge­ braists that were better than me. And if you hadn't interfered, I would have sue-

� Springer

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ceeded. I would have been the great­ est living algebraist in the world!" "What a bunch of . . . of hooey, Balony. You amaze me with your naivete. Don't you think the rest of the world would have said, yes he's the greatest living algebraist, but only be­ cause all the other algebraists died. What kind of recognition is that?" "It would have been good enough for me, Mangum. Good enough for me . " 0 0 0

Balony was hooked on charges of mur­ der, attempted murder, and using an overhead projector as a dangerous weapon. He's at the type of educational

institution that takes 20 years to gradu­ ate from; 1 5 with good behavior. And Gauss's Last Lemma? Ultimately, it turned out there was an error in one of the propositions that was used to con­ struct the proof, a twenty-year-old result first published by . . . you guessed it . . . Hal Balony. I believe the reason Balony was unaffected by the proposed proof was that he knew, either consciously or unconsciously, that his result, upon which it was built, was in error. So Gauss's Last Lemma is still out there. Like a black hole that can suck you in. It's still open. It's still waiting. But be very careful. It can be addict­ ing . . .

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© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3 , 2007

29

ljf1(9·i.(.i

David E . Rowe , Editor

Fermat Comes to Amer ica : H arry Schultz Vand ive r and F LT

(1914-1963)

LEO CORRY

I

F

ermat's Last Theorem (FLT) was long known as the most famous of all unsolved mathematical problems. Familiar even to laymen, it attracted the attention of both profes­ sional mathematicians and amateurs, many of whom thought they held the key to its solution. Ferdinand Linde­ mann, who was suddenly vaulted to fame when he proved the transcen­ dence of 1r in 1882, tried to solve FLT several times over the remainder of his career, only to come up with one faulty proof after another. Nor was he alone in this regard; indeed, FLT stood in a league by itself when it came to the number of incorrect proofs that found their way into print. As for failed at­ tempts by amateurs who sent their "so­ lutions" to mathematicians all over the world but which (thankfully) remained unpublished, the number cannot even be estimated. These circumstances make it easy to understand the general excitement that surrounded this story when Andrew Wiles finally completed his general proof of Fermat's conjecture in 1994. As news of this impressive achievement rippled through the mathematical com­ munity, it also made headlines that at­

Send submissions to David E. Rowe, Fachbereich 08-lnstitut fur Mathematik, Johannes Gutenberg U niversity, 055099 Mainz, Germany.

30

tracted the attention of broad lay audi­ ences who had no chance of grasping the ideas behind his work. Public in­ terest in Wiles's personal story and his long quest to solve FLT added an un­ usual sense of drama to the accom­ plishment itself. His curiosity about the problem from childhood, his eight years of self-imposed seclusion that led to the breakthrough, the tension that followed discovery of a non-trivial mistake in his initial proof, and the final resolution eight months later in collaboration with Richard Taylor, all these elements only enhanced interest in this appealing story. Still, from a broader perspective, in­ formed individuals might well shake their heads when reading the overly dramatized popular versions of the quest to solve FLT (a tendency occa-

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+Business Med1a, Inc.

Figure I.

Harry Schultz Vandiver and son, Frank, ca. 1930 (HSV). sionally found even in accounts written for more professional audiences). The most striking example of this is surely Simon Singh's best-selling boo k, Fer­ mat's Enigma, a book that conveyed to very broad audiences the excitement human beings feel about doing mathe­ matics. Its readers learn that Fermat's conjecture "tormented lives" and "ob­ sessed minds" for over three centuries, and thus constituted "one of the great­ est stories imaginable. " On the front flap of some editions one reads that FLT be­ came the Holy Grail of mathematics and that Euler "had to admit defeat" in his attempts to find a proof, while: Whole and colorful lives were de­ voted, and even sacrified, to finding a proof. . . . Sophie Germain took on the identity of a man to do re­ search in a field forbidden to fe­ males . . . . The dashing Evariste Ga­ lois scribbled down the results of his research deep into the night before venturing out to die in a duel in

1832. Yutaka Taniyama . . . tragically killed himself in 1 958. Paul Wolfskehl, a famous German indus­ trialist, claimed Fermat had saved him from suicide. On opening that hook, one reads that "The Last Theorem is at the heart of an intriguing saga of courage, skulldug­ gery, cunning, and tragedy, involving all the greatest heroes of mathematics. " Not surprisingly, a closer and more sober examination of the actual histor­ ical evidence surrounding research on Fermat's problem brings to light a far less dramatic version of these events. This is not to say that the history of FLT lacks interest, but the story hardly war­ rants the sense of high drama that re­ cent writers have brought to it.1 None of the mathematicians who appear in Singh's account (except for Wiles him­ self) ever devoted sustained research ef­ forts purely focused on an attempt to solve this famous problem. In fact, many of those mentioned in his book showed only the slightest interest in solving it, whereas only one mathe­ matician prior to Wiles took a similar passionate lifelong interest in FLT. This was Harry Schultz Vandiver (18821973), a figure who does not even ap­ pear in Singh's hook and who is only mentioned marginally in most other re­ cent accounts. Vandiver devoted nearly all of his professional life to resolving this famous problem, a quest that set him apart from his fellow mathematicians. For although FLT aroused curiosity among number­ theorists, the problem remained on the margins of the field for decades. Re­ markably few serious efforts were de­ voted to it during Vandiver's lifetime. Moreover, his research program in­ volved the kind of massive calculations of individual cases that most number­ theorists consciously avoided. Aided by electro-mechanical and, later on, elec­ tronic devices for making such calcula­ tions, Vandiver emerged as a prominent exponent of a research style that many of his contemporaries would have con­ sidered unworthy of a true mathemati-

nected with FLT. An interesting, though somewhat elusive historical question that I will attempt to answer at least par­ tially concerns Vandiver's professional status in the eyes of his contemporaries. 0 0 0

Figure 2. Maude, Frank and Harry S. Vandiver (HSV).

ciao's time. Nor has his reputation ben­ efited posthumously from the lavish praise that Andrew Wiles's work re­ ceived. Since Wiles's general proof came from a completely different di­ rection that bore little relationship with Vandiver's train of ideas, the latter's contributions have either been over­ looked or are seen today as devoid of direct interest for actual research in number theory. Nonetheless, from a historical point of view, Vandiver is a figure of consid­ erable interest, not only because of his intense involvement with FLT hut also for the role he played within the Amer­ ican mathematical community through­ out his long and, in many ways, exotic career. The present article offers a por­ trait of that career, along with a brief sketch of Vandiver's activities in con­ nection with FLT. In a follow-up to this article I will describe some other aspects of Vandiver's work not directly con-

A self-trained mathematician, Harry Schultz Vandiver was born October 2 1 , 1882, in Philadelphia. H e never com­ pleted high school, and the little col­ lege- and graduate-level mathematics he studied at the University of Penn­ sylvania in 1904-06 was undertaken in a rather haphazard, non-systematic manner. Thus he never obtained a col­ lege degree, except for an honorary doctorate that the University of Penn­ sylvania bestowed upon him in 1945 at the age of 63. 2 In 1900 he started submitting solu­ tions to problems posed in the Ameri­ can Mathematical Monthly, especially on topics in algebra and number the­ ory.3 This activity seems to have been his gateway to studying mathematics, and it was certainly how he got to know George David Birkhoff (1884-1944) well before the latter became the most influential American mathematician of his generation. The young Birkhoff wrote to Vandiver in 1 901 commenting on the latter's contributions to the Monthly while telling him about his own interest in FLT.4 Thus began a sub­ stantial correspondence that lasted sev­ eral years, though they did not meet un­ til 1913. Their joint paper, in 1904, was Birkhoff's first publication. 5 Between 1905 and 1917 Vandiver was working as a customs house bro­ ker and freight agent for his family's firm. In his letters to Birkhoff-written on stationery with the letterhead of the "John L. Vandiver, Custom House Bro­ ker" in Philadelphia-he openly ex­ pressed his admiration for his friend's knowledge and "boundless enthusiasm for mathematics. " Occasionally he sug­ gested ideas intended for possible ad­ ditional joint publications, but these never actually materialized. Birkhoff's juvenile interest in number theory and

1 [Corry 2008]. 2There are various sources of information about Vandiver's life, sometimes containing contradictory information. I have drawn here mainly on documents found at the Van­ diver Collection, Archives of American Mathematics, Center for American History, The University of Texas at Austin. See also [Greenwood et a/., 1 973], [Lehmer 1 973].

3See Am. Math. Mo. 7 (May 1 900), p. 1 46. His first number-theoretical problem appears in the Am. Math. Mo. 8 (Aug. 1 901), p. 1 80. A more significant one, dealing

with properties of Mersenne numbers appeared in Am. Math. Mo. 9 (Feb. 1 902), pp. 34-36. 4[Vandiver 1 963, 271]. 5[Birkhoff & Vandiver 1 904].

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

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who was preparing his monumental History ofthe Theory ofNumbers. In par­ ticular, Vandiver was actively involved in writing the chapter on FLT, and in 1928 he was co-author of a supple­ mentary volume to Dickson's work.8 Al­ gebraic Numbers was produced on Dickson's recommendation as the offi­ cial report of the Committee on Alge­ braic Numbers of the National Research Council, a committee Vandiver chaired between 1923 and 1928. This collabo­ ration with L. E. Dickson left a deep im­ print on Vandiver. Throughout the years, he continually referred to the spirit of Dickson's work as an example that should be followed in all of math­ ematics, and he attempted to implement several initiatives along these lines. Such undertakings included detailed bibliographies of various individual mathematical domains as well as pro­ posals for significant reforms in mathe­ matical reviewing and the refereeing systems in the USA. Strongly recommended by Dickson, Vandiver was appointed in 1 924 to a professorship at the University of Texas, Austin. This became his academic home until the end of his life, though his re­ lationship with colleagues at this insti­ tution can hardly be described as one of peaceful coexistence. Above all, his personal and professional relations with the almighty Robert Lee Moore (1882-1974) were a source of constant strain that reached remarkable peaks of mutual animosity.9 Beyond solitary re­

a network of Moore's academic de­ scendants found positions at depart­ ments throughout the country, and their efforts helped make point-set topology a leading field of research in the USA. Vandiver's network remained far more circumscribed, reflecting the more lim­ ited interest in FLT and other research topics he pursued throughout the years. Nevertheless, Vandiver traveled ex­ tensively and took repeated leaves of absence to pursue his research. Much of his correspondence with university authorities revolved around requests re­ lated to these leaves. Thus, it was with a touch of irony that in its sympathetic Memorial Resolution of 1973, Vandiver was remembered by the Faculty Coun­ cil as a distinguished former UT pro­ fessor, whose colleagues "bemoaned the fact that he did not staj' around very much. 12 He was constantly apply­ ing for research grants provided by a number of institutions, including the National Science Foundation, the Carnegie Institution, and the Guggen­ heim Foundation. In 1 934, he was the first mathematician ever to apply for support from the American Philosophi­ cal Society.13 In 1 953, at the age of sev­ enty-one, he requested (for the sixth time) funding from the Guggenheim Foundation for a planned six-month leave of absence. Surprisingly the Foun­ dation granted him approval , but then Vandiver decided to withdraw his ap­ plication. 14 Even at the age of 76 he re­ ceived a research grant from the NSF.

erable time for research. "Perhaps times

search, Vandiver's main strengths were

Although those within his close circle

will become so bad, " he added, "that I will be compelled to look for some teaching position."7 From 1 9 1 7 to 1 9 1 9 Vandiver served as yeoman in the U.S. Naval Reserve, but soon afterward, aided by Birkhoff's active endorsement, he accepted a teaching position at Cor­ nell. With his arrival at Cornell, Vandiver began collaborating with Chicago's Leonard Eugene Dickson (1875-1954),

clearly not in classroom teaching, but rather in direct and personal inter­ changes.10 While his work involved ac­ tive collaboration with several younger mathematicians, and particularly gradu­ ate students, he formally directed only five Ph.D. dissertations at Austin. 1 1 One cannot help but compare him in this re­ gard with Moore, who devoted a great deal of his energy to advising many promising Ph.D. candidates. Eventually

of friends would invariably write the warmest letters of recommendation in support of his many applications, this was not always the case with others. Birkhoff, for instance, advised the Guggenheim Foundation that it would be better to devote its resources to sup­ port younger men, and he saw no rea­ son why Vandiver could not continue to pursue his research at his home in­ stitution. 1 " Still, Birkhoff and others con-

During times of intense research effort, he would isolate h imselffrom all distractions. elementary geometry soon began to re­ cede in favor of his mature pursuits in analysis and applied mathematics. Van­ diver remained strongly focused on number theory and on related algebraic disciplines and consistently tried to pull Birkhoff back into these fields. In 1 9 1 5 Vandiver wrote to him: I am particularly anxious that you become interested in number the­ ory. If I can induce you to take up the subject I am sure you will never regret it. Your position in the math[ematical] world is now as­ sured, and I think you should be able to give considerable time to these things which virtually consti­ tuted the life work of such men as Gauss, Kummer, Kronecker, Dirich­ let-after your present work is com­ pleted.6 Birkhoff, in turn, consistently encour­ aged his friend to pursue his mathe­ matical interests beyond what his free time in business would allow him. Van­ diver eventually found himself in a real dilemma, but it took some time before he finally decided to embrace mathe­ matics as a profession. At one point he wrote to Birkhoff, half joking-half seri­ ous, that with business slackening be­ cause of the war, he now had consid­

6Vandiver to Birkhoff: March 1 8, 1 9 1 5 (HUG). 7Vandiver to Birkhoff: May 1 7, 1 9 1 5 (HUG). 8[Vandiver & Wahlin 1 928].

91 will deal with this in the follow-up to this article; part of the story is told in [Parker 2005, 226-231 ] . 1 0[Lehmer 1 973] describes Vandiver as a "poor lecturer. " Some of Vandiver's students expressed similar views elsewhere. 1 1 They are: Ferdinand Biesele ( 1 94 1 ) , Olin Faircloth ( 1 95 1 ) , Charles Nicol (1 954), Milo Weaver (1 956), and Richard Kelisky (1 957). 12http://www.utexas.edu/faculty/council/2000-2001 /memorials/SCANNED/vandiver. pdf. 13Edwin G. Conklin to Vandiver: July 6, 1 934 (APS). 1 4Vandiver to Henry Allen Moe: June 1 7, 1 953 (HSV). 15Birkhoff to Foundation: March 7, 1 953 (GFA).

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tinued to stress Vandiver's status as the acknowledged world-leading expert on FLT and to praise his (not totally unre­ lated) work on cyclotomic fields. In later years, many expressed admiration for the fact that a man of Vandiver's age could still be an active researcher. J (, This admiration was also often conveyed to him directly in personal letters 17 Vandiver's frequent travels were part of a somewhat nomadic lifestyle that of­ ten took him, his wife Maude ( nee Folms­ bee), and their son Frank 0926--2 005) around the country and also to Europe. Frank was home-schooled, as experience had hardened Vandiver's strong distrust of public schools.1H Frank reported that in his childhood the family never had a permanent home nor did they ever own a house. They would move from one apartment to another, renting the home of a colleague on leave while Vandiver was preparing to go on leave himself, then going back to another rented apatt­ ment and so on. For many years Van­ diver also had a "permanent" room at the Alamo Hotel.19 During times of in­ tense research effort (and these were not infrequent), he would isolate himself from all distractions by checking into this hotel room. He always kept a suitcase in his office, just in case such an eventual­ ity might arise. Alternatively, he some­ times preferred to lock himself in at home and remain incommunicado for several days. 20 He could single-mindedly concentrate on his work, sometimes for­ getting even to eat, thereby bringing him­ self to the brink of physical collapse. Two extra-mathematical topics sur­ face fairly often in his letters: classical music and baseball. Vandiver seems to have owned a remarkable collection of records that he managed to carry along with him as he moved from place to place. He especially loved Mozart, and in various letters he mentioned plans to write a mathematician's guide to listen­ ing to this genial composer's music.

When it came to baseball, he prided himself in taking original views that went against the flow of mainstream opinion. Commenting on a recent game played by his "favorite team," the New York Giants, against their powerhouse rivals, the Yankees, Vandiver noted that the Giants' "present line up includes many wonderful fielders,'' adding that "defense seems to be paid little atten­ tion to these days by the public. They prefer to look at cheap homeruns. "21 Above all, what he enjoyed was the ex­ citement of a tight contest, as "a game which ended with a one-sided score was not to his taste."22 All kinds of additional oddities were associated with this somewhat leg­ endary figure on (and off) the Texas campus, as reflected in a sympathetic account written by his Austin colleague Robert Greenwood: H.S. Vandiver was hardly the athletic type. There is no record of his ever having owned a car, and so he must have walked a lot. Between semes­ ters in the winters the University buildings frequently were heated up to about 40°F. But Professor Vandiver would walk up to campus, go into his office where he had a portable electrical heater with a spherical or parabolic reflecting surface. He would then work away on a mathematical theorem of current interest to him. Usually he would, on these occasions, work in his top coat with the electric heater warming his feet and legs 23 Many friendly exchanges of letters among his posthumous papers offer a closer glimpse into his unique personality. Richard Bellman C l920-19H4) was a bril­ liant and versatile mathematician whose fields of interest were much broader and quite different hom Vandiver's. In 19'59, Bellman created the Journal qf Mathe­ matical Arzalvsis and Applications and in­ vited Vandiver to join the editorial board. Vandiver initially declined:

I am now on a basic research grant of the N.S.F. , which does not expire for two years, and as I am now 76 years old, I do not feel that I can take on any mathematical work aside from what the Foundation ex­ pects me to do, namely, to do re­ search on number theory. 24 Bellman did not give up, however, and replied immediately in order to assure Vandiver that he would "have the re­ sponsibility of looking over the papers of only one mathematician, namely, yourself. " And he added: As far as your rather young age of 76 is concerned, I distinctly remem­ ber that you are the person who in­ sisted that he wanted to live to be 95 and to be shot by a jealous husband. Consequently, if you divided your re­ maining activities between number theory and these pursuits, I feel we have the best of the bargain 2� Vandiver's later years were affected by poor health. Still, he continued to work under a "Modified Service" appointment until the age of 80; only then did he be­ come Professor Emeritus. When he died on january 4, 1973, he was 89 years old.

Vandiver and FLT in Context Vandiver did not develop new concepts or overarching theories to deal-from completely novel perspectives-with FLT. Rather, his approach was that of a meticulous technician who fully ex­ hausted the unexploited potential of ex­ isting theories and refined them further where necessary. At the center of Van­ diver's work one finds extensive, highly complex calculations of particular cases, along with innovations aimed at im­ proving existing computational tech­ niques. He was apparently undaunted by even the most demanding computa­ tions, in part because he was willing to use a variety of tools (both material and conceptual) to achieve his task. In order to understand the context of

1 6Andre Weil to Foundation: June 1 953 (GFA). 17 Alfred Brauer to Vandiver: Dec. 23, 1 958 (HSV). Brauer assured Vandiver that he would support his request for a grant with the NSF, and the latter was indeed granted. 1 8Frank Vandiver later became a distinguished professor of American history and, among other things, Provost of Rice and President of Texas A&M University. 1 9See ]Greenwood et at., 1 973, 1 09321.

2°Frank Vandiver, interview with Ben Fitzpatrick and Albert C. Lewis, June 30, 1 999 (MOHP). 21 Vandiver to W. L. Ayres: August 31 , 1 951 (HSV). 22[Greenwood et at., 1 973, 1 0932].

23Robert Greenwood, "The Benedict and Porter Years, 1 903-1 937", unpublished oral interview (March 9, 1 988), (MOHP), p. 26. 24Vandiver to Bellman: April 20, 1 959 (HSV). 25Bellman to Vandiver: April 28, 1 959 (HSV).

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Vandiver's work, a few words must be said both about the status of FLT at the turn of the twentieth century as well as to the standing of research in number­ theory in the United States. During the first third of the twentieth century, the American community of number-theo­ rists was relatively small and not ex­ tremely prominent. As a rough measure, one might note that the index to the first ten issues of the Transactions ofthe AMS (1909) lists only two articles under the heading of number theory, and in the following decade, despite a noticeable increase, there were still only thirteen. In his correspondence, Vandiver often spoke about the lack of interest in num­ ber theory in the US. Looking through this correspondence for names of math­ ematicians actively involved in research in number theory prior to 1940, one finds above all foreign figures, some of whom Vandiver also visited in Europe: Rudolf Fueter, Edmund Landau, Emmy Noether, Helmut Hasse, Kurt Hensel, Philipp Furtwangler, Nikolai Grigore­ vich Chebotarev, Dmitry Mirimanoff, Taro Morishima, Trygve Nagell, Felix Pollaczek, and Arnold Walfisz. We also find a number of American mathemati­ cians, mainly active on the West Coast: Eric Temple Bell, Hans Frederik Blich­ feldt, Derrick Norman Lehmer (and later on Derrick Henry and Emma Lehmer), Robert Daniel Carmichael, Al­ bert Cooper, Leonard Eugene Dickson, Morgan Ward, and Aubrey Kempner. Unlike the case with their foreign coun­ terparts, this latter group essentially ex­ hausts that of number-theorists active in the American community. After 1940 the field became more active, though it con­ tinued to remain somewhat on the mar­ gins of the research community for some years.26 Vandiver's almost exclu­ sive focus on this field, along with the unusual circumstances surrounding his early mathematical career, helps to ac­ count for his rather unique situation within the American mathematical com­ munity. His decision to devote so much of his professional life to FLT made him a truly singular figure. Although even a mini-history of Fer­ mat's problem is well beyond the scope of this article, a bit of historical back-

ground is necessary in order to under­ stand Vandiver's work. This sketch will also underscore the fact that rather few significant contributions to solving the problem were made following Ernst Ed­ uard Kummer's work in the 1850s. As part of his long-standing efforts to deal with questions related to higher reci­ procity laws, Kummer developed many important concepts and techniques that turned out also to be relevant for FLT. Thus, it was Kummer who introduced the notions of regular and irregular primes, a distinction based on a prop­ erty of the "class number" hp of a cy­ clotomic field k(?). He then developed original methods that enabled him to prove that FLT is valid for all regular primes. In addition, he introduced three somewhat complex conditions which, when satisfied by an irregular exponent p, implied the validity of FLT for that ex­ ponent. Then, by means of long and te­ dious computations, he identified all ir­ regular primes under 164, obtaining these eight numbers: 37, 59, 67, 101, 103, 131, 1 49, and 1 57. Applying his criteria to the three cases of irregular primes un­ der 100 (37, 49, 67), he achieved his well-known result of 1857 that FLT is valid for all exponents less than 100. Be­ yond this, however, the calculations be­ came prohibitively complex and, in ad­ dition, it became clear to Kummer that some of the criteria he had developed would not apply for certain irregular prime exponents, such as p = 1 57. Kummer also proved that a prime p is regular if and only if it does not di­ vide the numerators of any of the Bernoulli numbers Eo, Bz, . . . Bp-3, which appear as coefficients in the ex­ pansion _x_ (/C -

1

=I

n=O

Bnx" n!

At that time values for the Bernoulli numbers had been calculated up to 13(,2 .27 The ability to identify higher val­ ues of regular or irregular primes would later come to depend on the possibil­ ity of calculating higher values of these numbers, an effort in which Vandiver was directly involved. Kummer was both an avid calculator and a gifted theorist, but it was the !at-

ter aspect of his work that most influ­ enced the development of number the­ ory in the decades to come, especially through the efforts of Richard Dedekind. By the turn of the twentieth century, par­ ticularly in the wake of Hilbert's influ­ ential Zahlbericht, the emphasis on a "conceptual" perspective (as opposed to the more algorithmic approach favored by Leopold Kronecker) became domi­ nant in the discipline. Results based on special calculations for particular cases were not favored under this view. These general trends in research account to a large extent for the remarkable fact that Kummer's results relating to FLT were not essentially improved or extended for almost sixty years. Prior to Kummer, Sophie Germain had proved an important result, as a consequence of which the proof of Fer­ mat's problem can be reduced to deal­ ing with two separate special cases. Case I asserts that for p > 2, xP + yP + zP 0 has no integer solutions for x, y, z relatively prime to the odd prime p. Case II asserts the same when one and only one of the three numbers x, y, z is divisible by p. The little progress that did take place in proving the the­ orem between Kummer and Vandiver dealt almost exclusively with case I. The most important single result was ob­ tained in 1909 by Arthur Wieferich (1884-1954). Wieferich proved that if three integers x, y, z relatively prime to p actually did satisfy x P + y P = zP, then the congruence z p- l = 1 (mod p2) must be satisfied. Dimitry Mirimanoff (18611945) extended this result in 1910 by proving that the same p would also sat­ isfy 3 P- l = 1 (mod p2). These two re­ sults, and some similar ones that were sporadically added later on, helped es­ tablish a lower bound for the value of the integers for which the Diophantine equation x P + y P = zP could be satis­ fied under the conditions of case I (and this, moreover, only by considering p, and irrespective of the values of x,y,z that may satisfy the equation). Based on these results, and directly motivated by the additional encourage­ ment provided by the creation in 1 908 of the Wolfskehl Prize, several mathe­ maticians decided to attack the problem =

26A detailed analysis of the internal structure of this community and its development (along the lines of [Goldstein 1 994] for the case of the French community of number-theorists in the second half of the nineteenth century) seems to be an interesting open task for historical research. 27[0hm 1 840].

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anew, producing several additional re­ sults along the same line of ideas. Thus one finds contributions by such leading figures as Philipp Furtwangler and Georg Ferdinand Frobenius, but also by the then unknown Vandiver. In 1914, in his first article on FLT,28 he proved a Wieferich-like congruence for ')P- 1 Vandiver's first truly substantial result came in 1920, when he identified a mis­ take in Kummer's article of 1857 and went on to correct one of its main ar­ guments.29 He continued to refine and develop his ideas on FLT over the next few years. A summary of his achieve­ ments appeared in his authoritative arti­ cle of 1929, 30 for which he was awarded the AMS's first Cole Prize in number the­ ory two years later. This award, honor­ ing the AMS's long-time secretary Frank Nelson Cole, was established for out­ standing work in this field.

