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Letter to the Editors Re: the Locker Puzzle ERIC GRUNWALD The Mathematical Intelligencer encourages comments about the...

Author: M.L. Senechal | C. Davis (Editors in Chief)

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Letter to the Editors

Re: the Locker Puzzle ERIC GRUNWALD

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.

aving been astonished by the Locker Puzzle and its winning strategy (Curtin and Warschauer, Mathematical Intelligencer 28 (2006), no. 1), I was delighted to read A. S. Landsberg’s letter (Mathematical Intelligencer 31 (2009), no. 2) with its charming variant on the original problem. I would like to point out that Curtin and Warschauer’s strategy is so impressive that, amazingly, a third person can be introduced into Landsberg’s game without reducing the overall probability of success. Thus there are three curtains, behind which are hidden a car, a car key, and a GPS system. Three people, the carmaster, the key-master, and the GPS-master, have two attempts each to find their designated object. After each one has tried, the curtains are pulled closed again, and once they have started looking, there is no communication between them. The strategy ensures a probability of 2/3 that all three people find their own object.

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Mathematical Capital Limited 187 Sheen Lane London SW14 8LE UK e-mail: [email protected]

Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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Letter to the Editors

Response to a Review of My Book 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.

his is my response to the review by Osmo Pekonen of The Artist and the Mathematician, in Vol. 31 (2009), No. 3. I found the review of my book The Artist and the Mathematician, which details the story of the fictitious mathematician Nicolas Bourbaki somewhat unfair. Mr. Pekonen disagrees with my characterization of Bourbaki as having played a significant role in the structuralist movement in mid-twentieth century France. Suffice it to say (as indeed I do in the book) that some of the members of the structural movement were also mathematicians—prominent among them Raymond Queneau—and that most mathematicians would agree that the approach to mathematics taken by Bourbaki was essentially ‘‘structuralist’’ in its very nature. Pekonen writes, ‘‘Aczel tends to overemphasize the impact of Bourbaki both inside and outside the mathematical community.’’ Perhaps Mr. Pekonen is much younger than myself. Most mathematicians of my age and older, and especially those with an international outlook, know quite well how influential Bourbaki really was—during the time of ‘‘his’’ greatest productivity. As I point out in the book, the whole ‘‘New Math’’ movement in the United States was a direct outcome of Bourbaki’s work and immense influence on mathematics.

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THE MATHEMATICAL INTELLIGENCER Ó 2010 Springer Science+Business Media, LLC

But the unkindest cut of all is when Pekonen writes: ‘‘He [Aczel] most often fails when he needs to explain a mathematical concept. The book is infested with errors and misconceptions.’’ Now, I’ve been called many things in my long career as a writer of books on mathematics, but nobody has, until now, accused me of failing to explain mathematical concepts. In a fair review—by a book-review editor, no less!— I would have expected that an outrageous statement such as this one be followed by at least one example of how I ‘‘failed’’ to explain a mathematical concept. None was given. Nor did Mr. Pekonen give a single example of the ‘‘errors and misconceptions’’ that he supposedly found in my book. If he found any, I would appreciate being informed of exactly what these ‘‘errors and misconceptions’’ were. Such unsubstantiated accusations have no place in a review in a professional journal. Finally, let me point out a ludicrous (and self-serving) element of Pekonen’s review. Part of my book was devoted to the misadventures of a key member of the Bourbaki group, the mathematician Andre´ Weil, in wartime Finland. This part was based largely on Weil’s own recollections in his autobiography, The Apprenticeship of a Mathematician (Boston: Birkhaüser, 2002), and my book makes ample references to Weil’s published book. During the Second World War, Weil was arrested as a spy by the Finnish police, and he felt that there was a good chance that he would be executed by Finnish officials. Mr. Pekonen, whose address is in Finland, seems to have taken offense to my quotations from Weil’s autobiography. Furthermore, he found it appropriate to quote an article he himself has written, in which he purportedly ‘‘proves’’ (how does one prove such a thing?) that Weil, in fact, was not in danger of execution. I fail to see the importance of Pekonen’s argument. Does it matter at all whether Weil was or was not about to be executed? What matters to me is how Weil felt while a wartime prisoner in Finland. Whether or not an execution would have actually taken place if he had remained in custody is a completely moot question.

Amir D. Aczel Center for the Philosophy and History of Science Boston University Boston, Massachusetts USA e-mail: [email protected]

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Answer in Search of a Question CHANDLER DAVIS

he cover of this issue of The Mathematical Intelligencer displays the full solution set of a combinatorialgeometric problem, to be explained below. The solution, and the problem, are due to Anthony Hill. Together they supply an answer to a problem in poetry. The poet in the story was Hill’s friend the late R.C. Kenedy, whose day job—poets tend to need day jobs—was as a librarian at the National Art Library in the Victoria & Albert Museum. Kenedy had completed a poem, ‘‘The Legend of the Changeless Girl’’, consisting of 66 six-line stanzas, and the inspiration struck him that each stanza should be accompanied by a different geometric design. All of the same sort, of course. Hill being a devotee of this kind of combinatorics, he accepted the challenge of supplying 66 designs of the same sort but not isomorphic; and as mathematicians often will, he set himself the complementary demand that the list of designs be not only without duplications but also exhaustive. Why settle for ‘‘if’’ when you can have ‘‘if and only if’’, after all!

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Failing to find a problem in his domain of mathematics whose complete list of solutions comprised exactly 66 objects, he set about trying to pose one. In an ambitious effort at still greater elegance, he demanded that the objects be graphs made up of exactly six lines each, in accord with the poem’s six-line stanzas. Some readers will want to glance back at our cover and identify what those doodles are. Maybe even to verify that they are non-isomorphic and are all the instances of what they are. They are all the embeddings in the plane triangular lattice of the smallest planar graph with no symmetries. This graph had been found by G. Po´lya some years before, and Hill knew a way to find all its embeddings. (See his article ‘‘The Continuous Charms of Discrete Mathematics— Some Grassroot Problems in Ars Combinatoria’’ in Proble`mes combinatoires et theórie des graphes, CNRS, 1976, in particular p. 231.) The poem by Kenedy for which this problem came into existence was never published, and is lost. Some young mathematician poet will perhaps be inspired by Hill’s example to compose a poem of 66 six-line stanzas worthy of adornment by these emblems.

Department of Mathematics New College, University of Toronto Toronto, Ontario, M5S 3J6 Canada e-mail: [email protected]


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Note

Double Interrogation ROBERT J. MACG. DAWSON

Department of Mathematics & Computer Science St. Mary’s University Halifax, NS B3H 3C3 Canada e-mail: [email protected]

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Snowflake Caleb Emmons

Department of Mathematics and Computer Science Pacific University Forest Grove, OR 97116 USA e-mail: [email protected]


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Viewpoint

Mathematics and Finance: An Ethical Malaise

Note: A preliminary version of the present article was rejected by the Gazette des Mathe´maticiens, published by the Socie´te´ Mathe´matique de France, on the basis that ‘‘a certain boredom with questions of mathematical finance may be setting in among our readers.’’ There is malaise, but talking about it may be boring … .

MARC ROGALSKI

A Long-Standing Malaise Grows Sharply Worse

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-inchief endoreses or accepts responsibility for them. Viewpoint should be submitted to either Editor-in-Chief, Chandler Davis or Marjore Senechal.

he recent financial crisis, and the ensuing economic and social crisis, have brought attention to the role that mathematics has been playing since the 1970s in national and international finance. Are mathematicians responsible for the crisis? This question, raised by several media, has put onto the mathematicians’ agenda a debate on the ethics of their discipline. But this debate has been sotto voce for the present, there have been few public 1 critiques; most of the comments concern technical aspects of the role of mathematics in finance. My aim here is to put forward some reflections emphasizing the political and social role of financial mathematics; I refer to the French example but, likely, the analysis holds for many Western countries.

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The malaise caused among many French mathematicians by the application of mathematics to finance is not a new thing. One of the first signs goes back a good many years now. At a session in March 1997 sponsored by, among others, the Socie´te´ Mathe´matique de France, Ivar Ekeland, presumably wanting to show how effective mathematics can be, made essentially the following assertion: ‘‘In the last 15 years, the flow of financial assets around the world has increased 10,000-fold.’’ I must say I was flabbergasted by this claim, and still more by the absence of any reaction from the eminent mathematicians there present! After all, if the exchanges of goods and services had increased 10,000-fold during the 15 years in question, then for sure we would have noticed it! It must, therefore, have been a matter of a parasitic growth of the world financial sector—and this, according to Ekeland, ‘‘thanks’’ to mathematics. Since then, the financial crisis and the economic crisis came along, and were followed by numerous polemics in newspapers and other French publications about the role and the responsibility of mathematicians. I did my best to read these discussions—which is not always easy for one with no knowledge of the techniques of financial mathematics, and little enough of finance. I must say that most of the questions raised and most of the arguments made seemed to me to be quite off the principal issue. For, as Ivar Ekeland says in a remarkable report, ‘‘Mode`les ećonomiques et morale’’ in the January 2009 issue of Pour la Science, economics (hence finance, albeit mathematized) is just a technique, and it is politics which chooses whom it serves. And whom it serves is the crucial question.

In Whose Interests Was Financial Mathematics Developed? I turn to a public lecture at a conference on financial mathematics held in February 2005 by the French Acade´mie des

1 But see, in France, N. Bouleau, ‘‘Malaise dans la finance, malaise dans la mathe´matisation,’’ Esprit, vol. 120 (February 2009), 37–50; A. Lambert and L. Mazliak, ‘‘E la nave va?’’ Gazette des Mathe´maticiens, 120 (April 2009), 103–105.; and J.-P. Kahane, ’’La science, les lumie`res et les ombres, le cas des mathe´matiques financie`res,’’ Bulletin de l’Association des Professeurs de Mathe´matiques de l’Enseignement Public (APMEP), vol. 486 (Jan. 2010), 51–59.

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THE MATHEMATICAL INTELLIGENCER 2010 Springer Science+Business Media, LLC

Sciences for the following account2 of the role of mathematics in finance: When one takes a look at the European financial market as described by the fluctuation of the EuroStoxx index during the last 12 months, one sees a very irregular trajectory …. Given this uncertainty, the market offers many possibilities of placing a financial bet on the future behavior of the index. For instance, one can buy or sell a contract which gives the right to the value of the index six months later. In fact there is an increasing number of derivative products such as options and certificates which allow to configure arbitrarily financial bets. What is the role of mathematics, and, in particular, of probability theory, in this financial context? As with games of chance, mathematics cannot help to win a financial bet. In particular, it does not offer any means to compute in advance the outcome. On the other hand, mathematics may help to understand the nature of a given bet by providing methods to decide whether the bet is advantageous, fair or acceptable, and to quantify its risk. Moreover, nowadays mathematics intervenes more and more in the construction of such bets, that is, in the design of new financial products. So one sees that the principal subject of financial mathematics is speculation. It should be recalled that the forerunner of this domain, Louis Bachelier, was honest enough to entitle his 1900 thesis ‘‘Theórie de la spećulation’’. It is thus inescapable that the question, ‘‘In whose interests was financial mathematics developed?’’ has only one possible answer: Financial mathematics in the forms now used, developed and taught is directed essentially toward acquiring techniques for increasing the rate of financial profits, with greater security for the speculative operations of financial actors (stockholders, banks, markets, managers of portfolios, funds, insurers,…).

AUTHOR

......................................................................... MARC ROGALSKI was for many years at

the Universite´ Lille 1 (Sciences et Technologies), France, in the Laboratoire Paul Painleve´. He is now Emeritus. His mathematical research has been largely on infinitedimensional convexity. He continues to be active at Jussieu in Paris, in the group founded by Gustave Choquet, who years ago was his thesis director. Another major interest is the teaching of mathematics, at both the secondary and the university levels. Equipe d’Analyse fonctionnelle Institut Mathe´matique de Jussieu 75013 Paris France e-mail: [email protected]

To be sure, one might propose other answers: Permitting insurers to cover new risks, facilitating loans to private individuals, companies, financial organizations and so on. But dig a little and you always find that the point is increasing speculative profits and making them more secure. In all the texts on financial mathematics there is rarely mention of this question of the objectives; the talk is always on technical questions: Are the models developed true to reality? Are they reliable, and within what limits? Should they be ‘‘improved?’’ Have they failed to do justice to more basic prior investigations? Are their users sufficiently aware of their limitations? Now those questions to me seem secondary in light of the principal effect financial mathematics has had: It has permitted financial organizations and their stockholders to reap exorbitant—‘‘double-digit’’—profits, and that has been and still is its principal objective. This is altogether plain from reading the press—before the crisis. When we read in Le Monde of June 13, 2007 that, ‘‘In 2006 bankers and insurers endowed 13 university chairs’’ and that ‘‘French traders are found in the markets of London and New York,’’ and bearing in mind the salaries and bonuses those traders were getting then, we can figure that the banks knew what they were doing—and that what they were about then and still are is profit, not philanthropy!

What Is Mathematicians’ Responsibility in This? The passage from Hans Fo¨llmer I quoted is clear enough: How can one speculate ‘‘better,’’ more securely, and therefore with more profit. That’s just what the various theories perfected by mathematicians are modelling. Can one retreat behind the excuse, ‘‘Oh, but they aren’t the ones directly using them, there’s a whole chain: Mathematicians, then ‘quants’ (financial analysts), then traders?’’ Surely not: Looking at the programs for the master’s degree in Financial Mathematics of various French universities, one sees clearly what the mathematicians in them are teaching: They are training the quants to use the models, up to and including trading. For example, here’s a passage found on the Internet3 addressed to prospective students in Paris aiming to become quants: … in the work of the Quant one can distinguish two types of activity: First, the development of products called derivatives (options, warrants, Swaps …) In the design of the derivative products he often participates along with the other agents—traders, structurers or … clients— essentially as an expert to assess amenability to mathematical treatment. … The second essential activity of the Quant … is global risk management. He must therefore design (in part) and calculate an array of indicators of short- and medium-term risk…. Other areas are currently being developed such as energy and climate derivatives. Recruiting Quants for these is important. New perspectives are seen in the creation of markets in polluting rights. There can be no doubt of the responsibility of mathematicians in the explosion of financial speculation of the

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Hans Fo¨llmer, in Aspects of Mathematical Finance, ed. Marc Yor, Springer, 2008. Nicole El Karoui and Gilles Page`s, ‘‘Quel parcours scientifique en amont du Master 2 pour devenir ‘Quant?’’’ http://www.maths-fi.com/devenirquant.html, accessed November 24, 2009.

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2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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last 30 years. Ivar Ekeland was surely right, it was ‘‘thanks’’ to mathematics that it could go so far. With what social consequences?

Historical Excursus on Class Struggle for Division of Surplus Value It is in order to recall the general economic context, as to France and Western Europe generally: The 30 ‘‘glory years’’ 1945–1975, when the economy, run under political ideas inherited from the Second World War, allowed some social development to the benefit of wage-workers and restriction of profit margins of the capitalists, were followed by 25 years in which the class struggle for share of surplus value turned sharply to the detriment of the wage-workers: About 10 points of the value added in France was transferred from salaries to owners’ profits. What role did financial mathematics play in this change in the division of surplus value? The financial sector began reaping ‘‘superprofits’’ (thanks largely to financial mathematics), such as to arouse envy in the productive sector, whose stockholders (some of whom were the same as in the financial sector) wanted to enjoy double-digit rates of profit too. The intensity of exploitation of labor rose as a consequence: Productivity of labor rose sharply (and not only by virtue of technical advances, as is shown by many studies of ‘‘work stress’’); wages were stagnant; even flourishing concerns felt obliged to trim payrolls to cut costs; firms outsourced, subcontracted. European political agencies (like national and international financial agencies) encouraged the reduction of labor costs (by deregulation and abandonment of social welfare provisions); unemployment grew, precarious and temporary employment shot up, poverty spread, social costs (deferred benefits) shrank. … It is well known4 where this sort of process leads: The loss of purchasing power by an important component of the workers (the unemployed, those with low-paid or intermittent employment, and then also some of the middle classes) causes a drop in consumption, hence a crisis of demand, overproduction, bankruptcies, and, soon enough, a classical crisis of capitalism, with a temporary drop in rates of profit in some sectors and the destruction of part of the capital. The classical countermeasure by capital (in play also before the depression of 1929) has been the expansion of indebtedness: Individuals, firms, even states are encouraged to go into debt to absorb the goods and services produced by the economy. This was a ‘‘retreat forward,’’ a ‘‘dynamic instability’’ whose possible outcome nobody could know with confidence (though there were, especially after the bursting of the ‘‘technology bubble,’’ plenty of

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economists sounding the alarm). Here also, the techniques of mathematical finance were widely used to support this unprecedented jump in credit and to mitigate the risks to the lenders by means of an apparatus of complex ‘‘derived products’’ which got beyond anyone’s real awareness in their spread and their toxicity. The immediate cause of the crash was the ‘‘subprime’’ crisis, and the deep (unregulatable) working of capitalism was the ultimate cause of this collapse (and of the last, and of all those that will follow if nothing is done to replace the capitalist model), but we still see that financial mathematics played more than a negligible role by putting itself in the service of one side in the class struggle.

Should Mathematics Be on the Owners’ and Stockholders’ Side in the Class Struggle? This is the deep ethical dilemma. Ought the present financial mathematics, intended essentially to increase rates of profit for stockholders, to be under the control of the mathematical community? Is it ethically defensible, in a country like France where research and higher education are essentially public activities, for everything—the ministry of education, the Centre National de la Recherche Scientifique, universities, engineering schools, the Agence Nationale pour la Recherche, and all—to sanction by degree-granting, research grants, government salaries and training of students5 a frank partisanship in the class struggle?6 Can the various mathematical societies, devoted to maintaining the image of mathematics before the public, continue to ignore this question? So far, in France, these societies, by such things as their pamphlet, Les me´tiers des mathe´matiques, seem to have implicitly assumed that intervention of our profession on one side of the class struggle was acceptable—in that it offers job possibilities to some of our students. I have spoken of France, but evidently the same considerations apply to every country where financial mathematics is practised. To be sure, mathematicians have the right individually, like other citizens, to defend the rise of stockholders’ profit and workers’ exploitation. They have a perfect right to put their knowledge and mathematical skills at the owners’ disposition via financial mathematics. But must the nation pay for that? Must universities and research institutions pay for this partisan activity and steer students toward it? To me it seems clear that the only answer consistent with our professional ethics is NO! Any mathematical activity which takes sides in the class struggle must be of a different sort and in different arenas than the activities we formally organize, teach and reward in our classrooms and laboratories.

This was very well explained in a celebrated book by an equally celebrated nineteenth-century author who may be regarded by financial mathematics mavens as outdated. Maybe they should reflect on the book’s ranking third in the number of editions printed in all history, after the Bible and Euclid’s Elements. 5 See above. 6 The challenge I am proposing has no bearing on applied mathematics in general, for it need not be partisan in this sense. This is a big subject into which I do not enter.

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THE MATHEMATICAL INTELLIGENCER

Viewpoint

Response to Rogalski IVAR EKELAND

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here are three fundamental theorems in mathematical finance.

T HEOREM 1 If you take no risk, you get the riskless rate. A risk is whatever can get in the way of your getting your money back: You buy stocks, or bonds, and prices go down (market risk), or the company defaults (credit risk), or the government nationalizes it without compensation (political risk), or all financial markets stop functioning (systemic risk).

T HEOREM 2 If you take no risk, you get the riskless rate. There is no such thing as a totally riskless investment. Buying US government bonds, for instance, is risky because their resale value fluctuates (when interest rates go up, the prices of bonds go down). The closest approximation to the riskless rate is the day-to-day rate: Put your money in today, withdraw it tomorrow with accrued interest. It is very unlikely that the bank will go under during the night. At the present time, this rate is almost zero. In better times, it was around (annual) 2%.

T HEOREM 3 If you take no risk, you get the riskless rate. An immediate corollary, for anyone who has had some mathematical training, is that if you get more than the riskless rate, you are taking some risk. So this is the first lesson of the theory: If anyone, at any time, will guarantee you an interest rate of 10%, show him (or her) confidently the door. By Theorems 1, 2, and 3, a return rate of 10%, way above the riskless rate, earmarks a risky investment, and whoever says there is no risk is lying. Of course, the higher the expected rate of return, the higher the risk. If you entrust your money to someone who promises to double it within the year, you have a better than even chance of never seeing it again. Returns of 10% to 20% a year cannot be sustained in the long run: At some point, an

event is bound to occur which will wipe out at one stroke all of the gains of the preceding years. All the investors who enjoyed that kind of return, way above the riskless rate, during the Greenspan era, were taking a risk, and what they were risking finally occurred in 2009. Yes, but did they know? In the US, the Securities and Exchange Commission requires funds to warn investors that ‘‘past performance does not necessarily predict future results’’, and in the UK, promotional pamphlets must carry the mention ‘‘stock prices can go down as well as up’’. These warnings are in the small print that you never read, and salesmen certainly won’t draw your attention to them. Even in the financial industry, there is an irrational belief that good times will continue. I am reminded of Laplace, who reportedly was willing to bet d + 2 against 1 that the sun will rise tomorrow, based on the fact that it has risen without fail d times in the past: The longer the bull market, the more confident you are that it will continue. Unfortunately, this is not what the mathematical models say, nor is it what history teaches us. This, in my view, is a strong reason for more, not less, mathematical finance. If this iron law of finance were as well known in the general public as, shall we say, the law of gravitation is, then there would be fewer disappointments, and unscrupulous salesmen would find it harder to pass off risky wares as riskless. It is really an iron law: I know of no way to get around it. So Rogalski’s claim that ‘‘the point [of financial mathematics] lies in increasing speculative profits and making them more secure’’ is wrong. Financial mathematics tells you that if you make a financial operation more secure, you decrease the expected profit, and if you make it totally secure, you get the riskless rate. Every financial operation has an upside (things work out well) and a downside (things go sour), and there is no mathematical formula that will keep the upside while getting rid of the downside. In other words, if you bet on a sure thing, you will find no takers to bet against. In 1992, George Soros won $1.1 billion by betting that the pound sterling would be devalued: He won, but he could have lost, and if he had, you would never have heard of him again. He certainly got no help from mathematics, and, in fact, he has a negative view of the models in current use. If bets go sour, mathematicians cannot help, but governments can. When the chickens came home to roost, and the risk that had been accumulating in the banks for 10 years materialized, the governments bailed them out at taxpayers’ expense. Governments have behaved like the rich fathers of Victorian England, who pay off their sons’ debts when they become too large: Surely this is not a way to develop their sense of thrift and responsibility to society. It is not my purpose here to analyze why governments do that. I just want to

This response is to the Viewpoint ‘Mathematics and Finance: An Ethical Malaise’ by Marc Rogalski. The Mathematical Intelligencer, Summer 2010


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point out that if speculative profits have been made more secure, it is because the governments paid off the bad bets, not because some mathematical magic prevented the risk from materializing. The only way I can see to correct this situation is to regulate the banking profession and the financial markets, as governments did in the aftermath of the 1929 crisis. Unfortunately, most of these regulations, such as the Glass-Steagall act of 1932, were done away with under pressure from the bankers; the Glass-Steagall act itself was repealed by the Clinton administration in 1999. The growth of financial markets in the past 30 years has been truly stupendous, but to attribute it exclusively to the power of mathematics is certainly naı¨ve: The same period has seen the birth of the Internet (remember, it did not exist in 1980), the dematerialization of securities (stocks and bonds used to be pieces of paper which were held in bank vaults and had to be delivered physically by the bearer to the seller), the explosion of the Asian economies (China started its economic ‘‘reforms’’ in 1978). Financial markets grow with the economy not only because rich individuals and pension funds seek to invest their money, but also because of the necessities of international trade and industry. It is perfectly true that the expansion and globalization of the world economy has been accompanied by an enormous increase in wealth inequalities, but I think it is not realistic to blame the mathematicians, or even the finance industry, for that. My view is that it is an illustration of an old law of international trade, first pointed out by the Swedish economists Heckscher and Ohlin in 1933: As economic barriers between nations fall, and the prices of goods are equalized across the world, the prices of factors (nonspecialized labor for instance) are equalized as well, so that the laborers in our societies tend to get paid the same as laborers in China. It means an impoverishment for some in the ‘‘West’’, but an enrichment for the laborers in

AUTHOR

......................................................................... IVAR EKELAND is currently Canada Research

Chair in Mathematical Economics at the University of British Columbia, serving also as Director of the Pacific Institute of Mathematical Sciences from 2003 to 2008. He was formerly a Professor of Mathematics at the University of Paris–Dauphine, where he was President of the University from 1989 to 1994. Well known for his work in mathematics, economics and finance, he is also known to a wider public for accessible expositions like Mathematics and the Unexpected and The Best of All Possible Worlds. Department of Mathematics University of British Columbia Vancouver V6T 1Z1 Canada e-mail: [email protected]

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China. University professors are not subjected to this kind of competition (education cannot, up to now, be delocalized), so inequalities in our societies increase. My point here is that economic forces are at play which vastly outweigh the puny might of mathematicians, and even the much more formidable force of the finance industry. If inequalities are to be reduced, then it is a matter for government policy, such as introducing a measure of protectionism—which the developed countries, like the US and the European Union, have been doing for a long time, even while they encourage others to open their borders. One should also bear in mind that the same 30 years which have seen this increase in inequalities have also seen billions of people lifted out of extreme poverty and women receiving an education. So I do not share Rogalski’s black-and-white view of the world, although I do agree that the financial industry right now has a stranglehold on society (the ‘‘too-big-to-fail’’ syndrome), which we both find dangerous. He puts a valid ethical question to us all: To what extent should mathematicians participate in the training of young people for the financial industry? More generally, is financial mathematics a legitimate field of study? I think myself that financial mathematics plays the role that optimization and control played in the 1960s and 1970s: A field of mathematics which had industrial applications, so that students rushed to learn it for very mundane reasons (getting a good job), and researchers got interested in it because there was funding available, and, for a change, there were people out there interested in their results. Some of the people in control theory doubtless went to work for the military, in the US as well as in Europe or the USSR, building these wonderful missiles which can deliver a thermonuclear warhead within one meter of where you are standing, certainly much more deadly than anything a student of financial mathematics can do. I think, however, that the whole field of optimization and control had a positive effect on mathematics and society: Its ideas and techniques pervade science and technology, and are now part of our common intellectual background. Financial mathematics is part of a more general subject, which is economic theory. Clearly stock prices and bond prices are not physical processes: They arise from interactions between professionals acting within certain rules and institutions, and trying to achieve certain aims, usually to maximize profit. A true theory should try to encompass not only the price process, but also the system which generates it. There is now quite a body of work (including sophisticated mathematical models) in that direction, and it is bearing fruit—coming up, for instance, with definite proposals for the regulation of the banking profession and financial markets. In my view, it is crucial that mathematicians participate in these investigations: Finance is too important to be left to financiers. Freud used to say, ‘‘No, our science is not an illusion; the illusion would be to think we could get from somewhere else what science is not able to give us’’.1

Freud, Sigmund, Die Zukunft einer Illusion, Internationaler Psychoanalytischer Verlag (1928), last sentence (author’s translation).

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Mathematician as Artist: Marston Morse JOANNE E. SNOW

AND

COLLEEN M. HOOVER

he mathematician Marston Morse (1892–1977) argued in several papers and popular lectures that mathematics is ‘‘the sister as well as the servant of the arts’’ ([18], p. 55). Although he formally presented this position in a talk presented in 1950 at a conference at Kenyon College in honor of Robert Frost, his view evolved over the course of his mathematical life. In this article, we trace the development of his theory, present his arguments, show how the arguments are a reflection of his mathematical life up to that point, and indicate how his later work supported his claim.

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On June 20, 1922, Morse married Celeste Phelps with whom he had two children: Meroe and Dryden. This

A Glimpse of the Life of Morse Harold (Calvin Comfort) Marston Morse was born to Ella Phoebe Marston and Howard Calvin Morse in Waterville, Maine, in 1892, and graduated summa cum laude with a B.A. in mathematics from Colby College in Maine in 1914. On the eve of U.S. involvement in World War I, Morse pursued graduate work at Harvard University, completing his thesis in 1917 on ‘‘Recurrent geodesics on a surface of negative curvature,’’ under the direction of George D. Birkhoff (1884–1944). Then he enlisted in the American Expeditionary Force, serving as an ambulance driver in France from 1917–1919. After the war, Morse held a oneyear position as the Benjamin Peirce Instructor of Mathematics at Harvard, and then positions at Cornell and Brown. He returned to Harvard in 1926 and remained there until 1935, when he accepted a position as the sixth professor in the School of Mathematics of the Institute for Advanced Study in Princeton, New Jersey. In 1962, Morse retired from the Institute but, as Professor Emeritus, continued to work on mathematics until his death in 1977.

Marston Morse in 1946. Photo courtesy of Louise Morse 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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marriage ended in divorce in 1930. On January 13, 1940, Morse married Louise Jeffreys. They had five children: Julia, William, Elizabeth, Peter, and Louise. For his thesis topic, Morse wanted a problem that combined geometry (topology) and analysis ([3], p. 2). His desire to relate these two fields continued throughout his life and is reflected in his research topics. Morse was a prolific mathematician, authoring or co-authoring 176 papers and seven books. He focused on variational analysis and is best known for the body of work that bears his name— Morse theory. The significance of Morse’s contribution to mathematics is evident in the many honors he received, such as the Boˆcher Memorial Prize, recognition as the Gibbs Lecturer in 1952, and honorary degrees from several institutions. He was elected to the French Acade´mie des Sciences and its Italian counterpart. Despite the long hours Morse devoted to research, he also found time for service. During World War II, he was a consultant for the Office of the Chief of Ordnance and headed the War Preparedness Committee, a group of mathematicians who helped direct the contributions of mathematicians to the war effort. He held positions in the National Research Council, the National Academy of Science, and the National Science Foundation. The mathematical community as well benefited from his leadership as a journal editor, a participant in committee work, and a vice-president and president of the AMS. After Morse’s death, Stephen Smale wrote, I would place Morse with George D. Birkhoff, Norbert Wiener and Solomon Lefschetz among American (i.e., American trained) nonliving mathematicians as those whose work has had the biggest impact on mathematics. ... What distinguished Morse in particular was his singleminded persistence with one theme, now known as Morse theory (or calculus of variations in the large). ... I believe that Morse theory is the single greatest contribution of American mathematics (perhaps excluding more recent contributions for which time has been too short to assess sufficiently).

Marston Morse during World War I. Photo courtesy of Louise Morse.

The depth of the contribution of Morse theory is reflected today in the vitality and breadth of what is now called Global Analysis; that is, in the study of differential equations, ordinary and partial, from a global or topological point of view [24]. Over the course of his life, Morse demonstrated a strong interest in the arts. His musical appreciation and aptitude came naturally since his mother was an organist. Morse

AUTHORS

......................................................................... ............................................................................... is Professor of Mathematics at Saint Mary’s College, where she has taught since 1983. She received a doctorate in mathematics in 1979 from the University of Notre Dame and is the author, with Kurt Weller, of Exploratory Examples for Real Analysis. Her area of interest is the history of mathematics. Her husband is also a mathematician but none of their four children has followed their parents’ footsteps.

JOANNE SNOW

Saint Mary’s College, Notre Dame USA e-mail: [email protected] 12


COLLEEN HOOVER received a doctorate in

mathematics in 1999 from the University of Notre Dame and is now an Associate Professor at Saint Mary’s College, where she has taught since 1997. She is a co-author of the Visual Math Dictionary. Her area of interest is the history of mathematics. It is much too early to tell if her two young sons will become mathematicians. Saint Mary’s College Notre Dame USA e-mail: [email protected]

played both the organ and piano, practicing the piano daily. His love of the arts was so intrinsic to his personality that he chose to preface his 1965 film Pits, Peaks, and Passes with a personal performance at the piano [21]. He also enjoyed literature, the visual arts, and philosophy. Thus, it was natural for him to relate mathematics to the arts.

The Development of Morse’s View on the Arts and Mathematics In a talk in 1950 on the sisterhood of mathematics and the arts, Morse argued three points of similarity: • intuition, rather than logic, is critical to discovery, • a mathematician or artist is internally driven to create, and • the histories of mathematics and the arts share common themes. ([18], p. 55–56) We can trace the development of his thesis and arguments in earlier addresses and writings. Morse began the commencement address he delivered at Colby College in 1939 with the line ‘‘I wish to speak to you today on the topic ‘Men Must Think’’’ [13]. Using examples from mathematics and religion, he explained how some people ‘‘must think for their own joy and self-respect.’’ This led him to consider the uses of thinking. He regretted that, in the minds of some, the material uses of scientific thinking supersede the value of pure inquiry. He explained what drives most scientists: For the great majority of scientists the desire to understand is the impelling force. They seek to replace disorder by order, to integrate the past and predict the future. For the ordinary student, science opens the mind, banishes supposition, elevates the standards of truth. Science suggests methods which may eventually be extended to all forms of human thought. One should not overlook thinking as an art. For it is surely as much an art as music and painting. Its creation gives the same pleasure. The choices of its forms are dictated by taste. As an example let me point out that a large part of the opposition to relativity theory in its earlier days was esthetic. The engineering-minded scientist of the day found the finely balanced and abstractly perfect form of relativity theory new and distasteful. Recent developments in logic and positivistic philosophy have again shown how the esthetic enters into the choice of scientific forms. The creator wishes his theory to be simple and the final choice as to what is simple is a matter of taste. These immaterial evaluations of thinking call for an act of faith. One must believe that natural curiosity, the desire to understand, and the esthetic appreciation of simple forms are the best guides for research [13]. Here we see already the beginnings of his thesis and argument. Morse allies thinking with art. The theory that results from thought should be aesthetically pleasing, and the thinker is driven by his own curiosity. A year after his Colby address, Morse connected mathematical thinking, in particular, with art in an article ‘‘Twentieth Century Mathematics’’, which appeared in American

Scholar, a journal published by the Phi Beta Kappa Society [14]. Here Morse’s topic is mathematics in the twentieth century. He reiterates some of his previous points and further develops his arguments. First, he emphasizes the unity between science and art and the aesthetic aspects of the end result: Mathematics is both an art and a science, and the lack of appreciation of this fact is responsible for much misunderstanding. ... Objective advances must be revised in form to make them esthetically acceptable and logically comprehensible, while advances of a more subjective nature, if complete and harmonious, will not long remain unapplied ([14], p. 499) This second line introduces another argument for the sisterhood of mathematics and art: ideas and intuition precede the logical presentation of the theory. Morse concludes with the lines: We may explain and evaluate mathematics in many ways, yet the impulse to understand always remains and cannot be denied. Mathematics will advance regardless of its difficulties, and, however great the advances, mathematicians will find still greater fields to conquer ([14], p. 504). Morse’s use of the word ‘‘impulse’’ points to his later claim that a mathematician is driven to pursue the subject, just as the artist is driven to create. Morse continued to refine his premise and arguments. He was concerned that the artistic dimension of the nature of mathematics be understood and took advantage of opportunities to explain it. The 1941 ‘‘Report on the School of Mathematics’’, prepared by Morse and John von Neumann, begins with their claim that it would be impossible to describe the content of the research undertaken by those in the School of Mathematics because to do so requires the use of the language of mathematics, which is understood only by mathematicians. Thus, they will present ‘‘the general tendency and purpose of such research, and the spirit, the atmosphere, in which it is undertaken.’’ To evaluate the work of the mathematicians, Morse and von Neumann make the following suggestion: Thus when dealing with mathematics it is probably more useful to judge it by the same standards by which a creative art is judged—that is, by esthetic standards. The esthetic angle may escape the layman who does not speak the ‘‘foreign language’’ in which the intellectual effort goes on. It may also seem strangely disconnected with the application which may ultimately be made of mathematical results. But it is there nevertheless, and ignoring it would lead to a complete misunderstanding of mathematics [20]. Here we see the familiar point that mathematics is an art. Morse and von Neumann also speak of the tension between the pure and the applied character of any mathematical work; Morse will later compare this tension to a theme in the history of art. That Morse felt so strongly about these points is seen in his reaction to a 1941 draft of the Report of the Director on the School of Humanistic Studies at the IAS. In a letter written to Dr. Frank Aydelotte, dated June 5, 1941, he wrote 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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another of the main arguments of his 1950 talk—the ‘‘fiercely individualistic’’ nature of mathematical creativity.

