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Letter to the Editors

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.

Opinion Survey: The Best Mathematical Books of the Twentieth Century

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n The Mathematical Intelligencer, vol. 29 (2007), no. 1, I asked for suggestions for the best mathematical books of the twentieth century. I am very grateful to the seven readers who responded. There was widespread disagreement about which of my two categories—essentially ‘‘academic’’ mathematics and what I have now learned to call ‘‘paramathematics’’—the books fell into, so I have abandoned the distinction.

The table below shows the books that received more than one vote. Despite the small sample size, we have an interestingly varied list and a clear winner. If any readers (whether or not they are members of the van der Waerden fan club) feel sufficiently inspired or irritated by the results to e-mail me some more suggestions, I will happily incorporate their ideas and write another letter to the editors. Eric Grunwald Mathematical Capital 187 Sheen Lane, London SW14 8LE United Kingdom e-mail: [email protected]

Votes cast

Book

4

B. van der Waerden, ‘‘Modern Algebra’’

3

N. Wiener, ‘‘Cybernetics’’

2

E. T. Bell, ‘‘Men of Mathematics’’

2

N. Bourbaki, ‘‘E´le´ments de Mathe´matique’’

2

H. Cartan, S. Eilenberg, ‘‘Homological Algebra’’

2

W. Feller, ‘‘An Introduction to Probability and its Applications’’

2

A. Grothendieck, ‘‘E´le´ments de Geómetrie Alge´brique’’

2

D. R. Hofstadter, ‘‘Go¨del, Escher, Bach: an Eternal Golden Braid’’

2

D. Knuth, ‘‘The Art of Computer Programming’’

2

B. Mandelbroit, ‘‘Les Objets Fractals’’

2

J. von Neumann, O. Morgenstern, ‘‘Theory of Games and Economic Behavior’’

2

R. Thom, ‘‘Stabilite´ Structurelle et Morphogeńe`se’’

2

E. Whittaker, G. Watson, ‘‘A Course of Modern Analysis’’

Ó 2008 Springer Science+Business Media, LLC., Volume 31, Number 1, 2009

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Mathematically Bent

Colin Adams, Editor

Riot at the Calc Exam 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.

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

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here had been a lot of unrest in the classroom all semester. To a certain extent, I was to blame. I decided right at the beginning of the course not to waste any time. So the first day, I introduced the triple integral. It was quite a shock for students who had yet to see a single integral. But that’s what I wanted. Shock therapy. Shock calculus. Embed an idea so deeply in their brains it would never get out. Brand their brains with a hot branding iron that said $$$. That first day, the smell of burning brain matter was overwhelming. Slack-jawed students with bulging eyes gaped in disbelief and horror. Several flipped frantically through their class schedules to check if they were in the right room. I hit them with a Fourier series to get their attention. By the end of that first day, they looked like the morning after an all night dorm party of the ‘‘Oktoberfest Meets Mardi Gras’’ variety; the bowed heads, the bloodshot eyes, the looks of nausea and anguish. ‘‘It’s no use sobbing,’’ I told one woman as she staggered out of the classroom. ‘‘Either you can do it or you can’t. Toughen up or get stomped all over.’’ Then she really started bawling. But it’s the truth. Knowledge isn’t for the faint of heart. There is some nasty knowledge out there. You have to be able to take it. The citadel of learning isn’t for the lily-livered. By the next class at least half of the livers were gone. It was a big improvement; we were down to a much more workable group. Now I could practice the one-on-one intimidation at which I excelled.

THE MATHEMATICAL INTELLIGENCER Ó 2008 Springer Science+Business Media, LLC.

We covered about 75 years of mathematics that day, from about 1837 to 1912. But I still wanted three more students to drop. Any more than that and I would fall below the 18 student minimum for the course to run. So I gave them a pop quiz on complex analytic functors over quaternionic tangent bundles. At the end of the 15 minutes, 13 students rushed up with drop slips for me to sign. I signed three. The first few weeks of the semester, I didn’t assign any homework. Why should I collect assignments that I had no intention of grading? It would just clutter up my office. But the students started to get nervous. Homework began to appear in my mailbox, problems from sections in the text that we were supposed to be covering. It infuriated me, this unsolicited work. I would return it the next class without having looked at it, only to find it in my mailbox again several hours later. So, finally, I was forced to assign and collect homework. I would slash over it with red magic markers and crayons at random. Then I would put big red Fs at the top of each and write things like, ‘‘Have you considered a job at McDonald’s?’’ And so, we settled into the routine of the semester. Every few weeks, I would call the registrar to insist that the classroom was inadequate and had to be changed. It was a quick way to ensure that the number of students present was kept to a reasonable size. I knew that those few who really wanted to learn would find out where the class had gone. Optimal teaching conditions occur when the number of students per faculty is kept to the minimum possible. Logically, then, my goal was to lose them all. But students in pursuit of passing grades are not easily shaken. Every time the class moved, at least a few would show up. Usually, though, the search for the new classroom made them late, and if there is one thing I will not abide, it is a lack of respect for

time. At exactly 10:00, I would lock the door to the classroom. No matter how desperately the late students banged on the door, whimpering and pleading, I wouldn’t let them in. You have to learn discipline before you can learn mathematics. There was one student named Wattle who was consistently late. He didn’t even bother coming to the door. Each class meeting, at about 10:05, his head would appear at the window of whatever classroom we were in. He would stand outside and take notes for the entire hour. I considered having the class moved to the second floor, but after a while I became almost fond of that head bobbing outside the window. As the semester progressed, students began to hound me. I could no longer go to my office during my office hours for fear of running into them. It got to where I would find them milling around my door at all hours of the day and night, wearing those sad hangdog looks, asking me where the class had moved to and what were the assignments. The students, they just want to suck you dry. About the eighth week, I lost control of the class. The balance of power in a classroom is always tenuous, and keeping the respect and awe of the students requires a delicate hand. I realized I had lost that control when they began to catch the blackboard erasers I hurled at them and fling them back. The student–teacher relationship was breaking down. Things deteriorated quickly. When I strolled across campus, I was hit by too many Frisbees.Ò Of course, anyone strolling across a college campus has to expect to be hit by one or two Frisbees. It’s part of being a pedestrian in a youth-encumbered environment. But I was averaging nearer to 10 or 12

Frisbees, a couple of footballs, and a baseball or two. Those baseballs can really sting. And then, one day, I was late to class. I had been debating the correct definition of the word ‘‘tangential’’ with a colleague and, in my excitement, I lost track of the time. I arrived at the classroom at 10:05 and the door was locked. I could hear snickering inside. That was when I decided I had had enough. Wattle never saw me coming. I grabbed him in a headlock and said through the window, ‘‘Unlock it or Wattle arrives in hell earlier than expected.’’ There was some debate, as Wattle was not particularly popular on campus. But as he began to gurgle and wave his arms dramatically, they opened the door. From then on, the students left me alone. They kept their distance. The few who continued to come to class usually sat in the back and seemed inordinately skittish. On the day of the final, I passed out the exam. I was pretty pleased with myself. Instead of having to make up two exams, one for the graduate course I was teaching and one for the calculus course, I used the graduate exam for both. Perhaps that was unfair, but there are a lot of constraints on a faculty member’s time. One has to weigh all one’s responsibilities. In this case, making up the calc exam was outweighed by the Barbara Mandrell special on TV. Unfortunately, Wattle noticed that the exam was for the graduate course. I had forgotten to change the course number in the upper right hand corner of the front page. He stood up. ‘‘That’s it. We don’t have to take this anymore.’’ He picked up the exam and crumpled it in his fist. A cheer went up.

‘‘Sit down, Wattle. Sit down or you flunk,’’ I yelled. He took the balled-up exam and hurled it at me. Suddenly, all the students were up, screaming and yelling. Crumpled exams flew everywhere. A chair sailed across the room. I ran for the door. When the police arrived, two students were wrapping the cord from the overhead projector around my neck. Quite frankly, I am lucky to be alive. I have only the students’ penchant for alcohol and their resultant slowed motor skills to thank for my survival. At the trial, I explained how the students, disgruntled with the downward spiral in their grades, had decided to take action. But rather than hitting the books and working diligently to improve their minds, they chose instead to murder the professor, and, thereby, prevent the distribution of grades. It was an unsuccessful attempt at cold-blooded premeditated first-degree murder. I was particularly eloquent. The judge sentenced them all to 10 to 12 years in one of America’s oldest educational facilities, the Federal Penitentiary in Leavenworth, Kansas. It was difficult for me to decide on the grades that they deserved. I considered petitioning the college to introduce a new grade, Z, which would essentially mean, ‘‘This student did so abysmally in the course that they deserve to die, but we did the next best thing. We had them locked away.’’ In the end I gave them all Fs except for Wattle. In a moment of weakness, I gave him a D-. I can imagine exactly how his head must look, bobbing behind the barred window at Leavenworth. My next semester starts in a week. It is clear that I will have to run my classes more strictly this time. No more Mr. Nice Guy.


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Mathematics Is Not a Game But... ROBERT THOMAS

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s a mathematician I began to take an interest in philosophy of mathematics on account of my resentment at the incomprehensible notion I encountered that mathematics was ‘a game played with meaningless symbols on paper’ —not a quotation to be attributed to anyone in particular, but a notion that was around before Hilbert [1]. Various elements of this notion are false and some are also offensive. Mathematical effort, especially in recent decades—and the funding of it—indicate as clearly and concretely as is possible that mathematics is a serious scientific-type activity pursued by tens of thousands of persons at a professional level. While a few games may be pursued seriously by many and lucratively by a professional few, no one claims spectator sports are like mathematics. At the other end of the notion, paper is inessential, merely helpful to the memory. Communication (which is what the paper might hint at) is essential; our grip on the objectivity of mathematics depends on our being able to communicate our ideas effectively. Turning to the more offensive aspects of the notion, we think often of competition when we think of games, and in mathematics one has no opponent. Such competitors as there are are not opponents. Worst of all is the meaninglessness attributed to the paradigm of clear meaning; what could be clearer than 2 + 2 = 4? Is this game idea not irredeemably outrageous? Yes, it is outrageous, but there is within it a kernel of useful insight that is often obscured by outrage at the main notion, which is not often advocated presumably for that reason. I know of no one that claims that mathematics is a game or bunch of games. The main advocate of the idea that doing mathematics is like playing a game is David Wells [3]. It is the purpose of this essay to point to the obscured kernel of insight.

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Mathematics Is Not a Game Mathematics is not a collection of games, but perhaps it is somehow like games, as written mathematics is somehow like narrative. I became persuaded of the merit of some comparison with games in two stages, during one of which I noticed a further fault with the notion itself: there are no meaningless games. Meaningless activities such as tics and obsessions are not games, and no one mistakes them for games. Meanings in games are internal, not having to do with reference to things outside the game (as electrons, for example, in physics are supposed to refer to electrons in the world). The kings and queens of chess would not become outdated if all nations were republics. The ‘meaningless’ aspect of the mathematics-as-game notion is self-contradictory; it might be interesting to know how it got into it and why it stayed so long. Taking it as given then that games are meaningful to their players and often to spectators, how are mathematical activities like game-playing activities? The first stage of winning me over to a toleration of this comparison came in my study of the comparison with narrative [4]. One makes sense of narrative, whether fictional or factual, by a mental construction that is sometimes called the world of the story. Keeping in mind that the world of the story may be the real world at some other time or right now in some other place, one sees that this imaginative effort is a standard way of understanding things that people say; it need have nothing at all to do with an intentionally creative imagining like writing fiction. In order to understand connected speech about concrete things, one imagines them. This is as obvious as it is unclear how we do it. We often say that we pretend that we are in the world of the story. This pretence is one way—and a very effective way—of indicating how we imagine what one of the persons we are hearing about can see or hear under the circumstances of the story. If I want to have some idea what a person in

certain circumstances can see, for example, I imagine myself in those circumstances and ask myself what I can see [5]. Pretending to be in those circumstances does not conflict with my certain knowledge that on the contrary I am listening to the news on my radio at home. This may make it a weak sort of pretence, but it is no less useful for that. The capacity to do this is of some importance. It encourages empathy, but it also allows one to do mathematics. One can pretend what one likes and consider the consequences at any length, entirely without commitment. This is often fun, and it is a form of playing with ideas. Some element of this pretence is needed, it seems to me, in changing one’s response to ‘what is 2 + 2?’ from ‘2 + 2 what?’ to the less concrete ‘four’ [6]. This ludic aspect of mathematics is emphasized by Brian Rotman in his semiotic analysis of mathematics [7] and acknowledged by David Wells in his comparison of mathematics and games. Admitting this was the first stage of my coming to terms with games. The ludic aspect is something that undergraduates, many of whom have decided that mathematics is either a guessing game (a bad comparison of mathematics and games) or the execution of rigidly defined procedures, need to be encouraged to do when they are learning new ideas. They need to fool around with them to become familiar with them. Changing the parameters and seeing what a function looks like with that variety of parameter values is a good way to learn how the function behaves. And it is by no means only students that need to fool around with ideas in order to become familiar with them. Mathematical research involves a good deal of fooling around, which is part of why it is a pleasurable activity. This sort of play is the kind of play that Kendall Walton illustrates with the example of boys in woods not recently logged pretending that stumps are bears [8]. This is not competitive, just imaginative fooling around. I do not think that this real and fairly widely acknowledged—at least never denied—aspect of mathematics has much to do with the canard with which I began. The canard is a reductionistic attack on mathematics, for it says it is ‘nothing but’ something it is not: the standard

AUTHOR

......................................................................... ROBERT THOMAS was an undergraduate

at the University of Toronto when Chandler Davis arrived there and obliged him to get acquainted with quantifiers among other things. He later studied also at Waterloo and Southampton. He has been at the University of Manitoba since 1970. He is editor of Philosophia Mathematica (www.philmat.oxfordjournals.org). A new hobby is grandfathering. St. John’s College and Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 e-mail: [email protected]

reductionist tactic. In my opinion, mathematics is an objective science, but a slightly strange one on account of its subtle subject matter; in some hands it is also an art [9]. Having discussed this recently at some length [10], I do not propose to say anything about what mathematics is here, but to continue with what mathematics is like; because such comparisons, like that with narrative, are instructive and sometimes philosophically interesting. The serious comparison of mathematics with games is due in my experience to David Wells, who has summed up what he has been saying on the matter for twenty years in a strange document, draft zero of a book or two called Mathematics and Abstract Games: An Intimate Connection [3]. Wells is no reductionist and does not think that mathematics is any sort of a game, meaningless or otherwise. He confines himself to the comparison (‘like a collection of abstract games’—p. 7, a section on differences—pp. 45-51), and I found this helpful in the second stage of my seeking insight in the comparison. But I did not find Wells’s direct comparison as helpful as I hope to make my own, which builds on his with the intent of making it more comprehensible and attractive (cf. my opening sentence).

Doing Mathematics Is Not Like Playing a Game Depending on when one thinks the activities of our intellectual ancestors began to include what we acknowledge as mathematics, one may or may not include as mathematics the thoughts lost forever of those persons with the cuneiform tablets on which they solved equations. The tablets themselves indicate procedures for solving those particular equations. Just keeping track of quantities of all sorts of things obviously extended still farther back, to something we would not recognize as mathematics but which gave rise to arithmetic. Keeping track of some of the many things that one cannot count presumably gave rise to geometrical ideas. It does seem undeniable that such procedural elements are the historical if not the logical basis of mathematics, and not only in the Near East but also in India and China. I do not see how mathematics could arise without such pre-existing procedures and reflections on them— probably written down, for it is so much easier to reflect on what is written down. This consideration of procedures, and of course their raw material and results, is of great importance to my comparison of mathematics and games because my comparison is not between playing games and doing mathematics. I am taking mathematics to be the sophisticated activity that is the subject matter of philosophy of mathematics and research in mathematics. I do not mean actions such as adding up columns of figures. Mathematics is not even those more complicated actions that we are happy to transfer to computers. Mathematics is what we want to keep for ourselves. When playing games, we stick to the rules (or we are changing the game being played), but when doing serious mathematics (not executing algorithms) we make up the rules—definitions, axioms, and some of us even logics. As Wells points out in the section of his book on differences between games and mathematics, in arithmetic we find prime numbers, which are a whole new ‘game’ in themselves (metaphorically speaking). Ó 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

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While mathematics requires reflection on pre-existing procedures, reflection on procedures does not become recognizable as mathematics until the reflection has become sufficiently communicable to be convincing. Conviction of something is a feeling, and so it can occur without communication and without verbalizing or symbolizing. But to convince someone else of something, we need to communicate, and that does seem to be an essential feature of mathematics, whether anything is written down or not— a fortiori whether anything is symbolic. And of course convincing argument is proof. The analogy with games that I accept is based on the possibility of convincing argument about abstract games. Anyone knowing the rules of chess can be convinced that a move has certain consequences. Such argument does not follow the rules of chess or any other rules, but it is based on the rules of chess in a way different from the way it is based on the rules of logic that it might obey. To discuss the analogy of this with mathematics, I think it may be useful to call upon two ways of talking about mathematics, those of Philip Kitcher and of Brian Rotman.

Ideal Agents In his book The Nature of Mathematical Knowledge [11], Kitcher introduced a theoretical device he called the ideal agent. ‘We can conceive of the principles of [empirical] Arithmetic as implicit definitions of an ideal agent. An ideal agent is a being whose physical operations of segregation do satisfy the principles [that allow the deduction in physical terms of the theorems of elementary arithmetic].’ (p. 117). No ontological commitment is given to the ideal agent; in this it is likened to an ideal gas. And for this reason we are able to ‘specify the capacities of the ideal agent by abstracting from the incidental limitations on our own collective practice’ (ibid.). The agent can do what we can do but can do it for collections however large, as we cannot. Thus modality is introduced without regard to human physical limitations. ‘Our geometrical statements can finally be understood as describing the performances of an ideal agent on ideal objects in an ideal space.’ (p. 124). Kitcher also alludes to the ‘double functioning of mathematical language—its use as a vehicle for the performance of mathematical operations as well as its reporting on those operations’ (p. 130). ‘To solve a problem is to discover a truth about mathematical operations, and to fiddle with the notation or to discern analogies in it is, on my account, to engage in those mathematical operations which one is attempting to characterize.’ (p. 131). Rotman is at pains to distinguish what he says from what Kitcher had written some years before the 1993 publication of The Ghost in Turing’s Machine [7] because he developed his theory independently and with different aims, but we readers can regard his apparatus as a refinement of Kitcher’s, for Rotman’s cast of characters includes an Agent to do the bidding of the character called the Subject. The Subject is Rotman’s idealization of the person that reads and writes mathematical text, and also the person that carries out some of the commands of the text. For example, it is the reader that obeys the command, ‘Consider triangle

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THE MATHEMATICAL INTELLIGENCER

ABC.’ But it is the Agent (p. 73) that carries out such commands as, ‘Drop a perpendicular from vertex A to the line BC,’ provided that the command is within the Agent’s capacities. We humans are well aware that we cannot draw straight lines; that is the work of the agents, Kitcher’s and Rotman’s. We reflect on the potential actions of these agents and address our reflections to other thinking Subjects. Rotman’s discussion of this is rich with details like the tenselessness of the commands to the Agent, indeed the complete lack of all indexicality in such texts. The tenselessness is an indication of how the Subject is an idealization as the Agent is, despite not being blessed with the supernatural powers of the Agent. The Agent, Rotman says, is like the person in a dream, the Subject like the person dreaming the dream, whereas in our normal state we real folk are more like the dreamer awake, what Rotman calls the Person to complete his semiotic hierarchy. Rotman then transfers the whole enterprise to the texts, so that mathematical statements are claims about what will result when certain operations are performed on signs (p. 77). We need not follow him there to appreciate the serviceability of his semiotic distinctions. The need for superhuman capacities was noted long ago in Frege’s ridicule [2] of the thought that mathematics is about empty symbols: ([...] we would need an infinitely long blackboard, an infinite supply of chalk, and an infinite length of time— p. 199, § 124). He also objected to a comparison to chess for Thomae’s formal theory of numbers, while admitting that ‘there can be theorems in a theory of chess’ (p. 168, § 93, my emphasis). According to Frege, The distinction between the game itself and its theory, not drawn by Thomae, makes an essential contribution towards our understanding of the matter. [...] in the theory of chess it is not the chess pieces which are actually investigated; it is a question of the rules and their consequences. (pp. 168-169, § 93)

The Analogies Between Mathematics and Games Having at our disposal the superhuman agents of Kitcher and Rotman, we are in a position to see what is analogous between mathematics and games. It is not playing the game that is analogous to mathematics, but our reflection in the role of subject on the playing of the game, which is done by the agent. When a column of figures is added up, we do it, and sometimes when the product of two elements of a group is required, we calculate it; but mathematics in the sense I am using here is not such mechanical processes at all, but the investigation of their possibility, impossibility, and results. For that highly sophisticated reflective mathematical activity, the agent does the work because the agent can draw straight lines. Whether points are collinear depends on whether they are on the agent’s straight lines, not on whether they appear on the line in our sketch. We can put them on or off the line at will; the agent’s results are constrained by the rules of the system in which the agent is

working. Typically we have to deduce whether the agent’s line is through a point or not. The agent, ‘playing the game’ according to the rules, gets the line through the point or not, but we have to figure it out. We can figure it out; the agent just does it. The analogy to games is two-fold. 1. The agent’s mathematical activity (not playing a game) is analogous to the activity of playing a game like chess where it is clear what is possible and what is impossible—the same for every player—often superhuman but bound by rules. (Games like tennis depend for what is possible on physical skill, which has no pertinence here.) 2. Our mathematical activity is analogous to (a) game invention and development, (b) the reflection on the playing of a game like chess that distinguishes expert play from novice play, or (c) consideration of matters of play for their intrinsic interest apart from playing any particular match—merely human but not bound by rules. It is we that deduce; the agent just does what it is told, provided that it is within the rules we have chosen. Analogous to the hypotheses of our theorems are chess positions, about which it is possible to reason as dependably as in mathematics because the structure is sufficiently precisely set out that everyone who knows the rules can see what statements about chess positions are legitimate and what are not. Chains of reasoning can be as long as we like without degenerating into the vagueness that plagues chains of reasoning about the real world. The ability to make and depend on such chains of reasoning in chess and other games is the ability that we need to make such chains in mathematics, as David Wells points out. To obtain a useful analogy here, it is necessary to rise above the agent in the mathematics and the mere physical player in the game, but the useful analogy is dependent on the positions in the game and the relations in the mathematics. The reflection in the game is about positions more than the play, and the mathematics is about relations and their possibility more than drawing circles or taking compact closures. Certainly the physical pieces used in chess and the symbols on paper are some distance below what is importantly going on. I hope that the previous discussion makes clear why some rules are necessary to the analogy despite the fact that we are not bound by those rules. The rules are essential because we could not do what we do without them, but it is the agent that is bound by them. We are talking about, as it were, what a particular choice of them does and does not allow. But our own activity is not bound by rules; we can say anything that conveys our meaning, anything that is convincing to others. Here is objectivity without objects. Chess reasoning is not dependent upon chess boards and chess men; it is dependent on the relations of positions mandated by the rules of the game of chess. Mathematics is not dependent on symbols (although they are as handy as chess sets) but on the relations of whatever we imagine the agent to work on, specified and reasoned about. Our conclusions are right or wrong as plainly as if we were ideal agents loose in

Plato’s heaven, but they are right or wrong dependent on what the axioms, conventions, or procedures we have chosen dictate. Outside mathematics, we reason routinely about what does not exist, most particularly about the future. As the novelist Jim Crace was quoted on page R10 of the 2007 6 2 Toronto Globe and Mail, ‘As a good Darwinist, I know that what doesn’t confer an advantage dies out. One advantage [of narrative (Globe and Mail addition)] is that it enables us to play out the bad things that might happen to us and to rehearse what we might do.’ In order to tell our own stories, it is essential to project them hypothetically into the future based on observations and assumptions about the present. At its simplest and most certain, the skill involved is what allows one to note that if one moves this pawn forward one square the opponent’s pawn can take it. It’s about possibilities and of course impossibilities, all of them hypothetical. It is this fundamental skill that is used both in reflection on games and in mathematics to see what is necessary in their respective worlds. I must make clear that David Wells thinks that entities in maths and abstract games have the same epistemological status but that doing mathematics is like (an expert’s) playing a game in several crucial respects, no more; he disagrees with the usefulness of bringing in ideal agents, indeed opposes doing so, apparently not seeing the advantage of splitting the analogy into the two numbered aspects above. This section is my attempt to outline a different but acceptable game analogy—a game-analysis analogy.

Conclusion Games such as bridge and backgammon, which certainly involve strategy, have a stochastic element that prevents long chains of reasoning from being as useful as they are in chess. Such chains are, after all, an important part of how computers play chess. The probabilistic mathematics advocated by Doron Zeilberger [12] is analogous to the analysis of such a stochastic game, and will be shunned by those uninterested in such analysis of something in which they see nothing stochastic. Classical (von Neumann) game theory, on the other hand, actually is the analysis of situations that are called games and do involve strategy. The game theory of that current Princeton genius, John Conway, is likewise the actual analysis of game situations [13]. Does the existence of such mathematical analysis count for or against the general analogy between mathematics and game analysis? On my version of the analogy, to identify mathematics with games would be one of those part-for-whole mistakes (like ‘all geometry is projective geometry’ or ‘arithmetic is just logic’ from the nineteenth century); but identification is not the issue. It seems to me that my separation of game analysis from playing games tells in favour of the analogy of mathematics to analysis of games played by other—not necessarily superhuman—agents, and against the analogy of mathematics to the expert play of the game itself. This is not a question David Wells has discussed. For Wells, himself an expert at abstract games such as chess and go, play is expert play based unavoidably on analysis; analysis is just Ó 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

7

part of playing the game. Many are able to distinguish these activities, and not just hypothetically. One occasionally hears the question, is mathematics invented or discovered?—or an answer. As David Wells points out, even his game analogy shows why both answers and the answer ‘both’ are appropriate. Once a game is invented, the consequences are discovered—genuinely discovered, as it would require a divine intelligence to know just from the rules how a complex game could best be played. When in practice rules are changed, one makes adjustments that will not alter the consequences too drastically. Analogously, axioms are usually only adjusted and the altered consequences discovered. What use can one make of this analogy? One use that one cannot make of it is as a stick to beat philosophers into admitting that mathematics is not problematic. Like mathematicians, philosophers thrive on problems. Problems are the business of both mathematics and philosophy. Solving problems is the business of mathematics. If a philosopher came to regard the analogy as of some validity, then she would import into the hitherto unexamined territory of abstract games all of the philosophical problems concerning mathematics. Are chess positions real? How do we know about them? And so on; a new branch of philosophy would be invented. What use then can mathematicians make of the analogy? We can use it as comparatively unproblematic material in discussing mathematics with those nonphilosophers desiring to understand mathematics better. I have tried to indicate above some of the ways in which the analogy is both apt and of sufficient complexity to be interesting; it is no simple metaphor but can stand some exploration. Some of this exploration has been carried out by David Wells, to whose work I need to refer the reader.

[2] Frege, Gottlob. ‘Frege against the formalists’ from Grundgesetze der Arithmetik, Jena: H. Pohle. Vol. 2, Sections 86-137, in Translations from the philosophical writings of Gottlob Frege. 3rd ed. Peter Geach and Max Black, eds. Oxford: Blackwell, 1980. I am grateful to a referee for pointing this out to me. [3] Wells, David. Mathematics and Abstract Games: An Intimate Connection. London: Rain Press, 2007. (Address: 27 Cedar Lodge, Exeter Road, London NW2 3UL, U.K. Price: £10; $20 including surface postage.) [4] Thomas, Robert. ‘Mathematics and Narrative’. The Mathematical Intelligencer 24 (2002), 3, 43–46 [5] O’Neill, Daniella K., and Rebecca M. Shultis. ‘The emergence of the ability to track a character’s mental perspective in narrative’, Developmental Psychology, 43 (2007), 1032–1037. [6] Donaldson, Margaret. Human Minds, London: Penguin, 1993. [7] Rotman, Brian. Ad Infinitum: The Ghost in Turing’s Machine. Stanford: Stanford University Press, 1993. [8] Walton, Kendall L. Mimesis as Make-Believe: On the Foundations of the Representational Arts. Cambridge, Mass.: Harvard University Press, 1990. Also Currie, Gregory. The Nature of Fiction. Cambridge: Cambridge University Press, 1990. [9] Davis, Chandler, and Erich W. Ellers. The Coxeter Legacy: Reflections and Projections. Providence, R. I.: American Mathematical Society, 2006. [10] Thomas, Robert. ‘Extreme Science: Mathematics as the Science of Relations as such,’ in Proof and Other Dilemmas: Mathematics and Philosophy, Ed. Bonnie Gold and Roger Simons, Washington, D.C.: Mathematical Association of America, 2008. [11] Kitcher, Philip. The Nature of Mathematical Knowledge. New York: Oxford University Press, 1984. [12] Zeilberger, Doron. ‘Theorems for a price: Tomorrow’s semi-rigorous mathematical culture’. Notices of the Amer. Math. Soc. 40 (1993), 978-981. Reprinted in The Mathematical Intelligencer 16 (1994), 4, 11-14. [13] Conway, John H. On Numbers and Games. 2nd ed. Natick,

REFERENCES

[1] Thomae, Johannes. Elementare Theorie der analytischen Functionen einer complexen Vera¨nderlichen. 2nd ed. Halle: L. Nebert, 1898 (1st ed. 1880), ridiculed by Frege [2].

8


Mass.: AK Peters, 2001. (1st ed. LMS Monographs; 6. London: Academic Press, 1976). Also Berlekamp, E.R., J.H. Conway, and R.K. Guy. Winning Ways for your Mathematical Plays. 2 volumes. London: Academic Press, 1982.

+

+

=

Formulas of Brion, Lawrence, and Varchenko on Rational Generating Functions for Cones MATTHIAS BECK, CHRISTIAN HAASE,

O

ur aim is to illustrate two gems of discrete geometry, namely formulas of Michel Brion [7] and of James Lawrence [15] and Alexander N. Varchenko [16], which at first sight seem hard to believe, and which— even after some years of studying them—still provoke a slight feeling of mystery in us. Let us start with some examples. Suppose we would like to list all positive integers. Although there are many, we may list them compactly in the form of a generating function: X x xk ¼ : ð1Þ x1 þ x2 þ x3 þ ¼ 1 x k[0

Let us list, in a similar way, all integers less than or equal to 5: X xk þ x 1 þ x 0 þ x 1 þ x 2 þ x 3 þ x 4 þ x 5 ¼ k5 5

x ¼ : 1 x 1

ð2Þ

AND

FRANK SOTTILE

Adding the two rational function right-hand sides leads to a miraculous cancellation x x5 x x6 x x6 ¼ þ þ ¼ 1 1x 1x 1x x1 1x 2

3

4

ð3Þ

5

¼xþx þx þx þx : This sum of rational functions representing two infinite series collapses into a polynomial representing a finite series. This is a one-dimensional instance of a theorem due to Michel Brion. We can think of (1) as a function listing the integer points in the ray [1,?) and of (2) as a function listing the integer points in the ray (-?,5]. The respective rational generating functions add up to the polynomial (3) that lists the integer points in the interval [1, 5]. Here is a picture of this arithmetic. + =

1

2

3

4

5

2008 Springer Science+Business Media, LLC., Volume 31, Number 1, 2009

9

Let us move up one dimension. Consider the quadrilateral Q with vertices (0, 0), (2, 0), (4, 2), and (0, 2).