Vandiver's Contributions to FLT

Vandiver's article of 192 1 went well be­ yond Kummer's results by proving FLT for exponents up to p = 2 1 1 . Even be­ fore it appeared in print, Vandiver re­ alized that his arguments could be used to extend the results to p < 269. Under his direction, specific calculations of various ranges were performed sepa­ rately by various M.A. students at Austin, including Samuel Wilks 0 906-1964) and Elizabeth Stafford 09022002). All calculations of values p, 100 < p < 2 1 1 , were performed by Wilks "using Monroe and Marchant electrical computing machines." Van­ diver and his team proved that if p di­ vides only one of the numbers B2, B4, . . . , Bp- 3 , and if this single Bernoulli number is not divisible by p, then FLT is valid for p. This allowed further cal­ culations for exponents p < 307. As the difficulty of the calculations increased,

Vandiver devised further methods to simplify and speed up the procedures, and also to allow for double-checking. At the same time, however, he was skeptical about the general validity of Case II of FLT. Eric Temple Bell wrote to him in 1929, If I remember rightly, you once said that you would not be surprised if the second case turned out to be false . . . . You give the limit five hun­ dred for exponents to be tried. I have no idea of the actual amount of com­ putation required for such an under­ taking, but I should think it would be terrific. There is no doubt in my mind that anyone who knows any­ thing about the Theory of Numbers would say that this work ought to be done while there is a man not only able to do it, but also willing. If in one of these exponents the compu­ tations should give a negative result, you will set a problem to exasperate generations of arithmeticians. I rather hope that it does turn out that way.31 In the next few years, however, Van­ diver would surpass the exponent 500 and would continue to confirm the va­ lidity of FLT for ever higher values of p, including those covered by case II.32 In an article in 1 934 he remarked that much of his "work concerning FLT is tending toward the possible conclusion that if the second factor of the class number" hp of K( �) is prime to p, "then FLT is true. " This is the famous "Van­ diver conjecture, " about which he had begun to speculate much earlier. Its im­ portance for algebraic number theory in general gradually gained recognition over the years, albeit in somewhat mod­ ifie d versions 53 This was by no means the only original conjecture that ap­ peared in his articles, however, as was pointed out in later research.34

The most significant progress in cal­ culations related to FLT resulted from Vandiver's work with the couple Der­ rick Henry Lehmer 0905-1991) and Emma Lehmer ( 1906--2 007). Their col­ laboration started in 1932, though the first joint publication did not appear un­ til 1939, when they proved the validity of FLT for 2 < p < 619 5'i Above 619, the calculations became prohibitively long and laborious to be carried out with the kind of desktop calculators available to the Lehmers. But in 1953 when elec­ tronic computers became available, the three mathematicians took a great leap forward, proving that FLT was true for all exponents p < 2000.36 Throughout their correspondence, Vandiver stressed that, beyond the specific results ob­ tained, this research had an enormous value for advancing research on cyclo­ tomic fields. In subsequent papers, he further refined the Kummer criteria for irregular primes, and this led to an ex­ tension of the results on FLT to 2000 < p < 2520 in 1954 and then, in 1 955, to p in the range 2520 < p < 4002 .37 This work of Vandiver and his col­ laborators using electronic computers did not, however, alter mainstream re­ search in number theory, at least not in the short run. Nor did it lead to renewed research efforts in connection with FLT. Still, seen in retrospect, these pioneer­ ing efforts opened the door to a new direction of research that remains active today. Additional results along similar lines have continued to confirm FLT for exponents exceeding one billion, and in Case I for higher values still. In fact, even after Wiles's general proof of FLT, new ranges of exponents are still being tested with ever improved techniques. 58 In 1946, following a request from the editors of the A merican Mathematical Monthly, 39 Vandiver published a de-

28[Vandiver 1 9 1 4] . 29[Vandiver 1 920, 1 922]. 30[Vandiver 1 929]. 31 Bell to Vandiver: Jan 1 5, 1 929 (HSV). 32For a detailed account of Vandiver's works during these years and how they eventually led to the use of electronic computers to solve FLT, see [Corry 2007].

33See, for instance, [lwasawa & Sims 1 965]. [Lang 1 978, 1 42] pointed out that the conjecture had originally been formulated by Kummer [Col/. Vol. 1 , 85]. Lang indi­ cated that "Vandiver never came out in print with the statement: "I conjecture etc . . . . ", but "the terminology 'Vandiver conjecture' seemed appropriate to me. In any case I believe it". 34[Herstein 1 950, Denes 1 952]. 35[Vandiver 1 937, 1 937 a].

36[Vandiver, Lehmer & Lehmer 1 954].

37[Vandiver 1 954, Vandiver, Selfridge & Nicol 1 955].

38[Wagstaff, 1 978; Buhler et at. 1 992; Buhler et a/. 2001]. 39Lester R. Ford to Vandiver: February 2, 1 945 (HSV).

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tailed exposition of the state of the art in research on Fermat's problem.40 This article became a classical locus of ref­ erence for many years to come. In sum­ marizing his opinion about the general validity of the conjecture, about which he was frequently asked, Vandiver drew a clear distinction between the two clas­ sic cases. He was convinced of the va­ lidity of case I, but not merely because it had been proved for very high val­ ues. Rather, his confidence in this case stemmed from some important theo­ rems he had proved along the way on trinomial congruences-a topic to which he had devoted many efforts. Case II involved a much more complex situation; thus, while he believed it would ultimately be proven, he did not think he had any compelling evidence to support it. Furthermore, he stated, he felt less sure than in 1934 about the va­ lidity of the Vandiver conjecture, pre­ cisely because of its close relationship with and possible dependence on the validity of FLT. Commenting on the fre­ quency with which apparently promis­ ing conjectures in number theory are eventually abandoned, he added: When I visited Furtwangler in Vienna in 1928 he mentioned that he had conjectured the same thing before I had brought up any such topic with him. As he had probably more ex­ perience with algebraic numbers than any mathematician of his generation, I felt a little more confident. (p. 576) While mentioning some additional re­ sults presented in his current account, he echoed the opinion voiced several years earlier in Bell's letter: However it would probably be best if I were wrong about this. I can think of nothing more interesting from the standpoint of the development of number theory, than to have it turn out the Fermat relation has solutions, for a finite number > 0, of primes l. Concluding, he wrote, Many mathematicians are often in­ terested in ascertaining how a par­ ticular topic connects up with other parts of mathematics. In case of Fer­ mat's Last Theorem it is well known that Kummer's attempts to prove it gave rise to the theory of ideals

which is now of fundamental impor­ tance in many parts of mathematics. The remarkable character of Kum­ mer's achievement has tended, how­ ever, to minimize the great number of connections which the theorem has with other subjects. Efforts on my part to clear up the question have led me into the following topics: Bernoulli numbers and polynomials and generalizations; Euler and Gen­ nochi numbers; Euler and Mirimanoff polynomials; partitions modulo m; finite fields and rings, including a great many types of congruences: the Dirichlet Zeta Function and the related Dedekind Function; the La­ grange resolvent and Jacobi 4> num­ ber and various generalizations in­ cluding generalized Gauss sums; the theory of Kummer fields, class num­ ber, class fields, power characters and laws of reciprocity in the theory of algebraic fields; Fermat's quotients and other arithmetic quotient forms; congruence theories as applied to power series; abstract algebra includ­ ing, particularly, group theory and semi-groups; and many types of Dio­ phantine equations aside from the Fermat relation itself. It seems Vandiver was rather carried away with enthusiasm. Only some of these topics have substantial, direct con­ nections with FLT. On the other hand, Vandiver himself was led to explore many of these potential payoffs, partly because of his interest in questions that arose from, or were thought to be use­ ful for solving FLT. In fact, in 1 952-53 he published a two-part article on as­ sociative algebras and the algebraic the­ ory of numbers, a paper he regarded as more important and innovative than any of his work directly connected with FLT. He was disappointed that this article was seldom cited: As far as I know, only one person has studied thoroughly this paper, and he is Alonzo Church.41 In private correspondence Church raised some interesting criticisms re­ garding the axiomatic debate devel­ oped by Vandiver, and Vandiver was quick to include Church's comments in a follow-up to this article.

40[Vandiver 1 946]. 41Vandiver to Mientka: March 1 3. 1 964 (HSV). See also, Vandiver to Herstein, April 2, 1 960.

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As late as the early 1 960s Vandiver was still publishing new results related to FLT. He also continued to work on a book about FLT and related topics in number theory, a project he pursued for many years. His archive contains hun­ dreds of typewritten pages with whole chapters nearly ready for publication, but for some reason this book was never published.

Correspondence on FLT As Vandiver came to know, having your name publicly attached to Fermat's problem could impose a considerable burden on a mathematician. Yet some experts found efficient ways to duck the unwelcome task of reading the steady stream of faulty proofs submitted by rank amateurs. In the early twentieth century, Edmund Landau came up with a nearly ideal solution to this corollary to Fermat's problem. As Gottingen's leading authority on number theory, Landau was officially entrusted with handling all correspondence related to the Wolfskehl Prize, which offered 100,000 Marks to anyone who could solve Fermat's conjecture. Landau had little interest in the problem, so he took on this duty with little enthusiasm. When the flow of incoming correspon­ dence from amateurs eventually be­ came unbearable, he became openly disgusted. So he prepared a form letter that looked something like this: Dear . . . . . . . . . , Thank you for your manu­ script on the proof of Fermat's Last Theorem. The first mistake is on: Page . . . . . . Line . . . This invalidates the proof. Professor E. M. Landau An assistant read through the manu­ scripts and filled in the missing details in the form letter. Fortunately for Vandiver, no rich oil­ man came along to establish a similar prize fund at the University of Texas for a successful proof of FLT. So this cir­ cumstance surely diminished the num­ ber of would-be problem solvers who might have written to him. Nevertheless he did receive enormous quantities of mail that only grew from year to year.

In fact, many American mathematicians (perhaps all of them?) saw in Vandiver the default address to which any letter on the topic should be redirected. Van­ diver's attitude toward these corre­ spondents was essentially positive, per­ haps because he had been something of an amateur mathematician himself. In response to attempted proofs by rank amateurs, he sent a pre-written, but rather polite reply. His archives contain no fewer than 225 such answers, sent between 1 934 and 1966. To those he considered qualified mathematicians he usually answered in some detail, though even in these cases the task became in­ creasingly onerous with time. An interesting letter from 1949 attests to this problem in the case of a math­ ematician, Taro Morishima, whose con­ tribution Vandiver truly appreciated. He was forewarned that Morishima was about to submit an article on FLT to an American journal. This prompted him to take preemptive action by contacting several editors (Aurel Wintner of the American journal of Mathematics. Rudolph E . Langer, Saunders Mac Lane, and Leonard Carlitz) to request that the article not be sent to him. He would certainly like to read this paper, he said, but at his leisure and not under pres­ sure to finish within some given period of time, however reasonable. Number theory, he added, "seems to be getting popular," but Vandiver felt he was drowning under the enormous corre­ spondence he now had to handle.42 A few months later, he again complained bitterly about this to another colleague, while requesting that no further letters be sent to him. Only if he received a manuscript from Siegel, Hasse, or Rademacher would he he willing to ex­ amine the work in detail. To which he added: "After nearly forty years of look­ ing at such manuscripts, good and bad, don't I deserve a respite?"43 One revealing interchange took place in 1961 around a proposed proof of FLT by Lucien Hibbert, then Execu­ tive Director of the Inter-American Bank for Development in Washington, D.C. Hibbert had been directed to Vandiver by Israel Herstein and Marshall Stone.

Born in 1899 in Haiti, he had received in 1937 a Ph.D. at the Sorbonne, work­ ing with Arnaud Denjoy (1 884-1974). In Haiti he had been professor of mathe­ matics and physics, and director of the Haitian Statistical Institute, and had en­ joyed a very successful career in the fi­ nance sector. He was Ambassadorial Representative of Haiti to the Organi­ zation of American States, and later served in the Ministry of External Af­ fairs. As usual, Vandiver read the man­ uscript fairly carefully and replied po­ litely and in some detail. In his answer, he summarized his general attitude to­ ward this matter: Next October 21 I shall start my 80th year of age. Beginning in the year 1 9 1 4 I published several articles on the Fermat problem which received attention from readers to the extent that many of them wrote letters to me, generally containing their opin­ ions . . . about the problem, and also what they regarded as proof of Fer­ mat's statement or contributions to that encl . For some years I made a practice of replying to such letters and giving my estimates of the value of their work. However, as I con­ tinued to publish from time to time through the years articles pertaining to the Fermat problem, my corre­ spondence along that line became so heavy that if I had continued to do this it would have taken most of my time. . . As an example of this, l have received four letters within the last few weeks and about seven since the first of the year pertaining to the Fermat problem . . . one of them . said the full proof of the theorem coven.:d about 50 pages. In his resume of the paper given me in his letter, he made a number of statements I could not understand at all: . . . so I told him I was sorry I had to refuse to help him . . . In my 50 years' experience with the problem I have often been con­ vinced for a time that I had a proof of the theorem using only the tools of elementary number theory and al­ gebra, but I found in every such case

that I had an error in my argument. After a time I became more and more skeptical of any apparent proof that I found using such ele­ mentary means, as I felt that if an elementary proof existed, it could hardly have escaped the attention of such great mathematicians as Euler, Legendre, Lame, Abel, Gauss, Cauchy, and Kummer, all of whom worked at the problem! . . . I have looked over the general character of your argument . . . and as far as I can see . . . you have used nothing but elementary alge­ bra therein, hence I cannot help be­ ing skeptical as to the accuracy of your work. if you regard this as a disparagement of your work, please note that I have just di!>paraged above all my own efforts ofthis char­ acter.14 Vandiver further advised Hibbert to write up full proofs for various specific cases, to see if they worked. And he added: "In giving you this advice, I am assuming that you would prefer to find the error yourself, if one exists, than to have someone else find it." Then, in a letter to Stone he explained what he really feared about cases like this one: For many years, in connection with "proofs" of FLT sent to me and which I examined it turned out in nearly every case that if I called the author's attention to an error in his work, soon after I would receive an­ other ms. which he assumed was a correction of his original paper. Also, if instead of pointing out an error I would merely state to him that there was a step in his argu­ ment which I did not understand, then the author would reply that I did not understand his [entire] argu­ ment. These things would be the be­ ginning of a long correspondence that I would have with him.45 Some years prior to Hibbert, Vandiver was involved in another notewor­ thy exchange with a Pakistani air force officer named Quazi Abdul Moktader Mohd Yahya, who was formerly "Pro­ fessor of Mathematics at Brajali Acad­ emy, East Pakistan." In various letters

42Vandiver to various: December 8, 1 949 (HSV}. 43Vandiver to J. R. Kline: January 1 2 , 1 950 (HSV}. 44Vandiver to Hibbert: June 23, 1 96 1 (HSV}. Emphasis in the original. 45Vandiver to Stone: June 26, 1 96 1 (HSV}.

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37

written to colleagues about this man, Vandiver referred to him as X, noting that "I do not wish to be sued for libel, in case the information in this letter somehow reaches him." He received a manuscript from him on FLT that the au­ thor wished to submit to the Proceed­ ings of the National Academy of Sci­ ences. As in other cases, Vandiver answered the initial letter politely, but this led to a lengthy and futile series of interchanges. Vandiver tried to put an end to this by suggesting that Yahya send his manuscript to a "regular math­ ematical journal," one "preferably in Switzerland or Germany where they seem to have more interest in number theory than in the U.S." He feared that this "dangerous character" might "write me a threatening letter, as some of these birds have done in the past. "46 Eventu­ ally Yahya was able to publish his (ob­ viously flawed) proof in a Portuguese journal in 1976.47 A third interesting correspondence over FLT took place in 1960-61 when Vandiver was contacted by a junior high school pupil from Baltimore named Joel Weiss. After learning the names of the three persons in the US who had re­ cently done work on the Fermat prob­ lem, Joel wrote to Vandiver (and also to the Lehmers) for advice on this topic, which he had chosen for a school term paper. He was willing to work hard, and so he explained his choice as follows: This theorem, which originally was a curiosity to me, turned out to be a stimulating research project well worth the 45 hours of work neces­ sary to complete it. I hope that my conclusion will start a new train of thought leading to an eventual proof of Fermat's Last Theorem. Joel later indicated at the end of his fin­ ished paper what this desired train of thought might be: I conclude that Fermat's Last Theo­ rem has been proven all this time, and that its entire proof is that of n 3. I have reached this conclu­ sion from an analyzation of a suc­ cession of cases of the theorem with exponents 3 through 9. After study=

ing these cases, it is apparent that the deviation between the sum of the terms in the left-hand member of the equation and that of the right­ hand member increases steadily with higher exponent value. There­ fore, I feel that it is only necessary to prove n = 3 because this is the point of lowest deviation. Any ex­ ponent value above this is immedi­ ately ruled out as a result of the fact that the deviation is greater than that of the third power thus making it impossible to suit the equation.48 Vandiver, who had written several po­ lite and possibly helpful letters to Joel along the way, also reacted politely to Joel's conclusion: most mathematicians, he kindly remarked, would not agree with the closing statement of his paper.

Figure l. Harry S. Vandiver (Creator: Walter Barnes Studio (HSV).

Recognition and Oblivion In the currently available literature, Van­ diver's name is barely mentioned in connection with FLT. For instance, in the popular "MacTutor History of Math­ ematics Archive" website, Vandiver barely rates a very short entry of his own. His name appears only in passing in the site's article on FLT, and he is not mentioned at all in the article on Dick Lehmer. From the point of view of current mathematical research asso­ ciated with the problem, especially fol­ lowing Wiles's dramatic breakthrough, this may be understandable. But from the point of view of the history of the problem, this lack of recognition is com­ pletely unjustified, though the reasons for this are not difficult to find. Although Vandiver was the undis­ puted world's leading expert on FLT during his lifetime, contemporaries of­ ten took an ambivalent attitude toward him and his passionate quest. Certainly he was well-known and respected both within the American mathematical com­ munity and abroad, but his interests were also viewed as exotic, and evi­ dence abounds that he was viewed as more bizarre than brilliantly original. Thus, it is not surprising that when his friend G. D. Birkhoff prepared a list of the 10 most prominent American math-

ematicians in 1926 for the Rockefeller foundation, Vandiver, then 44 years old, was not on his list. 49 Even during his most creative phase as a researcher he seems to have received less recognition than he probably deserved. Yet Vandiver received several high honors, including the Cole Prize and an honorary doctorate from the University of Pennsylvania; and, of course, he was the recipient of many research grants. Harry Vandiver was the only American mathematician whose work received mention in Edmund Landau's 1927 clas­ sic textbook on number theory. He was elected vice-president of the AMS for the term 1933-1935, and in 1935 he was an AMS Colloquium Lecturer. He served as assistant editor of the Annals of Math­ ematics from 1926 to 1939, and in 1934 he was elected to the National Academy of Sciences. Still, he always remained part of a small and rather marginal sub-community within the larger Amer­ ican mathematical research enterprise. Strongly fixated on his own work, he was certainly not a shaker and mover. He would not manage to attract large numbers of young researchers to his chosen field; he did not establish a re­ search school, nor did he develop an influential network of contacts with like-

46Vandiver to Hayman: April 3, 1 958 (HSV). 47Mathematica/ Reviews

lists a "private edition" by the author [Yahya 1 958], and three additional articles in

Portugaliae Mathematica

(1 973, 1 976 and 1 977).

48The entire correspondence appears in HSV: File 1 6-3. 49See [Siegmund-Schultze 2001 , 51]. Birkhoff's list included only mathematicians from three leading centers: Cambridge (Birkhoff, Morse, Osgood, Wiener, Whitehead); Chicago (Bliss, Dickson, E. H. Moore, Moulton); and Princeton (Alexander, Eisenhart, Lefschetz, Veblen).

38

THE MATHEMATICAL INTELUGENCER © 2007 Springer Science+ Business Media, Inc.

minded mathematicians. Nor was he an organizational talent who excelled when it came to promoting journals or organizing professional meetings. The honors conferred on Vandiver occasionally betray ambivalence. For example, only after Vandiver himself applied some direct pressure on uni­ versity authorities was he named Dis­ tinguished Professor at TU in Austin, in 1947. But his title, "Distinguished Pro­ fessor of Applied Mathematics and As­ tronomy," was certainly odd given his research expertise. More telling still is the context sur­ rounding a Festschrift published in his honor. In 1966 Bellman's journal of Mathematical Analysis and Applica­ tions brought forth the special issue dedicated to Vandiver on his eighty­ third birthday. The editors wished to honor him not only for his contributions to FLT and algebraic number theory but also because "he has profoundly influ­ enced the development of American mathematics for a period of over sixty years." And yet the American contribu­ tions to this volume were all written by his former students and close collabo­ rators. Side by side with these papers one finds a score of others written by leading number-theorists from abroad, figures such as Mordell, Hasse, Erdos, Szemeredi, Gel'fond and Morishima. It's odd that such a collection appeared in a journal far removed from Vandiver's own fields of interest. Evidently the de­ cision to publish such a Festschrift came from close friends who wanted to pay long-overdue tribute to the man and his work, yet sensed that no one outside Vandiver's inner circle would ever un­ dertake it. The honoree, then in deli­ cate health after undergoing surgery, was deeply touched by this gesture.so Vandiver's lifetime endeavor was characterized by remarkable indepen­ dence and a willingness to pursue self­ styled, original research programs. As a researcher, his style was marked by an indefatigable appetite for endless cal­ culations, by a peculiar style of collab­ oration with small groups of people who were dose to him, and by his pi­ oneering use of electronic computers in his fields of expertise. While Vandiver's contributions played no direct role in shaping the train of ideas that eventu-

Figure 4. AM S MAA meeting in Washington D.C. (HSV). Source: Capi­ tol Photo Services, Inc. -

Figure 5. joel Weiss with

a poster presentation of his work on FLT (HSV).

ally led to the general proof of FLT, and while opinions may vary as to the in­ trinsic mathematical significance of the ideas deve lo ped in his work, one can­ not make sense of the history of FLT without giving prominence to the story of this man, the only one ever to de­ vote his entire professional life to solv­ ing the problem. ACKNOWLEDGMENTS

Albert C. Lewis and David Rowe read earlier versions of this article. I thank them for the critical remarks which led to significant improvement. I have used archival material found in several institutions. I thank the archivists for assistance in locating and copying the originals, and for granting rermission to

quote. Pictures are reproduced and sources are quoted with pennission, us­ ing the following abbreviations: HSV: Vandiver Collection, Archives of American Mathematics, Center for American History, The University of Texas at Austin. MOHP: Oral History Project, The Legacy of R.L. Moore, Archives of American Mathematics, Center for American History, The University of Texas at Austin. HUG: George David Birkhoff Papers, Harvard University Archives: Call Num­ ber HUG 4213.2, Box 3, Folder "T-V " . APS: American Philosophical Society Archive. GFA: The John Simon Guggenheim Memorial Foundation Archive.

50Dorothy W. Baker to Bellman: September 29, 1 965 (HSV).

© 2007 Springer Science+Business Media, Inc., Volume 29, Number 3, 2007

39

REFERENCES

Kummer, Ernst E. (Co/� Collected Papers (ed.

Birkhoff, George David, and Harry S. Vandiver

by Andre Weil), Berlin, Springer-Verlag (1 975).

(1 904), "On the integral divisors of ah-bh,"

Lang, Serge (1 978), Cyclotomic Fields, New

Ann. Math. (2) 5, 1 73-180.

Lehmer, Derrick H. (1 973), "Harry Schultz

polski (1 992), "Irregular primes to one mil­

Vandiver. 1 882-1 973," Bull. AMS 80, 8 1 7-

lion," Math. Camp. 59, 71 7-722.

81 8.

Lewis, Albert C. (1 989), "The Building of the

sankyla, and M. Shokrollahi (200 1 ), "Irregu­

University of Texas Mathematics Faculty,

lar primes and cyclotomic invariants to 1 2

1 883-1 938," in Peter Duren (ed.) A Century

million," J. Symbolic Comput. 31 , 89-96.

of Mathem atics in Americ a-Part /11 , Provi­

dence, Rl, AMS, pp. 205-239. Ohm, Martin (1 840), "Etwas uber die Bernoul­

Theory," Annals of History of Computing

li'schen Zahlen," Jour. reine u. angew. Math.

(Forthcoming).

20, 1 1 - 1 2 .

--

Parker, John (2005), R.L. Moore. Mathemati­

Theory. Computers and Number Theory.

cian & Teacher, Washington DC, Mathemat­

Computers and Number Theory from Kum­

ical Association of America.

mer to SWAC," Archive for History of Exact Denes, Peter (1 952), "Beweis einer Vandi­ ver'schen Vermutung bezuglich des zweiten Falles des letzten Fermat'schen Satzes, " Acta Sci. Math. Szeged 1 4, 1 97-202.

Goldstein, Catherine (1 994), "La theorie des nom­ bres dans les notes aux Comptes Rendus de I'Academie des Sciences (1 870-1 9 1 4): un pre­

mier examen," Riv. Star. Sci. 2, 1 37-160. Greenwood, Robert E. et a!. (1 973), "In Memoriam. Harry Schultz Vandiver, 1 8821 973," Memorial Resolution, Documents

and Minutes of the General Faculty, The Uni­ versity of Texas at Austin, 1 97 4, 1 0926Herstein, Israel. (1 950), "A Proof of a Conjec­ ture of Vandiver," Proc. AMS 1 , 370-371 . lwasawa, Kenkichi, and Charles Sims (1 965), "Computation of Invariants in the Theory of

569-584.