The Formal Presentation of his Argument and Its Reflection in His Work

Albert Einstein and Marston Morse in 1938. Photo courtesy of Louise Morse

... I shall quote two sentences from the report: ‘I rejoice that the members of our School of Mathematics are at present engaged on various important and confidential tasks connected with ballistics and aviation, of direct use in the prosecution of the war. It seems to me that all the disciplines which are included under the term humanities have a value which, while entirely different, is nevertheless of supreme importance.’ I do not believe that the traditional influence of mathematics, its purpose, or its associations, are ‘‘entirely different’’ from the influence, purposes or associations of the humanities. ... Mathematics is both an art and a science. ... In spirit we mathematicians at the Institute would cast our lot in with the humanists. ... Mathematicians are the freest and most fiercely individualistic of artists. They are subject to no limitations of materials or instruments. Their direction at any time is largely determined by their tastes and intellectual curiosity. Their studies are really the studies of the human mind. To me the work of Einstein is even more important as a free and beautiful expression of the creative imagination of an individual than as a part of the science of physics [9]. In other words, mathematics is an art and mathematicians pursue what they feel driven to study. Morse also hinted at

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In preparing his talk for the conference at Kenyon College in honor of Robert Frost, Morse had the opportunity to synthesize and fully develop his thesis. The subject of the conference was ‘‘The Poet and Reality.’’ The five speakers, including Morse, addressed an audience of poets. Commenting on the Frost Conference, J. Donald Adams wrote in his New York Times column ‘‘Speaking of Books,’’ But the bombshell of the conference, and the contribution which, I think, spread the brightest light, was dropped, believe it or not, by a mathematician. Marston Morse of the Institute for Advanced Study at Princeton read a paper on ‘Some Reflections on Evaluations in Mathematics and the Arts’ which struck sparks, I believe, from the most sluggish minds in attendance. I hope to make his remarks the subject of another article [1]. Morse’s talk was published in the Yale Review in 1951 [15] and was reprinted in the Bulletin of the Atomic Scientists. [18] (We reference the latter source in this article.) Morse began his talk by presenting some personal background on his love of music. He gave examples of the utility of mathematics in the visual and musical arts, but he was not content with the role of mathematics as handmaiden. His thesis was: ‘‘But mathematics is the sister, as well as the servant of the arts. ... the basic affinity between mathematics and the arts is psychological and spiritual and not metrical or geometrical.’’ ([18], p. 55, 57) Morse’s arguments suggest how he had and would continue to pursue his research in mathematics.

The Critical Role of Intuition in Discovery Morse’s first argument for the sisterhood of mathematics and art is that the process of creation is the same. He states, ... discovery in mathematics is not a matter of logic. It is rather the result of mysterious powers which no one understands, and in which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty’s sake, and pulls it down to earth, no one knows how. Afterwards the logic of words and of forms sets the pattern right. Only then can one tell someone else ([18], p. 56–57). Discovery, then, is the result of intuition and inspiration. In support of his argument, Morse compares the creative processes of the poet Frost and the mathematician Poincare´. Both Frost and Poincare´ experienced bursts of creativity— ideas that came to them, they knew not from where. Indeed, Morse’s own creative process reflects his claim. According to Morse, the creative process begins with inspiration (the mysterious powers) and intuition (choosing one path for beauty’s sake). Following the insight, one finds a proof of the idea, often by hashing it out with others. Finally, one puts the ideas in a form that is logical and comprehensible to others.

From the earliest moments of his career as a mathematician, one sees evidence that this is how Morse himself created mathematics. After completing his thesis, he was awarded the Sheldon Traveling Fellowship, which he resigned to enlist in the American Expeditionary Forces. While in the service, he still thought about mathematics and publishing his thesis. In a letter to his mother, in the spring of 1917, he wrote I have written my thesis all out once and am now revising it. I hope to get it finished by May 1. It has taken me much longer than I expected. It is much more of a task to put your thoughts on paper than to get them. It has taken me the longer because I changed my plan in the winter regarding the logical make-up of the whole thesis. I had the thesis about done the way I had first planned it but found a better way to do it [12]. Here we see the ideas came first; the logical make-up was secondary to the ideas. Later, as a mature mathematician, Morse expressed the same point in a lecture in 1958 at the Fordham University Summer Institute for High School Teachers. The talk was subsequently revised and appeared in the journal Thought published by Fordham University Press. In this article, he describes an instance of his own creative process. Last winter a young mathematician at Princeton University by the name of Mazur broke through the barriers of mathematics of more than fifty years’ standing to solve an important case of the famous ’Schoenflies problem.’ ... I had worked on the ’regular’ problem some years previously without obtaining other than special solutions. However, my mind was filled with material accumulated over a period of twenty years of work on related problems. On hearing of Mazur’s solution I immediately posed a corresponding ’regular’ problem. ... My own first impression was that the methods which I had used, combined with the idea of Mazur were inadequate. A few moments of thought were given to the problem on each of the following days. On about the fourth day I had an intuition pointing to a solution of the problem. In the presence of a colleague, and in the course of about an hour a solution was roughly outlined. The details were not there but the main structure of the proof was clear. The method consisted of four successive transformations of the original problem into equivalent problems, the last of which appeared to admit a geometric solution. The most noteworthy point in this account is the following. It took three months to translate the original intuition into a logical form. Two times during this period of verification what seemed to be logical roadblocks appeared. It was even necessary to entertain the hypothesis that my first conceptions had been in error. In each such case I hastened to recall the original idea, then slept on the problem and started the next morning anew, keeping as close as possible to the first intuition. Finally a proof came through in exact accord with the original idea. What had apparently happened was this. In relying on logical methods to complete the proof, a point was reached where there were several paths which could be followed.

Each path was logically permissible, but only one was fruitful. It was necessary to recall the original intuition to find the path which led to the goal ([19], p. 20–21). Again, Morse makes clear that logic is subservient to intuition. This quote also reveals how verbalizing his thoughts was an intrinsic part of Morse’s work method. Some of those who worked with Morse offer testimony to his use of discussion as a means of discovery. Said Bill Transue, a frequent collaborator with Morse, Unlike many mathematicians who collaborate with others, Marston Morse preferred, when he could get it, a teˆte-a´-teˆte discussion of the work under way. ... his dedication to mathematics was extraordinary. Our discussions took place in a great variety of places—in the car (to get away from the children), in his bedroom (for the same reason), at the dinner table [25]. Morse loved to talk about his work. One of his assistants, Paul Halmos, reported that ‘‘Morse’s assistant’s duties were to do research with him eight hours a day sitting with Morse and listening to him talk and talk.’’ ([2] , p. 125) Those who met Morse saw his inspiration—his creative impulses, those ‘‘mysterious powers which no one understands.’’ Herbert Robbins, who had contact with Morse while at Harvard, said, ... Marston Morse impressed me deeply. Even though what he was talking about meant nothing to me ... I could see that he was on fire with creation. There was something going on in his mind of a totally different nature from anything I’d seen before. That’s what appealed to me ... there was something that attracted me to Marston that transcended anything I knew. I suppose it was his creative driving impetus—this feeling that your house could be on fire, but if there was something you had to complete, then you had to keep at it no matter what. ([23], p. 7–8) Morse’s children certainly had many opportunities to observe how their father worked. William Morse shares the following memory of his father. I once read that he was ‘‘afire with creation.’’ That is accurate. He had ideas; they were in his mind long before he got them onto paper. And until he died (of a heart attack on our front lawn), he worked as fast and as hard as humanly possible to get his ideas into writing. He wrote in pencil, neatly, on yellow lined sheets of paper. In the last 10 years, it was a race against the clock and about age 83 he told me he had too many ideas in his head and feared he would never get them written down, they would die with him. He worked around the clock because the ideas drove him, excited and inspired him [22]. Certainly intuition and inspiration drove Morse’s discoveries. The Drive to Create Morse’s next argument was, ‘‘A second affinity between mathematicians and other artists lies in a psychological necessity under which both labor. Artists are distinguished from their fellows who are not artists by their overriding instinct of self-preservation as creators of art.’’ ([18], p. 57) There were two components to this argument: 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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• an obsession with doing original work that will be lasting and important and • the instinct to preserve what one has created. As evidence, Morse provided examples from both the arts and mathematics. He noted that Carl Philipp Emmanuel Bach rejected the musical style of his genius father Johann Sebastian Bach. In the mathematical world, Morse points to the mathematical legacy of Poincare´. To make their own mark, Poincare´’s followers would have to invent new techniques. For example, Lebesgue, one of Poincare´’s young colleagues, made significant contributions to the emerging field of set theory. In his own life, we see Morse’s respect for his mathematical ancestors—Poincare´ and Birkhoff—as well as the need to distinguish himself. Poincare´ worked on many problems, but it was his work in dynamical systems and differential equations that interested G. D. Birkhoff. Birkhoff, Morse’s thesis advisor, solved a problem posed by Poincare´. Then Morse carried on this work. In an honorary degree citation from the Sorbonne in 1946, Morse was termed the ‘‘mathematical grandson’’ of Poincare´. Yet, just as C. P. E. Bach and Lebesgue had to set themselves apart from their predecessors, so did Morse. A mathematician makes his mark by recognizing that something is missing or incomplete and by purposefully remedying that want. Morse explained this point, as well as his distinctive contributions, in a letter dated February 4, 1966, to Brother John G. Driscoll of the Department of Mathematics at Iona College: In answer to your question as to the nature of mathematical inventions I would say that they may seem capricious to the outsider and for the lesser mathematicians they may be capricious. Inventions, however, of the major mathematicians are anything but capricious. It would be better to say they are individual and self-oriented. Each great mathematician senses some lack in the total mathematical picture. ... Poincare´ is the best example that I know of because he sensed the lack of unity in mathematics, the need for the combination of topology, differential equations, algebraic geometry, etc. Birkhoff took off from Poincare´. If you read Birkhoff’s life you will see this clearly. Birkhoff wrote a paper, ’Quelques theóre`mes sur le mouvements des syste`mes dynamiques,’ Bull. Soc. Math. France, 40 (1912), 305-323), where he outlined problems neither he nor Poincare´ could solve. I took off from them. I sensed another entirely different lack in mathematics. The rigidity in topology with its simplices. This seemed unnatural and inadequate for the applications of topology to analysis and differential topology. I believe that in these fields one can with profit get rid of much of this combinatorial structure and replace it by local analysis, globally integrated with the aid of group theory. In fact, my critical-point theory goes in this direction. That it may have been regarded as capricious is obvious from the fact that today for the first time many of my original discoveries are being recognized. ... 16


To sum up, every real mathematician senses a great need which others have not sensed. Otherwise they would have done something about it. He has to prove that the need can be satisfied, in part at least, and to justify his choice of aim by the developments which he adds to mathematics [7]. As Morse said, the lack he observed was ‘‘the rigidity in topology.’’ Some mathematicians saw only how topology could be of aid to analysis, but Morse felt analysis could help solve problems in topology as well. What is added to mathematics should be lasting and important. Morse felt pure research met these qualifications and was disinclined to do anything that did not. His army work obviously resulted in some neglect of his own research. Writing to Dr. Aydelotte in April, 1943, he said, As you know, I have been loath to leave my own research here for war work. ... My present task involves scientific investigations for which no one else is properly available. While this work is not likely to stand out through the years as does pure research, I cannot in good conscience refuse to do it. It does require originality and judgment, which are not available in anything like the degree needed. ... If this work ever becomes mediocre or ineffective I shall quit it at once [8]. As for the lasting nature of his research, Morse replied to a questionnaire from the IAS Board on January 6, 1976, My own field is shared in spirit by several of my colleagues. Astronomically, geographically, intellectually, and politically, the world seeks a global understanding of the local. My mathematical objective is to relate topology globally to local analysis and geometry. This research will go on. It depends on all of mathematics [11]. The artist wishes not just that his impact be lasting but that his own creations be preserved by those who follow him. Morse offered the example of Poincare´ criticizing his younger colleagues who invented the new technique of set theory. We see Morse’s instinct to preserve what he created by his obsession with one area of mathematics and with one set of techniques. Smale emphasizes this aspect in his description of Morse (cited earlier): What distinguished Morse in particular was his singleminded persistence with one theme, now known as Morse theory (or calculus of variations in the large) ([24], p. 33). Stewart Cairns explains why Morse did not use the new methods of his younger colleagues. Morse conducted his major research largely with the tools and techniques of classical analysis, supplemented by homology theory and geometric topology. These represent the most natural methods during approximately the first half of his career. Gradually thereafter, other mathematicians developed innovative concepts and procedures applicable not only in Morse theory but in a wide variety of other areas of mathematics. In terms of these powerful new methods, major portions of Morse theory were rewritten and valuable new results were proved. ... Morse continued his research with his own classical tools, in the firm knowledge that he could thus make more extensive progress than in any other way. To have done

otherwise would, for him, have been a fruitless tour de force and a source of delay in carrying out the ambitious program he had set for himself. It would be a profitable service to mathematics for someone to put Morse’s more recent results into the mathematical language in which younger mathematicians have been and are being trained. It might also be beneficial for some of the latter to study the work of Morse as written by him, for his writing is characterized by meticulously careful exposition and a conscientious concern with mathematical rigor ([4], p. 124). Just as Poincare´ criticized set theory, Morse found fault with some of the new techniques. To Morse, a new technique should be introduced or used only if it adds clarity to explanations, never simply because of a desire for something new. In a letter to the mathematician Arnaud Denjoy, dated January 14, 1971, we also see how he prized originality and beauty of mathematical work. I agree with you that many of the young mathematicians are devising algebraic abstractions which are obvious when they are relevant, and in general take more space to explain abstractly than it takes to establish the desired theorems without their use. I think there is a little return to mathematical realism in the U.S. The inflation and scarcity of jobs are among the causes. You and I have lived through exciting but difficult times. I go on the assumption that mathematical work, such as yours, is to be measured by its depth, beauty and originality, not abstractness and verbal novelty [6]. This tension that Morse felt between newer algebraic methods and his own methods is also an example for his third argument.

Historical Similarities Having discussed psychological similarities between an artist and a mathematician, Morse turned to the similarity in the history of art and mathematics. ‘‘The third type of evidence is found in the comparative history of the arts. The history of the arts is the history of recurring cycles and sharp antitheses. These antitheses set pure art against mixed art, restraint against lack of restraint, the transient against the permanent, the abstract against the nonabstract. These antitheses are found in all of the arts, including mathematics’’ ([8], p. 57–58). Morse gave examples of artists and mathematicians who aligned themselves with one of two antithetical positions. He did not take a global stance on these issues, and it may appear that he was straddling the fence. Or maybe he had resolved the tensions, guided by a desire to achieve beautiful and unified mathematics. At one time, Morse noted, music was not considered a pure art, but only an accompaniment to poetry. There is a parallel in the struggle between algebra and geometry. Some would like to keep the disciplines separate, whereas others would have one as a tool for the other. In the previous quote from the letter to Denjoy, it appears that Morse felt algebraic structures were sometimes unnecessary and unhelpful. Morse made this point clear when he

referred to his communication with Denjoy in a 1977 interview with Thomas Banchoff. Morse regretted that researchers ‘‘today emphasize algebra and computation more than geometry. ... And I think that geometric understanding should be increased for the good of mathematics. One example I got from Denjoy railed at the overemphasis on algebra. ... It’s been the problem since the days of ... Lebesgue’’ [3]. On the other hand, Morse believed that geometry and analysis were equals in the sense that analysis could be used to answer questions in geometry and vice versa. Morse knew there was a tension between pure mathematics and applied mathematics, and he strove to achieve balance in these opposing ideas. A biographer of Morse stated that he is ‘‘one of America’s leading theoretical mathematicians.’’ Morse circled the word theoretical and added the note ‘‘I have written 100 papers in applications’’ [17]. On the other hand in the following quote from ‘‘Science in the Modern World’’ one sees that he did not align himself with the applied mathematicians either. Small wonder, then, that a large proportion of the young mathematicians become technicians in limited fields mostly connected with the foundations. Some leap over the foundations and proceed at once to the front as represented by the material world; these are the ones whom we call applied mathematicians. They have good courage, but are frequently inadequately equipped, although with genius they may compensate for this deficiency. Then there are the few—all too few—who aim to build the whole edifice of mathematics, neither lingering too long over the foundations, nor too hastily testing their strength at the front. ... And there is Poincare´, whose topological analysis has not yet come to its full fruition, but which I believe contains the germs of an unexpected and startling unity for analysis and parts of physics ([16], p. 211). Morse sought to achieve balance between building the foundation and moving to the front. Like Poincare´, Morse wanted unity in mathematics. This leads to another of the antitheses Morse discussed: restraint vs. lack of restraint. Contrasting the restraint of Grecian art with the unrestraint of Berlioz’s The Last Day of the World, Morse said mathematicians are unrestrained when their work is ‘‘based neither on nature or logic’’ ([15], p. 58). On the other side are those who are so restrained by logic that their work is diminished: ‘‘Forever the foundation and never the cathedral’’ ([15], p. 58). By not leaping over the foundations, he could approach the unrestrained with the appropriate background and achieve unity in his mathematical ideas. His extension of critical-point theory to infinite-dimensional spaces can be seen as an example of lack of restraint. In Morse’s view, the contrast between the transient and the permanent takes the form of pursuing popular research topics instead of devoting oneself to a topic of personal or lasting interest. In the article ‘‘Science in the Modern World,’’ he bemoaned, ‘‘It is sad to see men of talent in science, superficially convinced of the importance of a topic in science, abandon this topic for another within a few years.’’ ([16], p. 210) 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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Finally, Morse addressed the interaction of the abstract and the nonabstract. Pythagoras and Kronecker considered the natural numbers, which are concrete, to be the only numbers; Morse contrasted this with Weierstrass, who developed a theory of the abstract real and complex numbers. In a letter dated November 9, 1965, to Professor Arthur E. Hallerberg, Chairman of the Editorial Committee of the NCTM History Yearbook, Morse criticized an essay on Modern Mathematics, which he found ‘‘inadequate and incomplete.’’ The abstract is required in proportion to the depth of the discovery. The more you discover the more you need a clear formulation. I am at present giving lectures in New York called ‘‘An Introduction to Analysis and Geometry in the Large.’’ I am using more abstraction than was ever used to present this theory, more for example than Milnor uses in his book. But I use them in ‘retrospect’ as the poet Frost says, so that I may be understood by my students. [10] On the other hand, Morse defended the necessity of abstraction while cautioning against it as a means of discovery. In ‘‘Twentieth Century Mathematics,’’ he wrote Modern mathematics is particularly characterized by abstraction. This is partly a reflection of the taste of mathematicians and partly a matter of necessity. It is associated by some with the political and economic troubles of the day but I am more inclined to find its cause in the nature of mathematics itself. Popular opinion to the contrary, abstract studies are often simplest. The process of abstraction rules out the irrelevant and permits greater generality. Mathematicians abstract in order to unify, simplify, comprehend and extend. Mathematicians differ as the value of abstraction as a mode of discovery. ... It is true that the tendency to start with abstraction is often an escape from the necessity of mastering the difficult and more concrete problems of classical mathematics. But one must distinguish between the tendency to abstract as an end and the use of abstraction as a means to an end ([14], p. 500–501). Morse termed his paper ‘‘Mathematics and the Arts’’ his ‘‘intellectual swan song to the world,’’ said his daughter Louise [5]. The philosophy expressed in this work and in his other works on mathematics and the arts reflects his own experience in a mathematical career that spanned nearly 70 years.

[6] Papers of Marston Morse. Correspondence and Other Papers, 1922–1978. Letter, Morse to Arnaud Denjoy (January 14, 1971), in folder ‘D’. HUGFP106.10, Box 4. Harvard University Archives. [7] Papers of Marston Morse. Correspondence and Other Papers, 1922–1978. Letter, Morse to Brother Driscoll (February 4, 1966), in folder ‘I’. HUGFP106.10, Box 7. Harvard University Archives. [8] Letter, Morse to Doctor Aydelotte (April 5, 1943). Stern Vertical File, Box 4, Folder 2, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA. [9] Letter, Morse to Doctor Aydelotte (June 5, 1941). Stern Vertical File, Box 4, Folder 3, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA. [10] Papers of Marston Morse. Correspondence and Other Papers, 1922–1978. Letter, Morse to Professor Arthur E. Hallerberg (November 9, 1965), in folder ‘M’. HUGFP106.10, Box 10. Harvard University Archives. [11] Papers of Marston Morse. Correspondence and Other Papers, 1922–1978. Morse’s response to questionnaire from the IAS board, in folder ‘I’. HUGFP106.10, Box 7. Harvard University Archives. [12] Marston Morse. Letter to his mother. Personal Materials from Louise Morse, 1917. [13] Marston Morse. Men Must Think. Personal Materials from Louise Morse, 1939. [14] Marston Morse. Twentieth Century Mathematics. American Scholar, 9:499–504, 1940. [15] Marston Morse. Mathematics and the Arts. The Yale Review, 40:604–612, 1951. [16] Marston Morse. Science in the modern world. Mathematics Magazine, 28(4):209–211, Mar–Apr 1955. [17] Current Biography. H.W.Wilson, March, 1957, Personal Materials from Louise Morse, [18] Marston Morse. Mathematics and the Arts. Bulletin of the Atomic Scientists, 15:55–59, February 1959. [19] Marston Morse. Mathematics, the Arts, and Freedom. Thought, 34:16–24, 1959. [20] Marston Morse and John von Neumann. Report on School of Mathematics, (1941). Stern Vertical File, Box 4, Folder 2, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA. [21] Marston Morse; Amram Nowak; Stewart S Cairns. Pits, Peaks, and Passes, a lecture on critical-point theory, Mathematical Association of America. Committee on Educational Media, 1965,

REFERENCES

[1] J. Donald Adams. Speaking of Books. New York Times.October 22, 1950.

1993. [22] William Morse. Letter to Joanne Snow. Personal Materials of Joanne Snow. August 23, 2001.

[2] Donald J. Albers and G. L. Alexanderson, editors. Mathematical

[23] Warren Page. An Interview with Herbert Robbins. The College

People Profiles and Interviews. Birkhau¨ser, Boston, MA, 1985. [3] Thomas Banchoff. Notes from Thomas Banchoff’s interview with

Mathematics Journal, 15(1):2–24, January 1984. [24] Stephen Smale. Obituary of Marston Morse. Math Intelligencer,

Marston Morse. 1977. [4] Stewart S. Cairns. Letter to the editor. Math Intelligencer, 1(3), 1978. [5] Louise Morse. Interview with Louise Morse and her daughter. July, 2001.

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1:33–34, 1978. [25] William Transue. Letter to Joanne Snow. Personal Materials of Joanne Snow. August 17, 2001.

Mathematically Bent

Colin Adams, Editor

Immortality COLIN ADAMS

The proof is in the pudding.

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 I?’’ Or even ‘‘Who am I?’’ This sense of disorientation 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.

â Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected]

the end of a dimly lit dead-end street in New Orleans, I found myself facing a massive oak door embedded in a dark gothic stone edifice. In its day, the house must have been a glorious sight, but the years had worn it down. Age and neglect had allowed soot to settle in every crack and crevice, of which there were now many, giving the house a weathered and ominous look. A weak moon illuminated the front door enough for me to find the iron knocker, which I proceeded to thump on the door. I waited a good minute before hearing a shuffling step approach from inside. The door unlocked and swung open slowly, revealing a slight, elderly man dressed in a style at least five decades out of date. He looked me over deliberately and then beckoned me inside. I entered a marble tiled rotunda smelling of mold and decay. He swung the door closed behind me, locking it with a large key that he pocketed. Then he walked me across the rotunda and ushered me into what appeared to be the library. He left me there, sliding the double doors closed behind him. As my eyes adjusted to the low light, I realized that there was someone in a chair facing away from me, seated at a desk across the room. He appeared to be writing. I stood waiting patiently until he stopped and swiveled his chair around to look at me. ‘‘Ah, Mr. Kindler,’’ he said. He stood but made no move to come shake my hand. He was tall and gaunt with a receding hairline. Although at least 60, he appeared to be in excellent shape and graceful in his movements. He also was dressed in the style of an earlier time, but on him it somehow seemed an appropriate fashion statement. ‘‘It is a pleasure to make your acquaintance, Mr. DeMoligne,’’ I said. ‘‘I have looked forward to meeting you for some time.’’ ‘‘And I you,’’ replied DeMoligne. ‘‘Please, take a seat.’’ He motioned to a wooden chair near where I stood. I sat down, but he continued to stand. ‘‘I believe that the first time I became aware of your work was your paper with Phelps on direct products of semilocal rings,’’ he said. ‘‘I was particularly interested in your method of proof of the main result. Collapsing out the entire collection of associated prime ideals was innovative, something I had not seen before. I mean that as a very high compliment.’’ ‘‘Thank you,’’ I replied. ‘‘That technique has proved useful in a variety of other contexts since then.’’ ‘‘Yes, I know. I have read all your papers.’’

A


19

‘‘Really? Then besides me, you are probably the only one who has,’’ I said with an attempt at a self-deprecatory smile. His expression did not change. DeMoligne rang a bell on his desk, and his servant re-entered. ‘‘Bring us beverages,’’ he said. Once the servant had left, he turned to me again. ‘‘You asked to meet with me. Why?’’ ‘‘Well,’’ I said, trying to think how best to word my response. I knew DeMoligne was a recluse of the first order, and I was frankly surprised that he had agreed to the meeting at all. ‘‘I read some of your papers, and I became intrigued. First of all, the mathematics is excellent. You have such an ability to tie together disparate fields. You seem to know such a broad spectrum of mathematics.’’ He nodded acceptance of this compliment. ‘‘Can I ask where you received your degrees?’’ I asked. ‘‘No one I have spoken to seems to know.’’ ‘‘I have no degrees,’’ he replied. ‘‘I am self-taught.’’ ‘‘All the more remarkable then,’’ I said. ‘‘But you did not come here simply to compliment me.’’ ‘‘No, I did not. It is really curiosity that brought me here. First of all, how to put this? You seem to be working in such overlooked areas. You choose to research subjects that are either out of date, or obscure backwaters that no one else finds interesting. The work you do in these areas is truly remarkable, but you don’t receive the attention you deserve because so few researchers are interested in these subjects.’’ DeMoligne laughed and waved a dismissive hand. ‘‘Why should I care if others find this work interesting? It is not contemporary mathematicians who determine what is important and what is not. That is determined by the ages. Perhaps 100 years hence, these theorems that I am proving will be critically important.’’ ‘‘Perhaps,’’ I acceded. ‘‘When mathematicians were proving theorems in number theory in the 1930s, how could they possibly have foreseen their use in computer cryptography 60 years later? So it has always been with mathematics, and so it will continue to be.’’ The servant slid open the library door and set one glass on the table next to me and one on the desk next to DeMoligne. Then he again retired, closing the door behind him. The glass contained a brown liquid that I could not identify. I held it up and took a whiff. ‘‘Try it,’’ said DeMoligne. ‘‘You will like it.’’ He lifted his glass and took a swallow. I took a small sip. It had a strange taste with which I was not familiar, with hints of vanilla and some root extract. But it was definitely interesting, and it warmed my throat as it went down. ‘‘It’s good,’’ I said. ‘‘Yes, it is,’’ said DeMoligne, ‘‘Now please continue.’’ ‘‘What is so interesting about your work,’’ I said, ‘‘is that it reminds me of someone else’s.’’ ‘‘Yes?’’ he said. ‘‘Who is that?’’ ‘‘Actually, I know this sounds funny, but it reminds me of Galois.’’

20


DeMoligne smiled. ‘‘Evariste Galois? But I am not working in algebra. I am working in geometry. And Galois died almost two centuries ago. His work is completely unrelated to my own.’’ ‘‘I know,’’ I said. ‘‘But somehow the methods used are still reminiscent of his. It is the style of proof much more than the substance itself.’’ He regarded me without speaking. ‘‘Also, sometimes, your work reminds me of Riemann,’’ I continued. I took another sip of the drink. The alcohol, assuming that’s what it was, was helping me to relax. ‘‘Bernhard Riemann. He worked in completely different areas, analysis, differential geometry. His work couldn’t be more dissimilar from that of Galois.’’ ‘‘Yes, but again, there is something about his audacity, his ability to see connections that reminds me of your work. Did you study their work? Am I right?’’ ‘‘You are a perceptive fellow,’’ said DeMoligne. ‘‘Extraordinarily perceptive. Perhaps dangerously so.’’ I smiled. ‘‘No one has ever considered my perceptive abilities dangerous before. More likely my lack of perception, if you were to ask my wife.’’ DeMoligne was staring at me carefully, sizing me up, as if considering options. There was a pause, and then he seemed to make up his mind. ‘‘Why do you do mathematics?’’ he asked. ‘‘Me? I fell in love with math when I was 19. The professor in a college course mentioned a minor open problem in graph theory. That night, I solved it. And the feeling of satisfaction was incredible. I had solved a problem no one in the history of humankind had ever solved before. It made me feel special. It seems a little silly now. It really was a minor result. But it made me feel as if I had carved for myself a little piece of immortality. My name would be attached to that result from then on, forever.’’ ‘‘Immortality,’’ he repeated, grimacing. ‘‘It is overrated.’’ I laughed. ‘‘Does my mathematics remind you of anyone else?’’ he continued. ‘‘Um, no, I don’t think so.’’ ‘‘How about Maclaurin, 1698 to 1746? Or perhaps Pascal, 1623 to 1662?’’ ‘‘Well, of course, they were exceptional mathematicians. Everyone knows of their work. But I haven’t read it in the original.’’ ‘‘You should. I am certain it would remind you of my own.’’ ‘‘Are these mathematicians that you studied?’’ I asked. ‘‘Mathematicians whose styles and approaches you have incorporated into your own?’’ ‘‘No,’’ he replied as he suddenly set his drink on the desk and strode swiftly across the room, settling into a chair only inches from my own. The movement was so quick and agile, it reminded me of a large cat. I involuntarily jerked back in my chair. ‘‘I am those mathematicians,’’ he said, staring at me intently. ‘‘What?’’ I replied, completely confused.

‘‘I am those mathematicians. I published their papers under their names. I am Maclaurin. I am Pascal. I am Galois. I am Riemann. Or at least I was. And now I am DeMoligne. ‘‘I’m, ah, I’m…’’, I stuttered. ‘‘I am them and they were me,’’ he repeated. ‘‘And there were others, as well.’’ I looked to the door, considering how I might escape from this person whom I had initially assumed to be eccentric but it now appeared was a madman. ‘‘I am extremely impressed that you noticed the similarities between my work and that of Galois and Riemann. No one else over the last four centuries has made the connection.’’ ‘‘Um, I am still a bit lost.’’ DeMoligne leaned forward close enough for me to feel his breath on my face. It felt cold. ‘‘It’s very simple, Mr. Kindler. I am not mortal. You spoke of immortality. Well, I am that. I am immortal. I once was mortal, but all that changed four centuries ago.’’ ‘‘Really? Sounds fascinating, but you know, I should probably be going.’’ I started to stand, but DeMoligne grasped my arm and pulled me back down into my chair. He kept his hand tightly gripped around my forearm. ‘‘You see, my friend, I had an unfortunate encounter those many years ago, an encounter that changed my life dramatically. An encounter with a vampire. And subsequent to that encounter, I myself became a vampire, one of the socalled undead.’’ Although this convinced me all the more he was insane, it also made me worry of what he was capable. ‘‘Oh,’’ I said hoping to mollify him. ‘‘Of course….’’ ‘‘And once a vampire, you become immortal. Barring a few unfortunate circumstances, your life will unravel before you forevermore.’’ ‘‘Okay, well, isn’t that great.’’ ‘‘No, it is not great. In fact it is exactly the opposite of great.’’ ‘‘What do you mean? Doesn’t everyone want to be immortal?’’ ‘‘Only if they haven’t tried it,’’ he said, a look of sadness crossing his face. ‘‘Try to imagine the tedium of it, as your life rolls by decade after decade. You see the same human stupidities repeated time and time again. You cannot form relationships, fall in love, because inevitably the one you love dies. And then what? Do it over again? ‘‘And as the decades tick by, each seems shorter than the last, since as a proportion of the life you have so far lived, they are a smaller and smaller fraction. Mortals come and go in what seems the blink of an eye. ‘‘Life becomes a pure monotonous agony. Something to dread, a long never ending, never changing ordeal.’’ Then he stopped and looked me straight in the eye. ‘‘At least it was until I discovered mathematics.’’ ‘‘What do you mean?’’ ‘‘I spent the first 100 years of my new existence suffering the intense boredom that is the life of a vampire. Nothing interested me anymore. I would have welcomed death heartily at that point.

‘‘But then I stumbled upon mathematics. At a monastery where I dined, I happened to notice a version of Pacioli’s text Summa de arithmetica, geometria, proportioni et proportionalita. I took it home with me. Over the next few months, as I read through it, I realized I could perhaps push much of the mathematics further. And I did. In those early days, I published under my own name. I remember well that immense satisfaction you mentioned in knowing I had discovered something no other creature had ever discovered before me, mortal or immortal. It was the first time in a long time that I looked forward to waking each evening. ‘‘And so began my pursuit of mathematical knowledge. The pure essence of knowledge. The elixir that I still crave. ‘‘At that time, I was a full member of the mathematical community. I knew Gerolamo Cardan and Gerardus Mercator. I was recognized as one of the great minds of that era. I experienced your kind of immortality. The immortality that comes from having your name attached to a result that will go down in the annals of mathematics. ‘‘But then my contemporaries began to grow old and die, while I was still young and vibrant. What a huge loss that was. Every time a mathematician dies, that storehouse of knowledge they have collected over a lifetime dies with them.’’ He paused, looking down for a moment, before again looking intently into my eyes. ‘‘I realized I had to hide the truth about myself. And my mathematics was doing just the opposite. It was bringing me attention and fame. And people were noticing that I seemed to not get any older as time passed. The life of a vampire is not so easy. There are those who seek to discover us and destroy us. Very few of us have lived anywhere near our full potential. ‘‘Before discovering mathematics, I would not have cared if someone destroyed me. My life was that miserable. But all that changed with mathematics. ‘‘So I had to kill off my persona. Fake my own death and move to a new place to create a new identity. It is easier to do than you might expect. Create a back story, fake some documents, and there you are. But more often than not, I had to change fields as well. Move from algebra to analysis, so as to leave as little connection as possible between the subsequent identities.’’ ‘‘And you are saying you were Riemann, Galois, Pascal, Maclaurin?’’ I asked, incredulous. ‘‘And many more.’’ He smiled, swelling a bit with pride. ‘‘And in all this time, I have had to hide it. You are the first to know the truth.’’ ‘‘Why tell me?’’ ‘‘You discovered it on your own. You just didn’t realize the magnitude of the connection between myself and these mathematicians.’’ As I listened open-mouthed to this incredible tale, I suddenly realized that a strange sensation of pins and needles was crawling slowly up my legs. DeMoligne released my arm, and I found I could not sit up. ‘‘I don’t feel right,’’ I said. ‘‘I need to leave.’’ ‘‘No, you do not need to leave,’’ he replied. ‘‘You need to stay. That drink has a substance in it that paralyzes the drinker. You see, although I am a vampire, I am a civilized


21

vampire. I do not believe in violent displays. This way, it is much easier for both of us.’’ ‘‘What do you mean…,’’ I uttered with difficulty as I attempted to lift my arm, to no avail. ‘‘I would like to have someone to talk mathematics with. It is not so gratifying working alone. Having no one with whom to share the beauty of the results. I am in search of a collaborator. A long-term collaborator, worthy of our collaboration.’’

22


‘‘But I, I…,’’ was all I could get out. ‘‘I have chosen you. It is a great honor. To be chosen to be the collaborator of Galois, of Riemann. I am looking forward to a long and fruitful association, a very long and fruitful association.’’ This is the last thing I remember him saying before he leaned forward one last time, his lips parting to reveal two long and extremely sharp teeth.