1 y2 þ ð1 xÞð1 yÞ ð1 xÞð1 y 1 Þ þ

x 4 y2 x2 þ ð1 x 1 Þð1 x 1 y1 Þ ð1 xyÞð1 x 1 Þ

¼ y2 þ xy 2 þ x 2 y 2 þ x 3 y2 þ x 4 y2 þ y þ xy þ x 2 y þ x 3 y þ 1 þ x þ x2:

Analogous to the generating functions (1) and (2) are the generating functions of the cones at each vertex generated by the edges at that vertex. For example, the two edges touching the origin generate the nonnegative quadrant, which has the generating function X X X 1 1 : x m yn ¼ xm yn ¼ ð1 xÞ ð1 yÞ m;n 0 m0 n0 The two edges incident to (0, 2) generate the cone ð0; 2Þ þ R 0 ð0; 2Þ þ R 0 ð4; 0Þ; with the generating function X

x m yn ¼

m 0;n 2

2

y : ð1 xÞð1 y1 Þ

The third such vertex cone, at (4, 2), is ð4; 2Þ þ R 0 ð4; 0Þ þ R 0 ð2; 2Þ, which has the generating function ð1

x4 y2 : x 1 y 1 Þ

x 1 Þð1

Finally, the fourth vertex cone is ð2; 0Þ þ R 0 ð2; 2Þ þ R 0 ð2; 0Þ; with the generating function x2 : ð1 xyÞð1 x 1 Þ Inspired by our one-dimensional example above, we add those four rational functions:

The sum of rational functions again collapses to a polynomial, which encodes precisely those integer points that are contained in the quadrilateral Q: Brion’s Theorem says that this magic happens for any polytope P in any dimension d; provided that P has rational vertices. (More precisely, the edges of P have rational directions.) The vertex cone Kv at vertex v is the cone with apex v and generators the edge directions emanating from v. The generating function X xm rKv ðxÞ :¼ m2Kv \Zd

for such a cone is a rational function (again, provided that P has rational vertices). Here we abbreviate x m for x1m1 x2m2 xdmd : Brion’s Formula says that the rational functions representing the integer points in each vertex cone sum up to the polynomial rP ðxÞ encoding the integer points in P: X rKv ðxÞ: rP ðxÞ ¼ v a vertex of P

A second theorem, which shows a similar collapse of generating functions of cones, is due (independently) to James Lawrence and to Alexander Varchenko. We illustrate it with the example of the quadrilateral Q: Choose a direction vector n that is not perpendicular to any edge of Q; for example we could take n = (2, 1). Now at each vertex v of Q; we form a (not necessarily closed) cone generated by the edge directions m as follows. If w n [ 0,

AUTHORS

......................................................................................................................................................... MATTHIAS BECK was an undergraduate

at Wu¨rzburg, Germany, where he also had a brief sideline as a street musician. He received his Ph.D. from Temple University. Before coming to San Francisco State University, he held postdoctoral positions at SUNY Binghamton, MSRI, and the Max Planck Institute. His research is in discrete and combinatorial geometry and number theory—and more particularly in enumerating integer points in polyhedra. Department of Mathematics San Francisco State University San Francisco, CA 94132, USA e-mail: [email protected] URL: http://math.sfsu.edu/beck

10


CHRISTIAN HAASE was born and raised in Berlin. After a respectable apprenticeship as an algebraic topologist, he ventured into polyhedral geometry, presaging his slide into lattice-point addiction (2000–date). At Berkeley and Duke he also contracted a serious case of algebraic geometry. Since 2005 he is in Berlin with an Emmy Noether Award from the German Research Foundation.

Fachbereich Mathematik & Informatik Freie Universita¨t Berlin, 14195 Berlin, Germany e-mail: [email protected] URL: http://erhart.math.fu-berlin.de

then we take its nonnegative span, and if w n \ 0, we take its strictly negative span.

For example, the edge directions at the origin are along the positive axes, and so this cone is again the nonnegative quadrant. At the vertex (2, 0) the edge directions are (-2, 0) and (2, 2). The first has negative dot product with n and the second has positive dot product, and so we obtain the half-open cone ð2; 0Þ þ R\0 ð2; 0Þ þ R 0 ð2; 2Þ ¼ ð2; 0Þ þ R [ 0 ð2; 0Þ þ R 0 ð2; 2Þ: At the vertex (4, 2) both edge directions have negative dot product with n and we get the open cone ð4; 2Þ þ R [ 0 ð0; 4Þ þ R [ 0 ð2; 2Þ; and at the vertex (0, 2) we get the half-open cone ð0; 2Þ þ R 0 ð2; 0Þ þ R [ 0 ð0; 2Þ: The respective generating functions turn out to be 1 x3 ; ; ð1 xÞð1 yÞ ð1 xÞð1 xyÞ 6 3

x y ; ð1 xyÞð1 yÞ

3

and

y : ð1 xÞð1 yÞ

Now we add them with signs according to the parity of the number of negative (w n \ 0) edge directions w at the vertex. In our example, we obtain 1 x3 ð1 xÞð1 yÞ ð1 xÞð1 xyÞ þ

x 6 y3 y3 ð1 xyÞð1 yÞ ð1 xÞð1 yÞ

¼ y2 þ xy 2 þ x 2 y2 þ x 3 y2 þ x 4 y 2 þ y þ xy þ x 2 y þ x 3 y þ 1 þ x þ x2 :

......................................................................... FRANK SOTTILE earned his Ph.D. at the University of Chicago in 1994. After appointments at the University of Toronto, the University of Massachusetts Amherst, and elsewhere, he landed at Texas A&M in 2004. His research interests include real algebraic geometry, Schubert calculus, and geometric combinatorics. He was a member of the editorial board of the Young Mathematicians Network 1994–1999.

Department of Mathematics Texas A&M University, College Station TX 77843, USA e-mail: [email protected] URL: http://www.math.tamu.edu/~sottile

This sum of rational functions again collapses to the polynomial that encodes the integer points in Q: This should be clear here, for the integer points in the nonnegative quadrant are counted with a sign ±, depending upon the cone in which they lie, and these coefficients cancel except for the integer points in the polytope Q: The identity illustrated by this example works for any simple polyope—a d-polytope where every vertex meets exactly d edges. Given a simple polytope, choose a direction vector n 2 Rd that is not perpendicular to any edge direction. Let Evþ ðnÞ be the edge directions w at a vertex v with w n [ 0, and Ev ðnÞ be those with w n \ 0. Define the cone X X R 0w þ R\0 w: Kn;v :¼ v þ w2Evþ ðnÞ

w2Ev ðnÞ

This is the analogue of the cones in our previous example. The Lawrence–Varchenko Formula says that adding the rational functions of these cones with appropriate signs gives the polynomial rP ðxÞ encoding the integer points in P: X ð1ÞjEv ðnÞj rKn;v ðxÞ: rP ðxÞ ¼ v a vertex of P

Here, rKn;v ðxÞ is the generating function encoding the integer points in the cone Kn;v : An interesting feature of this identity, which also distinguishes it from Brion’s Formula, is that the power series generating functions have a common region of convergence. Also, it holds without any restriction that the polytope be rational. In the general case, the generating functions of the cones are holomorphic functions, which we can add, as they have a common domain (the common region of convergence).

Proofs Brion’s original proof of his formula [7] used the Lefschetz–Riemann–Roch theorem in equivariant K-theory [3] applied to a singular toric variety. Fortunately for us, the remarkable formulas of Brion and of Lawrence–Varchenko now have easy proofs, based on counting. Let us first consider an example based on the cone K ¼ R 0 ð0; 1Þ þ R 0 ð2; 1Þ: The open circles in the picture on the left in Figure 1 represent the semigroup Nð0; 1Þ þ Nð2; 1Þ; which is a proper subsemigroup of the integer points K \ Z2 in K: The picture on the right shows how translates of the fundamental half-open parallelepiped P by this subsemigroup cover K: This gives the formula X 1 þ xy rK ðxÞ ¼ rP ðxÞ x m ðx 2 yÞn ¼ ; ð1 xÞð1 x 2 yÞ m;n 0

Figure 1. Tiling a simple cone by translates of its

fundamental parallelepiped. 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

11

as the fundamental parallelepiped P contains two integer points, the origin and the point (1, 1). A simple rational cone in Rd has the form ( ) d d X X K :¼ v þ ki wi j ki 2 R 0 ¼ v þ R 0 wi ; i¼1

i¼1

where w1 ; . . .; wd 2 Z are linearly independent. This cone is tiled by the ðNw1 þ þ Nwd Þ-translates of the halfopen parallelepiped ( ) d X P :¼ v þ ki wi j 0 ki \1 : d

w1 ¼ ð1; 0; 1Þ; and

w2 ¼ ð0; 1; 1Þ;

w3 ¼ ð0; 1; 1Þ;

w4 ¼ ð1; 0; 1Þ:

If we let K1 be the simple cone with generators w1 ; w2 ; w3 ; and K2 be the simple cone with generators w2 ; w3 ; w4 ; then K1 and K2 decompose K into simple cones. If s ¼ ð18 ; 0; 13Þ; then (4) holds, and no facet of s þ K1 or of s þ K2 contains any integer points. We display these cones, together with their integer points having z-coordinate 0, 1, or 2.

i¼1

The generating function for P is the polynomial X xm ; rP ðxÞ ¼ m2P\Zd

and so the generating function for K is rK ðxÞ ¼

X

x a rP ðxÞ ¼

a2Nw1 þþNwd

rP ðxÞ ; ð1 x w1 Þ ð1 x wd Þ

which is a rational function. This formula and its proof do not require that the apex v be rational, but only that the generators wi of the cone be linearly independent vectors in Zd : A general rational cone K with apex v and generators w1 ; . . .; wn 2 Zd has the form K ¼ v þ R 0 w1 þ þ R 0 wn :

ð4Þ

and no facet of any cone s þ K1 ; . . .; s þ Kl contains any integer points. This gives the disjoint irrational decomposition K \ Zd ¼ ðs þ K1 Þ \ Zd t t ðs þ Kl Þ \ Zd ; and so rK ðxÞ ¼

X

xm ¼

m2K\Zd

l X

rsþKi ðxÞ

ð5Þ

i¼1

is a rational function. For example, suppose that K is the cone in R3 with apex the origin and generators 12


xz þ xz 2 ; ð1 yzÞð1 y1 zÞð1 xzÞ 1þz ; and rsþK2 ðxÞ ¼ ð1 yzÞð1 y1 zÞð1 x 1 zÞ ð1 þ zÞ2 ð1 zÞ rK ðxÞ ¼ : ð1 yzÞð1 y1 zÞð1 xzÞð1 x 1 zÞ

rsþK1 ðxÞ ¼

If there is a vector n 2 Rd with n wi [ 0 for i = 1,...,n, then K is strictly convex. A fundamental result on convexity [2, Lemma VIII.2.3] is that such a K may be decomposed into simple cones K1 ; . . .; Kl having pairwise disjoint interiors, each with apex v and generated by d of the generators w1 ; . . .; wn of K: We would like to add the generating functions for each cone Ki to obtain the generating function for K: However, some of the cones may have lattice points in common, and some device is needed to treat the subsequent overcounting. An elegant way to do this is to avoid the overcounting altogether by translating all the cones [5]. We explain this. There exists a short vector s 2 Rd such that K \ Zd ¼ ðs þ KÞ \ Zd ;

The cone s þ K1 contains the 5 magenta points shown with positive first coordinate, whereas s þ K2 contains the other displayed points. Their integer generating functions are

Then rsþK1 ðxÞ þ rsþK2 ðxÞ ¼ rK ðxÞ, as ðxz þ xz 2 Þð1 x 1 zÞ þ ð1 þ zÞð1 xzÞ ¼ 1 þ z z 2 z 3 ¼ ð1 þ zÞ2 ð1 zÞ: While the cones that appear in the Lawrence–Varchenko formula are all simple, and those in Brion’s formula are strictly convex, we use yet more general cones in their proof. A rational (closed) halfspace is the convex subset of Rd defined by fx 2 Rd j w x bg; where w 2 Zd and b 2 R: Its boundary is the rational hyperplane fx 2 Rd j w x ¼ bg: A (closed) cone K is the interection of finitely many closed halfspaces whose boundary hyperplanes have some point in common. We assume this intersection is irredundant. The apex of K is the intersection of these boundary hyperplanes, which is an affine subspace. The generating function for the integer points in K is the formal Laurent series X xm : ð6Þ SK :¼ m2K

It is a priori less clear how to interpret this formal series as a rational function if K is not strictly convex, that is, if its apex is not a single point. The apex is a rational affine subspace L, and the cone K is stable under translation by any integer vector w that is parallel to L. If m 2 K \ Zd ; then the series SK contains the series X x nw xm n2Z

as a subsum. As this can converge only for x = 0, the series SK can converge only for x = 0. We relate these formal Laurent series to rational functions. The product of a formal series and a polynomial is another formal series. Thus the additive group C½½x11 ; . . .; xd1 of formal Laurent series is a module over the ring C½x11 ; . . .; xd1 of Laurent polynomials. The space PL of polyhedral Laurent series is the C½x11 ; . . .; xd1 -submodule of C½½x11 ; . . .; xd1 generated by the set of formal series fSK j K is a simple rational coneg: Since any rational cone may be triangulated by simple cones, PL contains the integer generating series of all rational cones. Let Cðx1 ; . . .; xd Þ be the field of rational functions on Cd ; which is the quotient field of C½x11 ; . . .; xd1 : According to Ishida [11], the proof of the following theorem is due to Brion.

case. (A d-dimensional simplex is the intersection of d + 1 halfspaces, one for each facet.) For a face F of the simplex P; let KF be the tangent cone to F, which is the intersection of the halfspaces corresponding to the d - dim(F) facets containing F. Let ; be the empty face of P; which has dimension -1. Its tangent cone is P:

T HEOREM 9 If P is a simplex, then 0¼

X ð1ÞdimðF Þ SKF ;

ð10Þ

F

the sum over all faces of P.

P ROOF . Consider the coefficient of xm for some m 2 Zd in the sum on the right. Then m lies in the tangent cone KF to a unique face F of minimal dimension, as P is a simplex. The coefficient of xm in the sum becomes X ð1ÞdimðGÞ : GF

But this vanishes, as every interval in the face poset of P is a Boolean lattice. Now we apply the evaluation map u of Theorem 7 to the formula (10). Lemma 8 implies that uðSKF Þ ¼ 0 except when F ¼ ; or F is a vertex, and then uðSKF Þ ¼ rKF ðxÞ: This gives X rKv ðxÞ; 0 ¼ rP ðxÞ þ v a vertex of P

T HEOREM 7 There is a unique homomorphism of C½x11 ; . . .; xd1 -modules u : PL ! Cðx1 ; . . .; xd Þ; such that uðSK Þ ¼ rK for every simple cone K in Rd : P ROOF . Given a simple rational cone K ¼ v þ hw1 ; . . .; wd i with fundamental parallelepiped P; we have d Y ð1 x wi Þ SK ¼ rP ðxÞ: i¼1

Hence, for each S 2 PL, there is a nonzero Laurent polynomial g 2 C½x11 ; . . .; xd1 such that gS ¼ f 2 C½x11 ; . . .; xd1 . If we define uðSÞ :¼ f =g 2 Cðx1 ; . . .; xd Þ; then u(S) is independent of the choice of g. This defines the required homomorphism. The map u takes care of the nonconvergence of the generating series SK when K is not strictly convex.

L EMMA 8 If a rational polyhedral cone K is not strictly convex, then uðSK Þ ¼ 0:

P ROOF . Let K be a rational polyhedral cone that is not strictly convex. Then there is a nonzero vector w 2 Zd such that w þ K ¼ K; and so x w SK ¼ SK : Thus x w uðSK Þ ¼ uðSK Þ: Since 1 x w is not a zero-divisor in Cðx1 ; . . .; xd Þ; we conclude that uðSK Þ ¼ 0: We now establish Brion’s Formula, first for a simplex, and then, using irrational decomposition, for the general

which is Brion’s Formula for simplices. Just as for rational cones, every polytope P may be decomposed into simplices P 1 ; . . .; P l having pairwise disjoint interiors, using only the vertices of P : P ¼ P1 [ [ Pl : Then there exists a small real number [ 0 and a short vector s such that if we set P 0 :¼ s þ ð1 þ ÞP

and P 0i :¼ s þ ð1 þ ÞP i

for i ¼ 1; . . .; l; then P 0 \ Zd ¼ P \ Zd ; and no hyperplane supporting any facet of any simplex P 0i meets Zd : If we write KðQÞw for the tangent cone to a polytope Q at a vertex w; then for v a vertex of P with v0 = (1 + )v + s the corresponding vertex of P 0 ; we have KðP 0 Þv0 \ Zd ¼ KðPÞv \ Zd and so this is an irrational decomposition. Then X X rKðPÞv ðxÞ ¼ rKðP 0 Þv ðxÞ v a vertex of P 0

v a vertex of P

¼

l X

X

i¼1 v a vertex of P 0i

¼

l X

rKðP 0i Þv ðxÞ

rP i ðxÞ ¼ rP 0 ðxÞ ¼ rP ðxÞ:

i¼1

The second equality holds because the vertex cones KðP 0i Þv form an irrational decomposition of the vertex cone KðP 0 Þv ; and because the same is true for the polytopes. This completes our proof of Brion’s Formula. 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

13

Consider the quadrilateral Q; which may be triangulated by adding an edge between the vertices (2, 0) and (0, 2). Let ¼ 14 and s ¼ ð 12 ; 14Þ: Then ð1 þ ÞQ þ s has vertices ð 12; 14Þ;

ð2; 14Þ;

ð 12; 2 þ 14Þ;

along [0, 1) and (2, 3], the value 2 along [1, 2], and vanishes everywhere else.

ð4 þ 12; 2 þ 14Þ :

We display the resulting irrational decomposition.

Q

We use the map u to deduce a very general form of the Lawrence–Varchenko formula. Let P be a simple polytope, and for each vertex v of P choose a vector nv that is not perpendicular to any edge direction at v. Form the cone Knv ;v as before. Then we have X ð1ÞjEv ðnv Þj rKnv ;v ðxÞ: ð11Þ rP ðxÞ ¼ v a vertex of P

Brion’s formula is the special case when each vector nv points into the interior of the polytope. We establish (11) by showing that the sum on the right does not change when any of the vectors nv is rotated. Pick a vertex v and vectors n, n0 that are not perpendicular to any edge direction at v such that n w and n w0 have the same sign for all except one edge direction m at v. Then Kn;v and Kn0 ;v are disjoint and their union is the (possibly) half-open cone K generated by the edge directions w at v such that n w and n0 w have the same sign, but with apex the affine line v þ Rm: Thus we have the identity of formal series

Already this simple example shows that our generators do not form a basis: they are linearly dependent. For P 0 ¼ ½0; 3 and Q0 ¼ ½1; 2; we get the same sum.

But this is the only thing that can happen.

T HEOREM 12 ([10, 18]) The linear space of relations among the indicator functions ½P of convex polyhedra is generated by the relations ½P þ ½Q ¼ ½P [ Q þ ½P \ Q; where P and Q run over polyhedra for which P [ Q is convex. A valuation is a linear map m: V ! V , where V is some vector space. Some standard examples are

SKn;v SK ¼ SKn0 ;v :

V R

mðPÞ volðPÞ

PL

SP ðxÞ

Cðx1 ; . . .; xd Þ

rP ðxÞ

R

1

Applying the evaluation map u gives rKn;v ðxÞ ¼ rKn0 ;v ðxÞ; which proves the claim, and the generalized Lawrence– Varchenko formula (11).

Valuations Valuations provide a conceptual approach to these ideas. Once the theory is set up, both Brion’s Formula and the Lawrence–Varchenko Formula are easy corollaries of duality being a valuation. We are indebted to Sasha Barvinok who pointed out this correspondence to the second author during a coffee break at the 2005 Park City Mathematical Institute. Let us explain. Consider the vector space of all functions Rd ! R. Let V be the subspace that is generated by indicator functions of polyhedra: 1 if x 2 P; ½P: x 7! 0 if x 62 P: We add these functions pointwise. For example, if d = 1, and P ¼ ½0; 2; Q ¼ ½1; 3; then ½P þ ½Q takes the value 1 14


That rP ðxÞ is a valuation is a deep result of Khovanskii– Pukhlikov [12] and of Lawrence [14]. The last example is called the Euler characteristic. This valuation is surprisingly useful. For example, it can be used to prove Theorem 13 below. The most interesting valuation for us comes from the polar construction. The polar P _ of a polyhedron P is the polyhedron given by P _ :¼ fx j hx; yi 1 for all y 2 Pg: It is instructive to work through some examples.

(1)

The polar of the square … is the diamond.

(2)

+ − + − + − + − +

=

The polar of a cone K … is the cone K_ :¼ fx j hx; yi 1 for all y 2 Kg: (3) Suppose that P is a polytope whose interior contains the origin and F is a face of P:

=

+

+

+

−

−

−

−

+

If we apply polarity to (14), we get the Brianchon–Gram Theorem [6, 9]. X ½Kv ½P ¼ v vertex

tangent cones of faces of positive dimension: ð15Þ Then the polar of the tangent cone KF … is the convex hull of the origin together with the dual face F _ :¼ fx 2 P _ j hx; yi ¼ 1g; which is a pyramid over F _ : For this last remark, note that if x 2 F _ and y 2 KF ; then hx; yi hF _ ; F i ¼ 1: Conversely, if x 2 K_F ; then hx; :i is maximized over KF at F by example (2), and it is at most 1 there. In these examples, the polar of the polar is the original polyhedron. This happens if and only if the original polyhedron contains the origin. (4) The polar of the interval [1, 2] is the interval [0, 1/2], but the polar of [0, 1/2] is [0, 2]. Now, we come to the main theorem of this section.

T HEOREM 13 (Lawrence [14]) The assignment ½P 7! ½P _ defines a valuation. This innocent-looking result has powerful consequences. Suppose that P is a polytope whose interior contains the origin. Then we can cover P _ by pyramids convð0; F _ Þ over the codimension-one faces F _ of P _ . The indicator functions of P and the cover differ by indicator functions of pyramids of smaller dimension. ½P _ ¼

This is essentially the indicator-function version of Theorem 9, but for general polytopes. If we now apply the valuation r, and recall that r evaluates to zero on cones that are not strictly convex, we obtain Brion’s Formula. Next, suppose that we are given a generic direction vector n. On a face F of P; the dot product with n achieves its maximum at a vertex vn ðF Þ: For a vertex v of P; we set [ relint F _ : F _n ðvÞ :¼ F :vn ðF Þ¼v

(The relative interior, relintðPÞ; of a polyhedron P is the topological interior when considered as a subspace of its affine hull.) In words, we attach the relative interior of a low-dimensional pyramid convð0; F _ Þ to the full-dimensional pyramid convð0; v _ Þ that we see when we look in the n-direction from convð0; F _ Þ: In this way, we obtain an honest decomposition X ½convð0; F _n ðvÞÞ: ð16Þ ½P _ ¼ v

For the polar of the square, this is

X ½convð0; F _ Þ lower dimensional pyramids: F_

ð14Þ The Euler–Poincare´ formula for general polytopes organizes this inclusion-exclusion, giving the exact expression ½P _ ¼

X

ð1Þcodim F

_

þ1

½convð0; F _ Þ:

We illustrate this when P is the square.

To compute the polar of the half-open polyhedron convð0; F _n ðvÞÞ; we have to write its indicator function 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

15

½convð0; F _n ðvÞÞ as a linear combination of indicator functions of (closed) polyhedra. If P is a simple polytope, then all the dual faces F _ are simplices. It turns out that the polar of convð0; F _n ðvÞÞ is precisely the forward tangent cone Kn;v at the vertex v. So the Lawrence–Varchenko formula is just the polar of (16). This gives a fairly general principle for constructing Brion-type formulas: Choose a decomposition of (the indicator function of) P _ ; and then polarize. We invite the reader to set up his or her own equations this way.

An Application Brion’s Formula shows that certain data of a polytope—the list of its integer points encoded in a generating function— can be reduced to cones. We have already seen how to construct the generating function rK ðxÞ for a simple cone K: General cones can be composed from simple ones via triangulation and either irrational decomposition or inclusion-exclusion. Given a rational polytope P; Brion’s Formula allows us to write the possibly huge polynomial rP ðxÞ as a sum of rational functions, which stem from (triangulations of) the vertex cones. A priori it is not clear that this rational-function representation of rP ðxÞ is any shorter than the original polynomial. That this is indeed possible is due to the signed decomposition theorem of Barvinok [1]. To state P Barvinok’s Theorem, we call a rational d-cone K ¼ v þ di¼1 R 0 wi unimodular if w1 ; . . .; wd 2 Zd generate the integer lattice Zd : The significance of a unimodular cone K for us is that its fundamental (halfopen) parallelepiped contains precisely one integer point p, and so the generating function of K has a very simple and short form rK ðxÞ ¼

xp ð1

x w1 Þ ð 1

x wd Þ

:

In fact, the description length of this is proportional to the description of the cone K:

T HEOREM 17 (Barvinok) For fixed dimension d, the generating function rK for any rational cone K in Rd can be decomposed into generating functions of unimodular cones in polynomial time; that is, there is a polynomial-time algorithm and (polynomially many) uniP modular cones Kj such that rK ðxÞ ¼ j j rKj ðxÞ; where j 2 f1g: Here polynomial time refers to the input data of K; that is, the algorithm runs in time polynomial in the input length of, say, the halfspace description of K: Brion’s Formula implies that an identical complexity statement can be made about the generating function rP ðxÞ for any rational polytope P: From here it is a short step (which nevertheless needs some justification) to see that one can count integer points in a rational polytope in polynomial time. We illustrate Barvinok’s short signed decomposition for the cone K :¼ ð0; 0Þ þ R 0 ð1; 0Þ þ R 0 ð1; 4Þ; ignoring cones of smaller dimension. 16


Although K is the difference of two unimodular cones, it has a unique decomposition as a sum of four unimodular cones.

In general the cone ð0; 0Þ þ R 0 ð1; 0Þ þ R 0 ð1; nÞ is the difference of two unimodular cones, but it has a unique decomposition into n unimodular cones. Arguably the most famous consequence of Barvinok’s Theorem applies to Ehrhart quasipolynomials—the counting functions LP ðtÞ :¼ # tP \ Zd in the positveinteger variable t for a given rational polytope [4] P: One P can show that the generating function t 1 LP ðtÞ x t is a rational function, and Barvinok’s Theorem implies that this rational function can be computed in polynomial time. Barvinok’s algorithm has been implemented in the software packages barvinok [17] and LattE [8]. The method of irrational decomposition has also been implemented in LattE; considerably improving its performance [13]. ACKNOWLEDGMENTS

Research of Beck supported in part by NSF grant DMS0810105. Research of Haase supported in part by NSF grant DMS-0200740 and a DFG Emmy Noether fellowship. Research of Sottile supported in part by the Clay Mathematical Institute and NSF CAREER grant DMS-0538734. REFERENCES

1. A.I. Barvinok, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res. 19 (1994), 769–779. 2. A.I. Barvinok, A course in convexity, Graduate Studies in Mathematics, vol. 54, American Mathematical Society, Providence, RI, 2002. 3. P. Baum, Wm. Fulton, and G. Quart, Lefschetz-Riemann-Roch for singular varieties. Acta Math. 143 (1979), no. 3–4, 193–211. 4. M. Beck and S. Robins, Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, New York, 2007. 5. M. Beck and F. Sottile, Irrational proofs of three theorems of Stanley, 2005, European J. Combin. 28 (2007), 403–409. 6. C.J. Brianchon, Theóre`me nouveau sur les polye`dres, J. Ećole (Royale) Polytechnique 15 (1837), 317–319.

7. M. Brion, Points entiers dans les polye`dres convexes, Ann. Sci. Ećole Norm. Sup. 21 (1988), no. 4, 653–663.

13. M. Koeppe, A primal Barvinok algorithm based on irrational

8. J.A. De Loera, D. Haws, R. Hemmecke, P. Huggins, and R.

decompositions, SIAM J. Discrete Math. 21 (2007), no. 1, 220– 236.

Yoshida, A user’s guide for LattE v1.1, software package

14. J. Lawrence, Valuations and polarity, Discrete Comput. Geom. 3

LattE (2004), electronically available at http://www.math. ucdavis.edu/*latte/. 9. J.P. Gram, Om rumvinklerne i et polyeder, Tidsskrift for Math. (Copenhagen) 4 (1874), no. 3, 161–163.

(1988), no. 4, 307–324. 15. J. Lawrence, Polytope volume computation, Math. Comp. 57 (1991), no. 195, 259–271. 16. A.N. Varchenko, Combinatorics and topology of the arrangement

10. H. Groemer, On the extension of additive functionals on classes of

of affine hyperplanes in the real space, Funktsional. Anal. i Pri-

convex sets, Pacific J. Math. 75 (1978), no. 2, 397–410. 11. M.-N. Ishida, Polyhedral Laurent series and Brion’s equalities,

lozhen. 21 (1987), no. 1, 11–22. 17. S. Verdoolaege, software package barvinok (2004), electroni-

Internat. J. Math. 1 (1990), no. 3, 251–265.

cally available at http://freshmeat.net/projects/barvinok/.

12. A.G. Khovanskii and A.V. Pukhlikov, The Riemann-Roch theorem

18. W. Volland, Ein Fortsetzungssatz fu¨r additive Eipolyhederfunk-

for integrals and sums of quasipolynomials on virtual polytopes,

tionale im euklidischen Raum, Arch. Math. 8 (1957), 144–

Algebra i Analiz 4 (1992), 188–216.

149.

2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

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Four Poems from When She Was Kissed by the Mathematician Sandra M. Gilbert After He Expounds the Different Infinities Half the night sleepless, dreaming infinities— the countable set and the unaccountable— she listens to his breathing, sometimes even, placid and perfectly divisible, the way she imagines certain numbers are, sometimes stopped by odd irregular murmurs, listens and counts his breaths, his unintelligible words as if she wrapped a rosary of integers around her wrists, his wrists, linking them both in the smaller infinity, the kind you can count and maybe comprehend, the one that Zeno scorned. His chest with its mane of gray rises and falls as she counts, crawls nearer, wishing he’d explain again or else embrace her, silence this abacus of prayer that ticks in her head: O God, whatever you are, let this one be —and the bed swells in the heat, in the dark, and the single sheet that holds them close winds round and round like the great enfolding spaces through which arrows fly and breaths and prayers on their eccentric route toward Zeno’s black incalculable target.