Fermat's last theorem (second paper) , " Duke Math. J. 3, 4 1 8-427. --

(1 946), "Fermat's Last Theorem, " Am.

Math. Mo. 53 (1 946), pp. 555-578. -- (1 954), "Examination of methods of attack

on the second case of Fermat's last theorem," Proc. Nat!. Acad. Sci. USA 40, 732-735. (1 963), "Some of my recollections of

George David Birkhoff," Jour. Math. Analysis and Applications 7, 271 -283.

Vandiver, Harry S., Derrick H. Lehmer, and

Siegmund-Schultze, Reinhard (2001 ), Rocke­

Emma Lehmer (1 954), "An application of

feller and the Internationalization of Mathe­

high-speed computing to Fermat's last

matics between the Two World Wars, Basel

theorem," Proc. Nat!. Acad. Sci. USA 40, 25-33.

and Boston, Birkhauser. Vandiver, Harry S. (1 9 1 4), "Extensions of the

Vandiver, Harry S., John L. Selfridge, and

criteria of Wieferich and Mirimanoff in Con­

Charles A. Nicol (1 955), "Proof of Fermat's

nection with Fermat's Last Theorem, " Jour.

last theorem for all prime exponents less than

reine u. angew. Math. 1 1 4, 3 1 4-31 8.

4002," Proc. Nat!. Acad. Sci. USA 41 , 970-

--

(1 920), "On Kummer's Memoir of 1 857

Concerning Fermat's Last Theorem," Proc. Nat!. Acad. Sci. USA 6, 266-269. --

(1 922), "On Kummer's memoir of 1 857,

concerning Fermat's last theorem (second paper)," Bull AMS. 28, 400-407. --

(1 929), "On Fermat's Last Theorem, "

--

973.

Vandiver, Harry S., and George E. Wahlin ( 1 928), Algebraic Numbers -II. Report of the Committee on Algebraic Numbers, Wash­

ington, DC, National Research Council. Wagstaff, Samuel S. (1 978), "The irregular primes to 1 25000," Math. Camp. 32 (1 42), 583-591 .

Trans. AMS 31 , 61 3-642.

1 0940.

(1 937), "On Bernoulli Numbers and Fer­

mat's Last Theorem, " Duke Math. J. 3,

--

(2008), "Number Crunching vs. Number

Science (Forthcoming).

ber," Bull AMS 40, 1 1 8-1 26.

-- (1 937a), "On Bernoulli numbers and

Buhler, J . P . , R. Crandall, R. Ernvall, T. Met­

diver, the Lehmers, Computers and Number

(1 934), "Fermat's last theorem and the

second factor in the cyclotomic class num­ --

York, Springer-Verlag.

Buhler, J.P. , R. E. Crandall, and R . W. Sam­

Corry, Leo (2007), "FLT Meets SWAC: Van­

--

(1 930), "Summary of results and proofs

Yahya, 0. A. M . M . (1 958), Complete proof

on Fermat's last theorem (fifth paper) , " Proc.

of Fermat's last theorem. With a foreword

Nat!. Acad. Sci. 1 6, 298-305.

by Dr. Razi-Ud-Din Siddiqui. Available from

--

(1 930a), "Summary of results and proofs

the author, Pakistan Air Force, Kohat, West

Cyclotomic Fields," J. Math. Soc. Japan 1 8,

on Fermat's last theorem (sixth paper)," Proc.

Pakistan (1 4 pp. Mimeographed appendix,

86-96.

Nat!. Acad. Sci. USA 1 7, 661 -673.

3 pp.).

LEO CORRY is head of the Cohn lnstrtute. His latest book,

and the Axiomatization of Physics, 1 898- 1 9 1 8

David Hilbert

(Kiuwer), was published

in 2004. His current research interests include the history of FLT and of computational approaches to number theory.

Cohn Institute for History and Philosophy of Science and Ideas

Tei-Aviv University 69978 Tei-Aviv Israel

e-mail: [email protected]

40

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media, Inc.

Spectral Variation, N ormal M atrices, and Finsler G eometry

RAJENDRA BHATIA

ow did two matrix-theorists who had never worked together before come to prove a theorem which has had consequences throughout the field and beyond? I will try to put together the personal and the mathemati­ cal sides of the Hoffman-Wielandt Theorem, its prehistory, and attempts (both successful and unsuccessful) to gener­ alise it. Wielandt was really trying to do the thing for operator nonns and the Frobenius nann was his second choice. Thus begins Alan Hoffman's commentary on his joint paper with Helmut Wielandt [HW] , one of the best known in linear algebra. The paper is less than three pages long and, of a piece with that brevity, Hoffman's commentary con­ sists of just one paragraph. It continues, Infact, he had a proof ofHW with a constant bigger than 1 in front. It was quite lovely, involving a path in matrix space, and I hope someone else has found a usefor that method. Since linear programming was in the air at the National Bureau of Standards in those days, it was nat­ ural for us to discover the proof that appeared in the pa­ per. The most difficult task was convincing each other that something this short and simple was worth publish­ ing. In fact, we padded it with a new proqf qf the Birk­ ho.ff theorem on doubly stochastic matrices. I think the reason for the theorem 's popularity is the publici�y given it by Wilkinson in his hook on the algebraic eigenvalue

problem U. H. Wilkinson, The Algebraic Eigenvalue Prob­ lem, Clarendon Press, Oxford, 1965). In this article I will explain what it was that Wielandt was really trying to do, why he wanted to do it for oper­ ator norms, and what some others had done before him and have done since. Wielandt's mathematical works [Wie l] straddle two dif­ ferent fields: group theory and matrix analysis. He began with the first, was pulled into the second, and then hap­ pily continued with both. The circumstances are best de­ scribed in his own words: The group-theoretic work was interruptedfor severalyears while, during the second half of the war, at the G6ttin­ gen Aerodynamics Research Institute, I bad to work on vibration problems. I am indebted to that time for valu­ able discoveries: on the one band the applicability of ab­ stract tools to the solution of concrete problems, on the other band, the-for a pure mathematician-unexpected dif.ficul�y and unaccustomed responsibility of numerical evaluation . It was a matter of estimating eigenvalues of non-se{fadjoint differential equations and matrices. I at­ tacked the more general problem of developing a metric spectral theory, to begin with forfinite complex matrices. The links between all parts of our story are contained in the two paragraphs I have quoted from Hoffman and from Wielandt.

© 2007 Springer Science+ Business Media, Inc, Volume 29, Number 3, 2007

41

By the time Wielandt came to Gi:ittingen in 1 942, Her­ mann Weyl had left. Thirty years earlier Weyl had pub­ lished a fundamental paper [We] on asymptotics of eigen­ values of partial differential operators. Among the several things Weyl accomplished in that paper are many interest­ ing inequalities relating the eigenvalues of Hermitian ma­ trices A, B, and A + B. One of them can be translated into the following perturbation theorem: If A and B are n X n

Hennitian matrices, and their eigenvalues are enumerated as a1 ::::: a2 ::::: a n, and {31 ::::: {32 ::::: ::::: f3n, respectively, then •

•

•

•

•

•

(1) Here IIAI I stands for the norm o f A a s a linear operator o n the Euclidean space e n; i.e., (2)

ll ll ii ll I IAI I = max l Ax : X E e n, x = 1 ) .

Apart from the intrinsic mathematical interest that Weyl's inequality (1) has, it soothes the analyst's anxiety about "the unaccustomed responsibility of numerical evaluation. " If one replaces a Hermitian matrix A by a nearby Hermitian matrix B, then the eigenvalues are changed by no more than the change in the matrix. Almost the first question that arises now is whether the inequality remains true for a wider class of matrices, and for a mathematician interested in "estimating eigenvalues of non-selfadjoint differential equations and matrices" this would be more than mere curiosity. The first wider class to be considered is that of normal matrices. (An operator A is normal if AA* = A*A. This is equivalent to the condi­ tion that in some orthonormal basis the matrix of A is di­ agonal. The diagonal entries are the eigenvalues of A, and A is Hermitian if and only if these are all real.) The eigenvalues of a normal matrix, now being com­ plex, cannot be ordered in any natural way, and we have to define an appropriate distance to replace the left-hand side of (1). If Eig A = {a�, . . . , aJ and Eig B = {{3�, . . . , f3 J are the unordered n-tuples whose elements are the eigen­ values of A and B, respectively, then we define the opti­

mal matching distance

(3)

d(Eig A, Eig B) = min max laJ - f3cr(J)I , u l $.j$. n

RAJENDRA BHATIA has been associated with

where a varies over all permutations of the indices { 1 , 2, . . . , n}. The question raised by Weyl's inequality is: if A and B are any two normal matrices, then do we have

ll d(Eig A, Eig B) :::; II A - E ?

(4)

This is what Wielandt, and several others over nearly four decades, attempted to prove. We will return to that story later. The operator norm (2) is the one that every student of functional analysis first learns about. Its definition carries over to all bounded linear operators on an infinite-dimen­ sional Hilbert space. That explains why this norm would have been Wielandt's first choice. There are other possible choices. The Frobenius nann of an n X n matrix A is defined as

IIAIIF = (tr A* A)1 12 =

(5)

(6)

dF(Eig A, Eig B) = m�n

42

THE MATHEMATICAL INTELLIGENCER

] 112 .

Hoffman credits ]. H. Wilkinson [Will with the publicity responsible for the theorem's popularity. Wilkinson writes

Tbe Wielandt-Hoffman theorem does not seem to have attracted as much attention as those arising from the di­ rect application of nonns. In my experience it is the most useful resultfor the error analysis of techniques based on orthogonal tranifonnations in floating-point arithmetic.

He also gives an elementary proof for the (most interesting) special case when A and B are Hermitian. In spite of Wilkinson's reversal of the order of names of its authors, the theorem is known as the Hoffman-Wielandt theorem. Unknown, it would seem, to Hoffman and Wielandt, and

151 Delhi most of the

Rao,

KR

whom we thank) shows him on his anival at University of Califomia

India

a1 - f3crcJ)I 2

(7)

Parthasarathy. The photograph of him here (taken by George Bergman,

e-mail: [email protected]

l

THEOREM 1 Let A and B be any two nonnal matrices. Tben

nealogy Project, he is scientifically a direct descendant of Arthur Cayley,

Indian Statistical Institute Delhi

L�

=

Instead of ( 4), Hoffman and Wielandt proved the following.

via A. Forsyth, E. Whittaker, James Jeans, RA. Fisher, C.R

New Delhi, I I 00 I 6

1.]

This norm arises from the inner product (A, B) tr A* B, and, for this reason, it has pleasant geometric features. It can be easily computed from the entries of A. If we replace the norm (2) with (5), then we must make a similar change in the distance (3) and define

time since his graduate student days. According to the Mathematics Ge­

Berkeley on a post-doctoral fellowship, 1 979.

(L l a;1 12 r2 .

to Wilkinson, the Hermitian special case of (7) had been announced several years earlier, by Karl Lowner in 1 934 [Lo]. This paper is very well-known for its deep analysis of operator monotone functions. Somewhat surprisingly, there is no reference to it in most of the papers and books where the inequality (7) is discussed. (Incidentally, Lowner was at the University of Berlin between 1922 and 1928. Wielandt came to study there in 1929 and obtained a Ph.D. in 1935. Lowner's original Czech name was Karel but, because his education was in German, he was known as Karl. Later, when he had to move to the United States, he adopted the name Charles Loewner.) Lowner does not offer a proof and says that the inequality can be established via a simple vari­ ational consideration. One such consideration might go as follows. When x = (x1 , . . . , x,J is any vector with real coordinates, let x t = ext, . . . ' xf) and xi = cxl, . . . x �) he the decreasing and increasing rearrangements of x. This means that the numbers x1, . . . , Xn are rearranged as xf ::::: ::::: x� and as x I :5 :5 x �· Then for any two vectors x and y, we have '

·

·

·

·

for every differentiable curve U(t) with U(O) = equivalent to saying

_!]_I dt

PROPOSITION 2 Let A and B ces. Then (9) (Eig l (A) , Eig i (B))

:5

tr

be

n

X

n Hermitian matri­

AB :5 (Eig l (A),

Eig 1 (B)).

OUs

n

X

n unitary matrices, and let

= { U BU* : U E U(n)},

be the unitary orbit of B. If we replace B by any element of OUs, then the eigenvalues of B are not changed, and hence neither are the two inner products in (9). Consider the function j(X) tr AX defined on the compact set OUs. The two inequalities in (9) are lower and upper bounds for j(X). Both will follow if we show that every extreme point Xo for f commutes with A. A point Xo on OUs is an extreme point if and only if =

_!]_I dt

tr A U( t)Xo U(t)*

t=O

=0

=

0

=

0.

The trace of a product being invariant under cyclic per­ mutation of the factors, this is the same as saying tr K(XoA - AXo)

=

0.

Since (K, L) - trKL is an inner product on the space of skew-Hermitian matrices, this is possible if and only if X0A - AXo 0. 0 =

=

Using the second inequality in (9) we see that

I lA - Ell} = IIAII} + II BII} ::::: IIAII} + II BII}

-

n

2trAB 2(Eig t (A), Eig t (B))

= '\' L lA ;t (A) - A .!t cB)I2

( 10)

j= l

This proves the inequality (7) for Hermitian matrices. The same argument, using the first inequality in (9), shows that

II A - B lli :::; I

(11)

j= l

}

}

I A CA) - A CB)I 2.

There is another way of proving Proposition 2 that Lowner would have known. In 1923, Issai Schur, the ad­ viser for Wielandt's Ph.D. thesis at Berlin, proved a very in­ teresting relation between the diagonal of a Hermitian ma­ trix and its eigenvalues. This says that if d = (d1, . . . , d,J and A = (A1 , . . . , A,J are, respectively, the diagonal en­ tries and the eigenvalues of a Hermitian matrix A, then d is majorised by A. This, by definition, means that k k '\' d.1l :::; L '\' A .Jl ' for 1 :5 k :5 n (12) L j= l

f= l

and

Proof If A and B are commuting Hermitian matrices, this reduces to (8). The general case can be reduced to this special one as follows. Let U(n) be the set of all

(AKXo - AXoK)

tr

·

To see this, first note that the general case can be reduced to the special case n = 2. This amounts to showing that whenever x1 ::::: x2 and Y1 ::::: Yz, then XtYI + XzYz ::::: XtYz + XzY1 . The latter inequality can be written as (x1 - x2) (y1 - y2) ::::: 0 and is obviously true. A matrix analogue of this inequality is given in the fol­ lowing proposition. If A is a Hermitian matrix we denote by Eig l (A) = (A iCA), . . . , A �(A)) the vector whose coor­ dinates are the eigenvalues of A arranged in decreasing or­ der. Similarly Eig i (A) = (A I (A), . . . , A �(A)) is the vector whose coordinates are the same numbers arranged in in­ creasing order. The bracket (x, y) stands for the usual scalar product IJ�1 XJYI

tr Ae 1KX0 e- tK

This is

for every skew-Hermitian matrix K. Expanding the expo­ nentials into series, this condition reduces to

·

(8)

t=O

I.

n

n

j= l

f= l

I df = I A f .

(13)

The notation d < A is used to express that all o f the rela­ tions ( 1 2) and (13) hold. Schur's theorem has been gener­ alized in various directions (see, e.g., the work of Kostant [K] and Atiyah [A]), and it provided a strong stimulus for the theory of majorisation [MO, p4]. A good part of this theory had been developed by the time Hardy, Littlewood, and P6lya wrote their famous book [HLP] in 1934, the same year as that of Lbwner's paper. The condition d < A is equivalent to the condition that the vec­ tor d is in the convex hull of the vectors A
}

tr

AB = I Aj (A)dtCB) = j= l

(Eig l (A),d(B))

© 2007 Springer Science-t Business Media, Inc., Volume 29, Number 3, 2007

43

where d(B) = (d1(B), . . . , dn(B)) is the diagonal of B. By Schur's theorem, this vector is in the convex set 0 whose vertices are Au(B). On this set the function f(w) IJ'= 1 A } CA) w1 is affine, and hence attains its maximum and min­ imum on vertices of 0. Now the inequalities (9) follow from =

(8).

The ideas occurring in this proof are extremely close to those in the paper of Hoffman and Wielandt. I now give their argument in a simpler version due to Ludwig Elsner. A matrix S is said to be doubly stochastic if its entries siJ are nonnegative, IJ'= 1 siJ 1 , and I�1 siJ = 1 . The set 0 consisting of n X n doubly stochastic matrices is convex. A famous theorem, attributed to Garrett Birkhoff [Bl, says that the vertices of n are the permutation matrices. Now let A and B be normal matrices, and choose unitary matrices U and V such that UAU* = D1 , and V BV* = Dz, where D1 and Dz are diagonal matrices whose diagonal entries are a1 , . . . , an, and {31 , . . . , f3 n, re­ spectively. Then =

IIA - El l} = II U*D1 U - V*Dz VII} = IID1W - WDzl \}, where W = UV* is another unitary matrix. The second (14)

equality in ( 1 4) is a consequence of the fact that the Frobe­ nius norm is unitarily invariant; i.e., that \\X1Y\\F = II T\\F. for all T, and all unitary X, Y. If the matrix W has entries wiJ, then the equality (14) can be expressed as

\ \A - B\ 1} =

I

i.j=l

I a; - f311z l wiJiz .

The matrix Cl wiJI2) is doubly stochastic, and the function f(S) = Ii,Jia; - {3112 siJ on the set 0 consisting of doubly sto­ chastic matrices is an affine function. So, the minimum of f is attained at one of the vertices of 0, and by Birkhoff's theorem this vertex is a permutation matrix P = (piJ). Thus

n

\\A - B\\} 2: I

i,j= l

I a ; - f3il2 PiJ·

If the matrix P corresponds to the permutation inequality says that

u,

then this

I lA - B\\J; 2: I Ia ; - f3u(i)l 2 . i= l

This is exactly the Hoffman-Wielandt inequality (7). Let me interject here that ideas very similar to these lead to a quick proof of Schur's theorem about the diagonal. Let A be a Hermitian matrix and let A UA U* be its spectral representation, where A is a diagonal matrix. If d and A are the vectors corresponding to the diagonals of A and A , respectively, then we have d = SA, where S is the matrix with entries siJ = l uiJI2 . This matrix is doubly stochastic. Hence, we have d < A. Now let us return to inequality (4) involving operator norms, the thing Wielandt and Hoffman wanted. Apart from Hermitians, dealt with by Weyl's result (1), there is another equally important subclass of normal matrices: the unitary matrices. Thirty years after [HW], R. Bhatia and C. Davis [BD) proved that the inequality (4) is true when A and B are uni­ tary. There were other papers a little earlier proving the in­ equality in special cases. One by this author [B1) showed that (4) is true when not only A and B but also A - B are =

44

THE MATHEMATICAL INTELLIGENCER

normal. The case of Hermitian A, B is included in this. V. S. Sunder [S] proved the inequality when A is Hermitian and B skew-Hermitian. In 1983 R. Bhatia, C. Davis, and A. Mcintosh [BDM] proved that there exists a number c such that for all normal matrices A and B (of any size n) we have

(15)

d(Eig

A,

Eig B) ::::; c

IIA - B\\.

A few years later R. Bhatia, C . Davis, and P . Koosis [BDK] showed that this number c is no bigger than 3. Thus it came to be believed, more strongly than before, that the inequality (4) is very likely true, in general, for normal A and B. Surprise: in 1992 ]. Holbrook [H] published an example of two 3 X 3 normal matrices A and B for which d(Eig A, Eig B) > 1 \A - B\\. (When n = 2, this is not possible.) Hol­ brook found his example by a directed computer search. As a sidelight, I should mention that a namesake of Wielandt, Helmut Wittmeyer [Wit], claimed that he had proved (4) for all normal A, B. For a proof he referred the reader to his Ph.D. thesis at the Technical University, Darm­ stadt, written in 1935, the same year as Wielandt's. There is no mention of this in Wielandt's papers, and so he must have been unaware of Wittmeyer's claim. Hoffman mentions, without any detail, that Wielandt had something "quite lovely, involving a path in matrix space. " An argument using paths in the space of normal matrices was discovered by this author [Bl]. This led to some new results and some new proofs. It also raises an intriguing problem in differential geometry. Let me explain these ideas. Though the inequality (4) fails to hold "globally," it is true "locally" in a small neighbourhood of a normal matrix A, even when B is not normal. More precisely, we have the following.

THEOREM 3 Let A be a normal matrix, and B any matrix such that \ A - B\\ is smaller than half the distance between each pair of distinct eigenvalues ofA. Tben d (Big A, Big B) :s

\\A - B\\.

l

Proof Let e = IIA - E l. First I show that any eigenvalue {3 of B is within distance 8 of some eigenvalue a1 of A. By applying a translation, we may assume that {3 0. If no eigenvalue of A is within a distance 8 of this, then A is in­ vertible. Since A is normal, we have IIA- 111 = 1/minl a1\ < 1/8. Hence =

I + A- 1(B - A) is invertible, and so is A (I + A- 1 (B - A)) = B. But then {3 = 0 could not have been an eigenvalue of B, and we have a contradiction.

This means that

Now let a1, . . . , ak be all the distinct eigenvalues of

A, and let D1 be the closed disk with centre a1 and radius 8 = \\A - B\\. By the hypothesis of the theorem, the disks D1, 1 :s j :s k, are disjoint. By what we have seen above, all

the eigenvalues of B lie in the union of these k disks. The rest of the proof consists of showing that if the eigenvalue ai has multiplicity m1, then the disk D1 contains exactly mi eigenvalues of B counted with their respective multiplici­ ties. (It is clear that this implies the theorem.) Let A(t) = (1 - t)A + tB, 0 :s t :s 1, be the straight line segment joining A and B. Then \ \A - A(t)\\ = t8, and so all

eigenvalues of A(t) also lie in the disks DJ- By a well-known continuity principle, as t moves from 0 to 1 the eigenval­ ues of A(t) trace continuous curves starting at the eigen­ values of A and ending at those of B. None of these curves can jump from one of the disks D1 to another. So if we start with mi such curves in Di, then we must end up with exactly as many. This proves the theorem. 0 You may recognize the reasoning in the second part of the proof above as an idea much used in complex analy­ sis around the Argument Principle. Can the local estimate of Theorem 3 be extended to a global one? Let N be the set of all normal matrices of a fixed size n. If A is in N, then so is tA for any real t. Thus N is a path-connected set. Let y(t) , 0 :::; t :::; 1 , be a contin­ uous curve in N, and let y(O) = A, y ( l ) B. We say y is a normal path joining A and B. The length of y with re­ 11 11 spect to the norm · is defined, as usual, by =

£11 · 11( y) = sup

m- 1

L k�o

li l l y C tk+ 1 ) - y( tk) ,

where the supremum is taken over all partitions of [0, 1] as 0 = to < t1 < · · · < tm = 1 . If this length is finite, y is said to be rectifiable. If y(t) is a piecewise C1 function, then £11·11( y)

=

f0 ll i(t)ll dt.

From Theorem 3 it is not difficult to obtain, using familiar ideas in differential geometry, the following.

THEOREM 4 Let A and B he normal matrices� and let y be

a rectifiable normal path joining A and B. Tben

( 1 6)

d(Eig A, Eig

B) :::; £1 1 ·11( y).

If we could find the length of the shortest normal path joining A and B, then ( 16) would give a good estimate for d (Eig A, Eig B) . The set N does not have 11 11 an easily tractable geometric structure, and the norm · is not Euclidean. So we are dealing here with non-Riemannian geometry (Finsler geometry) of a complicated set. Nevertheless, interesting in­ formation can be extracted from ( 1 6). In a variety of special cases Theorem 4 leads to the in­ equality (4). For example, this works when A and B lie in a "flat" part of N. By this I mean that the entire line seg­ ment y(t) = (1 - t)A + tB is in N. A small calculation shows that this is the case if and only if A, B, and A - B are nor­ mal-in particular, when A and B are Hermitian. Much more interesting is the fact that there are sets in N that are not affine but are "metrically t1at. " We say that a subset S of N is metrical�y flat if any two points A and B of S can he joined by a path y that lies entirely within II S and has length A - B ll . An interesting example is given by the following theorem.

THEOREM 5 Let S consist qf all n X n matrices of theform

zU where z is a complex number and U is a unitary matrix. Tben S is a metrically .flat subset C!fN .

Proof Any two elements o f S can be represented as r1 U1 , where r0 and r1 are nonnegative

Ao = t() Uo and A 1

real numbers. Choose an orthonormal basis in which the unitary matrix � U0 1 is diagonal:

U1 U() 1 = diag(e;e\ . . . , e;a,),

where

l i en :::; · · ·

:::;

j l e1 :::;

7T.