Platonic Stars ALEXANDRA FRITZ

AND

HERWIG HAUSER

B

ut of beauty, I repeat again that we saw her there shining in company with the celestial forms; and coming to earth we find her here too, shining in clearness through the clearest aperture of sense. For sight is the most piercing of our bodily senses; though not by that is wisdom seen; her loveliness would have been transporting if there had been a visible image of her, and the other ideas, if they had visible counterparts, would be equally lovely. But this is the privilege of beauty, that being the loveliest she is also the most palpable to sight Plato, Phaedrus

E X A M P L E 1 Figure 1 shows the zero-set of the following equation, f ðu; vÞ ¼ ð1 uÞ3

5 3 cu þ cv; 27

with c 6¼ 0;

ð1Þ Figure 1. Dodecahedral star with parameter value c = 81.

where uðx; y; zÞ ¼ x 2 þ y2 þ z 2 ; vðx; y; zÞ ¼ zð2x þ zÞðx 4 x 2 z 2 þ z 4 þ 2ðx 3 z xz 3 Þ þ 5ðy 4 y2 z 2 Þ þ 10ðxy 2 z x 2 y2 ÞÞ:

ð2Þ

For any value c [ 0, the zero-set of this polynomial, such as the one displayed in Figure 1, is an example of a surface that we want to call a ‘‘Platonic star’’. This particular example we call a ‘‘dodecahedral star’’ because it has its cusps at the vertices of a regular dodecahedron and has the same symmetries. We refer to the familiar Platonic solid with 12 regular pentagons as faces, 30 edges, and 20 vertices. See Figure 2e. This article deals with the construction of surfaces such as the one in Figure 1. We will always use polynomials such as u

and v in (2). Their role will become clear when we introduce some invariant theory. The general task is to construct an algebraic surface, that is, the zero-set X = V(f) of a polynomial f 2 R½x; y; z, with prescribed symmetries and singularities.1 By ‘‘prescribed symmetries’’ we mean that the surface should be invariant under the action of some finite subgroup of the real orthogonal group O3 ðRÞ . Most of the time we will consider the symmetry group of some Platonic solid S R3 . The symmetry group of a set A R3 is the subgroup of the orthogonal group O3 ðRÞ, formed by all matrices that transport the set into itself, that is, SymðAÞ ¼ fM 2 O3 ðRÞ; MðaÞ 2 A for all a 2 Ag O3 ðRÞ. (Often the symmetry group is defined as a subgroup of SO3 instead of O3.

Citation of Phaedrus from [8]. Supported by Project P-21461 of the Austrian Science Fund FWF. Figures 12 and 13 are generated with Wolfram Mathematica 6 for Students. All the other figures are produced with the free ray-tracing software Povray, http://www.povray.org. 1

Of course a lot of people have been working on construction of surfaces with many singularities, also via symmetries. We want to mention, for example, Oliver Labs and Gert-Martin Greuel.


23

The subgroup of O3 we consider here is referred to as the full symmetry group.) A Platonic solid is a convex polyhedron whose faces are identical regular polygons and the same number of polygons meet at each vertex. At each vertex of a Platonic solid the same number of faces meet. There are exactly five Platonic solids, the tetrahedron, octahedron, hexahedron (or cube), icosahedron, and dodecahedron. See Figure 2. Two Platonic solids are dual to each other if each is the convex hull of the centers of the faces of the other. The octahedron and the cube are dual to each other, as are the icosahedron and the dodecahedron. The tetrahedron is dual to itself. Dual Platonic solids have the same symmetry group. For a more rigorous and more general definition of duality of convex polytopes see [7, p. 77]. The Platonic solids are vertex-transitive polyhedra: their symmetry group acts transitively on the set of vertices. This means that for each pair of vertices there exists an element of the symmetry group that transports the first vertex to the second. One says that all vertices belong to one orbit of the action of the symmetry group. A convex polyhedron that has regular polygons as faces and that is vertex-transitive is either a Platonic solid, a prism, an antiprism, or one of 13 solids called Archimedean solids.2 One can extend the notion of duality as we defined it to Archimedean solids. Their duals are not Archimedean any longer; they are called Catalan solids3 or just Archimedean duals. Each Archimedean solid has the same symmetries as one of the Platonic solids, but with this proviso: in two cases we do not get the full symmetry group but just the rotational symmetries. Here we will deal with just three groups: the symmetry group of the tetrahedron Td, that of the octahedron and cube Oh, and that of the icosahedron and dodecahedron Ih. The Catalan solids are not vertex-transitive but are obviously face-transitive. By ‘‘prescribing singularities’’ of a surface we mean that the zero-set should have a certain number of isolated

singularities of fixed type, at prior chosen locations. A singular point, or singularity, of an algebraic surface is a point where the surface is locally not a manifold. This signifies that the first partial derivatives of the defining polynomial vanish at the point. Isolated means that in a neighborhood of the singularity there are no other singular points. An isolated surface singularity is said to be of type A2 if it has (up to local analytic coordinate transformations) the equation x3 + y2 + z2 = 0. The corresponding zero-set is a two-dimensional cusp Y as displayed in Figure 3a. Note that the cusp, in these coordinates, is a surface of rotation. Its axis of rotation is the x-axis. We call that axis the tangentline of the cusp Y at the origin. (Clearly it is not the tangentline in the usual, differential-geometric sense. The origin is a singularity of the cusp, that is, the surface is not a manifold there, so that differential-geometric methods fail there.) One can also view this tangent-line as the limit of secants of Y joining one point of intersection at the singular point 0 to another point of intersection moving toward 0. Now if X is any variety with a singularity of type A2 at a point p, then we define the tangent-line at this point analogously. Note that we are no longer dealing with a surface of rotation. We will choose the location of the singular points so that they all form one orbit of the action of the selected group. If we use the symmetry group of a Platonic solid, we can choose, for example, the vertices of the corresponding Platonic or Archimedean solid. Now we are ready to define our ‘‘object of desire’’, the ‘‘Platonic star’’. We want to emphasize that the following is not a rigorous mathematical definition. Let S be a Platonic (Archimedean) solid and m the number of its vertices. Denote its symmetry group in O3 ðRÞ by G. An algebraic surface X that is invariant under the action of G and has exactly m isolated singularities of type A2 at the vertices of the solid, is called a Platonic (Archimedean) star. We require that the cusps point outward, otherwise we speak of an anti-star. In both cases for all singular points p the tangent-line of X at p should be the line joining the origin to p.

AUTHORS

......................................................................................................................................................... ALEXANDRA FRITZ is a Master’s Degree

HERWIG HAUSER studied in Innsbruck and

candidate at the University of Innsbruck. She spent last year at the University of Vienna, where she worked on algebraic stars, as reported here, under the supervision of Herwig Hauser.

Paris; he is now a Professor at the University of Vienna. He has done research in algebraic and analytic geometry, especially in resolution of singularities. Among his efforts in presenting mathematics visually is a movie, ‘‘ZEROSET – I spy with my little eye’’.

Fakulta¨t fu¨r Mathematik Universita¨t Wien A-1000 Vienna Austria e-mail: [email protected]

2

Fakulta¨t fu¨r Mathematik Universita¨t Wien A-1000 Vienna Austria e-mail: [email protected]

Often the Archimedean solids are defined as polyhedra that have more than one type of regular polygons as faces but do have identical vertices in the sense that the polygons are situated around each vertex in the same way. This definition admits (besides the Platonic solids, prisms, and antiprisms) an additional 14th polyhedron called the pseudo-rhombicuboctahedron. This is a fact that has often been overlooked. The sources we use, namely [3, p. 47–59] and [4, p. 156 and p. 367], are not very clear about it. See [4]. 3 Named after Euge`ne Charles Catalan, who characterized certain semi-regular polyhedra.

24


(a)

(b)

Tetrahedron.

Octahedron.

(c)

(d)

Hexahedron.

Icosahedron.

(e)

Dodecahedron.

Figure 2. The five Platonic solids.

(a)

The A2 -singularity, x3 + y 2 + z 2 = 0.

(b)

Dodecahedral star and dodecahedron.

Figure 3. The two-dimensional cusp and the dodecahedral star.

(a) c = −300.

(b) c = −30.

(c) c = −15.

(d) c = −3.

(e) c = −1.5.

(f) c = −27 /32.

(g) c = −0.6.

(h) c = −0.3.

(i) c = −0.03.

(j) c = 0.

(k) c = 0.003.

(l) c = 0.03.

(m) c = 0.3.

(n) c = 3.

(o) c = 30.

(p) c = 81.

(q) c = 300.

(r) c = 3000.

Figure 4. Dodecahedral star with varying parameter value c; for c B -27/32 the surfaces are clipped by a sphere of radius 4.5. 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

25

Later we will see that the algebraic surfaces defined by equation (1) from the introductory example satisfy by construction the conditions of the definition above. For now, consider the illustrations, especially Figure 3b, that suggest that this claim is true. If we choose c [ 0, we get stars, for c \ 0 anti-stars. The choice c = 0 yields an ordinary sphere. See Figure 4 for the effect of varying the parameter c. Note that the singularities stay fixed on a sphere of radius 1 for all parameter values, so for c \ 0 we have to zoom out to be able to show the whole picture. For c = -27/32 the anti-star has a point at infinity in the direction of the z-axis, which is among the normals of the faces of the dodecahedron. By symmetry it will also have points at infinity in the direction of the normals of the remaining faces. The pictures suggest that for c [ -27/32 the dodecahedral anti-stars and stars are bounded while they remain unbounded for c \ -27/32. It might be interesting to refine the definition of stars and anti-stars by requiring that the surfaces be bounded. In this article we shall not consider this question.

Some Basics from Invariant Theory In order to explain our construction of the equations for the stars we need a few results from invariant theory. Those who are familiar with the topic can proceed to the next section; those who want to know more details than we give can refer to [10]. For ease of exposition, we work over the complex numbers C. Let there be given a finite subgroup G of GLn ðCÞ. Typically, this will be the symmetry group of a Platonic solid, allowing also reflections. The group G acts naturally on Cn by left-multiplication. This induces an action of G on the polynomial ring C½x1 ; . . .; xn , via p f(x) = f(p x). A polynomial f is called invariant with respect to G if p f = f for all p [ G. For instance, if G is the permutation group Sn on n elements, the invariant polynomials are just the symmetric ones. The collection of all invariant polynomials is clearly closed under addition and multiplication, and thus forms the invariant ring C½xG :¼ ff 2 C½x; f ¼ p f ;

for all p 2 Gg:

In the nineteenth century it was a primary goal of invariant theory to understand the structure of these rings. Hilbert’s Finiteness Theorem asserts that for finite groups, C½xG is a finitely-generated C-algebra: There exist invariant polynomials g1 ðxÞ; . . .; gk ðxÞ such that any other invariant polynomial h is a polynomial in g1, …, gk, say hðxÞ ¼ Pðg1 ðxÞ; . . .; gk ðxÞÞ. Said differently, C½xG ¼ C½g1 ; . . .; gk : In general, the generators may be algebraically dependent, that is, may satisfy an algebraic relation R(g1, . . ., gk) = 0 6 0. It is important to for some polynomial Rðy1 ; . . .; yk Þ 4

understand these relations. As a first result, it can be shown that C½xG always contains some n algebraically independent elements, say u1, . . ., un. These need not generate the whole ring. But it turns out that u1, . . ., un can be chosen so that C½xG is an integral ring extension of its subring C½u1 ; . . .; un . This is Noether’s Normalization Lemma. In particular, C½xG will be a finite C½u1 ; . . .; un -module. A theorem that probably first appeared in an article by Hochster and Eagon [5] asserts that for finite groups G, the invariant ring is even a free C½u1 ; . . .; un -module (one says that C½xG is a Cohen-Macaulay module). That is to say, there exist elements s1 ; . . .; sl 2 C½xG such that C½xG ¼ l aj¼1 sj C½u1 ; . . .; un . This decomposition is called the Hironaka decomposition; the ui are called primary invariants4 and the sj secondary invariants.5 Therefore each invariant polynomial f has a unique decomposition f ¼

l X

sj Pj ðu1 ; . . .; un Þ;

j¼1

for some polynomials Pj 2 C½x1 ; . . .; xn . Things are even better if G is a reflection group. An element M 2 GLðCn Þ is called a reflection if it has exactly one eigenvalue not equal to 1. A finite subgroup of GLðCn Þ is called a reflection group if it is generated by reflections. In a reflection group, C½xG is even generated by n algebraically independent polynomials u1, . . ., un and vice versa (Theorem of Sheppard-Todd-Chevalley) – so that the decomposition reduces to f ¼ Pðu1 ; . . .; un Þ for a uniquely determined polynomial P. Here is how we shall go about constructing the equations for our Platonic stars: Find a polynomial in the invariant generators such that f has the required geometric properties. (Remember that when we speak of symmetry groups we do not restrict to proper rotations. The symmetry groups of the Platonic solids as we defined them are reflection groups. By the Sheppard-Todd-Chevalley Theorem, this can be checked by calculating the primary and secondary invariants.) Even though, for each f, the polynomial P is unique, there could be several f sharing the properties. This phenomenon will actually occur; it is realized by a certain flexibility in choosing the parameters of our equations. The families of stars which are thus obtained make certain parameter values look more natural than others. This is the case for the plane symmetric star with four vertices, where only one choice of parameters yields a hypocycloid, the famous Astroid (see example 9). For surfaces, the appropriate choice of parameters is still an open problem. This raises also the question of whether (in analogy to the rolling small circle inside a larger one for the Astroid) there is a recipe for contructing the Platonic stars with distinguished parameter values. We don’t know the answer.

In the following chapter on the construction and in the examples, we write u; v; w instead of u1 ; u2 ; u3 . Note that sometimes we do not need all three of them, as in the introductory example of the dodecahedron; but a general invariant polynomial may depend on all three. 5 There exist algorithms to calculate these invariants. One is implemented in the free Computer Algebra System SINGULAR. See http://www.singular.uni-kl.de/ index.html for information about SINGULAR and http://www.singular.uni-kl.de/Manual/latest/sing_1189.htm#SEC1266 for instruction.

26


Construction of Stars In this section the group G is once again one of the three real symmetry groups of the Platonic solids. If the scalars of the input of the algorithms for the calculation of primary and secondary invariants are contained in some subfield of C, then the scalars of the output are also contained in this subfield, see [10, p.1]. In our examples the inputs are real matrices (the generators of G) and the outputs are the primary and secondary invariants that generate the invariant ring as a subring of C½x1 ; . . .; xn . They even generate the real invariant ring, R½x1 ; . . .; xn G . See the last section ‘‘Technical Details’’ for a proof. The symmetry groups of the Platonic solids are reflection groups. This implies that we have primary invariants fu; v; wg R½x; y; z such that R½x; y; zG ¼ R½u; v; w. In the following we always assume that we have already constructed a set of homogeneous primary invariants fu; v; wg R½x; y; z. Our aim is to construct a polynomial f in the invariant ring of G with prescribed singularities. By the results from the previous section we may write the polynomial uniquely in the form X aijk ui v j wk ; ð3Þ f ðu; v; wÞ ¼ id1 þjd2 þkd3 d

where d1 ¼ degðuÞ; d2 ¼ degðvÞ; d3 ¼ degðwÞ, and aijk 2 R. Such a polynomial has the desired symmetries, so we may move on and prescribe the singularities. They should lie at the vertices of a Platonic or an Archimedean solid. Let S be a fixed Platonic (or Archimedean) solid. In the introduction we mentioned that these solids are vertex-transitive. This implies that the algebraic surface corresponding to the polynomial (3), which is an element of the invariant ring of the symmetry group of S, has to have the same local geometry at each vertex of S. Therefore it is sufficient to choose one vertex and impose conditions on f (u, v, w) guaranteeing an A2-singularity there. We can always suppose that S has one vertex at p := (1, 0, 0), otherwise we perform a coordinate change to make this true. Having a singularity is a local property of the surface, so we have to look closer at f at the point p. We do that by considering the Taylor expansion at p, that is, substitute x + 1 for x in f (u(x, y, z), v (x, y, z), w (x, y, z)). We have the following necessary condition for a singularity of type A2, with c1 and c2 being real constants not equal to 0; see [1, p.209].

E X A M P L E 2 (Octahedral and Hexahedral Star). The octahedron (the Platonic solid with 6 vertices, 12 edges, and 8 faces) and its dual the cube (or hexahedron - with 8 vertices, 12 edges, and 6 faces) have the same symmetry group Oh, of order 48. We choose coordinates x, y, and z of R3 such that in these coordinates the vertices of the octahedron are (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1). Then Oh is generated by two rotations r1, r2 around the x- and the y-axes by p/2, together with the reflection s in the xy-plane: 0 1 0 1 1 0 0 0 0 1 B C B C r1 ¼ @ 0 0 1 A; r2 ¼ @ 0 1 0 A; 0

0 1 1 0

B s ¼ @0 1 0 0

F ðx; y; zÞ :¼ f ðu1 ðx þ 1; y; zÞ; u2 ðx þ 1; y; zÞ; u3 ðx þ 1; y; zÞÞ ¼ c1 ðy2 þ z 2 Þ þ c2 x 3 þ higher order terms:

coefficients of f from (3), that is, in our notation the parameters aijk. Additionally we obtain inequalities that give us information about whether we will obtain a star or an anti-star. In general this system of equations will be underdetermined. We will be left with free parameters, as we already saw in the introductory example of the dodecahedral star. Evidently, in this construction we have to choose the degree d of the undetermined polynomial f. If we choose it too small, the system of equations may not have a solution; but we want d to be as small as possible subject to this. The degree d has to be greater or equal to 3, clearly. It depends on the degrees of the primary invariants ui, as we will see in the examples. The same construction should work for any dimension n. The case of plane curves, n = 2, is easier to handle. Even there the results are quite nice, as we will see in the section on ‘‘plane dihedral stars’’. An interesting generalization for n = 4 would be to calculate ‘‘Schla¨fli stars’’, corresponding to the six convex regular polytopes in four dimensions, which were classified by Ludwig Schla¨fli, [2, p. 142]. We now conclude this section with a detailed example. More examples will follow in the next section, namely, the remaining Platonic stars and some Archimedean stars. We will also present some selected surfaces with dihedral symmetries in real 3-space.

ð4Þ

‘‘Higher order terms’’ here refers to all terms that have weighted order, with weights (1/3, 1/2, 1/2), greater than 1— that is, all monomials x i y j z k with i/3 + j/2 + k/2 [ 1. If c1 and c2 have the same sign, the cusps will ‘‘point outward’’, that is, we obtain a star. If they have different signs, the cusps will ‘‘point inward’’. Now expanding F (x, y, z) and comparing the coefficients of x, y, and z with the right-hand side of equation (4), we obtain a system of linear equations in the unknown

0 1 0 C 0 A: 1

1 0

0

These matrices are the input for the algorithm implemented in SINGULAR that computes the primary and secondary invariants. In this example the primary invariants that generate the invariant ring are the following (although it is easy to see that these three polynomials are invariant, it is not evident that they are primary invariants, that is, generate the invariant ring as an algebra): uðx; y; zÞ ¼ x 2 þ y 2 þ z 2 ; vðx; y; zÞ ¼ x 2 y2 þ y2 z 2 þ x 2 z 2 ;

ð5Þ

wðx; y; zÞ ¼ x 2 y2 z 2 :


27

Now how low can the degree be of our indeterminate polynomial? (Here and in the rest of this article degree means the usual total degree in x, y, z.) Clearly it must be even. A degree four polynomial yields no solvable system of equations. Let us try a polynomial of degree six, f ðu; v; wÞ ¼ 1 þ a1 u þ a2 u2 þ a3 u3 þ a4 uv þ a5 v þ a6 w: We substitute x + 1 for x and expand the resulting polynomial F (x, y, z) = f (u (x + 1, y, z), v (x + 1, y, z), w (x + 1, y, z)). Next we collect the constant terms and the linear, quadratic, and cubic terms, and compare them with the right-hand side of (4). This yields the following system of linear equations: Constant term of F : Coefficient of x : Coefficient of x 2 : Coefficient of y2 and z 2 : Coefficient of x 3 :

1 þ a1 þ a2 þ a3 ¼ 0; 2a1 þ 4a2 þ 6a3 ¼ 0; a1 þ 6a2 þ 15a3 ¼ 0; a5 þ a1 þ a4 þ 2a2 þ 3a3 ¼ c1 ; 4a2 þ 20a3 ¼ c2 : ð6Þ

Since the monomials y, z, xy, xz, yz do not appear, we do not obtain further equations from them. Solving the first three equations from the above system yields the polynomial (7) with three free parameters. In addition we get an inequality from the condition that the coefficient of x3 must have the same sign as the coefficient of y2 and z2 if we want to obtain a star. Substituting the solution of the first three equations yields c1 = a4 + a5 and c2 = -8. We impose a4 + a5 = 0 to obtain a star or an anti-star: f ðu; v; wÞ ¼ ð1 uÞ3 þ a4 uv þ a5 v þ a6 w; with a4 þ a5 6¼ 0: ð7Þ If we allowed all three parameters to be zero we would obtain the sphere of radius 1. We have already made clear that for a4 + a5 = 0 the zero-set of (7) cannot have singularities of type A2, so it must either be smooth or have singularities of a different type. If we choose a4 = c, a5 = 0 and a6 = -9c, c = 0, the zero-set is again not an octahedral star, for it has too many singularities; we will describe this phenomenon in more detail after the example of the hexahedral star. For the other choices of parameters the corresponding zero-sets are octahedral stars for a4 + a5 \ 0, or anti-stars for a4 + a5 [ 0. See Figure 5a. Sometimes additional components appear and the stars or anti-stars become unbounded. In all examples presented in this article, especially when there is more than one free parameter, special behaviors (such as additional components, unboundedness, or maybe more singularities than expected) may occur for special choices of the free parameters. Most pictures presented are merely based on (good) choices of parameters. As we already mentioned, it would be interesting to find conditions that prevent this behavior so that we could prescribe boundedness as well as irreducibility in the definition of a star. Now we turn to the Platonic solid dual to the octahedron, namely the cube. If we use the same coordinates as before, it has vertices at ð p1ffiffi3 ; p1ffiffi3 ; p1ffiffi3Þ. But as we already mentioned, we prefer to have a vertex at (1, 0, 0),

28


(a)

Octahedral star, a4 = −100,

(b)

Hexahedral star, a1 = −100,

a5 = 0, a6 = 0.

a2 = 0, a3 = 0.

Figure 5. Octahedral and hexahedral star.

so we perform a rotation to achieve this, and write the invariants in the new coordinates. With these invariants we can proceed as in the example of the octahedron. Again we get no solution with degree four and must use a polynomial of degree six. After solving the system of equations we perform the inverse coordinate change and obtain the following polynomials (8) as candidates for hexahedral stars or anti-stars: f ðu; v; wÞ ¼ 1 3u þ a1 u2 þ a2 u3 þ a3 uv þ ð9 3a1 Þv ð8Þ 9ð3 þ a3 þ 3a2 Þw; with 3a1 + 9a2 + 2a3 = 0. For a1 = 3, a2 = -1, and a3 = 0 we obtain the sphere. Other choices such that 3a1 + 9a2 + 2a3 = 0 may give singularities but cannot give A2-singularities. Again there is a choice of parameters, namely a1 = 3, a2 = -1 and a3 = c = 0, for which the surface has too many singularities. We obtain the same object as in the example of the octahedral star, with equation (9) below. In the other cases we obtain a hexahedral star for 3a1 + 9a2 + 2a3 \ 0 (Figure 5b), or anti-star for 3a1 + 9a2 + 2a3 [ 0, even though, as in the example of the octahedral stars, additional components may appear. Before proceeding, let us say more about the surface (9) that emerged as a special case both of the octahedral and the hexahedral stars. It has 14 singularities, exactly at the vertices of the octahedron and the cube, see Figure 6, f ðu; v; wÞ ¼ ð1 uÞ3 þ cuv 9cw;

with c 6¼ 0:

ð9Þ

We will call this object a 14-star or 14-anti-star for c \ 0 or c [ 0, respectively. The parameter value c = 0 yields obviously a sphere. See Figure 7 for an illustration of the dependence on the parameter. This star does not correspond to a Platonic or Archimedean solid, but to the polyhedron S that is the convex hull of the vertices of a hexahedron and an octahedron that all have the same Euclidean diameter. This polyhedron has 14 vertices, 36 edges, and 24 faces, which are isosceles triangles. See Figure 6b. It is remarkable that it appears here, for the symmetry group Oh does not act transitively on its vertices! The vertices of the hexahedron form one orbit, the vertices of the octahedron another. If we followed the program of this paper and sought such a star, we would need to fix two points, one in each orbit, and

Here a degree three polynomial yields no solution but degree four already suffices: f ðu; v; wÞ ¼ 1 2u þ cu2 þ 8v ð3c þ 1Þw; with c 6¼ 1:

ð11Þ

For c \ 1 we obtain a star, for c [ 1 an anti-star. Its singular points (for c = 1) are (1, 1, 1), (-1, -1, 1), (1, -1, -1), and (-1, 1, -1). If we choose c = 1 in (11) the polynomial f has four linear factors, see Figure 8j:

(a)

(b)

14-star, c = −50.

Polyhedron S corresponding to the 14-star.

Figure 6. 14-star and the corresponding convex polyhedron.

prescribe singularities at both. This would lead to a larger system of linear equations. Note, by the way, that if the vertices of the octahedron and the cube have pffiffiffidifferent Euclidean norms of a certain ratio, namely 2= 3, the convex hull is a Catalan solid, called the rhombic dodecahedron (14 vertices, only 12 faces because the triangles collapse in pairs into rhombi, and 24 edges). This is the dual of the Archimedean solid called the cuboctahedron that will be discussed later.

Further Platonic and Archimedean Stars E X A M P L E 3 (Tetrahedral star). The tetrahedron is the Platonic solid with 4 vertices, 6 edges, and 4 faces. Its symmetry group Td has 24 elements. If we choose coordinates x, y, z such that one vertex is (1, 1, 1), the invariant ring is generated by the primary invariants displayed in (10). One could also choose (1, 0, 0) as a vertex to avoid a coordinate change, but then the invariants would be more complicated. Note how different the primary invariants are from those of the octahedron and the hexahedron (5). uðx; y; zÞ ¼ x 2 þ y2 þ z 2 ; vðx; y; zÞ ¼ xyz; 2 2

2 2

ð10Þ 2 2

wðx; y; zÞ ¼ x y þ y z þ z x :

(a) c = −10000.

(g) c = 0.

(b) c = −1000.

(h) c = 0.1.

f ¼ ðx 1 þ z yÞðx 1 z þ yÞðx þ 1 z yÞ ðx þ 1 þ z þ yÞ: For very small c-values there seem to appear four additional cusps at the vertices of a tetrahedron dual to the first one; but these points stay smooth for all c 2 R. For 0 \ c \ 1 the zero-set of our polynomial has additional components besides the desired ‘‘star shape’’. For c [ 1 we get anti-stars, see Figure 8. Note that for c [ 0 the surfaces are unbounded. So unlike the previous examples, there are no bounded anti-stars.

E X A M P L E 4 (Icosahedral star). The icosahedron is the Platonic solid with 12 vertices, 30 edges, and 20 faces. The symmetry group Ih of the icosahedron and its dual, the dodecahedron, has 120 elements. Its invariant ring is generated by the polynomials (2) from the first example, together with a third one (12), wðx;y;zÞ ¼ 4x 2 þz 2 6xz z 4 2z 3 x x 2 z 2 þ2zx 3 þx 4 25y 2 z 2 30xy 2 z 10x 2 y2 þ5y4 z 4 þ8z 3 x þ14x 2 z 2 8zx 3 þx 4 10y 2 z 2 10x 2 y2 þ5y4 Þ:

ð12Þ

We point out that both invariants v and w factorize (over R) into six, respectively ten, linear polynomials. The zerosets of these linear polynomials are related to the geometry. To explain this, we introduce a new terminology: Given a Platonic solid P, we call a plane through the origin a

(c) c = −100.

(d) c = −10.

(e) c = −1.

(f) c = −0.1.

(i) c = 1.

(j) c = 3.5.

(k) c = 4.

(l) c = 5.

Figure 7. 14-star and anti-star; for c C 4 the surfaces are clipped by a sphere with radius 5. 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

29

(a)

(b) c = −1000.

(c) c = −100.

(d) c = −3.

(e) c = −1.

(f) c = 0.

(h) c = 0.6.

(i) c = 0.98.

(j) c = 1.

(k) c = 1.02.

(l) c = 3.

c = −100000.

(g) c = 0.3.

Figure 8. Tetrahedral star (and anti-star) with varying parameter value c; for c [ 0 the images are clipped by a sphere with radius 5.

centerplane of P if it is parallel to a face of the solid. The dodecahedron has twelve faces and six pairs of parallel faces, so it has six centerplanes. They correspond to the six linear factors of the second invariant v. Analogously the icosahedron has ten centerplanes, which give the linear factors of w. We have written out the factorization in the last section, see (38). For the dodecahedral and the icosahedral star the ‘‘smallest possible degree’’ is six. The third invariant has degree ten, so we do not use it in either case. An equation for the icosahedral star is the following: f ðu; v; wÞ ¼ ð1 uÞ3 þ cu3 þ cv;

with c 6¼ 0:

ð13Þ

Figure 9 shows icosahedral stars (c \ 0) and anti-stars (c [ 0) for various c-values. For c = 0 we get a sphere of

radius 1. For all c = 0 the 12 singularities lie on this sphere. For c = 27/32 the surface has points at infinity in the direction of normals to the faces of the corresponding icosahedron. Note that this is just the negative value of c for which the dodecahedral stars are unbounded. The illustrations suggest that for c [ 27/32 the surfaces become unbounded, while they are bounded for c \ 27/32.

E X A M P L E 5 (Cuboctahedral star). The cuboctahedron is the Archimedean solid with 14 faces (6 squares and 8 equilateral triangles), 24 edges, and 12 vertices. See Figure 11b. Its symmetry group is that of the octahedron and cube. We use the invariants (5). Our construction yields a polynomial of degree six, with three free parameters:

(a) c = −1000.

(b) c = −100.

(c) c = −10.

(d) c = −0.1.

(e) c = 0.

(f) c = 0.1.

(g) c = 0.5.

(h) c = 0.8.

(i) c = 27/ 32.

(j) c = 0.9.

Figure 9. Icosahedral star and anti-star, with varying parameter c; for c [ 27/32 the surfaces are clipped by a sphere with radius 11. 30


Figure 10. The zero-set of x3 + y2 - z2 = 0.

f ðu; v; wÞ ¼ 1 3u þ au2 þ ð12 4aÞv þ bu3 ð4 þ 4bÞuv þ cw;

ð14Þ

f ðu; v; wÞ pffiffiffi pffiffiffi 128565 þ 115200 5 49231296000 5 93078919125 c3 þ ¼1þ 1295029 15386239549 c4 c1 3c2 3 uþ pffiffiffi pffiffiffi 230400 5 257130 238926989250 126373248000 5 þ c3 þ 1295029 15386239549 c4 þ 3c1 þ 8c2 þ 3 u2 þ pffiffiffi pffiffiffi 115200 5 þ 128565 91097280000 5 172232645625 c3 þ þ 1295029 15386239549 c4 3c1 6c2 1 u3 þ pffiffiffi pffiffiffi 121075 51200 5 102400 5 242150 þ c3 þ 2c3 c4 v þ 11881 11881 uv þ c1 u4 þ c2 u5 þ c3 u2 v þ c4 w;

with a + b = 2 and 8(a + b) - c = 16. For a = 3, b = -1, and c = 0 we obtain a sphere. In this example we have a new kind of behavior. We always got inequalities from the condition that the coefficients of x3 and y2 + z2 in the Taylor expansion of f in (1, 0, 0) should have the same sign. In this case the coefficient of x3 is -8, but y2 and z2 have different coefficients, namely a + b - 2 and 16 - 8(a + b) + c, respectively. So if both are negative we obtain stars, see Figure 11a; if both are positive, anti-stars; but if they have different signs, we will have a ‘‘new’’ object, whose singularities have, up to local analytic coordinate transformations6, equation x3 + y2 - z2 = 0. See Figure 10. The singularities always lie on a sphere of radius 1.

E X A M P L E 6 (Soccer star). The truncated icosahedron is the Archimedean solid which is obtained by ‘‘cutting off the vertices’’ of an icosahedron. It is known as the pattern of a soccer ball. It has 32 faces (12 regular pentagons and 20 regular hexagons), 60 vertices, and 90 edges. See Figure 11d. Its symmetry group is the icosahedral group Ih. For this example we do need the third invariant, because the first polynomial that yields a solvable system of equations is of degree ten. We obtain the following equation with four free parameters, in the invariants (2) and (12):

(a)

Cuboctahedral star,

(b)

a = 0, b = 0, c = −100.

Cuboctahedron.

with c4 6¼ 0 and bðc1 ; c2 ; c3 ; c4 Þ pffiffiffi :¼ ð991604250 419328000 5Þc4 þ 20316510c3 þ pffiffiffi þ ð135776068 121661440 5Þc2 pffiffiffi pffiffiffi þ ð33944017 30415360 5Þc1 þ 30415360 5 33944017 6¼ 0:

ð15Þ

We obtain stars if we choose c1, c2, c3, and c4 such that c4 and b(c1, c2, c3, c4) have the same sign. Otherwise we obtain anti-stars. See Figure 11c.

Plane Dihedral Stars Analogous to the Platonic and Archimedean stars in three dimensions, we will define plane stars. Let P be a regular polygon with m vertices. Its symmetry group in O2 ðRÞ is the dihedral group denoted by Dm. It is of order 2m. A plane m-star is a plane algebraic curve that is invariant under the action of the dihedral group Dm and has exactly m singularities of type A2 (that is, with equation x3 + y2 = 0) at the vertices of P, ‘‘pointing away form the origin’’; otherwise, that is, if the cusps ‘‘point towards the origin’’, we speak of an m-anti-star. In the examples presented here the singularities will be at the mth roots of unity. If we consider the dihedral groups as subgroups of O2 ðRÞ, they are reflection groups. This is not true if we view them as subgroups of O3 ðRÞ, as we do in the next batch of examples.

(c)

Soccer star,

c1 = −100, c2 = −100,

(d)

Truncated

icosahedron.

c3 = −100, c4 = −100.

Figure 11. Two Archimedean solids and stars. 6

But if we allow complex local analytic coordinate transformations, the singularities are still of type A2.


31

1.0

0.5

0.5

0.5

0.0

0.0

0.0

y

1.0

y

y

1.0

0.5

0.5

0.5

1.0

1.0 1.0

0.5

0.0

0.5

1.0

1.0 0.5

0.0

x

0.5

1.0

1.0

0.5

0.0

x

(a) A 2-star (19), c1 = −4.

(b) The Deltoid.

0.5

1.0

x

(c) The Astroid.

Figure 12. Some plane dihedral stars.

There is another way to construct plane stars, namely as hypocycloids. A hypocycloid is the trace of a point on a circle of radius r that is rolling within a bigger circle of radius R. If the ratio of the radii is an integer, R : r = k, then the curve is closed and has exactly k cusps but no selfintersections. Hypocycloids have a quite simple trigonometric parametrization (16): u 7! ðk 1Þr cos u þ r cos ðk 1Þu; ðk 1Þr sin u r sin ðk 1ÞuÞ; u 2 ½0; 2p: ð16Þ

Figure 12a. For c1 [ 0 it is an unbounded anti-star. In both cases it has two singularities, at (±1 , 0). The hypocycloid for k = 2 is parametrized by (2r cosu, 0) where u is in [0, 2p]. So it is not an algebraic curve but an interval on the x-axis.

There are algorithms for the implicitization of trigonometric parametrizations; see [6]. It turns out that hypocycloids are stars in our sense: they have the correct symmetries and singularities of type A2. In the construction of stars via primary invariants we always try to find a polynomial of minimal degree that has these properties. We will see that sometimes the hypocycloids coincide with the stars we obtain that way. In one of the examples presented here, namely the 5-star, the degree of the implicitization of the hypocycloid is higher than the degree of the polynomial our construction yields. This suggests that we define a ‘‘star’’ as the zero-set of the polynomial of minimal degree satisfying all other conditions.