18


They Debate Triangles and Medians

She Grapples With Operations Research

That they are is obvious to him, remarkable to her. She grants the points, their dark

... the by now famous problem of the jeep... concerns a jeep which is able to carry enough fuel to travel a distance d, but is required to cross a desert whose distance is greater than d (for example 2d). It is to do this by carrying fuel from its home base and establishing fuel depots at various points along its route so that it can refuel as it moves farther out... [But] in general, the more jeeps one sends across, the lower the fuel consumption per jeep.

necessity, each a moment brimming with its own being— and the lines, well, given points and given time, no doubt there must be lines, those fateful journeyings from here to there, from this to that. But the vertices where journeys meet, the angles, wide or narrow, yearning for closure and then letting go— aren’t these, she asks, unlikely as the medians that cling together at the center of each triangle, knotting altitudes and perpendiculars into a single web of possibility? And maybe Euclid got it halfway right: in luminous sections, intersections, everything is joined and rational, at least for a while, as if somebody had suddenly conjectured yes, it can make sense— and the triangles and medians of you and me and them last and glow till one by one the fastenings unclasp and that which must be linear sheds the comforts of shape, each line going its lonely distance to the non-Euclidean place where parallels diverge in darkness.

—David Gale, ‘‘The Jeep Once More or Jeeper by the Dozen’’ The mathematician is crossing the desert, his fine high features creased with thought. One tank of fuel at this depot, another stashed at that. How many caches needed in between? She worries. It’s all too Zeno for her liking. And what if he insists on the Sahara? No, he promises, he’ll only try the kindlier Mojave this time, with its rainstruck buds and rare new blossoms rising while his jeep, his squad of jeeps, moves slowly on the trip through sand, through quarks and quirks of sand, their particles an endless series as she waits and hates his danger. The mathematician crosses, curses, blesses, the infinite regressions of the desert: and the desert sun storms down like thunder, like a roar of light against his beard, his temples clenched with calculations and desire. At stated stations palms, dates, springs of comfort will appear. And there he’ll prudently

[Editor’s Note. David Gale of the University of California Berkeley, a long-time Intelligencer collaborator (and my friend for a much longer time) died 7 March 2008. Longtime readers will remember his lively and inventive columns for this magazine, many of which were collected in Tracking the Automatic Ant (Springer, 1998). Mathematicians everywhere value his contributions to convexity, combinatorics, and applications (of games, inequalities, etc.) to social sciences. One of the major landmarks here was his Theory of Linear Economic Models (McGraw-Hill, 1960). The story will be told at length in a tribute to Gale to appear as a special issue of Games and Economic Behavior. And many of you have followed his admirable venture into a ‘‘math museum‘‘ on the Web, see http://mathsite.math.berkeley.edu/main.html. Those who have been especially attentive will have noticed an unusual and touching gesture: David Gale dedicated a theorem to his partner, Sandra Gilbert! See The Intelligencer, vol. 15 (1993), no. 4, 61. After all, he said, this is only reciprocity, for she dedicated poems to me. And here, with a delay, are some of her poems. They appeared earlier in her Kissing the Bread: New and Selected Poems, 1969–1999, W.W. Norton, 2000, and are reprinted here by permission. —Chandler Davis]

Ó 2008 Sandra M. Gilbert, Volume 31, Number 1, 2009

19

sequester further energies. Blank sky and melting gold, keen blade of lemmas roaring through his engine. She stands on the sidelines in the shade. She stirs a pitcher of gin and lemonade. Astute, her body manufactures leafy murmurs as she turns herself into a crystal dish of peaches. The mathematician is crossing the desert, crossing, journeying past Zeno, past the infinite. She wants to be the first oasis that he reaches.

He Explains the Book Proof The shadowy clatter of the cafe´ frames the glittering doorway. A white cup and a blue bowl inscribe pure shapes on the table. The mathematician says, Let’s turn the pages and find the proof in the book of proofs. He says, It’s as if it’s already there, somewhere just outside the door. as if by sitting zazen in a coffee house, someone could get through or get ‘‘across," or as if the theorems had already all been written down on sheer sheets of the invisible, and held quite still, so that to think hard enough is simply to read and to recall— the way the table remembers the tree, the bowl remembers the kiln.

Department of English, University of California Davis, Davis, CA 95616, USA. e-mail: [email protected] 20


Years Ago

David E. Rowe, Editor

Jakob Steiner’s Systematische Entwickelung: The Culmination of Classical Geometry VIKTOR BLA˚SJO¨

Send submissions to David E. Rowe, Fachbereich 08—Institut fu¨r Mathematik, Johannes Gutenberg University, D55099 Mainz, Germany. e-mail: [email protected]

Introduction

S

teiner’s Systematische Entwickelung of 1832 was a monumental unification of classical geometry based on a new conception of projective geometry and a new approach to conic sections. We shall study this work in some detail. I shall also claim that Steiner was committed to the unification and reverence of classical geometry, and that his work was a remarkable success by these standards, but that later generations imposed different standards of success, downplaying historical continuity and favouring intrinsically motivated programmatic agendas—in the case of projective geometry epitomised by von Staudt (1847)—which caused a lasting and undeserved depreciation of Steiner’s work. It ought to be uncontroversial that the Systematische Entwickelung was intended as a unification of classical geometry since, first, it is packed with classical theorems and references to classical geometers, and, second, Steiner says so. In his preface, Steiner explains that the goal of his work is to ‘‘extract a thread of continuity and a common root’’ from classical geometry by uncovering ‘‘fundamental properties that contain the germ of all theorems, porisms, and problems of geometry, so generously made available to us in older and modern times.’’ Classical geometry has thus far produced ‘‘a collection of separate tricks, however clever, but no organically connected whole. This work tries to uncover the organism by which the most varied features of the spatial world are connected. A small number of very simple fundamental relationships make up the schema by which the remaining mass of theorems can be developed consistently and without difficulty.’’ Elsewhere he said: ‘‘As a teacher I tried whenever possible to treat each subject as consisting of a single idea, and to see the individual

theorems as mere consequences left as footprints in the development of this one idea. Almost unconsciously, this led me to the actual genetic viewpoint, as it must have appeared to the ancient geometers, although I approached it in the opposite way. Since I had a wealth of solved problems and theorems available to me, my task was to focus not on individual theorems but rather the general principles of synthetic construction from which all these inventions follow, to present them in this capacity and treat them exhaustively according to these principles.’’ (Graf (1897, p. 12–13); for more of Steiner’s own words on these matters, see Lange (1899, pp. 19–21, 23–24)). Finally, I cannot help but perceive as symbolic an observation made by Jacobi (1843) in a letter to his wife written when Jacobi and Steiner were both in Rome: ‘‘Steiner has an aptitude for finding walled-in old Doric columns in old stables and worn-down buildings . . . through which one is clearly reminded that one is walking on classical soil.’’ Nevertheless, many commentators miss this point and consequently fail to appreciate Steiner’s work. Back in the old days the Systematische Entwickelung used to be called ‘‘epoch making’’ (Ko¨tter (1901, p. 252)), a ‘‘masterpiece’’ (Zacharias (1912, p. 41)), ‘‘a model of a complete method and execution for all other branches of mathematics’’ (Jacobi (1845), quoted in Burckhardt (1976, p. 18)), and so on. Not so today, however. As a framework for this discussion, let us state a principle which should be a truism but which is in fact violated in many commentaries on Steiner. Suppose a mathematician X writes a work aiming to achieve Y. Anyone wishing to criticise this work could reasonably be expected to argue that either (i) X does not achieve the objective Y; or

Ó 2008 SPRINGER SCIENCE+BUSINESS MEDIA, LLC., Volume 31, Number 1, 2009

21

(ii) the objective Y is not worth pursuing. A charge of the type ‘‘X does not deal with Z’’ is plainly irrelevant. Still, this is precisely the type of criticism offered by many commentators. Leading the crusade is Klein (1979, p. 118), who writes in his discussion of the Systematische Entwickelung: ‘‘in retreating from the ground won by Mo¨bius and rejecting the principle of signs from synthetic geometry, [Steiner] deprived himself of the possibility of more general formulations. Thus, when dealing with cross-ratios he was forced always to fix the order of the elements; but, above all, he lost the occasion of mastering the imaginary. He never really understood it and fell into the use of such terms as ‘the ghost’ or ‘the shadow land of geometry.’ And of course his system had to suffer from this selfimposed restriction. Thus, though there are two conics x12 þ x22 x32 ¼ 0 and x12 þ x22 þ x32 ¼ 0 from a projective point of view, in Steiner’s system there is no room for the second. Von Staudt was the first to liberate synthetic geometry from these and other imperfections.’’ The ‘‘Steiner: bad—von Staudt: good’’ dichotomy is very prevalent. For example, Coolidge’s discussion (1934, pp. 222–223) of the Systematische Entwickelung is mostly concerned with finding ‘‘slips’’ and is immediately followed by an ecstatic discussion of von Staudt, ‘‘this deep thinker,’’ who ‘‘perceived two essential weaknesses in the synthetic geometry of his predecessors. (a) The basis of projective relations was the cross-ratio. This is projectively invariant but, as previously given, was based on distances and angles which are not in themselves unalterable. (b) What are imaginary points anyway? What can be said about them, except that they are imaginary?’’ Although Klein, as a proud disciple of Plu¨cker, is certainly biased in this issue, his point of view has nevertheless taken hold quite widely. Laptev & Rozenfel’d (1996, p. 37), for example, do not hesitate to state that Steiner’s disregard for negative and complex numbers is ‘‘an important defect.’’ Even the ‘‘Geometry’’ article in the Encyclopædia Britannica (Heilbron (2007)), supposedly an objective source, takes a Kleinian stance: ‘‘Poncelet’s followers

22


realized that they were hampering themselves, and disguising the true fundamentality of projective geometry, by retaining the concept of length and congruence in their formulations, since projections do not usually preserve them. . . . Efforts were well under way by the middle of the 19th century, by . . . von Staudt . . . among others, to purge projective geometry of the last superfluous relics from its Euclidean past.’’ These commentators miss the point. Steiner achieves exactly what he sets out to achieve—a systematic unification of classical geometry—whereas von Staudt of course never comes close to anything of the sort. The intrinsic theory point of view naturally makes Steiner’s work look flawed, but the real issue is whether the ideas of von Staudt et al. would have helped Steiner further his objective. I say that the answer is almost always no. Steiner is not ‘‘forced’’ to fix the order of the elements in the crossratio; he chooses to use this classical notion of the cross-ratio and it serves him well. It is not that there is ‘‘no room’’ for the conic x12 þ x22 þ x32 ¼ 0; there is in fact no point in it since Steiner’s only interest is the classical theory. His work does not ‘‘suffer’’ from exclusion of imaginary elements, because they are not needed, and therefore Steiner could not care less ‘‘what can be said about them.’’ As for using metric notions where purely projective ones are possible—so what? If our goal is to unify classical geometry, metric notions are by no means an ‘‘essential weakness.’’ To insist on purely projective methods is a ‘‘self-imposed restriction’’ if there ever was one. What was ‘‘purged’’ by von Staudt et al. was not ‘‘superfluous relics’’ but historical continuity. Klein (1979, p. 118) also claims that ‘‘further imperfections affect the basic definitions of Steiner’s system, so that many more theorems have exceptions than Steiner was aware of.’’ To support this claim, Klein refers to Baldus (1923). But this seems to me to be based on a misunderstanding due to a too modern reading of Steiner. Baldus does not so much attack Steiner’s work, per se, but rather contemporary authors using his definition of projectivity, his argument being that it cannot properly handle pencils centred at infinity (p. 87). But Steiner himself

does not use pencils centred at infinity; indeed, doing so would have caused him great difficulty, since, as the above gentlemen so eagerly remarked, his definition of the cross-ratio is metrical.

Projective Geometry in the Systematische Entwickelung I shall now provide an overview of the mathematical content of the Systematische Entwickelung. First, we study the cross-ratio, which is the foundation of the entire theory. This immediately yields swift proofs of the classical theorems of Pappus and Desargues. The real triumph, however, is Steiner’s theory of conic sections, which we shall study subsequently. The Cross-Ratio We shall now see how Steiner arrived at the cross-ratio. Consider a line A and a pencil at B, and let a; b; c; . . . be the points where the line is intersected by the lines a; b; c; . . . of the pencil (Figure 1). The line and the pencil are said to be in perspective. If we move the line or the pencil, then a; b; c; . . . will no longer correspond to a; b; c; . . ., but there will be some definite relation between the two entities (i.e., knowing the lengths ab, bc, etc., means knowing the angles \ab, \bc, etc.). We find this relation as follows. Let p be the line of the pencil perpendicular to A, and draw the perpendicular d1 a to d. Then Bpd and ad1 d are similar, giving Bp ad1 ¼ ; Bd ad or, since ad1 ¼ Ba sinð\adÞ, ad Ba Bd ¼ : sinð\adÞ Bp By the same argument, ac Ba Bc ¼ ; sinð\acÞ Bp bc Bb Bc ¼ ; sinð\bcÞ Bp bd Bb Bd ¼ ; sinð\bdÞ Bp which combine to give . ad bd sinð\adÞ sinð\bdÞ . ac bc ¼ sinð\acÞ sinð\bcÞ

a2 , the third Pappus point, located on A2, as was to be shown. Desargues’ Theorem (§21) Steiner’s proof of Desargues’s theorem begins with the following lemma (Figure 27). Let A, A1, A2 be three projectively related lines, i.e., there are projections 7 a1 ;b1 ;c1 ;...; A ! A1 : a;b;c;... ! 7 a2 ;b2 ;c2 ;...; A ! A2 : a;b;c;... ! A1 ! A2 : a1 ;b1 ;c1 ;... ! 7 a2 ;b2 ;c2 ;...;

Figure 1.

or ad . ac bd

bc

¼

sinð\adÞ . sinð\acÞ sinð\bdÞ

sinð\bcÞ

;

which no longer depends on the positions of the line and the pencil, so we have found the relation we were looking for. This is the cross-ratio. An immediate consequence ðxx5; 10Þ is the theorem that for any line cutting a pencil, the points a; b; c; d of the line corresponding to the lines a, b, c, d of the pencil always have the same crossratio (since the right-hand side in the above expression stays the same), which is a more conventional statement of the projective invariance of the cross-ratio. In particular, since the cross-ratio is preserved when a line is projected onto another line, such a projection is determined by its effect on any three points ðxx6; 10Þ.

the triangle efl1 is inscribed in the triangle BB1 B2 . This means that e, f, and l (the intersection of A and Bl1 ) all come back onto themselves when sent throughout the series of projections A ! A1 ! A2 ! A defined by the projection points B; B2 ; B1 . So, by the three-point determinacy, this cycle of projections is the identity. Following the course of a through these projections traces out a triangle aa1 a2 , with

and let the lines meet in a point e;e1 ;e2 . The lemma says that the three points of projection B;B1 ;B2 are on a line. Proof: The line BB1 is a projection line of A ! A1 as well as A ! A2 , so it goes through three corresponding points d;d1 ;d2 . But d1 d2 must be a projection line of the third projection A1 ! A2 , so B2 must also be on this line. Thus B;B1 ;B2 are collinear, and the lemma is proved. Desargues’s theorem may now be proved as follows. Let aa1 a2 and bb1 b2 be two triangles in perspective, i.e., the lines A, A1, A2 connecting corresponding vertices meet in a point e. Let B;B1 ;B2 be the intersections of the extensions of corresponding sides of the triangles, and

Pappus’ Theorem (§23) We are given a hexagon B1 BB2 a1 eaB1 with vertices alternately on two lines B1 eB2 and a1 Ba (Figure 30; I shall reproduce Steiner’s figures with their original numbering). We wish to show Pappus’s theorem: The three points of intersection of opposite sides are on a line. Draw the lines ae (A) and BB2 , meeting in f, and draw the lines a1 e (A1) and BB1 , meeting in l1 . The points f and l1 are the first two Pappus points; we need to show that the third point, the intersection of a1 B2 and aB1 , is on the same line (A2). To do this, we note that

Figure 30. Ó 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

23

Figure 27.

use these points to project A ! A1 , A ! A2 , and A1 ! A2 . Under these projections, a;b;e, and a1 ;b1 ;e, and a2 ;b2 ;e correspond by construction, so by the three-point determinacy of projections, the three lines are projectively related, so, by the result just proved B;B1 ;B2 must be on a line, which is Desargues’s theorem.

Conic Sections in the Systematische Entwickelung

parallel to A, A1; thus if we let q be the point at infinity of A, then q1 will be the intersection of A1 and q; and similarly, if we let r1 be the point at infinity of A1, then r will be the intersection of A and r. Consider now two other tangents a, b. To show that the correspondence is a projection we need to show that the cross-ratio of a; r; b; q is the same as the cross-ratio of a1 ; r1 ; b1 ; q1 , i.e., ar . aq a1 r1 . a1 q1 ¼ : br bq b1 r1 b1 q1

As promised above, we shall now see how a remarkably unified and simple approach to conic sections is made possible by the basic theory of projections.

Since q and r1 are the points at infinity, this simplifies to .a q ar . 1 1 1¼1 br b1 q1 or

The Fundamental Theorem on Conic Sections (§§37–39) Steiner’s fundamental theorem on conic sections is, according to himself, ‘‘more important than all the previously known theorems about them, for it is the true fundamental theorem, since it is so comprehensive that almost all other properties of these figures follow from it in the simplest and clearest way, and the method by which they will be deduced surpasses any known point of view in terms of simplicity and convenience’’ ðx39Þ. We shall discuss the theorem first in terms of circles. It extends to general conics by projection, of course. Consider a circle with center M and two of its tangents A, A1 (Figure 38). Any other tangent a pairs a point of A with a point of A1. We shall prove that this correspondence A ! A1 is a projection. First, let q,r be the two tangents 24


ar a1 q1 ¼ br b1 q1 : Therefore, to show that the correspondence is a projection, we need to show that the quantity ar a1 q1 is independent of the choice of a. To do this, we note first that rq1 cuts the circle

Figure 38.

in half, making an equilateral triangle drq1 with base angles a = a1. Connecting a to M obviously bisects the angle at a, and, similarly, a1 M bisects the angle at a1 . Also, by comparing the angle sums of the triangle aMa1 and the quadrilateral arq1 a1 , we see that the angle \aMa1 is equal to a. Therefore, the triangles aMa1 , arM, Mq1 a1 are similar, giving ar=rM ¼ Mq1 =q1 a1 , or ar a1 q1 ¼ Mr Mq1 , so ar a1 q1 is indeed independent of a, as we needed to show. Thus we have proved the following theorem ðx38:IVÞ: Any two projectively related lines define a conic section to which they and all their projection lines are tangents (Figure A(i)). Or, dually, any two projectively related pencils define a conic section through the centers of the pencils as the locus of intersections of corresponding lines (Figure A(ii)). The two dual forms of the fundamental theorem subsume, as we shall prove below, two of the most prominent earlier attempts at systematic approaches to conic sections, namely those of Pascal and Newton. Pascal (1640) envisioned a unification of the theory of conic sections based on his ‘‘hexagrammum mysticum.’’ Immediately after having introduced his theorem he says: ‘‘[W]e propose to give a complete text on the elements of conics, that is to say, all the properties of diameters and other straight lines, of tangents, etc., to construction of the cone from substantially these data, the description of conic sections by points, etc.’’ (Pascal (1640), quoted from Struik (1969, p. 165)). This program saw a revival with the discovery of Brianchon’s theorem.

Figure A. The two dual ways of generating conic sections by the fundamental theorem (From Courant & Robbins (1941, pp. 208, 205)).

Newton (1667/68) unified much conic section theory by the following construction (Figure B): Given two rules HFG and RKS with fixed angles \HFG and \RKS, and fixed points F and K, if we make one intersection, S, move along a line, then the other intersection, R, traces out a conic. This is a very efficient tool, not least for solving construction problems involving conics. Newton later put some of this theory in the Principia (1687, Book I, Section V), although it does

not do much good there, appearing, as it does, in a section having ‘‘but little connection with the rest of the Principia’’ (Ball (1893, p. 81)). A more systematic account was later provided by Maclaurin (1720). The fundamental theorem also has several interesting immediate corollaries ðxx4041Þ. If, for instance, the two generating lines are similar, then the line at infinity is a projection line and thus a tangent to the conic, so therefore the conic must be a parabola. The

Figure B. Newton’s organic construction of conic sections. (From Newton (1967–1981, Vol. 2, p. 118).)

fundamental theorem also shows that a conic is determined by five tangents (or, dually, five points): Two tangents are taken as the generating lines and the other three determine the projective relation between them, by the three-point determinacy of projective transformations. Furthermore, the fundamental construction of conic sections extended to space becomes a construction of onesheeted hyperboloids: For any two projectively related lines in space, their projection lines generate a onesheeted hyperboloid ðx51:IVÞ. I should also mention that Steiner’s fundamental theorem is commonly called ‘‘Steiner’s definition’’ of conic sections, and is sometimes criticised as such; e.g., ‘‘Steiner’s definition assigns a special role to two points on the conic, obscuring its essential symmetry’’ (Coxeter (1993, p. ix)). Also, Kline (1972, pp. 847–848), in his discussion of the Systematische Entwickelung, speaks of Steiner’s ‘‘now standard projective method of defining the conic sections’’ and claims that ‘‘he did not identify his conics with sections of a cone,’’ which is false (they are projective images of circles, as we have seen), and, I might add, fundamentally inconsistent with Steiner’s commitment to classical geometry. The freedom to define the objects of study as one pleases comes only when a theory matures into the stage of intrinsic motivation. Pascal’s and Brianchon’s Theorems (§42) We shall prove Brianchon’s theorem. Pascal’s theorem, of course, follows dually. Consider a hexagon B1 B2 kaa1 l1 B1 circumscribing a conic (Figure 31; the conic is not shown). Take the two sides ak (A) and a1 l1 (A1) as the two generating tangents of the fundamental theorem. Then the fundamental theorem says that these two sides A and A1 are projectively related and that the other four sides are projection lines connecting corresponding points of A and A1. Now project A onto kl1 (A2) from B1 , and project A1 onto A2 from B2 . These two projections agree on three points—the images of b and b1 , k and k1 , and l and l1 —so, by the threepoint determinacy, they must also agree on the image of a and a1 , i.e.,

Ó 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

25

Let l be such a line. For every point on l, there is a polar line through the circle, as above. We claim that all these polar lines have one point in common, so that this point is the natural pole of l. Monge ð1799; x39Þ proves this by cleverly bringing in the third dimension. Imagine a sphere that has the circle as its equator. Every point on l is the vertex of a tangent cone to this sphere, where the two tangents to the equator are part of this cone and the polar line is the perpendicular projection of the circle of intersection of the sphere and the cone. Now consider a plane through l tangent to the sphere. It touches the sphere at one point P. Every cone contains this point (because the line from any point on l to P is a tangent to the sphere and so is part of the tangent cone). Thus, for every cone, the perpendicular projection of the intersection with the sphere goes through the point perpendicularly below P, and this is the pole of l, and l is the polar of this point.

Figure 31.

B1 a and B2 a1 intersect A2 at the same point. So the three diagonals B1 a; B2 a1 ; A2 meet at a point (a2 ), which is Brianchon’s theorem. Newton’s Organic Construction of Conic Sections (§46) Newton’s organic construction of conics follows immediately from the fundamental theorem. Recall from Section 3.1 that we have two rules with fixed angle, and we wish to show that as one intersection moves along a line the other traces out a conic. The fact that the first intersection moves along a line means in our language that we have two projective pencils, B and B1 (Figure 50). The second intersection is the intersection of two other pencils, B2 and B3 , which, since the angles of the rules are fixed, are in fact just rotated copies of the previous pencils. So, by the fundamental theorem, these two projectively related pencils define a conic section. Pole and Polar Theory (§44–45) In this section, we shall see how the theory of poles and polars may be approached through Steiner’s fundamental theorem on conic sections. This is particularly interesting since this theory was a precursor of the modern notion of duality, which, although it had been recognised before, Steiner was the first to give a prominent place at the very foundation of projective geometry ðx1Þ. 26


Figure 50.

Poles and Polars Before Steiner Polar reciprocation with respect to a circle associates a line with every point and a point with every line, as follows. Consider a line that cuts through the circle. It meets the circle in two points. Draw the tangents to the circle through these points. The two tangents meet in a point. This point is the pole of the line. Conversely, the line is the polar of the point. Thus, we can deal with lines that cut through the circle and, equivalently, points outside the circle. But what about a line outside the circle (or, equivalently, a point inside the circle)?

Polar Theory and Duality Polar reciprocation thus suggests a duality between points and lines. Brianchon (1806) was perhaps the first to use this idea in his proof of ‘‘Brianchon’s theorem,’’ the dual of Pascal’s theorem. The polar reciprocation approach to duality was systematised by Poncelet (1822). In this tradition, Brianchon’s theorem can be derived as follows. We start with Pascal’s theorem, i.e., we have a hexagon inscribed in a conic. When we apply polar reciprocation, the conic goes to a conic, because through any point off the conic there are two tangents to it, and they go to two collinear points on the new curve, so the new curve has degree two, so it is a conic. And the vertices of the hexagon, being points of the conic, go to tangents to the new conic, and thus the hexagon goes to a circumscribed hexagon, and the sides of the original hexagon go to the vertices of the new, and the line where extensions of opposite sides meet goes to the point where lines connecting opposite vertices meet. It is, however, not necessary to approach duality through polar reciprocation. Gergonne (1825/26) argued that the same effect is achieved ‘‘by merely exchanging the two words points and lines with one another.’’

This suggests itself from an extension of the above ideas beyond the domain of conics: Any curve can be assigned a dual curve, namely the curve enveloped by the polars of all its points. These two contrasting views led to a dispute between Poncelet and Gergonne; see, e.g., Gray (2007, Chapter 5). Steiner notes in his preface that this dispute is put in perspective by his work, concluding that while ‘‘Gergonne’s principle proves, in retrospect, to be more primitive, closer to the source, Poncelet has made an equally valuable contribution, in his development and furthering of synthetic geometry, so that this field may no longer be disregarded, as it has been all too often and all too frivolously in modern times.’’ Steiner’s Approach to Polar Reciprocation Steiner’s approach to polar reciprocation may be illustrated by his proof of the theorem of Monge studied above. The proof uses the harmonic property of the complete quadrilateral, so we shall prove that first. A harmonic set of points is four points with crossratio 1, ad . ac ¼1 bd bc (or -1 according to many other authors, since, in the typical case, bd is ‘‘negative’’). A complete quadrilateral is a figure determined by four lines. Figure 26 shows a complete quadrilateral with sides a,b,a1,b1 and diagonals AE; a1 E; aC. We wish to show that each diagonal is divided harmonically by the other two, i.e., aDb1 C and a1 DbE and ACBE are all harmonic sets. Consider the three lines a,b,c through a, and consider the fourth line d that makes a, d, b, c harmonic. In the same way, consider the line d1 that makes a1, d1, b1, c harmonic. When the four lines a, d, b, c or the four lines a1, d1, b1, c are intersected by the diagonal aC, the result is then a harmonic set of points. It must be the same set in both cases, by the three-point determinacy, since the two sets have three points in common: a and a1 meet at a, and b and b1 meet at b1 , and c is shared. Therefore, d and d1 must meet at some point D0 that makes aD0 b1 C harmonic. Applying the same argument

Figure 26.

with a1 C in place of aC shows that d and d1 must also meet at some point D00 that makes a1 D00 bE harmonic. But by construction, D0 must be on the diagonal aC, and D00 must be on the diagonal a1 E, so both D0 and D00 must in fact be D, the intersection of these two diagonals. Thus aDb1 C and a1 DbE are harmonic sets, as was to be shown. Now we are ready for Steiner’s proof of Monge’s theorem above. Consider Figure 43. We wish to show that as the intersection f of the two tangents A1, A3 at the points a1 ; a3 moves in a line y, then the line a1 a3 turns around the point y. Draw two other tangents A, A2 to make a complete quadrilateral. By the harmonic property of the complete quadrilateral, e; y; d; z is a harmonic set of points. Projecting from f, we see that a1 ; y; a3 ; v is a harmonic set of points. Projecting from z, we see that a; y; a2 ; u is a harmonic set of points. But as f moves, a; a2 ; u stay the same. Therefore, y must stay the same also, as was to be shown.

Conclusion Let me now revisit my thesis that Steiner was committed to historical continuity. We have seen that there is good reason to interpret the Systematische Entwickelung in this way. We have also seen that some remarks of Steiner himself point in this direction. This is the sum of my evidence. Biographical accounts of Steiner—of

which there are very few, and none of full length—do not offer direct support for the thesis. On the contrary, the thesis appears to contradict a standard characterisation of Steiner. Criticism of Steiner goes hand in hand with biographical accounts of him as intuitively gifted but unscholarly and somewhat flimsy. Lampe (1900, p. 138), for example, says that ‘‘it is very probable that he never studied the writings of other mathematicians, but merely looked through them to compare his results with those of his predecessors.’’ How, then, is one to explain that the Systematische Entwickelung reads like a monumental unification of classical geometry? The biographers propose an easy solution: They simply maintain that the whole work was more or less ghostwritten by Jacobi, ‘‘who, unlike Steiner, read an extraordinary amount and was well versed in the mathematical literature’’ (Graf (1897, p. 13–14)). Despite never reading any books, Steiner still provides frequent historical references in the Systematische Entwickelung, including, on one occasion, references to 15 mathematicians on a single page (Werke, I, p. 340). Geiser (1874, p. 249) thinks that ‘‘one may assume’’ that it was Jacobi who ‘‘made the careful literary references possible,’’ an opinion shared by Obenrauch (1897, p. 253). Lampe (1900, pp. 138–139) and Biermann (1963, p. 40) also emphasise


27

Figure 43.