Let K = diag (ie1, . . . , ieJ. Then K is a skew-Hermitian matrix whose eigenvalues are in (- i7T, i7T] . We have

IIAo

- A1ll

ll

II li ll ro UoI - r1 � = ra! -j r1 IU1 U o 1 j ra - r1 exp(ie1) . = max ra - r1 exp( ie1) =

=

}

This last quantity is the length of the straight line joining j j the points ra and r1 exp(ie1) . q e1 < 7T, this line segment can be parametrised as r(t) exp(ite1) , 0 :::; t :::; 1 . The equa­ tion above can then be expressed as

II

{0 J[r(t) exp(itel)J ' j dt j l = f r'(t) + r(t) ie1 dt. 0

ll Ao - A 1 =

Let A(t) = r(t)e1KU0 , 0 :::; t :::; 1 . This is a smooth curve in S joining Ao and A1 , and its length is

r0 IIA'(t)ll dt = f0 ll r'(t)e1KUo + r(t)Ke1KUoll dt 11 II = { r'(t) I + r(t)K dt, 1

0

since e'KU0 is unitary. But ll ! j l j II r'(t)I + r(t)K max r'(t) + ir(t) Bi. = r' (t) + ir(t)e1 . =

}

The last three equations show that the path A(t) joining Ao IIAo ll and A1 - A1 . l j has length is not needed. In this case II If e1 = 7T,l the argument above j Ao - A1 l l = ro - r1 exp(iel ) = ra + r1. This is the length of the piecewise linear path joining Ao to 0 and then to A1. 0 Theorems 4 and 5 together show that the inequality ( 4) is true when A and B are scalar multiples of unitaries. The­ orem 4, in a more general form and with a different proof, was given in [Bll. Theorem 5 was first proved in [BH] . When n = 2, the entire set N is metrically t1at. This can be seen as follows. Let A and B be 2 X 2 normal matrices. The eigenvalues of A and those of B lie either on two par­ allel lines, or on two concentric circles. In the first case, we may assume that the lines are parallel to the real axis. Then the skew-Hermitian part of A - B is a scalar, and hence A - B is normal. We have seen that in this case the line segment joining A and B lies in N. In the second case, if a is the common centre of the two circles, then A and B are in the set a! + S, which is metrically t1at. Since the inequality ( 4) is not always true for 3 X 3 nor­ mal matrices, the set N must not be metrically t1at when n � 3. I have pointed out some metrically flat subsets of N. There may well be others. An intriguing problem, that seems hard, is that of find­ ing a "curvature constant" for the set N. For each n, let k(n) be the smallest number with the following property. Given any two n X n normal matrices A and B there exists a nor­ mal path y joining them such that

=

© 2007 Springer Science +Business Media, Inc., Volume 29, Number 3, 2007

45

We know that k(2) 1 , and k(3) > 1 . Is the sequence k( n) bounded? If so, is the supremum of k( n) some familiar number like, say, n/2? It will be appropriate to end with a related story in which Wielandt played an important role. In 1950, V. B. Lidskii [Li] published a short note in which he gave a matrix­ theoretic proof of a theorem that arose in the work of F. Berezin and I. M . Gel'fand on Lie groups. Lidskii's theorem says that if and are Hermitian matrices, then the vec­ tor Eig 1 Eig J, lies in the convex hull of the vec­ tors obtained by permuting the coordinates of Eig(A In another formulation, it says that for all 0 :::; k :::; n, and < ik :::; n, we have indices 1 :::; i1 < i2 < =

(A)

A

-

B (B) ·

07)

B) .

·

·

k

k

�1

�1

k

I AtCA + B) :::; I A f;CA) + I A J(B). .

�I

Wielandt [Wie2] discovered a remarkable maximum princi­ ple from which he derived these inequalities as he "did not succeed in completing the interesting sketch of a proof given by Lidskii. " The inequalities (1) and ( 10) of Weyl and Lbwner are subsumed in (a corollary of) Lidskii's theorem. A norm 1 1 1 · 1 11 on matrices is said to be unitarily invariant if U V = for all unitary matrices U and V The operator norm (1) and the Frobenius norm (5) have this property. It follows from Lidskii's theorem that if and are Hermitian matrices, then

Il A I l I IAI I

A

t411 · 111(Eig

( 18)

B

A, Eig B) :::; l i lA - �I I

for every unitarily invariant norm. Fascinated by the inequalities (17), several mathemati­ cians discovered more such relations. This led to a con­ jecture by Alfred Horn in 1 962 specifying all possible lin­ ear inequalities between eigenvalues of Hermitian matrices B. Horn's conjecture was proved towards the and end of the twentieth century by Alexander Klyachko, and Alan Knutson and Terence Tao. In the intervening years it was realised that the problem has ramifications across sev­ eral major areas of mathematics. The interested reader can find more about this from the expository articles [B3l, [Fl, [KT].

A, B,

A+

[B2] R. Bhatia, Matrix Analysis, Springer-Verlag, 1 997. [B3] R. Bhatia, Linear algebra to quantum cohomology: the story of Al­ fred Horn's inequalities, Amer. Math. Monthly 1 08 (2001 ) , 289-31 8. [BD] R . Bhatia and C. Davis, A bound for the spectral variation of a uni­ tary operator, Lin. Multi/in. Algebra 1 5 (1 984), 7 1 -76. [BDK] R. Bhatia, C. Davis, and P. Koosis, An extremal problem in Fourier analysis with applications to operator theory, J. Funct. Anal. 82 (1 989), 1 38-1 50.

[BDM] R. Bhatia, C. Davis, and A. Mcintosh, Perturbation of spectral subspaces and solution of linear operator equations, Lin. A/g. Appl. 52/53 ( 1 983), 45-67.

[BH] R . Bhatia and J. A. R. Holbrook, Short normal paths and spectral variation, Proc. Amer. Math. Soc. 94 (1 985), 377-382. [B] G. Birkhoff, Tres observaciones sabre el algebra lineal, Univ. Nac. Ucuman Rev. Ser. A, 5 (1 946), 1 47-1 5 1 .

[F] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schu­ bert calculus, Bull. Amer. Math. Soc. 37 (2000), 209-249. [HLP] G. H. Hardy, J. E. Littlewood, and G. P61ya, Inequalities, Cam­ bridge University Press, 1 934. [HW] A. J. Hoffman and H. W. Wieland!, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1 953), 37-39. [H] J. Holbrook, Spectral variation of normal matrices, Lin. A/g. Appl. 1 74 (1 992), 1 31 -1 44.

[KT] A. Knutson and T. Tao, Honeycombs and sums of Hermitian ma­ trices, Notices Amer. Math. Soc. 48 (2001 ) , 1 75-186. [K] B. Kostant, On convexity, the Weyl group and the lwasawa de­ composition, Ann. Sci. E. N. S. 6 (1 973), 4 1 3-455. [Li] V. B. Lidskii, The proper values of the sum and product of sym­ metric matrices, Dokl. Akad. Nauk SSSR 75 (1 950), 769-772. [Lo] K. Lowner, O ber monotone Matrixfunctionen, Math. Z. 38 (1 934), 1 77-2 1 6.

[MO] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorisation and Its Applications, Academic Press, 1 979.

[S] V. S. Sunder, Distance between normal operators, Proc. Amer. Math. Soc. 84 (1 982), 483-484.

[We] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte lin­ earer partieller Differentialgleichungen, Math. Ann. 71 (1 9 1 2), 441 479. [Wie1 ] H. Wielandt, Mathematical Works, B. Huppert and H. Schneider eds., W. de Gruyter, 1 996. [Wie2] H. W. Wielandt, An extremum property of sums of eigenvalues,

REFERENCES

[A] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 1 4 (1 982), 1 -1 5.

[B1 ] R. Bhatia, Analysis of spectral variation and some inequalities, Trans. Amer. Math. Soc. 272 (1 982), 323-332.

46

THE MATHEMATICAL INTELLIGENCER

Proc. Amer. Math. Soc. 6 (1 955), 1 06-1 1 0 .

[Will J . H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford Uni­ versity Press, 1 965. [Wit] H. Wittmeyer, Einfluss der Anderung einer Matrix . . . , Zeit. Angew. Math. Mech. 1 6 (1 936), 287-300.

i&ftii . i§i.@ih£11$.1 .. '1.111.Jhi¥J

Mathematical T ravelers GERALD l. ALEXANDERSON, MONIKA CARADONNA, AND LEONARD f. KLOSINSKI

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just

as

unrestricted.

We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

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

M arj orie Senec h a l , Editor

athematicians attending profes­ sional meetings or on holiday, in our experience, often get to­ gether to sign postcards to send to col­ leagues who are absent. We have cir­ culated such postcards ourselves, often at the urging of friends. A few inquiries made to colleagues in other disciplines indicate, however, that this practice may be considerably less common among other groups. In mathematics the cus­ tom continues to this day. It is by no means a recent practice, as became clear when we were looking at a cache of postcards collected by George P6lya over many years, some of which he in­ herited from Adolf Hurwitz when P6lya edited his collected papers after Hur­ witz's death in 1919. Most of these postcards have little or no mathematical or historical signifi­ cance, and most are of the "having­ wonderful-time-wish-you-were-here" variety. Some of the early ones do not really fall into this class, being ti·om a single sender and concerning a variety of topics. We could not resist, however, including some of these exceptions here because those writing them are so il­ lustrious-Weierstrass, Kronecker, Haus­ dorff, and Hadamard, for example. Sometimes the translations are not so easy to provide: it is clear that not all important mathematicians got A's in penmanship. To be kind, we shall as­ sume that some of these postcards were written on a train or on a boat ride dur­ ing rough weather. In the earliest (Figure 1), Leopold Kronecker writes from Berlin on May 14, 1882, to Adolf Hurwitz, in Gottin­ gen: "Esteemed Doctor, I shall not de­ lay a single moment sending you my best wishes on the occasion of the suc­ cessful completion of the matter of your Habilitation, a matter in which I played a heartfelt part. You will receive the reprints of my papers one of these days. Sincerely, L Kronecker"

[

Next (Figure 2) Karl Weierstrass writes from Berlin in June 19, 1883, again to Hurwitz, when the latter was still in Gbttingen as a Privatdozent: "Most Honored Doctor, I certainly do not object if, in your forthcoming pub­ lication, you wish to make reference to a proof I presented in the Seminar, even Jess so, since in the near future, my pa­ per regarding Lindemann's research will be published in the Sitzungsherichte of our Academy. With the friendliest greet­ ings, Weierstrass." Alfred Pringsheim, whose work was mainly in complex analysis, but who is probably better known as a philan­ thropist and the father-in-law of the novelist Thomas Mann, wrote to Hur­ witz (Figure 3): "Dear Colleague, Many thanks for your friendly information. But permit me a question. You state that integrability of f(x) and g(x) for - 7r s x s 7T is sufficient to support the result in question. Is this to mean that from the beginning you restrict the re­ sult to functions f(x) and g(x) remain­ ing finite? If NOT, then one would have to require at least the integrability of (a) j�x)g(x) , but it seems to me, in addi­ tion, the integrability of (b) ( f(x))2 and (g(x))2. The latter requirement ((b)) is exactly the one I am interested in. I am currently attempting an extension of the convergence considerations based on Harnack's theorem (cf. Munch. Berichte 30 (1900), p. 6 1 ff. ) in my pa­ per "On the behavior of power series on the circle of convergence. " In my re­ search I came across condition (b), which is not very pleasant at all, and I really wish I could eliminate it. Would this really be possible with your result? Please excuse my taking such advan­ tage of your time. Best greetings. Sin­ cerely yours, Alfred Pringsheim . " And all of this is on one postcard! In Figure 4, we see the first of the "picture postcards, " showing the Chateau de Chillon on Lac Leman in

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

47

Po

Figure I.

TK

R

E.

Card from Leopold Kronecker to Adolf Hurwitz

Switzerland and the Dents du Midi in the background. Hermann Minkowski is writing to Hurwitz from the Hotel Bristol in Territet (a small town just up the road from Chillon in the direction of Montreux). Writing on April 13, 1 905, he says: "Dear Friend, For a few days now we have been here and like it ex­ tremely well. Even though we are not sovereigns, I send you heartfelt greet­ ings from your homeland. I hope to be able to be in Zurich for one day at the end of next week and to see you and your loved ones in good health and happy. Best wishes from my wife. Sin­ cerely, H. Minkowski" A postcard (Figure 5) showing a car­ toon of a drinking party is signed by Carl Runge, Minkowski, and David Hilbert, mailed from Gottingen on Feb­ ruary 15, 1907, to Hurwitz, by now at the Polytechnicum (later to be called the Eidgenossische Technische Hochschule

(ETH) in Zurich). It says: "Dear Hur­ witz, Gathered at the Christmas drink­ ing party of the Mathematische Verein [Mathematical Society), we send you friendly greetings. " A picture (in color!) o f the Bracken­ hahn (a mountain railway) (Figure 6) is accompanied by a rhyming verse by Hilbert-it rhymes in German-that says "Here we sit on the Bracken/The sun beckons/Now we cool our throats which are dry/And Mrs. M. changes her socks. " Further, on the left side, are greetings from E. Buschke and Kathe Hilbert, and running up the other side, "Best wishes" from Hermann and Susie Minkowski. Across the top in what ap­ pears to be Minkowski's hand it reads: "I was very happy to receive your let­ ter . . . and will answer it soon. We are constantly on day trips." All of the above are written in Ger­ man, but in a postcard (Figure 7) from

(

Figure 2.

48

Card from Karl Weierstrass to Adolf Hurwitz

THE MATHEMATICAL INTELLIGENCER

Bellagio on the Lago di Como, proba­ bly written by Vito Volterra, we have a message to Hurwitz in French, saying "Our thoughts come to you from the shore of Lake Como where a happy en­ counter has united us all. The memory of you is even more beautiful than the blue dome of the sky gleaming in the mid-day sun. Vito Volterra, David Hilbert, Kathe Hilbert." From another drinking party in Got­ tingen, Hurwitz received a picture post­ card (Figure 8) showing Bismarck's house and saying "Greetings from the Christmas drinking party of the Mathe­ matische Verein, Hilbert, [Felix) Klein, Minkowski, [Constantin] Caratheodory" On a postcard (Figure 9) showing a view of Heidelberg from the Philoso­ pher's Lane, Hilbert wrote in 1905 to Hurwitz: "Dear Friend, we are stationed here at the Schlosshotel but are not tak­ ing our meals here because we are on

, .. , .

•

1.'.-if

•""

·�

- -,.- -1 1-· (I

.,

�� 4'

/!.

f. lo

.. .. .

,.4.. J-

� /.

I

"'"'

1-f.

_ ....._

A.. ,/, � ,

�•

Figure 3. Card from Alfred Pringsheim to Adolf Hurwitz

CARTE P O S TALE

J

l F igure 4. Chateau de Chillon on Lac Leman: Hermann Minkowski t o Adolf Hurwitz

Figure 5. Cartoon showing drinking party: Runge, Minkowski, and Hilbert to Hurwitz

© 2007 Springer Science + Business Media, Inc., Volume 29. Number 3, 2007

49

rra r r

PT

Figure 6. Brockenbahn: David and Kathe Hilbert, Buschke, Hermann and Susie Minkowski to Hurwitz

•

Figure 7. Bellagio on the Lago di Como: Volterra, David and Kathe Hilbert to Hurwitz

Figure 8. Bismarck's house, Gottingen: Hilbert, Klein, Minkowski, and Caratheodory to Hurwitz

50

THE MATHEMATICAL INTELLIGENCER

Figure 9.

Heidelberg from the Philosopher's Lane: Hilbert to Hurwitz

the road almost the whole day. Today we made a lunch date with Weber on the . . . Yesterday he came looking for us at the hotel-in vain. Hopefully, you did not only get home safely hut are also experiencing the refreshing effect of your trip to Baden-Baden. We often reminisce about the stimulating days we spent with you. Best regards, also from my wife, Hilbert" Again, this time from Weesen, a summer resort at the western end of the Wallensee, not far from Zurich, Hilbert wrote to Hurwitz (Figure 10): "Dear Friend, We sit here at the Lake in the most brilliant sunshine having our morn-

Figure I 0.

ing coffee after a full 1 0 hours of rest last night. It was really wonderful to chat with you for several hours but we did­ n't get to our old reminiscences of our times in KC'migsherg and its people. We hope we can still catch up on that. If this should work for you, we will choose our location so that on our return trip you can visit (for example, just before Stuttgart). Sincerely, D. Hilbert" From the Hotel Beau Rivage in Lo­ carno, Felix Hausdorff wrote on March 24, 1926, to P6lya at the Polytechnicum in Zurich ( Figure 1 1 ): "Dear Colleague, My wife and I are staying here for the legendary Spring of Ticino. However, at

the moment it leaves its warming power to the central heating system. Are you perhaps coming to this region as well? I would enjoy very much hearing some amusing mathematics (after two years' worth of work on the new edition of Set Theory). The problem collection hy you and Szeg6 is excellent! Do you know this problem? P,lx) [the] Legendre polynomial, xk its zeros: :":: 1/[P'nCxk)F = 2/3 , n :=::: 2. Best wishes to you and your wife. Sincerely, F. Hausdorff'' This postcard shows in color the pilgrimage church Madonna del Sasso, which sits high above the town on a wooded, rocky hill with views over

Weesen: Hilbert to Hurwitz

© 2007 Springer Science+Business Media, Inc., Volume 29, Number 3 , 2007

51

Figure I I .

Hotel Beau Rivage, Locarno: Hausdorff to P6lya

the Lago Maggiore. Both sender and re­ cipient would know the church for its famous Bramantino altarpiece. We could conclude from these cards that mathematicians of the period knew how to travel in style not only to loca­ tions with good views and good hik­ ing-Italy's Lake Country, Switzerland, Baden Baden, and the Rheinland-but when they went there they stayed at good hotels. The Hotel Beau Rivage in Locarno, for example, got the highest recommendation for that city in a Baedeker Guide of the period. And from Paris we have a note of

Figure 1 2.

52

thanks (Figure 1 2) showing the Jardin du Luxembourg in Paris, from Jacques Hadamard, dated in the 1930s. In the upper left-hand corner he thanks P6lya for "the information regarding the Tay­ lor series. I 'll send it to [Szolem] Man­ delbrojt whom it concerns in particu­ lar." The card was sent not to the P6lyas' flat in Zurich but to their chalet in En­ gelberg in the Swiss Alps. From France we move on to Eng­ land to a card (Figure 1 3) dated March 1 1 , 1954, and addressed to P6lya in Palo Alto, extending greetings from L. J. Mordell, Carl [Ludwig] Siegel, Mabel

Jardin du Luxembourg, Paris: Hadamard to P6lya

THE MATHEMATICAL INTELLIGENCER

Mordell, Margie Wigner Dirac, and P. A. M. Dirac. It shows the Main Entrance to St. John's College, Cambridge. Slightly earlier that year (February 13, 1954), there is a card (Figure 1 4) say­ ing "Herzlichste Grusse aus Gottingen" showing the City Hall and the Johan­ niskirche in Gottingen. The card is signed by Siegel and Harold Davenport. It is addressed to the P6lyas in Zurich, at the residential hotel where they of­ ten stayed after World War II. L. ]. Mordell obviously arranged to have another card (Figure 1 5) sent to P6lya, this time from Budapest, and

Figure 1 3. St. John's College, Cambridge: Louis and Mabel Mordell, Siegel, P .A.M. and Margit Dirac to P61ya

s� . ZO IC

Figure 1 4. City Hall and johaniskirche , G6ttingen: Siegel and Davenport to P6lya

Figure I S . Chain Bridge, Budapest: Mordell, F. Riesz, Turan, Ohlath, Erd6s, Foldes, and Peter to P6lya

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

53

Figure 1 6. Fisherman's Bastion, Budapest:

Szeg6, Riesz, Fejer, Obl ath, Turan, Renyi to P6lya

ES means ...

Figure 1 7. Ballantine Ale coaster: Eilenberg, Leibler, Martin, Montgomery,

though some signatures are not legible, one can easily pick out, in addition to Mordell's, those of Frigyes Riesz, Pal Turin, Richard Oblath, Pal Erdos, Istvan Foldes, and R6sza Peter. A second post­ card (Figure 16) from Budapest is signed by Gabor and Veronica Szego, Frigyes Riesz, Lip6t Fejer, Richard Oblath, Pal Turan, and Alfred Renyi, among others. The first shows the Chain Bridge, the second the Fisherman's Bas­ tion, both in Budapest. A variant on the postcard theme is a Ballantine ale cardboard coaster (Figure 17), probably dating from the 1 950s, which some mathematicians

Erdos, Brunings r o Virginia Halmos

sent home with Paul Halmos from a mathematical meeting, to express greetings to Paul's wife Virginia. They wrote in a spiral on the reverse of the coaster: "Dear Ginger, I hope to see a lot of you during the next year. Why don't you get a big apartment and keep a guest room for me . Sammy [Eilenbergl. Wish you were here. I have lost all of the golf balls . Dick [Leiblerl. Sorry not to have gotten to see you to say good-bye. If you want to teach Oil* in the College see Bill O'Meara. Dick Martin. I miss you a lot and so do Lucy, Ray, m. See you soon. Deane [Montgomery] . Viszontlatsasra,

"One of Robert Hutchins's courses at the University of Chicago: "Organization. Interpretation and Integration".

54

THE MATHEMATICAL INTELLIGENCER

miszelyer iidvoslettel [Au revoir, with heartfelt greetings]. E. P. [Erdos Paull Kind regards. Have a good time. Josephine Brunings]" In 2002 the Mathematical Sciences Research Institute (MSRI) in Berkeley sent out a card with season's greetings showing printed signatures of David Eisenbud, Robert Osserman, and Robert Megginson, among many oth­ ers. On the reverse was a watercolor showing a view of the Institute from above, with San Francisco Bay in the background and, in the far distance, the Golden Gate Bridge. It was a fine math­ ematical card.

I.�Mj.t§j:@hi£11#fh§4fii.'l,f§!'d

P it Your Wits aga inst Young Minds ! JAMES TANTON

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford U niversity, Department of Mathematics, Bldg. 380, Stanford , CA 94305-2 125, USA e-mai l: [email protected] .edu

M i chael Kleber and Ravi Vaki l , Ed itors

l

am not the author of this article! True, I may be the person writing it, but the approaches and flashes of insight presented here belong to the young stu­ dents (ages 8 to 1 8) with whom I have worked over the past few years. Mathematics is really an exercise of creative play, flexibility of the mind, and adaptability, and who is better suited for such play than the young and in­ tellectually unadulterated? So I invite you to match your wits with the par­ ticipants of the Boston Math Circle (www . themathcircle.org) and with the middle-school and high-school students attending the St. Mark's Institute of Mathematics ( www . stmarksschool.org). I offer here a small sample of favorite math problems these students have worked on and have made their own. Although the results themselves might not be new, their approaches to them are novel and, in many cases, represent significant simplifications of the stan­ dard published proofs! Try these problems yourself and see what innovative approaches you can develop. Attempt to use nothing more than the tools of high-school mathe­ matics. Free your mind from the clutter of high-level ideas, and see what power that can bring' 1. Dots on a Circle Figure 1 is a familiar exercise warning against the temptation of inductive rea­ soning. Of course the answer to the question posed by Figure 1 is not 32, which is a surprise. (What is the an­ swer?) But here is the joy of youngsters' creative thinking and the real challenge for you: Who said the lines have to be straight? Consider a diagram of (a finite num­ ber oO paths within a circle connecting boundary points and intersecting at a fi­ nite number of points within the circle (Figure 2). Develop a general theory that successfully counts the number of re­ gions obtained in terms of the number of paths and the number of intersection points present. Permit concurrent paths.

Use your theory to find a formula for the sequence of numbers in the classic straight-line case presented by Figure 1 . (Comment: Don't be distracted by Euler's formula for planar graphs. You'll have a much easier time pretending you don't know it!) 2. Theon's Ladder Theon of Smyrna (ca. 140 C.E.) knew a

that if b is a rational approximation to

Vz with a and b positive integers, then ·

a + 2b a+ b

· h Iess th an -- 1s a better one (w1t

half the error). Starting with initial value

l, iteration of this transformation gives 1 the famous sequence of approximations known as Theon's Ladder: 1 1

3 2

7 5

- � - � - �

�

41 29

17 12 �

99 70

239 169

� -- � . . .

Many patterns lie hidden in this se­ quence-the denominators alternate in parity, the numerators squared differ from twice their corresponding denom­ inators squared by ± 1 , for instance­ but a truly surprising pattern lies within every odd term of the sequence. Notice: 1 1

0+ 1 and 02 + 12 = 1 2 1

7 5

3+4 and Y + 42 = 5 2 5

41 29 239 169

20 + 2 1 and 202 + 2 1 2 = 292 29 1 19 + 1 20 and 169 1 192 + 1 202 = 1692

1. Prove that every odd term of Theon's encodes ladder a Pythagorean triple with legs that dif­ fer by one. Does every such Pythagorean triple appear? 2. Surely there is an equally surprising pattern lurking within the even terms. What?

© 2007 Springer Science +Business Med1a, Inc., Volume 29, Number 3, 2007

55

a') '·

1 piece

Z pieces

4 pieces

8 pieces

16 pieces

??

What's the next number? (Assume that the dots are appropriately spaced so that no three line-segments are concurrent within the circle.) Figure I.

lines

=

3

intersections

pieces

Figure 2.

=

9

:

lines

5

:

ished. Tie the three loose ends to a wooden spoon as three separate knots. Show that, no matter what braid you create (with the middle strand tied to the middle position), it is possible to completely detangle the braid simply by maneuvering the spoon back and forth between the strands. (Try it!) This shows that your braid produced with free ends could just as well have been produced with ends fixed in place! What can be said about a braid if the yellow strand is tied, instead, at one end of the spoon? 4

intersections

pieces

How many intersections?

=

9

=

?

3. Quickie: Sums of Two Squares Pythagoras's theorem shows that a number N can be writ­ ten as a sum of two squares if, and only if, there is a lat­ tice square of area N. •

•

•

•

•

•

•

•

5. Integer Triangles With 15 matchsticks, it is possible to make seven different tri­ angles (of positive area) using a whole number of matchsticks per side: 7-7-1, 7-6-2, 7-5-3, 7-4-4, 6-6-3, 6-5-4, and 5-5-5. It is a surprise to discover that if one adds an extra matchstick to the mix the count of permissible triangles decreases: there are only five integer triangles of perimeter 16! What's going on? Develop a general formula for the number of integer tri­ angles of perimeter n. (And avoid high-powered mathe­ matics as you do it!) 6. Counting Square Factors Let d( n) denote the number of square factors a number n possesses (for example, d(20) 2 and d(36) 4), and let d(l) + d(2) + + d(n) . . Ave(n) . Fmd ltm,._.coAve(n) , ==

•

==

·

·

==

·

n

the average number of square factors for any number.

•

<><><> •

Answers

• I) :

Figure 3.

�. )2

The math equation.