In this case a degree four polynomial suffices to generate a star; see Figure 12b. The polynomial (21) is completely determined; we have no free parameters. It coincides with the hypocycloid for k = 3, which is also called the Deltoid:

E X A M P L E 7 (2-star). The group D2 has primary invariants

f ðu; vÞ ¼ ð1 uÞ3 þ c1 v þ c2 uv;

2

uðx; yÞ ¼ x ;

ð17Þ

vðx; yÞ ¼ y2 :

Our construction yields the degree six polynomial with six free parameters: f ðu; vÞ ¼ð1 uÞ3 þ c1 v þ c2 uv þ c3 v 2 þ c4 uv2 þ c5 u2 v þ c6 v 3 ;

with c1 þ c2 þ c5 6¼ 0:

ð18Þ

The choice c1 = 0 and the remaining parameters equal to 0 yield the simple equation f ðu; vÞ ¼ ð1 uÞ3 þ c1 v;

with c1 6¼ 0:

ð19Þ

For c1 \ 0 we obtain a 2-star. The corresponding curve 1 runs through the points ð0; pffiffiffiffiffiffi c1 Þ and is bounded. See 32


E X A M P L E 8 (3-star). The primary invariants of D3 are uðx; yÞ ¼ x 2 þ y2 ; vðx; yÞ ¼ x 3 3xy 2 :

f ðu; vÞ ¼ 1 6u 3u2 þ 8v:

ð20Þ

ð21Þ

E X A M P L E 9 (4-star). The dihedral group of order eight, D4, has primary invariants uðx; yÞ ¼ x 2 þ y2 ; vðx; yÞ ¼ x 2 y2 :

ð22Þ

Our construction yields the following polynomial of degree six with two free parameters: with c1 þ c2 6¼ 0;

ð23Þ

For c1 + c2 \ 0 we obtain stars, for c1 + c2 [ 0 antistars. In both cases additional components might appear. The curves become unbounded for c2 [ 4. The hypocycloid with four cusps is also called the Astroid. Its implicit equation is (1 - u)3 - 27v = 0. So if we choose c1 = -27 and c2 = 0 in (23) we obtain the same curve. See Figure 12c.

E X A M P L E 10 (5-star). The primary invariants of D5 are uðx; yÞ ¼ x 2 þ y 2 ; vðx; yÞ ¼ x 5 10x 3 y2 þ 5xy4 :

ð24Þ

If we try a degree five polynomial, we obtain (25) with no free parameters. It only permits anti-stars.

f ðu; vÞ ¼ 1

10 8 u þ 5u2 v: 3 3

type A1, that is, they have, up to analytic coordinate transformations, equation x2 + y2 = 0. One could call this curve an algebraic pentagram. For c [ 80 the curve has two components; see Figure 13e. The implicit equation of the hypocycloid with five cusps is already of degree eight, while the polynomial we found with our construction has degree six. The two equations cannot coincide for any choice of the free parameter c.

ð25Þ

So let us use degree six. This yields the following polynomial for plane 5-stars or anti-stars: f ðu; vÞ ¼ 1

c þ 10 8 u þ ð2c þ 5Þu2 ð1 þ cÞv þ cu3 ; 3 3 with c 6¼ 1; 5: ð26Þ

Here, as the parameter value c varies we observe a quite curious behavior. For c \ -1 one obtains a star, the smaller c gets, the smaller is its ‘‘inner radius’’; see Figure 13a. The choice c = -1 yields a circle with radius 1— the circle containing the five singularities of (26) for other c. For -1 \ c \ 5 the cusps of (26) point inward, that is, we have anti-stars. For - 1 \ c \ 0 the curve has one bounded component; for c = 0, it is unbounded with five components, Figure 13b; for 0 \ c \ 5 the curve is again bounded, but has five components, like drops falling away from the center, Figure 13c. For c = 5 only finitely many points satisfy the equation, the five points that are singular in the other cases. If we choose c [ 5 we obtain stars again, that is, the cusps point outward, even though for 5 \ c \ 80 the curve also has five components, like drops falling towards the origin, Figure 13d. The curve we obtain for c = 80 is special in that it has self-intersections, that is, five additional singularities. They lie on a circle with radius , on a regular pentagon. These ‘‘extra singularities’’ are of

Dihedral ‘‘Pillow Stars’’ in R3 If we consider the dihedral groups Dm as subgroups of O3 ðRÞ, they cease to be reflection groups, so we have to consider the secondary invariants as well. The number of secondary invariants depends on the order of the group and the degrees of the primary invariants; see [10, p.41]. In the examples we give here there are always two secondary invariants. The first one, s1, is always 1, so we do not mention it every time but just give the second one, s2. In this section our aim is to construct surfaces that are invariant under the action of Dm with singularities at the mth roots of unity in the xy-plane, and which in addition pass through the points (0, 0, ±c) with 0 6¼ c 2 R. Intuitively the resulting surface should look like a pillow. To lead to such a shape, more conditions, such as boundedness and connectedness, would have to be imposed. We do not have a systematic theory, but our experimental results seem promising. In these examples we have a large

1.0

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(a) 5-stars.

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Figure 13. 5-stars and anti-stars. 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

33

(a)

D 3 -star.

(b)

(c)

D4 -star.

D5 -star.

Figure 14. Pillow stars.

1 þ c1 c4 þ c4 c6 u 3v þ c1 u2 þ c2 uv þ 3v 2 c2 þ c3 w þ c4 u3 v 3 þ c5 uw þ þc6 vw þ c7 uv2

f ðu; vÞ ¼ 1

þ c8 u2 v;

ð32Þ

Figure 15. Zitrus for c = -4.

number of free parameters, unfortunately. We have tried to choose values giving attractive results.

with c3 + c6 \ 0 and -(1 + c1c4 + c4c6) + c2(c2 + c7) \ 0. See Figure 14b for the resulting surface, where we chose c = 1/3, c3 = -27 and set all the other parameters equal to 0.

E X A M P L E 11 (D3). The primary invariants of D3 O3 ðRÞ are

E X A M P L E 13 (D5). The primary invariants of D5 are

uðx; y; zÞ ¼ z 2 ; vðx; y; zÞ ¼ x 2 þ y 2 ;

ð27Þ

wðx; y; zÞ ¼ x 3 3xy 2 ;

uðx; y; zÞ ¼ 0z 2 ; vðx; y; zÞ ¼ x 2 þ y2 ; 5

ð33Þ 3 2

4

wðx; y; zÞ ¼ x 10x y þ 5xy :

its secondary invariant is s2 ðx; y; zÞ ¼ 3x 2 yz y3 z:

ð28Þ

A polynomial of degree three yields no solution. The general equation of a degree four polynomial in the invariant ring of D3 is f1(u, v, w) + b s2, where f1(u, v, w) is an indeterminate polynomial of degree four in R½u; v; w as in the previous examples, and b is a constant. A degree four polynomial suffices to obtain a solvable system of equations. It yields the following polynomial with three free parameters: f ðu; v; wÞ ¼ 1

1 þ c1 c4 u þ c1 u2 þ c2 uv 6v 3v 2 þ 8w; c2 ð29Þ

with -(1 + c1c4) + c2c2 \ 0. Note that the secondary invariant s2 does not appear in the above polynomial, its coefficient b is 0. We obtain a nice result for c1 = c2 = 0, c = 1/3; see Figure 14a.

E X A M P L E 12 (D4). The group D4 has the following primary and secondary invariants: uðx; y; zÞ ¼ z 2 ; vðx; y; zÞ ¼ x 2 þ y 2 ;

Its secondary invariant is s2 ðx; y; zÞ ¼ 5x 4 yz 10x 2 y3 z þ y5 z:

ð34Þ

A degree five polynomial already produces a solvable system of equations, but the resulting polynomial with three free parameters only permits anti-stars. So we choose a polynomial of degree six; again s2 does not appear: 1 þ c1 c4 þ c3 c6 10 þ c4 u v þ c 1 u2 c2 3 8 þ c2 uv þ ð5 þ 2c4 Þv2 ð1 þ c4 Þw þ 3 þ c3 u3 þ c4 v 3 þ c5 uv 2 þ c6 u2 v:

f ðu; v; wÞ ¼ 1

ð35Þ

The zero-sets of these polynomials are stars for c4 þ 1\0 and ð1 þ c1 c4 þ c3 c6 Þ þ c2 ðc2 þ c5 Þ\0; 4

6

or

2

c4 5 [ 0 and ð1 þ c1 c þ c3 c Þ þ c ðc2 þ c5 Þ [ 0: ð36Þ A nice choice for the free parameters is c = 1/3, c4 = -3, setting all other parameters equal to 0. See Figure 14c.

ð30Þ

2 2

wðx; y; zÞ ¼ x y ;

E X A M P L E 14 (Zitrus). The last example we want to ð31Þ

present is the surface Zitrus. It is the plane 2-star rotated around the x-axis (Figure 15). Its equation is

Our construction yields a degree six polynomial; as in the previous example, the secondary invariant s2 happens to drop out:

f ðx; y; zÞ ¼ ð1 ðx 2 þ y2 þ z 2 ÞÞ3 þ cðy2 þ z 2 Þ; with c\0: ð37Þ

s2 ðx; y; zÞ ¼ x 3 yz xy 3 z:

34


Outlook In all the examples presented above we observed ‘‘unwanted’’ behavior for special choices of the free parameters: the surfaces became unbounded at some point, or additional components appeared. Sometimes we even had more singularities, or singularities of a different type than we expected. Further investigations would be necessary to find conditions preventing such behavior. After this is done, one could refine the definition of ‘‘stars’’ and ‘‘anti-stars’’ by demanding that the surfaces be bounded and irreducible. Dual (Platonic) solids have the same symmetry group, hence the same primary invariants were used to construct the corresponding stars. But there seems to be no obvious duality between the stars such as occurs for dual (Platonic) solids.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffi pffiffiffi pffiffiffi 5y 5 3 þ 5 3y þ 2 75 þ 30 5z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffi pffiffiffi x 75 þ 30 5 þ x 75 þ 30 5 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffi pffiffiffi pffiffiffi þ 5y 5 3 5 3y þ 2 75 þ 30 5z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffipffiffiffi pffiffiffipffiffiffi x 75 30 5 þ x 75 30 5 5 þ 5y 5 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi þ 5 3y 2 75 30 5z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffi pffiffiffipffiffiffi pffiffiffi x 75 30 5 þ x 75 30 5 5 5y 5 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 3y 2 75 30 5z : ð38Þ

Technical Details Factorization of the primary invariants of Ih In example (4) of the icosahedral stars, we claimed that two of the primary invariants of Ih factor into linear polynomials corresponding to the centerplanes of the dodecahedron and icosahedron, respectively, and we promised to give the factorizations explicitly. Here they are: vðx; y; zÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1 ¼ zð2x þ zÞ ð 5 þ 1Þx 10 2 5y 2z 16 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ð 5 þ 1Þx þ 10 2 5y 2z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ð 5 1Þx 10 þ 2 5y þ 2z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ð 5 1Þx þ 10 þ 2 5y þ 2z ;

pffiffiffi pffiffiffi 1 wðx; y; zÞ ¼ 3x þ x 5 þ z 3x þ x 5 z 20250000 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffi pffiffiffi pffiffiffi 2x 75 þ 30 5 þ x 75 þ 30 5 5 þ 5 3y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 75 þ 30 5z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi 2x 75 þ 30 5 þ x 75 þ 30 5 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 3y 75 þ 30 5z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffipffiffiffi 2x 75 30 5 þ x 75 30 5 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 3y þ 75 30 5z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffipffiffiffi 2x 75 30 5 þ x 75 30 5 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi þ 5 3y þ 75 30 5z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffipffiffiffi x 75 þ 30 5 þ x 75 þ 30 5 5

The invariant ring R½x1 ; . . .; xn G Let G GLðRn Þ be a finite subgroup. Then there exist n homogeneous, algebraically independent polynomials u1 ; . . .; un 2 C½x1 ; . . .; xn (called the primary invariants of G) and l (depending on the cardinality of G and the degrees of the ui) polynomials s1 ; . . .; sl 2 C½x1 ; . . .; xn (the secondary invariants of G) such that the invariant ring l decomposes as C½x1 ; . . .; xn G ¼ aj¼1 sj C½u1 ; . . .; un . There are algorithms to calculate these primary and secondary invariants; see [10, p.25]. Also in [10, p.1] it is claimed that if the scalars of the input for these algorithms are contained in a subfield K of C, then all the scalars in the output will also be contained in K. So in our case with G GLðRn Þ, the primary and secondary invariants will be real polynomials: u1 ; . . .; un ; s1 ; . . .; sl 2 R½x1 ; . . .; xn . Now the claim is, in the notation above: l R½x1 ; . . .; xn G ¼ aj¼1 sj R½u1 ; . . .; un . l The first inclusion R½x1 ; . . .; xn G aj¼1 sj R½u1 ; . . .; un is trivial. We prove the opposite inclusion: Let f 2 R½x1 ; . . .; xn G C½x1 ; . . .; xn G be an invariant polynomial. l As C½x1 ; . . .; xn G equals aj¼1 sj C½u1 ; . . .; un , we can write f uniquely in the following form: f ðx1 ; . . .; xn Þ ¼

l X X sj cja ua ; j¼1

a2A

where cja = dja + ieja are complex constants, and A is some finite subset of Nn . Then ! l X X X a a f ðx1 ; . . .; xn Þ ¼ sj dja u þ i eja u a2A

j¼1

¼

l X j¼1

sj

X a2A

dja ua þ i

a2A l X j¼1

sj

X

eja ua

ð39Þ

a2A

¼ f1 ðx1 ; . . .; xn Þ þ if2 ðx1 ; . . .; xn Þ: Here f1 and f2 are real polynomials. Since f is also contained in the real polynomial ring, f2 must be equal to 0. But P P P P from f2 ðx1 ; . . .; xn Þ ¼ lj¼1 sj a2A eja ua ¼ a2A ð lj¼1 sj eja Þ P ua ¼ 0 it would follow that for all a [ A the sum lj¼1 sj eja must


35

be equal to 0, because the ui are algebraically indepenP P l dent. Hence f ¼ f1 ðx1 ; . . .; xn Þ ¼ lj¼1 sj a2A dja ua 2 aj¼1 sj R½u1 ; . . .; un . ACKNOWLEDGMENTS

We thank Frank Sottile for valuable suggestions and C. Bruschek, E. Faber, J. Schicho, D. Wagner, and D. Westra for productive discussions. We also thank Manfred Kuhnkies and all of FORWISS, University Passau, for their motivating enthusiasm. (We recommend looking at some of the beautiful 3D-prints of algebraic surfaces produced by FORWISS and Voxeljet http://www.forwiss.uni-passau.de/ de/projectsingle/64/main.html/).

REFERENCES

[1] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singularities of differentiable maps. Vol. I, volume 82 of Monographs in Mathematics. Birkhaüser, Boston, 1985.

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[2] H. S. M. Coxeter. Regular polytopes. Methuen, London, 1948. [3] P. R. Cromwell. Polyhedra. Cambridge University Press, Cambridge, 1997. [4] B. Gru¨nbaum. An enduring error. Elemente der Mathematik, 64(3):89–101, 2009. [5] M. Hochster and J. A. Eagon. Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. American J. of Mathematics, 93(4):1020–1058, 1971. [6] H. Hong and J. Schicho. Algorithms for trigonometric curves (simplification, implicitization, parametrization). J. Symbolic Computation, 26:279–300, 1998. [7] J. Matousˇek. Lectures on discrete geometry. Springer, 2002. [8] Plato. Phaedrus. Project Gutenberg, http://www.gutenberg.org/ etext/1636, October 2008. Translated by B. Jowett. [9] T. Roman. Regula¨re und halbregula¨re Polyeder. Kleine Erga¨nzungsreihe zu den Hochschulbu¨chern fu¨r Mathematik; 21. Deutscher Verlag der Wissenschaften, 1968. [10] B. Sturmfels. Algorithms in Invariant Theory. Springer, second edition, 2008.

Mathematical Entertainments

Michael Kleber and Ravi Vakil, Editors

An Amazing Mathematical Card Trick ARTHUR T. BENJAMIN

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, surprising, Figure 1. A random-looking arrangement of cards.

or appealing that one has an urge to pass them on. Contributions are most welcome.

magician gives a member of the audience 20 cards to shuffle. After the cards are thoroughly mixed, the magician goes through the deck two cards at a time, sometimes putting the two cards face to face, sometimes back to back, and sometimes in the same direction. Before dealing each pair of cards into a pile, he asks random members of the audience if the pair should be flipped over or not. He goes through the pile again four cards at a time and before each group of four is dealt to a pile, the audience gets to decide whether each group should be flipped over or not. Then the cards are dealt into four rows of five cards. The audience can decide, for each row, whether it should be dealt from left to right or from right to left, producing an arrangement like the one shown in Figure 1. Finally, the magician ‘‘folds’’ the cards as directed by the audience. For instance, in Figure 1, if the audience requests that the right edge be folded, each card in the right column would be flipped over on top of the card on its immediate left. This would result in the cards looking like Figure 2.

A

â

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail: [email protected]

Figure 2. After folding the right row, the cards look like this.

The audience repeats the folding requests until eventually all the cards are in one pile. When the magician spreads the cards, there are 15 face-down cards and five face-up cards, and the five face-up cards form a royal flush! As we’ll explain, this trick practically works by itself, and the underlying mathematics is simple and elegant. To best explain its secret, we begin with a warm-up trick, which is even easier to perform and also gets a strong audience reaction. 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

37

are placed on the main diagonal. The trick is now guaranteed to work by itself. After all the folding is done, the four aces will be facing one direction and the other twelve cards will be facing the opposite direction. If the aces are face down, then the magician simply turns the stack of cards over before spreading them out.

Figure 3. The initial layout of cards (with the four aces hidden on the main diagonal).

Four Ace Surprise A magician deals 16 cards face down into four rows of four cards apiece, then turns four of the cards face up, as in Figure 3. Next, the magician asks the audience to ‘‘fold in’’ the edges of the cards, as in the previous trick. At the end of this procedure, after six folds, all 16 cards will be stacked into one pile. When the magician spreads the sixteen cards, there will be 12 face down cards, and four face up cards. The audience will be surprised to see that the face up cards are the four aces. The Secret The magician makes sure that the four aces begin on the main diagonal (top left corner to the bottom right corner). To achieve this, the magician could start with the four aces in the first, sixth, 11th, and 16th card in the deck, and then deal the four rows of cards from left to right. An easier (and more deceptive) approach is to stack the four aces at the top of the deck, then haphazardly deal cards into ‘‘random’’ positions in the four-by-four grid, taking care that the aces

AUTHOR

......................................................................... ARTHUR T. BENJAMIN earned his Ph.D.

from Johns Hopkins University in 1989, and has taught at Harvey Mudd College ever since. In 2000, he received the Haimo Award for Distinguished Teaching by the Mathematical Association of America. His books include ‘‘Proofs That Really Count,’’ ‘‘Biscuits of Number Theory,’’ and ‘‘Secrets of Mental Math.’’ He is also a magician who performs his mixture of math and magic to audiences all over the world. Harvey Mudd College Claremont, California USA

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Why This Trick Works This trick is a beautiful application of parity. Imagine the 16 cards are dealt onto a four-by-four checkerboard with black and white squares. In the initial configuration of Figure 3, the white squares are occupied by the four face-up cards and the four aces (face down on the main diagonal), and the black squares are occupied by the remaining eight facedown cards. After the cards are folded, all the cards will be stacked on a single ‘‘destination’’ square. Suppose the destination square is black. Now, since each ace begins face-down on a white square, every ace will be flipped an odd number of times on its way to the destination, and will therefore end face up. By similar reasoning, the four face-up cards and the other eight facedown cards will be face down at the destination. Likewise, if the destination square is white, then it will contain four face-down aces and all other cards will be face up.

Shuffled to Your Heart’s Content We are now ready to explain the trick described at the beginning of the article. It is based on an idea presented in the card magic book, Dear Mr. Fantasy, by John Bannon. In this trick, I use the Ten, Jack, Queen, King, and Ace of each suit from the deck, and ask my assistant to ‘‘Please shuffle these 20 cards to your heart’s content.’’ After the cards are returned to me, I spread the cards face up and deal the cards, two at a time, into a pile on the table. But as I deal the cards, I put some of these pairs face to face, some pairs back to back, and leave some pairs facing the same direction. Next, I deal the cards into four rows of five. The cards look like a random mix of face up and face down cards, like the one shown in Figure 1. We then go through the folding procedure that was done in the Four Ace Surprise. After all the cards have been folded into a single pile, I say, ‘‘Now it would be amazing if, after all this shuffling, flipping and folding, all the cards were now facing the same direction. It didn’t happen here, but it would have been amazing. The reason they aren’t all in the same direction is because you started this trick by shuffling these cards to your heart’s content. And you did quite a nice job of it, because look what we have here…’’ Next I spread the cards to reveal that there are precisely five face-up cards, namely 10 , J , Q , K , A (in some order) as I say, ‘‘…a royal flush in hearts!’’ The Secret After the cards are shuffled and I go through the cards two at a time, each pair has three possibilities: (i) Neither card is a heart; (ii) Both cards are hearts; (iii) Exactly one card is a heart.

Figure 4. The original shuffled 20 cards.

Figure 5. The new arrangement after the magician deals the cards, two at a time, into a pile.

In case (i), the cards are dealt face to face. In case (ii), the cards are dealt back to back. In case (iii), the cards are both dealt face up, but the heart card must go second. With case (iii), if the heart is the first of the face-up pair, then deal the two cards as a pair; if the heart is in the second position, then deal the two cards singly. Either way, the heart will be the second card dealt into the pile. The trick now works by itself, just like in the Four Ace Surprise. Example Suppose that after the assistant shuffles the cards, the cards appear in the order given in Figure 4. Then, after being dealt into a pile, the cards would be in the order given in Figure 5. After being dealt into four rows of five, the cards appear as in Figure 1. Why This Trick Works After the shuffled cards are dealt, two at a time, into the pile, let’s say that the cards are in positions 1 through 20, as

we go from top to bottom. For example, in Figure 5, the King of Spades is face down in position 1, the Ten of Spades is face up in position 2, the Ten of Diamonds is face down in position 3, and so on. As a result of our dealing rules, regardless of whether we see case (i), (ii) or (iii), the heart cards must either be face up in an odd position or face down in an even position, and the nonhearts have the opposite orientation (face down in an odd position or face up in an even position). Notice that this situation remains true even if a pair of cards is flipped over during the deal. Thus, to make the trick more interesting, as I deal each pair of cards, I will ask my assistant (or random members of the audience) if she wants the cards to be flipped or not. After dealing all the cards two at a time, I then go through the cards four at a time and ask the assistant, for each group of four cards, whether the cards should be flipped or not. It still remains true that the hearts (and only the hearts) will be face up in an odd position or face down in an even position. 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

39

Finally, as the magician deals the cards into four rows of five on an invisible checkerboard (with a white square in the upper left corner), each heart will either be face up on a white square or face down on a black square, and the nonhearts will have the opposite orientation. In fact, as long as you deal the cards onto the checkerboard so the colors alternate, your hearts and nonhearts will be in the right positions. (I let my assistant decide if she wants the cards dealt left to right, right to left, in zig-zag order or as a spiral. Note that the trick would not work if you dealt the cards from top to bottom.) As in the Four Ace Surprise, the folding procedure will automatically result in all the hearts facing one direction and the nonhearts facing the other direction. You can imagine variations of this trick using alphabet cards, number cards, or more complicated instructions. It’s a fun trick to perform for any audience, and I enjoy performing it (and sometimes explaining it) for my discrete mathematics class. Either way, this trick is surely one that your audience should flip over.

40


ACKNOWLEDGMENT

I am grateful to John Bannon for giving me permission to describe his trick in this article. The ideas behind this trick were developed and discussed by Henry Dudeney, Bob Hummer, Steve Freeman, Martin Gardner, John Bannon and Lennart Green, as described in Bannon’s book, Dear Mr. Fantasy, published in 2004. For more applications of these ideas, see the online article, ‘‘Many Fold Synergies,’’ http://www.maa.org/columns/colm/cardcolm200603.html by Colm Mulcahy. REFERENCE

Bannon, John, Dear Mr. Fantasy (2004), available at http://John BannonMagic.com Harvey Mudd College Claremont, CA USA e-mail: [email protected]

Years Ago

David E. Rowe, Editor

The Early History of the Brick Factory Problem LOWELL BEINEKE

AND

ROBIN WILSON

ew mathematical concepts can trace their origins to the labor camps of World War II and to the mathematical jottings of an abstract artist. In this article we trace the origins of the crossing number of a graph, the minimum number of crossings that arise when the graph is drawn in the plane, with particular reference to the war-time experiences of the Hungarian number-theorist Paul Turań and to the geometrical explorations of the British artist Anthony Hill.

F

Origins In July, 1944 the danger of deportation was real in Budapest and a reality outside Budapest. We worked near Budapest, in a brick factory. There were some kilns where the bricks were made and some open storage yards where the bricks were stored. All the kilns were connected by rail with all the storage yards. The bricks were carried on small wheeled trucks to the storage yards. All we had to do was to put the bricks on the trucks at the kilns, push the trucks to the storage yards, and unload them there. We had a reasonable piece rate for the trucks, and the work itself was not difficult; the trouble was only at the crossings. The trucks generally jumped the rails there, and the bricks fell out of them; in short, this caused a lot of trouble and loss of time which was rather precious to all of us (for reasons not to be discussed here). We were all sweating and cursing at such occasions, I too; but nolens-volens the idea occurred to me that this loss of time could have been minimized if the number of crossings of the rails had been minimized. But what is the minimum number of

â

Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik, Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail: [email protected]

Figure 1. Paul Turań in a war-time labor camp.

crossings? I realized after several days that the actual situation could have been improved, but the exact solution of the general problem with m kilns and n storage yards seemed to be very difficult… This description of the brick factory problem was written by Paul Turań for the first issue of the Journal of Graph Theory [25]. Earlier, a slightly different version had appeared in a letter to Richard Guy in February 1968 and was recounted in [12] (see below). Figure 1 shows Turań in the labor camp during the war-time years. Paul Turań’s letter to Richard Guy In 1944 our labor combattation had the extreme luck to work —thanks to some very rich comrades—in a brick factory near Budapest. Our work was to bring out bricks from the ovens where they were made and carry them on small vehicles which run on rails in some of several open stores which happened to be empty. Since one could never be sure which store will be available, each oven was connected by rail with each store. Since we had to settle a fixed amount of loaded cars daily it was our interest to finish it as soon as possible. After being loaded in the (rather warm) ovens the vehicles run smoothly with not much effort; the only trouble arose at the crossing of two rails. Here the cars jumped out, the bricks fell down; a lot of extra work and loss of time arose. Having this experience a number of times it occurred to me why on earth did they build the rail system so uneconomically; minimizing the number of crossings the production could be made much more economical. A related problem, the houses-and-utilities problem, is of unknown origin. It was described by the mathematical puzzler Henry Dudeney [4, 5] as ‘as old as the hills’, and as 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

41

Figure 2. A ‘solution’ of the utilities problem.

Figure 3. A utilities problem on a torus.

Figure 4. The complete bipartite graph K4,5.

Figure 5. Plane drawings of K1,5 and K2,5.

Figure 6. Two drawings of K3,3. 42


‘an extinct volcano [that] bursts into eruption in a surprising manner’ which is ‘much older than electric lighting, or even gas’. The puzzle concerns three neighbours A, B, and C who wish their homes to be connected to the utilities of water, gas and electricity (W, G, and E) in such a way that no connections cross. According to Sam Loyd [20], his father, the American puzzler Sam Loyd, described it as a ‘hoary old chestnut … which I first brought out in 1900’. A number of entertaining variations on the problem are described by David Kullman [18]. It turns out that no solution exists unless we flout the rules (as Dudeney does in [4] and in Problem 251 of [5]) and allow one of the connections to pass through a house, as shown in Figure 2. In 1961, the Scottish mathematician T. H. O’Beirne [22] described a utilities problem involving four utilities and four houses situated on a torus; this version does have a solution (see Figure 3). We can describe these problems in mathematical terms. The complete bipartite graph Km,n is obtained by joining each of a set of m vertices to each of a set of n other vertices (see Figure 4, which shows K4,5); for Turań’s brick factory problem, the m vertices correspond to the kilns and the n vertices to the storage yards. The problem asks how many crossings are required if the graph Km,n is drawn in the plane. It is easily seen that if m or n is 1 or 2, then Km,n can be drawn in the plane without any edges crossing; Figure 5 shows plane drawings of K1,5 and K2,5. However, no such drawing is possible if both m and n are 3 or more. In particular, the ‘utilities graph’ K3,3 has no plane drawing, although a drawing exists with just one edge-crossing (see Figure 6). In general, we define the crossing number cr(G) of a graph G to be the minimum number of edge-crossings needed when G is drawn in the plane (assuming only two edges appear at each crossing), and we see that cr(K1,5) = cr(K2,5) = 0 and cr(K3,3) = 1. Turań’s brick factory problem asks for the crossing number cr(Km,n), for any natural numbers m and n. The following table lists some values of cr(Km,n); for example, cr(K4,5) = 8. m=n

3

4

5

6

3

1

2

4

6

7 9

4

2

4

8

12

18

5

4

8

16

24

36

6

6

12

24

36

54

Some time after the end of the war, according to Guy [11], Turań communicated the brick factory problem to other mathematicians. In October 1952, during his first visit to Poland, he posed it in lectures in Wrocław and Warsaw. Solutions were proposed almost simultaneously by the probabilist Kazimierz Urbanik [26], who had attended the Wrocław lecture and who described his solution at a topological seminar there on November 12 1952, and by the topologist Kazimierz Zarankiewicz [28], who was present at the Warsaw lecture.

Zarankiewicz subsequently submitted his solution [29] to Fundamenta Mathematicae on December 15 1952.1 His statement of the brick factory problem follows; note that he makes explicit the restriction that no three edges may intersect at an internal point.

THEOREM I. If (a) in the Euclidean plane two sets of points, A and B, are given, A consisting of p points a1, a2, a3, …, ap, and B consisting of q points b1, b2, b3, …, bq (p and q are natural numbers); (b) for each pair of points ai, bj, where i = 1, 2, 3, …, p, j = 1, 2, 3, …, q, there exists a simple arc lying in the plane and having the points ai, bj as its end points; (c) the arcs lie in such a way that no three arcs have an interior point (i.e., a point that is not an end point) in common; (d) K(p, q) denotes the smallest number of intersection points of arcs; then the following formulas hold: K ð2k; 2nÞ ¼ k 2 k n2 n ;

ð1Þ

K ð2k; 2n þ 1Þ ¼ k 2 k n2 ;

ð2Þ

K ð2k þ 1; 2n þ 1Þ ¼ k 2 n2 :

ð3Þ

In an endnote, Zarankiewicz mentions Urbanik’s interest in the problem: As has been found by K. Urbanik and noticed by A. Reńyi and P. Turań, independently of one another, formulas (1), (2) and (3) can be written in the form of a single formula, K ðp; q Þ ¼ ðp 1 E ðp=2ÞÞE ðp=2Þðq 1 E ðq=2ÞÞE ðq=2Þ; where E(x) denotes the greatest integer B x. So Zarankiewicz’s claim is the following: Zarankiewicz’s conjecture: The minimum number of crossings in any drawing of the complete bipartite graph Kp,q is

Figure 7. The crossing number of K4,5.

The standard approach to crossing-number problems is to find a drawing with a certain number of crossings, and then to show that no drawing can have fewer. As Zarankiewicz observed, the above number of crossings can be attained by the following construction: Divide the p vertices into two sets of equal (or nearly equal) sizes and place the two sets equally spaced on the x-axis on either side of the origin. Do the same for the q vertices, placing them on the y-axis, and then join appropriate pairs of vertices by straight-line segments. Figure 7 illustrates the construction for K4,5, showing how it can be drawn with 8 crossings. Zarankiewicz’s proof of the case p = 3 was correct, but, as observed independently by the graph-theorists Paul Kainen (1965) and Gerhard Ringel (1966) (see Guy [12]), his inductive argument was deficient; it is a ‘one-legged induction’ that works easily when going from odd to even values of p or q, but not from even to odd. Thus, his formula yields only an upper bound for the minimum number of crossings. Since then, Daniel Kleitman [16], who learned of the problem from Richard Guy, has shown that the formula yields the correct minimum number of crossings of Kp,q when p or q is at most 6. Douglas Woodall [27] has extended these results to include the crossing numbers of K7,q and K8,q, for q = 7, 8, 9, and 10. The problem remains unsolved in general.

½p=2ðp ½p=2 1Þ ½q=2ðq ½q=2 1Þ; where [] is the ‘integer part’. Note that his formula can be rewritten more conveniently as ½p=2½ðp 1Þ=2 ½q=2½ðq 1Þ=2 or as ðp 1Þ2 =4 ðq 1Þ2 =4 ; for example, the minimum number of crossings in any drawing of K4,5 is ½4=2½3=2 ½5=2½4=2 ¼ 32 =4 42 =4 ¼ 8:

1

Developments Without any formal training in higher mathematics, the British artist Anthony Hill (Figure 8) conducted his own explorations into a wide range of geometrical and combinatorial objects (see box). In particular, unaware of the brick factory problem, he drew a number of points in the plane, joined them all in pairs by curves, and investigated how many times these curves must cross one another. In mathematical terms, the problem concerns the complete graph Kn, obtained by taking n vertices and joining each pair by an edge; Figure 9 shows drawings of K4, K5,

A contemporary paper, On a Problem of K. Zarankiewicz [17] by Toma´s Ko¨vari, Vera So´s, and Paul Turań, refers to a different problem.


43

Figure 9. The complete graphs K4, K5, and K6.

Finding the crossing numbers of the complete graphs has a more confused history than that of the complete bipartite graphs. For one thing, the complete graph problem seems to be a more natural place to start and various people may have considered it until its difficulty discouraged them from pursuing it. Those who heard Turań describe the brick factory problem may also have thought about this problem; certainly Paul Erd} os claimed in 1960 to have been looking at the problem for at least 20 years, but uncharacteristically seems to have told no one else about it. However, it does seem as though serious investigations into the complete graph problem originated with Anthony Hill around 1958, and that the earliest (albeit unpublished) records exist in the form of his notes and correspondence dating back to that time. After consulting some colleagues as to whether the problem was known, and following a great deal of experimentation, Hill found drawings of K6 with 3 crossings, K7 with 9 crossings, K8 with 18 crossings, and K9 with 36 crossings. Figure 10 shows some of his geometrical jottings. After a great deal of experimentation, Hill also produced a construction that can be described as follows:

Figure 8. Anthony Hill with geometrical objects.

Anthony Hill Anthony Hill (b. 1930) describes himself as a ‘constructivist working as a geometric formalist’ and has been described by the architect Yona Friedman (see [15, p. 84]) as follows: Anthony Hill is a ‘discoverer-artist’: Each work of his is an act of discovery, either of an abstract mathematical structure (which he succeeds to translate into an artwork) or of an aesthetic structure (which he transposes into graph theory).

Label the vertices 1, 2, …, n, and arrange the odd numbered ones equally around the inner of two concentric circles and the even ones around the outer circle. Then join all pairs of odd vertices inside the inner circle, join all pairs of even vertices outside the outer circle, and join even vertices to odd ones in the region between the circles.

In 1952 Hill attended lectures by the philosopher Imre Lakatos in London, and in 1958 embarked on a collaboration with fellow-artist John Ernest on the crossing-number problem. He was later awarded a Leverhulme Fellowship to research on symmetry as an Honorary Research Fellow in the Department of Mathematics at University College, London, and was an invited speaker at the International Conference on Combinatorial Mathematics at the New York Academy of Sciences in 1970. He has written a dozen papers in graph theory.

As he recalled: ‘‘Looking at diagonals of polygons—some inside and some outside—that’s how I came to it’’. Figure 11 illustrates his construction for K7. This construction led Hill to make the following conjecture in the late 1950s, probably for the first time:

and K6. It can be shown that if n B 4, then Kn can be drawn in the plane without any crossings, whereas K5 needs at least one crossing and K6 needs at least three; these drawings confirm that cr(K4) = 0, cr(K5) = 1 and cr(K6) = 3 (Figure 9). The following table lists the known values of cr(Kn) for n C 5 (the last two values were determined by Pan and Richter [23]): n

5

6

7

8

9

10

11

12

cr(Kn)

1

3

9

18

36

60

100

150

As Friedman [6] has remarked: ‘‘These crossovers are like rabbits . . . they have a tendency to multiply at a terrifying rate’’.