Jacobi’s role in teaching the ignorant Steiner about the literature. I believe Jacobi’s role is overstated by these authors. Surely Steiner would have discussed such matters with Jacobi, and indeed there are such indications in the the excerpts from their correspondence published by Jahnke (1903b, e.g., p. 271). But in the same letters Steiner also brings up many references himself. Also, the references in the Systematische Entwickelung are not very careful at all (not up to Jacobian standards); most of the time, Steiner just gives a name with an occasional ‘‘bekanntlich’’ thrown in here and there. And if the references were primarily supplied by Jacobi, then it would perhaps seem strange for Steiner to write 28


explicitly in his preface that ‘‘all important theorems already discovered by others I have, to the extent of my knowledge, credited to their original discoverers.’’ The fact that Steiner gives more frequent references in the Systematische Entwickelung than elsewhere could easily be accounted for without assuming the interference of Jacobi by the fact that the Systematische Entwickelung is fundamentally a unification of classical geometry, so references are highly relevant, whereas many of his other works are, in essence, self-contained, so references would not contribute to the purpose of the work. This is also consistent with the common claim that Steiner did not always give proper

references when he borrowed ideas from others—e.g., Klein (1979, p. 116); apparently Jacobi also had reservations, see Steiner (1833) in Jahnke (1903b, p. 272)—but that he did do so when it lent credibility to his work. For example, as I have noted elsewhere (Bla˚sjo¨ (2005, x1)), in his work on the isoperimetric problem, Steiner (1842b, xx1314) solves a particular subproblem treated unsuccessfully by the Greeks, whereupon he promptly refers to Pappus and others; see also the preface to Steiner (1842a). Furthermore, although some of the references may be considered cosmetic (e.g., pointing out origins of terminology) and could easily have been added by Jacobi, almost all of them are very naturally linked to the mathematical content. Since the standard view is that Steiner ‘‘did not care much for literature study [and] book knowledge’’ and that ‘‘he created ‘his’ geometry from himself, from his exceptional intuition’’ (Biermann (1963, p. 31)), it must be a remarkable coincidence then that his work happens to contain a slew of classical theorems in every section, all set for Jacobi to go over and pen in the references. To sum up, the caricature of Steiner as unscholarly is often propagated but rarely, if ever, backed up by evidence. In fact, this view is so plainly inconsistent with Steiner’s work that its proponents need an elaborate and unsubstantiated ghostwriter theory to protect it. Thus, I feel justified in not regarding this biographical material as disproof of my thesis. Finally, if I may venture a generalisation, I think the factor of adherence to historical continuity is important for understanding 19th-century mathematics beyond Steiner. I feel that the notion of what constitutes legitimate research underwent a quite radical transformation in the late 19th-century, to some extent motivated mathematically but perhaps most of all institutionally, prompted by a great increase in the number of mathematicians, doctoral students, and publication quantity and pace, a process in which historical continuity was largely sacrificed. Indeed, nowadays popular mathematicians such as Riemann and Cantor are celebrated for the revolutionary character of their work, whereas the likes of Steiner are very much

depreciated. Perhaps this is due to a failure to recognise the proper historical setting for these works and, in particular, a lack of appreciation of their vision of mathematics. I, for one, do not believe that the demise of Steiner and that of historical continuity coincide by accident.

Heilbron, J. L. (2007). ‘‘Geometry’’ in Encyc-

mit besonderer Beru¨cksichtigung ihrer

lopædia Britannica Online. Retrieved August 1, 2007. Jacobi, C. G. J. (1843). Letter to his wife, 14

Begru¨ndung in Frankreich und Deutschland und ihrer wissenschaftlichen Pflege in O¨sterreich. Carl Winiker, Bru¨nn.

December 1843. Excerpts in Ahrens

Pascal, B. (1640). Essay pour les coniques.

REFERENCES

Ahrens, W. (1907). Briefwechsel zwischen C.

Jahnke, E. (1903a). Schreiben Jacobis an den Staatsminister v. Eichhorn betreffend

Paris. Smith, D. E., ed. (1959). A Source Book in

G. J. Jacobi und M. H. Jacobi. No. 22 in

Jakob Steiner. Archiv der Mathematik

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und Physik, 4, 277–280.

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Jacobi, C. G. J. (1845). Letter to Staatsminister v. Eichhorn, 8 July 1845. In Jahnke (1903a).

zur

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Jahnke, E. (1903b). Auszu¨ge aus drei Briefen Steiners an Jacobi. Archiv der Mathematik und Physik, 4, 268–277.

Teubner, Leipzig. Baldus, R. (1923). Zur Steinerschen Definition der Projektivita¨t. Math. Ann., 90(1–2), 86–

163–167) and Smith (1959, pp. 326–330). Poncelet, J. V. (1822). Traite´ des proprie´te´s

Steiner, J. (1832). Systematische Entwickelung der Abha¨ngigkeit geometrischer Gestalten von einander, mit Beru¨cksichti-

Klein, F. (1928). Vorlesungen u¨ber die Ent-

gung der Arbeiten alter und neuer

wicklung der Mathematik im 19. Jahrhundert. Teil I. Springer-Verlag, Berlin.

Ball, W. W. R. (1893). An essay on Newton’s

Klein, F. (1979). Development of Mathematics in

Geometer u¨ber Porismen, ProjectionsMethoden, Geometrie der Lage, Transversalen, Dualita¨t und Reciprocita¨t, etc..

‘‘Principia’’. Macmillan and Co., London.

the 19th Century. Math Sci Press, Brook-

G. Fincke, Berlin. In Steiner (1881, pp.

line, Mass. Translation of Klein (1928).

229–460) and Ostwald’s Klassiker der

102.

Biermann, K.-R. (1963). Jakob Steiner: Eine biographische Skizze. Nova acta Leopoldina, 27, 31–45. Bla˚sjo¨, V. (2005). The isoperimetric problem. Amer. Math. Monthly, 112(6), 526–566. Brianchon, C. J. (1806). Sur les surfaces courbes du second degree´. Journal de l’ećole Polytechnique, 17, 297–302. Eng-

Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford University Press, New York.

exakten Wissenschaften, nos. 82–83. Steiner, J. (1833). Letter to Jacobi, 31 December 1833. In Jahnke (1903b).

Kolmogorov, A. N. & Yushkevich, A.-A. P., eds. (1996). Mathematics of the 19th Century, Vol. II: Geometry, Analytic Function Theory. Birkhaüser, Basel.

Steiner, J. (1842a). Sur le maximum et le minimum des figures dans le plan, sur la sphe`re et dans l’espace en geńe´ral. Premier me´moire. Journal fu¨r die reine

lish excerpt in Smith (1959, pp. 331–336). Burckhardt, J. J. (1976). Jakob Steiner. Dic-

Ko¨tter, E. (1901). Die Entwickelung der syn-

und angewandete Mathematik, 24, 93–

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Geiser, C. F. (1874). Zur Erinnerung an Jakob Steiner. Verhandlungen der Schweizerischen Naturforschenden Gesellschaft, 56,

Monge, G. (1799). Geóme´trie descriptive.

de l’e´tendue. Annales de mathe´matiques pures et appliqueés, 16, 209–231.

construction of curves. In Newton (1967–

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215–251. Gergonne, J. (1825/26). Considerations philosophiques sur les e´le´ments de la science

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5(2),

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Lage. Bauer und Raspe, Nu¨rnberg. Zacharias, M. (1912). Einfu¨hrung in die projektive Geometrie. B. G. Teubner, Leipzig. Department of Philosophy, Logic and Scientific Method London School of Economics London United Kingdom e-mail: [email protected]


29

Mathematical Entertainments

The World’s Tallest Cryptic

Michael Kleber and Ravi Vakil, Editors

T

all, isn’t it?’’ ‘‘It tries to be. Choose your entries right, and you can head up as high as you want.’’ ‘‘But then I’ll never reach a final answer!’’ ‘‘Well, that you have to do in the altogether different way described.’’

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11: Emotionless jerk comes after mug and ring (7) 13: Note about comedienne Margaret’s academic organization (6)

Entertainment Editors’ Note To the right sort of puzzle fan, January means one thing: the MIT Mystery Hunt. This annual event challenges teams of MIT students and enthusiasts from around the world to solve a succession of puzzles of all shapes, sizes, and styles, some never before seen. Answers to individual puzzles somehow fit together to reveal the higher-order metapuzzles a team must solve to win—earning them the right to run the hunt the following year. This puzzle is a cryptic crossword which appeared in the 2008 Mystery Hunt, reproduced here with the kind permission of its constructor, Kevin Wald. The ‘‘final answer’’ to this puzzle is a single word, which may not be obvious even if you have solved the crossword part of the puzzle: Kevin‘s directions and clues contain the only hints you’ll get. Readers not familiar with cryptic-style (also called British-style) crossword clues are invited to peruse the National Puzzler’s League’s online guide,1but it is an acquired skill. As an alternative, on page following Clues of this issue you can find a list of the Answers to each of the sets of Across and Up clues. Even with these answers in hand, filling in the grid and discovering the final answer poses a challenge. If all else fails, the puzzles, solutions, and explanations for the entire 2008 MIT Mystery Hunt can be found at http://web.mit.edu/puzzle/www/08/. The Entertainments Editors welcome submissions of crosswords and other puzzles with similar appeal at all levels of accessibility. 14: Our lady, in shackles, reveals the exact length of the final answer to this puzzle (4, 7) 16: Revolutionary gun (4)

obtains

submachine

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1 http://www.puzzlers.org/guide/index.php?expand= cryptics1

28 + 45n: Change caused by an effect of the moon spinning (4) Norse God in love with noise (4) Behead cod; otherwise, it stinks (4) 31 + 45n: Network showing Cavemen, a cartoon about cavemen (3) Do I fill the role of Pierre’s friend? (3) In Arles, you mostly work (3) 32 + 45n: One who is in love, dear — or crazy (6) One who positions things in real confusion (6, var. spelling) Druggie starting to snort printer ink (6) 33 + 45n: Fired after getting old, misguided Dan bombed (10) Hauled around a nob that’s tossed back bubbly (10) Vehicles containing Mr. Weasley, fruit drink, and guns (10) 34 + 45n: Spots headless boys (3) Desire for the riches of the East (3) 37 + 45n: He’d misread an Old English letter (3) Upset Playboy’s founder with expression of disdain (3) The ultimate in hyacinths (3)

38 + 45n: Lunatic erases things for relaxation (6) Lets Ed hurt the presumptive heir (6) Trees filled with, um, a kind of glue (6) 39 + 45n: Metal shirts and shades (5) Ultimately, not suspicious of the Lone Ranger’s sidekick (5) 41 + 45n: Fabrics of the first of seven types (5) Each bird that sings a Weird Al parody (3, 2) 42 + 45n: Evacuated Ferrara with one Italian’s animals (5) Eye part of over-trimmed veal (5) Perform numbers with one actress named Reed (5) 43 + 45n: Humongous nude actor named McKellen (3) Attention: This is a serving of corn (3) Californian airport is negligent (3) 45 + 45n: A mass of ice secondarily absorbs a ten-millionth of a joule (4) Is in the wrong English train, son (4) Shamuses mentioned in issues of an MIT paper (4) 47 + 45n: One unbelievably long time (3) Stimpy’s pal is almost torn apart (3) Where you permanently store data for a Gypsy (3) 49 + 45n: Set is little help (4) I run around, becoming a wreck (4) Ashen after getting red alert (4) 52 + 45n: Some bread, left out for a dullard (3) A cur’s remark is far inferior (3) 54 + 45n: 1,101 here in Quebec (3) Electronic chips is complicated [sic] (3, abbr.) 55 + 45n: Stare with disgust, essentially, at a scripting language (4) Exhibit surprise by staring at the tailbiting snake (4) 56 + 45n: Heard about Damon’s pad (3) Needlefish found in hangars (3)

Ó 2007 Kevin Wald, Volume 31, Number 1, 2009

31

57 + 45n: Not-quite-subdued Scotsman’s cap (3) Leg Maxim perhaps flipped over (3) Lass is to delay heading back (3)

6: Organic compound a biblical book discussed (5)

58 + 45n: Tail wild female singers (4) Legendary king of the Huns, at Long Island (4) Each hurt! (4) Shaft put MDMA into beer (4)

8: Reason for a suit to reflect at both ends (4)

59 + 45n: Prefix on ‘‘form,’’ ‘‘phyll,’’ or ‘‘loch,’’ oddly (6) Dance in a medical facility, wearing a string tie (6)

10: Lump of uranium extracted from mass of condensed vapor (4)

60 + 45n: Variant lyre modified to serve the purposes of a bard (11) Disturbed rest isn’t his motivation for drinking (11) That senior’s remarkably husky vocal quality (11)

15: Three or four centers from the local football team (4)

63 + 45n: Made certain act involving $5 pasta perverse (11) Petey and Gertie Minuit will get drunk in cheap bars (11) Zeroes in Met damaged coins again (11) 65 + 45n: Bed in a small house (3) Mr. Serling’s punishment (3) Posed as madcap Tesla (3) 67 + 45n: Author and she almost get engaged (4) Metro Goldwyn Meyer initially thinks they run things (4, abbr.) Pat’s failing a high school exam (4, abbr.) 68 + 45n: Weapon in a room (3) Seldom ignoring every odd character with a teaching degree (3, abbr.) Despot exhibits psychic power (3, abbr.)

Up 1: Cold, reddish outer layer (5) 3: Wind ought to exist, with oxygen in it (4) 4: Twisted and hurt (5) 5: Scrabble piece shows * but not D (4) 32


7: Hears about flightless birds (5)

9: Socks a biblical prophet (5)

12: As announced, bishopric had to yield (4)

17, 62 + 45n: Italian wines damage dateless veggie dishes (8) Surveillance agents capturing one criminal with dead body parts (8) Lab I rave about is not always the same (8)

One who pledges has abused coke, tritium, and a Bic, say (6, 7) 27 + 45n: Immoderate iron magnate John Jacob, by reputation, has pull (5, 2, 6) The Riga banker is corrupt? Very sad (13) Oy, Cain’s plants adapted proteins used in photosynthesis (13) 29 + 45n: Legendary queen of Carthage finally uttered wedding vow (4) A bird rendered extinct by party after party (4) Deer takes small amount of drugs (4) 30 + 45n: Embarrassed after mostly unnecessary vehicle is frequently punctured (6–7) Philosopher and seer can’t see Dr. Awkward (4, 9) Crusading group and mad doctor reunite (8, 5) 32 + 45n: Allows a bit of discussion of this school (6) Powerful ditty about Roosevelt (6)

18, 64 + 45n: Spilled dirt from large books (6) Former Philippine president’s manuscript about an oil company (6) Mistakenly put Roman ‘‘I’’ in John Wayne’s first name (6)

35 + 45n: Mentioned eatery in an Indian city (5) Discourage retrospective about Senator Kennedy (5) One who catches long fish U.S. Grant’s opponent tossed back (5)

20, 66 + 45n: Naiad frolicking with goddess (5) Weeping leaves Rory disheartened (5) Cooked tater and fish (5)

36 + 45n: Poorly written ‘‘if’’ ran further down the page (5) Ruled by German and British Queen (5)

21, 67 + 45n: Men’s sad, pathetic, irrationality (7) Ms. Zadora left fashionable street musician (7) Shamus turned down pierced Carya glabra seeds (7) 23, 68 + 45n: Top-notch text from Mao, nevertheless (1–3) Cry of rage in Dublin and environs (4) 24, 69 + 45n: Ares, keep gripping a grim, tawdry Caltha palustris (5, 8) This part has no misdirected arrows fired while retreating (8, 5) 25 + 45n: Wild rice I got can be created by prokaryotes (13) Cause excitment in Paris’s summer with 999 (about a thousand) spacecraft components (6, 7)

40 + 45n: The Way Grain Spills (3) Using your tongue, draw a digit (3) Also shouted at (3) 44 + 45n: In the style of Somerville’s leadership, unfortunately (4) A tavern turned Egyptian, perhaps (4) That’s cute—a youngster’s heading out (4) 46 + 45n: Six-legged critter infesting them Mets (5) Send money found in clock back (5) Nonsense about (for example) somersaulting thesaurus writer (5) 48 + 45n: No, I’m (gevalt!) the guy who played Spock (5)

Bloom and Minderbinder’s Greek island (5) Caroming truck follows Monsieur Marcel Marceau et al. (5) 50 + 45n: Prize is a reversible tie (5) ‘‘Flying Ur’’ is a former name for a flying company (5) 51 + 45n: Morse code symbol had rotated (3) That item following ‘‘D’’ is a Morse code symbol (3) Name the Venetian ‘‘Zilch’’ (3)

52 + 45n: Dramatic sequence within farce (3) Turned to codeine, originally not requiring a prescription (3, abbr.) Sphere a male sib spun (3) 53 + 45n: Assemble like soldiers entirely enthralled by Flipper (4, 2) Leaf is folded up to form a bra pad (6) 57 + 45n: Reportedly obtain water from the French Quarter of a city (6)

Leave unfinished number with Faust’s author (6) Band featuring Matt Johnson and wildly het twins (3, 3) 61 + 45n: Poetically superior to ‘‘Titania’’ or ‘‘cinnabar’’ in sound (1’2) Rabbi and former actor Harrison (3) Criticize inscription on a tombstone (3)

(Answers on next page)


33

Tallest Cryptic Clue Answers and Brief Explanations In the explanations, ‘‘ana’’ = anagram, ‘‘rev’’ = reversal, ‘‘hom’’ = homophone, and ‘‘double def’’ = double denfinition.

Across 2 DOWN TO EARTH (DARTH ‘‘comprehends’’ WOE NOT ana) 11 ROBOTIC (TIC ‘‘comes’’ after ROB + O) 13 SCHOOL (SOL ‘‘about’’ CHO) 14 FOUR LETTERS (OUR + L ‘‘in’’ FETTERS) 16 STEN (NETS rev) 18 FEDERATED (FETED ‘‘getting’’ DER + A) 19 TODD ([tha]T + ODD) 21 PERI (RIPE ana) 22 SEAM (SEEM hom) 25 + 45n BREW (HEBREW - HE), PEEK (PIQUE hom), REIN (RAIN hom) 26 + 45n LAHR (LA + H[ono]R), LIFT (LI + FT), RAPS (SPAR rev) 28 + 45n EDIT (TIDE rev), ODIN (O + DIN), ODOR ([c]OD + OR) 31 + 45n ABC (A + BC), AMI (AM I), TOI (TOI[l]) 32 + 45n ADORER (DEAR OR ana), ALINER (IN REAL ana), STONER (S[nort] + TONER) 33 + 45n CANNONADED (CANNED ‘‘after getting’’ O + DAN rev), CARBONATED (CARTED ‘‘around’’ A + NOB rev), CARRONADES (CARS ‘‘containing’’ RON + ADE) 34 + 45n ADS ([l]ADS), YEN (double def) 34


37 + 45n EDH (HE’D ana), FEH (HEF rev), NTH (‘‘in’’ hyaciNTHs) 38 + 45n EASERS (ERASES ana), ELDEST (LETS ED ana), ELMERS (ELMS ‘‘filled with’’ ER) 39 + 45n TINTS (TIN + TS), TONTO ([no]T + ONTO) 41 + 45n SILKS (S[even] + ILKS), EAT IT (EA + TIT) 42 + 45n FAUNA (F[errar]A + UNA), FOVEA (OF rev + VEA[l]), DONNA (DO + NN + A) 43 + 45n IAN ([g]IAN[t]), EAR (double def), LAX (double def) 45 + 45n BERG ([a]B[sorbs] + ERG), ERRS (E + RR + S), TECS (TECHS hom) 47 + 45n EON (ONE ana), REN (REN[t]), ROM (double def) 49 + 45n LAID (L + AID), RUIN (I RUN ana), WARN (WAN ‘‘after getting’’ R) 52 + 45n OAF (LOAF - L), ARF (FAR ana) 54 + 45n ICI (I + CI), ICS (SIC ana) 55 + 45n GAWK ([dis]G[ust] + AWK), GASP ([starin]G + ASP) 56 + 45n MAT (MATT hom), GAR (‘‘found in’’ hanGARs)

ana ‘‘in’’ DIVES), REMONETIZES (ZEROES IN MET ana) 65 + 45n COT (double def), ROD (double def), SAT (AS ana + T) 67 + 45n MESH (ME + SH[e]), MGMT (MGM + T[hinks]), PSAT (PAT’S ana) 68 + 45n ARM (A + RM), EDM ([s]E[l]D[o]M), ESP (what dESPot ‘‘exhibits’’)

Up 1 CRUST (C + RUST) 3 OBOE (O + BE, ‘‘with’’ O ‘‘in it’’) 4 WOUND (double def) 5 TILE (TILDE - D) 6 ESTER (ESTHER hom) 7 RHEAS (HEARS ana) 8 TORT (TO + R[eflec]T) 9 HOSEA (HOSE + A) 10 CLOD (CLOUD - U) 12 CEDE (SEE’D hom) 15 TRIO ([pa]TRIO[ts])

57 + 45n TAM (TAM[e]), GAM (MAG rev), GAL (LAG rev)

17, 62 + 45n MARSALAS (MAR + SALADS - D), TOENAILS (TAILS ‘‘capturing’’ ONE ana), VARIABLE (LAB I RAVE ana)

58 + 45n ALTI (TAIL ana), ATLI (AT + LI), ACHE (EACH ana, & lit), AXLE (‘‘put’’ X ‘‘into’’ ALE)

18, 64 + 45n FOLIOS (SOIL + OF rev), MARCOS (MS ‘‘about’’ ARCO), MARION (ROMAN I ana)

59 + 45n CHLORO (OR LOCH ana), BOLERO (ER ‘‘wearing’’ BOLO)

20, 66 + 45n DIANA (NAIAD ana), TEARY (TEA + R[or]Y), TETRA (TATER ana)

60 + 45n NARRATIVELY (VARIANT LYRE ana), THIRSTINESS (REST ISN’T HIS ana), THROATINESS (THAT SENIOR’S ana)

21, 67 + 45n MADNESS (MEN’S SAD ana), PIANIST (PIA + IN rev + ST), PIGNUTS (PI + STUNG rev)

63 + 45n DEFINITIZED (DEED ‘‘involving’’ FIN + ZITI rev), DIMINUTIVES (MINUIT

23, 68 + 45n A-ONE (‘‘from’’ mAO NEvertheless), EIRE (IRE hom)

24, 69 + 45n MARSH MARIGOLD (MARS + HOLD ‘‘gripping’’ A GRIM ana), PARTHIAN SHOTS (THIS PART HAS NO ana) 25 + 45n BACTERIOGENIC (RICE I GOT CAN BE ana), ROCKET ENGINES (ROCK + ETE + NINES ‘‘about’’ G), POCKET LIGHTER (PLIGHTER ‘‘has’’ COKE ana + T) 27 + 45n FEAST OR FAMINE (FE + ASTOR + FAME ‘‘has’’ IN), HEARTBREAKING (THE RIGA BANKER ana), PLASTOCYANINS (OY CAIN’S PLANTS ana) 29 + 45n DIDO ([uttere]D + I DO), DODO (DO ‘‘after’’ DO), DOSE (DOE ‘‘takes’’ S) 30 + 45n NEEDLE-SCARRED (RED ‘‘after’’NEEDLES[s] + CAR), RENE DESCARTES(SEER CAN’T SEE DR ana), TEUTONIC ORDER (DOCTOR REUNITE ana)

32 + 45n ADMITS (A + D[iscussion] + MIT’S), STRONG (SONG ‘‘about’’ TR)

50 + 45n AWARD (A + DRAW rev), USAIR (UR IS A ana)

35 + 45n DELHI (DELI hom), DETER (RE TED rev), EELER (R. E. LEE rev)

51 + 45n DAH (HAD rev), DIT (IT ‘‘following’’ D), NIL (N + IL)

36 + 45n INFRA (IF RAN ana), UNDER (UND + ER)

52 + 45n ARC (‘‘within’’ fARCe), OTC (TO rev + C[odeine]), ORB (BRO rev)

40 + 45n TAO (OAT rev), TOE (TOW hom), TOO (TO hom) 44 + 45n ALAS (ALA + S[omerville]), ARAB (A + BAR rev), AWAY (AW + A + Y[oungster]) 46 + 45n EMMET (‘‘infesting’’ thEM METs), REMIT (TIMER rev), ROGET (ROT ‘‘about’’ E.G. rev) 48 + 45n NIMOY (N + I’M + OY), MILOS (double def), MIMES (SEMI rev ‘‘follows’’ M)

53 + 45n FALL IN (ALL ‘‘enthralled by’’ FIN), FALSIE (LEAF IS ana) 57 + 45n GHETTO (GET EAU hom), GOETHE (GO + ETHE[r]), THE THE (HET ana + HET ana) 61 + 45n O’ER (ORE hom), REX (R + EX), RIP (double def) Ab Initio Software 201 Spring Street Lexington, MA 02421 USA e-mail: [email protected]


35

The Search for Quasi-Periodicity in Islamic 5-fold Ornament PETER R. CROMWELL Introduction

T

he Penrose tilings are remarkable in that they are non-periodic (have no translational symmetry) but are clearly organised. Their structure, called quasiperiodicity, can be described in several ways, including via self-similar subdivision, tiles with matching rules, and projection of a slice of a cubic lattice in R5 . The tilings are also unusual for their many centres of local 5-fold and 10fold rotational symmetry, features shared by some Islamic geometric patterns. This resemblance has prompted comparison, and has led some to see precursors of the Penrose tilings and even evidence of quasi-periodicity in traditional Islamic designs. Bonner [2] identified three styles of self-similarity; Makovicky [20] was inspired to develop new variants of the Penrose tiles and later, with colleagues [24], overlaid Penrose-type tilings on traditional Moorish designs; more recently, Lu and Steinhardt [17] observed the use of subdivision in traditional Islamic design systems and overlaid Penrose kites and darts on Iranian designs. The latter article received widespread exposure in the world’s press, although some of the coverage overstated and misrepresented the actual findings. The desire to search for examples of quasi-periodicity in traditional Islamic patterns is understandable, but we must take care not to project modern motivations and abstractions into the past. An intuitive knowledge of group theory is sometimes attributed to any culture that has produced repeating patterns displaying a wide range of symmetry types, even though they had no abstract notion of a group. There are two fallacies to avoid: 36

THE MATHEMATICAL INTELLIGENCER 2008 Springer Science+Business Media, LLC.

• abstraction: P knew about X and X is an example of Y therefore P knew Y. • deduction: P knew X and X implies Y therefore P knew Y. In both cases, it is likely that P never thought of Y at all, and even if he had, he need not have connected it with X. In this article I shall describe a tiling-based method for constructing Islamic geometric designs. With skill and ingenuity, the basic technique can be varied and elaborated in many ways, leading to a wide variety of complex and intricate designs. I shall also examine some traditional designs that exhibit features comparable with quasi-periodic tilings, use the underlying geometry to highlight similarities and differences, and assess the evidence for the presence of quasi-periodicity in Islamic art. A few comments on terminology. Many of the constructions are based on tilings of the plane. A patch is a subset of a tiling that contains a finite number of tiles and is homeomorphic to a disc. I use repeat unit as a generic term for a template that is repeated using isometries to create a pattern; it is not so specific as period parallelogram or fundamental domain. A design or tiling with radial symmetry has a single centre of finite rotational symmetry. The other terminology follows [8] for tilings, supplemented by [33] for substitution tilings.

Islamic Methods of Construction Although the principles of Islamic geometric design are not complicated, they are not well-known. Trying to recover the principles from finished artwork is difficult, as the most

conspicuous elements in a design are often not the compositional elements used by the designer. Fortunately medieval documents that reveal some of the trade secrets have survived. The best of these documents is manuscript scroll MS.H.1956 in the library of the Topkapi Palace, Istanbul. The scroll itself is a series of geometric figures drawn on individual pages, glued end to end to form a continuous sheet about 33 cm high and almost 30 m long. It is not a ‘how to’ manual, as there is no text, but it is more than a pattern book as it shows construction lines. A halfsize colour reproduction can be found in [25], which also includes annotations to show the construction lines and marks scored into the paper with a stylus, which are not visible in the photographs. References in this article to numbered panels of the Topkapi Scroll use the numbering in [25]. Islamic designs often include star motifs. These come in a variety of forms but, in this article, we need only a few simple shapes that correspond to the regular star polygons of plane geometry. Taking n points equally spaced around a circle and connecting points d intervals apart by straight lines produces the star polygon denoted by {n/d}. This, however, is the star of the mathematician; it is rare for an artist to use the whole figure as an ornamental motif. More often, the middle segments of the sides are discarded. Many of the early Islamic designs are created by arranging 6-, 8- or 12-point stars at the vertices of the standard grids of squares or equilateral triangles. The more general rhombic lattice allows other stars to be used. An example based on {10/3} is shown in Figure 1(a). The angles in the rhombus are 72 and 108, both being multiples of 36—the angle between adjacent spikes of the star. Draw a set of circles of equal radius centred on the vertices of the lattice and of maximal size so that there are points of tangency. Place copies of the star motif in the circles so that spikes fall on the edges of the lattice. This controls the spacing and orientation of the principal motifs, but the

AUTHOR

......................................................................... may be known to readers through his contributions to the Intelligencer on Celtic art (vol. 15, no. 1) and the Borromean rings (vol. 20, no. 1), or his books on polyhedra and knot theory, both published by CUP. He is interested in anything 3-dimensional with a strong visual element, and also combinatorial and algorithmic problems. He was recently awarded a research fellowship by the Leverhulme Trust to work on the mathematical analysis of interlaced patterns.

PETER CROMWELL

Pure Mathematics Division, Mathematical Sciences Building University of Liverpool, Peach Street Liverpool L69 7ZL England e-mail: [email protected]

design is not yet complete. There are some spikes of each motif that are not connected to a neighbouring motif but are free and point into the residual spaces between the circles. The lines bounding these free spikes are extended beyond the circumcircle until they meet similarly produced lines from nearby stars. This simple procedure bridges the residual spaces and increases the connectivity of the star motifs. The same pattern of interstitial filling should be applied uniformly to all the residual spaces and the symmetry of the design as a whole should be preserved as far as possible. The result is shown in Figure 1(c). In this case the kites in the interstitial filling are congruent to those in the star. This pattern is one of the most common decagonal designs, and we shall name it the ‘stars and kites’ pattern for reference. This basic approach produces a limited range of periodic designs with small repeat units and it only works for stars with an even number of points. A more general method that can be used with all stars, and also enables combinations of different stars to be used in a single design, is based on edge-to-edge tilings containing regular convex polygons with more than four sides. Figure 1(b) shows a tiling formed by packing decagons together, leaving nonconvex hexagonal tiles between them. After placing {10/3} stars in each decagon tile, we use the same kind of interstitial filling procedure as before to develop the pattern in the hexagons. This change from circle to polygon may seem minor, but it gives rise to a range of generalisations. We are no longer restricted to a lattice arrangement of the stars—any tiling will suffice. The tiling may contain regular polygons of different kinds allowing different star motifs to be combined in the same design; the tiling naturally determines the relative sizes of the different stars. We can even discard the regular star motifs that initiate the interstitial filling and seed the pattern generation process from the tiling itself. In this last case, we place a pair of short lines in an X configuration at the midpoint of each edge, then extend them until they encounter other such lines—this is similar to applying interstitial filling to every tile. The angle that the lines make with the edges of the tiling, the incidence angle, is a parameter to be set by the artist and it usually takes the same value at all edges. There is no requirement to terminate the line extensions at the first point of intersection; if there are still large empty regions in the design, or it is otherwise unattractive, the lines can be continued until new intersections arise. This technique, known as ‘polygons in contact’ (PIC), was first described in the West by Hankin [9–13], who observed the polygonal networks scratched into the plaster of some designs, while working in India. Many panels in the Topkapi Scroll also show a design superimposed on its underlying polygonal network. Although the purpose of the networks is not documented, it does not seem unreasonable to interpret them as construction lines. Bonner [2, 3] argues that PIC is the only system for which there is evidence of historical use by designers throughout the Islamic world. The method is versatile and can account for a wide range of traditional patterns, but it is not universally applicable. An alternative approach is used by Caste´ra [5], 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

37

(a)

(b)

(c) Figure 1. The ‘stars and kites’ pattern.

who arranges the shapes seen in the final design without using a hidden grid. The PIC method is illustrated in the next four figures. Figure 2 shows two designs produced from a tiling by regular decagons, regular pentagons, and irregular convex hexagons. In part (b), a star motif based on {10/4} is placed in the decagon tiles, which gives an incidence angle of 72 for the other edges; the completed design is one of the most widespread and frequently used of all star patterns. Part (c) shows a design that is common in Central Asia and based on {10/3} with an incidence angle of 54. A {10/2} star and an incidence angle of 36 reproduces the stars and kites pattern. The design in Figure 3 is from [14] and contains star motifs based on {7/3}; in the tiling the 7-gons are regular but the pentagons are not. Figure 4 is based on a 38


tiling containing regular 9-gons and 12-gons. I have chosen an incidence angle of 55 to make the convex 12-gon elements in the design into regular polygons and some line segments inside the non-convex hexagonal tiles join up without creating a corner, but, as a consequence, neither star motif is geometrically regular. Plates 120–122 in [4] are traditional designs based on the same tiling. Figure 5 shows a design with 10-fold rotational symmetry based on panel 90a of the Topkapi Scroll, which Necipog˘lu labels as a design for a dome [25]. The original panel shows a template for the figure containing one-tenth of the pattern with the design in solid black lines superimposed on the tiling drawn in red dotted lines. Notice that some of the tiles are two-tenths and three-tenths sectors of a decagon. Domes were also decorated by applying PIC to polyhedral

(a) Underlying polygonal network

(b) Incidence angle 72°

(c) Incidence angle 54°

Figure 2. A tiling and two star patterns derived from it. The petals of a rose motif

in each pattern are highlighted. networks. Patterns with a lower concentration of stars were produced by applying PIC to k-uniform tilings composed of regular 3-, 4-, 6-, and 12-sided polygons—see plates 77, 97, and 142 in [4] for some unusual examples. The two designs of Figure 2 display another common Islamic motif. In each design, a set of hexagons surrounding a star has been highlighted in grey. The enlarged star motif is called a rose and the additional hexagons are its petals. In this case, the rose arises because the decagon in the underlying tiling is surrounded by equilateral polygons, but they can also be constructed using a set of tangent circles around the circumcircle of the star [16] and used as compositional elements in their own right.