The set of numbers that can be so represented begins: 1 , 2, 4, 5 , 8, 9, 10, 13, 16, 17, 18, 20, . . . Use geometry to show that this set is closed under multiplication. That is, show that the product of any two numbers in this list also ap­ pears in the list. (Avoid algebra at all costs!) 4. Braids with No Free Ends Take three strings, two colored red, say, and one yellow, and tie them to the back of a chair so that the yellow strand lies in the middle position. Now braid the three strands in any manner you choose. That is, cross adjacent strands over or under each other in turn in any organized or disorga­ nized fashion of your choice. Make sure that the yellow strand is returned to the middle position when you are fin-

56

THE MATHEMATICAL INTELLIGENCER

1. A joy for young students is to count and to look for pat­ terns. When a group of 8-10-year-olds were given this chal­ lenge at the Boston Math Circle in 200 1 , they decided to count everything: the number of circles C (always 1 !), the number of dots along the boundary, the number of paths drawn L (lines), the number of intersections I, and the number of re­ gions P (pieces). It soon appeared that C + L + I P and "CLIP theory, " as they called it, was born. They noted that the theory was true for very simple diagrams, and seemed to remain true if an extra path is added to a diagram. They reasoned as follows: As soon as this line intersects a pre-existing path, two things occur: the count of intersections increases by one and a region is split in two, increasing the count of pieces by one. The formula C + L + I P remains balanced. This equation also remains valid when the path eventually re­ turns to the circumference of the circle: the path is com­ pleted thereby increasing the number of lines by one, which is balanced by the fact that a final region is split in two. ==

==

In 2005, ninth-grade geometry students at St. Mark's School developed the same argument and noted further that this line of reasoning remains valid even if paths self-in­ tersect and pass through pre-existing intersection points (counting an intersection with multiplicity k if k paths lie atop a first path). These students also noted that, in the straight-line case, each line determines, and is determined by, a choice of two boundary dots and each intersection determines, and is de­ termined by, a choice of four boundary dots. Thus, if there are n dots along the circumference of the circle,

and so

Theon's ladder yield numbers that are both triangular and square! For example:

3 2

-

17 12 99

7

0

(3-1 2) -- , 2 2

�

�

�

=

. ( 1 , 1) and � = 51 = 1 1s square

. and tnanguI ar

( 17 - 1 1 2 ) --- , 2 2

= (8,6)

( 99

= (49 , 35) and

1 70 ) ,2

--

2

.

and Ti3 = 56 = 36 1s square . and tnangu I ar . 49 = 535 = 1 225 1s . square and tnangu 1ar

CLAIM. Theon's method produces every Pythagorean triple whose legs differ by 1 and every positive integer that is both square and triangular. PROOF. We will show that every reduced fraction

This corresponds to the sum of the entries for the first few rows of Pascal's triangle,

P=

(n

� 1 ) + (n � 1 ) + (n � 1) + (n � 1) + (n � 1}

explaining the initial appearance of the powers of two.

FURTIIER. What is the connection between CLIP-theory and

Euler's graph-theoretic formula?

Let a1 = 1 and � = 1 and set an+l = a, + 2hn and bn + l = a, + bn so that the nth term of Theon's ladder is

2.

given by

a,

. Note that the entries an are always odd. bn One checks that an+ l2 - 2 bn+ J2 = - (a,? - 2bn2), and since a12 - 2 h1 2 = - 1 , we have: a,/ - 2b,,2

=

( - 1) n

(Incidentally, it is now clear that

( a )2 " does indeed ap­ bn

proach the value 2 as n increases.) For n odd, the entries of Theon's ladder correspond to integer solutions of the equation x2 - 2y2 = - 1 with x an odd number, or, equivalently, to

(x

; 1y

+

(x

; 1y

=

y2, thus yielding Pythagorean triples whose legs differ by 1 . For n even, the entries correspond to integer solutions of x2 - 2y2 = 1 , or, equivalently, to the equation T�_1 = Sv, 2 n(n + 1) 2 where Tn = ---- is the nth triangular number and 2

Sn = n2 is the nth square number. Thus the even terms of

0 �:

.

�

.

�··

.

. .

..

.

.

10

:

.

.

.

.

::t: 1

� with

appears in Theon's

ladder. First note that if !!:_ is such a fraction with b = 1 ,

b

1

then it follows that a = 1 also, and so we have the term -, 1 the start of the ladder. Suppose instead that

a2 - 2 b2 =

::t: 1

� is a reduced fraction satisfying

with b > 1 . If this is to be in Theon's ladder,

2b - a is a- b the appropriate entry. Moreover, algebra shows that 2 b - a is positive, as is a - h, and that a - b is a denominator smaller than b. (For example, to establish that 2 b > a, square both sides and make use of the relation a2 - 2 b2 = 2:: 1 .)

what must the term before it be? One checks that

Repeated use of this backwards approach must eventually produce an expression with denominator equal to 1 . As this entry belongs to Theon's ladder, so too then must the expression

a --;;·

0

FURTHER. Start Theon 's ladder instead with the fraction 5 + 12 17

= - . Show that every second term corre�ponds to 13 13 a Pythagorean triple with legs that differ by 7. (Do we get every such triple?) W'hat do the even terms of the sequence give? Can we generate all Pythagorean triples via Theon 's method? W'hat are the appropriate "seed" terms? W'hat do we obtain if we run Theon 's ladder backwards, infinitelyJar to the left? What can be garneredfrom the equa­ tions x2 - 3.l = :t 1 ? (Approximations to the square root of 3? Variants of Pythagorean triples?)

---

Of course a tremendous wealth of information lies hid­ den in Fell's equations and its simple variants. Students of the Boston Math Circle have spent many an hour delving into its riches. 3.

Is Figure 4 enough? Have we, in fact, established al-Khazin's algebraic identity

(a2 + h2)(c2 + d2) = (ac + bdi + (ad - bc)2?

FURTIIER. J.fN and kN are both sums of two squares, must

.

.

Figure 4. Multiplying areas.

a >0 and b > 0 satisfying a2 - 2 b2 =

k he too?

A1·ea 50

4. One can read a braid from top down as a sequence of crossings. Let L denote the crossing of a left strand over or under the middle strand, and R the crossing of the right two strands.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3 , 2007

57

LLRLR

Observe that it does not matter if an L is an under or an over crossing. Quick experimentation shows that one can convert any L under-crossing to an over-crossing, and vice versa, by pushing the wooden spoon between the right two strands just below the crossing. (Try it!) The same is true for any R crossing. Thus any braid is encoded in a well­ defined manner as a string of Ls and Rs.

Note, moreover, that a braid possessing two consecutive L-crossings is equivalent to a braid with those two crossings removed: convert one to an under-crossing and the other to an over-crossing. Similarly, we can remove any two consecu­ tive Rs from a braid. Thus, any braid with fixed ends is equiv­ alent to a braid encoded by an alternating string of Ls and Rs.

and to an L if it was an R. (For example, the code LRLL­ RLRLL becomes LLLRRRRRL .) Count the number ofLs in the modified code and, from it, subtract the number of Rs. If this difference is a mul­ tiple of three, then the original braid can be untangled. Ifthe difference is congruent to 1 modulo 3, then the braid is equivalent to a single L, and equivalent to R if it is con­ gruent to - 1 .

This provides a wonderfully swift technique for analyzing three-braids!

FuRTHER. Take a rectangularpiece offelt, cut in it two slits,

and make the braid RLRLRL with fixed ends. Notice that, with the appropriate under and over crossings, the strands of this braid can be made 'Jlat" (that is, there are no internal twists in any of the individual strands).

Classify all flat three-braids with fixed ends that can be so produced. (In 2005, high-school students attending the St. Mark's Institute of Mathematics research group made sig­ nificant progress with this tough question.) 5. (My thanks to Elizabeth Synge, seventh-grade home­ schooler, for her help in writing the details of this proof.) Let T(n) denote the number of triangles with integer side­ lengths ("integer triangles") of perimeter n. The first 20 val­ ues of T(n) are

0, 0, 1 , 0, 1 , 1 , 2 , 1 , 3 , 2 , 4, 3 , 5 , 4,

Also, if a braid ends in either sequence LRL or RLR, then a 180° rotation of the spoon deletes the sequence from the

braid.

=

Thus every braid with fixed ends reduces to one of five possibilities: LR, RL, R, L, or the "empty braid," the untan­ gled state. But the braid LR is equivalent to just R (via LR = RRLR = R), and the braid RL to L. Thus any braid reduces to either R, L, or the untangled state. Neither of the first two options has yellow strand returning to the middle position. Thus, the only permissible option for a braid with the mid­ dle strand ending in the middle position is for the braid to be equivalent to the untangled state! High-school students from 2005 St. Mark's Institute of Mathematics went further and established:

Given a string ofLs and Rsfor the code ofa braid, change every second entry of the code-to an R if it was an L,

58

One striking pattern of note is that the values of the se­ quence seem to repeat after a shift of three places. Specif­ ically:

CONJECTURE 1. T(2n) LRL

THE MATHEMATICAL INTELLIGENCER

7, 5, 8, 7, 10, 8

=

T(2 n - 3) for n > 1 .

Thus the sequence {T } appears to be two intertwined copies of the sequence 0, 0, 1 , 1 , 2, 3, 4, 5, 7, 8, . . . If this conjecture is indeed true, we need only focus on this se­ quence of even terms in our analysis. We will need the following result: LEMMA 2. Given a positive integer n, three positive integers a, b, and c with a 2: b 2: c and a + b + c = n are the side­

lengths of an integer triangle ofperimeter n if, and only if, a is strictly less than half of n. This quickly follows from the triangle inequality.

CoROllARY 3. No integer triangle of even perimeter pos­

sesses a side of length 1 .

We can now establish the conjecture:

PRooF OF CONJECI1JRE 1. Let (a, b, c), with a 2: b 2: c, be

a triple of integers representing the side-lengths of a triangle of perimeter 2 n. (Notice that c > 1 .) Then (a - 1 , b 1 , c 1) are the side-lengths of an integer triangle of perimeter 2n - 3. This correspondence is one-to-one and onto. D -

-

In 2005 students of the St. Mark's School Institute of Math­ ematics came to this point very quickly (within two hours of playing with the problem!) and were excited by the cor­ respondence of simply "adding one" to each side length. In their musings they explored the option of "adding two" and "adding four" (so as to keep within the class of triangles of even perimeter). This led to the following key result: LEMMA 4. For n

even

with n > 1 2 , T( n) - T( n - 12)

=

.!!. - 3.

2

PROOF OF LEMMA 4. Let (a, b, c), with a 2: b 2: c, be a triple

of integers representing the side-lengths of a triangle of perimeter n - 1 2 . Then (a + 4, b + 4, c + 4) is a triple representing a triangle of perimeter n, with a ::S .!!. - 3, and

2

every triangle of even perimeter n > 1 2 with longest side at most this length arises this way. The correspondence to this subset is one-to-one and onto. Our correspondence "misses" the triangles of perimeter n with longest sides of lengths .!!._ - 1 and .!!._ - 2 . A simple counting argument

2

2

n shows that there are precisely - - 3 of these triangles. D 2 If we set T(O) = 0, then this formula is valid for all n even with n 2: 1 2 . Set n 1 2k + a for k 2: 0 and a = 0, 2 , 4, 6, 8, or 10. We have : =

T(l 2 k + a)

=

=

I ( T(12r a) - T(l2(r I (6r + 2 3) + T(a) k

+

r= l

1) + a)) + T(a)

1 = 3k2 + - ka + T(a) 2

(12k + a)2 48

_

a2

48

48

.l. Thus it is precisely the fractional amount needed 2 (12k + a)Z .

to round

up or down to the nearest mteger,

48

namely T(l2k + a). Thus, for n even, we have:

T(n)

=

� :; )

=

T(n + 3)

/ ( n + 3)2 = \ 48

=

·

Given integers n and k, the quantity

).

l �J

counts the num­

ber of times k appears in a list of all the factors of all

l �J +

the numbers 1 through n and the (finite) sum

l

l �J ;J + +

·

·

·

counts the total number of times a

square number appears in the list of all factors of num­ bers 1 through n. We have:

l l + ; + � (l � J � J J . . ) n - I -n l k2 J 1 k k2 - --;; ( : IL : JI) l IAve(n) - I k 1 1 -n I (-nk - -nk J ) I ::::; Ave( n) =

_

·

+

l v;; l

1

k= l

=

tv;; J 1

2

l

-

2

·

Notice that

IVn l

1

2 -

=

1

IVnl

k= l

1 [v;;j

limn-->oc Ave( n)

=

I -k21 00

k= l

-n k=l I1=

=

lVnl � o -

n

1f2 . 6

CoMMENT. Advanced students at the Boston Math Circle were oc

aware of the value of I

1

2

(and could recount Euler's k clever approach to finding it) . They were able to develop essentially the same limit argument presented here. k= l

One can go quite far with the general idea of this ap­ proach and prove, for instance, that if a series of positive terms

where angled brackets indicate rounding to the nearest in­ teger. For n odd:

T( n)

r

. This turns out to be correct and 6 9 we can make the argument rigorous as follows:

4

as n � oo, from which it follows that

+ T( a) .

Since T(a) = 0, 0, 0, 1 , 1 , 2 for a = 0, 2 , 4, 6, 8, 10, notice a2 that T(a) is a fractional quantity with modulus less

than

1 + ..!._ + ..!._ + . .

k= l

!!:_ -

r= J

6. Every number possesses 1 as a factor and, speaking loosely, one quarter of the numbers possess 4 as a factor, one-ninth the factor 9, and so on. This suggests that the average count of square factors among all numbers is

I _!__ converges to a finite value L then, on average, ak

k=J

a number possesses L factors from the set {a1 , a , a3 , . . . ) . 2 Thus, for example, a number possesses, o n average, two factors that are a power of two, two factors that are trian­ gular numbers, and e - 1 factors that are factorials.

FuRTIIER. Let S(n) be the number a.(scalene integer trian­

FURTIIER. Show that, on average, a number possesses ln2

COMMENT. In 2002, students of the Boston Math Circle in­

St. Mark's Institute of Mathematics 25 Marlborough Road Southborough MA 01 772

gles ofperimeter n. What do you notice about the sequence of these numbers? dependently discovered this result, as well as some nice con­ nections to partitions of integers. They published their work in FOCUS ( Vol. 22, no. 5, 4-6).

more oddfactors than even factors.

USA e-mail: JamesTanton@stmarksschool. org

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

59

i;i§iil§i,'tJ

Osmo Pekonen , Ed itor

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

us

us,

your expertise and your

predilections.

I

Solving Mathematical Problems: A Personal Perspective

by Terence Tao

NEW YORK, OXFORD UNIVERSITY PRESS, 2006,

xii +

103 PP. , PAPERBACK, US $24.95,

ISBN 0-19-920560-4

REVIEWED

Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyviiskylii, Finland e-mai l: [email protected]

60

BY JOHN J. WATKINS

ike many other professional math­ ematicians I have spent a good portion of my mathematical life actively involved with the sort of prob­ lems posed in mathematical competi­ tions such as the various international Mathematical Olympiads and the William Lowell Putnam Mathematical Competition, and so I have devoted a great deal of effort, time, and thought not only to solving many of these prob­ lems, but also to the far more elusive task of helping students learn how to do so for themselves. Thus, it was with a great deal of anticipation that I picked up Terence Tao's new book Solving Mathematical Problems. After all, Tao had just been awarded the Fields Medal earlier in the summer at the International Congress in Madrid, and here he was taking the time to write a book for young people on how to solve competitive math problems. I knew his perspective would be a tremendously interesting and useful one for me and my students. It turned out I was right, though not for exactly the reason I thought. In fact, I became utterly confused by the very first sentence in Tao's preface to this second edition of his book: "This book was written 1 5 years ago; literally half a lifetime ago, for me. " But, as I mentioned, I am a professional mathe­ matician and am perfectly capable of digging myself out of a hole when nec­ essary. I quickly recalled that Terence was 3 1 when he received the Fields Medal in Madrid in the summer of 2006

THE MATHEMATICAL INTELLIGENCER © 2007 Spnnger Science+Business Mecia, Inc.

and since I could see that the preface had been written from his home insti­ tution at UCLA in December 2005, he must have been 30 at the time; so it sud­ denly all made sense. The book in my hands had not been written, as I had thought, by a brilliant 30 year-old Fields Medal winner, but instead by a rather extraordinary 1 5-year-old prodigy. What I most hope to do in this re­ view is convince you that this remark­ able work is the fine book it is precisely because it was written by the 1 5-year­ old prodigy and not by the mature math­ ematician the author would later be­ come. Tao himself seems to be fully aware of this, and indeed resisted the urge to bring to bear his formidable later experiences and current level of insight in order to eliminate what he now de­ scribes as "a certain innocence, or even naivety" from the original exposition. He wisely recognized that "my younger self was almost certainly more attuned to the world of the high-school problem solver than I am now." It is the voice of the younger Tao speaking directly to today's would-be problem-solvers. By the time Terry was 1 1 , in Adelaide, Australia, he was already participating in international competition. By 1989, he became the youngest gold medal win­ ner ever in the International Mathemati­ cal Olympiad-he had previously won the silver medal in 1988 and the bronze medal in 1987. Perhaps a completely nat­ ural next step for this highly precocious teenager was then to write a "How-To" book on problem solving. What is quite surprising, however, is that at that young age he had the maturity to do such a good job of it. In this book Terry-and I'll call him Terry as a way to remind you that the author is 15 and not yet a Fields Medal­ ist-shows us how to solve only 25 problems, which works out to about four pages per problem; he spends a lot of time discussing his thoughts about each of them. He begins with the fol­ lowing problem: A triangle has its lengths in an arith­ metic progression, with difference d. The area of the triangle is t. Find the lengths and angles of the triangle. (See Figure 1 .)

a rea t

b+d

Figure

He uses this first problem to lay down a few basic fundamentals of problem solving such as understanding the problem, the data, and the objective, as well as selecting good notation and writing down what you know in the selected notation and drawing a dia­ gram. He offers three reasons why it is helpful to put everything down on pa­ per: a) you have an easy reference for later on; b) the paper is a good thing to stare at when you are stuck; and c) the physical act of writing down what you know can trigger new inspirations and connections. My first reaction when I read (b) was to laugh, but then-and this was the moment it re­ ally hit me that this 1 5-year-old had some important things to teach me about problem solving and about teaching-! realized how many times in my life I have stared at a diagram and bits of notation on a piece of pa­ per while waiting to get unstuck on a problem (and because I trained myself to do it on paper I now without much effort can do it in my "mind's eye" and can work on problems at almost anytime and almost anywhere) . For the next page or two he dis­ cusses ways to modify this problem, a theme he returns to again and again, and always there is a sense of motion, never linear, but always searching, cast­ ing a net of connecting ideas and facts, looking for a promising route towards a solution. Eventually he settles on an early idea, Heron's formula, that was

previously jotted down and looked es­ pecially promising. In this problem the semiperimeter s simplified to s = 3 b/2 (applying his axiom on good notation, Terry had chosen b to represent the middle length among the three lengths b - d, b, and b + d in the triangle) . The solution then falls out routinely. There are details to be worked out, to be sure; one has to use the quadratic formula on a quartic polynomial and then know how to use the law of cosines to eval­ uate the angles in the triangle once the sides are known. But this happens quickly and matter-of-factly in the text. The real action and attention is in the exploratory stage of solving the prob­ lem. In many ways the entire book is contained in this one example. Tao re­ peats the procedure for us on one beautiful problem after another, grad­ ually allowing his thought processes to sink in for the reader. I couldn't help but notice that Terry began his book with a geometry prob­ lem that was based in a fundamental way on arithmetic progressions. Al­ though Terence Tao is now widely re­ spected for his work in many areas of mathematics including the n-dimen­ sional Kakeya problem, wave maps in general relativity, Horn's conjecture, and nonlinear Schroedinger equations (with a group of four other mathemati­ cians known as the "I-team"), he is most celebrated at the moment for settling in 2004, with Ben Green of the University of Bristol in England, one of the most

famous conjectures in number theory, a conjecture concerning arithmetic pro­ gressions! It had long been conjectured that there exist arbitrarily long, but finite, arithmetic progressions of prime num­ bers. (It is a fairly easy exercise to see that any infinite arithmetic progression contains infinitely many composite numbers.) For example, 47, 53, 59 is an arithmetic progression of length 3 consisting entirely of primes, and 25 1 , 257, 263, 269 is an arithmetic progres­ sion of length 4 consisting entirely of primes. But longer sequences are quite difficult to find. A recently discovered sequence of 10 consecutive primes be­ gins with a prime number having 93 digits and the numbers in the sequence have a common difference of 2 1 0 . The text I used when I last taught number theory, an excellent 2002 edition, claimed, with seemingly justifiable con­ fidence: "Finding an arithmetic pro­ gression consisting of 1 1 consecutive primes is likely to be out of reach for some time." The author clearly had not anticipated a math prodigy from Aus­ tralia who had honed his skills in in­ ternational mathematics competitions, for in 2004 Tao and Green proved that the prime numbers contain arithmetic progressions of any finite length what­ soever, thus putting to rest a centuries­ old conjecture. One of my favorite problems in the book begins with a rectangle, con­ structs the intersection of its diagonals, and then extends two of its sides to create two more points and then asks the problem-solver to show that three ratios, respectively, of six of the given or constructed line segments are equal. What I enjoyed most about this prob­ lem was watching Terry solve it, much in the way I recently watched in awe a virtuoso performance by the young violinist Joshua Bell. As always, Terry looks for ways to reformulate the prob­ lem. He first plays a bit with the ratios to get them as simple and symmetric as possible, but he then considers other rearrangements including one which multiplies them out to get products in­ stead. This doesn't seem to help much but gives him a slight opening since it looks a little familiar to him (though not to me). Here is a formal statement of the problem (see Figure 2):

© 2007 Springer Science +Business Media, Inc., Volume 29, Number 3, 2007

61

c Figure 2

Let ABFE be a rectangle and D the intersection of the diagonals AF and BE. A straight line through E meets the extended line AB at G and the extended line FE at C so that DC = DG. Show that AB!FC = FC!GA = GA!AE. And after awhile here is what he is looking at: FC X BC = AG X BG. And here is what it reminds him of. If a point P is outside of a circle and a line from P cuts the circle at two points Q and R, then the product PQ X PR is called the power of the point P. This terminology was not used by Terry but was introduced by Jacob Steiner in 1 826. The astonishing thing-and I urge you to draw a diagram-is that this product is independent of the line itself; that is, the product depends only on P and not on Q and R. So, in par­ ticular, if the line is chosen to be a tan­ gent to the circle, then Q and R coin­ cide at T, the point of tangency, and PQ X PR = PT2, and in turn PT2 can easily be expressed in terms of the radius of the circle and the distance from P to the center of the circle. (See Figure 3.) So, that's how Terry, with the flour­ ish of a virtuoso, could polish off this problem. There is not a circle in sight in the statement of the problem, but along the way he is vaguely reminded of a beautiful result from high-school geometry (not that I recall ever seeing it), and so then he notices that a circle

62

THE MATHEMATICAL INTELLIGENCER

with center in the middle of the rectan­ gle conveniently passes through points A, B, and F, and suddenly he is done. Besides the fun of watching Terry solve problems such as these, there is much to learn from his book. One of the lessons of course is that it helps to know things. It helps to know Heron's formula or to know about the "power of the point". But the more important lesson that Terry is so adept at show­ ing repeatedly is that it is absolutely vi­ tal to be on alert for some subtle clue to turn up in the problem that is telling us that one of these marvelous facts lodged somewhere deep within our brains might actually be relevant to the problem at hand.

I think this book is destined to be­ come a classic, to find its place on our bookshelves alongside P6lya's How To Solve It. But I also think this book is likely to be especially effective for its target audience, the young problem­ solver. In his preface-his first preface, that is-Terry tells us why the Greek philosopher Proclus believed we should like mathematics, and then Terry tells us "but I just like mathe­ matics because it is fun . " The entire book is filled with this youthful exu­ berance and it runs through every problem and solution like a splashing river. He speaks of "gung-ho algebraic attacks", "the first sneaky thing to be done" , "our equation is a mess", "hack­ and-slash coordinate geometry", "we should just play around with it", "the general proof smells heavily of induc­ tion", "geometry is full of things like this". The reader is just swept along by the sheer joy of it all. Terence Tao had an amazing year in 2006. In addition to the Fields Medal, he was also awarded several other im­ portant prizes, but most notably he re­ ceived one of the year's MacArthur Fel­ lowships which comes with a "no strings attached" $500,000 stipend. The year 2006 crop of MacArthur Fellows not surprisingly included only one mathematician; however, it also hap­ pened to include an author of chil­ dren's books, David Macauley. I won­ der if anyone on the MacArthur selection committee realized that they also selected among their Fellows for the year 2006 a truly remarkable child author, Terry Tao.

Figure 3

REFERENCES

1 . D. M . Burton, Elementary Number Theory,

fifth edition, McGraw-Hill, 2002. 2. G. P61ya, How To Solve It, new edition,

Princeton University Press, 2004. Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail: [email protected]

Introduction to Circle Packing: The Theory of Discrete Analytic Functions by Kenneth Stephenson NEW YORK, CAMBRIDGE U NIVERSITY PRESS, 2005, 356 PP. $60.00, HARDCOVER, ISBN-10: 0-521-82356-0 REVIEWED BY J. W. CANNON, W. J. FLOYD, AND W. R. PARRY

his book describes the rich math­ ematics associated with patterns of tangent circles in two-dimensional surfaces; the theory is a full discrete ver­ sion of the classical theory of one com­ plex variable. Classical theorems have discrete analogues that are visually strik­ ing. The size of an individual circle is generally proportional to the absolute value of the derivative of some analytic mapping, so that the distortions of a con fo rmal mapping are visible. The au­ thor, Ken Stephenson, describes the the­ ory, gives applications, and illustrates

T

Figure 2.

The Apollonian gasket.