44


Hill’s conjecture: The minimum number of crossings in any drawing of the complete graph Kn is ð1=64Þ ðn 1Þ2 ðn 3Þ2 2

ð1=64Þnðn 2Þ ðn 4Þ

for n odd; for n even:

He also noted that the formula in the odd case n = 2r + 1 is the square of the triangular number 1/2 r(r – 1), while that in the even case n = 2r is the product of the consecutive triangular numbers 1/2 (r – 1)(r – 2) and 1/2 r(r – 1). Using a variation of Hill’s construction, with the vertices placed on the two ends of a tin can, J. Blazˇek and M. Koman [2] confirmed Hill’s conjectured results as upper bounds that can be combined into a single formula as crðKn Þ ð1=4Þ½n=2½ðn 1Þ=2½ðn 2Þ=2½ðn 3Þ=2;

Figure 10. Some jottings from Anthony Hill’s notebooks.

Figure 11. The crossing number of K7.

or equivalently, crðKn Þ ð1=4Þ ðn 1Þ2 =4 ðn 3Þ2 =4 ; these are now known to yield the correct values for all n B 12 (see [23]). It follows from these bounds that crðKn Þ ð1=64Þnðn 1Þðn 2Þðn 3Þ: On the other hand, as Guy [12] observed, each copy of K5 in Kn gives rise to at least one crossing, and such a crossing occurs in n – 4 copies of K5 (since the crossing involves only four vertices and there are n – 4 possibilities for the fifth vertex); thus, crðKn Þ ð1=120Þnðn 1Þðn 2Þðn 3Þ:

It follows that Hill’s construction yields the right order of magnitude for cr(Kn). By the spring of 1959, Hill and his friend the American artist John Ernest had arrived at the above formulas and approached some professional mathematicians about the problem. Among these was the geometer Bernard Scott of King’s College, London, who offered to try to learn whether the problem was known and whether it had been solved. Scott suggested contacting Andrew Booth at Birkbeck College, University of London; Booth, a computer pioneer in the 1940s, had been programming a computer to work on the classification of knots, and a student of his was sent to Hill to investigate the possibility of finding a suitable program for the complete graph problem. Hill also paid the first of a series of visits to the Dutch mathematician and philosopher L. E. J. Brouwer in April of that year. Brouwer was of the opinion that the crossingnumber problem might be like the four-colour problem and present great difficulties, in spite of its simple sounding nature. In May 1959, Hill communicated the problem to Professor Ambrose Rogers of University College, the geometer John Todd in Cambridge, and the combinatorialist Richard Rado at the University of Reading. At Rado’s suggestion, he wrote also to the French graph-theorist Claude Berge. Rado believed the problem to be difficult, but no one could shed any light on it. In November 1959, Richard Guy gave a seminar at University College on unsolved elementary problems, attended by Hill. Hearing about the complete graph problem, probably via Rogers, Guy wrote the first paper [11] on it in Nabla, the Bulletin of the Malayan Mathematical Society—a natural place for him to publish since for 10 years he had taught in Singapore (then part of Malaya). In the same paper, Guy also investigated the corresponding problem for drawings of complete graphs on a torus.


45

Figure 12. Anthony Hill explaining some graph drawings2.

In May 1961, H. P. Goodman, a student of Booth’s, wrote a letter to Nature [9] from the Department of Numerical Automation at Birkbeck College, describing their attempts to solve this seemingly intractable problem using a computer: This problem does not appear tractable analytically, so it was programmed for the University of London Mercury computer. The programme was written on the assumption that a minimum for n + 1 points can be obtained by adding an extra point in a suitable place on a minimum solution [with m crossings] for n points. However, the computations have proved that this apparently natural assumption is false: Two different minimum configurations for n = 7, m = 9 were taken, and one led to the true minimum n = 8, m = 18, while the other led to n = 8, m = 19. By December 1960, their computer had yielded the values of 60 for n = 10 and 100 for n = 11, results that have since been proved correct. After reading Goodman’s letter in Nature, two chemists, Joseph P. Manfreda and Martin B. Sheratte of North Haven, Connecticut, wrote to Hill offering a proof. They submitted their paper to Nature, whose editors sent it to Paul Erd} os and the graph-theorist Frank Harary. Hill was witness to a consultation between Erd} os and Harary in which, after a great deal of uncertainty, they arrived at the conclusion that Manfreda and Sheratte had been unsuccessful in obtaining a correct argument. Around this time, Hill communicated his results to Harary who (according to Hill) ‘took no interest in my algorithm, and so it has not been properly exposed’. Nevertheless, Harary and Hill produced a joint paper [13], summarizing the progress on the two crossing-number problems that had been made up to that point.

Variations Although the main problems in the area remain unsolved, there have been a number of other directions for research. We conclude by briefly summarizing three of these. 2

This photograph was taken by Je´rome Ducrot and appears in [15, p. 85].

46


Straight-Line Drawings In the 1930s and 1940s, Klaus Wagner and Istvań Fa´ry proved independently that every graph that can be drawn in the plane without crossings can be so drawn in such a way that all the edges are straight lines. Following from this, in 1958 Anthony Hill defined the straight-line crossing number cr(Kn) (later called the linear or rectilinear crossing number) to be the smallest possible number of crossings needed when the complete graph is drawn with straight lines in the plane. It is tempting to believe that the values of cr(Kn) and cr(Kn) are equal for all values of n, but surprisingly this is not the case. These values are indeed equal for n B 7 and for n = 9, but for n = 8 we have cr(K8) = 18 and cr(K8) = 19; a drawing of K8 with 18 crossings (due to Hill’s colleague John Henderson) and a straight-line drawing with cr(K8) = 19 crossings appear in Figure 13 (see Harary and Hill [13]). For n = 10, cr(K10) = 60 and cr(K10) = 62 (see Brodsky, Durocher, and Gethner [3]). Dan Bienstock and Nate Dean [1] proved some interesting results on the straight-line crossing numbers of graphs with low crossing numbers. Extending the result of Wagner and Fa´ry mentioned above, they showed that if cr(G) B 3, then cr(G) = cr(G). They also showed that, counter-intuitively, there are graphs with cr(G) = 4 but arbitrarily high straight-line crossing numbers. Products of Cycles In addition to the complete bipartite graphs and the complete graphs, much attention has been given to the products of cycles. If Cp and Cq are cycles, then their Cartesian product Cp 9 Cq is the result of taking q copies of Cp and joining corresponding vertices in a cyclic manner; the graph C4 9 C5 is shown in Figure 14. The original motivation for studying these graphs was that they can all be drawn on a torus without any crossings, but have arbitrarily large crossing numbers in the plane, as shown by Harary, Kainen, and Schwenk [14]. They noted that the general version of Figure 14 yields the inequality cr Cp Cq ðp 2Þq; for 3 p q; and conjectured that equality always holds. As with the complete graph and complete bipartite graph conjectures, a proof has been elusive. Attempts to prove it have led to quite a rich theory beyond this particular problem—see the survey of crossing numbers by Richter and Salazar [24]. Through the successive efforts of a number of mathematicians (see Myers [21] for a survey of early results), equality has been established for all q when p B 7; it has also been and Salazar [8] for all values of p and confirmed by Glebskii q when q C p(p + 1). Finding Crossing Numbers Efficiently As M. R. Garey and D. S. Johnson [7] observed in 1983, crossing number problems have practical applications—for example, for providing lower bounds on the amount of chip area required by a graph in a VSLI (very large scale integration) circuit layout (see also Leighton [19]). It is

Figure 13. The usual and straight-line crossing numbers of K8.

geometrical notebooks. We should also like to thank Richard Guy, Bruce Richter, Marjorie Senechal, and David Rowe for their helpful comments.

REFERENCES

[1] D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993), 333–348. [2] J. Blazˇek and M. Koman, A minimal problem concerning complete plane graphs, In: Theory of Graphs and Its Applications (ed. M. Fiedler), Czechoslovak Academy of Sciences (1964), 113–117.

Figure 14. The product C4 9 C5.

[3] Alex Brodsky, Stephane Durocher, and Ellen Gethner, The rectilinear crossing number of K10 is 62, Electron. J. Combin. 8, No. 1 (2001), Research Paper 23, 1–30.

therefore important to know whether the crossing number of a given graph can be found efficiently. It is well known that there are efficient linear-time algorithms for testing whether a given graph is planar—that is, for testing whether its crossing number is 0. Furthermore, Martin Grohe [10] showed in 2004 that, for any fixed value of k, there is a quadratic-time algorithm for determining whether a given graph has crossing number k. However, Garey and Johnson [7] have shown that determining the crossing numbers of graphs in general is an NP-complete problem, so that no polynomial-time algorithms are likely.

Aftermath As we have seen, there has recently been a great deal of progress in our knowledge and understanding of crossing numbers. In spite of this, the two basic challenges of proving Zarankiewicz’s conjecture and Hill’s conjecture remain. In spite of massive efforts by many people, these conjectures have withstood all attempts, remaining unproved for over 50 years. ACKNOWLEDGEMENTS

We wish to express our thanks to Vera So´s, Paul Turań’s widow, for supplying the photograph of him, and to Anthony Hill for many helpful conversations and access to his

[4] H. E. Dudeney, Perplexities, Strand Magazine 46, No. 271 (July 1913), 110; solution in Strand Magazine 46, No. 272 (August 1913), 221. [5] H. E. Dudeney, Amusements in Mathematics, Thomas Nelson and Sons (1917), Problem 251 and solution, 73, 200. [6] Yona Friedman, Towards a Scientific Architecture (transl. Cynthia Lang), Cambridge, MA: MIT Press (1975), 63. [7] M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Discrete Methods 4 (1983), 312–316. [8] L. Glebskeii and G. Salazar, The crossing number of Cm 9 Cn is (m – 2)n for all but finitely many n for each m, J. Graph Theory 47 (2004), 53–72. [9] H. P. Goodman, The complete n-point graph, Letter to Nature, 190, No. 4778 (27 May 1961), 840. [10] M. Grohe, Computing crossing numbers in quadratic time, J. Comput. System Sci. 68 (2004), 285–302. [11] Richard K. Guy, A combinatorial problem, Nabla (Bull. Malayan Math. Soc.) 7 (1960), 68–72. [12] Richard K. Guy, The decline and fall of Zarankiewicz’s theorem, In: Proof Techniques in Graph Theory (ed. F. Harary), New York: Academic Press (1969), 63–69. [13] Frank Harary and Anthony Hill, On the number of crossings in a complete graph, Proc. Edinb. Math. Soc. (II) 13 (1962–63), 333– 338.


47

[14] F. Harary, P. C. Kainen, and A. J. Schwenk, Toroidal graphs

[24] R. B. Richter and G. Salazar, Crossing numbers, Topics in

with arbitrarily high crossing numbers, Nanta Math. 6 (1973), 58–69.

Topological Graph Theory (eds. L. W. Beineke and R. J. Wilson), Cambridge University Press (2009), 133–150. [25] Paul Turań, A note of welcome, J. Graph Theory 1 (1977), 7–9. [26] K. Urbanik, Solution du proble`me pose´ par P. Turań, Colloq.

[15] Anthony Hill, Catalogue of the Retrospective Exhibition, Arts Council of Great Britain, Hayward Gallery, 1983. [16] D. J. Kleitman, The crossing number of K5,n, J. Combin. Theory 9 (1970), 315–323. [17] T. Ko¨vari, V. So´s, and P. Turań: On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50–57. [18] David E. Kullman, The utilities problem, Math. Magazine 52 (1979), 299–302. [19] F. T. Leighton, New lower bound techniques for VSLI, In: Pro-

Math. 3 (1955), 200–201. [27] D. R. Woodall, Cyclic-order graphs and Zarankiewicz’s crossingnumber conjecture, J. Graph Theory 17 (1993), 657–671. [28] K. Zarankiewicz, The solution of a certain problem on graphs of P. Turan, Bull. Acad. Polon. Sci. Cl. III. 1 (1953), 167–168. [29] K. Zarankiewicz, On a problem of P. Turan concerning graphs, Fund. Math. 41 (1954), 137–145.

ceedings of the 22nd Annual Symposium on Foundations of Computer Science, Washington, D. C.: IEEE Computer Society (1981), 1–12. [20] Sam Loyd, Jr., Sam Loyd and His Puzzles, New York, Barse (1928), 6, 87–88. [21] Nadine C. Myers, The crossing number of Cm 9 Cn: a reluctant induction, Math. Magazine 71 (1998), 350–359. [22] T. H. O’Beirne, Christmas puzzles and paradoxes, 51: For boys, men and heroes, New Scientist 12, No. 266 (21 December 1961), 751–753. [23] Shengjun Pan and R. Bruce Richter, The crossing number of K11 is 100, J. Graph Theory 56 (2007), 128–134.

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Department of Mathematical Sciences Indiana University — Purdue University Fort Wayne Fort Wayne, IN 46805 USA e-mail: [email protected] Department of Mathematics and Statistics The Open University, Walton Hall Milton Keynes MK7 6AA UK e-mail: [email protected]

The Basics of Bases AIDEN A. BRUEN

AND

TREVOR C. BRUEN

t is a rare undergraduate student in science and engineering who does not take a course in linear algebra. Such courses progress quickly to the heart of the matter, namely to the concepts of a spanning set and a basis of a given vector space V. Instructors will be at pains to emphasize that every spanning set contains at least one basis of V. Unfortunately it ends there. To us it seems high time to explore the following general question: How many bases are contained in a spanning set? This apparently fundamental question seems never to be raised. Intuitively one might expect that the more vectors there are in the spanning set the larger the number of bases contained in the spanning set. But can this be quantified? Let us be more explicit. Suppose G is an n 9 m matrix of rank n over some field F. Then, necessarily, m Cn. Here, we think of m as usually being much larger than n. In these terms, we are thinking of a large collection of n-component column vectors and studying choices of independent ntuples. Equality of row rank to column rank is often proven by reducing G to row-echelon form; this procedure finds a basis but obscures the issue of the number of bases. That is, it shows that some set of n columns of G is linearly independent; but we are asking how such sets there are. many m Clearly there can be at most of them, but when can n the bound be attained? The columns of any matrix G comprise a spanning set T of the ‘‘column space’’ of G; here we are assuming the column space is all of V = V (n, F). We will study the number of bases f (T ) that are contained in the spanning set T of V. So far we have only that m 1 f ðT Þ : n

I

Although the lower bound for f (T) has not received attention, the upper bound and the structure of T in the case of equality have been intensively studied over finite

fields for many years. The original motivation was statistical (see Bruen, et al. [2]). Subsequent to that the impetus was in connection with MDS codes, which MacWilliams and Sloane [9] call ‘‘one of the most fascinating chapters in all of coding theory’’. We will return to this application at the end of this article, but let us set the stage now. In general, the maximum length of MDS codes for a given field and dimension, and their structure in the optimal case, remain unknown. However, at least asymptotically, the Reed-Solomon codes (RS codes) are optimal in the family of MDS codes (see Bruen, et al. [2]). These RS codes are fundamental in technological applications ranging from computer drives to CD and DVD players to all manner of digital imaging, such as the amazing pictures transmitted by Voyager II (see Cipra [6]) . The decoding of RS codes relies on algorithms with a long history (see Barg [1]), beginning long before the invention of the codes by Reed and Solomon in the 1960s [10]. The first success was due to a French engineer in public service named Gaspard Clair François Marie baron Riche de Prony (1755–1839). After that the mathematical problem was considered in a short paper by Ramanujan. Subsequently the Berlekamp–Massey decoding algorithm (which was used by Voyager II) came into prominence; new decoding algorithms continue to be discovered. The generator matrix for an RS code (see below) is a matrix of size n 9 (q + 1) whose columns are of the form (1, t, t2, ..., tn-1), together with the column (0, 0, ..., 1). Here each t is an element of the finite field of order q over which we are working. Geometrically, the columns yield a normal rational curve in the underlying projective space of projective dimension n - 1, called a ‘‘higher-dimensional conic’’. The main points in the present article are the following. 1. A brand-new lower bound on f (T). We show that this lower bound is the best possible, over any field, and characterize the case of equality. These results were absolutely unexpected to us. 2. A discussion of the difference in behaviour between finite fields and infinite fields.

Received by the Editors 18 February 2009.


49

3. A new result concerning f (T) when f (T) is within 1 of m : The proof highlights the difficulty of getting n control of f (T). 4. A brief discussion of the importance of f (T) to algebraic coding theory.

The following section takes up the upper bound. In particular, using a little algebraic geometry we sketch an embedding result for finite fields. Two of the applications of the main result, relating to error-correcting codes, are briefly discussed in the final section.

A main theme is translation of vector-space questions into the language of projective geometry, which turns out to be more amenable and to give insight into the true difficulty of calculating f (T) in general. The point is that if T is a spanning set, then f (T) is unaltered if we multiply each vector in T independently by a non–zero scalar. In other words, our problem really concerns not vectors but 1-dimensional vector subspaces. Thus, although our method of proof is combinatorial it is informed by the esoteric art of projective geometry, which seems to cast the investigations into a much clearer light. Take even the humble case of a 3–dimensional vector space over the reals. It is not clear, a priori, how difficult it is to estimate the number of bases contained in a spanning set T. However, the true nature of the difficulty emerges when we reduce the question to determining the number of triangles in an arbitrary (spanning) set of points in the plane, which can be taken to be the Euclidean plane rather than the real projective plane. You will see that even in the simplest case, Theorem 1, naı¨ve reliance on mathematical induction does not suffice. There is little leeway in the proofs— there is danger at every turn! The proof of Theorem 1 provides the blueprint for the general case. This article is organized as follows. In the second section, Vector Spaces and Bases, we obtain the fundamental lower bound for the number of bases in a spanning set in a vector space. This uses the results of the preliminary section Geometric Results.

Geometric Results Let V = V (n + 1, F), n C 1, denote the vector space of dimension n + 1 over a field F. Associated with V is the projective space R ¼ PGðn; F Þ of (projective) dimension n, which is the lattice of non–zero subspaces of V. Thus the points of R are the 1–dimensional subspaces of V, the lines of R are the 2–dimensional subspaces of V, and so forth. Each non–zero vector v in V gives rise to a unique point P in R consisting of all non–zero scalar multiples of v: Then a basis (or spanning set) of V yields a basis (or spanning set) of R: Conversely, given a base or spanning set of R, then by choosing a non–zero representative vector for each point, we obtain a basis (or spanning set) in V. In the geometric context, a basis (or base) is often called a simplex. As a preview to the main result of this section we prove the special case when R ¼ PGð2; F Þ: In this case, a base is simply a triangle, that is, a set of 3 non–collinear points.

T HEOREM 1 Let S be a spanning set of t distinct points in R ¼ PGð2; F Þ: Then the number of bases of R contained in t1 S is at least : Moreover, equality occurs if and only 2 if S consists of t - 1 points on a line together with a point off that line. P ROOF . The case t = 3 being clear, we proceed by induction on t; so assume that t C 4: S is not a simplex.

AUTHORS

......................................................................... ..... ......................................................................... AIDEN A. BRUEN after university studies

TREVOR C. BRUEN was born, raised, and

in Dublin, moved to Toronto to do his doctoral work with F. A. Sherk, working also with H. S. M. Coxeter, E. Ellers, and A. Lehman. After some time at the Los Alamos National Laboratory, he settled in Calgary. He works on discrete geometries and their applications and on information theory. He has won many tennis tournaments, and has taken many hikes in the Rockies with his wife Katri, but these days he is especially a devotee of the Montreal Canadiens hockey team.

educated in London, Ontario, until going to Montreal to obtain his PhD. There followed two years on fellowship in Berkeley, California; then his recent appointment to the faculty at St. Francis Xavier University. He works in projective geometry and discrete mathematics. He enjoys music, literature, and travel; in his youth he was a highly ranked tennis player. This is his first joint article with his father.

Department of Electrical and Computer Engineering University of Calgary Calgary, Alberta T2G 5P9 Canada e-mail: [email protected] 50


Department of Mathematics University of California Berkeley, California 94720 USA e-mail: [email protected]

If no 3 points of S are collinear, then the number of t t1 bases in S is [ ; since t C 4. 3 2 Now let x C 3 denote the largest number of points of S on any line of R: Let l be any line containing exactly x points of S, and let P be any point of S on line l. Then the number of triangles of S containing P is at least (x - 1)(t - x), where |S| = t C 4. We have 3 B x B t - 1, the last inequality coming from the fact that S is a spanning set so that not all points of S may lie on a line. Examining the above quadratic, we see that it takes its least value at the endpoint x = t - 1, where it equals t - 2. Deleting P from S yields a set S1. S1 is also a spanning set, for x C 3 and there exists a point of S off l. Now |S1| = t - 1. By induction the number of triangles t2 contained in S1 is at least : Thus the total number of 2 t2 t1 triangles in S is at least þ ðt 2Þ ¼ : The 2 2 construction has shown that equality occurs iff S consists of t - 1 points on a line together with a point off that line. We will express our n-dimensional result in terms of the following definition.

D EFINITION 1 In R ¼ PGðn; F Þ; an extended simplex is a spanning set S, with |S| = t, such that some line l of R contains exactly t + 1 - n points of S. We note that, since t C n + 1, t + 1 - n C 2. The definition for the underlying vector space V = V (n + 1, F) is analogous. Our main result in this section is as follows.

T HEOREM 2 Let S be a spanning set of t distinct points in R ¼ PGðn; F Þ; n 2: Then the number of bases of R con tþ1n tained in S is at least : Equality occurs if and 2 only if S is an extended simplex in R: It happens that the formula is correct for n = 1 as well, but there it is obvious. Before proceeding to the proof of Theorem 2, we require some further results.

L EMMA1 Let t, n be positive integers with t C n + 2. Then

t nþ1

[

tþ1n : 2

P ROOF . The quotient of the two sides is tðt 1Þðt 2Þ. . .ðt n þ 2Þ : ð3Þð4Þ. . .ðnÞðn þ 1Þ Because t - 1 [ n, this is a product of quotients each of which is [ 1. This proves the Lemma.

L EMMA 2 Let S be a spanning set of t distinct points in R ¼ PGðn; F Þ with t C n + 2. Assume that the number of bases of tþ1n R contained in S is less than or equal to : 2

Then there exists a hyperplane H in R containing at least n + 1 points of S such that H \ S is a spanning set for H.

P ROOF . First, suppose if possible that every (n + 1)subset of S spans R ¼ PGðn; F Þ; and so forms a base of R: Then the number of bases of R contained in S is exactly t : But by Lemma 1, this would contradict our nþ1 hypothesis on the number of bases contained in S; so this is excluded. This entitles us to assume X is an (n + 1)-subset of S which does not span R: But S is a spanning set. Therefore, by enlarging X with points of S if necessary, we end up with a set X1 spanning a hyperplane, which we call H. Then indeed H \ S = X1 spans H, while |H \ S| C|X| = n + 1, as required. In what follows, a base (or spanning set) of a projective space C contained in a set S will be called an S–basis (or S–spanning set) of C:

P ROOF OF T HEOREM If t = n + 1 then the conclusion is true because S is then a basis. So assume t C n + 2. For n = 2 the result has been proved as Theorem 1. So we may assume that n C 3, and proceed by induction on both n and t. We may assume further that the number of tþ1n S–bases of R is at most : The object then is to 2 prove that S is an extended simplex. From Lemma 2, there exists a hyperplane H of R and a subset S1 of S lying in H, spanning H and containing exactly x points of S, where x C n + 1. We now claim that there exists a point P of S1 in H that belongs to more than (x + 1 - n) S1–bases of H. To see this, let y denote the average number of S1–bases of H containing a point of S1 in H. Denote by J the set of incidences of points of S1 with S1–bases of H. xþ2n By induction on n, there are at least bases 2 of H contains of H contained in S1. Since any S1–base xþ2n exactly n points of S1, we have j J j n: On 2 the other hand, | J| = xy. If there were no point P lying on more than x + 1 - n of the S1–bases of H, then we would have y B x + 1 - n so that xþ2n xðx þ 1 nÞ n: ðAÞ 2 However, a simple calculation reveals that the right-hand side is greater than the left-hand side of (A), using the fact that x C n + 1 and n C 3. This proves our claim that there is a point P in S1 lying on at least (x + 2 - n) S1–bases of H. Now, for each S1–base of H we get an S–base of R by adjoining any point of S not in H. Thus z, the number of S–bases of R containing P, satisfies Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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z ðx þ 2 nÞðt xÞ:

ðBÞ

Examining this quadratic in x, using the fact that x C n + 1 and that x B t - 1, we get z t þ 1 n:

ðCÞ

Set T ¼ SnP: We now consider two possibilities: (a) The set T spans R (b) The set T does not span R Assume (a). By induction on t, there exist at least tn T –bases and none of them contains P. In addi2 tion, from (C) above, there are at least (t + 1 - n) S–bases containing is at least P. So the total number of S–bases tn tþ1n : þ t n þ 1 which is greater than 2 2 This contradicts our initial assumption on the number of S–bases of R: So assume (b). Then the set T lies in a hyperplane R1 that onn, there are at least does not contain P.By induction ðt 1Þ ðn 1Þ þ 1 tnþ1 ¼ T –bases of R1 : 2 2 By adjoining P to each of these T–bases we obtain an S–base of R: By our initial assumption on the total number of S–bases of R we conclude that this accounts for all the S–bases of R: tnþ1 T– By induction on the dimension n, the 2 bases of R1 form an extended simplex in R1 : Adjoining P we see from the definition that S = T [ {P} is an extended simplex. Conversely, an extended simplex of R contains exactly tnþ1 S–bases of R; each base being obtained by 2 augmenting a pair of distinct points on the line l in Definition 1 to a basis of R (which needs all the points of R off l and hence is unique). This completes the proof of the Theorem.

Vector Spaces and Bases Remember that we promised to convert our estimated count of bases (simplices) in projective space to an estimated count of bases of a vector space. Let R ¼ PGðn; F Þ: Then each point P of R is identified with a 1–dimensional subspace U of the vector space V = V (n + 1, F). Conversely, each non–zero vector v in V corresponds to a unique point in R; namely the subspace of V spanned by v: Let T be a set of non–zero vectors in V that span V. Then, as above, T gives rise to a set S, say, of points in R that spans R: Assume now that no vector in T is a scalar multiple of another. Then there is a 1-1 correspondence between the vectors in S and the points in T. Moreover, a base of R in S corresponds uniquely to a basis of V contained in T. Theorem 2 translates to the following.

T HEOREM 3 Let T be a spanning set of t distinct non–zero vectors in the vector space V = V (n, F) of dimension n over the field F. Assume that no vector in T is a scalar multiple of another. Then 52


(a) If n= 2, the number of bases of V contained in T is t : 2 (b) If n C 3, then of bases of V contained in T the number tnþ2 is at least : Moreover, equality occurs iff T 2 is an extended simplex consisting of a set of t - n + 2 vectors in a 2–dimensional vector space, together with n - 2 residual vectors such that T spans V.

P ROOF . Part (a) is clear. Part (b) follows from Theorem 1. In projective space, the definitive answer to our quest was given in the last section, and it has now been transferred to the vector space. Yet we are not through with the minimal count, for we have not found the consequences of allowing vectors in our set which may be multiples of one another. For this, we need to extend the notion of simplex in a different direction.

D EFINITION 2 Let T be a set of t distinct vectors in V = V (n, F), n C2. Then T is said to be a multiple simplex if n o T ¼ a1 e1 ; a2 e1 ; . . .; atnþ1 e1 ; e2 ; e3 ; . . .; en ; where a1, a2, ..., at-n+1 are distinct non–zero elements in F, and fe 1 ; e 2 ; . . .; e n g is a basis of V. We say that a T–basis of V is a basis of V contained in T. The multiple simplex above contains exactly (t - n + 1) T–bases. Here is the 2-dimensional case of our main result, serving also as a lemma for the general case which follows.

T HEOREM 4 Let T be any set of t distinct non–zero vectors in V = (2, F), such that T spans V. Then the number of T-bases of V is at least t - 1. Equality holds if and only if T is a multiple simplex.

P ROOF . Let T consist of t1 multiples of the vector u1 t2 multiples of the vector u2 .. . tn multiples of the vector un where none of the vectors ui is a multiple of one of the others. Then t = t1 + t2 + ... + tn. We assume that t1 C t2 C ... tn C 1. The number of T–bases is then given by t1(t2 + ... + tn) + t2(t3 + ... + tn) + ... + tn-1tn. Thus the total number of T–bases of V is at least t1 ðt2 þ . . . þtn Þ ¼ t1 t t1 2 : If t1 = 1, then t2 = t3 = ... = tn = 1; that is, T is a set of t distinct vectors in V none of which is a scalar multiple of t another. Then the number of T–bases of V is at least ; 2 which is greater than or equal to t - 1, the desired estimate. Equality holds iff t = 2 and t1 = 1 = t - 1, which does indeed give a multiple simplex in V, since n = 2.

Otherwise t1 C 2. The total number of bases from the above is at least t1 t t12 : We compute that—because t1 C 2 and t C t1 + 1— this is at least t - 1, as desired. Equality holds if and only if t1 = t - 1, yielding a multiple simplex in V.

T HEOREM 5 Let T be any set of t non–zero vectors in V = V (n, F), n C 2, such that T spans V. Then the number of T-bases of V is at least t - n + 1. Equality holds if and only if T is a multiple simplex. P ROOF . We have t C n, so t - n + 1 [ 0. If t = n, then T is a simplex (which is a special case of a multiple simplex). So we may assume that t [ n. The preceding Theorem gives our result for the case n = 2, so we use induction on n, with n C 3. If no vector in T is a scalar multiple of another vector in T, then we also know the answer, from Theorem 2: the number of T–bases of V is at tnþ2 least : But because t [ n, this is greater than 2 t - n + 1, and we are done. It remains to allow vectors u; v in T with v ¼ ku: Then, we can construct an (n - 1)–dimensional subspace V1 and a subset T1 of V1 spanning V1 and containing vectors u; v; such that T1 consists of x of the vectors in T, say, with x Cn and x B t - 1. By induction, the number of T1–bases of V1 is at least x - (n - 1) + 1 = x - n + 2. Each of these bases gives rise to a T–basis of V by adjoining any vector of T not in T1. Thus the number of T–bases of V is at least (x - n + 2)(t - x). Examining this quadratic in x, bearing in mind that x C n and x B t - 1, we conclude that the number of T–bases of V is at least t - n + 1. If equality holds, then we must have t = x + 1, so there is just one vector in T that is not in T1, and the number of T1 bases of V1 must be exactly x - n + 2. By induction this forces T1 to be a multiple simplex, which in turn forces T itself to be a multiple simplex. Conversely, as pointed out earlier, any multiple simplex contains exactly t - n + 1 bases of V. Thus the theorem is proved. We have not used any special properties of the field F, and the results are best possible in any field. There is one exception: In Theorems 1 and 2 we took for granted that there could be a line containing t + 1 - n points. Over a field of order q, the total number of points on a line is q + 1. So we needed the assumption that q C t - n.

The Number of Bases over Various Fields Let us now give inequalities in the other direction. First we deal with infinite fields F; just to make life simple, let R ¼ PGð2; F Þ: Let S be a set of t points in R: What is the biggest possible number of triangles contained t in S ? We already know it can’t be more than : For an 3 example attaining this bound we may choose any t points on a non–degenerate conic. It is almost as easy to construct

t 1: 3 Namely, choose a set S1of t - 1 points on a conic; they t1 give us our first triangles. Next let l be the line 3 joining two points P =Q of S1. The number of points on l is infinite. Thus we can find a point R on l such that if A, B are two points of S1, both different from P and Q, then the line AB meets l in a point =R. Then the set S1 augmented by R provides the desired example, for R forms a triangle with any pair of points in the set except for (P, Q). This observation can be generalized and extended to higher dimensions (using normal rational curves instead of conics). The facts are much more complicated for finite fields. Let us recall that any finite field F has exactly q elements where q = pm is a prime power. Any two finite fields of order q are isomorphic, and the unique field of order q is denoted by GF (q) where GF denotes ‘‘Galois Field’’ in honor of Galois [7]. GF (q) is constructed using an irreducible polynomial of degree m over the prime field of order p [7]. For m = 2 the process is analogous to the construction of the complex numbers as an extension of the field of real numbers. So, let us suppose that F = GF (q), the Galois field of order q, with R ¼ PGð2; F Þ ¼ PGð2; qÞ: Let S be a set of t points in R such that every set of 3 points in S forms a triangle. Let q be odd. A celebrated result of B. Segre asserts that then t B q + 1, with equality if and only if S is the set of points on a non–degenerate conic [11]. If q is even then it is known that t B q + 2. An example is obtained by adjoining a point (‘‘the nucleus’’) to a non– degenerate conic. There are other known examples, due to Segre and other mathematicians [2, 11]. It is a major problem to find, for arbitrary projective dimension n, the largest t for which there exist spanning sets S of size t in R ¼ PGðn; qÞ such that every (n + 1)-subset of S is a base of R: The problem first arose in statistics and has remained unsolved, despite strenuous efforts, for over fifty years. We refer to [2] for further details. In R ¼ PG ð2; q Þ we can construct a spanning set S of size t such that the number of triangles in S is exactly t 1 by the procedure used for infinite fields, pro3 t3 vided \q 1; because each line of R contains 2 exactly q + 1 points. But when t is large, we will show that there is no such S. First we need a little algebraic geometry. Recall that each point in R is a triple (x, y, z) = (0, 0, 0). Alternatively we can regard a triple as determining a line. Such a triple— now written as [a, b, c] = [0, 0, 0] — denotes the set of all points (x, y, z) such that ax + by + cz = 0. The set of lines [a, b, c] for (a, b, c) satisfying a homogeneous equation of degree n, say, is said to form a ‘‘dual curve’’. Just as any line has at most n points in common with a curve of degree n (unless the whole line lies on the curve), so also each point P lies on at most n lines of a dual curve (unless every line on P lies in the dual curve). This is a special case of Be´zout’s Theorem in algebraic geometry. examples where the number of triangles is


53

We will now examine the situation when q is even (an analogous proof holds when q is odd); we use another result of B. Segre [11], which goes as follows.

T HEOREM 6 Let X be a set of k points in R ¼ PG ð2; qÞ; with q even. (i) Assume that k B q + 1 and that no 3 points of X are collinear. Then the lines of R meeting X in exactly one point —the tangents to X—form a dual algebraic curve E of degree q + 2 - k. (ii) Suppose that jXj ¼ k [ 12ðq þ 2Þ: Then E is unique and no secant of X (i.e., no line of R containing exactly 2 points of X) lies in E. This equips us to study a set S of t points spanning R t such that S contains exactly 1 triangles. This 3 assumption means that there exists exactly one triple P, Q, R of points of S which fail to form a triangle— that is, which are collinear. Let X = S - {P}. Then X is a set of t - 1 points with no 3 collinear. The number of tangents of X on P is t - 3. Also, from Theorem 6 the tangents to X form a dual curve E of degree q + 2 - (t - 1) = q + 3 - t. Moreover P lies on t - 3 of these tangents. Now suppose that jXj ¼ t 1 [ 12q þ 2, so that t - 3 [ q + 3 - t. As before, then, the number of tangents of X on P is t - 3. All these tangents lie in E, and E has degree q + 3 - t which is less than t - 3. It follows from Be´zout’s theorem as outlined above that each line on P —in particular, the line PQR— is in E. Now if t - 3 [ q + 3 - t, so that t [ q2 þ 3; it follows that t [ 12ðq þ 2). From Segre’s result above, no secant of X lies in E. In particular, the line PQR is not a line of E. This contradiction proves the following result.