You can see the PIC method in action and design your own star patterns using Kaplan’s online Java applet [34]— you select a tiling and the incidence angles of the star motifs, then inference logic supplies the interstitial pattern. The tilings used as the underlying networks for the PIC method of construction often have a high degree of symmetry, and they induce orderly designs. Islamic artists also produced designs that appear to have a more chaotic arrangement of elements with local order on a small scale but little long-range structure visible in the piece shown. Panels in the Topkapi Scroll reveal that these designs, too, have an underlying polygonal network assembled from copies of a small set of equilateral tiles (see Figure 6) whose angles are multiples of 36: 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

39

(a)

(b)

Figure 3. Design containing regular 7-point stars.

(a)

(b)

Figure 4. Design containing 9- and 12-point stars.

• • • •

a rhombus with angles 72 and 108 a regular pentagon (angles 108) a convex hexagon with angles 72 and 144—the bobbin a convex hexagon with angles 108 and 144—the barrel • a non-convex hexagon with angles 72 and 216—the bow-tie • a convex octagon with angles 108 and 144 • a regular decagon (angles 144). The motifs on the tiles are generated using the PIC method with an incidence angle of 54. The barrel hexagon and the 40


decagon have two forms of decoration. One decagon motif is just the star {10/3} and its constituent kites are congruent to those on the bow-tie; the other decagon motif is more complex and the symmetry is reduced from 10-fold to 5-fold rotation. The shapes of the tiles arise naturally when one tries to tile with decagons and pentagons. The bow-tie and barrel hexagons are familiar from the previous figures. The octagon and the remaining hexagon are shapes that can be obtained as the intersection of two overlapping decagons. The motif on the hexagon resembles a spindle or bobbin wound with yarn. This distinctive motif is easy to locate in a

Figure 5. Design from panel 90a of the Topkapi Scroll.

design, and its presence is a good indication that the design could be constructed from these tiles. The promotion of irregular tiles from supplementary shapes to compositional elements in their own right marked a significant development in Islamic design. Regarding the tiles as the pieces of a jigsaw allows a less formal approach to composition. A design can be grown organically in an unplanned manner by continually attaching tiles to the boundary of a patch with a free choice among the possible extensions at each step. This new approach gave artists freedom and flexibility to assemble the tiles in novel ways and led to a new category of designs. It seems to have been a Seljuk innovation as examples started to appear in Turkey and Iran in the 12th–13th centuries. The widespread and consistent use of these decorated tiles as a design system was recognised by Lu and Steinhardt [17]; similar remarks appear in Bonner [2] and the tiles are also used by Hankin [10]. Figure 7 shows small patches of tiles. There are often multiple solutions to fill a given area. Even in the simple

Rhombus

Bow-tie

Pentagon

Bobbin

combination of a bobbin with a bow-tie shown in part (a), the positions of the tiles can be reflected in a vertical line so that the bow-tie sits top-right instead of top-left. The patch in part (d) can replace any decagonal tile with a consequent loss of symmetry, as the bow-tie can point in any of ten directions. Patches (b) and (c) are another pair of interchangeable fillings with a difference in symmetry. Figure 8 shows some traditional designs made from the tiles. Parts (a) and (b) are from panels 50 and 62 of the Topkapi Scroll, respectively; in both cases the original panel shows a template with the design in solid black lines superimposed on the tiling drawn in red dots. The designs in parts (b), (c), and (d) are plates 173, 176, and 178 of [4]. The designs of (e) and (f) are Figures 33 and 34 from [16]. The edges of the tilings are included in the figures to show the underlying structure of the designs, but in the finished product these construction lines would be erased to leave only the interlaced ribbons. This conceals the underlying framework and helps to protect the artist’s method. The viewer sees the polygons of the background outlined by the ribbons, but these are artifacts of the construction, not the principal motifs used for composition. The internal angles in the corners of the tiles are all multiples of 36 so all the edges in a tiling will point in one of five directions—they will all lie parallel to the sides of a pentagon. Fitting the tiles together spontaneously produces regular pentagons in the background of the interlacing, and centres of local 5-fold or 10-fold rotational symmetry in the design. This symmetry can be seen in some of the configurations of Figure 7. However, in patterns generated by translation of a template, this symmetry must break down and cannot hold for the design as a whole. This is a consequence of the crystallographic restriction: the rotation centres in a periodic pattern can only be 2-, 3-, 4- or 6-fold. This was not proved rigorously until the 19th century but it must surely have been understood on an intuitive level by the Islamic pattern makers. Perhaps these tilings were appealing precisely because they contain so many forbidden centres; they give the illusion that one can break free from this law of nature. Unfortunately, when a large enough section of a tiling is shown for the periodicity to be apparent, any (global) rotation centres are only 2-fold, and

Barrel (1)

Decagon (1)

Barrel (2)

Octagon

Decagon (2)

Figure 6. An Islamic set of prototiles. 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

41

(a)

(b) (f)

(c)

(d)

(g)

(e) Figure 7. Small patches of tiles.

the symmetry type of the (undecorated) tiling is usually one of pgg, pmm or, more commonly, cmm. Figure 9(b) shows the design on one wall of the Gunbad-i Kabud (Blue Tower) in Maragha, north-west Iran; similar designs decorate the other sides of the tower. At first sight the design appears to lack an overall organising principle but it fits easily into the framework shown in Figure 9(a). Centred at the bottom-right corner of the panel is the patch of Figure 7(g) surrounded by a ring of decagons. A similar arrangement placed at the topleft corner abuts the first, leaving star-shaped gaps. The rings of decagons are filled with the patch of Figure 7(d) with the bow-ties facing outwards, except for the one on the bottom edge of the panel, which is filled with a 42


decagonal tile. The star-shaped gaps are filled with the five rhombi of Figure 7(b). The design does contain irregularities and deviations from this basic plan, particularly in the bottom-left corner of the panel. Also the decagon in the top-left corner is filled with Figure 7(d) rather than a decagonal tile. Figure 9(a) can also be taken as the foundation of the design shown in Figure 10. The centres of the rose motifs in the centre of the figure and in the top-left corner are diagonally opposite corners of a rectangle that is a repeat unit for the design. The underlying framework in this rectangle is the same as that of the Maragha panel. The full design is generated from this cell by reflection in the sides of the rectangle. Note that it is the arrangement of the tiles that is

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 8. Periodic designs.

(a)

(b)

Figure 9. Design from the Gunbad-i Kabud, Maragha, Iran. 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

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Figure 10. Design from the Karatay Madrasa, Konya, Turkey.

reflected, not the tiles with their decorative motifs; the interlacing of the full design remains alternating. The boundaries of the unit rectangle are mostly covered by the sides of tiles or mirror lines of tiles, both of which ensure continuity of the tiling across the joins. However, in the topright and bottom-left corners (the cell has 2-fold rotational symmetry about its centre), the tiles do not fit in the rectangle but overhang the edges. This is not a problem with this method of generating designs: the overhanging tiles are simply cut to fit and the reflections take care of the continuity of the ribbons. In Figure 10 this is most obvious in the middle near the bottom where pairs of bow-ties and bobbins merge. The centre of the tiling can be filled with the 44


patch shown in Figure 7(g) but this has been discarded in favour of a large rose motif. A different construction for this pattern is presented by Rigby in [26]. When experimenting with the tiles of Figure 6, one soon learns that those in the top row are more awkward to use than the others—the 108 angles must occur in pairs around a vertex and this limits the options. Indeed many designs avoid these tiles altogether and are based solely on the three shapes in the bottom row. The design in Figure 11 is unusual in that it is largely composed of awkward tiles (rhombi, pentagons, and octagons) together with a few bobbins. The large star-shaped regions in the tiling can be filled with the patch shown in Figure 7(f),

Figure 11. Design from the Sultan Han, Kayseri, Turkey.

(a)

(b)

(c)

Figure 12. Subdivisions ofpthree ffiffiffi tiles into smaller copies of the same three tiles. 1

The scale factor is

2

7þ

5 4:618.

continuing the use of the same set of tiles, but instead this motif is replaced by the star {10/4}. Once a design has been constructed, it can be finished in different ways according to context and the materials used. In some of the accompanying figures, the regions have been given a proper 2-colouring (chessboard shading), in others the lines have been made into interlaced ribbons. The basic line drawing can also be used by itself as when it is inscribed in plaster.

What is Quasi-Periodicity? The discovery of crystalline metal alloys with 5-fold symmetry in their diffraction patterns caused great excitement in the 1980s. Sharp spots in a diffraction pattern are evidence of long-range order which, at that time, was synonymous with periodicity, but 5-fold rotations are incompatible with the crystallographic restriction so a new kind of phenomenon had been observed. The novel solids

became known as quasi-crystals and the underlying order as quasi-periodicity. For crystallographers, the production of sharply defined points in a diffraction pattern is a defining characteristic of quasi-periodicity. In the study of the decorative arts, however, the term ‘quasi-periodic’ is used somewhat informally and does not have an agreed definition. Readers should be aware of this potential source of confusion when comparing papers. For the tilings and the related geometric designs discussed in this article, one option is to impose a homogeneity condition on the distribution of local configurations of tiles (this is weaker than the crystallographic definition). This and other properties will be illustrated through the following example. The example is constructed from the patches shown in Figure 12. The patches are chosen only to demonstrate the technique and not for any artistic merit—the unbalanced distribution of bow-ties leads to poor designs. Any patch tiled by bow-ties, bobbins, and decagons can be converted 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

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Figure 13. A step in the construction of a quasi-periodic tiling.

into a larger such patch by subdividing each tile as shown in the figure and then scaling the result to enlarge the small tiles to the size of the originals. This process of ‘subdivide and enlarge’ is called inflation. Each side of each composite tile is formed from two sides of small tiles and the major diagonal of a small bobbin; in the inflated tiling the half-bobbins pair up to form complete tiles. Let P0 be a single decagon and let Pi+1 be the patch obtained by inflating Pi for all i 2 N . Figure 12(b) shows P1 and Figure 13 shows P2. We can iterate the inflation process to tile arbitrarily large regions of the plane. Furthermore, because P1 contains a decagon in the centre, Pi+1 contains a copy of Pi in the middle. Therefore Pi+1 is an extension of Pi, and by letting i go to infinity we can extend the patch to a tiling, P?, of the whole plane. Notice that the symmetry of the initial patch is preserved during inflation so P? will have a global centre of 10-fold symmetry and hence cannot be periodic. In general, inflation only provides the ability to create arbitrarily large patches that need not be concentric, so some work is required to show that the limit exists and it is a tiling of the plane [19]. Two tilings are said to be locally indistinguishable if a copy of any patch from one tiling occurs in the other tiling, and vice versa. The family of substitution tilings defined by the prototiles and subdivisions shown in Figure 12 is the set of all tilings that are locally indistinguishable from P?. There are, in fact, an 46


uncountable number of tilings in the family but any patch in any one of them will be contained in some Pn. The basic combinatorial properties of a substitution tiling based on a finite set of n prototiles T1,...,Tn can be encoded in an n 9 n matrix: the entry in column j of row i is the number of small Ti in a composite Tj. For the example here with the tiles in the order bow-tie, bobbin, decagon, this substitution matrix is ! 10 5 20 7 11 25 : 0 2 11 A matrix is said to be primitive if some power of it has only positive non-zero entries. If a substitution matrix is primitive then the patch of tiles produced by repeated inflation of any tile will eventually contain copies of all the prototiles. Properties of the tiling can be derived from the algebraic properties of a primitive matrix. For example, the largest eigenvalue is the square of the scale factor of the inflation and the corresponding eigenvector contains the relative frequencies of the prototiles in a full tiling of the plane; the corresponding eigenvector of the transposed matrix contains the relative areas of the three prototiles. Inffiffiffi our p example the frequency eigenvector is p ffiffiffi 5 þ 5 5; 5 þ 7 5; 4 . Since some of the ratios between the entries are irrational, any substitution tiling made from these subdivisions is non-periodic [30, 31].

Although our substitution tilings have no translational symmetry, they do share some properties with periodic tilings. First each tiling is edge-to-edge; it is constructed from a finite number of shapes of tile, each of which occurs in a finite number of orientations; there are finitely many ways to surround a vertex. The tiling is said to have finite local complexity. For primitive substitution tilings this has an important consequence: given any patch X in the tiling there is some number R such that a disc of radius R placed anywhere on the tiling will contain a copy of X. A tiling with this property is called repetitive. This means that copies of any finite portion of the tiling can be found evenly distributed throughout the tiling. You cannot determine which part of the tiling is shown in any finite diagram of it. For the purposes of this article, a tiling is called quasiperiodic if it is non-periodic, has finite local complexity, and is repetitive. By extension we can call an Islamic design constructed using the PIC method quasi-periodic if its underlying polygonal network is a quasi-periodic tiling. Unfortunately, it is impossible to tell from any finite subset of a tiling whether it is quasi-periodic or not. So, in order to assert that a tiling could be quasi-periodic, we need to identify a process such as inflation that could have been used to generate the piece shown and can also be used to generate a complete quasi-periodic tiling.

Multi-level Designs Some panels of the Topkapi Scroll show designs of different scales superimposed on one another. This interplay of designs on multiple scales is a feature of some large Islamic designs found on buildings where viewers experience a succession of patterns as they approach. From a distance,

(a) Panel 28

(c) Panel 32

large-scale forms with high contrast dominate but, closer in, these become too large to perceive and smaller forms take over. Early methods to achieve this transition from big and bold through medium range to fine and delicate were simple, often just a matter of progressively filling voids in the background to leave a design with no vacant spaces. (There is a secondary pattern of this form on the Gunbad-i Kabud.) Differences in size and level of detail were expressed using variation in density, depth of carving, colour and texture. Later designs are more ambitious and use the same style on more than one scale. It is even possible to re-use the same pattern. Designs that can be read on several scales are often referred to as self-similar but this term itself has multiple levels of meaning. In its strictest sense it means scale invariant: there is a similarity transformation (an isometry followed by an enlargement) that maps the design onto itself. The transformation can be weakened to a topological equivalence—for example the homeomorphisms in iterated function systems leading to fractals. In a weaker sense still, it means only that motifs of different scales resemble each other in style or composition but are not replicas. We shall use the term hierarchical for multi-level designs of this latter form. In panel 28 of the Topkapi Scroll three drawings are superimposed on the same figure: a small-scale polygonal network is drawn in red dots, the corresponding smallscale design is drawn in a solid black line, and a large-scale design is added in a solid red line. The polygonal network corresponding to the large-scale design is not shown but can be deduced—the two polygonal networks are shown superimposed in Figure 14(a). The other parts of the figure

(b) Panel 31

(d) Panel 34

Figure 14. Underlying 2-level polygonal networks of panels from the Topkapi

Scroll. 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

47

(a)

(d)

(b)

(e) (c) Figure pffiffiffi15. Subdivisions derived from the Topkapi Scroll. The scale factor is

3þ

5 5:236.

show the polygonal networks underlying three more 2level designs from the scroll, but neither of the networks is shown in these panels, only the finished 2-level designs in black and red. Superimposing the large- and small-scale polygonal networks of these panels reveals subdivisions of some of the tiles: a rhombus in panel 28, two pentagons in panel 32, and a bobbin in panel 34. In all cases, the side of a composite tile is formed from the sides of two small tiles and the diagonal of a small decagon. We can also identify the fragments of the large-scale polygons cropped by the boundaries of the panels. These panels are not arbitrarily chosen parts of a design—they are templates to be repeated by reflection in the sides of the boundary rectangle. Although a superficial glance at Figure 14(d) might suggest that the large-scale network is a bobbin surrounded by six pentagons, a configuration that can be seen in the smallscale network, reflection in the sides generates rhombi, pentagons, and barrels. The large-scale design generated by panel 31 is shown in Figure 8(g). Panel 28 appears to be truncated on the right and is perhaps limited by the available space. If it had 2-fold rotational symmetry about the centre of the large rhombus, the large-scale design would be that of Figure 8(h). A consistent choice of subdivision emerges in all four panels and the subdivisions of the five 48


tiles used are shown in Figure 15. I believe this has not been reported before. Figure 16 shows my 2-level design based on panel 32. The composite tiles generate the large-scale design (shown in grey) and the small tiles generate a small-scale design (black and white) that fills its background regions. The barrel tile has two forms of decoration: I have used the simple motif for the large-scale design and the other motif on the small-scale design. Completing the small-scale design in the centre of a composite pentagon is problematic. For a pentagon of this scale, only a partial subdivision is possible: once the half-decagons have been placed, one is forced to put pentagons at the corners; only a pentagon or a barrel can be adjacent to the corner pentagons, and both cases lead to small areas that cannot be tiled. The grey area in Figure 15(b) indicates one such essential hole. I have chosen a slightly different filling from the one in the Topkapi Scroll. The large-scale design is that of Figure 2(c). Figure 17 gives a similar treatment to panel 34. It contains four copies of the template rectangle shown in Figure 14(d), two direct and two mirror images. In this case, the large-scale pattern is expressed using shading of the regions. Examples of both styles can be found on buildings in Isfahan, Iran.

Figure 16. A 2-level design based on panel 32 of the Topkapi Scroll.

The bow-tie is notable by its absence from Figure 15. It suffers the same fate as the pentagon: the tiles at its two ends are forced and its waist cannot be tiled. (The largescale polygonal network underlying panel 29 of the scroll has a quarter of a bow-tie in the top right corner surrounded by pieces of decagons, but it is not based on subdivision in the same way as the others.) In Figure 16 the visible section of the large-scale design can also be found as a configuration in the small-scale design. However, larger sections reveal that the pattern is not scale invariant. This is a general limitation of these subdivisions. It is not possible to use the subdivisions of Figure 15 as the basis of a substitution tiling because,

without subdivisions of the pentagon and bow-tie, the inflation process cannot be iterated.

A Design from the Alhambra The design illustrated in Figure 18 forms the major part of a large panel in the Museum of the Alhambra—see [24] for a photograph. The panel has been assembled from fragments uncovered in 1958, but the original would have been from the 14th century. The lower part of the figure shows the finished design and the upper part shows a polygonal network that I propose as the underlying framework. The principal compositional element of the framework is a decagon surrounded by ten pentagons, which gives rise to 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

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Figure 17. A 2-level design based on panel 34 of the Topkapi Scroll.

the 10-fold rose recurring as a leitmotif in the final design. Copies of this element are placed in two rings, visible in the top left of the figure—an inner ring of ten and an outer ring of twenty; adjacent elements share two pentagons. The connections between the inner and outer rings are of two kinds. The shaded rhombi contain the translation unit from the familiar periodic design of Figure 2(b). The construction of the design in the remaining spaces is shown in Figure 19: in part (b) the design is seen to be a subset of the configuration of pentagonal motifs of part (a), whereas (c) shows the same design over a network that includes halfbarrels and one-tenth decagons—the polygons used in Figure 18. The edges in the resulting polygonal network are of two lengths, which are related as the side and diagonal of a pentagonal tile. The final design can be generated from this network using a generalisation of the PIC method: the short edges have incidence angle 72 and the long edges have incidence angle 36. A 20-fold rose is placed in the centre; the tips of alternate petals meet 10fold roses, and lines forming the tips of the intermediate petals are extended until they meet other lines in the pattern. The reconstructed rectangular panel also has quadrants of 20-fold roses placed in the four corners, a common feature of such panels that reflects the fact that most are subsets of periodic patterns. However, the quadrants are misaligned and are also the most heavily restored areas of the panel. I have omitted them from the figure. 50


This design is unusual in the large number of straight lines it contains that run across the figure almost uninterrupted. The marks in the bottom right corner of Figure 18 indicate the heights of horizontal lines; there are five families of parallels separated by angles of 36. In some quasiperiodic tilings it is possible to decorate the prototiles with line segments that join up across the edges of the tiling to produce a grid of continuous straight lines that extend over the whole plane. These lines are called Ammann bars. The intervals between consecutive parallel Ammann bars come in two sizes, traditionally denoted by S and L (short and long). They form an irregular sequence that does not repeat itself and never contains two adjacent Ss or three adjacent Ls. The lines in Figure 18 are not genuine Amman bars. Those marked with an asterisk do not align properly across the full width of the piece shown but deviate so that the S and L intervals switch sides. (Structural defects of this kind have been observed in quasicrystals, where they are known as phasons). The periodic design in Figure 2(b) has similar lines but its sequences repeat: the vertical ‘Ammann bars’ give sequence SLSL, the lines 36 from vertical give SLLSLL, and those 72 from vertical are not properly aligned. Makovicky et al [24] propose Figure 18 as an example of a quasi-periodic design. They try to find a structural connection between it and the cartwheel element of the

Figure 18. Construction of panel 4584 in the Museum of the Alhambra.

(a)

(b)

(c)

Figure 19.

Penrose tiling. After acknowledging that attempts to match kites and darts are problematic, they try to match it with a variant of the Penrose tiles, one discovered by Makovicky

[20] as he studied the Maragha pattern shown in Figure 9(b). Their boldest assertion is Conclusion 6 [24, p. 125]: The non-periodic cartwheel decagonal pattern from the excavations in the Alhambra and from the Moroccan localities is based on a modified Penrose non-periodic tiling derived recently as ‘PM1 tiling’ by Makovicky… We conclude that a symmetrized PM1-like variety of Penrose tiling must have been known to the Merinid and Nasrid artesans (mathematicians) and was undoubtedly contained in their more advanced pattern collections. Elsewhere in the paper, the authors are more cautious and realistic about the nature of their speculation. They offer an alternative construction based on an underlying radially symmetric network of rhombi whose vertices lie in the centres of the decagonal tiles [24, Fig. 23]. 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

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In order to classify a pattern as periodic or radially symmetric, we must have a large enough sample to be able to identify a template and the rules for its repetition. Similarly, to classify a pattern as quasi-periodic, we must describe a constructive process that allows us to see the given patch as part of a quasi-periodic structure covering the whole plane. It is not sufficient that geometric features of a design, such as rotation centres, can be shown to align with those of a familiar quasi-periodic tiling within a finite fragment. We need to find a procedure built on elements of the design. The set of tiles underlying Figure 18 and the set P2 shown in Figure 13 are both large patches with 10-fold symmetry, but only in the second case do we know how to extend it quasi-periodically. In my opinion, the design strategy underlying the Alhambra pattern does not require an understanding of Penrose-type tilings, and is based on little more than the desire to place large symmetric motifs (roses) in a radially symmetric pattern and fill the gaps. The construction outlined at the start of this section produces the complete design using methods and motifs believed to have been used by Islamic artists. The general structure has the same feel as Figure 5. The ‘Ammann bars’ are an artifact of the construction, although the structure of the design may have evolved and been selected to enhance their effect. They would also have helped to maintain accurate alignment of elements during its construction.

Designs from Isfahan Figure 20 shows a 2-level design that, like the Topkapi Scroll examples above, is based on subdivision. The largescale design is the stars and kites pattern derived from the bow-tie and decagon tiling of Figure 1(b). The subdivisions of the bow-tie and decagon used to generate the smallscale design are shown in Figures 21(a) and (c) with

the large-scale pattern added in grey. The side of a composite tile is formed from the diagonals of two bobbins and one decagon. The pattern cannot be scale invariant: the polygonal network for the large-scale design contains a bow-tie surrounded by four decagons but this local arrangement does not occur in the small-scale network. These subdivisions were derived by Lu and Steinhardt [17] from three hierarchical designs found on buildings in Isfahan. The grey areas in Figure 22 mark out the sections of the large-scale polygonal network underlying these designs: the rectangular strip runs around the inside of a portal in the Friday Mosque, the triangular section is one of a pair of mirror-image spandrels from the Darb-i Imam (shrine of the Imams), and the arch is a tympanum from a portal, also from the Darb-i Imam—see [17, 35] for photographs. Bonner [2] gives an alternative subdivision scheme for the Darb-i Imam arch using the tiling of Figure 2(a) as the basis for the large-scale design. The mosaic in the Darb-i Imam tympanum differs from the symmetrically perfect construction of Figure 20 in several places. For example a bow-tie/bobbin combination like Figure 7(a) in the top right corner of the central composite bow-tie is flipped; bow-tie/bobbin combinations in the corners of the upper composite decagon are also flipped; a decagon at the lower end of the curved section of the boundary on each side is replaced by Figure 7(d). The modifications to the composite decagon appear to be deliberate as the same change is applied uniformly in all corners. Replacing the small decagons may make it easier to fit the mosaic into its alcove. The bow-tie anomaly is possibly a mistake by the craftsman. If we want to use the Isfahan subdivisions as the basis of a substitution tiling, we need to construct a companion subdivision of the bobbin tile. In doing so we should

Figure 20. A 2-level design modelled on the Darb-i Imam, Isfahan. 52


(b)

(a)

(c) Figure 21. Subdivisions (a) and (c) are pffiffiffi derived from designs on buildings in

Isfahan [17]. The scale factor is 4 þ 2 5 8:472.

emulate the characteristics of the two samples—properties such as the mirror symmetry of the subdivisions, and the positions of the tiles in relation to the grey lines. Notice that focal points such as corners or intersections of the grey lines are always located in the centres of decagons, and the interconnecting paths pass lengthwise through bow-ties. Figure 21(b) shows my solution: it satisfies some of these criteria, but it is spoilt by the fact that some of the corners of the grey lines are so close together that decagons centred on them overlap, and there is a conflict between running the path through a bow-tie and achieving mirror symmetry

at the two extremes. This extra subdivision enables the inflation process to be performed, but the resulting tilings are probably of mathematical interest only. The large scale factor for the subdivisions yields a correspondingly large growth rate for the inflation. After two inflations of a decagon the patch would contain about 15000 tiles; for comparison, the patch shown in Figure 13 contains about 1500 tiles. Lu and Steinhardt use the Isfahan patterns in their discussion of quasi-periodicity. Commenting on the spandrel, they say [17, p. 1108]: 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

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as such it would have been very familiar to medieval viewers and recognised even from a small section.

Connections with Penrose Tilings

Figure 22. Sections of the bow-tie and decagon tiling

used in the Isfahan patterns. The Darb-i Imam tessellation is not embedded in a periodic framework and can, in principle, be extended into an infinite quasiperiodic pattern. By this they mean that the visible fragment of the largescale design is small enough that no translational symmetry is immediately apparent and so the patch could be part of a non-periodic tiling. If we only have access to a finite piece of any tiling, it is impossible to decide whether it is periodic without further information on its local or global structure. Although the lack of conspicuous periodicity in the Darb-i Imam design could be interpreted as a calculated display of ambiguity on the part of the artist, to me it seems more likely to be the result of choices influenced by aesthetic qualities of the design, and the relative sizes of the tesserae in the small-scale pattern and the area to be filled. The fact that the same periodic tiling is a basis for all three Isfahan designs makes it a good candidate for the underlying organising principle. Translation in one direction is visible in the Friday Mosque pattern. Lu and Steinhardt also observe that the medieval artists did not subdivide a single large tile but instead used a patch containing a few large tiles arranged in a configuration that does not appear in the small-scale network. They then remark [17, p. 1108]: This arbitrary and unnecessary choice means that, strictly speaking, the tiling is not self-similar, although repeated application of the subdivision rule would nonetheless lead to [a non-periodic tiling]. This gives the impression that, if the medieval craftsmen had wanted to, they could have started with a single tile and inflated it until it covered the available space. But we must beware of seeing modern abstractions in earlier work. There is no evidence that medieval craftsmen understood the process of inflation. The mosaics require only one level of subdivision, and they do not contain a subdivision of the bobbin that would be needed to iterate the inflation. In my opinion the Isfahan patterns, like the 2-level designs in the Topkapi Scroll, are best explained as an application of subdivision to generate a small-scale filling of a periodic large-scale design. Furthermore, the choice of the large-scale design seems far from arbitrary: it is one of the oldest and most ubiquitous decagonal star patterns, and 54


The use of subdivision and inflation to produce quasiperiodic tilings with forbidden rotation centres came to prominence in the 1970s with investigations following the discovery of small aperiodic sets of tiles, the Penrose kite and dart being the most famous example. Penrose tilings have local 5-fold and 10-fold rotation centres and the fact that some Islamic designs share these unusual symmetry properties has prompted several people to explore the connections between the two [1, 17, 20, 24, 27]. Figure 23 shows subdivisions of the kite and dart into the bow-tie, bobbin, and decagon tiles. As in earlier examples, the sides of the kite and dart lie on mirror lines of the tiles. Using this substitution, any Penrose tiling can be converted into a design in the Islamic style [27]. Furthermore, because the kite and dart are an aperiodic set, such a design will be non-periodic. The transition can also proceed in the other direction. Figure 24 shows subdivisions of the three Islamic tiles into kites and darts. Two of the patches are familiar to students of Penrose tilings: (a) is the long bow-tie component of Conway worms and (b) is the hub of the cartwheel tiling. Notice also that (b) is assembled from (a) and (c) in the manner of Figure 7(d). Kites and darts come with matching rules to prohibit the construction of periodic tilings when the tiles are assembled like a jigsaw. In Figure 24 the two corners at the ‘wings’ of each dart and the two corners on the mirror line of each kite are decorated with grey sectors; the matching rule is that grey corners may only be placed next to other grey corners. This prevents, for example, the bow-tie and the decagon in the figure from being assembled in the stars and kites pattern: it is not possible to place two bow-ties on opposite corners of a decagon. The markings on the kites and darts in Figure 24 endow the composite tiles with a matching rule of their own. Each side of a composite tile has a single grey spot that divides its length in the golden ratio; we decorate each side with an arrow pointing towards the short section. Instead of defining the matching rule at the vertices of the tiling, as

Figure 23. Subdivisions of the Penrose kite and dart.