Figure I . The owl and its uniformization (p. 1 2). A page number accompa­ nying a figure caption refers to a figure in the book, reprinted with permis­ sion from Cambridge University Press.

everything with explicit packings cre­ ated by his marvelous suite of computer programs titled CirclePack (available at no cost from the website http://www. math.utk.edu/�kens/). Ken's trademark is the owl as shown in the accompany­ ing illustration. (Figure 1 ) The most famous classical circle packing is the Apollonian gasket: insert a fourth tangent circle in the triangu­ lar gap formed by three mutually tan­ gent circles and, iteratively, continue to insert tangent circles into the trian­ gular gaps formed, ad infinitum. (Fig­ ure 2) In 1 934, Paul Koebe [Koebel, whose 1907 uniformization theorem is a cen­ tral result in the theory of conformal mappings, employed circle packings as a tool in conformal mapping. In 1938, Lester R. Ford [Ford] described the Farey fractions as the tangency points on the real line in a version of the Apollonian gasket. E. M. Andreev [Andreev] stud­ ied circle packings in the construction of convex polyhedra in Lobachevskii

space. William P. Thurston rediscov­ ered Andreev's theorems in the late 1970s, gave a new proof, and employed circle packings in his famous orbifold trick, which he used in the proof of his Geometrization Conjecture for Haken

Figure 3.

manifolds. (See [Thurs1), [Morgan] , and [Kapovich].) In 1985 Thurston awakened general interest in the theory of circle packing by discussing his elementary approach to Andreev's theorem at the de Branges Symposium, and using the theorem to give a constructive, geometric approach to the Riemann Mapping Theorem [Thurs2] . Thurston's talk has spawned a literature with well over a hundred pa­ pers, and now Stephenson's book of 356 pages. A circle packing, in its simplest form, is a collection of circles in the plane or 2-sphere such that each gap between circles is triangular, abutting three pair­ wise tangent circles. Every child is fa­ miliar with the fundamental packing of the plane by pennies, in which circle centers form the vertices of the planar tiling by equilateral triangles. We would call this penny-packing the equilateral packing, but it is more commonly known as the standard hexagonal pack­ ing. In the last section of this review, we will consider a sequence H; of hexagonal penny-packings, where the size of the pennies in packing H; is 1/ i. (Figure 3) Stephenson's book considers the rigidity and flexibility of circle packings,

The penny-packing and its dual triangulation.

© 2007 Springer Science+ Business Media. Inc., Volume 29, Number 3, 2007

63

Figure 4.

vertex).

The Dual Triangulation (outward pointing arrows all go to the same

• • •

•

•

•

•

• • •

Figure 5.

Schwarz Reflection (p. 1 93).

Figure 6.

Convergence to Riemann mapping (p. 278).

64

THE MATHEMATICAL INTELLIGENCER

•

•

and the approximation of conformal maps by circle packings. Rigidity de­ scribes the discrete version of the clas­ sical uniformization theorem for Rie­ mann surfaces. Flexibility mirrors, in discrete terms, the huge variety of con­ formal and analytic mappings used in mathematics and other sciences. Ap­ proximation joins the discrete with the classical by showing that many of the most important classical mappings can be approximated by circle packings. Thurston once carefully constructed a pattern of tangent circles for the cover illustration of one of his publications and sent the layout to a graphic artist for professional rendering. The artist came to him in frustration. "I can't get the circles to match the pattern, " he said. "I used my plastic template with circles of all sizes and started to lay out the pattern, but I couldn't make the cir­ cles fit!" "I had forgotten to tell him," said Thurston, "that, when the outer cir­ cle of the pattern has been chosen, and when its tangency points with three of the inner circles have been fixed, then the combinatorial tangency pattern de­ termines the exact size and position of every circle. The entire configuration is unique up to linear fractional transfor­ mation of the plane. " The reader might try constructing some packings by hand in order to appreciate the difficulties. Stephenson has managed, by means of his program CirclePack, to construct circle packings with prescribed patterns of tangency that involve more than 50,000 circles. His program is modelled on Thurston's algorithms. The existence of circle packings with any plausible pat­ tern of tangencies is assured by the fun­ damental theorem of the book, the Dis­ crete Uniformization Theorem. This is an exact analogue of the classical uniformi­ zation theorem for Riemann surfaces. Plausible tangencies are described by triangles. If three circles are to be mu­ tually tangent, with no other circles in the triangular gap, then one assigns to the three circles an abstract triangle whose vertices correspond to the cen­ ters of the three circles. An edge of such a triangle represents a path from one center to another, the edge passing through the point of tangency. Such an edge is said to be dual to the tangency; it might be an edge of another such tri­ angle as well. We require that these tri­ angles triangulate a surface S. The re-

suiting configuration of triangles is called a complex triangulating S. In the illus­ trated example, the complex triangulates the 2-dimensional sphere. (Figure 4) There is great freedom and flexibil­ ity in choosing an appropriate complex. The Discrete Uniformization Theorem says that each corresponds to an es­ sentially unique circle packing .

(" •

•

• , D

• •

./

II • ("

THEOREM (DISCRETE UNIFORMI­ ZATION THEOREM) Let K be a com­

Figure 7.

plex that triangulates a topological sur­ face S. Then there exist a Riemann surface SK homeom01phic to S and a cir­ cle packing P for K in the associated in­ trinsic spherical, Euclidean, or h;perbolic metric on SK such that P is univalent and fills SK. The Riemann surface SK is unique up to conformal equivalence and P is unique up to conformal autom01phisms ofSK

Dessin d'enfants (p. 289).

I

llo

F igure 8.

Convergence to the Riemann mapping (p. 27).

The proof is leisurely and instructive, though it occupies about eighty pages and comprises Chapters 4 through 9. Despite this rigidity, circle packings are flexible. The rigid universal cover­ ings can be mapped with great freedom if one allows exotic branching, covering, and boundary behavior. The results be­ come essentially as flexible as classical analytic functions. Stephenson shows how to model Blaschke products, the functions of the disc algebra of analytic functions, all sorts of boundary behavior, discrete entire functions, discrete poly­ nomials, discrete rational functions, and discrete exponentials. He describes at­ tempts at a discrete error function. A few of his examples are shown in the three accompanying graphics. (Figures 5-7) Among the classical notions that have interesting circle packing ana­ logues (as theorems, techniques, con­ jectures, or problems) are extremal length, the type problem, Schwarz re­ flection, the Schwarz-Pick Lemma, Liou­ ville's Theorem on bounded entire func­ tions, Koebe's 1/4-Theorem, normal families and convergence, the maxi­ mum principle, and the monodromy theorem. As the author says, "Discrete analytic functions not only mimic their classical counterparts, . . . but actually approxi­ mate them. . . . It turns out that given the slightest chance, circle packings will almost trip over themselves in their rush to converge" (p. 247).

© 2007 Springer Science+Business Media, Inc., Volume 29, Number 3, 2007

65

Thurston's approach to the Riemann Mapping Theorem via circle packings was completed by Rodin and Sullivan [Rodin-Sulll. Given a bounded, simply­ connected domain [! in the complex plane, the goal is to approximate a Rie­ mann mapping that takes the open unit disk D onto [! by circle packing map­ pings. One chooses a penny size, say 1/ i, and fills [! as nearly as possible with a connected portion P; of the hexago­ nal penny packing H;, where the un­ derlying triangulation K; defined by P, is a closed topological disk. The Dis­ crete Uniformization Theorem yields a packing PK; of the open unit disk D by circles whose tangencies correspond ex­ actly to the edges in K; and whose outer, or boundary, circles are all tangent to the boundary circle of D. The corre­ spondence between the circles of PK; and those of P; can be used to define a mapping from most of D to most of f!. The Radio-Sullivan Theorem claims that, after a minor normalization, these par­ tial mappings converge to a Riemann mapping from D onto f!. (Figure 8) The original proof made strong use of the combinatorics of the hexagonal penny-packing. Other authors have re­ moved many of the restrictions in­ volved. The best result to date seems to be that of He and Schramm (see the references in this book). REFERENCES

Press, Orlando, San Diego, San Francisco, New York, London, Toronto, Montreal, Syd­ ney, Tokyo, Sao Paulo (1 984): 37-1 25. [Rodin-Sull] Rodin, Burt, and Sullivan, Dennis, "The Convergence of Circle Packings to the Riemann Mapping," J. Differential Geometry 26 (1 987): 349-360. [Thurs1] Thurston, William P., "The Geometry and Topology of 3-Manifolds," Princeton Uni­ versity Notes, preprint.

[Thurs2] Thurston, William P . , "The Finite g Mapping Theorem," invited talk (An Interna­ tional Symposium at Purdue University in Celebration of de Branges' Proof of the Bieberbach Conjecture, March 1 985). J. W. Cannon Department of Mathematics Brigham Young University Provo, Utah 84602 USA e-mail: [email protected] W. J. Floyd Department of Mathematics Virginia Tech Blacksburg, VA 24061 USA e-mail: [email protected] W. R. Parry Department of Mathematics Eastern Michigan University Ypsilanti, Ml 481 97 USA e-mail: [email protected]

[Andreev] Andreev, E. M . , "Convex Polyhedra in Lobacevskii Space," Mat. Sbornik 81 , No. 1 23 (1 970a). [Russian] "Convex Polyhedra in Lobacevskii Space," Math. USSR Sbornik 1 0 (1 970b): 41 3-440

[English]. "Convex Polyhedra in Lobacevskii Space," Mat. Sbornik 83 (1 970c): 256-260 [Russian] .

"Convex Polyhedra of Finite Volume in Lobacevskii Space," Math. USSR Sbornik 1 2 (1 970d): 255-259 [English] . [Ford] Ford, Lester R . , "Fractions," American Mathematical Monthly 45 (1 938): 586-601 .

[Kapovich] Kapovich, Michael, Hyperbolic Man­ ifolds and Discrete Groups, Birkhauser,

Boston (200 1 ) : 467 pages. [Koebel Koebe, P., "Kontaktprobleme der kon­ formen Abbildung , " Ber. Sachs. Akad. Wiss.

Introduction to Cryptography with Coding Theory, Second Edition by Wade Trappe and Lawrence Washington SADDLE RIVER, NJ, PRENTICE-HALL, 2006, 592 PP., US$ 90.20, HARDCOVER, ISBN 0-13-186239-1 REVIEWED BY MICHAEL ANSHEL AND KENT D. BOKLAN

Leipzig Math.-Phys. Kl. 88 (1 936): 1 41 -1 64.

[Morgan] Morgan, John W., "On Thurston's Uni­ formization Theorem for Three-Dimensional Manifolds, " in The Smith Conjecture, Morgan, John W., and Bass, Hyman, eds., Academic

66

THE MATHEMATICAL INTELLIGENCER

ost ten-year-old boys and girls run around a lot. Many play video games. Some accidentally download computer viruses. And quite

a few invent secret codes, their very own means of disguising their communica­ tions from parents and peers. Children quickly learn the rules of cryptography: their techniques must be efficient and their methods must be able to be un­ done, too. (Budding cryptanalysts, who spend their efforts breaking the systems of their classmates, are scarcer than young cryptographers.) It's the bread and butter of cryptography, the en­ crypting, and there's a popular mythol­ ogy to Top Secret ciphers and spy in­ trigue-with the television shows with the strong encryption that somehow al­ ways manages to get broken. Today we are inundated with media pronounce­ ments of strong (or strongest!) protec­ tions with such ubiquitous phrases as, " 1 28 bit encryption." It seems that every­ one does it or claims to do it. Even I can do it, with the Captain Midnight de­ coder badge that I bought on e-bay. But exactly how does it all work? Cryptog­ raphy is not just the latest trend, like the hula hoop, Betamax, and the Spice Girls. It's here, it's not going away, and some­ one needs to know how it really works-and if it's really strong.

Ah, but a man 's reach should exceed his grasp, or what's an SSL for? An excellent first step toward the un­ derstanding of the black boxes of (com­ mercial) encryption is to work through

Introduction to Cryptography with Cod­ ing Tbeory, second edition, by Wade

Trappe and Lawrence Washington (which we dub WaTr for purely metri­ cal purposes). Read it and you'll learn the answer to that mysterious question, "What's [in] that SSL thing?" You may still fumble, though, when your friends ask you, "Should I really trust ama­ zon.com with my credit card number?" An introduction to cryptology, the sum of cryptography and the cryptanalysis, usually starts with a fundamentals class at the elective undergraduate or early graduate level. WaTr fills about two of these courses, two semesters worth, and it's aimed at an audience of computer science, engineering and mathematics students. But this text is not just replete with the classical ciphers-the Vi­ generes and the Enigma's-but is full of the flotsam and jetsam that fill the ether about them, those cryptographic prim­ itives and applications (like the key dis­ tribution protocols and the digital sig-

natures) that are the backbone of the few high-profile protocols upon which so much of today's data security rests. WaTr provides pedagogy in two dis­ tinct voices. Unfortunately, this duality is often distinguished by the strengths of the expositions. As we are told in the preface, WaTr plans to "cover a broad selection of topics from a math­ ematical point of view." To be com­ prehensive is a near Herculean labor. Some volumes, like [2] and [6], do very well to touch upon almost all of the notable features of cryptography today, but they are not texts for a first lesson in the mechanics of how and why and the mathematics behind it all. WaTr fea­ tures a sound balance of methods and attacks; it is a pleasure to read. There are the occasional proofs of the math­ ematical statements, when the proofs are elementary, but WaTr is about a wider introduction. Certainly, a lot lurks hidden beneath the surface, including the hidden Markov models. Trappe and Washington fittingly point to many of the deeper ideas, especially in the the­ ory of elliptic curves, and they keep the reader both aware and enticed for further study. On the down side, the privation of implementation details and implementation issues in WaTr is a real loss for the student who wants to run with the encryption ball; the devil, after all, is in the cryptographic small print. There are exceptions, though, and WaTr does include the very clever work of [4]. But this is a first text and, as such, serves well. This second edition of WaTr features several important additions to the first edition, including identity-based public key cryptogra phy, an elegant construc­ tion which holds substantial promise in future applications. Significant cryptan­ alytic advances in the theory of hash functions are included. The study of hash functions (and collision-finding) is experiencing a revival due to exciting work beginning with [3] and culminat­ ing in [7]. Thanks to these efforts, we're now looking for new hash algorithms because our faith in the old ones has been ruffled. Also new to the second edition of WaTr is a chapter on lattice methods (including the Lenstra-Lenstra­ Lovasz method for finding short vec­ tors). Notable by its absence, though, in both the first and second editions of WaTr is the Merkle-Hellman Knapsack

scheme, the Icarus of public key cryp­ tosystems. Knapsack was all the rage in the late 1970s: it was elegant and based upon a known NP-hard problem (unlike the RSA system) . Shamir, in 1982, shook the foundation and cer­ tainly the confidence of the young field of modern cryptography by cracking it-and in so doing changed the face of (public key) cryptography and crypt­ analysis to this day. The story of Knap­ sack is part of the history. It makes for great reading, it's high drama, and it provides a strong lesson. But it's not in WaTr and it should be. What is in WaTr and is a highlight of the text is the gentle introduction to DES, (which was) the Data Encryption Standard. WaTr presents it slowly, a few rounds at a time. Block ciphers, like DES, 3DES ("triple DES") and Rijndael (the new standard, the Advanced En­ cryption Standard) are the load-bearers of data encryption. They are each com­ posed of rounds, and a single round is a structured shaking of the input. Start­ ing with the plain text input to the first round and repeated on the output of the previous round, a block cipher is designed with enough rounds so that the result, the cipher text, is jumbled enough-meaning that the influence of the input on the output (and vice versa) is fully diffused. The mixing steps are usually a combination of permutations and substitutions and some non-linear lookups (DES has some famous S-boxes that do this) all the while being de­ signed to be invertible so plain text can be recovered. WaTr explains the work­ ings of DES in parallel with the crypt­ analytic method of differential crypt­ analysis. By so doing, it becomes clear(er) why DES has 16 rounds. The discourse does get a bit technical. How­ ever, the parallel presentation is well worth the effort of careful study. The best cryptographic algorithm designs are structured around what attacks are known and then laid out to be resistant to them. For symmetric key protocols, like block ciphers, it's all about the mud­ dling and the repetition of the

The paradig m of "easy to do but hard to undo " lies at the heart of e1yptography.

processes. For asymmetric schema, as in public key cryptography, designs are predicated upon mathematical prob­ lems that are "easy" (computationally efficient) to perform but believed to be "hard" (computationally infeasible) to invert-without some extra piece of knowledge, a key. These problems are called trapdoor one-way functions and are not to be confused with one-way functions for which there is no key to undo them. (Hash functions are one­ way functions.) There is no real proof that trapdoor one-way functions exist since obtaining lower bounds for these kinds of complexities seems near im­ possible. There is faith, based upon many years of very limited success, in a few select problems: the discrete log­ arithm problem and that of finding, for some e, an e-th root modulo a number of unknown factorization. Discrete logarithm problems (dip's), as involved in, for example, the classi­ cal Diffie-Hellman key agreement pro­ tocol, take a form such as: find x if

7x � 2434711235764822669040730

(mod 4083497104378553871 280549).

WaTr's treatment of the discrete log problem and approaches to solve it, the Pohlig-Hellman algorithm and the index calculus, are exemplary for first-year stu­ dents. The hard (but nevertheless toy­ sized) dip above can be solved on your laptop. When the modulus has several hundred digits, things get very tough. (The index calculus approach, while sub-exponential complexity, does not scale well enough to be efficient.) The RSA architecture involves raising a message, m, to a fixed known power, e, modulo a number n whose factor­ ization is a secret (and e and n are rel­ atively prime). A result may look some­ thing like this (for e = 31): m3 1

�

1970517852344637324142632145 5642097240677633038639787310457 022491789 (mod 495960937377360 604920383605744987602701 1013993 99359259262820733407167).

Breaking RSA is about finding m. If n can be factored, this is easily ac­ complished. (Raise both sides of the equation above to the power d where

d-1

�

e (mod rp(n)).

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

67

That recovers m. The proof is a simple application of Euler's generalization of Fermat's Little Theorem.) It is unknown if there is a way to find m that is more efficient than factoring the modulus. Since factoring special types of large numbers is believed to be hard, the RSA system, with a sufficiently large, hard modulus is currently considered secure. (The example presented here is, hope­ fully, a small step towards dispelling the common misconception that an RSA modulus need be the product of two large primes. For efficiency, it ought to be-or close. It need not be, though.) WaTr gives a quick overview of some factoring techniques; enough to convince the would-be RSA code­ breaker that things can be tremen­ dously challenging. The questions of the (computational) equivalence of the RSA problem and of factoring-and of the discrete logarithm problem and the Diffie-Hellman proto­ col-are amongst the most important open issues in cryptology today. Progress has been made on the latter question (see [5]) in the affirmative di­ rection. On the former, the recent work is less than convincing. WaTr really shines in its initiation into the world of elliptic curves and el­ liptic curve cryptography. From a sim­ ple introduction to the group law to H. Lenstra's beautiful elliptic curve factor­ ing method (which can solve the RSA question in this review) to the elliptic curve analogue of the discrete logarithm problem and the Diffie-Hellman proto­ col, WaTr provides a fine rendering for the initiate. The elliptic curve one-way trapdoor function is simple: given an el­ liptic curve E defined over a finite field, a point P on E and k some positive integer, find k given kP (where kP = P + P + P + · · + P, k times). Again, it seems (computationally) difficult-or infeasible-to do this for large enough carefully chosen examples. There's a lot of data security (and commercial prod­ ucts) banking on that. There are unfortunate omissions in WaTr. There's no real discussion of (al­ gorithmic) complexity where it may have been well-placed to provide the reader with a sense of appropriate key sizes and protocol (and attack) strength. And there's the sporadic lack of the sense of largeness, the why's of why are things like this?. More generally, ·

68

THE MATHEMATICAL INTELLIGENCER

what is incidental and what is of real import in the digital world is not always clear. These gaps, though, are alleviated in part by an assortment of excellent (and detailed) end-of-chapter exercises and computer problems that allow and encourage the reader to identify some of the subtleties and gain a deeper ap­ preciation of the why's. WaTr skips almost the whole field of stream ciphers. That's a shame. Stream ciphers are a major component of encryption technology today. And WaTr features only a cursory look at linear feedback shift registers, the pri­ mary constituent of most stream ciphers over the past century and many very good random-number generators. Lin­ ear feedback shift registers (and their associated tap polynomials) are rich in mathematical theory and can be designed and combined to provide very satisfactory output. Bad random­ number generators, at least for crypto­ graphic purposes, are based upon sim­ ple linear congruential generators of the form X11

=

AXn- l + B (mod m)

where m is fixed and A and B (and �) are unknowns (but chosen so that the period of the generator is large). One can easily deduce the next "randomly" generated number from knowledge of the previous three-and this predictabil­ ity makes for a very bad random-num­ ber generator. (Yet this is how many rand functions work!) Stream ciphers are needed for real time encryption when you can't wait for a whole block of plain text to arrive before you use your block cipher. Stream ciphers aren't just for voice communication anymore. Cryptography sells, from the great propaganda of "the only provably se­ cure system" (one-time pads) to the in­ troduction of quantum cryptography. Using principles of quantum mechanics for cryptographic applications is an idea now a few decades old-and remains ever intriguing. It also makes for great press. Most notable among quantum methods is the key exchange protocol introduced by Bennett and Brassard which allows legitimate participants to (probabilistically) recognize the exis­ tence of an eavesdropper on their communication. It's a lovely idea that requires substantial overhead (a chan­ nel so clean that a photon in transit is

undisturbed). WaTr presents the ideas, this glimpse of a possible future with quantum cryptography and with quan­ tum computers (if substantial ones can ever be built). Then, with Shor's algo­ rithm, the pre-eminent quantum com­ putational cryptanalytic tool, most everything would change-for security, for the Internet, and for cryptology­ leading us to wonder, in the words of Buffy, "Where do we go from here?" A world of post-quantum cryptogra­ phy is being studied in anticipation of one plausible future. Non-abelian ap­ proaches have been suggested which do not seem to succumb to quantum attacks. Though such ideas are not in WaTr, for they are still in their early de­ velopment, these considerations are providing new avenues of investigation. ([1] offers an analogue of the Diffie-Hell­ man key establishment protocol wherein, instead of a discrete logarithm problem, the restricted conjugacy search problem serves as the trapdoor one-way function.) WaTr tries to cover a lot: the past, the present, and the (uncertain) future. It is occasionally uneven in its mathe­ matical level, the knowledge expected of the reader. The Information Theory and the Error Correcting Codes chap­ ters are not as carefully composed as much of the rest of the book and do not have the same (encouraging) instructional rhythm. The latter part could benefit from some compression and reordering, and the Information Theory section could afford some ex­ panded coverage of language recogni­ tion. (How does your computer know an acceptable decryption when it finds one?) WaTr is the best book of its kind. Appendices of Matlab, Maple, and Mathematica exercises support the rhetoric of the individual chapters be­ cause in cryptology small examples can give a false sense of security. We can quibble with what's not in WaTr, but you can't do it all at once. And what WaTr does is almost always done well. To do it all-that would be as daunt­ ing as the task of breaking " 1 28 bit en­ cryption," whatever that is. REFERENCES

[1] I. Anshel, M. Anshel, D. Goldfeld, An alge­

braic method for public-key cryptography. Math. Res. Lett. 6 (1 999), 287-291 .

[2] Menezies, van Oorschot and Vanstone, Handbook of Applied Cryptography, CRC

Press 1 997.

t was an uninteresting assignment, except for the tenth problem: Evalu­ ate

[3] A. Joux, Multicollisons in iterated hash func­ tions. Application to cascaded construc­ tions, Advances in Cryptology-CRYPTO 2004, Lecture Notes in Computer Science

31 52, Springer-Verlag, 2004, 306-31 6. [4] P. Kocher, Timing attacks on implementa­ tions of Diffie-Hellman, RSA, DSS, and other systems, Advances in Cryptology-CRYPTO 96, Lecture Notes in Computer Science

1 1 09, Springer-Verlag, 1 996, 1 04-1 1 3 . (5] U. Maurer, Towards the Equivalence of Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms, Advances in Cryptology- Crypto '94, Lecture Notes in Computer Science 839, Springer-Verlag,

1 994, 271-281 . (6] B. Schneier, Applied Cryptography, 2nd edition, John Wiley, 1 996. [7] X. Wang, Y. Yin, H . Yu, Finding collisions in the Full SHA-1 , Advances in Cryptology­ Crypto 2005, Lecture Notes in Computer Science 362 1 , Springer-Verlag, 1 7-36.