T HEOREM 7 Assume that, in R ¼ PG ð2; qÞ; with q even, t [ 12q þ 3: Then there does not exist a set S of t points in R t containing exactly 1 triangles. 3 It turns out that this bound on t is sharp. That is, for t ¼ 12q þ 3; such a set S of t points does exist. The corresponding theorem for vector spaces is

T HEOREM 8 Let V be a vector space of dimension 3 over the finite field of order q, with q even. Let T be a spanning set of t vectors in V with t [ 12q þ 3: Then the number of t bases contained in T cannot be equal to 1: 3

Error–correcting Codes Let us recall the basic idea of linear error–correcting codes (see also [3] and [9]). They were originally used for encryption (see Barg [1]). Suppose m is a ‘‘message’’ consisting of a k–tuple of entries from some field, for example the binary field. If m1 54


is a message differing only slightly from m; there is a certain probability that, if m is transmitted, m1 will be received due to transmission errors. The gravity of the error is measured by the Hamming distance between the tuples, defined as the number of entries where they differ. To ameliorate the situation we ‘‘encode’’ each message m as mG; where G is a suitable ‘‘generator matrix’’ of size k 9 n. Then, depending on G; mG and m1 G will differ in several positions and the error will be more readily detected. From this the receiver can correctly decode a codeword to obtain the original message. In the formal theory, we always insist that the generator matrix be of full rank k. Each vector m; which is a row vector with k entries, is mapped to the codeword c ¼ mG; which is a row vector with n entries. Since G has rank k, this mapping is injective. The code C is the set of all such codewords c for all possible vectors m: Thus C consists of all possible linear combinations of the rows of G. Such a code C is called a linear (n, k) code. A fundamental idea of coding theory is that if n is large, then the Hamming distance between two codewords is much greater than the Hamming distance between the corresponding messages, and this affords better error–correction possibilities upon the receipt of the message. A fundamental parameter for C, denoted by d, is the minimum distance between any two codewords. Clearly, increasing d will allow better error–correction. The Singleton bound asserts that d Bn - k + 1 [9]. If equality holds, C is called an MDS (MaximumDistance Separable) n code. It will follow that each of the sets of distinct k columns of C forms a basis for the column space [9]. To design the MDS code to maximize d, for a given k, we want to maximize n. By and large (see [2]), it seems that the maximum can be attained only if C is a Reed–Solomon code. As in the beginning of the paper, think of the columns of G as a set T of n vectors in V = V(k, q). It is very important to determine those sets of k columns of G which form a basis, that is, a T-basis, of V. The reason is that such a submatrix is k 9 k non-singular. Thus once we know the entries of a codeword in those k positions, we can uniquely recover the codeword and thus the message, even if the remaining entries of the codeword are lost in transmission. Because of this, the entries in the k positions corresponding to a basis are called information sets for G or C. In practice we are mainly concerned with projective codes, that is, with linear codes whose generator matrix G is projective. This means that no column of G is a scalar multiple of another. Casting Theorem 3 above in this terminology we have the following.

T HEOREM 9 A linear (n, generator matrix k) code whose is projective has at least

nkþ2 2

information sets.

We mention briefly another application relating to the dual code C\ of C above. The code C\ is a linear (n, n - k) code and consists of all row vectors of length n over the field

F whose dot product with each and every row vector in G (and thus in C ) is zero. To indicate why the dual code is of interest, we first define the Hamming weight of a codeword u to be the number of non–zero entries in u: Then, associated with any code B is its weight enumerator polynomial f (B): the polynomial whose coefficients are the number of codewords of a given weight. From the MacWilliams identities (see MacWilliams and Sloane [9]) we can combine partial information on f (C), f (C\) to recover full information on f (C). There is a another basic application for C\. If, as sometimes happens, we have C ,C\, then we sharply limit the structure of C using methods of invariant theory due to A. M. Gleason and others (see Assmus and Mattson [8]). It was this approach that paved the way for the ‘‘computer proof of the century’’, the proof of the non–existence of a projective plane of order 10, which involved the work of several authors including the senior author of this article (see Lam, et al. [4]). The existence of a vector u in C\ of weight k, whose non–zero entries lie in a given k column positions, clearly implies that those positions do not give an information set, so that the k vectors formed from that set of k columns of G do not form a basis for the column space of G— that is, do not form a T–basis. Thus, Theorem 9 has implications also for the number of codewords in C\ of weight k and also their structure. Many of the results on projective planes can be generalized to other structures such as matroids. Another generalization relates to the ‘‘matrix tree’’ theorem concerning the number of spanning trees in a graph [5]. There are various other potential applications to block designs,

incidence structures, permutation groups, and (in addition to the linear codes treated here) non–linear codes.

REFERENCES

[1] Alexander Barg. At the dawn of the theory of codes. Mathematical Intelligencer 15(1993), no. 1, 20–28. [2] A. A. Bruen, J. A. Thas, and A. Blokhuis. On M. D. S. codes, arcs in PG(n, q) with q even and a solution of three fundamental problems of B. Segre. Inventiones Math. 92(1988), 441–459. [3] Aiden A. Bruen and Mario A. Forcinito. Cryptography, Information Theory and Error–Correction. Wiley, 2005. [4] L. Thiel, C. W. H. Lam, and S. Swiercz. The non-existence of finite projective planes of order 10. Canadian Journal of Mathematics 41(1989), 1117–1123. [5] David Carlson, David C. Lay, Charles R. Johnson, and A. Duane Porter, editors. Linear Algebra Gems. Mathematical Association of America, 2002. [6] Barry Cipra. The ubiquitous Reed–Solomon Codes. SIAM News 26(1), 1993. [7] I. N. Herstein. Topics in Algebra. Xerox College Publishing, Second edition, 1975. [8] E. F. Assmus Jr. and H. F. Mattson. Coding and combinatorics. SIAM Review 16(1974), 349–388. [9] F. J. MacWilliams and N. J. A. Sloane. The Theory of Error– Correcting Codes. North–Holland, Amsterdam, 1983. [10] I. S. Reed and G. Solomon. Polynomial codes over certain finite fields. Journal of SIAM 8(1960), 300–304. [11] B. Segre. Lectures on modern geometry. Roma: Cremonse, 1961.


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The Mathematical Tourist

Dirk Huylebrouck, Editor

In Lazare Carnot’s Footsteps ALAIN JUHEL 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.

till underestimated as a mathematician, Lazare Carnot is overshadowed by . . . himself, by the great figure he was in the history of France. Today, his works are being rediscovered: in projective geometry, he is considered a milestone between Desargues and Poncelet, who claimed Carnot’s spiritual fatherhood in the preface of his treatise. We can, at last, read his ‘‘Geóme´trie de Position’’ online, whereas it could only be found in a few libraries in the past. It is an opportunity for reminiscing in some places marked with his presence. Carnot, Which One? Almost every town and every village in France has a Carnot street. The first question is who is honoured, for at least three members of the family achieved national celebrity: Lazare (1753–1823), a mathematician and a key figure in the French Revolution; his son Sadi (1796–1832), an outstanding physicist who settled the fundamental principles in thermodynamics; his grandson, who

S

â

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

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THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

became president of France, and whose first name was also Sadi (1837–1894). A ‘‘Carnot’’ street without a reference to a first name is generally a tribute to Lazare, and a ‘‘Sadi Carnot’’ street to the (murdered) president. France has not forgotten her famous physicists in favour of her mathematicians, no; the truth is that very few streets have been named after scientists. Famous writers have received more attention, although the ones most cited had a major political influence, such as Hugo, Lamartine, or Zola. The same is true of Lazare Carnot, much honoured as a political personality of the French Revolution. But, for the Eiffel tower Lazare the mathematician was chosen by the engineer Gustave Eiffel as one of 72 prominent French scientists honoured by plaques around the first floor. The plaques disappeared when the tower was painted at the turn of the nineteenth century, but were reestablished in 1986; Carnot’s name is on the side facing the centre of Paris (Fig. 1). Even restricting ourselves to Lazare, the question ‘‘Carnot, which one?’’ remains: The officer in the Engineers Corps? The statesman? The mathematician? Lazare Carnot was all three, and he also was a scholar of classical literature all his life, a fine connoisseur of Greek and Roman classical authors. He wrote poems [5] and named his son for the Persian poet Saaˆdi, who lived in Shiraz in the XIIIth century. The statues of Lazare Carnot all pay tribute to the great figure in the French Revolution, but the Mathematical Tourist should not walk by them! As we shall see, small details show Carnot the scientist has not always been forgotten by the sculptors. Also, his different qualities interact in a very close way and explain the many breaks in his scientific career: he was forced to become a politician by circumstances, and he entered the ‘‘Committee of Public Safety’’ as an expert in military questions, where his celebrated efficiency in organizing the army may be regarded as mathematical rigour. Finally, a trip to Burgundy is a charming enterprise. Nolay. This small Burgundy village is situated off the main road near Beaune (20 km), in the famous vineyards. Lazare was born there on May 3, 1753. Surely the village’s description in today’s tourist guides differs from those of the past [1]: Nolay is the first stop of people from the Morvan, when they go down their granite and snowy mountains at the beginning of fall, their bagpipe players leading the way, as they come to work at the wine harvest in Coˆte d’Or. SaintMartin, patron saint of its church, is the protector of the wine growers and, despite his Episcopal dignity, of the drunkards; neither the first group nor the second could be absent on the saint’s day. The family house has changed a bit since the late 19th century (compare Fig. 2 to the drawing, made around 1900, shown on http://gallica.bnf.fr/ark:/12148/btv1b7742518q).

Figure 1. (a) The ring of scientists’ names on the Eiffel Tower; (b) Carnot’s place.

Figure 2. The front of the Carnot family house, with the statue by sculptor Jules Roulleau (1882).

The style of the statue is typical for its time: the aim was to sum up the functions and qualities of the glorified hero. (Compare it with François Rude’s well known sculpture dedicated to Monge in Beaune [15]). Roulleau’s statue shows Carnot holding a map displaying a scene of military operations, celebrating the strategist, and a smaller ‘‘winged victory’’ statue, his inspiration (Fig. 3). A closer look reveals a compass, a tool he holds together with the map, illustrating his role as staff-officer. It is also a geometer’s tool and a highly symbolic one. The whole establishes the balance between war and peace in Carnot’s life. The map has been laid down on a pile of books, as is easily seen from behind: these may be books he wrote, or some of his favourites, to remind us of his taste for Greek or Roman masterpieces. His firm and resolute face shows the strong determination and rectitude of the man—a quality that turned out to be responsible for many tribulations in his life. The dedication on the pedestal (Fig. 4) evokes Carnot’s most famous title: ‘‘Organizer of the Victory’’ (Organisateur de la Victoire). He received it, rather surprisingly, on the day he had to face an impeachment procedure! His son Hippolyte

tells the story, as told by a deputy of the National Convention. Carnot had entered the ‘‘Committee of Public Safety’’ (at that time, the government of France) on August 14, 1793, and he had proved himself indispensable in charge of the equipment and at the head of the armies and continued in his position after Robespierre’s downfall (July 27, 1793, or Thermidor 9, Year II in the French revolutionary calendar). However, on May 27, 1795, some deputies suggested that former members of the Committee should be tried to determine whether they had a role in the Great Terror [1]: It happened that someone brought an accusation against Carnot. When this name was mentioned, everyone in the Assembly was seized with so strong an emotion they were afraid to raise their voices; then there were a few moments of a deep, anxious silence. All of a sudden, a voice from the upper ranks in the centre shouted: ‘‘Could you dare to lay hands on the man who organized the victory of the French Armies?’’ The successful words: ‘‘He organized the victory’’ were repeated by everyone with a thrill of enthusiasm. Then they acclaimed: ‘‘Back to the agenda! Back to the agenda!’’ and the accusation was given up. From the Me´zie`res Engineers Military College to the Committee of Public Safety. The story of this dedication has led us too far in his career; a short flashback is necessary. Carnot drew d’Alembert’s attention when he prepared for the Me´zie`res Engineers Military College (Fig. 5). That was also the time he discovered his taste for mathematics through reading Bossut’s and Be´zout’s textbooks. His first attempt to pass the competitive examination ended in failure. The following year (1770), he took the entrance exam again and came out third. At Me´zie`res, he became Gaspard Monge’s pupil. Though only seven years older than Carnot, Monge was the leading man in the school and had a great influence among his students. Carnot’s first posting was Calais, in 1773, and then in several other northern towns in France, near the border, where the main fortified places were located—a defence system largely originated under King Louis XIV by Vauban, whose noteworthy eulogy was written by Carnot at the Academy in Dijon in 1784. One of the postings was in Arras, the birthplace of Robespierre, and the two men were 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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Figure 3. (a, b) Details of the statue.

Figure 5. The Me´zie`res school (archives of the Ećole Polytechnique).

Figure 4. The dedication to the Organizer of the Victory.

introduced to one another as members of the same literary circle. Another was Saint-Omer, where Carnot met Sophie Du Pont, who became his wife. It is therefore not surprising that Carnot was elected as a deputy of the department of Pas du Calais, where he lived, and not of the Coˆte d’Or, where he was born, as a member of the Legislative Assembly (in the constitutional monarchy, 58


1791), then of the National Convention (First Republic, 1792). Lazare sat in the education board, where he met Condorcet. Both were fond of mathematics and wanted to introduce a major reform, but they did not share the same approach: Condorcet’s was more theoretical, Carnot’s more pragmatic. The revolutionary storm, mainly the proclamation of ‘‘The Endangered Country’’ (‘‘La Patrie en Danger’’) turned him away from mathematics. In the summer of 1793, the Committee of Public Safety was expanded from 9 members to 12, and a trio of scientists, Carnot, Prieur de la Coˆte d’Or, and Lindet joined the committee that had been previously led by lawyers and barristers. Chemist Guyton de Morveau, the one and only scientist in the Committee up to this time, suggested that a ‘‘board of four citizens, learned in chemistry

and mechanics’’ should be created inside the government. Within a few days, the Carnot, Prieur, and Lindet trio was nicknamed ‘‘the Hard Working Three’’. As for Carnot himself, at that time he spent twelve hours a day at work, sometimes more: he told his son sixteen or eighteen hours were no exception. He was the first to arrive at the Committee in the morning, and he stayed there late into the night (‘‘He is a bloody rascal who keeps staying all night long at the Committee in order to open every letter or parcel’’, a Robespierrian opponent would say.) He had published only a few works at that time, and obviously the titanic amount of work he had to face left no time for doing mathematics. But massproduction of weapons required a scientific organization, and there Carnot’s exceptional qualities showed their best. At the time, Paris looked like a giant factory, and no passerby could miss the massive preparations for defence. But just in case, Carnot organized some shows in the Assembly when he presented his report on that subject on November 3, 1793. It was more than a century before Taylor and Ford carried out his principles. In Carnot’s mind, there was no difference between warfare and mathematics, at least, considered from the planner’s point of view: he tackled the problem of building new trigonometric tables for cadastral use in the same way. The need for it came from a fundamental principle of the Metric System: a quadrant must be divided into one hundred parts (named grads) instead of ninety. De Prony, who was responsible for the computations, told Arago: A member of the Convention summoned me to his office. Someone told me he was Carnot – I was not acquainted with him at the time – and he gave me very precise explanations of what my work would be. The job was to build the most accurate tables ever seen, the widest and biggest monument in the Computing Art. However skilfully I set it, I never could complete such an awesome task without the brilliant idea of applying the principle of ‘division of labour’, as if it were a production process. (De Prony, quoted in [1]). From the very beginning of the Committee (April, 1793), Guyton de Morveau had initiated a mobilization of scientists comparable to the leveé en masse of the troops; Carnot extended and amplified this work. He asked his colleagues to write down clear and useful textbooks, and they started work with enthusiasm: in 1793, Monge joined his forces with Vandermonde and chemist Berthollet to publish an Advice to the Metalworkers about Puddled Steel Production and then in 1794 wrote, singlehandedly, a Description of the Art of Producing Guns (Fig. 6). Carnot was very attentive to new technologies and their applications for national defence. He was the first secretary of defence planning, and he set up an . . . Air Force! He had been impressed by the brothers Montgolfier’s experience in 1784 and, ten years later, he launched the Ećole Nationale d’Ae´rostation. Observation balloons were used for the first time on the battlefield on June 2, 1794, near Charleroi, and they turned out to be decisive, a few weeks later, at the battle of Fleurus (June 26, 1794). Claude Chappe’s optical telegraph—a semaphore system—is another example of an invention that he quickly put to use: on September 1, 1794, Parisians were informed about the victory over the Austrians in Conde´-sur-l’Escaut less than an hour after the event.

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Figure 6. Monge: ‘‘Description de l’Art de Fabriquer des Canons’’ (‘‘Description of the art of producing cannons’’; archives of the Ećole Polytechnique).

The Convention received the news at the beginning of its session and had time enough to enact a change in the name of the town, from Conde´ to Nord-Libre, and to send it back before the evening, to the enemies’ great surprise. On the Northern Border. In March 1793, Carnot suggested the Convention should send members from its ranks as government representatives to the departments, whose mission was to account for the local situation and to write reports to the Assembly. Carnot set an example by going himself to the Northern departments he knew from his officer’s work. He not only stayed with the staff far away from the battlefield, he galvanized the soldiers with his tricolour plumed hat balanced on the point of the sword. On October 15 and 16, he engaged, together with general Jourdan, in the key battle to keep control of the town of Maubeuge: losing it would have meant an open road to Paris for the enemy. Later, Emperor Napoleon called this victory ‘‘the most extraordinary feat of arms in the Revolution’’, and asked all the officers around him: ‘‘Do you know, Gentlemen, who did that? Carnot.’’ In Maubeuge, a group of statues by sculptor León Fagel celebrates the military leader; Carnot’s central place is highly symbolic; Jourdan stands on his left. In 1893, for the centenary celebrations, his grandson himself came, as President of the Republic, and inaugurated the monument (Fig. 7). Reign of Terror. Despite its apparent cohesiveness, the Committee of Public Safety was far from being the homogeneous body it tried to appear. Heated discussions took place daily, but the members of the Committee had made the rule that, without exception, the final decision would be taken unanimously. Carnot may have suffered frequently and deeply from this iron discipline, but he fully agreed with this rule, as he thought there was no other way for an efficient government and no other way to prevent the country from collapsing. The Committee handled a range of problems, and each member was more or less specialized; as a result, a high level of confidence was needed. Carnot was a fair, upright man, but it was quite impossible to keep 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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Figure 7. (a) Statue in Maubeuge, general view; (b) Carnot, between Duquesnoy and Jourdan.

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control of everything he had to sign his name to: One could not sign except by trusting the people who studied the case, for one had to sign about five or six hundred papers a day. [2]. This might have led to great changes in the history of science, for one can find in a document in the archives in Auxerre (and in [13]) that the man under arrest was . . . Joseph Fourier! The order was signed on July 15, 1794, less than two weeks before the end of Robespierre’s dictatorship; therefore, it could have resulted in tragedy. It seems almost certain that Carnot had no acquaintance with Fourier at that time; Fourier’s saving grace was the date, rather than his innocence. However, many sources show that Carnot, despite his harshness and his solidarity with the Committee, listened to the voice of his conscience: Here is sitting among you, Gentlemen, a venerable member of this Academy, who is equally experienced in Pure and Applied Mathematics; […] he went through a long career without making an enemy of anybody or even deserving it! However, one day a few wretches asked for this wonderful head, in which a scientific monument was conceived, which was one of the best honoured in the Revolution’s honour. Our colleague became aware of the danger from an anonymous letter: the threatening storm upon him was away, it said, but it could as well constantly come back. This mysterious friend’s hand suggested a plan, a behaviour guide, some safety rules, the imperious need for a refuge. […] This anonymous writer, Gentlemen, was Carnot; the Geometer he wanted to preserve for the sake of science and our affection was M. de Prony. At that time, M. de Prony and Carnot were not acquainted. [9] (Fig. 8) M. Arago stopped for a short break in his lecture at the end of the tale; everyone’s eyes turned to that renowned old man (M. de Prony was eighty-two years old), and he got warm applause as a strong emotion moved his face. [1] An Honest Life. Carnot’s guiding line in politics must be regarded as a model of honesty. He had to face difficult choices and never evaded his duty. The price he had to pay was a double banishment, and a double striking off the Academy, but he turned these times of disfavour to a plain, productive scientific activity. Instead of feeling depressed

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Figure 8. Gaspard de Prony, 1755–1839 (in the library of the Ećole Polytechnique).

by these failures, he reacted by taking new directions in his life. On July 22, 1796, Carnot was elected to the Institute, succeeding the late Vandermonde, in the Mechanics Section. In this he was supported by Monge, De Prony, and Laplace, who paid tribute to his first work, the Essai sur les machines en geńe´ral (1778), which dealt with Physics and Engineering; it included deep thoughts about the nature of energy and laws of conservation and turned out to be one of Sadi’s leading influences. Incidentally, Sadi’s birth occurred a few days before, on June 1, 1796, in the Palais du Luxembourg; Carnot lived there at the time, as a member of the Directoire. (In 1932 a commemorative plaque was put on the wall, and for the ceremony physicist

Jean Perrin and mathematician E´mile Picard were invited speakers). But 1797 turned out to be a very bad year: on September 3, Carnot had to escape quickly from Luxembourg at night to avoid being sentenced to penal servitude in Cayenne, French Guiana. Thus began his first banishment, an adventurous, hazardous life: he had to run away from Geneva, dressed up as a laundry errand boy… On October 17, he was banned from the Institute, by order of the Executive Directory; very few members made any protest. It must be noted that such interference had never happened before in France. December 25 was a very special Christmas Day for general Bonaparte: he was given Carnot’s seat in the Institute as a present! It is noteworthy that Monge was the main architect of his election; one could have expected a greater respect for his former pupil and colleague at the time. But Monge was so fascinated by the winner of the Italian campaign that nothing could have stopped him: neither scruples lying heavily on his conscience, nor the surprisingly empty bag of the applicant’s scientific works. Doubtless Bonaparte had some taste for mathematics, but his only known opus was part of his spoils in Italy: the celebrated geometrical construction of the centre of a given circle was a work by Mascheronni; after being stolen by a clever tactician, it went down in history as Napoleon’s construction. On March 26, 1800, Carnot, back in France after the founding of Consulate, was elected to the Institute a second time. But this was not a redress to the previous unfairness: it only happened because a member in the Academy passed away… During the Napoleonic times, Carnot retired from politics. He had shown himself a firm, resolute but dramatically solitary opponent to Consulate for Life, then to Empire (see [3, 4]). He only offered his services when Napoleon was defeated in Leipzig (1813). Feeling some hope of change in the Emperor’s mind, he accepted the charge of Home Secretary during ‘‘The Hundred Days’’. In 1815 the French Monarchy was restored. On July 24, a list of 56 banned

people was published; Carnot was among them, for ‘‘regicide’’, an unforgivable ‘‘crime’’. He started a journey to Poland, passing near Wattignies; this was his last view of France before his long exile ending in Magdeburg, where he died on August 2, 1823. His last dwelling place is commemorated with a bust in the town. On March 21, 1816, Carnot was crossed off from the Institute a second time, by a royal ordinance. Mathematics, at Last! Science! He had a vocation, an enthusiasm for it; it was his personal fortress, his Eden. Though taken away from it a hundred times by his public duties, he came back to it as quickly as possible; then he devoted himself entirely to his research, resuming work effortlessly at the point he had dropped it. He showed a beautiful and original mind, and sometimes, seeking for some new road, he was fortunate enough to come ahead of those he was pleased to call his masters, and Lagrange above all. [1] And Lagrange wrote to him, speaking of the Me´taphysique du Calcul Infinite´simal: If I had heard about your work, I should not have tried to begin writing mine. Carnot’s mathematical works were published in a rather short period, during the breathing spaces in his political action: Me´taphysique du Calcul Infinite´simal (1797), De la Corre´lation des Figures en Geóme´trie (1801), Geóme´trie de Position (1803), Essai sur la theórie des transversales (1806) (Fig. 9). In these last two books one can find his own and famous generalization of Menelaus’ theorem and a proof, by means of his cross sections (transversales), of Brianchon’s theorem for a circle. Then he noticed that it could be extended to every conic section, considered as a central perspective of a circle: he was not very far from projective geometry! Finally, an undoubtedly essential part of his work setting up some colleges where the best students, selected only on their own abilities, would be trained at the highest level to become the leaders of the nation in research, teaching, or administration. The ‘Organizer of Victory’ had succeeded in choosing, according to their merit, the young generals that

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Figure 9. (a) Title page of the Geometry treatise; (b) some figures from the book. 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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were victorious on the battlefields and he planned to apply the same methods in selecting executives for civil purposes. Arago reminds us of that point: If I were spared a few moments more, I would have to list, among all the great institutions to which Carnot made a handsome contribution, the first Ećole Normale, the Ećole Polytechnique, the Museum d’Histoire Naturelle, the Conservatoire des Arts et Me´tiers; and among the works that he promoted, the Measure of the Earth, the settling of our new system of weights and measures, the great, unrivalled tables of the Cadastre. These are rather beautiful claims, Gentlemen, in a so-called destructive era. [9]. Launching the Ećole Polytechnique was, according to his friend Prieur, one of the very last matters he had to deal with; however, he worked as carefully on it as if it were his first days in the Committee: Many times, we had a chat about it. […] We felt compelled to set up a college where every kind of engineer should be trained; it was one of our favourite topics. But a torrent of home and foreign affairs swept us away; we lived under the tyranny of emergency. After Thermidor 9, we

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talked one more time about it. Carnot stayed in the Committee, I was out now; so he told me I could take advantage of my forced leisure to think it over and go deeper into this matter; I followed his advice. As soon as the project was fairly elaborated, we conferred with Monge, our former teacher in Me´zie`res, who embraced it with his usual exuberance and became, at once, the kingpin of the group in charge of the teaching schedule. I applied to it all the ardour I was capable of, and [Carnot] devoted his last days in the Government to taking steps to work out the founding decree for this new college. Moreover, after he left the Committee, he remained influential enough to watch over the destinies of an institution we felt so interested in. Prieur, quoted in [1]. The Ećole Polytechnique has retained a tender memory of his efforts, as we can read on the lower part of the frame of his portrait; and a bust of Carnot can be seen in the college, in its main hall (Fig. 10). But he did not restrict his project to the elite; he also suggested to the Emperor, during the Hundred Days, some ideas to set up a Primary Education that could satisfy at the

Figure 10. (a) Carnot’s bust; (b) Carnot’s portrait; (c) detail on the frame. 62


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Figure 11. (a) Pediment of the Pantheón; (b) close-up on Carnot’s rank.

same time the requirements of mass education and, from a qualitative point of view, at a rather good level; his project included a Committee of experts and an experimental school (ećole d’essai d’e´ducation primaire) to test the proposals of the Committee. Here are the first words of his letter (1815, April 27): Your Highness, We have got an example for the progress of Reason; it comes from a New-World country, that became civilized more recently, maybe however in a better way than most others in what we usually call the Old-World. When the Americans from the United States choose a place to settle a town, or even a hamlet, their first care is to bring there a teacher, together with some agricultural tools, because these sensible men, followers of Franklin and Washington, think that improving everyone’s mind is quite as pressing as reclaiming lands, tiling roofs on houses or clothing oneself, considering the first-rank needs of human beings. [2]. Such a long-term program was not reintroduced in France until 1880, and how many political changes did it have to go through before its rebirth—no fewer than two monarchies, one republic, a new empire, and, at last, a third republic! Paris, a Posthumous Homage. On August 4, 1889, for the centennial celebration of the French Revolution, Carnot’s ashes were carried to the Pantheon. One more time, the statesman was celebrated rather than the scholar; in the same ceremony were also transferred the remains of some Soldiers of Year II made famous through Victor Hugo’s verses. Nor did Andre´ Malraux see things otherwise when he made his famous speech in honour of the symbol of the French Resistance during World War II, Jean Moulin, inviting his ashes to enter the monument ‘‘beside Carnot together with the Soldiers of Year II ’’. However, it can be said that, in some other way, the scholar was in the Pantheon before the politician or the officer. When Louis-Philippe, ‘‘King of the French’’, restored the Pantheon as a secular monument (probably to assert his difference from the preceding kings, who made it into a church, as a token gesture for the nation), he asked sculptor David d’Angers, a notorious republican, to carve the

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pediment as we can see it today. With some help from Hippolyte Maindron, d’Angers worked from 1831 to 1837 to carve an allegory: the Country gives laurel wreathes to her heroes; the scholars stand on her right (which means, on an observer’s left) and soldiers on her left. Carnot stands in the second rank among scientists, before Berthollet and Laplace (Monge is the third in the first rank). On the opposite side, the first man is the young hero that he chose himself to be the commander of the Italian Army, General Bonaparte. The government tried to have La Fayette’s figure removed from the group, and, as the sculptor refused to obey, boycotted the unveiling; but it seems that no objection had been made about the presence of Carnot (Fig. 11).

REFERENCES

Carnot’s works, letters, and speeches: These references are available online, although probably without translation in English. Readers of French will be able to taste Carnot’s beautiful style. The author admits the translations in the present article are but pale reflections of the originals. [1] Me´moires sur Carnot, by his son Hippolyte: http://books.google. fr/books?id=_BP2SDG2AlYC&pg=PP11&dq=memoires+carnot &source=gbs_selected_pages&cad=0_1#PPP11,M1 [2] Me´moires historiques et militaires sur Carnot, introduced by P.-F. TISSOT: http://books.google.fr/books?id=gwOzGT6RIiQC&print sec=titlepage&dq=memoires+carnot&source=gbs_summary_ r&cad=0 [3] Against the Empire settlement (text on my Web page): http:// home.nordnet.fr/*ajuhel/Carnot/discours_empire.pdf [4] Against the Empire settlement (images, facsimile): http://gallica. bnf.fr/ark:/12148/bpt6k84022w [5] Poetical Works: http://gallica.bnf.fr/ark:/12148/bpt6k840237 [6] Mathematical Works: Essai sur les Machines en Geńe´ral, Re´flexions sur la Me´taphysique du Calcul Infinite´simal: http://gallica.bnf. fr/ark:/12148/bpt6k62609f [7] Geóme´trie de Position:

http://imgbase-scd-ulp.u-strasbg.fr/

displayimage.php?album=768&pos=0


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[8] Essai sur la theórie des transversales: http://gallica.bnf.fr/ark:/ 12148/bpt6k62584x

OTHERS

[13] J. Dhombres, J.-B. Robert, Fourier (Belin, 1998) [14] J.-V. Poncelet, Traite´ des Proprie´te´s Projectives (Bachelier, 1822), http://imgbase-scd-ulp.u-strasbg.fr/thumbnails.php?album=433

BIOGRAPHIES

[9] F. Arago, Lazare Carnot, notice biographique in Œuvres comple`tes,

[15] J. Brooke, The Monge statue in Beaune, France, The Mathematical Intelligencer 10(4) (1988), 44

tome 1, pp. 511–632: http://gallica.bnf.fr/ark:/12148/bpt6k92707n [10] J. Dhonbres, N. Dhombres, Carnot (Fayard, 1977) [11] C.C. Gillespie, Lazare Carnot, savant (1971, in English) Princeton University Press [12] on my Web page: http://home.nordnet.fr/*ajuhel/Carnot/Carnot. html

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Lyceé Faidherbe 9, rue Armand Carrel 59034 Lille France e-mail: [email protected]

Reviews

Osmo Pekonen, Editor

Mathematics in Ancient Iraq: A Social History by Eleanor Robson PRINCETON AND OXFORD: OXFORD UNIVERSITY PRESS, 2008, XXVII + 472 PP., BIBLIOGRAPHY, INDEXES OF TABLETS, AND SUBJECTS. US$49.50/£34.95, ISBN: 978-0-691-09182-2 REVIEWED BY JENS HØYRUP

ver since Moritz Cantor, typical general histories of mathematics have contained an initial presentation of ‘‘Babylonian mathematics’’. As a rule, the tenor of these presentations has followed the author’s overall view of what constitutes mathematics, and what constitutes history of mathematics. Accordingly, Morris Kline [1], strongly engaged in ancient Greek and post-Renaissance demonstration-based mathematics, dismissed anything pre-Greek as not really mathematics; Dirk Struik [2], highly aware of the connection between mathematics and its technological context, was much more balanced; Struik even started with a chapter on what would today be labelled ‘‘ethnomathematics’’ (he himself called it ‘‘Stone Age mathematics’’). Still, for the mathematical substance of the Babylonian tradition, Struik only skimmed the 1941 edition of Neugebauer’s Exact Sciences in Antiquity, and his general historical framework dissolved any specificity of the Babylonian world in a postulated ‘‘continuity and affinity of the Oriental civilizations,’’ regarding the distinction between ‘‘Egyptian, Babylonian, Chinese, Indian, and Arabian cultures’’ as ‘‘mechanical divisions’’ (p. xii). The majestic source collections of Neugebauer [3], Thureau-Dangin [4], and Neugebauer and Sachs [5] certainly give a much more differentiated picture of Babylonian mathematics (I do not include in this list E. M. Bruins’ and M. Rutten’s publication of the important mathematical texts from Susa [6], since the picture this publication offers is often terribly distorted by Bruins’ editorial commentary). However, none of them tries to present a history (neither ‘‘internal’’ nor in broader context)—as Neugebauer says explicitly [3: III, 79], ‘‘development of the consequences that can be drawn from the text material is not among the aims I have set myself ’’. In recent decades, much work has also been published about the mathematics of single periods or about specific aspects of Mesopotamian mathematics (now transcending the period where the concept ‘‘Babylonian’’ is adequate). All in all, the only genuine history of Babylonian/Mesopotamian mathematics was published by Kurt Vogel in 1959 [7] and aimed at the gymnasium level (possibly A. A. Vajman’s book from 1961 [8] should also be understood as history). Given both this target group (with all respect for the German ho¨here Schule

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

â Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyva¨skyla¨, Finland e-mail: [email protected]


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and for Vogel’s historical insight) and the immense progress in Assyriological knowledge and understanding attained since then, Vogel’s book is obviously outdated by now. This situation has now been happily changed by Eleanor Robson. Her book, though claiming to be a ‘‘social history’’, is indeed more than that—it is better characterized as a basic history of mathematics as embedded in its social context and function. Omitting discussion of the most sophisticated topics, she gives the reader a good account of almost everything mathematical going on in ancient Iraq between the outgoing fourth and the late first millennium (BCE, as all unqualified dates in what follow) in as far as it is documented in surviving sources—the most obvious exception being the techniques of mathematical astronomy, examination of which might indeed be an overwhelming task. In her delimitation of ‘‘mathematics’’, Robson comes closer to Dirk Struik than to Morris Kline; often, she broadens the subject to ‘‘numeracy’’ and presents, for instance, the contents of private household computations. This notwithstanding, a large part of her evidence consists of those texts that are conventionally considered as ‘‘properly mathematical’’: those connected to the school training of scribes. Robson’s Iraq is neither ‘‘Babylonia’’—which is only meaningful from the moment Babylon becomes an important polity in the early second millennium and would in any case exclude the Assyrian north—nor ‘‘Mesopotamia’’, a word borrowed from classical Greek (with a cognate in Biblical Hebrew) and unconnected to the geographical realities of our own days. Instead, she emphasizes that present-day Iraq corresponds roughly to a region that was as much of a cultural unity some 4000 years ago as was ‘‘Italy’’ around 1100 CE. I suspect (and sympathize with) an implicit argument against those who claim Iraq to be merely an artificial, post-WWI creation that is better cut into three statelets whose oil resources it would be easier to grab. The large majority of known ‘‘properly mathematical’’ texts are from the Old Babylonian period (2000–1600, according to the ‘‘middle chronology’’)—with a handful of exceptions, from its second half. Until the 1970s, the only other known mathematical texts were from the Seleucid era (third and second centuries), apart from a few tables of reciprocals tentatively ascribed to the Ur III period (21st century). As a rule, these texts had been bought by museums on the antiquity market, and neither place of origin nor precise date were known—which obviously contributed to making any writing of history impossible (just imagine how it would be to write the history of early modern mathematics from nothing but a pell-mell of undated and unlocated mathematical manuscripts ranging from Cardano to Abel, and from school to academy level!). Since then, a small number of mathematical texts from the late fourth, third, and mid-first millennia have been discovered; some terminological grouping of the Old Babylonian corpus has been achieved; and Eleanor Robson herself has found firm evidence that the tables of reciprocals in question are indeed of Ur III date. Still, most presentations of ‘‘Babylonian/Mesopotamian mathematics’’ concentrate on the periods from which many texts are known—these, at least, allow us to make a portrait of the mathematical culture of specific moments or periods (actually, only of the Old Babylonian period). 66


Robson has chosen a radically different way, forcing the presentation into a different scheme allotting more or less equal space to each approximate half-millennium. Occasionally, this compels her to treat a topic that might just as well or more naturally have been dealt with elsewhere under a half-millennium for which other material is scarce (e.g., ‘‘tabular accounting’’, reaching from the third into the first millennium, dealt with in the context of the later second millennium). On the whole, however, the system works astonishingly well; this is evidently only possible because Robson is not a historian of mathematics who has specialized in cuneiform mathematics but a fully trained all-round Assyriologist (first trained as a mathematician, however, which is also important). Robson offers a picture in which numeracy (and, when this word is appropriate, mathematics) is always bound up with the scribal function—the treatment of symmetry and general visual culture as aspects of geometry being partial exceptions. On the other hand, the use of numeration as a literary device in epics and royal inscriptions hardly counts as an exception to the rule, both literary genres being products of scribal culture. Until the mid-second millennium this compound is further linked to state administration and to the idea of mathematically determined social justice guaranteed by the king—even much later, the iconography of rulers exhibits the measuring rod and rope as royal insignia, but how long their role in the division of land was really remembered is unclear (not to speak of ‘‘just’’ distribution). Venerated symbols, as we know, often survive their original meaning—how many Christians remember today that ‘‘God’s lamb’’ does not stand for the supposed tenderness of the Saviour but for the butchered sacrificial animal whose blood buys off the wrath of the heavenly Father? Contrary to Robson, I actually doubt that Old Babylonian scribal calculators saw work on advanced ‘‘algebraic’’ problems dealing with areas as connected to their administration of royal ‘‘justice’’. No second-millennium source I know of (and none cited by Robson) contains any hint of such a view. From the Old Babylonian period through the earlier first millennium, documents also reflect the application of scribal mathematics in private merchant households. However, in the final phase, first in the Assyrian north, everyday administration, statal as well as private, was increasingly performed on ephemeral supports (wax tablets, papyrus, parchment), and probably in Aramaic; when its outcome needed duration, however, for instance in contracts, it might be transferred to clay and written in either Akkadian cuneiform or in alphabetic Aramaic, albeit without the calculations. (This omission was no new trend: intermediate calculations are almost exclusively known from the school genre—‘‘good scribes never showed their working’’, as Robson observes (p. 78) in the context of the 21st century, where wax tablets are first spoken of.) In the south, Robson can still point to a mid-first–millennium cuneiform curriculum encompassing metrological lists and tables of square numbers (and hardly much more mathematics) as well as tablets reflecting household numeracy (money, interest, metrology, land measurement). As the two (Assyrian and Babylonian) dialects of the Akkadian language died as vernaculars after the mid-first millennium, cuneiform culture was maintained only by a narrow environment of

scholar-scribes, identifying themselves as exorcists, copyists of omen series, incantation- and lamentation-priests, and the like. As Robson shows, they belonged to a small, interconnected circle of (real, namely blood, not apprenticeship-cumadoption) families. This environment also created mathematical astronomy; finally, it produced the small number of surviving Seleucid non-astronomical mathematical texts. An appendix of 46 pages lists all published mathematical texts. Robson’s work is amply richer in detail than can be rendered in a review—and history can only be written on the basis of documented details, hopefully to be fitted together into a meaningful pattern of general conclusions. Nothing comparable has been done before, and it has been a great pleasure to read the book, from which I have learned much. But although clay tablets survive the millennia better than most other media, readers should remain aware that only a small part of the evidence we would like to have has survived, that much less has been excavated—and that many of the tablets that have been excavated are still waiting to be read. As Assyriologists sometimes say, the best place to dig is in the Museum cellars. Any pattern that can be constructed on the basis of the evidence we do possess is therefore a reconstruction, an extrapolation building in part on what its author sees as reasonable assumptions. It is therefore no wonder that some of Robson’s general conclusions can be disputed, just as she herself objects in her book to conclusions drawn by previous workers—including some of mine. Detailed discussions of such doubts belong in the context of Assyriological or historical journals if not in private letter exchanges—just explaining to a nonspecialist audience what they are about would require pages and pages of background information. I shall therefore only air such doubts as are of general character, without going into details with my arguments. Robson concentrates on what can be documented in the cuneiform record itself. Although she does recognize the existence of nonliterate or at least not cuneiform-literate numerate activities in the area, she tends to not take them into account in her historiography, implicitly supposing that the development of the literate mathematical tradition took place inside a closed scribal environment. This is of course no different from the way the history of other mathematical periods is mostly written; but the approach might be unduly restrictive here no less than elsewhere. When discussing possible links to Greek mathematics, Robson restricts herself, on one hand, to sweeping arguments (of a kind which might prove that Descartes did not ultimately draw on medieval abbacus algebra, and which in any case collapse in front of the indubitable connection between Seleucid and Egyptian Demotic mathematics)—on the other, to rather unspecific references to renowned publications that take the pertinence of these for granted in a way that unwittingly supports the myth of the Greek genius that invented everything on its own without interaction with other cultures. The indubitable links to Islamic practical geometry and to Jaina geometry go wholly unmentioned; there is only a general exhortation (p. 288) to ‘‘explore

[the place of cuneiform mathematics] within the sciences in the Middle East, and Asia more generally’’. This almost sounds like Struik’s ‘‘affinity of the Oriental Civilizations’’.