2.

3.

(a)

(b)

(c)

Figure 24. Patches of Penrose kites and darts.

4.

(a)

(b)

Figure 25. Subdivisions of marked tilespthat ffiffiffi preserve

the markings. The scale factor is

1 2

3þ

5 2:618.

with the Penrose example previously described, we place constraints on the edges of the tiling: the arrows on the two sides forming an edge of the tiling must point in the same direction. With these markings and matching rule, the bowtie and bobbin are an aperiodic set. To prove this note that the subdivisions in Figure 25 show that we can tile the plane by inflation, and that any periodic tiling by bow-ties and bobbins could be converted into a periodic tiling by kites and darts but this is impossible. The substitution matrix for these marked tiles is associated with the Fibonacci sequence and the ratio of bobbins to bow-ties in a substitution tiling is the golden ratio. Notice that a horizontal line running through the centre of a composite bow-tie passes lengthwise through the small bow-ties and short-ways across the small bobbins. Inflation produces a longer line with the same properties, and a substitution tiling will contain arbitrarily long such lines. Any infinite lines must be parallel as they cannot cross each other. These lines inherit their own 1-dimensional substitution rule.

Conclusions In the preceding sections I have described methods for constructing Islamic geometric patterns, given a brief introduction to the modern mathematics of substitution tilings, and analysed some traditional Islamic designs. The conclusions I reached during the course of the discussion are isolated and summarised here: 1. It is possible to construct quasi-periodic tilings from the set of prototiles used by Islamic artists (Figure 6). Examples can be generated as substitution tilings based

5.

on inflation or using a matching rule with marked versions of the tiles. Islamic artists did use subdivision to produce hierarchical designs. There are examples illustrating the method in the Topkapi Scroll, and three designs on buildings in Isfahan can be explained using this technique. Indeed, their prototiles are remarkable in their capacity to form subdivisions of themselves in so many ways. There is no evidence that the Islamic artists iterated the subdivision process—all the designs I am aware of have only two levels. This is to some degree a practical issue: the scale factor between the small-scale and large-scale designs is usually large and the area of the design comparatively small. With the subdivisions used in the Topkapi Scroll, iteration is impossible as composite versions of the pentagon and bow-tie do not exist. There is no evidence that the Islamic artists used matching rules. Ammann bars are the nearest thing to a form of decoration that could have been used to enforce nonperiodicity. Similar lines that appear on some designs are a by-product of the construction, not an input to the design process, although the designs may have been selected because this feature was found attractive. The designs analysed in this article do not provide evidence that Islamic artists were aware of a process that can produce quasi-periodic designs. They are periodic, generated by reflections in the sides of a rectangle, or are large designs with radial symmetry. The multi-level designs are hierarchical, not scale invariant.

In this article I have concentrated on designs with local 5-fold symmetry. In Spain and Morocco there are analogous designs with local 8-fold symmetry, including some fine 2-level designs in the Patio de las Doncellas in the Alcazar, Seville—see [22] for photographs. The geometry of the polygonal networks underlying these designs is pffiffiffi grounded on the 2 system of proportions rather than the golden ratio. Plans of muqarnas (corbelled ceilings built by stacking units in tiers and progressively reducing the size of the central hole to produce a stalactite-like dome) sometimes display similar features. These networks have a strong resemblance to the Ammann–Beenker quasiperiodic tiling composed of squares and 45–135 rhombi [33]. This tiling is another substitution tiling that can be generated by subdivision and inflation; the tiles can also be decorated with line segments to produce Ammann bars. Similar claims to those assessed in this article have been made for some of the Islamic 8-fold designs [2, 6, 22, 23]. To me, it seems most likely that the Islamic interest in subdivision was for the production of multi-level designs. Islamic artists were certainly familiar with generating designs by applying reflection, rotation and translation to repeat a template. They probably had an intuitive understanding of the crystallographic restriction and a feeling that global 5-fold and 10-fold rotation centres are somehow incompatible with periodicity. They did have the tools available to construct quasi-periodic designs but not the theoretical framework to appreciate the possibility or significance of doing so.


55

ACKNOWLEDGMENTS

I would like to thank Paul Steinhardt for clarifying some statements in [17] and Peter Saltzman for sharing a draft of his article [29]. I am also very grateful to the following people for their critical reading of an early draft of this article and for suggesting improvements: Helmer Aslaksen, Elisabetta Beltrami, Jean-Marc Caste´ra, Dirk Frettlo¨h, Chaim Goodman-Strauss, Emil Makovicky, John Rigby, Joshua Socolar, and John Sullivan.

17. P. J. Lu and P. J. Steinhardt, ‘Decagonal and quasi-crystalline tilings in medieval Islamic architecture’, Science 315 (23 Feb 2007) 1106–1110. 18. P. J. Lu and P. J. Steinhardt, ‘Response to Comment on ‘‘Decagonal

and

quasi-crystalline

tilings

in

medieval

Islamic

architecture’’, Science 318 (30 Nov 2007) 1383. 19. F. Lunnon and P. Pleasants, ‘Quasicrystallographic tilings’, J. Math. Pures et Applique´s 66 (1987) 217–263. 20. E. Makovicky, ‘800-year old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired’, Fivefold Symmetry, ed. I. Hargittai, World Scientific, 1992, pp. 67–86.

BIBLIOGRAPHY

1. M. Arik and M. Sancak, ‘Turkish–Islamic art and Penrose tilings’, Balkan Physics Letters 15 (1 Jul 2007) 1–12. 2. J. Bonner, ‘Three traditions of self-similarity in fourteenth and fifteenth century Islamic geometric ornament’, Proc. ISAMA/Bridges: Mathematical Connections in Art, Music and Science, (Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 1–12.

21. E. Makovicky, ‘Comment on ‘‘Decagonal and quasi-crystalline tilings in medieval Islamic architecture’’, Science 318 (30 Nov 2007) 1383. 22. E. Makovicky and P. Fenoll Hach-Alı´, ‘Mirador de Lindaraja: Islamic ornamental patterns based on quasi-periodic octagonal lattices in Alhambra, Granada, and Alcazar, Sevilla, Spain’, Boletıń Sociedad Espanõla Mineralogıá 19 (1996) 1–26.

manuscript. 4. J. Bourgoin, Les Ele´ments de l’Art Arabe: Le Trait des Entrelacs,

23. E. Makovicky and P. Fenoll Hach-Alı´, ‘The stalactite dome of the Sala de Dos Hermanas—an octagonal tiling?’, Boletıń Sociedad Espanõla Mineralogıá 24 (2001) 1–21. 24. E. Makovicky, F. Rull Pe´rez and P. Fenoll Hach-Alı´, ‘Decagonal

Firmin-Didot, 1879. Plates reprinted in Arabic Geometric Pattern and Design, Dover Publications, 1973. 5. J.-M. Caste´ra, Arabesques: Art Dećoratif au Maroc, ACR Edition,

patterns in the Islamic ornamental art of Spain and Morocco’, Boletıń Sociedad Espanõla Mineralogıá 21 (1998) 107–127. 25. G. Necipog˘lu, The Topkapi Scroll: Geometry and Ornament in

3. J. Bonner, Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Derivation, unpublished

1996. 6. J.-M. Caste´ra, ‘Zellijs, muqarnas and quasicrystals’, Proc. ISAMA, (San Sebastian, 1999), eds. N. Friedman and J. Barrallo, 1999, pp. 99–104.

Islamic Architecture, Getty Center Publication, 1995. 26. J. Rigby, ‘A Turkish interlacing pattern and the golden ratio’, Mathematics in School 34 no 1 (2005) 16–24. 27. J. Rigby, ‘Creating Penrose-type Islamic interlacing patterns’,

7. G. M. Fleurent, ‘Pentagon and decagon designs in Islamic art’,

Proc. Bridges: Mathematical Connections in Art, Music and

Fivefold Symmetry, ed. I. Hargittai, World Scientific, 1992, pp. 263–281.

Science, (London, 2006), eds. R. Sarhangi and J. Sharp, 2006,

8. B. Gru¨nbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, 1987.

28. F. Rull Pe´rez, ‘La nocioń de cuasi-cristal a trave´s de los mosaicos a´rabes’, Boletıń Sociedad Espanõla Mineralogıá 10 (1987) 291–

9. E. H. Hankin, ‘On some discoveries of the methods of design employed in Mohammedan art’, J. Society of Arts 53 (1905) 461– 477. 10. E. H. Hankin, The Drawing of Geometric Patterns in Saracenic Art,

pp. 41–48.

298. 29. P. W. Saltzman, ‘Quasi-periodicity in Islamic ornamental design’, Nexus VII: Architecture and Mathematics, ed. K. Williams, 2008, pp. 153–168.

Memoirs of the Archaeological Society of India, no 15, Government of India, 1925.

30. M. Senechal, Quasicrystals and Geometry, Cambridge Univ.

11. E. H. Hankin, ‘Examples of methods of drawing geometrical ara-

31. M. Senechal and J. Taylor, ‘Quasicrystals: The view from Les Houches’, Math. Intelligencer 12 no 2 (1990) 54–64.

besque patterns’, Math. Gazette 12 (1925) 370–373.

Press, 1995.

12. E. H. Hankin, ‘Some difficult Saracenic designs II’, Math. Gazette 18 (1934) 165–168. 13. E. H. Hankin, ‘Some difficult Saracenic designs III’, Math. Gazette 20 (1936) 318–319. 14. C. S. Kaplan, ‘Computer generated Islamic star patterns’, Proc. Bridges: Mathematical Connections in Art, Music and Science, (Kansas, 2000), ed. R. Sarhangi, 2000, pp. 105–112. 15. C. S. Kaplan, ‘Islamic star patterns from polygons in contact’, Graphics Interface 2005, ACM International Conference Proceeding Series 112, 2005, pp. 177–186. 16. A. J. Lee, ‘Islamic star patterns’, Muqarnas IV: An Annual on Islamic Art and Architecture, ed. O. Grabar, Leiden, 1987, pp. 182–197.

56


INTERNET RESOURCES

32. ArchNet. Library of digital images of Islamic architecture, http://archnet.org/library/images/ 33. E.

Harriss

and

D.

Frettlo¨h,

Tilings

Encyclopedia,

http://tilings.math.uni-bielefeld.de/ 34. C. S. Kaplan, taprats, computer-generated Islamic star patterns, http://www.cgl.uwaterloo.ca/*csk/washington/taprats/ 35. P. J. Lu and P. J. Steinhardt, Supporting online material for [17], http://www.sciencemag.org/cgi/content/full/315/5815/1106/DC1 36. D. Wade, Pattern in Islamic Art: The Wade Photo-Archive, http://www.patterninislamicart.com/

Geometric Constructions with Ellipses ALISKA GIBBINS

T

AND

LAWRENCE SMOLINSKY

he geometric problems of trisecting a general angle and doubling the cube cannot be solved by the use of a straightedge and compass alone. These beautiful results were a triumph of modern algebra, first published by Pierre Laurent Wantzel [1]. The constructible points are those that are in iterated quadratic extensions of the base field. A consequence, the Gauss-Wantzel Theorem, states that a regular n-gon is classically constructible if and only if /(n) is a power of 2; here / is the Euler / function, which counts how many integers are less than and relatively prime to n. To the modern reader the question of what are the possible numerical magnitudes (lengths) is answered by the process of the topological completion of the rational numbers, i.e., the real numbers. The ancient Greeks showed the existence of numerical magnitudes by affirmative use of the axioms—proving existence is a plausible motive for why the Ancients engaged in constructions [2, 3]. Straightedge-and-compass constructions are a rigorous application of Euclid’s first three postulates. However, there are other procedures the Greeks used for constructions. For example, a procedure not grounded in the postulates is the rotation of a plane figure to produce a solid, or the intersection of a plane with a solid figure (p. 29 [4]). Pappus of Alexandria described a classification of methodology for geometric problems—one which he attributed to those he called Ancients. A construction is called planar if it is done with straightedge and compass alone, solid if it uses conic sections, and linear if it uses higher order curves [2]. Solid solutions do exist to the classical problems. Pappus gave two trisection constructions with hyperbolas that are possibly due to Apollonius.

Menaechmus, the discoverer of conic sections, is supposed to have made his discovery while working on the problem of doubling the cube, and he gave a construction using parabolas. Those constructions have been described in The Intelligencer [5]. In 1895 James Pierpont essentially gives the analysis of which numbers are constructible using conic sections. In two pages at the end of a Bulletin paper, Pierpont remarks that ‘‘Greek geometers frequently allowed the use of the conic sections in a geometric construction,’’ and he determines that a regular n-gon allows a solid construction if and only if /(n) has only factors of 2 and 3 [6, 7]. Recent work by Carlos R. Videla explores solid constructions, and Videla gives a complete and more modern version of Pierpont’s result [5]. Videla allows the construction of a conic when the focus, directrix, and eccentricity are constructible. The conically-constructible numbers may be obtained by circles, parabolas, and hyperbolas alone. We will consider construction with ellipses. The elliptic constructions in this paper are from an undergraduate project by the first author and directed by the second. Patrick Hummel also gives a treatment of elliptic constructions in a paper from his undergraduate project [8]. While the abstract theory is similar, the constructions are different. The authors are grateful to the referee for directing them to Hummel’s paper and other sources. We will primarily be concerned with solid constructions, but it is worthwhile to note that ancient Greek constructions were extremely rich and varied. A beautiful demonstration of this variety is a construction to double the cube by Archytas of Taras, who came out of Plato’s Ó 2008 Springer Science+Business Media, LLC., Volume 31, Number 1, 2009

57

If P is a point of intersection of the cylinder, cone, p and ffiffiffi degenerate torus, then the distance of P to the origin is 3 b:

1 2

B

ðx 2 þ y 2 þ z 2 Þ2 ¼ x 2 þ y2 by equation ð3Þ ¼ x by equation ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ b x 2 þ y2 þ z 2 by equation ð2Þ:

O

A 1

1

3

4

2

4

1 2

Figure 1. Example with b = 1/2.

Academy. We now give a version of this construction [9]. Let O be the origin of the xy-plane, and consider the circle with center (1/2, 0) and radius 1/2, i.e., x 2 þ y 2 ¼ x:

ð1Þ

Take a point B on the circle and let b be the length of the segment OB. Let A = (1, 0), so OA is a diameter. See Figure 1. Archytas can find the cube root of b. The construction requires three dimensions. Start with the right circular cylinder with axis parallel to the z-axis containing our circle, which is given by equation (1). Take the cone with vertex O obtained by rotating the line containing OB about the x-axis. This cone is given by b2 ðx 2 þ y2 þ z 2 Þ ¼ x 2 :

ð2Þ

The third and surprising construction is to take the circle in the xz-plane of radius 1/2 and diameter OA and rotate it about the z-axis. The result is a degenerate torus—a circle rotated about a tangent line. See Figure 2. The equation of the degenerate torus is ðx 2 þ y2 þ z 2 Þ2 ¼ x 2 þ y2 :

ð3Þ

The planar and solid constructions can be built from a limited number of well-defined constructions with lines, circles, and other conics, and so can be analyzed by modern algebra a` la Gauss, Wantzel, and Pierpont. In general, the rich variety of constructions introduced by the ancient Greeks requires more analysis before allowing algebra to come to bear. Some constructions allow neusis (sliding a straightedge on which a line segment is marked). An analysis allowing neusis on lines is discussed by Martin [7]. Constructions using neusis between a circle and a line are discussed by Arthur Baragar [10]. Wilbur Richard Knorr, in his discussion of angle-trisection methods by ancient geometers in [11] (page 216), comes up with about a dozen solutions and writes, ‘‘In view of the massive extinction of documentation from antiquity, we can hardly presume that this list would exhaust the entire range of ancient solutions.’’ There are a lot of interesting constructions and questions to explore. In the modern treatment of construction problems, one first translates a geometric question into an algebraic question by use of the Cartesian or Gaussian plane, and then analyzes the question using the power of algebra and number theory. This first step was already started—as it were—in the beginning with Descartes’s analytic geometry. Rene´ Descartes gave a trisection construction in his 1637 La Geóme´trie, the monograph in which he introduced analytic geometry. This construction uses a parabola and a circle, and relies on the the triple-angle formula [12]. A version of Descartes’s trisection for the angle h = p/3 is shown in Figure 3. The construction uses the parabola defined by y = 2x2 and the circle through the origin with center (1/2 cos (h),1). The x-coordinates of the points of intersection satisfy the equation x(4x3 - 3x -cos (h)) =0 or x(x - cos (h/3))(x - cos (h/3 + 2p/3)) (x - cos (h/3 + 4p/3)) = 0, by the triple-angle formula

AUTHORS

.........................................................................................................................................................

58

ALISKA GIBBINS is a graduate student at

LAWRENCE SMOLINSKY started in topol-

Ohio State, studying geometric group theory with Mike Davis. She spent a year teaching literacy in New Orleans, and is an accomplished Cajun cook. She got her B.S. from Tulane University in New Orleans, except for moving to Louisiana State for a semester while Tulane was recovering from Hurricane Katrina. This paper grew from a collaboration during that semester.

ogy, getting his doctorate at Brandeis under the late Jerry Levine. More recently he has worked in integrable systems and representation theory. While at LSU (as Chair since 2004), he has worked with students in many activities. For one, the LSU Mathematics Contest for high school students, which annually draws about 200–300 contestants. For another instance, the present article!

Department of Mathematics, Ohio State University, Columbus, OH 43210, USA e-mail: [email protected]

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA e-mail: [email protected]


Figure 2. Archytas construction with b = 1/2.

2

1

1

1

1

2

2

1

Figure 3. Descartes’s trisection of h ¼ p3 .

cos (3h) = 4 cos3(h) - 3cos (h). In Figure 3, there are four distinct points of intersection although two are very close together.

Classical Constructions Start with an initial set of points P in the plane. The initial set of points should include (0, 0) and (1, 0) (and

may include additional points, e.g., to form a general angle). The set of points one may derive using only a straightedge and compass will be called classically constructible points derived from P. The set of all numbers that arise as the ordinate or abscissa of classically constructible points is the set of classically constructible numbers. We recall some of the facts about classically constructible numbers, taking some of the background notions from Hungerford [13]. It is one of the founding observations that starting with an initial set of points P the classically constructible numbers form a field, i.e., using straightedge and compass constructions one may start with two numbers and construct the sum, difference, product, and quotient. Furthermore, if (x, y) is a classically constructible point, it is a simple exercise to show that (y, x) is a classically constructible point. It is useful to introduce the notion of the plane of a field. If F is a subfield of the real numbers R; then the plane of F is the subset F F R2 : Suppose P and Q are distinct points in the plane of F. Then the line determined by P and Q is a line in F. Similarly, the circle with center P and containing Q is a circle in F. It is a straightforward calculation that the intersection points of two lines in F are points in the plane of F. Furthermore, if a circle in F is intersected with either a line in F or a circle in F, then the intersection points are in the plane of F(r), where r is the Ó 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

59

square root of an element of F and F(r) is the field formed by adjoining r to F [13].

Elliptic Constructions We mention three approaches for constructing ellipses that were known to the Ancients and can be used to extend straightedge-and-compass constructions to constructions with ellipses. The first approach is to use the fact that an ellipse is the locus of points whose distance from the two foci is a constant sum. This property was known at least as far back as Apollonius [2]. The second approach is to use the interpretation of an ellipse as the motion of a point on a line segment whose endpoints slide along perpendicular lines. This construction was reported by Proclus, who was a head of the Academy [2]. This construction could be accomplished with knowledge of the line containing the foci and the lengths of the major and semimajor axis. The third approach is to allow the construction of an ellipse given its directrix, focus, and eccentricity. Each of the ellipse construction techniques ostensively would determine a different set of constructible points in the plane, but these three sets all turn out to be the same. This fact follows from part (2) of Proposition 3 below. We use the first method, which may be accomplished with pins and string. Our fundamental constructions are: (C1) Given three points, one may insert pins in the three points, tighten the string around the pins, and remove one pin. Keep the string taut, and use a pen to draw the ellipse around the two pins as foci and passing through the third. (C2) Given two points, one may insert pins in the two points, tighten the string around the pins, remove one pin, and use a pen to draw the circle with center one point and the other on the circle. (C3) Given two points, one may draw the line through the two points. Which points can be reached by constructions with a straightedge and pins and string? Start with a set of points P, which include (0, 0) and (1, 0). The elliptically constructible points derived from P are all the points of intersection obtained from the ellipses and lines constructed by the operations above. We call the obtained coordinates the field of elliptically constructible numbers. Analogous to the previous definitions for classical constructions is the following definition. Suppose F is a subfield of the real numbers R: If O, P, and Q are distinct points in the plane of F, then the ellipse containing O with foci P and Q is an ellipse in F.

R EMARK 1 If an ellipse is in standard position then its equation can be given as Ax 2 þ Cy 2 F ¼ 0; qffiffiffi qffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C [ A [ 0. Let a ¼ AF ; b ¼ CF and c ¼ : a2 b2 . Then the foci of the ellipse are at (± c, 0), and (a, 0) is a point on the ellipse. This ellipse is constructible by pins and string, if and only if a and c are elliptically 60


constructible numbers. To construct this ellipse with the the sliding-line-segment approach, the segment will have length a + b, and the distinguished point will separate the segment into parts of length a and b. The ellipse is constructible with a sliding segment if and only if a and b are constructible. Next note that the directrix is the line 2 x ¼ ac and the eccentricity is e ¼ ac : The ellipse is constructible by the focus-and-directrix approach if and 2 only if ac and ac are constructible.

L EMMA 2 Suppose that F is a subfield of R in which every positive number has a square root. If cos (h) [ F, then rotation of the plane by h induces a bijection on the plane of F. If (r, s) is in the plane of F, then translation of the Cartesian plane by (r, s) induces a bijection on the plane of F. P ROOF . Note that if cos (h) is in F, then sin (h) is in F since sin2h = 1 - cos2h. The formulas for rotation of the plane by ±h and translation of the plane by ± (r, s) show that they are bijections of the plane of F. The main lemma is the following.

P ROPOSITION 3 Suppose that F is a subfield of R in which every positive number has a square root. (1) Consider an ellipse E described by the equation ax2 + bxy + y2 + dx + ey + f = 0. The ellipse E is in F if and only if a, b, d, e, and f are in F. (2) An ellipse E is in F if and only if its eccentricity, directrix, and foci are in F. An ellipse E is in F if and only if the lengths of its semi-major and semi-minor axes are in F and the line containing the foci is in F. (3) If E1 and E2 are ellipses in F, then the coordinates of the points of intersection of E1 and E2 are in a field F(R), where R is the set of real roots of a quartic polynomial with coefficients in F. If E is an ellipse in F and L is a line in F, then the coordinates of the points of intersection of E and L are in the field F.

P ROOF . Part 1. If E is in F, then the foci and center are in the plane of F. The sine and cosine of h are also in F, where h is the angle formed by the x-axis and the line containing the foci. The ellipse E 0 in standard position obtained by rotation by h and translation of E is again in F (Lemma 2). By Remark 1, the coefficients of the equation of E0 are in F. Undoing the translation and rotation shows the coefficients of E are in F (consult the formula for the transformation of the coefficients of a conic). The converse is similar, but rotate E into standard position using h, where cot ðhÞ ¼ ac b unless b = 0. (If b = 0 then h = 0 or p.) By use of trigonometric identities, sin (h) and cos (h) are seen to be in F. Part 3. First consider the intersection of two ellipses. Any two constructible ellipses E1 and E2 have equations of the form:

a1 x 2 þ b1 xy þ y 2 þ d1 x þ e1 y þ f1 ¼ 0

ð4Þ

a2 x 2 þ b2 xy þ y2 þ d2 x þ e2 y þ f2 ¼ 0:

ð5Þ

1

Solving for y, y¼

f1 f2 þ ðd1 d2 Þx þ ða1 a2 Þx 2 : ðb1 b2 Þx þ ðe1 e2 Þ

1

ð6Þ

1

Putting this expression back into equation (4) we get Ax 4 þ Bx 3 þ Cx 2 þ Dx þ G ¼ 0;

ð7Þ

1

where the coefficients are A ¼ e1 f2 þ f22 þ e2 f1 2f2 f1 þ f12 B ¼ d1 e2 d2 e1 b1 f2 þ 2d2 f2 2d1 f2 þ b2 f1 2d2 f1 þ 2d1 f1 C ¼ b1 d2 þ

d22

þ b2 d1 2d2 d1 þ

d12

2

þ a1 e2 a2 e1 þ 2a2 f2

2a1 f2 2a2 f1 þ 2a1 f1 D ¼ a1 b2 a2 b1 þ 2a2 d2 2a1 d2 2a2 d1 þ 2a1 d1 1.25

G ¼ a22 2a2 a1 þ a21 : Let R be the real roots of Equation (7). The proof of the other claim is similar. Part 2 is similar in spirit to Parts 1 and 3. Note that Part 2 implies that the three methods of construction given in the beginning of this section yield the same constructible numbers.

1.3 0.95

Trisection of the General Angle Suppose we start with an angle of measure h. Translate it to the congruent central angle \ABC with A = (cos (h), sin (h)), B = (0, 0), and C = (1, 0). To trisect this angle, we must construct the point A0 = (cos (h/3), sin (h/3)), and \A0 BC trisects the angle. Constructing A0 is equivalent to constructing the number cos (h/3), because sin (h/3) can pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi then be produced as 1 cos2 ðh=3Þ: Let q = cos (h). By a triple-angle formula, cos (h/3) is a real solution to the equation 3

4x 3x q ¼ 0:

1

Figure 4. Trisection of h ¼ p3 .

which factors as (4x3 - 3x - q)(x - 1) = 0. One of the solutions is cos (h/3). The example of the trisection of h ¼ p3 is shown in Figure 4. In equations (8), q ¼ 12 : There are four distinct points of intersection, although two are very close together.

Cube roots and Doubling the Cube The ability to construct a cube whose volume is double a given cubepffiffiis ffi the same as the ability to multiply a side length by 3 2.

The other roots are cos (h/3 + 2p/3) and cos (h/3 + 4p/3).

T HEOREMpffiffiffi5 If a is an elliptically constructible number,

T HEOREM 4 The general angle can be trisected.

then so is

P ROOF . Let F be the field of constructible numbers derived from (0, 0), (1, 0), and (cos (h), 0). Let q = cos (h) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and p ¼ 17 4q : Consider the following ellipses with coefficients in F: 2x 2 þ 4y 2 qx þ 2py þ 2 ¼ 0 6x 2 þ 4y2 þ ð2p 4Þy ð2 þ qÞx p 1 ¼ 0:

ð8Þ

These ellipses are in the plane of F by Proposition 3 part (1). Solving the system of equations (8) for the xcoordinates of the points of intersection, we obtain that they are the real roots of the equation 4x 4 4x 3 3x 2 þ ð3 qÞx þ q ¼ 0;

3

a.

P ROOF . Let F be the field of constructible numbers derived from (0, 0), (1, 0), and (a, 0). Consider the following equations of ellipses with coefficients in F: pffiffiffi 2x 2 þ y2 ax þ 2 2y þ 1 ¼ 0 ð9Þ pffiffiffi pffiffiffi 3x 2 þ y2 ax þ ð1 þ 2 2Þy þ 2 ¼ 0: These ellipses are in the plane of F by Proposition 3 part (1). Solving the system of equations (9) for the xcoordinates of the points of intersection, we obtain that they are the real roots of the equation x 4 ax ¼ 0: pffiffiffi The real roots are 0 and 3 a: Ó 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009

61

1

1

2

3

Figure 5. Constructing

ffiffiffi p 3 2.

trisections and real cube roots are exactly the constructions required to obtain all of F: The fields of elliptically constructible numbers and conically constructible numbers are the same. So the regular ngons that are elliptically constructible were determined by Pierpont [5, 6]. A regular n-gon is elliptically constructible if and only if /(n) = 2s3t for some s and t. For example, the 7and 9-sided regular polygons are elliptically constructible but not classically constructible. Note also that one only has to allow the construction of translations of ellipses in standard position to do elliptic constructions. Oblique ellipses are not required to obtain the field of elliptically constructible numbers, for they are not required for constructions in the proofs of Theorems 4 and 5. The type of ellipses may be further restricted: the ratio of the lengths pffiffiffiffiffiffiffi to the minor pffiffiffi pffiffiof ffi the major axis may be restricted to 1; 2; 3; and : 3=2. Can this be improved? REFERENCES

To double the cube, we need to let a = 2 in equations (9). This gives the two ellipses shown in Figure 5.

Concluding Remarks We can determine which numbers are elliptically constructible. Suppose P is a set of points and F R is the field of elliptically constructible numbers determined by P. Let F ¼ F þ iF : Then F is a subfield of the complex numbers and the constructible points in the Gaussian plane. Let F0 be the field generated by the rationals and the coordinates of the points in P.

[1] Wantzel, L. ‘‘Recherches sur les moyens de reconnaıˆtre si un proble`me de Geóme´trie peut se re´soudre avec la re`gle et le compas,’’ Journal de mathe´matiques pures et appliqueés Se´r. I 2 (1837), 366–372. Available free online through the gallica library (Bibliothe`que Nationale de France). [2] Knorr, Wilbur Richard, The Ancient Tradition of Geometric Problems, Dover Publications, New York, 1993. [3] Zeuthen, H.G., ‘‘Die geometrische Construction als ‘Existenzbeweis’ in der antiken Geometrie,’’ Math. Ann. 47 (1896), 222– 228. [4] Mueller, Ian, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements, The MIT Press, Cambridge, Massachusetts and London, England, 1981.

T HEOREM 6 (1) F is the smallest field containing F0, i, and the square roots, cube roots, and conjugate of each element. (2) F is the smallest field which contains F0 and the real roots of every fourth-degree polynomial with coefficients in F.