Michael Anshel Department of Computer Sciences The City College of New York, CUNY 1 38th Street and Convent Avenue New York, NY 1 0031 USA e-mail: [email protected] Kent D. Boklan Department of Computer Science Queens College, CUNY 65-30 Kissena Boulevard Flushing, NY 1 1 367-1 597 USA e-mail: [email protected]

Dr. Euler's Fabu lous Formu la: Cures Many Mathematical I l ls by Paul]. Nahin PRINCETON, NJ, PRINCETON UNIVERSITY PRESS, 2006. xxii + 380 PP. $29.95 ISBN: 978-0-69111822-2; 0-691-11822-1 REVIEWED BY PAMELA GORKIN

remember thinking, "You can't do that." Then I figured it out. It was one of those mathematical moments that makes you say "wow." Paul ]. Nahin's hook, Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, is filled with such moments. The book, like its title and cover, is clever, creative, and unique. It contains every short story Nahin can think of that uses e7Ti + 1 0, plus a few that don't. You might know that Euler's formula was one of the most frequently cited "great equations," according to a poll con­ ducted by Physics World in 2004. In 1990, readers of The Mathematical In­ telligencer voted it the most beautiful of 24 formulas, with a score of 7.7/10. Why did this equation receive such a high score? Many people cite its sim­ plicity and brevity, as well as the con­ nection between five important con­ stants in mathematics. Though there are dissenters, most mathematicians agree: this formula needs no introduction. Nahin has a gift for recognizing good stories and has put together a collection of mathematical "tales" about Euler's formula that would make a fine addi­ tion to a differential equations or com­ plex analysis class. The reader should, however, be forewarned: although the back cover informs us that the book is ''accessible to any reader with the equivalent of the first two years of col­ lege mathematics," to read and enjoy this book, most readers will need more mathematical maturity. In addition, though Euler's formula may need no in­ troduction, applications of Euler's for­ mula need motivation-and they don't always get it in this book. Many of the formulas and computa­ tions incluclecl are among the highlights of a typical complex analysis course (Wallis's formula, for example). There are also many stories that will be new to readers. There is an account of the Gibbs phenomenon, which is a story with a fas­ cinating history. (A longer version of this history, without Nahin's biography of the overlooked Henry Wilbraham, appeared in an article by Edwin Hewitt and Robert E. Hewitt in 1 979.) A wonderful =

chapter titled "Vector Trips" features R. Bruce Crofoot's story about his clog Rover. Crofoot runs a pretty compli­ cated path, which he sketches for the reader, each morning. He is the proud owner of a well-trained clog who always runs exactly one foot to his owner's right. Given the path, the owner, and the clog, it turns out that Crofoot runs farther than Rover. The question is: How much farther did Crofoot run? I liked the article when I read it in Math­ ematics Magazine and I liked it here too. It's not really an application of Euler's formula, but it is a nice use of complex numbers and vectors. On the other hand, the discussion of the vibrating string problem (as well as a development of a solution to the wave equation) really does use the fact that ffx = cos x + i sin x in an essential way. This serves as the introduction to the story of what "was probably (almost certainly) the first 'Fourier series' . " When you think o f Fourier series you probably don't think of funny stories, but in Nahin's hands they become amusing. He presents Euler's "remark­ able claim" that 7T -

- �

2

=

.

f'

_

t

----

n=l

SID

sin( nt)

-----

n

•\ sin(2 t) sin(3 t) + ( t; + --- + 2 3 ---

·

·

·

.

As Nahin points out, this is indeed re­ markable, in part because it is not true (check out what happens at t 0). But now Nahin has your attention; now you should want to know the story behind Euler's claim. Other stories would have benefitted =

from a little motivation. Nahin presents

the "beautiful formula" 00

L

n=l

C nn+11/ n2 -

=

�/1 2

and Euler's result "which made him world famous":

These are followed by more sums, in­ cluding one "dazzling result," a "spec­ tacular application of Parseval's for­ mula," a "pretty result, " and "an even more beautiful generalization" of it that will appear in the succeeding chapter. Now, "excited" is not the first word that comes to mind to describe my students

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007

69

when I cover series, particularly one af­ ter the other (as Nahin does here), no matter how enthusiastic I am when I present it. So, I would imagine that young readers would need more than a list of beautiful infinite sums to keep their attention. Consider the way Nahin motivates finding X

I m= l

( - l) m cos(mx) ( m + l)(m + 2)

·

After a very brief discussion in which we learn, primarily, that Ramanujan was a self-taught genius who was in­ terested in this question, we read: " this problem will be interesting because at one time it 'interested' a genius. " Often the motivation i s there but it follows the result. For example, we learn that "writing cos(wt) and sin(wt) in terms of complex exponentials is the key to solving difficult problems," which appear three chapters later. Then, in a discussion on Fourier series, we find out that we'll get examples of valuable results . . . soon. On p. 214, Nahin promises that we'll see how forming the product of two time func­ tions is essential to the operation of speech scramblers and radios. He ful­ fills this promise on p. 289. The wait for motivation is, occasionally, too long and the phrase ''I'll tell you later" ap­ pears far too often. Nahin's strength is his ability to draw the reader in. For this reason, it is dis­ appointing when he claims that his ex­ planation or proof is simply the result of cleverness. This happens frequently: in a discussion of how to evaluate "Dirichlet's discontinuous integral"

we learn that the solution will depend on an auxiliary integral. This idea, we are told, came from someone who was "very, very clever. " This is followed by the "clever trick" of evaluating a dou­ ble integral over a triangle by chang­ ing the order of integration. Then, to aid the discussion of Fourier's integral theorem, Nahin presents the "devilishly clever trick" of thinking of a function defined on the real line as periodic with infinite period. Many of these are not tricks at all, but rather insights, meth­ ods, or techniques. The author is an electrical engineer writing a book for a general audience.

70

THE MATHEMATICAL INTELLIGENCER

His approach is one that m1mm1zes technicalities and rigorous justification. The reader should be aware, therefore, that Nahin is a man who will, in his own words, "reverse the order of inte­ gration on a double integral as fast as you can snap your fingers," who never hesitates to interchange integrals and in­ finite sums, can't resist pulling deriva­ tives through integrals, rearranges series without comment, and, again in his own words, manipulates two "Fourier series equations in a pretty rough-and-ready way with little (if any) regard to justi­ fying the manipulations. " The author points out that, sometimes, being overly concerned with technicalities can "par­ alyze" a mathematician into inaction. In addition, Nahin's approach makes the book readable and helps to maintain the reader's interest (though there are some things that will be pretty hard for a pure mathematician to swallow, in­ cluding one derivation that Nahin him­ self admits, "many 'pure' analysts are truly aghast [at]"). While Nahin's style will appeal to a wide audience, a problem arises when the author holds others to a different standard. Nahin criticizes Philip Davis and Reuben Hersh for using the unique­ ness of factorization of the integers without mentioning it in their proof that Vz is irrational. In an early (and oth­ erwise beautiful) chapter, titled Tbe Ir­ rationality of r, Nahin tells us that he is following Carl Siegel's book, Tran­ scendental Numbers, which was written for graduate students at Princeton. He adds that Siegel often "leaped" over steps and some of the leaps are of "Olympic size." In fact, the holes Siegel left are standard fare for a good grad­ uate student, and the treatment here is quite close to that of Siegel except for the attention to relatively small details. At times it appears that Nahin is unable to decide whether Dr. Euler's Fabulous Formula is a rigorous, detailed treat­ ment of mathematical formulas or an ef­ fort to convey the beauty of the for­ mulas to a wide audience without regard to technicalities. Nahin offers us enthusiasm. He tries to surprise and, sometimes, startle the reader. When he offers his opinion on various subjects outside the main thrust of the book we are reminded of Strunk and White's advice: "Do not inject opin­ ion. . . . Opinions scattered indiscrimi-

nately about leave the mark of egotism on a work." Consider the following: Nahin begins his book with a discus­ sion of beauty, mathematical and oth­ erwise. This leads him to a discussion of Jackson Pollock, of whom Nahin says, "anybody who can observe the re­ sult of simply throwing paint on a can­ vas-what two-year-olds routinely do in ten thousand day-care centers every day . . . and call the outcome art . . . is delu­ sional or at least deeply confused (in my humble opinion). " Fortunately for the reader, the number of opinions ex­ pressed is a decreasing function of the page number. The book ends with a short biogra­ phy of Euler. Nahin treats Euler's life in stages; Euler's years in Switzerland, the years in St. Petersburg, the years in Berlin, and the return to St. Petersburg. Nahin's writing is entertaining, marred only by the curious statement that "while there is a steady stream of bi­ ographies treating famous persons . . . there is not even one book-length bi­ ography, in English, of Euler." At this point it would have been more appro­ priate for Nahin to focus on the large amount of work about Euler's mathe­ matics and life that does exist: There is William Dunham's book Euler, Tbe Mas­ ter of Us All and Varadarajan's recently published book, Euler Through Time: a new look at old themes. There are pa­ pers on Euler's years in St. Petersburg, a history of analysis beginning with his work, and a history of the logarithm ending with his work. Birkhauser Ver­ lag and the Euler Commission of Switzerland have published Leonbardi Euleri Opera Omnia, an enormous work containing Euler's mathematics and correspondences. There's an Euler society, an Euler archive, Sandifer's on­ line column How Euler Did It, and a big celebration planned in Basel for Euler's 300th birthday. Euler was even featured on the old Swiss 10-franc note, not a claim most "famous persons" can make. Frequent lack of motivation, strongly stated opinions, and overzealous at­ tempts to get the reader's attention de­ tract from an otherwise well-written, well-researched, and interesting idea. In other words, despite its flaws, Dr. Euler's Fabulous Formula is an exciting mathematical read. This book is ideal for readers who see themselves in Nahin's description of G. H. Hardy: "dis-

playing an unevaluated definite integral to Hardy was very much like waving a red flag in front of a bull . . . " But this is a mathematics book with a sense of humor and a lot of opinions. If that's not how you like your mathematics books, you probably won't be curling up by the fire with this one. In any case, one thing is certain: if you present the short stories here to a classroom full of students who have successfully com­ pleted their first course of differential equations and who still have a mathe­ matical twinkle in their eyes, and if you present them with Nahin's energy and gusto, you are surely going to be a very popular teacher. Department of Mathematics Bucknell University Lewisburg, PA 1 7837 USA e-mail: [email protected]

Tribute to a Mathe magician Edited by Barry Cipra, Erik D. Demaine, Martin

L.

Demaine,

and Tom Rodgers WELLESLEY, MA, A. K. PETERS, HARDCOVER, 350 PP., 2004, US$ 38.00, ISBN: 1568812043 REVIEWED BY CLIFF PICKOVER

ach year, as I begin to write my next popular mathematics book, I gaze at my bookshelves filled

with books by Martin Gardner, and I

chant to myself, "What has Martin not already done? What hasn't he done'" Many consider Gardner to be the father of recreational mathematics. He has brought mathematics to the general pub­ lic, and many mathematicians began their lifelong love of mathematics as a result of Gardner's influence. His mind has roamed far and wide. He is also an avid debunker of pseudoscience, and his "Mathematical Games" column in Scien­ tific American was key to introducing important and fascinating mathematical subjects to a wide audience. Some of his hottest and most memorable columns in­ cluded topics on flexagons, john Con­ way's Game of Life, polyominoes, the

soma cube, Penrose tiling, and fractals. Yes, Gardner is my hero and my inspi­ ration, and his articles, books, and kind and humble approach to life will leave a mark upon the world forever. Tribute to a Mathemagician is the third book in a series written to honor the mind, writings, and works of Mar­ tin Gardner. Each book is a collection of articles by seasoned and amateur mathematics and puzzle aficionados­ all of whom Gardner has inspired. The current book is based on dozens of ar­ ticles, many of which were presented at a "Gathering for Gardner" conference held in 2004. Sample puzzles and games in this book include blackjack, Chinese ceramic puzzle vessels, paper folding, Mongolian interlocking puzzles, rolling block puzzles, and sliding puzzles. The articles range from one-page teasers to full-length articles. The topics are organized in six parts: Braintreasures, Brainticklers, Brainteasers, Braintem­ plers, Braintaunters, and Braintools. Each part contains a variety of chal­ lenges, which vary in difficulty so that both students and veteran mathemati­ cians will find something to delight. For example, the chapter titled ·'Chinese Ce­ ramic Puzzle Vessels" contains a valu­ able history of a peculiar set of attrac­ tive puzzles, many of which are based on historically known laws of physics, such as the use of siphons. Another de­ lightful chapter describes a three-legged hourglass that lets users measure frac­ tions of the total time. The hourglass­ designed by M. Oskar van Deventer and illustrated proudly in the book-con­ tains eight minutes' worth of sand alto­ gether. The three lobes of the hourglass

Dissections," and "Rolling Block Mazes." Martin Gardner, now in his 90s, at­ tended only the first two Gatherings for Gardner meetings. But as a writer who has encouraged the field of recreational mathematics to blossom, and inspired thousands of careers, he remains the guiding spirit of both the book and the conference. Many readers of The Math­ ematical lntelligencer have been molded and remolded by Gardner, and we know that he has left a mark on us all.

are situated at the corners of an equi­

about them-what should I read?

lateral triangle. When it is turned, the sand in the top lobe flows equally into the lower two lobes. Operating instruc­ tions involving rotations are given to achieve various timings. Other chapters deal with polyomino number theory and Godelian puzzles. A brief sample of some favorite chapter titles gives a flavor of the diverse content and in­ cludes: "Configuration Games, " "Five Algorithmic Puzzles," "Mongolian Inter­ locking Puzzles," "Fold-and-Cut Magic," "The Three-Legged Hourglass," "The In­ credible Swimmer Puzzle," "Sliding­ Coin Puzzles," "Underspecified Puz­ zles," ''The Complexity of Sliding-Block Puzzles and Plank Puzzles," "Hinged

P.O. Box 549 Millwood, New York 1 0546-0549 USA e-mail: [email protected]

Elementary Set Theory with a U niversal Set by Melvin Randall Holmes LOUVAIN·LA·NEUVE, BRUYLANT-ACADEMIA, 1998, 241 PP. HARDCOVER, 1150 BF, ISBN 2-87209488-1 REVIEWED BY ROBERT JONES

c: A: s:

he following conversation was overheard during a stroll across the campus. Persons: Professor Calculus Professor Algebra one of their students.

s: In calculus this morning you were talking about sets. I don't know much c: You might begin by looking at Holmes's book on the universal set. s: The universal set? What's that? c : It's the complement of the null set. s: But that's not a set, that's everything! e : That's a popular myth. Of course the universal set is a set. The null set has a complement, every set does. A: You can't get along without alge­ bra! s: It still sounds paradoxical. c: No it's not. Holmes shows you how to avoid the paradoxes. A: Most standard introductions to set theory use a so-called relative comple­ ment notation to form set complements.

© 2007 Springer Science+ Business Media. Inc . . Volume 29. Number 3, 2007

71

c:

Take a look at Halmos's book

Naive Set Tbeory.

Why bother, if it's naive? It's anything but naive! s : Sophisticated naive set theory? A: Now you've got it. s : But why should you do away with relative set complements? A : They're a throwback to your fears about sets that are too big. With a uni­ versal set, we can use absolute set com­ plements. They're simpler to write down. Don't worry, there's no danger when using them. s : How did all this come about; what led up to it? c: Well, before you read Holmes, take a look at Grattan-Guinness. He ex­ plains why you need sets in a calculus course. s : Okay, I will, but tell me anyway, who thought up the universal set? c : Holmes reviews its history. It be­ gan with Quine. Quine showed how to have a universal set, yet avoid the set theory paradoxes. He first presented the material in a course in mathematical logic that he gave in his first year of teaching at Harvard. His system is known as "New Foundations," or just NF, after the title of his 1 937 paper. s: Wow, I suppose that mathemati­ cians got excited! c : Well, NF has a dramatic history. Ernst Specker, a mathematician in Zurich, derived the first really deep re­ sult about NF: he showed that it ex­ cludes, or is inconsistent with, the un­ restricted Axiom of Choice. A: That disappointed some re­ searchers, but it led to a better under­ standing of NF. c : Yes, his result showed everyone that they should not try to append the Axiom of Choice to the axioms of NF. s : And then? c: Rosser wrote a foundation for all of mathematics using NF. Jensen pro­ duced a set theory, NFU, that is a ver­ sion of NF, but which was proven, with­ out the AC, to be consistent. This gave mathematicians a choice of using either NF or, with similar proof methods but known consistency, NFU. A: Perhaps the best assessment of NF, as a foundation for mathematics, is in the book by Fraenkel, Bar-Hillel, and Levy. c : That was "the great non-special­ ized beginning of set theories with a s:

A:

72

THE MATHEMATICAL INTELLIGENCER

universal set," the time of the great am­ ateurs. s: But what's happened lately? Or did it just stop there? c: No, indeed. Holmes covers that in just the first 14 chapters. s: So it moved on? c: Yes, after that, the mathematicians claimed the field as their own. But the rest of the story is more specialized and too complex to summarize. In 1987 there was a seminar to honor Quine and report on the work he had brought about. By then he had become the grand old man of the subject. Oswald, who spoke in the seminar, said "W.V. Quine confessed to be amazed at the work that has been done on the sub­ ject he had fathered, and at seeing what a 'Pandoras's box' he had invented in

1937."

Maybe I should read the seminar proceedings? c: Holmes's book is a better intro­ duction; I think it is the best introduc­ tion now available. s: What does he say about this latter period? c: He makes his own choice of top­ ics. I think the most interesting part is the next-to-last chapter, which he wrote with Robert Solovay. It includes un­ published work. The last chapter is an excursion into the prospects for found­ ing mathematics upon functions with­ out appealing to sets to define them. A: Don't you have any criticisms? No book is perfect. c : One small point and a complaint. s: And those are? c: My small point is this: Holmes sug­ gests an alternative name for the theory customarily called "Morse-Kelley set theory." But why rewrite history? s: And your complaint? c: Holmes should have taken ad­ vantage of his collaboration with Robert Solovay to describe, at least briefly, Solovay's beautiful model of the real numbers, in which every subset of the real numbers is Lebesgue measurable. A: Not many calculus students know about Lebesgue measure. s: I don't. c: Mine do. You'll learn it next se­ mester. But if you want to know more about Solovay's model nowA: What person with at least a mod­ icum of interest in mathematics would not? s:

c: -then take a look at Kanamori's book on large cardinals. Anyway, I wish Holmes had discussed how Solovay's model fits in with NF and NFU. None of them requires the Axiom of Choice. s: I 'd like to buy Holmes's book. c: It's out of print. But Holmes promises a second edition on his web­ site. You might also read Forster's book-it's called Set Tbeory with a Uni­ versal Set-and Holmes's review of it in the journal of Symbolic Logic.

REFERENCES

Boffa, M . , and E. Specker (cochairs). Mathe­ matisches Forschungsinstitut Oberwolfach, Tagungsbericht, Number 9, 1 987: New Foundations, March 1 -3, 1 987.

Forster, T. E. Set Theory with a Universal Set. Exploring an Untyped Universe. Oxford logic guides, no. 20, Clarendon Press, Oxford Uni­ versity Press, Oxford and New York, 1 992, viii + 1 52 pp. Second edition, Oxford logic guides, no. 31 , Oxford University Press, 1 995. Fraenkel, A A., Y. Bar-Hillel, and A Levy. Foun­ dations of Set Theory, North-Holland, sec­ ond edition, 1 984, Studies in logic and the foundations of mathematics, 67. Grattan-Guinness, lvor. From Calculus to Set Theory, London, 1 980. Halmos, Paul. Naive Set Theory, van Nostrand, Princeton, 1 960. Holmes, M. Randall. review of Forster's book above. The Journal of Symbolic Logic, 58 (1 993), 725-728. Jensen, Ronald Bjorn. On the consistency of a slight (?) modification of Quine's 'New Foun­

dations' , " Synthese, 1 9 (1 969), 250-263. Kanamori, Akihiro. The Higher Infinite. Large Cardinals in Set Theory from their Begin­ nings, Springer-Verlag, Berlin and New York, 1 994, Section 1 1 . 1 . Kelley, John L. General Topology, Van Nos­ trand, New York, 1 955. Quine, W.V.O. "New foundations for mathe­ matical

logic,"

American

Mathematical

Monthly, 44 (1 937), 70-80.

Quine, W. V. 0. "The inception of 'New Foun­ dations' , " Bull. Soc. Math. Belg. -Tijschr. Be/g. Wisk. Gen 45 (1 993), 3, Ser. B,

325-327. Rosser, J. Barkley. Logic for Mathematicians, Second Edition, Chelsea, New York, 1 973. Specker, E. P.: "The axiom of choice in Quine's 'New Foundations for Mathematical Logic' ", Proceedings of the National Academy of Sci­ ences of the U. S.A., 39 (1 953), 972-5.

Rurweg 3 D-41 844 Wegberg Germany e-mail: ivanhoe491 [email protected]

l
synopses of Godel's 1931 incomplete­ ness theorems, the Erkenntnis piece of 193 1 due to Godel himself, and the text of a lecture delivered in Vienna in 1932 by Karl Menger. And although the au­ thors write in the introduction that the book is meant as an "easily digestible introduction" to Godel's life, work, and the Viennese culture in which he lived and flourished, it will undoubtedly be of interest to readers on every level, from newcomers to this area of the his­ tory of mathematics to seasoned logi­ cians and mathematicians.

WIESBADEN, VIEWEG, 2006, 225 PP. , WITH 200 PICTURES, €29,90, ISBN 3-8348-0173-9 REVIEWED BY JULIETTE KENNEDY

he Austrian logician Kurt Godel, a brilliant man beset by misfor­ tunes of all kinds, has drawn the attention of a number of biographers­ with mixed results. One wonders about the rights of the deceased in such mat­ ters. These lines from Othello's final so­ liloquy (from Act 5 of Shakespeare's play) seem quite apt: I have done the state some service and they know't. No more of that. I pray you, in your letters, When you shall these unlucky deeds relate, Speak of me as I am. Nothing ex­ tenuate, Nor aught set down in malice. Such being the case, Kurt Code!. The Album comes as a welcome addition to the literature on Godel. As the cat­ alogue of the exhibition which took place at the University of Vienna this year in conjunction with the celebra­ tion there on the occasion of Godel's centennial, it is to a large extent made up of reproductions of documents and photographs with accompanying com­ mentary. And while far too many of the letters and other documents are un­ dated, compromising the historical value of the volume, it is nevertheless one of the most fascinating and also one of the most corrective books to ap­ pear about Godel so far. The book is divided into three sec­ tions: a biographical section comprising roughly half of the book, a section de­ voted to Godel's work, a brief section titled ''G6del's Vienna. " These are fol­ lowed by an appendix consisting of two

Godel's Life

"Godel in a good mood and brilliant as usual. . . I like him infinitely much and no one, no one of my friends can stim­ ulate me as he does." So wrote Godel's friend Oskar Morgenstern after visiting with Godel, presumably sometime in the mid-1970s. A number of interesting documents are included in this first section. For ex­ ample, a few pages from one of Godel's schoolbooks are reproduced on page 18. They are undated, but Godel must have been a very young child when he wrote these lines. In a typical example of the old-fashioned rote method of teaching, in the workbook Godel had to write the symbols " + " and the num­ bers "2" and "3" a hundred (or more) times, presumably for practice, and then finally the equations " 1 + 1 2" and "2 + 1 = 3." But then his mind wan­ dered and he wrote " 1 + = 2 1 . " His re­ port card of February 10, 1917 is here too, on page 19, and there we see that the 1 0-year-old boy rated "sehr gut" in every subject-except mathematics!, in which he only rated a "gut. " A fascinating set of pictures and cap­ tions deal with Godel's first few years in Vienna, where he became a student at the University of Vienna in 1924. It seems to have been a golden time for Goclel, intellectually and in other ways too. One could hardly walk clown the street in Vienna in the 1920s without bumping into a major cultural or scien­ tific luminary of the twentieth century­ and the University of Vienna was no less blessed in this respect. Godel at­ tended lectures by Philipp Furtwangler (cousin of the great conductor), Hein­ rich Gomperz, Hans Hahn, Moritz Schlick, and Rudolf Carnap (all of whom are pictured here), to name just .