Post Scriptum I would like to add a personal historical note for the record. On page 7, Robson suggests that ‘‘it is perhaps no coincidence that ‘Algebra and naı¨ve geometry’, Jens Høyrup’s seminal work on the language of Old Babylonian algebra’’ was published in 1990, the year in which Neugebauer and Bruins died. Coincidence in fact it is. The manuscript was submitted in 1986 to one of the pertinent editors of Archive for the History of Exact Sciences (not van der Waerden, which in retrospect I regret); after a full year this editor refused it, accompanying the refusal with a question showing that he had not read up to line 7 on the first page. The manuscript was then invited in 1988 by Altorientalische Forschungen, but typographical composition was difficult and took a long time. Preliminary presentations of my results had already appeared in Erdem (1986) and Mathematische Semesterberichte (1989). I had also sent a first extensive but very preliminary university print to Neugebauer in 1984 and had received a kind but noncommittal postcard together with some offprints on the calendar topic on which he was working at that moment. I am happy to be able to say that I did not celebrate the departure of a giant by kicking his teeth while standing on his shoulders (that is certainly not what Robson wants to insinuate, I should say). Nor had I been so scared by Bruins that I dared not criticize him in writing while he was alive and spitting flames. REFERENCES

[1] Morris Kline, Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. [2] Dirk J. Struik, A Concise History of Mathematics. New York: Dover, 1948. [3] O. Neugebauer, Mathematische Keilschrift-Texte. I-III. (Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik. Abteilung A: Quellen. 3. Band, erster-dritter Teil). Berlin: Julius Springer, 1935, 1935, 1937. [4] F. Thureau-Dangin, Textes mathe´matiques babyloniens. (Ex Oriente Lux, Deel 1). Leiden: Brill, 1938. [5] O. Neugebauer, A. Sachs, Mathematical Cuneiform Texts. (American Oriental Series, vol. 29). New Haven, Connecticut: American Oriental Society, 1945. [6] E. M. Bruins, M. Rutten, Textes mathe´matiques de Suse. (Me´moires de la Mission Archeólogique en Iran, XXXIV). Paris: Paul Geuthner, 1961. [7] Kurt Vogel, Vorgriechische Mathematik. II. Die Mathematik der Babylonier. (Mathematische Studienhefte, 2). Hannover: Hermann Schroedel / Paderborn: Ferdinand Scho¨ningh, 1959. [8] A. A. Vajman, Sˇumero-vavilonskaja matematika. III-I Tysjacˇeletija do n. e. Moscow: Izdatel’stvo Vostocˇnoj Literatury, 1961. Section for Philosopy and Science Studies, Roskilde University P6, Box 260, DK-4000, Roskilde, Denmark e-mail: [email protected]


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Mathematics in India by Kim Plofker PRINCETON, NEW JERSEY, OXFORDSHIRE, UNITED KINGDOM: PRINCETON UNIVERSITY PRESS, 2009, 357 PP., US$39.50, £27.95, ISBN: 978-0-69112067-6. REVIEWED BY HOMER S. WHITE

ntil fairly recently, a mathematician seeking an appreciation of classical Indian mathematics without having first to master Sanskrit and other text-critical apparatus would have been well advised to begin with the surveys by B. Datta and A.N. Singh [2] and by T.A. Sarasvati Amma [1]. The former emphasizes algebra, and the latter geometry, but both works focus on results found in the socalled ‘‘root texts’’ of the tradition, arranged topically rather than chronologically and made over into modern concepts and notation so as to convey rapidly a sense of the impressive scope and achievements of Indian mathematics. In the last few years, it has also become possible for the nonspecialist to deepen his or her initial impression by means of direct encounters with commentaries on the root texts. Commentaries preserved and advanced the practice of mathematics in a number ways: by clarifying the results themselves, indicating the relationships between them, providing examples of their use, and—particularly during the later classical period—offering proofs for them. Especially valuable recent contributions include Agathe Keller’s translation [6] of the earliest known commentary on the mathematical chapter of the Aryabhat : iya, which is arguably the single most influential root text in Indian mathematics, and the long-awaited translation by K.V. Sarma of the Gan: itayuktibh a:sa [11]. Although this text does not, strictly speaking, fall in the commentarial genre, it is devoted to the of proofs (yuktis) of results in : a) exposition (in Sanskrit, bhas the mathematics (gan: ita, ‘‘that which is reckoned’’) of the important Kerala School. A third recently published sourcebased piece, drawing on both root texts and commentaries— and especially useful for those who are preparing undergraduate courses in the history of mathematics—is Kim Plofker’s Indian contribution to Victor Katz’ sourcebook on non-Western mathematics [5]. Mathematics in India supplements the earlier works without superseding them. However, it should be the recommended place to begin, not only for its treatment of roottext results combined with extracts from commentaries, but also because the author builds on previous general histories in three areas: context, text, and transmission. Context. Mathematics in India is arranged chronologically, from the viewpoint of what Plofker calls the ‘‘mainstream narrative’’, that is, ‘‘the narrative currently accepted by most mainstream historians as consistent with the textual record, linguistic and archaeological evidence, and the history of other mathematical traditions’’ [Preface, p. vii]. A chronological approach encourages consideration of the

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historical and social context of the discipline, as opposed to simply mining the texts for their ahistorical mathematical content. The Introduction contains an historical overview of South Asia that emphasizes the region’s connection with the rest of Eurasia, and a brief introduction to Sanskrit language and literature that helps the reader to understand the typical structure of Sanskrit texts. (Root texts were a sequence of either verses or sutras—highly compact aphorisms. Commentarial texts were usually in prose, although there do exist mathematical commentaries that attempt to present highly complex proofs in verse form.) Examples of context-setting in later chapters include: a discussion of what can be inferred, from the relatively sparse mathematical references in Vedic texts, about the state of calendrics and astronomy in the Vedic period; a treatment of the late-Vedic Súlbasutras (‘‘Rules of the Cord’’) that covers not only the rules that indicate knowledge of what we now call the Pythagorean theorem, but also their use in the construction of some actual altar arrangements, such as the three fires—Householder’s fire, Oblation fire, and Southern fire—common to most ceremonies, and the Great Altar used in soma rituals; a section on ‘‘Mathematicians and Society’’ that summarizes what little is known about how mathematicians earned their living and passed on their knowledge, the role of the mathematician in Indian society, and the place of the discipline within the ‘‘universe of Sanskrit learning.’’ Overall, Plofker manages to strike a fine balance in situating Indian mathematics within Indian culture generally. On the one hand, Indian mathematicians were for the most part educated Brahmins who—like the scholars in any other classical culture—possessed a solid grounding in the various types of knowledge—grammatical, philosophical, religious, and literary—that were considered prerequisites of a scholarly life. Thus, for instance, to interpret the concrete-number system (see the following), or to understand the finer points of Indian debates on the role, in the practice of astronomy, of textual authority versus empirical observation [7], or to follow discussions of whether the principle of proportional reasoning is an instance of syllogistic inference ([8] p. 54 and [10] pp. 204–209), it is helpful for the contemporary reader to be familiar with, respectively: the elements of Indian mythologies, the text-critical Mima¯msa, and the logic-oriented Nya¯ya schools of philosophy. Plofker’s appendix on Sanskrit language, verse, and literature provide some initial awareness of the extramathematical context of Indian mathematics. On the other hand, in classical India—again, as in other cultures—mathematics was more likely than other disciplines to develop rather independently of prevailing social, political, and intellectual trends. Thus, just as in the study of eighteenth-century developments in European mathematics, a knowledge of Newtonian mechanics is more relevant than an equally good command of, say, the French Revolution, so in the study of Indian mathematics during the first millennium CE—when astronomy drove mathematical innovation to such an extent that mathematics, astronomy, and astrology were typically regarded as a unified discipline astra, ‘‘star science’’)—it is vmore important to know (jyotih: s´ basic astronomy than to have expertise in the other nonmathematical disciplines. Accordingly Plofker devotes an entire chapter to the basic concepts of geocentric astronomy

that underlie the four major schools (paks: as) of Indian astronomy. Text. Plofker furnishes precis of the major texts of the tradition, roughly in chronological order (the Kerala School is treated in a separate chapter). She generally avoids repeating results covered in earlier texts, unless the author takes a noteworthy new approach. More than in previous histories, one is made aware of the variety of mathematical subgenres. Whereas many of the widely known texts, including those of A¯ryabhat: a and Brahmagupta, are mathematical sections of siddh antas (comprehensive works on mathematical astronomy), or— like Bha¯skara’s famous Lil avati and Bijagan: ita—are commonly treated as such, there also exist independent mathematical works. Among the lesser known of these works included by the author is the Bakhsha¯li Manuscript (700–1200 CE). This is the earliest surviving Sanskrit manuscript of mathematics, important as a source of information on how Sanskrit mathematics was actually written. Another text that deserves to be better known to nonspecialist readers is the later Gan: itakaumudi (‘‘Moonlight on Mathematics’’) a general treatise that is distinguished by a comprehensive discussion of permutations and combinations and by a section on various types of magic squares. A little-known but very interesting subgenre of Indian mathematics is the karan: a, a concise astronomical manual that provides shorthand formulas for approximating positions of celestial bodies. The rationales for these approximations are not given: according to one commentator quoted by Plofker (p. 108), the formulas of a karan: a are supposed to be ‘‘not apparent to others [but] easily used by the stupid.’’ Reconstruction of rationales can present a challenge, as Plofker, whose early research dealt with karan: as, illustrates in an ingenious discussion of a rule, appearing in Bha¯skara’s Karan: akutuhala, for approximating a terrestrial latitude from an equinoctial shadow and equinoctial hypotenuse. Herself a trained paleographer, Plofker apprises the reader (see Appendix A.3) of the characteristics of South Asian mathematical manuscripts themselves, especially as they are relevant to understanding published texts. In India, the abundance of paper, in the form of palm leaves and birch bark, contributed to the voluminous production of manuscripts: of the estimated thirty million extant manuscripts in Sanskrit, at least one million are believed to deal with jyoastra. On the other hand, paper deteriorates quickly in tih: s´ India’s warm climate, so Indian mathematical manuscripts are liable to be removed from their autograph ancestors by many more copies than would a European manuscript of, say, a work of Archimedes. Indian manuscripts also include few or no diagrams. Indeed, in some works (Nilakan::tha in [8], for instance), the author restricts himself entirely to verbal descriptions of even quite complicated geometrical arrangements. Perhaps because of the emphasis on the oral communication of knowledge in Indian culture, mathematicians devised various schemes by which to represent numbers in multiple or concise ways. Plofker introduces the reader to amkhya the charming tradition of bhutas (‘‘concrete numbers’’) by which a number may be represented by a kind of thing—any kind of thing—generally regarded as existing in

that number. The ability to reference a number in many different ways facilitated the communication of mathematics in verse form. Consider, for example (pp. 221–222), a verse attributed to Ma¯dhava of the Kerala School: Gods, eyes, elephants, serpents, fires, three, qualities, Vedas, naks: atras, elephants, arms: the wise have said that this is the measure of the circumference when the diameter of a circle is nine nikharvas. There are thirty-three gods in the standard Indian pantheon; humans have two eyes and two arms; there are eight kinds of serpents and eight kinds of elephants; there are four collections of Vedic scriptures; the three ritual fires were previously mentioned; the S amkhya school of philosophy holds that the manifest world is made up of the three gun: as (‘‘qualities’’) in various combinations; and, analogously to the constellations of the Zodiac, there are twenty-seven star groupings (naks: atras) lying roughly along the ecliptic that serve to locate celestial bodies. Numbers are reported in order of increasing place-value, and a nikharva is 1011, so the verse amounts to the claim that p

2; 827; 433; 388; 233 ; 900; 000; 000; 000

an approximation that is good to eleven decimal places. (The irrationality of p is explicitly asserted, though not proved, by some Indian mathematicians. See pp. 41–42 of [8], translated in Footnote 1 to p. 155 of [1].) Plofker discusses other number-representation systems that are equally intriguing, though less known among nonspecialists. In the kat: apayadi system, for example, each consonant in the Sanskrit alphabet is assigned a numerical value. Thus, in kat: apayadi notation, a six-verse poem (p. 246) by the same Ma¯dhava: The name of the most excellent one is best. The creator of knowledge is the Himalaya mountain. The knower of the Vedic hymns is the burning sun ... encodes a table of quite accurate Sine values! Transmission. The hallmark, arguably, of the History of Mathematics Department at Brown University, where Plofker took her Ph.D. under the direction of the late David Pingree, was research into the transmission of mathematical knowledge across cultures in premodern times. Accordingly, transmission is a concern for Plofker, whose linguistic training includes Greek, Latin, Persian, and Arabic, as well as Sanskrit. In Chapter 2 she assesses the evidence, based on examination of numerous extant Sumerian and Akkadian cuneiform clay tablets, for the much-disputed hypothesis that ‘‘at least some features of Indian mathematical astronomy at the close of the Vedic period’’ derive from Mesopotamia. In Chapters 3 and 4 she considers the extent to which Indian astronomy, including the concept of epicycles, the use of the Zodiac, and the casting of horoscopes, was influenced by contact with the Greeks early in the first millenium CE. (Interestingly, though, the Indians appeared to have relied on their own mathematics to perform computations in the new astronomy, rather than adopting Greek geometry and trigonometry.) Chapter 8 is entirely devoted to exchanges with the Islamic world, including late classical collaborations between roughly 1400 to 1750 CE, such as the introduction of the Islamic astrolabe into India and the Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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appearance of Sanskrit mathematical texts in Persian. Nevertheless the investigation of such interchange is still in its infancy, since, as Plofker notes (p. 271), ‘‘most of the known texts reflecting [these interchanges] are still unpublished and little studied, while there probably exist many others as yet unknown.’’ On the other hand, in the case of the celebrated Kerala School in Southern India (roughly 1400 to 1600 CE), in which power series representations of trigonometric functions were derived by techniques that vividly call to mind the infinitesimal calculus, and where much effort was expended in devising proofs for even quite elementary mathematical results, there appears to be, at present, no evidence for either an Islamic influence on Keralese mathematics or for the suggestion (see, e.g., [4] and [9]) that sixteenth-century Jesuit missionaries transmitted Keralese trigonometrical methods to the early developers of calculus in Europe. In the area of transmission in particular, we see that Plofker’s commitment to the ‘‘mainstream narrative’’ is connected to an effort to move beyond two unfortunate trends in the study of classical Indian mathematics. In Plofker’s own words (p. 2): ... the historiography of science in India has long been coopted for political purposes. Most notoriously, some nineteenth-century colonial officials disparaged local intellectual traditions, which they termed ‘‘native learning,’’ in order to justify Westernized education for future colonial servants. Many nationalists responded in kind by promoting various separatist or Hindu nationalist historiographies, often including extravagant claims for the autonomy or antiquity of their scientific traditions. The influence of all these attitudes persists today in politicized debates about history, religion and culture in Indian society, For my part, I welcome Plofker’s efforts, implicit throughout Mathematics in India, to get past politicized historiography. For quite a while it has been impossible for anyone reasonably acquainted with Indian mathematics to discount its scope, depth, original contributions, and attention to mathematical reasoning; consequently it is no longer necessary to correct such outdated disparagements by calling attention obsessively to every known instance in which an Indian mathematical result predates its discovery in Europe. ‘‘We all seem to agree’’, says Plofker (p. viii), ‘‘that Indian mathematics is brilliant and fascinating.’’ To come to terms with the ways in which this vibrantly human tradition developed along its own lines and at its own pace—and yet in connection with the mathematics of other cultures— serves only to increases our fascination with it. In addition, the examples of exchange between Hindu and Muslim scholars in such places as Jaipur and the later Mughal court are exemplary for all who today value peaceful cooperation across the boundaries of culture and religion. In an article on comparative philosophy [3], the late American Sanskritist David Ingalls writes: In comparing the Indian and Western traditions from this technical point of view one finds curious discrepancies. I have pointed out two of them, the discovery in India of a technique and a theory centuries before they were discovered in Europe. Such points are interesting but they should not be the goal of comparative studies. 70


What is of real importance is to notice how similar results can be achieved by wholly different methods .... To observe such phenomena is to increase one’s estimate of human possibilities. In this way comparative philosophy is one of the best methods of breaking away from the limitations of one’s environment and gaining a deeper impression of man. It is not too much to hope that a less politicized approach to the history of mathematics can yield similar benefits. Mathematics in India, the first book-length contribution from a still young but already quite accomplished scholar, from whom one hopes to hear much more in years to come, is a step in the right direction.

REFERENCES

[1] T.A. Sarasvati Amma, Geometry in Ancient and Medieval India, Dehli: Motilal Banarsidass Publishers Private Limited, Second Revised Edition, 1999. [2] B. Datta and A.N. Singh, History of Hindu Mathematics: A Sourcebook, 2 vols. Bombay: Asia Publishing House, repr. 1962. [3] D. Ingalls, ‘‘The comparison of Indian and Western Philosophy’’, Journal of Oriental Research Volume 22, pp. 1–11, 1954. [4] G.G. Joseph, The Crest of the Peacock, Princeton: Princeton University Press, 2000. [5] V. Katz (editor), The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook, Princeton, NJ, Oxford, UK: Princeton University Press, 2007. [6] A. Keller, Expounding the Mathematical Seed: A Translation of skara I on the Mathematical Chapter of the A¯ryabhat Bha : iya, Volume 1: The Translation and Volume 2: The Supplements, Basel, Boston, Berlin: Birkhaüser Verlag, 2006. [7] C.Z. Minkowski, ‘‘Astronomers and their reasons: Working paper stra,’’ Journal of Indian Philosophy Volume 30, No. 5, on jyotih : sá pp. 495–514, 2002. ca rya with the [8] K. Sa¯mbasívasá¯stri, The A¯ryabhat : iya of A¯ryabhat :a Bha:sya of Nilakan ::thasomasutvan, Part I: Gan : itapada, Trivandrum: Superintendent, Government Press, Trivandrum Sanskrit Series No. 110, 1930. [9] C.K. Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE (History of Science, Philosophy and Culture in Indian Civilization, vol. X, part 4), Delhi: Pearson Longman, 2007. skara ca rya with Kriya kramakari of vati of Bha [10] K.V. Sarma, Lila ra yan Sán_kara and Na a, Hoshiarpur: Vishveshvaranand Vedic : Research Institute, 1975. [11] K.V. Sarma, K. Ramasubramanian, M.D. Srinivas, and M.S. :sa (Rationales in Mathematical AstronSriram, Gan : ita-Yukti-Bha omy) of Jyes ::thadeva, Volume I: Mathematics, Delhi: Hindustan Book Agency, 2008.

Georgetown College Georgetown, KY 40324 USA e-mail: [email protected]

Mathematical Omnibus: Thirty Lectures on Classic Mathematics by Dmitry Fuchs and Serge Tabachnikov PROVIDENCE, RI: AMERICAN MATHEMATICAL SOCIETY, 2007, 463 + XV, US $59.00, ISBN-10: 0-8218-4316-8, ISBN-13: 978-0-8218-4316-1 REVIEWED BY JOHN J. WATKINS

he first thing I look at each week when my New Yorker magazine arrives in the mail is the cartoons. I did exactly the same thing recently when I got a copy of Mathematical Omnibus: Thirty Lectures on Classic Mathematics. Before reading even a single word, I found myself looking at the illustrations that begin each essay and each chapter in this book. The marvelous and mysterious drawings by Sergey Ivanov beautifully echo the mathematical content of these 30 essays. The fundamental idea underlying Chebyshev polynomials is captured perfectly by two men holding the ends of three very long snakes which, having escaped from a basket, have coiled themselves into perfect polynomial shapes of one, three and five arches. An essay on the Mo¨bius strip is introduced with a clever drawing of an extremely tall thin person embedded in a Mo¨bius strip reaching up to touch her own toes. The beautiful topic of Alexander’s horned sphere is introduced as the two horns of a Picasso-like bull linked with the two arms of a kneeling matador. A topological essay on cone eversion is accompanied by a spectacular drawing that could well have been executed by Salvador Dali. The illustration for the chapter on ellipses and ellipsoids, a drawing which also appears in color on the cover, predictably involves billiard balls but in an unexpectedly elegant setting where a waiter is poised with cue in hand to make his billiard shot on the top of an elliptical table as two seated dinner guests calmly peruse their menus. The 30 lectures in Mathematical Omnibus, by Dmitry Fuchs and Serge Tabachnikov, are based on articles these authors wrote for the Russian popular science magazine Kvant from 1970–1990, and also on lectures they have given in the Soviet Union and in the United States (where they have lived since 1990). They intend the book for a broad audience, ‘‘from high-school students to accomplished researchers,’’ and cover a diverse range of interesting mathematical topics with widely varying levels of difficulty. Some of the best material in the book is historical in nature, for example, a fascinating discussion that carefully follows Euler’s discovery of a formula for the infinite product

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f ðxÞ ¼ ð1 xÞð1 x 2 Þð1 x 3 Þð1 x 4 Þ ¼ 1 x x 2 þ x 5 þ x 7 x 12 x 15 þ x 22 þ x 26 x 35 x 40 þ in terms of what are now known as the Euler pentagonal numbers (3n2 ± n)/2. About 70 years after the discovery of

this formula, Gauss found an even more remarkable formula for the cube of Euler’s function: f 3 ðxÞ ¼ ð1 xÞ3 ð1 x 2 Þ3 ð1 x 3 Þ3 ð1 x 4 Þ3 ¼ 1 3x þ 5x 3 7x 6 þ 9x 10 11x 15 þ : The next power of f(x) to have a simple formula is f 8(x), which was discovered by Felix Klein. The authors continue their discussion of the powers of the Euler function with an excerpt from an article called ‘‘Missed Opportunities’’ presented by Freeman Dyson as the Josiah Willard Gibbs Lecture at the January 1972 meeting of the American Mathematical Society. Dyson began life as a number-theorist, sitting as an undergraduate at the feet of G. H. Hardy, but is better known now as one of the most prominent physicists of the twentieth century never to be awarded a Nobel prize. This cautionary article gently suggests that it would be a good thing if physicists and mathematicians talked to each other; it is well worth reading in its entirety. As the story is retold by Fuchs and Tabachnikov, Dyson found a truly beautiful formula for the Ramanujan tau function, a formula which in turn can also be written as a formula for the 24th power of the Euler function (in fact, Dyson was concerned with powers of a closely related function called the Dedekind eta function). Dyson noticed that the powers for which there is an elegant formula for f n(x) are the integers n ¼ 3; 8; 10; 14; 15; 21; 24; 26; 28; 35; 36; . . . : As a physicist Dyson had seen these same numbers (except 26) many times before as the dimensions of finite-dimensional simple Lie algebras. He lost the opportunity to see a deep connection between modular forms and Lie algebras because, in his words, ‘‘the number-theorist Dyson and the physicist Dyson were not speaking to each other.’’ I suspect Freeman Dyson would appreciate the exercises included with each lecture in Mathematical Omnibus. In his classic book on science, war, technology and ethics, Disturbing the Universe, Dyson writes, ‘‘The difference between a text without problems and a text with problems is like the difference between learning to read a language and learning to speak a language.’’ At the time, he was spending his Christmas vacation working through all the problems in a text on differential equations and trying to learn ‘‘to speak the language of Einstein.’’ The exercises in Mathematical Omnibus are usually quite good, but there is an occasional silly one such as in the lecture on impossible tilings where they ask whether a given polygonal chessboard can be tiled by dominoes. In fact, only three dominoes need be placed into position before the impossibility of a tiling is demonstrated. Its historical perspective is one of the great strengths of this book, but there are some inaccuracies. The authors exhibit a distinct Eurocentric bias in their discussion of rational approximations of p. They mention that the familiar approximation 22/7 was known to Archimedes, and that the approximation 355/113, which is correct to six decimals, was discovered four centuries ago by Adriaen Metius. In fact, this truly remarkable approximation was discovered in the fifth century by the Chinese astronomer and mathematician Zu Chongzhi. Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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Another strength of this book is that Fuchs and Tabachnikov manage to breathe new life into topics that at first glance appear to be old hat. In lecturing on the insolvability of the quintic polynomial, I typically show how equations of degree 4 can be reduced to equations of degree 3 and then present Cardano’s formula for the cubic, thus giving the impression that this completely settles the matter for equations of degree 3 and 4, just as the quadratic formula handles quadratic equations. The authors, however, show quite dramatically just how unreliable Cardano’s famous formula can be, with several well chosen examples where the formula does in fact produce a correct solution but not in a form that is recognizable as the simple solution it - 4p=ffiffiffi 0, actually is. For example, for the equation x3 + 3x p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Cardano’s formula produces the solution x ¼ 2 þ 5þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3 2 5; which must in fact equal 1 since x = 1 is clearly a solution and the monotone increasing function x3 + 3x - 4 must have exactly one real solution. Similarly, it is easy these days to be misled by the powerful Fundamental Theorem of Algebra and forget that the number of real roots of a polynomial has far less to do with the degree of a polynomial and far more to do with the number of nonzero coefficients. In this context, the authors use the dreadful term ‘‘fewnomial’’ to describe polynomials of large degree, such as x100 - 1, with only a few nonzero coefficients. ‘‘Sparse polynomial’’ might be a far more natural term to use for a polynomial most of whose coefficients are zero, since this coincides nicely with common usage elsewhere in mathematics (for example, sparse matrix and sparse graph). Still, they do go on to present a nice proof, using Rolle’s Theorem, that a polynomial with k nonzero real coefficients has at most 2k - 1 real roots, and also prove Descartes’s rule that the number of positive roots of a polynomial having real coefficients is at most equal to the number of sign changes in the sequence of nonzero coefficients. For the most part, the writing in Mathematical Omnibus is very clear, but once in awhile a phrase can seem a little off: ‘‘What percent of the numbers ... is odd?’’ Every so often this can cause confusion. They describe surfaces that can be obtained by bending sheets of paper (they call these developable surfaces) and specify the two crucial physical properties of paper involved: First, paper is not compressible or stretchable; second, paper is absolutely elastic. But the use of the word ‘‘elastic’’ is not appropriate here since in normal usage this connotes the very property they wish to avoid: Stretching (as in an elastic band or elastic wrap). They go on to ask a very interesting question about constructing a Mo¨bius strip from a rectangular piece of paper. Anyone who has ever done this knows that if the end result is to be attractive, the strip of paper needs to be rather long to begin with. The very natural question they ask is: What is the length of a strip of paper, one inch wide, such that any longer strip can be twisted into a Mo¨bius strip but no shorter strip can (without folding)? Surprisingly, the answer is not known. They give a very clever construction where, if folding is allowed, a Mo¨bius strip can be made from a one inch wide strip of paper of arbitrarily short length!

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The 30 lectures are accompanied by hundreds of figures that clarify the concepts quite well, but not always. In an essay on curvature and polyhedra the authors present an elegant proof by induction of the lovely theorem, giving the formula A = a1 + + an - (n - 2)p for the area of a convex n-gon on a unit sphere with angles a1 ; . . .; an . However, the proof of the base case for a spherical triangle is illustrated by a figure that, while technically correct, is very minimal and could make the heart of the argument rather difficult for some readers to discover. This is a case where a more elaborate figure—perhaps with shading—might make a fundamentally easy argument stand out more clearly. They begin this essay by asking how one defines the curvature of a polygonal line. (Would a high-school student know what they mean by a polygonal line?) They define the curvature of a plane wedge and then say the curvature of a polygonal line is the sum of the curvatures of its angles (leaving, I think, confusion in some minds as to exactly what angles and wedges are involved). It isn’t helpful that in the accompanying figure of a convex polygon the exterior angles don’t correspond in size very well to their counterparts in the figure showing that their sum is 2p. In addition to figures, almost every lecture contains photographs or images of the famous mathematicians mentioned in the essay. This certainly adds significantly to the historical interest; however, it is unfortunate that these images are all exactly the same size and merely collected together in rows and columns in a single batch at the end of each lecture. Several of the lectures involve topics that invite active participation on the part of the reader. One can assemble two noncongruent polyhedra from identical sets of polygons, with edges matched in the exact same way: For example, imagine a cardboard model of the standard convex icosahedron and another, nonconvex model made from identical triangular pieces but with a single vertex pushed in toward the center so that this vertex and the five surrounding triangles form an indentation in the polyhedron. The authors use Euler’s famous formula relating the numbers of vertices, edges and faces of a convex polyhedron to prove Cauchy’s theorem that if the corresponding faces of two convex polyhedra are congruent and adjacent in the same way, then the two polyhedra are congruent. Then they follow this with an extraordinary result, discovered in 1978, that there exist flexible polyhedra. They even include a figure so you can make your very own cardboard model of one with just 14 triangular faces!

REFERENCES

F. Dyson, Missed Opportunities, Bull. Amer. Math. Soc. 78 (1972), 635–652. F. Dyson, Disturbing the Universe, New York: Harper & Row, 1979.

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

Four Films by George Csicsery ¨ NTER M. ZIEGLER REVIEWED BY GU

}s N Is a Number. A Portrait of Paul Erdo Zala Films, Oakland, CA, 1993, 57 min., US $29.95, ISBN: 0-9724588-1-6 oon after Paul Erd} os died in 1996, two booklength biographies appeared: The Man Who Loved Only Numbers: The Story of Paul Erd} os and the Search for Mathematical Truth by Paul Hoffman (1998) and My Brain Is Open: The Mathematical Journeys of Paul Erd} os by Bruce Schechter (1998). I dare to say that neither of them captures the ‘‘true’’ Paul Erd} os. Just observe that on Erd} os’s Wikipedia page in German Hoffmann’s book is termed ‘‘a biography in the form of a novel,’’ while Schechter starts out his book by saying that he first heard about Erd} os in 1982, and since then always wanted to meet him, so when 14 years later he heard that Erd} os had died, he decided to write a book about him, and started interviewing people about him at his funeral in Budapest. This I consider absurd, in particular since Erd} os was not reclusive and not hard to meet. (This is different if you are interested in writing a biography of Euler, say, or of Grothendieck—though Winfried Scharlau, who is working on Grothendieck’s biography and published the first of three volumes in 2007, apparently has met the master.) If you want to get an authentic picture of Erd} os, then indeed the 1993 documentary ‘‘N Is a Number. A Portrait of Paul Erd} os’’ by George Csicsery is the best source we have. The one-hour documentary was filmed between 1988 and 1991 in four different countries, so in a very global way the camera follows Erd} os around. The film is a movie for a ‘‘general audience,’’ so the descriptions of Erd} os’s mathematical achievements are somewhat simplistic. But the combination of a camera ‘‘following him around,’’ conversations with Erd} os himself, and interview statements by many of his colleagues and friends altogether gives a manyfaceted portrait of an unusual mathematician.

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Julia Robinson and Hilbert’s Tenth Problem Zala Films, Oakland, CA, 2008, 54 min., US $29.95, ISBN 978-097245885-6 Julia Robinson was probably the most famous woman mathematician of her time. Her most important contributions were towards the solution of Hilbert’s Tenth Problem (‘‘Is there an algorithm for deciding whether a polynomial equation with integer coefficients has an integer solution?’’). She worked towards the solution of this problem for 22 years. Her 1961 Annals paper (joint with Martin Davis and Hilary Putnam) was one of several

major stepping stones that she provided towards the eventual negative solution of the problem by the 22-yearold Yuri Matiyasevich in 1971. Julia Robinson was also a pioneer: The first woman mathematician elected to the US National Academy of Sciences (1975), and the first woman president of the American Mathematical Society, 1983–1984. Her life and career are documented by a book-length biography written by someone who knew her very well: Her sister Constance Reid (Julia: A Life in Mathematics, Mathematical Association of America, 1996). Nevertheless, the movie version provided by George Csicsery is enlightening: It presents a wealth of black and white photographs from their childhood, and besides Constance Reid it features interviews with Robinson’s key co-authors Martin Davis, Hilary Putnam and also Yury Matiyasevich. It introduces a role model, illustrates how dedicated mathematicians (not remote geniuses, but real humans) collaborated towards the solution of a key mathematical problem, and tells an exciting story. The relationship between Julia Robinson and the much younger Matiyasevich comes to life. It is also a movie that could motivate you, your students, and the public to learn more about Julia Robinson.