[5] Videla, Carlos R., ‘‘On Points Constructible from Conics,’’ The Mathematical Intelligencer, 19 (1997), no. 2, 53–57. [6] Pierpont, James, ‘‘On an Undemonstrated Theorem of the Disquisitiones Arithmeticae,’’ Bull. Amer. Math. Soc. 2 (1895), 77– 83. [7] Martin, George E., Geometric Constructions, Springer-Verlag, New York, 1997. [8] Hummel, Patrick, ‘‘Solid Constructions Using Ellipses,’’ PME

P ROOF . Part (1) is shown in [5]. To see part (2), note that F ¼ Re F: By Cardano’s and Ferrari’s formulas, F contains the real roots of fourth-degree polynomials with coefficients in F. Conversely, by part (3) of Proposition 3, F is obtained from F0 by repeated iteration of adjoining real roots of polynomial of at most fourth degree. The main observation in the proof of part (1) is that taking a cube root of a complex number is trisecting an angle and taking the cube root of a real number, i.e., p ffiffiffi hi 3 R e 3 is a cube root of Rehi. Combining this observation with Cardano’s and Ferrari’s formulas shows that

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Journal 11 (2003), 429–435. [9] Heath, Thomas, Greek Mathematics Vol. 1, Oxford University Press, London, 1921. [10] Baragar, Arthur, ‘‘Constructions Using a Compass and TwiceNotched Straightedge,’’ Amer. Mathematical Monthy 109 (2002), 151–164. [11] Knorr, Wilbur Richard, Textual Studies in Ancient and Medieval Geometry, Birkha˜user, Boston, Inc., Boston, 1989. [12] Yates, Robert C., ‘‘The Trisection Problem’’ National Mathematics Magazine (continued as Mathematics Magazine) 15 (1941), 191–202. [13] Hungerford, Thomas W., Algebra, New York: Springer 1997.

The Mathematical Tourist

Abstract Neo-Plasticity and Its Architectural Manifestation in the Luis Barragan House/Studio of 1947 JIN-HO PARK, HONG-KYU LEE, YOUNG-HO CHO, KYUNG-SUN LEE

AND

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

Dirk Huylebrouck, Editor

Does your hometown have any mathematical tourists 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 included a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

L

uis Barragan (1902–1988), born and raised in Guadalajara, Mexico, was a modern architect whose works have influenced contemporary building designs in his native country and beyond. His architecture responds to the contextual and natural inheritance of Mexico, signifying a new residential dwelling predicated on modernity and indigenously rooted in the symbol of Mexican living. The manner in which his buildings are integrated within their given ‘‘place’’ is perhaps the key factor in his significance and renown. While drawing from cultural and regional references of Mexico, Barragan offered a utopian vision of the unification of the vernacular Mexican style with architectural purity and simplicity. Stucco walls with bricks, intense saturated colors, and natural illumination

possessing a spiritual quality defined Barragan’s designs. Barragan continues to exert a profound influence on contemporary architecture. (See [4], [6], [7] and [8].) His vision has inspired some of the best-known contemporary Mexican architects including Ricardo Legorreta, Andrea Casillas, and Enrique Norton of TEN (Taller Enrique 1 Norton) Arquitectos, among others. Ricardo Legorreta is among the disciples of Barragan who make use of his sense of color, spatial composition, and design vocabulary. Among Barragan’s work, his own house and studio stands out for its interplay of abstract planes and bold masses. Its colorful walls provide internal rooms and patios with pleasant filtered light. Barragan writes, ‘‘I have left large plane walls without window openings, both for plastic beauty… . By the use of large wall surfaces one can also obtain spaces with varying luminosity, which creates an ambience more comfortable and intimate.’’2 Barragan’s architecture is associated with two primary connections. The abstract neo-plasticity of De Stijl and Bauhaus strongly inspired the geometry of the house, whereas Barragan’s association with avant garde artistic circles, which included Diego Rivera, Frida Kahlo, and Jos Clemente Orozco, infused him with indigenous culture and regional principles. Mathematical Intelligencer readers may wonder why this house is the subject of an article. Although much has been written about the Luis Barrragan house/studio, most studies of the house are descriptive presentations lacking formal and mathematical

1 A Californian architect, Mark Mack, belongs to this group. In an interview with the author, Mack expressed, ‘‘Barragan for me was a very interesting character because he used very modern spatial articulation in his buildings. But when you look at the interior and the way the details are done, they are very traditional. However, the shapes overwhelm the tradition, becoming a new shape and a new form.’’ See Jin-Ho Park, ‘‘An Interview with Mark Mack,’’ in the Architectural Magazine POAR, Seoul: Ganhyang [13]. See Burri, R. (2000) Luis Barragań, London: Phaidon Press; Eggener, K, (2001) Luis Barragań’s Gardens of El Pedregal, New York: Princeton Architectural Press; Federica Zanco, F. (2001) Luis Barragań: The Quiet Revolution, Skira Editore; Julbez, J. and Palomar, J. (1997) The life and work of luis barragan, New York: Rizzoli; Pauly, D. and Habersetzer, J. (2002) Barragan: Space and Shadow, Walls and Colour, Basel: Birkhaüser. 2 ‘‘Luis Barragan,’’ Arts and Architecture, August [2], pp.24–25.

2008 SPRINGER SCIENCE+BUSINESS MEDIA, LLC., Volume 31, Number 1, 2009

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study. The present article reviews his architectural thought in an effort to grasp the abstract nature of Barragan’s achievement, and also explores the underlying logic applied to the house through an in-depth analysis and interpretation of the house. Among other aspects, Barragan’s works prominently feature the transformation of rigorous combinations of simple but timeless geometry into unique spatial compositions. The simple geometry not only serves as a key principle for the consistent and systematic quality underlying his work, but also provides an order for the formal elements that encompass the spatial composition. It expresses the clarity of his thought and insight, and achieves overly formal and unique architectonic space forms, thus providing a source of Barragan’s expression. Therefore, central to this article is the notion that Barragan’s house is characterized by a powerful commitment to the spirit of abstraction coupled with a strong geometrical basis. Above all, we would encourage interested Mathematical Intelligencer readers to visit the house to experience its unique formal quality and gain new architectonic insights.

The Barragan House The house and studio of Luis Barragan is located within Mexico City. Adjacent to an earlier home he designed, it was completed in 1947.3 The lot for the house is 100 feet across its front (about 30 m) and 140 feet (about 42 m) deep.4 The house was the point of departure for his subsequent works. Barragan’s work from this period focuses on colors and forms and the light that defines his buildings. The focus may also be on the emotional quality of the form and light in their abstract manifestations. Barragan lived and worked in the house alone until his death in 1988 from Parkinson’s disease. The Barragan house is known for its unique characteristics and the serene form of both the house and garden. The property is completely hidden from the 3

Figure 1. A view of the Barragan house, 1947, from General Francisco Ramirez Street.

outside neighborhood by a plain façade on a small narrow street.5 The entire house is screened and turned inwards to allow for greater privacy, creating colorful interior spaces and shaded courtyards. An exterior view is presented in Figures 1–5. Upon entering the house, a dark entrance hall with indirect lighting and volcanic lava floors extends to a vestibule facing a pink wall. In the vestibule, stairs link the volcanic lava floor stairway to the mezzanine above.6 To the right, one is led to the main living area. The main room includes the living room and a library that is a double-height space with dark exposed wooden girders. These style girders are typical in Barragan’s houses, one example being in the Lopez house of 1948. Here one finds a serene enclosed garden. High walls surround the garden with bastions set at intervals. These walls serve as a protective barrier to the outside world, bringing tranquility and comfort. Water and lush vegetation are utilized to regulate the temperature of both the garden and the building. The entire layout of the procession to the interior space and the garden vividly reflects the elaborate image of the Lahambra Moorish garden: ‘‘… [W]hile walking along the lava crevices, under the shadow of imposing ramparts of living stocks, I suddenly

discovered, to my astonishment, a small secret green valley— the shepherds call them ‘‘jewels’’ — surrounded and enclosed by the most fantastic, capricious rock formations … .’’7 The living room is partitioned by folded screens and lowered walls, which are movable according to functional needs. These elements create an uninterrupted flow of rooms. The exposed pine ceiling structure visibly extends beyond the boundaries of the individual partitioned rooms, so that, while remaining private, they are not completely isolated from one another. Reinforcing the dynamic quality of the high open space is its element, the stairs. The wooden stairs of the library lead to the mezzanine. The stairway without a handrail becomes a dynamic element through its expression of flowing movement. The doorway found at the top of the stairs seems a part of the stairway, because it is the same width and is made out of the same material. Barragan’s studio is located next to his house with one wall in common. The studio, with various offices, has direct access from the street through a vestibule. A typical patio is located on the west side of the studio, yet originally it overlooked an enclosed patio with a large window. The patio was enclosed with high walls on three sides, offering a

This house is currently known as the Ortega house realized in 1940. ‘‘Luis Barragan,’’ Arts and Architecture, August [2], pp.24–25. 5 Clive Bamford Smith, Five Mexican Architects, Architectural Book Publishing Co., Inc. New York, [16], p.74. 6 Barragan Foundation, Casa Luis Barragan Guide [3], Mexico. 7 In his official address, 1980 Pritzker Architecture Prize, see Paul Rispa, ed., Barragan, the Complete Work, New York: Princeton Architectural Press, [15], pp. 204–207. 4

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Figure 2. Floor Plans of the Barragan House, 1947. This drawing was reconstructed based on drawings from Casa Luis Barragan Guide, published by the Barragan Foundation in 2004.

Figure 3. One-quarter scale reconstructed model by Byung-in Yu.

Figure 4. Left: Mathias Goeritz’s sketch hung on the living room of the Barragan house. Right: Mathias Goeritz’s sculpture, ‘‘The Doors to Nowhere.’’

visual barrier to lend the serene quality of an enclosed space, decorated with traditional ceramic vases. Between the offices and the studio lies an outdoor space for cleaning. The second-floor plan includes two bedrooms, a guest room, a dressing room, a mezzanine for the house, and

two offices above the studio. The house and the studio are not linked on the second floor. Interestingly, these rooms rely on natural light. The high enclosed walls on the roof terrace provide the space with a sense of privacy and serenity. The terraced garden is blocked off from the street.

The walls help focus attention on the discontinuity between the roof terrace and the outside environment. The terrace becomes a totally isolated part of the house and offers no vista. Exposed only to the sky, the bold roof terrace brings to mind the light sculptures of James Turrell.8

8

Refer to The Life and Work of Luis Barragan, by Jose M. Buendia Julbez, Juan Palomar, and Guillermo Eguiarte. For example, James Turrell’s Skyspace is a freestanding enclosed chamber where one sits on a bench and views the sky and atmospheric changes through an opening in the roof.


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Figure 5. Similar paintings hung on the walls of the Luis Barragan house (reconstructed by the authors).

Sources of Barragan’s Abstract Neo-Plasticity From the construction of this house onward, Barragan begins to resolve his planar surfaces. Horizontal and vertical planes begin to link, trapping rectangular planes within. Barragan’s rectilinear designs fuse abstract neoplasticity with the Mexican landscape tradition. He promoted a rich vocabulary of local materials and a wide range of colors within formal plasticity innovations. Early on, while traveling in Europe in the 1920s, Barragan was inspired by the work of architects such as Gropius, Mies van der Rohe, and Le Corbusier. When he started his practice in 1927, his early designs reflected the Spanish-Mexican vernacular tradition. The year 1947 was generally regarded as the beginning of Barragan’s systematic development, a period that continued until his death in 1988. Unlike his early work, his later designs exhibit simple geometric forms. Barragan was particularly associated with European immigrants within the United States and Mexico. Among others, his association with Mathias Goeritz was of primary influence on his abstract and plastic work. Goeritz carried pure plastic forms to their most extreme limits in his designs. Filtering through Goeritz’s abstraction and influences of minimal art, Barragan incorporated Euro-American Modernist 9

design into the Mexican landscape and his color schemes, creating a unique and exhilarating new design style. Torres de Sate´lite, designed by Mathias Goeritz and Luis Barragan in 1957 and built in 1958, is an example of their collaboration. It is located in Ciudad Sate´lite, a middle-class zone, in the northern part of Naucalpan, Mexico. Josef Albers’s paintings also pay attention particularly to simple compositions and contrasting color. Barragan collected a few of Josef Albers’s paintings, and the series Homage to the Square was displayed on the walls of the Barragan house. A formal analogy between Albers’s paintings and Barragan’s architecture can be readily drawn. Albers’s series Homage to the Square are based on a grid system, drawn on both horizontal and vertical divisions of 20 units each. The first series of Albers’s paintings consists of four squares with four shades of one color. The squares within each painting are nested proportionally, according to their sizes. In basic composition (Figure 6a), the units to either side of the nested squares are twice as large as the units on the top and bottom. The proportional relationship between the squares is based on simple whole numbers such as 1, 2, 3, etc. Accordingly, their arrangement is bilaterally symmetrical along a vertical axis, but not strictly

See Josef Albers’s 1963 book, Interaction of Color, New Haven: Yale University Press.

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concentric, providing a dynamic result. Three more basic types are added later on in his series. Their composition relies on the same divisional technique but an individual square is removed from the four-square composition (Figures 6b–d). With this mathematically plotted framework, Albers experimented with the retinal effects of color within a series of nested squares.9 The squares are used to investigate color interaction with the adjoining colors where they contrast, recede, or pop out. The consecutive squares of color turn out to be perfectly harmonious and purely abstract, unlike anything in nature. There is no evidence that Barragan held a particular regard for mathematics. However, through the use of simple geometry associated with whole-number ratios and the chromatic colors of the paintings, it is evident that Barragan was influenced by Albers’s approaches. Perhaps Barragan takes Albers’s system to achieve harmony and proportion within his works. Barragan clearly appreciated Albers’s approach to exploring the potential of abstract values, shape, color, and texture. At the Bauhaus, Albers dealt principally with abstract, formal issues. He also stressed common materials and their inherent properties. For Albers, a deep knowledge of abstract composition enhanced

Figure 6. Albers’s different square compositions and their proportions (reconstructed by the authors).

the comprehension of materialistic quality and social suitability. Barragan’s approach was also along these lines. Barragan’s form, defined by planar walls of different heights and colors, involves explorations of three-dimensional depth using light and space.10 His work has been praised as having attained a degree of mystical and spiritual abstraction. An example is the color palette used within the Barragan house, which includes yellow, purple, pink, red, and an earthybrown, coupled with neutral grays and whites. Barragan did not follow any systematic color theory such as Johannes Itten’s color circle. Also, Barragan’s use of color is not based on material properties; instead, he sought to articulate color to influence and reinforce desired spatial effects. Within the Barragan house, most walls are colored white, which acts as a foreground element that defines the spatial extension outwards. Key walls

in certain rooms are meanwhile colored. The vestibule best reveals the use of pink to reinforce the spatial intent of the house. The reflected light landing on the pink surface leads visitors from the entrance hall to the vestibule, thereby reinforcing linear movement between spaces via transitional zones. In the living room, Barragan applied yellow to the floor. Counterbalancing the yellow floor is a dark brown wooden ceiling that continues to the adjacent rooms. In the dining room, Barragan applies red to the walls. Three-dimensional space combined with color is reminiscent of Gerrit Thomas Rietveld’s Schroder house, as well as the color drawings of Cornelis van Eesteren and Theo van Doesburg (see Figure 7). Typically, abstract expressionist painters intended to move beyond representation to pure form. In reality, these painters were inspired to create from patterns, shapes, and colors they found within the natural landscape.

Van Doesburg’s early window design illustrates a series of abstract processes that begin with a naturalistic image that is transformed, step-bystep, into an abstract composition of geometric shapes. This is a classic example of abstract expressionism.11 The abstract expressionist movement is described as being inclined heavily towards conceptualization, surpassing all that is to be perceived in material reality. This exponent of conceptualized abstraction influenced Barragan in terms of his abstraction of nature and his feelings about Mexican architecture. Denouncing the traditional image, Barragan searched for a new vision of Mexican architecture through abstraction. Barragan conceptualized the traditional image of the Mexican house and manifested it in a new plastic volumetric morphology that surpassed the traditional model, shedding all formal connotations and structural organization to trace inner force.

Figure 7. Left: Cornelis van Eesteren and Theo van Doesburg, ‘‘Contra-Construction’’ of 1923; middle: Cornelis van Eesteren and Theo van Doesburg, ‘‘Maison Particulie`re’’ of 1922; right: Barragan house color scheme (computer reconstructed). 10

Barragan also used ‘‘two-sided walls:’’ ‘‘One side of his walls, facing the viewer frontally, reveals the sun’s colors; the other side is always shrouded in shadows, suggesting absent presences who seem to await their call to enter the stage.’’ Emilio Ambasz, ‘‘Luis Barragan House and Atelier for Barragan, Tacubaya, Mexico, [1],’’ GA Houses, Tokyo, ADA EDITA. 11 See Allan Doig, Theo Van Doesburg, London: Cambridge University Press, [5].


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Geometrical Analysis of the House While Barragan was respectful of local and regional conditions, he aggressively pursued the abstraction of form, as in the abstract expressionists’ paintings. Barragan abstracted the naturalistic image of the Mexican landscape and then translated it into abstract geometric space and form, thus introducing a new sense of order, space, and form: Not that of organic nature but of simple geometry. Barragan’s entire house appears to be an experiment in simple geometry; there is no evidence that Barragan used a formal system. Nevertheless, when the ground plan or façade is examined, one first recognizes that spatial division is based on a rectangle. The floor plan is analyzed in an attempt to find a basic grid that will establish an underlying geometry for the design as a whole.12 Upon analysis, a 4.25 square meter unit module (M) best explains the house plan. Figure 8 shows the plan overlaid on the determined grid. This grid is essential in determining the proportional layout of the house. The spatial division might be explained by more functional reasons, however, it may also be due to the influence of abstract paintings, which go beyond functional form. For Barragan, the relation between the house and the garden is integral. He wrote, ‘‘In designing and planning these functional gardens it is of primary importance to invest effort in character and atmosphere, as well as in plastic beauty.’’ He continued, ‘‘We found that… if we were to create beautiful architectural forms that were in harmony with [the landscape], we would have to opt for extreme simplicity: Abstract quality, preferably straight lines, flat surfaces and primary geometric shapes.’’13 These ideas are clearly reflected within the house. When Barragan planned the house, he considered the garden as an empty volume related to the house in terms of its shape and design. The house is composed of three major zones: The residence, studio, and garden. When these zones are diagrammed with the primary geometric 12 13

Figure 8. Analysis of the floor plan: The diagram shows how the unit grid (1 M = 4.25 m) is carried through the floor plan (reconstructed by the author).

Figure 9. Plot plan analysis. Left: Three major spaces. Right: Three squares rearranged according to Albers’s painting.

shape, ignoring some minor areas, each zone forms a square and double square representing a spatial territory (Figure 9a). While the parti of the residential area and the garden relies on the square, the studio is on the double square. The size of the square garden

approximately measures 5 M 9 5 M, while the residential area is 4 M 9 4 M. The basic square of the studio is 2.5 M 9 2.5 M. When the three squares are rearranged to the proportions of one of Albers’s paintings, the compositions turn out to be similar, as shown in

The analysis of the floor plan and elevation is based on the drawings from the book Casa Luis Barragan Guide, by the Barragan Foundation. Paul Rispa, ed. Barragan, the Complete Work, New York: Princeton Architectural Press, [15], pp. 34–35.

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Figure 10. Window articulation of the street façade.

Figure 9b. Is this too speculative or accidental? What we can presume is that Barragan intuitively planned three major spaces according to their size relationships using the idea of Albers’s painting to manifest plastic form.

Facade Analysis The Spanish-Mexican traditional house face cannot be found in the Barragan house. In contrast with the plastered planar surfaces of the street and garden façade are various windows that appear to be randomly arranged and are far from being symmetrical. The alignment of the windows has little or no virtue on first impression. An examination of the window frames is significant, because their size, profile, and proportion are strongly related to the character and appearance of the Barragan house. Although collectively the windows are disorganized and of different sizes, each window is ordered using similar proportions with regard to a square. Upon closer examination, it is seen that this square element also dominates the street and garden façade. All window gratings and framings are formed according to the addition and subdivision of the square. Nevertheless, this square unit is not related to that of the floor plans. That is, the square unit of the plan (4.6 m) is not carried through to the elevations. Barragan apparently sought freedom from a single unit constraint.

The imposing façade of the house faces General Francisco Ramirez Street. In the façade, the window openings are all different and are not aligned repeatedly, as shown in Figure 10. Various shapes of window openings as well as gratings are created. These shapes are formed according to a square and a rectangle. That is, windows for the guestroom, dressing room, library, ventilation openings, garage opening, and bathroom are based on a square, but the other two windows for the studio offices are in a rectangular form: Approximately, one window is 7:6 and the other is 6:5 in proportion (Figure 11). In the garden façade, two separate planar walls are formed according to a square. One creates the living room and the other the sleeping and kitchen areas. When extended to chimney height, the dotted line of the living room plane forms a square (abce in Figure 12). The other planar wall forms a square as well (defg) as shown in Figure 12. Window shapes also appear on the garden façade in two forms: Square and rectangular. Square windows are further subdivided with simple ratios such as 1:1 and 2:1. The living room window looking out onto the garden relies on a half division (Figure 13a), and the window of Barragan’s own bedroom is divided into a tripartite form (Figure 13b). Rectangular windows form three types of ratios: A square and a third for

4:3, a square and a fifth for 6:5, and a square and a seventh for 8:7. They may be generated by either a square that is deducted from the rectangle, or a module square that is added to form rectangular windows (Figures 13c–f). For example, the 4:3 rectangle is composed by either assembling 108 square modules or subtracting a square from a rectangle, where the remainder is an undivided rectangle. This remainder can be further subdivided into square modules. It is remarkable that this procedure of making a window is very much like the classical ‘‘anthyphairesis.’’14 Following Fowler (1987), Lionel March [10, 11] provides a pictorial approach to the anthyphairetic procedures of Platonic mathematics. Fowler explained the approach as ‘‘a process of repeated and reciprocal subtraction which is then to generate a definition of ratio as a sequence of repetition numbers.’’ Here, March elaborated the notion by depicting a repetitive subtractive and additive composition. For example, an 11 9 4 rectangle is subtracted, thus leaving a 1 9 1 square as a unit remainder (Figure 14a). Also, based on a 1 9 1 square unit, various modules are added and concatenated (Figure 14b). This generates ‘‘a definition of ratio as a sequence of repetitions numbers, namely, the anthyphairesis.’’ Most of the windows are approximately measured in whole numbers in proportion. Unlike the complexity of

14

Lionel March, Architectonics of proportion: a shape grammatical depiction of classical theory Environmental and Planning B: Planning and Design, 1999, Vol. 26, pp. 91–100.


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Figure 11. Proportional design of window framing and grating.

Figure 12. The geometric design of the garden façade.

Figure 13. Window design derived from the addition and subdivision of a square.

the space forms, there involves surprisingly few room dimensions and corresponding ratios. In the house, six different ratios, 1:1, 2:1, 4:3, 6:5, 7:6, and 8:7, are collected. The noticeable character of the ratio is that most of the fractions advance by adding one to both the numerator and denominator. The ratio is equivalent to the classical sequence of superparticular numbers (March, [9]). All window proportions used in the Barragan house are obtained using this procedure. In addition, related window designs with different ratios can be further generated with the same method as a family group (Figure 15). The subdivision of the window frame is a unique practice of Barragan. In Barragan’s other houses, there lie a

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variety of subdivisions. The window acts as a picture frame that shows a given view of the natural landscape without. Albers’s and Piet Mondrian’s paintings come to mind in terms of their simple proportional divisions (Figure 16). In addition, each interior window shutter in the Barragan house is unique. The white wood panel shutters are carefully designed according to their purpose. Several different types of panel configurations are observed throughout the house. They include: A single panel system, a three panel system with one side containing a single panel and the other side two panels, and four panel systems where each is divided in different proportions. These panels are double hung

using hinges. The shutters in the private rooms operate in a unique way. For example, when the shutter is divided into four panels, the upper portion of the shutters is meant to be opened first; only then can the lower shutters be opened. Central to this idea is privacy: The upper portion is meant only to allow daylight without losing privacy. In other instances, two shutter panels are hinged together. Like windows, each shutter is proportionally divided (Figure 17). These divisions are derived from simple whole number ratios such as 1:1, 2:1, 3:2, 4:3, 7:6, etc. Barragan did not leave us detailed descriptions of how he designed, nor did he outline his strategy for controlling building geometry. In addition, specific geometric practices of the house do not appear among Barragan’s sketches or original working drawings. Due to the relative lack of documentary evidence, one can hardly delineate an exact geometrical principle within the Barragan house. Nevertheless, through studying and modeling of plans, elevations, sections and construction details, the author believes that Barragan, consciously and subconsciously,

Figure 14. The anthyphairesis for an 11 9 4 rectangle: A repetitive subtractive (above) and additive composition (below). (After Lionel March.)

Figure 15. Barragan’s rectangular window ratios, which correspond to the anthyphairetic procedure. (After Lionel March.)

Figure 16. Barragan’s window frame designs in terms of their proportional divisions. a: Living room window frame of the Barragan house; b: Living room window frame of the Lopez house; c: Water fountain entrance of the Galvez house; d. Living room window frame of the Galvez house.

strives towards an abstract neo-plasticity incorporating the use of the square to design the proportions of the façades and floor plans. Furthermore, various

interwoven themes that influence the development of the house clearly reflect Barragan’s approach. There is little argument that the house is part of

the beginning of the Barragan style that culminated in the highly interpretative work reflecting abstract artists such as Mathias Goeritz and Josef Albers. It appears that Barragan did not use a systematic computational method in designing the house but very much an intuitive procedure of addition and subtraction of the square. Superficially, the window layout of the street façade looks disorganized, devoid of any regularity. Closer observation, however, reveals that the manner in which they are designed is similar to the classical ‘‘anthyphairesis,’’ which involves a repetitive subtractive and additive composition. Therefore, it can be speculated that instead of mathematically computed harmony, the house is a manifestation of abstract neo-plasticity, where the design is a search towards a glimpsed subconscious conception. In Barragan’s oeuvre, the house was a turning point, establishing a new style in his architecture. The abstract form of geometry used in the Barragan house formed the groundwork of his future career and established the foundations of his developing ideas. Barragan continued to develop similar vocabularies and design elements in his later projects, most notably the


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Figure 17. Interior wooden window shutters. a: four subdivided panels are hinged for the mezzanine floor; b: three subdivided panels are hinged for the guest bedroom; c: four subdivided panels are hinged for the bedroom; d: a single panel for a small window.

Lopez house, Galvez house, and Gilardi house.

[7] Federica Zanco, F. (2001) Luis Barragań:

[15] Rispa, P. (ed.) (1995) Barragan, the

The Quiet Revolution, Milano: Skira Editore.

Complete Work, New York: Princeton

[8] Julbez, J., Palomar, J., and Eguiarte, G. ACKNOWLEDGMENT

This work was supported by an INHA University research grant.

(1997) The Life and Work of Luis Barragan, New York: Rizzoli. [9] March,

L.

(1998)

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Mack, POAR, Seoul: Ganhyang. Barragan: Space and Shadow, Walls and Colour, Basel: Birkhaüser.

Department of Architecture Inha University 253 Yonghyun-dong, Nam-gu Incheon 402-751 Korea e-mail: [email protected] Department of Architecture Daelim College 526-7 Bisan-dong, Dongan-gu Anyang 431-715 Korea College of Architecture Hongik University 72-1 Sangsu-dong, Mapo-gu, Seoul 121-791 Korea

Reviews

Osmo Pekonen, Editor

Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing by Daniela Calvetti and Erkki Somersalo HEIDELBERG, SPRINGER SCIENCE + BUSINESS MEDIA, 2007, 202 PP., EUR32.95 ISBN 978-0387-73393-7 REVIEWED BY URI ASCHER

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

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

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he application of scientific computing as a tool for understanding and gaining quantitative knowledge of physical processes typically has two phases. In the first phase, a mathematical model is generated, and in the second, the model is simulated on a computer using appropriate numerical methods. Now, assuming that the mathematical model is not so incredibly complex that it must be simplified, should these two phases be independent of each other? There is a lot to be said for such a phase separation. Many useful numerical methods for differential equations, for instance, have been derived, analyzed, and programmed without a specific application in mind. Thus, once a researcher in biological evolution has succeeded in formulating the propagation and control of a measles epidemic as a time-dependent system of ordinary differential equations (ODE), there are canned routines available that may be used for the subsequent simulation of the ODE system. These routines are typically both more efficient and more reliable than what the mathematical

biologist would write for a special-purpose end, even though the specific application was not taken into account during the design and implementation of the general-purpose software. Probably the extreme in this regard are the excellent general packages available for numerical linear algebra tasks such as solving a linear system of equations or finding the eigenvalues of a matrix [2, 5]. Under normal circumstances, such packages, and general numerical methods, should certainly be used rather than reproduced. The rationale for a full-phase separation breaks down, however, when the mathematical model to be simulated is significantly incomplete or in doubt. Such is often the case with ill-posed inverse problems, where an orthodox solution of the mathematical model initially presented is neither possible nor desirable. Note the difference between the objective notion of solvability of an ill-posed problem, considered by Hadamard more than 100 years ago, and the subjective notion of desirability. For example, the problem of deblurring a noisy image (say a police snapshot of one’s license plate when caught speeding) can be modeled as a singular or highly ill-conditioned linear system. The latter can be subsequently solved approximately, using some form of regularization [3, 4]. But the desirability of such a solution may well depend on the manner by which the unknown measurement noise has been handled or accounted for! A natural, although by no means only, way to account for lack of knowledge in modeling and for subjectivity, is the Bayesian probability framework. This establishes the connection between three separate areas that this book explores. Introduction to Bayesian Scientific Computing is a 200-page, easily accessible, pleasant introduction fusing Bayesian approaches with numerical linear algebra methods for inverse problems: A tutorial that one does not have to believe in all its details to enjoy. To make it so accessible, the


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authors often use informal language, a lot of motivation, and an excellent set of examples. They essentially avoid formal mathematics (e.g., no theorems and no proofs, although they do use formulae, and their underlying mathematics is carefully thought out). There is no attempt at completeness, nor of comprehensive referencing. The authors also avoid explicit mention of machine learning [1]. Having had some limited previous exposure to all three components of this mixture, namely Bayesian probability, numerical linear algebra and inverse problems, this book’s approach has worked for me. There are 10 chapters, each covering a ‘‘lecture’’ in a graduate course that the authors have given at universities in Italy, Finland, and the USA. An uninitiated graduate student probably would need a week or two to absorb the material in each of these lectures, so there is a blueprint here for a graduate course of a normal trimester length. However, as the authors acknowledge, this is not a self-contained textbook, and it must be supported by a reading list of other texts, which they supply. The first three chapters introduce the necessary essentials for Bayesian inference. In an inverse problem, we want to estimate an unknown quantity x from a set of indirect measurements y. The corresponding problem of statistical inference is to infer properties of an unknown probability density distribution given the data which have been generated from that distribution. Following essential definitions and a few basic theorems, one arrives at the Bayes formula that says that the posterior probability density of x given y is proportional to the likelihood, which is the density of y given x

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(corresponding to the forward problem in inverse problem parlance) times the density of the prior (corresponding to introducing a priori information such as past experience or image smoothness). A maximum a posteriori (MAP) estimator can subsequently be obtained for x. Chapter 4 then introduces the third link, numerical methods for linear systems of algebraic equations, with an eye towards ill-conditioned problems and the smoothing properties of truncated conjugate gradient-type iterations. This is followed in Chapter 6 with the probabilistic design of preconditioners, called here priorconditioners, which allow a few iterations towards the solution of an ill-conditioned linear system to capture more features of a desired solution. There is also a quick section on designing a prior based on a training set and on model reduction using principal component analysis. Chapters 7 and 8 are concerned with conditional Gaussian densities and yield some rather important formulae for noisy linear systems of algebraic equations. The basic task is to obtain the probability distribution of some components of a multivariate, normally distributed random variable with the values of the other components fixed. Here, there is a good emphasis and exploitation through examples of the additional information that the Bayesian probability framework yields, namely, not only a point solution (or a single output), but also means for assessing its worth and trustworthiness in terms of credibility envelopes. There usually are, after all, other, often simpler ways to incorporate prior information into a regularization method, if that were the only thing at stake.