=

a few; he also quickly found for him­ self a society of like-minded and, judg­ ing from the photographs, high-living, university friends, who, as was cus­ tomary then in Vienna, spent their clays in a series of roving discussions in cafes, some of which are pictured. Many of these friends were to go on to promi­ nent positions in post-war academia: Herbert Feigl, for example, a student of Moritz Schlick and one of Godel's best friends, was to become president of the American Philosophical Association­ but he is seen here as a young man ca­ vorting at the beach with the Schlick family. Other friends from this period who would later emigrate to the United States include Karl Menger, Olga Taussky, and Oskar Morgenstern. Schlick, a central figure in the Vien­ nese philosophical culture at the time, was, together with Hans Hahn, the leader of the so-called Vienna Circle, also called the Schlick Circle, a discus­ sion group Godel attended that quickly became identified with the doctrine of "logical positivism" (a term coined by Feigl and Blumberg in their 1931 "Log­ ical positivism: A new movement in Eu­ ropean philosophy" [1]). Godel was never drawn to logical positivism him­ self, but the exposure to the discussions that took place at those meetings must have been crucial to his development as a logician. Otto Neurath, Karl Menger, Gustav Bergmann, Rudolf Car­ nap, and Friedrich Waismann were just a few of the regular members; Ludwig Wittgenstein visited periodically, as did outsiders like Frank Ramsey and W.V.O. Quine. That Hahn and Schlick would come together to lead this historic seminar was characteristic of the hybrid nature of logic at the time. Godel commented on this to Hao Wang in the late 1970s ([4], p. 82): When I entered the field of logic, there were fifty percent philosophy and fifty percent mathematics. There are now ninety percent mathematics and only one percent philosophy . . . Godel began his studies in Vienna in 1924 in physics with Hans Thirring; and although he switched to matheThe sun was setting on this golden period of G6del's life

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matics two years later, studying first number theory under Furtwangler and then logic under Hahn and, informally, Carnap, he continued to take physics courses up to the time he graduated in 1929. The background in physics Godel obtained as a student may explain his being able to make a significant con­ tribution to relativity in 1947. (See [2].) In fact, Godel wrote this about his work in relativity in a 1955 letter to Carl Seelig (see [3] , p. 252): I have, however, in connection with certain philosophical problems, de­ voted myself for some time to a less difficult complex of questions from general relativity theory, namely cos­ mology. The fact that here I, as a newcomer to the field of relativity theory, could immediately obtain es­ sentially new results seems to me sufficient to demonstrate the unfin­ ished state of the theory. That the sun was setting on this golden period of Godel's, if not Vi­ enna's, if not indeed Europe's exis­ tence, is first alluded to on page 26, where it is mentioned that when Godel obtained his Austrian citizenship in 1929, the year he turned in his thesis containing the completeness theorem for first-order logic, the political situa­ tion in Vienna was becoming increas­ ingly violent, with riots in 1 927 leaving 89 people dead and the conservatives in essentially open war with the so­ cialist majority. The political situation manifested itself with particular vehe­ mence at the University of Vienna, which, being considered particularly "red," was increasingly targeted by the authorities-with, of course, disastrous results as the 1 930s wore on and Na­ tional Socialism eventually saturated the university. By May of 1 938, out of 258 emeritus and full professors, asso­ ciate professors, and lecturers, 94 had "retired" from the philosophical faculty; and those salaried employees who wished to remain had to endure a so­ called dejudification procedure in or­ der to keep their positions. The oath to the Fuhrer that univer­ sity personnel had to sign is reproduced here on page 57, as is a letter from the Studentenfuhrer from that month on page 56, apprising students of the new post-Anschluss decorum, which re­ quired them to stand and give the Hitler salute at the beginning and end of all

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lectures. Such an occasion is pictured in a chilling photograph on page 55 of the book. Finally, in a tragic symptom of the times, Schlick was assassinated at the university in 1936 by an ex-student named Nelbock, whom Godel knew from his mathematics and philosophy classes. After mounting a defense based on the idea that Schlick's atheism had caused him to become deranged, Nel­ bock got off after serving only two years for the crime. In a post-war coda to the Nelbock episode, we learn (on page 186) that Viktor Kraft, who wrote the first book about the Vienna Circle, was successfully sued by Nelbock in 1950, for Kraft's use of the term "persecution mania" to describe Nelbock at the time of the assassination-that is to say, Kraft had to withdraw the expression. Need­ less to say, deprived of one of its found­ ing members, the Vienna Circle broke up soon afterwards. The thought of Godel in the midst of all this gives one pause, although of course, not being Jewish, his difficulties paled besides those of his Jewish friends and teachers. Nevertheless Godel's dif­ ficulties during the 1930s-his most pro­ ductive decade mathematically-were substantial, and are documented here. He was hospitalized a number of times with nervous breakdowns, due perhaps not only to a natural inclination in this direction, but perhaps in part also due to outside circumstances. Hahn as well as Schlick had died, and his Dozentur, which he had gotten in 1932, was with­ drawn and could only be restored as a so-called "Dozent neuer Ordnung" upon his being found ethnically and politically acceptable to the Nazi regime. At the same time his problems with the au­ thorities were compounded by his three trips back and forth to the United States during that decade, each of which re­ quired a travel visa. Finally, he was roughed up in the street one day by some young men, reportedly members of the Hitler Youth. Regarding Godel's university posi­ tion, in a letter to the Rector (repro­ duced here on page 68), the Ministry of Interior and Cultural Affairs recom­ mended against Godel being granted the neue Dozentur, on the grounds of his "doubtful" political stance. This was not just an idle guess on the part of the Ministry, but the result of an investiga-

tion involving the interrogation of Godel's friends and colleagues. Never­ theless, Godel was eventually granted the Dozentur in 1940; but by that time he and his wife Adele had already moved to the United States. The Godels were never to return to Europe. In fact Godel would write to his mother (in a letter reproduced on page 87) that for a time he was plagued by nightmares of being trapped in Vienna and not be­ ing able to leave. As for Godel's life in the United States, one letter stands out particularly as a footnote to the Godel-Thirring con­ nection. Before Godel left Vienna for the United States in 1940, Thirring asked him to warn Einstein that Nazi Germany might be in a position to develop a nu­ clear weapon (the possibility of atomic fission having been discovered in Berlin in late December 1938, when for the first time fission was actually produced in the laboratory). Many years later (in 1972) Thirring wrote to Godel and asked him if he had ever passed on the warning. In this letter (reproduced on page 1 43) Godel replied that he had not passed it on. He gave a number of rea­ sons: at the time he had been out of contact with physics for a decade, and hadn't known about the development of fission; but then when he did finally hear about "these things," he was skep­ tical-not for any scientific reason, but for what he called a "sociological rea­ son": he didn't think the culture at the time was at such a point in its evolu­ tion for such a development to ensue. Rather this should come at the end of our "Kulturperiode. " Although i t i s not difficult t o imag­ ine Godel's real reasons for not con­ tacting Einstein about Thirring's warn­ ing-certainly the reasons Godel gives here do not seem very convincing-one may nevertheless be strongly inclined to take Godel to task over the matter. As it happens the physicist Leo Szilard had already visited Einstein in 1939 to convey the same warning. In any case, after the war Godel's and Einstein's views about the stockpiling of nuclear weapons and the pursuit of the ensu­ ing arms race between the USSR and the United States, and about matters of war and peace in general, were nearly indistinguishable. Godel had no hesita­ tion about expressing those views-at least in the 1950s:

Einstein warned the world not to try to attain peace by rearmament and intimidating the adversaries. He said that this procedure would lead to war and not to peace, and he was quite right. And the fact is well known that the other procedure (try­ ing to come to an agreement in an amiable way) wasn't even attempted by America, but refused from the first. It isn't the one and only ques­ tion as to who started matters, and for the most part it would be diffi­ cult to establish. But one thing is cer­ tain: under the slogan "democracy, " America i s waging a war for an ab­ solutely unpopular regime and un­ der the name of a "police action" for the UN and does things to which even the UN does not agree . . Godel's correspondence with his mother was being monitored by the FBI, presumably because of Godel's friendship with Einstein. Accordingly this passage of Godel's 1950 letter was included in a report written by one General Cornelius Moynihan of the U.S. armed forces, to none other than ]. Edgar Hoover (see page 1 45).

Godel's Work In this section of the book one sees most clearly that it is aimed at a gen­ eral audience; the expert may take is­ sue with the finer points of the de­ scriptions of Godel's theorems or of his philosophical work, or find those ac­ counts too sparse. But the mathemat­ ics here is in all particulars completely correct. Also the historical emphasis is right: Hilbert, Russell , Tarski, von Neumann, Turing-all are included, with no small amount of text devoted to their contributions to logic. As is Cantor-indeed a page from a letter he wrote to Hilbert is included here, written in Cantor's extravagantly florid handwriting. We also meet up with Husser!, and Leibniz, and even Gold­ bach. In pages headed "Time travel with Godel, " Godel's work in relativity is touched upon, with a long quote from Palle Yourgrau summanzmg Godel's work and why it implies that "one can travel to any region of the past."

To show how Godel's work in physics was seen by contemporary physicists, the authors include a note from the then Director of the Institute for Advanced Study, Harry Woolf (to himself, presumably) listing the topics to be spoken about at Godel's funeral, which took place in 1978: 1. set theory + the continuum hypoth­ esis 2. logic = incompleteness + consis­ tency 3. (Minor): relativity-not worth a talk-X. On the opposite end of the spec­ trum, so to speak, the subsection called "Theology" begins somewhat inauspi­ ciously with an undated notebook from the Godel Nachlass titled: "Errors in the Bible. " But then it goes on to discuss Godel's ontological proof and his wider views on exact theology in a very sen­ sible way. One of the most interesting docu­ ments in the book appears in this sec­ tion (page 1 59) : the receipt for the copy of the papal encyclical called "Mit bren­ nender Sorge" (in English ·'With burn­ ing worry") issued by Pope Pius XI.1 The encyclical, published (unusually) in German in March of 1937 but distrib­ uted secretly, condemns Nazism. The receipt indicates that Godel purchased his copy on December 20, 1937. Unfortunately, very short shrift is given to Godel's philosophical work. For example, in a section called "Plato's Shadow, " subtitled "An unadulterated Platonist,"2 we are only told that Godel's Platonist views "appear strange in the twentieth century."

Godel's Vienna Much to their credit, the authors cast a wide net in this section, including not only photographs and text devoted to Godel's scientific milieu, but to figures from the wider culture such as Robert Musil, whom Godel never met, and Her­ mann Brach, whom he did. Interest­ ingly, both wrote novels whose heroes are mathematicians; in fact, the authors remind us on page 1 94 that Brach's 1933 novel called The Unknown Quan­ tity features a hero who dreams of find­ ing a logic without axioms-not a little

reminiscent of the project of informal rigor due to Godel and also to Georg Kreisel. The book ends with two letters the poet Hans Magnus Enzensberger wrote to Godel in 1957 and in 1974, asking him for an interview. I was delighted to notice that the return address on the 1974 letter is 1 5 Commerce Street, New York City, which was then the home of Christiane Zimmer, nee von Hofmannstahl, daughter of the famous poet and Strauss librettist Hugo von Hofmannstahl, and herself a wartime emigre to the United States with her hus­ band the Indianologist Heinrich Zim­ mer. Most likely Enzensberger would have been staying with Mrs. Zimmer, who by 1974 had become the sine qua non of emigre German and Austrian life in New York City: she offered material and other support to untold numbers of artists-for example, the food served at her Sunday night salons, which I had the pleasure of attending in its last years, was often the only square meal an artist could have gotten in New York that week. She also offered hospitality to any and all visiting German and Austrian writers, such as Max Frisch, Gunter Grass, and Siegfried Lenz. Her picture should perhaps have been included here; not necessarily for her very faint connection to Godel, but because she demonstrated so beauti­ fully the idea that one could create a new and vibrant life in a new country, even after being forced out of one's own country-a lesson Godel was never to learn, scarred as he was by the events of the 1 930s, and perhaps by events from his childhood too. REFERENCES

[1] Herbert Feigl and Albert Blumberg, Logical positivism. a new movement in European philosophy. Journal of Philosophy 28:28 1 296, 1 931 . [2] Kurt Godel, "An example of a new type of cosmological solutions of Einstein's field equations of gravitation, " Reviews of mod­ em physics 2 1 :447-450, 1 949.

[3] Kurt Godel, Collected Works. V: Corre­ spondence H-Z. S. Feferman, et a!., eds.,

Oxford University Press. Oxford, 2003.

1 1 n a rare slip, the authors get the German title wrong, misidentify it as a bull, and attribute it to Pius XII.

2This is actually Russell's description of Gbdel in the second volume of Russell's Autobiography, on the occasion of his meeting Gbdel at Einstein's house in Prince­ ton in 1 943.

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[4] Hao Wang, A logical journey: From Godel to philosophy. Final edition and with an ad­

dition to the preface by Palle Yourgrau and Leigh Cauman. MIT Press, Cambridge, MA, 1 996. Juliette Kennedy Department of Mathematics and Statistics University of Helsinki Helsinki, Finland e-mail: [email protected]

Die Vermessung des U nendl ichen ( Measuring Infinity) an opera by Ingomar Griinauer REVIEWED BY JEAN-MICHEL KANTOR

art of mathematics consists in giv­ ing precise meanings to specific words. Take 'infinity': it first ap­ peared two thousand years ago when Anaximander of Miletus, a pre-Socratic philosopher, coined the word 'Ape­ iron'-an obscure notion of vagueness and unboundedness.1 It took centuries to give a mathematical meaning to in­ finity. And it is still not over!2 The first, crucial step was Aristotle's claim that there was no 'actual infinity', just the 'potential infinity' represented by one, two, three, . . . In fact, until re­ cent surprising discoveries concerning Archimedes, 3 general opinion held that for the Ancient Greeks there was no real infinity. The mathematics of infinity in the modern sense began in the middle of the nineteenth century with the priest Bernard Balzano ( 1781-1848). Balzano wrote a book, Paradoxes of the Infi­ nite, in which he tried to define a cal­ culus with all sorts of infinities (infi­ nitely small and large). When Georg Cantor ( 1845-19 18), studying the sets of unicity (now called exceptional sets) of trigonometric series, extended the work of his colleague Heinrich Heine (182 1-1881), he was led naturally to the

study of parts of the continuum, the set of all real numbers. Cantor used the language of sets and functions, and with his proof that the set of real num­ bers is non-denumerable he set the stage for the new mathematical theory of infinity. Then he introduced the main actors: transfinite numbers, cardinals and ordinals, and the Alephs. But Can­ tor was also hoping to achieve a sub­ lime goal-understanding the infinity of infinities. Not only was he aware of the religious dimension of his work, it was a strong stimulus, as it had been for other mathematicians, such as Pythagoras, or Luzin and the Name­ worshippers, or Godel. Cantor consid­ ered his theory of sets a revelation of truth inspired by God. Much later, set theory became the lingua franca of mathematics, and al­ though some important difficulties had been apparent from the beginning, the new math (as they called it) was taught in most schools from the 1 960s on. Cantor, who created set theory, was born in Saint Petersburg, but moved with his family to Germany when he was twelve. He was a professor at Halle University from 1 869 to 191 3.4 Cantor suffered early from manic depression, which increased after 1899, and he later spent periods in various psychiatric in­ stitutions. He is buried in Halle. The city of Halle was founded 1 200 years ago, but it has recently fallen on hard times: due to the economic crisis in the former German Democratic Re­ public, the city lost a third of its popu­ lation in five years. The city center has many old and well-preserved buildings, among them the house where Georg Friedrich Handel was born, but the outer parts of the city resemble a Russ­ ian provincial town. The Opernhaus of Halle is a nice building of classical eigh­ teenth-century style. For its Jubilaeum, the city council, to honor Cantor, the other of its two great men, commissioned an opera from In­ gomar Griinauer, a composer born in Vienna in 1938. Griinauer did not try to give a precise mathematical account of Cantor's work, but rather was loosely inspired by

1Simplicius quoting Anaximander in "Commentaries of Aristotle's Physics," 24,1 3. 2Woodin, Hugh W., The continuum hypothesis, Parts 1 and 2, Notices of the AMS. vol. 48, 200 1 .

-the resistance from the older gen­ eration of mathematicians to the new mathematics, -the resistance from younger ele­ ments like Cantor's friend Schwarz, -the efforts of Cantor to prove the Continuum Hypothesis (CH), one of the central questions of set theory even now, and -Cantor's mental problems and problems connected with his wife and daughter. On stage Cantor is euphoric when he thinks he has proven CH, and goes ahead with the hierarchy of the Alephs. During depressive periods he sits by a river-a feature of much German ro­ mantic literature (one cannot help thinking of Holderlin's poems). The Continuum Hypothesis is more than a leitmotif: it is the symbol of the opera. The formula 2t<�o c is written at the center of the stage curtain, and is often quickly flashed onto the wall, a hundred times during some scenes. It is even chalked by Cantor on his violin. Axel Kohler plays Cantor, a perfect role for this very gifted singer. He sings as a baritone when remembering his youth or the happy periods of his life, and as a counter-tenor when thinking about set theory. When Cantor opposes his enemies, like Kronecker, the singer speaks, while his depressive moods are represented by the sound of the violin he plays. The old mathematics is rep­ resented by old mandarins in wigs, sometimes even carrying plaster statues. The new mathematics is represented by four charming Alephs, dancing women who come and take off the wigs of the old teachers. The music is an oratorio with most of the themes inspired by the famous Fugue in B Minor from Bach's 'Well-tempered clavier'. This fugue is also well-known to musicians because it uses all twelve half-tones, and it in­ spired Schoenberg. Griinauer's post-se­ rial style is still quite popular among musicians in Germany. The orchestra, led by Roger Epple, a popular director there, received strong applause. The staging of the opera is quite elaborate, with part of the orchestra at the rear, sometimes visible, with a part

3cf. Reviel Netz and William Noel, The Archimedes Codex, May 2007, Weidenfeld & Nicolson. 4Stern, Manfred, Memorial Places of Georg Cantor in Halle, The Mathematical lnte!ligencer, v.1 0, n.3, 1 988, 48--49.

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of the chorus coming in front to sing its opposition to the new mathematics, and the rest of the choir standing on ei­ ther side of the balcony. Huge mirrors­ flat, plastic screen panels that move on the stage-sometimes close in on Can­ tor when he is mentally distressed. The set-up is complex, but the seven scenes follow quickly one after the other un­ til the final scene, where Cantor bor­ rows Icarus's wings from his daughter. Having come too close to the Sun­ the Absolute-Infinity of Infinities-the wings catch fire, and the opera ends. An interesting interpretation indeed of an important chapter in the history of mathematics. 0 0 0

Interestingly, conveying highly abstract matters through the medium of opera seems to be a trend. "La Passion de Si­ mone," an oratorio on the life of the fa­ mous philosopher Simone Wei! (sister of Andre Wei!, a great mathematician of the last century and one of the creators of the Bourbaki group) premiered in Vi­ enna on November 26, 2006, under the direction of Peter Sellars. The oratorio was composed by Kaija Saariaho, a Finn living in France; the libretto, by Amin Maalouf, a Lebanese writer also living in France. It will be performed in Los An­ geles in 2007 (see www.saariaho.org). Jean-Michel Kantor Universite de Paris VII , France e-mail: [email protected]

The Oxford M urders by Guillermo Martinez ABACUS, 2005, 197 PP., US $20.65, ISBN 0-349-11721-7 REVIEWED BY MARY W. GRAY

I

magine arriving from Argentina for a post-doc at Oxford and finding your landlady's body shortly thereafter. Enter Seldom, an eminent mathemati­ cian ("one of the four leading minds in the field of logic"), who has written a book with a chapter on serial killers as well as proved the fictional mathemat­ ical analog of Heisenberg's Uncertainty Principle. Then imagine the introduc­ tion of symbolism worthy of the

Da Vinci Code [1] mixed with a little Wittgenstein and a touch of Gi:\del. Fi­ nally, in a deus ex machina moment Seldom presents a solution. There you have the gist of The Oxford Murders. This is a mathematical mystery in which the mathematics, real and imag­ inary, is definitely more interesting than the myste1y. For setting the scene, char­ acter development, and posing an in­ tellectual puzzle, Dorothy Sayers in Gaudy Night [2] and Reginald Hill's In­ spector Morse still reign supreme in the Oxford milieu. Unlike those of the char­ acters created by many other authors, Seldom's fictional mathematical result is non-trivial, interesting, and not totally unbelievable. Translated from the Spanish ( Crimenes imperceptibles), Martinez's novel shows a hint of the magical realism of his fel­ low Argentinean, Jorge Luis Borges (in fact, Martinez has written in Borges y la Matemiitica [3] of the mathematics in his work). Keeping this heritage in mind will get the reader through some rather implausible bits. Also implausible, al­ though not in the same way, is that the mathematician-narrator passes up the chance to go hear Wiles announce his result in order to go on a tennis date, albeit with an attractive woman. In a twist to the current Poincare contro­ versy, the narrator's Russian office mate at the Mathematical Institute alleges that the Fermat proof was stolen from him. There is also a hint along the way that the deaths might be a warning to math­ ematicians and that Wiles might be in danger. This is said to relate back to Fermat's association with a latter-day Pythagorean group and the connection of his Last Theorem with a series of mysterious deaths, including those of Turing and Taniyama. The murder of the landlady, a WWII cryptologist, is announced in a note as "first in a series, " accompanied by a mysterious circle-to be followed by a fish and a triangle after subsequent deaths. What next? A Pythagorean would know, but even in Oxford would, as Martinez claims, a random member of the general public realize what symbol comes next? The central­ ity of Pythagorean thinking to the plot is clear from the German title: Die Pythagoras-Morde, although in fact the key to the mystery lies in the Ash­ molean's Assyrian frieze. That there is

little character development in the book may reflect the author's dedication to the elegance of a proof with no irrele­ vant distractions. However, it is the challenge of weeding out the irrelevant distractions that, for most readers, makes for a good mystery. Aside from the narrator's indifference to the Wiles seminar, the author does a good job of invoking the atmosphere of the Oxford Mathematical Institute. One could argue whether the attempt to ex­ plain Gi:\del's theorem is entirely effec­ tive, although the analogy to the in­ ability to prove guilt or innocence (the Scottish "not proven" verdict) is apt. However, the injection of mathematics into the story is more effective than in the recent Uncle Petros and Goldbach 's Conjecture [4] or The Parrot's Theorem (5]. Nonetheless, some of the explana­ tions can get a bit tedious, with pas­ sages such as the following occupying many pages: In those days I was a fervant Com­ munist and was very impressed by a sentence of Marx's, from The Ger­ man Ideology, I think, which said that historically humanity has only asked itself the questions it can an­ swer. For a time I thought this might be the kernel of an explanation: that in practice mathematicians might only be asking the questions for which, in some partial way, they had proof. . . . As I listened, I had a sudden in­ tuition, the knight's move, so to speak, that exactly the same kind of phenomenon [as the Heisenberg Principle] occurred in mathematics, and that everything was, basically, a qu est i on of scale. The indeter­ minable propositions that G6del had found must correspond to a sub­ atomic world, of infinitesimal mag­ nitudes, invisible to normal mathe­ matics. The rest consisted in defining the right notion of scale. What I proved, basically, is that if a mathe­ matical question can be formulated within the same "scale" as the ax­ ioms, it must belong to the mathe­ maticians' usual world and be pos­ sible to prove or refute. But if writing it out requires a different scale, then it risks belonging to the world-submerged, infinitesimal, but latent in everything-of what can be neither proved nor refuted.

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Martinez has an interesting characteri­ zation of the process of proof: In mathematics there's a democratic moment, when the proof is set out line by line. Anyone can follow the path once it's been marked out. But there is of course an earlier moment of illumination . . . . Only a few peo­ ple, sometimes only one person in many centuries, manage to see the correct first step in the darkness. The book's title will attract many non-mathematicians, and perhaps Martinez has the gift of helping peo­ ple follow the path. The Independent [6] said in its review "Read it, and be temporarily convinced that applied mathematics is suddenly within your grasp." The character Seldom appears in Re­ garding Roderer [7] (Acerca de Roderer), an earlier novella by Martinez. An ex­ tended discourse on proof and binary logic, it is an examination of genius and its destructive power. It concludes with its Argentinean narrator, unnamed as in Oxford Murders, going off to study un­ der Seldom at Cambridge [Cambridge is correct]. Martinez has also published another novel, La Mujer del Maestro [8], and several books of short stories. Roderer has been included in a collec-

tion of the best Argentinean books of the century. Oxford Murders has won the Argentine Planeta prize and has been translated into several dozen lan­ guages. Although he has been a fre­ quent participant in writers' workshops, Martinez's day-to-day job is directing the Mathematics Department at the School of the Sciences at the University of Buenos Aires. It is reported that Oxford Murders will be filmed in 2007, for release in 2008, with John Hurt playing the role of Seldom. It will be interesting to see whether it is rather faithful to the book or as different as the film of A Beau­ tiful Mind [9] was from Sylvia Nasar's book. Martinez's book has a limited appeal, but mathematicians, especially those who have spent some time at Oxford, will find it absorbing as an at­ mospheric novel if not necessarily as a mystery. I can see how the film could be made to attract a wider audience if there were to be more emphasis on the dramatic-perhaps an enactment of Wiles's announcement-and the ro­ mantic. The choice of Hurt seems to indicate that the focus of the film would be on Seldom, but if someone like Diego Luna of Y Tu Mama Tam­ bi{m [ 10] were selected to play the un-

�Springer 1M

named narrator, a whole new audience might develop an interest in mathe­ matics. REFERENCES

[1 ] D. Brown, The DaVinci Code, New York: Doubleday, 2003. [2] D. L. Sayers, Gaudy Night, New York: HarperTorch, 1 995. [3] G. Martinez, Borges y Ia Matematica, Buenos Aires: Eudebra, 2003. [4] A Doxiadis, Uncle Petros and Goldbach's Conjecture, New York, Bloomsbury USA,

2001 . [5] D. Guedj, The Parrot's Theorem, New York, Thomas Dunne, 2001 . [6] E. Hagestadt, The Independent, 27 Janu­ ary 2006. [7] G. Martinez, Regarding Roderer, New York, St. Martin's Press, 1 994. [8] G. Martinez, La Mujer del Maestro, Buenos Aires, Planeta, 1 998. [9] S. Nasar, A Beautiful Mind, New York, Si­ mon & Schuster, 1 998. [1 0] Y Tu Mama Tambi{m, IFC Films, 2002. Department of Mathematics and Statistics American University Washington DC 2001 6-8050 USA e-mail: [email protected]

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78

THE MATHEMATICAL INTELLIGENCER

k1fi,i.C9.h.l§i

Robin Wilson

The Philamath' s Alphabet-0 Octagonal stamp: Everyone is famil­

iar with the usual rectangular stamps issued by every country. However, sev­ eral countries have produced stamps whose shapes are more unusual. An early set of octagonal stamps was pro­ duced by Thessaly in 1 898; these came supplied with four corners that were removed before mailing.

Octant: An octant is an instrument that

was used for navigational purposes. Like the sextant, which is based on an angle of one-sixth of a circle (60°), the octant is based on one-eighth of a cir­ cle ( 45°). To measure an object's alti­ tude, the observer views it along the top edge of the instrument, and the po­ sition of a movable rod on the circular rim gives the desired altitude. Omar Khayyam: Omar Khayyam, or

al-Khayyami ( 1048-1 131) was a mathe­ matician and poet who wrote on the bi­ nomial theorem, algebra, and geome­ try. In algebra he presented the first systematic classification of cubic equa­ tions and discussed their solutions; such equations were not solved in general until the sixteenth century. In geometry he was concerned with the 'parallel postulate' of Euclid. In the West he is remembered mainly for his collection of poems known as the Rubazyat. Ortelius: In 1 570 the illustrious Belgian cartographer Abraham Ortelius ( 1 527-

Octant

Please send all submissions to the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics, The Open U niversity, Milton Keynes, MK7 6AA, England e-mail: r.j.wi [email protected]

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Oaear II

1 598), geographer to the king of Spain, produced his Theatrnm orbis terrarnm, a collection of seventy maps; this is usu­ ally considered to be the first atlas. The word 'atlas' was coined by his friend the cartographer Gerard Mercator. Oscar n, King of Sweden and Nor­ way ( 1829-1907) was an enthusiastic

patron of mathematics. On his sixtieth birthday he offered a prize of 2500 Swedish crowns for a memoir on math­ ematical analysis. The winner was Henri Poincare ( 1 854-1912), who wrote on the 'three-body problem' of determining the motion of the sun, earth, and moon. Ostrogradsky: Mikhail Ostrogradsky

( 1 801-1861) took up mathematics to fi­ nance his ambition of becoming an army officer. While studying in Paris, he worked in mathematical physics, proving the 'divergence theorem', often ascribed to Gauss. He later achieved his ambition by teaching mathematics in the Russian military academies.