Porridge Pulleys and Pi. Two Mathematical Journeys MSRI, Berkeley, CA, 2006, 28:30 min., US $40.00 ISBN 0-9639903-6-5 Meet the mathematicians: This film portrays two very different mathematicians, Vaughan Jones (of ‘‘Jones polynomial’’ knot theory fame) and Hendrik Lenstra (the number-theorist who invented the elliptic curve factoring algorithm). They are very different in their origins (Jones comes from New Zealand, Lenstra from Holland), in their family backgrounds (Lenstras father was a mathematician, and so are several of his brothers, while Jones comes from a nonacademic background), in their approach towards problems, in their styles—you never see Lenstra without a tie, except when he is swimming. Meet the mathematicians and their families: This short film includes interview sequences not only with Lenstra, but also with his younger brother Arjeh, and with their mother Ina. It shows Vaughan Jones with his family singing—and it tells a great number of interesting stories, in just half an hour, about Jones’s involvement with knotted DNA and about Lenstra’s work in the completion of the missing center in the Escher picture ‘‘Print Gallery.’’ Mathematics alive! Amazingly, both Jones and Lenstra come cross as quite ‘‘ordinary mathematicians,’’ dedicated to their work but approachable. The fact that both of them are outstanding mathematicians is mentioned in passing at best—and perhaps if you don’t know that Jones received a Fields Medal in 1990, and Lenstra has been the program chair for the International Congress of Mathematicians 2010 in India, then they seem even more approachable. Meet these mathematicians!


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Hard Problems. The Road to the World’s Toughest Math Contest MAA, 2008, 82 min. (feature), 45 min. (classroom version), US $29.95, ISBN 978-088385902-5 I must admit that this film moved me particularly, since it brought up memories from my own participation in the IMO—the International Mathematical Olympiad—in Washington, DC in 1981. I have only very few photos from that trip to the US, which of course meant a lot to me. (One out of focus snapshot shows me and the other members of the German team on the lap of the Einstein statue in the garden of the National Academy of Sciences.) Csicsery’s ‘‘Hard Problems’’ movie documents the selection of candidates and the path to success for the US team on the 2006 IMO, which took place in Ljubljana, Slovenia. It was a long and winding road to Ljubljana: The movie shows the whole process of tests, selection of teams, and training sessions for the competition. It follows a small group of aspiring and highly motivated students, including

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interviews with parents, brothers, sisters and friends. It reports about tests and training sessions, with interviews with the contestants, their trainers and former participants. It is present at the final selection of the IMO team and follows it to Ljubljana. It captures the hopes and fears before and between the two decisive 412 hour tests, and the tension during the tests. It gives glimpses into the grading and evaluation process, the subtle negotiations between team leaders and the international jury about single points given or not (in doubt most of the time: Not). Thus, it conveys the excitement of the competition and the excitement of dealing with ‘‘Hard Problems.’’ It conveys the excitement of Doing Mathematics.

Institute of Mathematics MA 6-2, TU Berlin 10623 Berlin Germany e-mail: [email protected]

The Housekeeper and the Professor by Yoko Ogawa PICADOR, NEW YORK, 2009, 192 PP., US$14.00, ISBN 0-312-42780-8 REVIEWED BY FANG-MEI LIN AND WANN-SHENG HORNG

athematical fiction has recently emerged as a new genre, and its influence on both mathematics education and literary writing has become increasingly significant, as shown by the website Mathematical Fiction. Not surprisingly, many of the math novels are written by authors who major in math. These novels use the form and techniques of story-telling to illustrate mathematical concepts and history. The Housekeeper and the Professor stands out among these novels for its double contribution to disseminating math knowledge and literary imagination. The author, Yoko Ogawa, majored in literature and the arts. Her works are popular not only in Japan, but also in the West. Many of her works have been adapted into films, including The Housekeeper and the Professor. Yoko Ogawa was born in 1962 in Japan and graduated from Waseda University, Japan. Since 1980, she has published more than 20 works of fiction and nonfiction. The Hakase no aishita sushiki, the original Japanese version of The Housekeeper and the Professor, was published in Japan in 2003, and has been translated into English with different editions and titles, including The Gift of Numbers and The Housekeeper and the Professor. In 2006 she coauthored An Introduction to the World’s Most Elegant Mathematics (in Japanese) with Masahiko Fujiwara, a mathematician, as a dialogue on the extraordinary beauty of numbers (see Wikipedia). She has also won many important literary awards in Japan; overall, she is an author able to maintain a balance between literary reputation and mass popularity. This novel is about a math professor whose memory, after a car crash, lasts for only 80 minutes. The story uses a first person point-of-view, starting with ‘‘I,’’ who is a female housekeeper hired by the professor’s sister-in-law to take care of him. There are only four persons in this story: the housekeeper, the professor, the housekeeper’s son, and the professor’s sister-in-law. None of them is named. Ogawa models the character of the professor on Paul Erd} os, whose biography by Paul Hoffman (1998) she includes in the bibliography. She also refers to some interesting episodes in the biography and transforms them into plots in the novel. Because of his short-term memory, the professor can only ‘‘recognize’’ the housekeeper by means of tape slips sticking to his jacket. These slips are reminders of things to do and information about the housekeeper. However he still retains his ability for mathematical thinking. The contrast between dysfunctional daily activities and mathematical competence leads the professor to begin every conversation with numbers and to associate everything with numbers. Every

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morning, the housekeeper comes to his house as a stranger, and he greets her by asking, ‘‘What is the size of your shoes?’’ ‘‘24.’’ ‘‘What a noble number it is! It is the factorial of 4.’’ ‘‘What is your telephone number?’’ ‘‘5761455? How incredible! It is the total amount of the prime numbers up to one billion.’’ Apparently the neutrality of number concepts protects the professor himself from misunderstanding other people. He is fond of the housekeeper’s 11-year-old son, giving him the nickname ‘‘Root’’ (his flat head resembles the square root pffi sign . ‘‘You are a Root,’’ the professor tells him. ‘‘You accept any number that comes your way, rejecting none. A truly generous symbol, Root.’’ Moreover, since the housekeeper’s birthday is February 20th, a date written as 220, the professor explains its ‘‘amicable’’ relation with the number, 284, which is engraved on his own watch, the 284th President Award for his brilliant achievement in mathematics in college years. Acceptance, friendliness, and caring are themes running through the entire novel. The author implies that mathematics should be friendly. She creates the situation in which a male math professor and a female caretaker exchange knowledge and loving care. The reciprocal relationship is made possible by the encounter of two lonely outcasts: the housekeeper is an unwed single mother, and the professor is dysfunctional in everyday routine. Actually, the sister-in-law is a lonely outcast, too. She and the professor had a love affair many years before. In their youth, the professor wrote Euler’s Equation eip = -1 on a piece of paper as a code attesting to their sweet and bitter love, with -1 representing the total situation. As readers, we are not sure whether the professor still remembers this. The author creates an ambiguous picture, with the sister-in-law reminiscent of old days and jealous of the housekeeper, and the professor holding a neutral attitude toward the sisterin-law. Ogawa uses Euler’s formula to explore conflict and harmony, a literary metaphor for the complexity of human feelings. In the words of the narrator (the housekeeper), ‘‘Euler’s formula shone like a shooting star in the night sky, or like a line of poetry carved on the wall of a dark cave.’’ Root, prime numbers, perfect numbers, amicable numbers, and Euler’s formula have become alluring symbols of mystery, eternity, and human nature. The novel also treats ‘‘mistake’’ in a positive way. The professor encourages the housekeeper and Root not to be afraid of mistakes when solving mathematical problems. In real life, he himself apparently made mistakes in his ‘‘unethical’’ affair with his sister-in-law. Likewise, the housekeeper is an unwed single mother who accepts the fact that the father of her child has left them. Other math novels emphasize the importance of mathematical proof, but this book stresses the courage to love and to make mistakes. In this work, ‘‘love’’ is used less often for human interaction than for learning mathematics. Ogawa weaves episodes of baseball games into numbers and memories. The professor has a collection of baseball cards from many years before; the housekeeper and the boy Root introduce him to baseball again. He is overexcited and becomes severely ill after their outing. Over the years his health deteriorates, and Root grows up to be a high-school math teacher. In the end, the narrator flashes back to Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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watching a baseball game with the professor and Root, concluding with a simple sentence: ‘‘And I can just see the number on the back of his pin-striped uniform. The perfect number 28.’’ The novel should not be read in a simple manner as a male professor teaching math to a female housekeeper and her son. We use the book as required reading in a freshman course, ‘‘Mathematical Thinking.’’ On first reading, the majority of students regard the professor as the ‘‘hero’’ and ignore the reciprocal relationship: the professor teaches math to the housekeeper and her son, while he is dependent on the assistance of the housekeeper in everyday life. When we ask students to think about what would happen if gender roles were reversed, they reply, ‘‘The story would be the same,’’ or ‘‘Women are not good at math; it’s hard to imagine a female math professor.’’ Without appropriate guidance, the novel might reinforce gender stereotype. When the novel was adapted to cinema, the narrator becomes the school math teacher, who flashed back to his childhood and his mother’s work as a housekeeper. We recommend both the novel and the film. The English edition appends questions for readers to think about. These questions are both mathematical and literary. The similarity between mathematics and narrative is receiving academic attention [cf. Thomas 2002]. Mathematics cannot avoid the use of language to represent its problems and solutions; how to describe thus becomes an important issue that can be approached as narratology, similar to the studies of literary works. With the growth of mathematical fiction, we welcome the emergence of this genre and look forward to more works that can lead to multidisciplinary

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dialogues among literary studies, mathematics education, and history of mathematics.

BIBLIOGRAPHY

Hoffman, Paul (1998). The Man Who Loved Only Numbers. New York: Hyperion. Mathematical Fiction: http://www.cofc.edu/*kasmana/MATHFICT/. Ogawa, Yoko (2003). Hakase no aishita sushiki (Japanese original edition of The Housekeeper and the Professor). Tokyo: Shinchosha. Ogawa, Yoko (2009). The Housekeeper and the Professor (Trans. by Stephen Snyder). New York: Picador. Scholes, Robert, James Phelan, and Robert Kellog (2006). The Nature of Narrative. Oxford/New York: Oxford University Press. Thomas, Robert S. D. (2002). ‘‘Mathematics and Narrative,’’ The Mathematical Intelligencer 24: 43-46. Wikipedia. http://en.wikipedia.org/wiki/Yoko_Ogawa.

Graduate Institute of Taiwan Culture, Language and Literature National Taiwan Normal University Taipei, Taiwan e-mail: [email protected] Department of Mathematics National Taiwan Normal University Taipei, Taiwan e-mail: [email protected]

Mathematics and the Aesthetic: New Approaches to an Ancient Affinity by Nathalie Sinclair, David Pimm and William Higginson (eds.) SPRINGER, NEW YORK: 2007, 288+XVI PP., US $64.95 ISBN 978-0-387-30526-4 REVIEWED BY BRIAN ROTMAN

his book is an edited collection of essays from nine contributors—the three editors and six others—most of which appeared as talks at a conference, Beauty and the Mathematical Beast, at Queen’s University in Ontario in 2001. It comprises an introduction, a concluding chapter, and three parts of three chapters each. The first part, The Mathematician’s Art, starts with ‘‘Aesthetics for the Working Mathematician,’’ by Jonathan Borwein. Embracing Emil Artin’s proclamation, ‘‘We believe that mathematics is an art,’’ Borwein argues that aesthetics comes before utility in mathematical research and that opportunities for invoking questions of economy and structural beauty abound not only in research but throughout mathematical education. In particular, Borwein is an impassioned advocate for experimental mathematics, for the importance of computational methods and the use of visualization/manipulation software as a source of intuition and fruitful hunches and a means of generating a ‘‘feel’’ for a problem that eludes purely symbolic analysis. Among his many examples from his own research and others are two gems of visualization: Tom Apostol’s ‘‘lovely new geometric proof’’ of the irrationality of H2—which, as he points out, delivers a new insight into this ancient result, and the Coxeter/Kelly visual demonstration of Sylvester’s conjecture about noncollinear points in the plane which, like Apostol’s, uses a minimal configuration argument. The next chapter is ‘‘Beauty and Truth in Mathematics,’’ by Doris Schattschneider. Observing that though there are certainly formulas and assertions that have been found to be aesthetically pleasing (e.g., eip = -1), it is proofs that are most likely to be labeled beautiful by mathematicians. She elaborates by listing characteristics of a proof that prompt mathematicians to designate it as beautiful: ‘‘Elegance’’ (cutting to the essential idea), ‘‘ingenuity’’ (has an unusual or surprising twist), ‘‘insight’’ (offers a revelation of why something is true), ‘‘connections’’ (enlighten a larger picture), ‘‘paradigm’’ (provides a widely applicable heuristic). Examples, for the most part accessible to high-school students, include the different Chinese and Western geometrical proofs of Pythagoras’s theorem and (another gem) Polya’s proof of Hab B (a + b)/2, which not only demonstrates the arithmetic-geometric inequality, but also shows why it’s true and why equality occurs only when a = b. Schattschneider ends with an insistence on the aesthetic of ‘‘doing mathematics,’’ which she illustrates from her

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own protracted efforts trying to prove a result she ‘‘knew’’ to be true, on tiling the plane with congruent polygons. The final chapter in this section, ‘‘Experiencing Meaning in Geometry,’’ by David Henderson and Daina Taimina, starts from the authors’ journeys (from opposite directions) between art and geometry. They too give primacy to intuition and stress that mathematical proofs are valuable not because they satisfy some list of criteria such as elegance, wide applicability and so on (though they might), but only when they provide insight, when they reveal not just that a result is true but why it is true By examining different conceptions of a ‘‘point’’ and a ‘‘straight’’ line and focusing on various physical models of the hyperbolic plane (including a crocheted model), as well as introducing certain nineteenth-century machines for drawing straight lines, they show how concrete models and mechanisms can illuminate otherwise obscure, or at least difficult to conceptualize, mathematical abstractions. They conclude: ‘‘We believe that the understanding of meanings in mathematics (often through aesthetic experiences) comes before an understanding of the analytic formalisms.’’ The next section, A Sense of Mathematics, offers a triple of increasingly more general perspectives. It starts with ‘‘The Aesthetic Sensibilities of Mathematicians,’’ by Natalie Sinclair, who asks what are ‘‘the animating purposes of mathematicians, why do mathematicians do mathematics? What impulses, what inclinations are responsible for producing the body of knowledge that is mathematics?’’ Sinclair’s approach is mainly descriptive and classificatory: She gathers quotations from the literature and seeks answers by interviewing a group of mathematicians. These lead her to propose a tripartite categorization of the aesthetic impulses underlying mathematics: The evaluative—the familiar and ever-present judgements mathematicians make about the significance and value of results and proofs; the generative— the dimension of the aesthetic pertaining to the creation of new ideas and insights into what mathematicians do; the motivational—the role of the aesthetic involved in attracting mathematicians to certain fields and stimulating interest in particular problems. After ranging widely over different areas, time periods and mathematicians, she concludes that the subject ‘‘satisfies the basic human impulse to find and describe pattern.’’ The final chapter in this section, ‘‘The Meaning of Pattern,’’ by Martin Schiralli, again emphasizes mathematicians’ search for and construction/appreciation of pattern, but looked at in a wider context of how pattern is conceived in biology (the writings of Gregory Bateson) and in visual art (Ernst Gombrich’s Art and Illusion). Both these thinkers, in Schiralli’s reading of them, point to an idea of pattern beyond a static, immediately apprehensible arrangement, a pattern behind the pattern as it were, what Bateson articulates as ‘‘a dance of interacting parts.’’ The ubiquity of pattern and its importance in diverse fields suggests that for mathematics the concept needs a more focused view, one that goes back to the roots of the subject, if it’s to be distinguished from its occurrence in biology and art. To this end, the essay looks at the school of Pythagoras and the understanding of pattern to be found in their notion of number (particularly in relation to arithmos) and the meaning of concept as this plays out in the work of Philolaus of Croton. Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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The third chapter, ‘‘Mathematics, Aesthetics and Being Human,’’ by William Higginson, is the most ambitious. It asks in relation to mathematics, ‘‘What does it mean to be human?’’ Higginson argues that the aesthetic drive is a ‘‘manifestation of a universal human ability to sense what ‘fits’ in a given situation and what does not.’’ He elaborates this over a wide terrain: Anecdotes and surveys of how mathematics and mathematicians have been (and still are) perceived by schoolchildren, the recent surge in popular interest and artistic production of films, novels and plays featuring mathematics, and claims by cognitive science about embodied origins of mathematics. It culminates in the suggestion that the thinking behind Ellen Dissanayake’s book Homo Aestheticus, on the origins and motives of art, might find an appropriate (and even more fundamental) formulation in a concept he designates as Homo Mathematico-Aestheticus. The final section, Mathematical Agency, starts with ‘‘Mechanism and Magic in the Psychology of Dynamic Geometry,’’ by R. Nicholas Jackiw. Jackiw is the designer of The Geometer’s Sketchpad, a well-known software program that allows users to create and manipulate mathematical constructions. His essay has two concerns, each with its own aesthetic dimension. One is to confirm the powerful and vivid mathematical experience—in mathematical research no less than education—afforded by what has come to be called ‘‘dynamic geometry’’ that such software and programs like it facilitate. The other seeks to embed the Sketchpad in a wider examination of the nature and importance of mechanical devices in relation to mathematical thought. Arguing against what he sees as the trivialization of devices as merely didactic and inessential aids, he sketches a deeper historical tradition which portrays ‘‘machines being conceived and received as embodiments, exemplars, repositories and demonstrations of profound scientific knowledge.’’ This allows him to conclude that devices such as the Sketchpad operate under a tension between two impulses: A conventional, explanatory, didactic one and a more ‘‘magical’’ mode when their purpose is to ‘‘astound and amaze’’ rather than produce a stabilizing enlightenment. Next in this section is ‘‘Drawing on the Image in Mathematics and Art,’’ by David Pimm. Aesthetics, Pimm declares at the outset, is to be interpreted in sensorial terms as ‘‘firmly rooted in the senses by means of which we perceive.’’ Further, perception here is overwhelmingly visual, involving what John Berger called ‘‘ways of seeing.’’ The combination allows him to put the question of aesthetics equally to (visual) art and mathematics, mobilizing a wealth of comment by artists, critics and art historians as well as by mathematicians. This in turn provides a natural platform for a focus on the visual-diagrammatic dimension of mathematics which leads to predictable issues: The nature of telling (the letter) versus showing (the image) concrete, demonstrative examples versus abstract axiomatic presentation, the significance of the attribution ‘‘modern’’ to mathematics, and the complete and deliberate expunging of all diagrams in the Bourbaki group’s set-theoretical axiomatization of mathematics that exerted such a forceful (and for many baneful) influence on the

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presentation of the subject for several decades. Pimm quotes Pierre Cartier (himself a member of the group) explaining that the ‘‘Bourbaki were Puritans and Puritans are strongly opposed to pictorial representation of truths of their faith.’’ From this Pimm dilates on iconoclasm and an anthropological understanding of ‘‘purity,’’ which allows him to see a proof as exerting a form of agency. The final essay of the section, ‘‘Sensible Objects,’’ by Dick Tahta, perhaps the most loosely focused contribution here, consists of an associatively linked series of responses, suggestions and remarks provoked by the question of what a sensible object is and how it might or might not relate to the kind of objects mathematicians refer to. Mixing and sampling eigth-century Christian iconoclasm, nineteenth-century Romantic poetry, several psychoanalytic theories, Renaissance art, mysterious Neolithic stone balls, vibrating strings and resonating ideas, the essay evades any definite conclusions, but instead ends with ‘‘the mystery of things.’’ Connecting all this is an insistence that interpretive practices together with their communities are important to how ‘‘sensible’’ and ‘‘object’’ are to be thought together. As I mentioned at the beginning, these contributions are sandwiched between an (historical) introductory chapter and a concluding chapter (‘‘Aesthetics and the ‘Mathematical Mind’,’’ which focuses on psychological themes). The authors insist that the oft-cited attributes of mathematics— detachment, certainty, abstraction and perfection—be seen not as ‘‘objective’’ characteristics or as elements of generic aesthetic theories (such as those of Kant, for example) but as questions about the individual subject, questions of motive, desire, psychological need and pleasure. This leads in two directions. One is the nature of the unconscious and the preconscious aspect of mathematical knowledge/intuition and its coming into consciousness (for example, as told famously by Poincare´ and subsequently systematized by Hadamard into stages of mathematical creation). The other points to the recognition of the ‘‘darker’’ aspects of the mathematical psyche. For example, the desire for detachment, and so on, can be seen as the mathematician’s fight against what one commentator describes as the ‘‘uncertainty, disorder, irrationality, being out of control’’ that haunts its practitioners. This is followed by a clutch of dark affects: ‘‘The melancholy disposition of the mathematical mind’’ (Albrecht Du¨rer’s engraving Melancolia is reproduced), and then, more extreme—further from Reason— aspects of the mathematical psyche in ‘‘blindness, solipsism and the ‘mathematical mind’’’ and ‘‘autism and the ‘mathematical brain.’’’ Mathematics and the Aesthetic is a richly varied collection of essays that will supply numerous leads, avenues, openings and provocations to anybody interested in the pleasures and rigors of mathematicial thinking. Department of Comparative Studies Ohio State University Columbus, OH 43210 USA e-mail: [email protected]

Plato’s Ghost: The Modernist Transformation of Mathematics by Jeremy Gray PRINCETON: PRINCETON UNIVERSITY PRESS, 2008,VIII + 515 PP., US$45.00, ISBN: 978-0-691-13610-3 REVIEWED BY JEREMY AVIGAD

t only takes a few minutes on Amazon.com or MathSciNet to make the case that Jeremy Gray is among the most prolific historians of mathematics working today. Winner of the 2009 AMS Albert Leon Whiteman Prize for notable exposition and exceptional scholarship in the history of mathematics, his books and articles, and the many collections of essays that he has edited, cover just about every aspect of mathematics in the nineteenth and early twentieth centuries. It is therefore no small assertion to say that Plato’s Ghost is his most far-reaching and ambitious work to date. Gray’s goal is to clarify the sense in which modern mathematics is ‘‘modern,’’ and to explore the historical process by which the subject attained that character: In this book I argue that the period from 1890 to 1930 saw mathematics go through a modernist transformation. Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated—indeed, anxious—rather than a naı¨ve relationship with the day-to-day world... This is about as close to a definition of ‘‘modernism’’ as Gray provides, but the rest of the introduction does manage to fill out the picture considerably. In art history, the term ‘‘modernism’’ is used to characterize a cultural movement in the late nineteenth and early twentieth centuries, with a strong tendency toward abstraction, which self-consciously aimed to distance itself from enlightenment views and values. In a review in the New Yorker (23 February 2009), Louis Menand conveys a sense of the movement: How ... did people like Picasso and Joyce change the game? They did it by shifting interest from the what to the how of art, from the things represented in a painting or a novel to the business of representation itself. Modern art didn’t abandon the world, but it made art-making part of the subject matter of art... Modernism was formally difficult and intellectually challenging. Its thrills were not cheap. This description comes fairly close to characterizing what Gray takes to be ‘‘modern’’ about modern mathematics as well. Many of his central themes have to do with formal aspects of mathematics, and he pays particular attention to developments in logic and our understanding of language, the rise of the axiomatic method, and mathematical and philosophical attempts to come to terms with the nature of mathematical reasoning itself. At the same time, he is keenly

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interested in the way modern mathematics gradually broke free of its empirical moorings, as, for example, one ceased to view geometry as the study of space but rather as an exploration of the many structures that could possibly serve as useful representations of space; and in the increasing focus on these representations, rather than what was being represented. By the middle of the twentieth century, it had become common to view mathematics as the study of abstract structures that stand independent of the empirical world but yet, paradoxically, play an essential role in our scientific theorizing. Gray ably traces the evolution of this viewpoint from the incipient glimmerings in the nineteenth century to its mature form. There are many ways to write the history of mathematics. Even if one felt that history proper should just be an assemblage of bare, unadorned facts (a view to which Gray does not subscribe), one would still have to make choices as to which facts are relevant or important. One can comb through archival material and letters to determine who first proved what theorem when, where an idea or method first originated, or who learned what from whom and how. One can focus on the lives of mathematicians, filling out their personalities and ambitions, and chronicling their struggles, hardships, rivalries, triumphs, and failures. One can, instead, write the history of mathematics as a self-standing history of ideas, describing the research agendas and central problems, and the ways that mathematical theories and methods developed in response to ‘‘internal’’ mathematical pressures. Or one can situate the mathematical ideas in a broader philosophical context, focusing on the ambient (either implicit or explicitly stated) views as to the nature and goals of the subject. With an even broader scope, one can focus on mathematics as an institution, and situate all the above in the context of the social, national, political, and economic factors that bear upon the mathematical profession and its research agendas. One can expand the circle of ideas even further, and view mathematics as a part of a broader cultural history, including developments in literature, the arts, and even theology. The striking thing is that in this book, Gray does it all, and the focus on the development of mathematics from 1890 to 1930 is perhaps the only sense in which the narrative is constrained. The Library of Congress classification characterizes the subject matter as follows: 1. Mathematics–History–19th century. 2. Mathematics–Philosophy. 3. Aesthetics, Modern–19th century. Even that characterization doesn’t do justice to the full scope of the book, whose topics include the history of projective geometry, from the late eighteenth century ‘‘descriptive geometry’’ of Gaspard Monge to the algebraic perspective of Klein’s Erlangen Program; the eighteenth century philosophical views of Immanuel Kant, post-Kantian interpreters from Herbart and Fries to Cassirer, and reactions to Kant from Frege to Helmholtz to Poincare´ and Russell; the history of nonEuclidean geometry, through the work of Riemann, and its bearing of the developments on the foundations of physics, through the writings of Poincare´, Duhem, Hertz, Minkowski, Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 2, 2010

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and Einstein; the rival methodological approaches to algebraic number theory by Dedekind and Kronecker; the history of set theory from Cantor to the set-theoretic paradoxes, and then on to Zermelo’s axiomatization; the development of the axiomatic tradition from the British algebraists through the American axiomatic school and Hilbert; developments in logic from Boole, Peirce, Frege, and Schro¨der to Russell and Hilbert; developments in linguistics; developments in psychology, and views on the relationship between psychology and logic, especially those of Helmholtz and Wundt; developments in the foundations of analysis, including the infinitesimals of Du Bois-Reymond and Stolz and the ‘‘five letters’’ among Baire, Borel, Lebesgue, and Hadamard; the early twentieth century ‘‘crisis of foundations,’’ and the manueverings of Brouwer, Weyl, and Hilbert, as well as many of the lesser players; analogies between Cauchy’s contributions to analysis and the history of music; comparisons to Catholic modernism, a late nineteenth century movement that tried to reconcile the view of the Church with postenlightenment science; and much, much more. Gray’s focus on the period from 1890 to 1930 may at first seem odd, since historians of mathematics typically take the ‘‘birth’’ of modern mathematics to have occurred in the late nineteenth century. The resolution to this apparent anomaly reminds me of a joke that made its rounds in the New York Jewish community in the mid-1980s, amid debates surrounding recent supreme court rulings on abortion. Question: according to Jewish tradition, when does a fetus become a viable human being? Answer: when it graduates from law school. Whatever the joke tells us about overbearing Jewish parents, it can also serve to remind us that the passage from birth to maturity is a tortuous process. Whereas mid-twentieth century mathematics would have been barely recognizable to most late nineteenth century mathematicians, today’s mathematics would seem perfectly familiar to any post World War II mathematician. To be sure, theories have become more complex and proofs have become longer, but today’s research agendas would seem familiar, and styles of argumentation are substantially the same. What Gray does in this book is chart the growth of mathematical modernism, from its first tentative steps in the nineteenth century, through its coming of age, to the point where it attained the mature character that is recognized as a hallmark of the subject today. After a brief introduction, Chapter 1, ‘‘Modernism and mathematics,’’ sets the stage for the narrative that follows, setting out some of the themes and issues that will play a role. The next three chapters follow a generally chronological order. Chapter 2, ‘‘Before Modernism,’’ describes some of the first glimmerings of modern ideas, in geometry, analysis, algebra, and philosophy. This includes the development of projective geometry, the rigorization of analysis, the appearance of algebraic number theory and algebraic logic, and the neo-Kantian positions of Herbart and Fries. Chapter 3, ‘‘Mathematical modernism arrives,’’ shows modernism attaining its mature form. We get descriptions of Klein’s Erlangen program; Poincare´’s work on non-Euclidean geometry; Riemann, Helmholtz, and Lie on the possible geometries of space; Cantor’s theory of the transfinite; modern theories of the continuum; elements of Dedekind’s structuralism; and Frege’s philosophy 80


of logic, among the many topics discussed. In Chapter 4, ‘‘Modernism avowed,’’ modern developments are solidified and firmly anchored, making it possible for the subject to settle down to business in the new mold. We get axiomatic geometry a` la Hilbert, the modern French analysis, modern set theory, modern algebra, twentieth century philosophy of mathematics (including discussions of Russell, Poincare´, Hilbert, Brouwer, Weyl, and Cassirer), and the crisis of foundations. Having traced the arc of the development in broad terms, Gray then devotes the last three chapters to exploring some aspects of the transformation in greater depth. Chapter 5, ‘‘Faces of mathematics,’’ explores the relationship between mathematics and physics (with views of Riemann, Duhem, Poincare´, Hertz, Hilbert, Minkowski, and Einstein); theories of measurement, the continuum, and infinitesimals; and historical and popular presentations of the subject throughout the transformation. Chapter 6, ‘‘Mathematics, language, and psychology,’’ brings linguistics and psychology into the picture. Finally, Chapter 7, ‘‘After the war,’’ considers the modern transformation in hindsight, focusing on postwar foundational and philosophical understandings of modern mathematics. Gray has a knack for making the mathematical ideas broadly accessible, say, to anyone with an undergraduate background in mathematics. For all that it does, his treatment is not exhaustive, and a more focused and less ambitious approach to the subject might have covered a number of topics in greater depth. Nor is it fully balanced; for example, the contents are biased towards geometry over algebra and analysis, that being a particular specialization of Gray’s. But the evolving views as to the nature of geometry and its relationship to the empirical world play a very important role in the transition to the modern view, so the extra attention given to the subject is justified. The book tells a number of stories that may not be familiar to contemporary mathematical readers, but should be, given the impact that they have had on the broader history of ideas. For example, many of us today know that Gottlob Frege famously railed against ‘‘psychologism,’’ that is, the view that the task of logic was to describe the psychological processes that underlie the laws of thought. The very notion sounds odd today, so much so that it is difficult to imagine why such a view ever seemed attractive. For that reason it is especially interesting to see people such as Helmholtz and Wundt grapple with the question of how best to study the nature of human language, knowledge, and thought. Gray manages admirably to provide a sense of the constellation of ideas at play. Some of the works Gray describes are further out of the mainstream, with lesser impact on the modern tradition, but interesting nonetheless. For example, he provides a brief account of a little-known work of 1907, Psychologie du nombre, by a Frenchman, S. Santerre, which undertakes to axiomatize ‘‘facts of consciousness’’ and use that to ground our knowledge of arithmetic. But even the familiar stories in the history of mathematics take on new life in Gray’s hands. For example, even readers who are tired of hearing of the early twentieth century ‘‘crisis of foundations’’ are likely to be moved by the drama of Zermelo’s well-ordering proof, and the reactions found in the ‘‘five letters’’ of Baire, Borel, Lebesgue, and Hadamard.

One of the interesting features of the book is that Gray not only gathers all the data, but valiantly tries to impose some kind of coherent order to the sprawling assemblage of ideas. But this is not an easy task, and some of the biggest and most obvious questions are ultimately left unanswered. For example, what is the connection between mathematical modernism and the aesthetic modernist movement? Did the latter influence the former, or can they be traced to a common cause? On page 8, Gray suggests that the biological model of ‘‘convergent evolution,’’ whereby unrelated species develop similar features in response to similar environmental pressures, may be appropriate; but, he admits, ‘‘the common features in the present case are hard to discern.’’ Is mathematical change driven by internal values and problems, or by a broader philosophical understanding of the nature and goals of the practice? The book offers evidence of both. What, in the end, are we to make of the modernist transformation? Was it ultimately a good thing? The tone of the book is generally positive—the transformation generally comes across as a march toward progress rather than a descent into meaningless abstraction—but the history itself, full of heated debates and opposing viewpoints, reminds us that with any change there are both gains and losses, and that the issues are not so clear cut. By the end of the book, one begins to suspect that there are no easy answers to the big questions. At best, we can make some progress by delimiting their scope, and the answers we get will be sensitive to how this is done. Although Gray is to be commended for not oversimplifying, this conclusion might come as a bit of a let-down to

those of us who like to see all our plot lines tied up in a neat package at the end, as in a murder mystery or a Victorian novel. But there is a wealth of valuable data here which, if not fully processed and pigeonholed, is at least tagged and cataloged in a helpful way. Plato’s ghost provides an insightful and informative resource for anyone doing mathematics today who has wondered how (and perhaps why) the subject has come to possess the features it has today. The book provides a lot to think about, which is exactly what a good history should do. By now, we have seen postmodern philosophy and postmodern art. Should we expect to see postmodern mathematics any time soon? Who knows—but if it comes to pass, we can only hope that Gray, or someone of his breadth and insight, will be around to help us make sense of what has occurred. ACKNOWLEDGMENTS

I am grateful to Spencer Breiner, Edward Dean, Solomon Feferman, Jose´ Ferreiro´s, and Paolo Mancosu for comments and corrections.

Department of Philosophy Carnegie Mellon University Pittsburgh, PA 15213 USA e-mail: [email protected]


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Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2003-2004 The Great Trigonometrical Survey This survey, extending from the southernmost tip of India to the foot of the Himalayas, was a major undertaking throughout much of the nineteenth century. Of great scientific value, it produced accurate measurements of the size of the Earth, the curvature of the Earth’s surface, and the height of Mount Everest.

Jantar Mantar Jantar Mantar (‘calculation instrument’) is an observatory in Jaipur, built around 1730 by the Maharajah Jai Singh II. Consisting of 14 massive geometrical instruments for measuring astronomical data, it includes the 90-foot Samrat Yantra, the world’s largest sundial. There are also Jantar Mantars in Delhi and other Indian cities.

Jorge Juan (1713–1773) Juan, mathematician and astronomer, received his scientific training at the Spanish military school for naval officers. In the 1730s, he joined La Condamine’s geodesic mission to Peru to take scientific measurements that showed the Earth

to be flattened at the poles (as Newton predicted), rather than elongated (as claimed by Descartes).

Mersenne Primes A Mersenne prime is a prime number of the form 2n - 1. Only 47 of them are known, the largest being 243,112,609 - 1, which has 12,978,189 digits. The one featured here, discovered in 2001, had about 4 million digits. They are named after the Minimite friar Marin Mersenne (1588– 1648), who found nine of them.

Jańos Andra´s Segner (1704–1777) Segner was born in Pressburg/Pozsony in Hungary (now Bratislava in Slovakia) and trained in medicine, physics and mathematics. He proved Descartes’s rule of signs, invented hydraulic turbines, introduced the axes of inertia of a solid body, and was the first professor of mathematics at Go¨ttingen University. A lunar crater is named after him.

John von Neumann (1903–1957) Born in Budapest, von Neumann spent most of his career at the Princeton Institute for Advanced Study. His enormous range of interests included set theory, Hilbert spaces, rings of operators, the theory of games, mathematical physics and automata theory. Often called ‘the father of modern computing’, he introduced stored programs to the subject. He was a consultant on the construction of the atomic bomb in Los Alamos.

Jorge Juan Great Trigonometrical Survey

Mersenne primes Jantar Mantar

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Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

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Jańos Segner

John von Neumann

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