Chapters 5 and 9 address the important issue of sampling from a given distribution in order to verify that the distribution is what we think it is (or to approximate integrals in many dimensions). This exposition culminates in the Markov Chain Monte Carlo (MCMC) sampling and the classical Metropolis-Hastings algorithm. Finally, Chapter 10 wraps it up by using concepts and methods from different previous chapters, introducing hypermodels and solving an example of deblurring a one-dimensional surface with discontinuities. What I like most about this book is the apparent enthusiasm of the authors and their genuine interest in explaining rather than showing off. This enthusiasm is contagious, and the result is very readable.

REFERENCES

[1] C. M. Bishop. Pattern Recognition and Machine Learning. New York: Springer Science + Business Media, 2006. [2] T. A. Davis, Direct Methods for Sparse Linear Systems. SIAM, 2006. [3] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic, 1996. [4] J. Kaipo and E. Somersalo, Statistical and Computational Inverse Problems. New York, Springer Science + Business Media, 2005. [5] Y. Saad, Iterartive Methods for Sparse Linear Systems. PWS Publishing Company, 1996. Department of Computer Science University of British Columbia Vancouver, BC V6T 1Z4, Canada e-mail: [email protected]

A Certain Ambiguity. A Mathematical Novel by Gaurav Suri and Hartosh Sing Bal PRINCETON, PRINCETON UNIVERSITY PRESS, 2007, US$ 27.95, 292 PP. ISBN-13: 978-0-69112709-3 REVIEWED BY TOM PETSINIS

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orks of fiction containing varying degrees of mathematics have burgeoned in recent years. They include historical fiction based on mathematicians, surreal works exploring mathematical ideas, and speculative fiction based on famous theorems and conjectures. Publishers see such books as demystifying the subject and perhaps making it accessible to a wider readership. Conversely, the general reading public must be interested in the subject matter, or new titles would not be emerging with such frequency. Co-authored by Suri and Bal, A Certain Ambiguity is a weighty addition to what is becoming a genre. The novel is narrated by Ravi, a young Indian whose interest in mathematics is ignited by his grandfather Vijay Sahni, who dies tranquilly at the beginning of the book. At 18, Ravi leaves India for an unnamed American university, intent on pursuing a career in finance. In his first semester, he enrolls in an elective unit called ‘‘Thinking about Infinity,’’ presented by the inspiring Nico Aliprantis, an unconventional lecturer whose skill in making baklava suggests a Greek background. Ravi soon learns that his grandfather visited America in the early decades of the 20th century, and this sets him off on a trail of detection. It turns out that as a visiting scholar in the fictional town of

Morisette, New Jersey, Vijay was imprisoned for expounding mathematical ideas that were construed as blasphemous. (He was prisoner number 1729—an obvious reference to the Indian mathematician Ramanujan who, on his death-bed, saw this number as the smallest number that could be expressed as the sum of two cubes in two different ways.) From newspapers and transcripts of the trial, Ravi discovers that Vijay developed a rapport with John Taylor, the presiding judge. The two men discussed at some length the question of God’s existence, Euclid’s axioms, and the nature of ultimate truth. In the end, the judge questioned his own religious views, secured Vijay’s early release, and the two men went on to maintain a lifelong friendship. As Ravi’s course progresses, he is forced to decide between a career in finance to repay his family’s investment in his education, or one in mathematics in honour of his beloved grandfather. The authors withhold his decision, though the reader feels the young man’s journey, his enquiring nature, and the influence of the charismatic Nico will draw him to mathematics. The relationship between Ravi and his grandfather recalls another novel with mathematical content: Uncle Petros and Goldbach’s Conjecture by Apostolos Doxiadis. The authors of A Certain Ambiguity could have used Doxiadis’s economy in structuring their novel. As it stands, their work is overly discursive, with long tracts of mathematical exposition that interrupt the narrative flow. At times it appears as though the narrative is nothing more than a vehicle for lectures on mathematics. This is often the problem with novels of this type: They fail to strike the right balance between the didactic and the dramatic. In this case, the novel’s didactic sections are lucid and engaging. The history of mathematical infinity is clearly outlined, beginning with Zeno’s Paradoxes through to convergent series. Cantor’s hierarchy of

infinity is expressed and illustrated in a manner accessible to the general reader. Euclid’s fifth postulate is discussed at some length, with interesting references to Gauss, Bolyai, Lobachevsky, Riemann, and Einstein. But despite the admirable collection of wellexplained ideas, the book falls short in many areas of literary fiction. Ravi, the main character, is thinly drawn and fails to grip the reader with his first-person voice. There is little sense of time and place. The mathematical interpolations are too lengthy and come at the expense of narrative, to the extent of diminishing the reader’s interest in the characters. Other material appears in the novel without preparation or justification. There a several gratuitous ‘‘diary’’ entries from mathematicians ranging from Pythagoras to Go¨del. In what is essentially a realistic novel, these entries are not sufficiently framed by the story. Had they been Ravi’s dreams or daydreams, there may have been some justification for them: As it is, they are simply further exposition without integration. The novel ends with a chapter-length diary record of Judge Taylor’s experiences and trip to India to visit Vijay. The literary writing is at its strongest here, with the Judge, more so than Ravi, emerging as the novel’s best-developed character. Authors who embark on novels of ideas, especially mathematical, face the challenge of making those ideas appear to come naturally from the characters; in other words, the ideas must be shown through flesh and blood. One of the attributes of good fiction is its power to pull readers into its world and keep them interested in its characters. A Certain Ambiguity is quite strong on mathematical exposition. Unfortunately, as a novel, it doesn’t fully draw the reader into its fictional landscape. Teaching and Learning Services Victoria University Footscray Park, Melbourne VIC 8001, Australia e-mail: [email protected]


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Symmetry and the Monster by Mark Ronan NEW YORK, OXFORD UNIVERSITY PRESS, 2006, 255 PP. US$27.00 ISBN 978-0-19-280722-9. REVIEWED BY KISHORE MARATHE

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ymmetry and the Monster recounts the story of an exceptional result in the history of mathematics: The classification of finite simple groups. The existence and uniqueness of the largest sporadic group, dubbed the Monster, was the last piece in the classification. The complete classification is arguably the greatest achievement of 20th century mathematics. In fact, it is unique in the history of mathematics: The result of hundreds of mathematicians working in many countries around the world for over a quarter century. This global initiative was launched by Daniel Gorenstein, whose book [5] is still an excellent general reference for this material. We now describe the highlights of this fascinating story. The first four chapters introduce groups and their application in Galois’s work. Recall that a group is called simple if it has no proper nontrivial normal subgroups. Thus, an Abelian group is simple if and only if it is isomorphic to one of the groups Zp, for p a prime number. This is the simplest example of an infinite family of finite simple groups. Another infinite family of finite simple groups is the family of alternating groups An, n [ 4 that we study in the first course in algebra. These two families were known in the 19th century. The last of the families of finite groups, called groups of Lie type, were defined by Chevalley in the mid 20th century. Chapters 5 to 9 discuss this material. By the early 20th century, the Killing–Cartan classification of simple Lie groups defined over the field C of complex numbers had produced four infinite families and five exceptional groups. This classification starts by classifying simple Lie algebras over C and then constructing 76

corresponding simple Lie groups. In 1955, using this structure but replacing the complex numbers by a finite field, Chevalley’s fundamental paper showed how to construct finite groups of Lie type. This work led to the classification of all infinite families of finite simple groups. However, it was known that there were finite simple groups, called sporadic groups, that did not belong to any of these families. Chapters 10 to 14 are devoted to the discoveries of the 26 sporadic groups. The first sporadic group was constructed by Mathieu in 1861. In fact, he constructed five sporadic groups, now called Mathieu groups. There was an interval of more than 100 years before the sixth sporadic group was discovered by Janko in 1965. Two theoretical developments played a crucial role in the search for new simple groups. The first of these appeared in Brauer’s address at the 1954 ICM in Amsterdam. It gave the definitive indication of the surprising fact that general classification theorems would have to include sporadic groups as exceptional cases. In fact, Fischer discovered and constructed his first three sporadic groups in the process of proving such a classification theorem. Brauer’s work made essential use of elements of order 2. The second came in 1961, when Feit and Thompson proved that every nonAbelian simple finite group contains an element of order 2. The proof of this one line result occupies an entire 255page issue of the Pacific Journal of Mathematics (Volume 13, 1963). Before the Feit–Thompson theorem, the classification of finite simple groups seemed to be a rather distant goal. This theorem and Janko’s new sporadic group greatly stimulated the mathematics community to look for new sporadic groups. John Leech had discovered his 24dimensional lattice while studying the problem of sphere packing. The Leech lattice provides the tightest sphere packing in 24 dimensions. (However, the sphere packing problem in other dimensions is still wide open.) Symmetries of the Leech lattice contained Mathieu’s largest sporadic group. It also had a large number of symmetries of order 2. Leech believed that the symmetries of his lattice contained other sporadic groups as well. Leech

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

was not a group-theorist and he could not get group-theorists interested in his lattice. But he did find a young mathematician (who was not a grouptheorist) to study his work. In 1968, John Conway was a junior faculty member at Cambridge. He quickly became a believer in Leech’s ideas. He tried to get Thompson, the great guru of group-theorists, interested in his work. Thompson told him to find the size of the group of symmetries and then call him. Conway later remarked that he did not know that he was using a folk theorem which says: The two main steps in finding a new sporadic group are (i) find the size of the group of symmetries, and (ii) call Thompson. Conway worked very hard on this problem and soon came up with a number. This work turned out to be his big break. It changed the course of his life and has made him into a world-class mathematician. He called Thompson with his number. Thompson called back in 20 minutes and told him that half his number could be a possible size of a new sporadic group and that there were two other new sporadic groups associated with it. These three groups are now denoted by Co1, Co2, Co3 in Conway’s honor. Further study by Conway and Thompson showed that the symmetries of the Leech lattice give 12 sporadic groups in all, including all five Mathieu groups. In the early 1970s, Conway started the ATLAS project to collect all essential information (mainly the character tables) about the sporadic groups and some others. The work continued into the early 1980s when all the sporadic groups were finally known. After Conway’s work, the next major advance in finding new sporadic groups came through the work of Berndt Fischer. Working under Baer, Fischer became interested in groups generated by transpositions. Recall that, in a permutation group, a transposition interchanges two elements. Fischer first proved that a group G generated by such transpositions falls into one of six types. The first type is a permutation group and the next four lead to known families of simple groups. It was the sixth case that led to three new sporadic groups, each related to one of the three largest Mathieu groups. The geometry underlying the

construction of G is that of a graph associated to generators of G. Permutation groups and the classical groups all have natural representations as automorphism groups of such graphs. Fischer’s graphs give not only some known groups, but also his three new sporadic groups. He published this work in 1971 as the first of a series of papers; no further papers in the series ever appeared. In fact most of his work is not published. Fischer continued studying other transposition groups. This led him first to a new sporadic group, now called the Baby Monster. By 1981, 20 new sporadic groups were discovered, bringing the total to 25. The existence of the 26th and the largest of these groups was conjectured independently by Fischer and Griess in 1973. Several scientists conjectured that this exceptional group must have relations with other areas of mathematics as well as with theoretical physics. The results that have poured in since then seem to justify this early assessment. Some strange coincidences noticed first by MacKay and Thompson were investigated by Conway and Norton. They called this group the Monster and their unbelievable set of conjectures ‘‘Monstrous Moonshine.’’ Their paper [2] appeared in the Bulletin of the London Mathematical Society in 1979. The same issue of the Bulletin contained three papers by Thompson discussing his observations of some numerology between the Fischer–Griess Monster M and the elliptic modular functions. Thompson stated his conjectures about the relation of the characters of the Monster and Hauptmoduls for various modular groups. He also showed that there is at most one group which possesses the properties expected of M and has a complex, irreducible representation of degree 196883 = 47.59.71 (47, 59 and 71 are the three largest prime divisors of the order of the Monster group). Conway and Norton had conjectured earlier that the Monster should have a complex, irreducible representation of degree 196883. Based on this conjecture, Fischer, Livingstone and Thorne (Birmingham notes 1978) computed the entire character table of the Monster. The construction of the Monster was announced by Griess in 1981, and the

complete details were given in [6]. Griess first constructed a commutative, nonassociative algebra A of dimension 196884 and then showed that the Monster group is its automorphism group. In the same year, the final step in the classification of finite simple groups was completed by Norton by establishing that the Monster has an irreducible complex representation of degree 196883 (the proof appeared in print later). Combined with the earlier result of Thompson, this proved the uniqueness of the Monster. So the classification of finite simple groups was complete. The various parts of the classification proof together fill thousands of pages. The project to organize all this material and to prepare a flow chart of the proof is expected to continue for years to come. The last three chapters give a brief account of the construction of the Monster and the Monstrous Moonshine Conjectures. We now give a mathematical formulation of these conjectures.

Monstrous Moonshine Conjectures 1. For each g [ M there exists a MacKay–Thompson series Tg(z) with normalized Fourier series expansion given by Tg ðzÞ ¼ q 1 þ

1 X

cg ðnÞq n ;q ¼ e 2piz :

1

ð1Þ There exists a sequence Hn of representations of M, called the head representations, such that cg ðnÞ ¼ vn ðgÞ;

ð2Þ

where vn is the character of Hn. 2. For each g [ M, there exists a Hauptmodul Jg for some modular group of genus zero, such that Tg = Jg. In particular, (a) T1 = J1 = J, the Jacobi Hauptmodul for the modular group C. (b) If g is an element of prime order p, then Tg is a Hauptmodul for the modular group Gp studied by Ogg. 3. Let [g] denote the set of all elements in M that are conjugate to gi, i [ Z. Then Tg depends only on the class [g]. Note that from Equation (1) and

(2), it follows that Tg is a class function in the usual sense. However, [g] is not the usual conjugacy class. There are 194 conjugacy classes of M but only 171 distinct MacKay–Thompson series. Conway and Norton calculated all the functions Tg and compared their first few coefficients with the coefficients of known genus-zero Hauptmoduls. Such a check turns out to be part of Borcherds’s proof, which he outlined in his lecture at the 1998 ICM in Berlin [1]. The first step was the construction of the Moonshine Modul. The entire book [3] by Frenkel, Lepowsky, and Meurman is devoted to the construction of this module, denoted by V \ : It has the structure of an algebra called the Moonshine vertex operator algebra (also denoted by V \ ). They proved that the automorphism group of the infinite dimensional graded algebra V \ is the largest of the finite, sporadic, simple groups, namely, the Monster. The second step was the construction by Borcherds of the Monster Lie algebra using the Moonshine vertex operator algebra V \ : He used this algebra to obtain combinatorial recursion relations between the coefficients cg(n) of the MacKay–Thompson series. It was known that the Hauptmoduls satisfied these relations and that any function satisfying these relations is uniquely determined by a finite number of coefficients. In fact, checking the first five coefficients is sufficient for each of the 171 distinct series. Thus all the ‘‘Monstrous Moonshine’’ conjectures are now parts of what we can call the ‘‘Moonshine Theorem.’’ Its relation to vertex operator algebras, which arise as chiral algebras in conformal field theory and string theory, has been established. In spite of the great success of these new mathematical ideas, many mysteries about the Monster are still unexplained. A recent update on the Moonshine may be found in the book by Gannon [4]. We conclude this summary with a comment, a modification of the remarks made by Ogg in [8] when the existence of the Monster group and its relation to modular functions were still conjectures (strongly supported by computational evidence). Its deep significance for theoretical physics is still emerging. So mathematicians and physicists, young

Ó 2008 The Author(s). This article is published with open access at Springerlink.com Volume 31, Number 1, 2009

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and old, should rejoice at the emergence of a new subject, guaranteed to be rich and varied and deep, with many new questions to be asked and many of the conjectured results yet to be proved. It is indeed quite extraordinary that a new light should be shed on the theory of modular functions, one of the most beautiful and extensively studied areas of classical mathematics, by the largest and the most exotic sporadic group, the Monster. That its interaction goes beyond mathematics, into areas of theoretical physics, such as conformal field theory, chiral algebras and string theory, may be taken as strong evidence for a new area of research which this reviewer has called in [7] ‘‘Physical Mathematics.’’ Symmetry and the Monster is written in nontechnical language and yet conveys the excitement of a great mathematical discovery usually accessible only to professional mathematicians. The author knew many of the contributors, and this brings a nice personal touch to the narrative. His use of nonstandard terminology seems quite unnecessary, however. The term ‘‘atom of symmetry’’ is not more illuminating than ‘‘simple group’’ for the lay reader and is annoying to anyone who has taken a first course in algebra. There are several factual errors and misstatements. The worst puts Newton and Leibniz developing calculus in the 16th century (p. 87) and again in the 17th century

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(p. 89). Janos Bolyai’s appendix is at the end of his father’s book on geometry and not in the book by Gauss (p. 195). Parts dealing with physics, especially the last chapter, contain misstatements. There is no evidence at this time that string theory combines quantum physics and general relativity (p. 72) or that it provides a model for elementary particles (p. 218). The level of material varies greatly. It is doubtful that a reader who needs to be reminded of the quadratic formula, golden ratio or p and e will take away much mathematics from this book. But in spite of these shortcomings, the book gives a good description of many aspects of an important event in the history of mathematics. OPEN ACCESS

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. REFERENCES

[1] R. E. Borcherds. Monstrous moonshine and monstrous Lie superalgebras. Inventi-

[3] I. Frenkel, J. Lepowsky, and A. Meurman. Vertex Operator Algebras and the Monster. Pure and App. Math., # 134. Academic Press, New York, 1988. [4] T. Gannon, Moonshine Beyond the Monster.

Cambridge

University

Press,

Cambridge, 2006. [5] D. Gorenstein, Finite Simple Groups. Plenum Press, New York, 1982. [6] R. Griess. The friendly giant. Invent. Math., 69:1–102, 1982. [7] Kishore Marathe. A Chapter in Physical Mathematics: Theory of Knots in the Sciences. In: B. Engquist and W. Schmidt eds., Mathematics Unlimited—2001 and Beyond, pp. 873–888, Berlin, 2001. Springer-Verlag. [8] A. P. Ogg, Modular functions. In Santa Cruz Conference on Finite Groups, Proc. Sympos. Pure Math., 37, pp. 521– 532, Providence, 1980. Amer. Math. Soc.

Department of Mathematics City University of New York Brooklyn College Brooklyn, NY 11210, USA e-mail: [email protected]; [email protected]

ones Math., 109:405–444, 1992. [2] J. H. Conway and S. P. Norton. Monstrous moonshine. Bull. London Math. Soc., 11(3):308–339, 1979.

Max Planck Institute for Mathematics in the Sciences Leipzig Dresden, Germany

Ernst Zermelo. An Approach to His Life and Work by Heinz-Dieter Ebbinghaus, in cooperation with Volker Peckhaus BERLIN, SPRINGER SCIENCE + BUSINESS MEDIA, 2007, XIV + 356 PP, €49.95, ISBN: 978-3-54049551-2 REVIEWED BY HENRY E. HEATHERLY

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rnst Zermelo was the most influential set-theorist of the first half of the 20th century. He is best known now for his work on the Axiom of Choice and for being the first person to give an axiomatic treatment of set theory [1]. However, his mathematical career began along quite different lines with a doctoral dissertation on the calculus of variations (Berlin, 1894), an assistantship with Max Planck at the Institute for Theoretical Physics in Berlin, and several important, albeit controversial, papers on thermodynamics. Ebbinghaus has given us a full biography of Zermelo the mathematician and scientist, as well as an insightful description of Zermelo’s professional and personal life, and his interactions with colleagues and adversaries. Zermelo left Planck’s Institute in 1897 for further study in theoretical physics at Go¨ttingen, which resulted in an Habilitation thesis on hydrodynamics in 1899. However, soon after he arrived in Go¨ttingen, Zermelo began to take serious interest in set theory and logic. He later wrote (in 1930) that this was due to the influence of David Hilbert. In the winter semester of 1900/ 1901, Zermelo gave the first course ever devoted entirely to set theory. By 1902, he had published his first paper on the subject, a short note on transfinite cardinal number addition. At the Third International Congress of Mathematicians (Heidelberg, August 1904), Julius Ko¨nig delivered a lecture where he claimed that Cantor’s Continuum Hypothesis was false and that the cardinality of the continuum is not an aleph. He also claimed to have refuted

the Well Ordering Principle (WOP). This caused considerable controversy in which Zermelo was actively involved. He quickly found an error in Ko¨nig’s argument, as did others. This affair focused Zermelo’s attention on the WOP. By late September 1904, Zermelo had a proof of the WOP and had explicitly stated his ‘‘principle of choice,’’ later called the ‘‘Axiom of Choice.’’ He communicated these results in a letter to Hilbert, and Hilbert had the relevant parts published [2, pp. 139–141]. At the end of the paper, Zermelo states that he owed the idea of using the principle of choice to Erhard Schmidt. Zermelo’s proof of the WOP became the object of intense criticism that arose from three main sources: An uneasiness with his new vehicle, the principle of choice; suspicion of any argument that seemed similar to those which had led to recently discovered paradoxes; and an old mistrust of Cantor’s set theory. Among those who were highly critical were Poincare`, Schoenflies, Borel, and Felix Bernstein. In 1908, Zermelo gave a second proof of the WOP, again using the principle of choice [2, pp. 183–198]. Also in this paper is a critique of the objections that had been raised against the first proof. To solidify the foundation for his proof of the WOP, further its comprehensibility, and make clear the role of the principle of choice, Zermelo formulated the first axiomatic treatment of set theory, which he published in 1908 [2, pp. 198–215]. He did this using seven axioms, with the Axiom of Choice (AC) being number six. He noted that he was unable to prove the consistency of these axioms, but he showed that several of the known paradoxes of set theory cannot be obtained from his axioms. Although Zermelo’s system was immediately used by some, the general response to it was ambivalent (see [1, Section 3.3]). It is telling that Hausdorff did not use the axiomatic point of view in his very influential book on set theory published in 1914. The first textbook on axiomatic set theory was by Fraenkel in 1919, who used Zermelo’s system as his base. In the early 1920s, both Fraenkel and Skolem refined Zermelo’s system.

In 1930, Zermelo reformulated his system, adding two more axioms. He called this system the ‘‘Zermelo– Fraenkel axiom system.’’ It included AC, so now we would call it ZF + AC, or ZFC. From 1902 until 1907, Zermelo was a Privatdozent at Go¨ttingen. His academic career was progressing slowly, possibly because of several interruptions due to illness. Using his very significant influence with the Prussian Ministry of Cultural Affairs, Hilbert was able to have Zermelo appointed to a lectureship in mathematical logic at Go¨ttingen. Because of health problems, Zermelo could not begin his lectures as scheduled in the winter semester 1907/ 1908, but in the summer semester of 1908, he gave the first course in mathematical logic ever offered at a German university. Zermelo’s plans to write a book on mathematical logic never achieved fruition, however; the lecture notes from his course on mathematical logic are in the Zermelo Nachlass, [3]. The lectureship in logic was only a temporary expedient. Zermelo wanted a permanent position. By 1909, he had impressive research credentials with a substantial list of published work in calculus of variations, thermodynamics, hydrodynamics, and set theory. He also was strongly supported by Hilbert. Yet he had been passed over several times for permanent positions. The reasons were illness and controversy. Zermelo’s early papers on thermodynamics contained some sharp disagreements with earlier work by Ludwig Boltzmann, which led to a public controversy between the two lasting until just after Boltzmann’s death in 1906. Ebbinghaus (see p. 25) illustrates the degree of acrimony in this affair with excerpts from a letter from Boltzmann to Felix Klein. In addition to the general controversy over Zermelo’s work on the WOP and AC, Zermelo was noted for the ‘‘polemical aspect’’ of his personality, an attribute that was still being commented on when Zermelo was in his sixties. Helmuth Gericke, who was Zermelo’s research assistant in 1934, later commented that ‘‘sometimes he even insulted his friends.’’ Among other things, this caused a personal controversy with Felix Bernstein, which in turn became linked to


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arguments within the Philosophical Faculty at Go¨ttingen. Zermelo’s ill health, extending back to his youth, frequently left him unable to satisfy his teaching commitments. By 1905, it had become a serious cause of interruptions in his career. He suffered from respiratory illnesses and was diagnosed in June 1906 with tuberculosis of the lungs. To this one must add his occasionally erratic mental condition. In 1910, Zermelo’s outstanding research record and the strong recommendations by Hilbert and others overcame the negatives and he was offered a professorship at the University of Zu¨rich. His time in Zu¨rich, 1910 to 1916, is the only period in his life that he held a paid university professorship. In Zu¨rich, Zermelo continued his research on set theory as well as working on measure theory, abstract algebra, and game theory. The latter concerned an application of set theory to the game of chess (he was an enthusiastic chess player). Zermelo’s only doctoral student at Zu¨rich was Waldemar Alexandrow, who completed a dissertation on the foundations of measure theory in 1915. Paul Bernays and Ludwig Bieberbach each made their Habilitation under Zermelo in Zu¨rich. In light of his later renown in set theory and logic, it is somewhat surprising that Bernays’s thesis in Zu¨rich was on modular elliptic functions; he later did a second Habilitationsschrift on logic at Go¨ttingen under the direction of Hilbert. Long interruptions due to illness hindered Zermelo in supervising the mathematical development of students. His health continued to deteriorate until he was forced to retire from the University in April 1916. For the next five-and-a-half years, Zermelo had no academic home base. In this interim he frequently stayed at health resorts, but he continued to work mathematically. In October 1916, he was awarded the Alfred AckermannTeubner Prize of Leipzig University. In March 1921, Abraham Fraenkel began a correspondence with Zermelo concerning the independence of Zermelo’s axioms for set theory. In one such letter, Zermelo formulated a second-order version of the axiom of replacement, yet he also criticized this because of its nondefinite character. He also took this 80


ambivalent position publicly, for example at a 1921 meeting of the German Mathematical Union in Jena. In October 1921, Zermelo moved to Freiburg in southwestern Germany, where he lived until his death in 1953. In 1926, he was appointed ‘‘full honorary professor’’ at the Mathematical Institute of the University of Freiburg. This carried no salary, but he was given the fees paid by participants in his courses. His health improved somewhat, allowing him to undertake modest teaching activities. He typically gave one course each semester, and these ranged from applied mathematics to foundations, as well as courses in complex analysis, real functions, and differential equations. His scientific activity increased remarkably. His work on applied mathematics included a paper on ‘‘navigation in the air as a problem of the calculus of variations’’ and one on evaluating the results of chess tournaments. Of course, he continued work on the foundations of set theory and on logic. This included work on infinitary languages and infinitary logic. During his Freiburg period, Zermelo became involved in several acrimonious controversies with leading figures in set theory and logic. The first was with Fraenkel. Zermelo served as editor for the publication of Cantor’s collected mathematical and philosophical works. This project, which Zermelo recommended to the Berlin publishing house Springer-Verlag in 1926, led to controversy and hard feelings with Fraenkel. The main point of contention concerned the biographical essay of Cantor that Fraenkel wrote for the collected works and Zermelo’s highly critical remarks concerning this in correspondence with Fraenkel. The next controversy Ebbinghaus calls ‘‘A ‘War’ Against Skolem.’’ In both his published comments and in personal correspondence, Zermelo reacted vigorously and even harshly to Thoralf Skolem’s 1929 paper which gave a version of definiteness that essentially corresponds to second-order definability. Zermelo felt that axiomatic set theory was threatened by Skolem’s results and that he had ‘‘a particular duty’’ to fight against it. Soon after this, Zermelo became involved in the controversy swirling around Go¨del’s Incompleteness Theorem

of 1930. This led to a lively personal correspondence with Go¨del, as well as public or published remarks by each of them. Zermelo conducted himself in his usual tactless style and seemed to be concerned with ‘‘plots’’ against him. During this period, he suffered a nervous breakdown from which he quickly recovered. Less than two years after the Nazis took power in Germany, Zermelo ran afoul of the regime. In January 1935, Eugen Schlotter, an assistant at the Mathematical Institute and an ardent Nazi, denounced Zermelo to the university authorities. Other accusations quickly followed, and a formal investigation was undertaken by the rector, who officially recommended that Zermelo give up his teaching duties. In March 1935, Zermelo resigned from his position at the University of Freiburg. The loss of his honorary professorship and his disappointment over the behavior of some of his former colleagues left Zermelo bitter. The year 1935 marks the beginning of a decline in Zermelo’s mental energy. He worked on a book on set theory which was never completed. By the end of the 1930s, he had withdrawn completely from the scene of foundations of mathematics. However, he continued some smaller mathematical projects, pure and applied, as evidenced by his Nachlass. In October 1943, he married Gertrud Seekamp, whom he had known for some time. In 1946, Zermelo was reappointed as honorary professor at Freiburg, but he was unable to return to lecturing because of increasing blindness and infirmities of age. He died on May 21, 1953. Gertrud lived to be over 100, outliving Ernst by more than 50 years. The biography under review contains numerous photographs of Zermelo, his family, and his colleagues. Some of these give insight into Zermelo’s personality, e.g., Zermelo at the dinner table with his dog (p. 175) and the small three-wheeled car that he drove in the early 1930s (p. 146). Of considerable interest, as well as being helpful in getting an accurate, unvarnished perception of events and personalities, are the many excerpts from letters, not only to or from Zermelo, but also correspondence between other major figures. There is an extensive list of references and a helpful chronological vita. The book is well edited, with only a few minor typographical errors. It is

highly recommended for university libraries and for those interested in the history of mathematics of the 20th century.

Influence, Springer-Verlag, New York,

by Kurt Grelling. Universita¨tsarchiv Frei-

1982. [2] Jean van Heijenoort, From Frege to Go¨del.

berg, C129/224 (Part I) and C129/215 (Part II), Freiberg.

A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, 1967.

REFERENCES

[1] Gregory H. Moore, Zermelo’s Axiom of Choice. Its Origins, Development, and

[3] Ernst Zermelo, Mathematische Logik. Vorlesungen gehalten von Prof. Dr. E. Zermelo zu Go¨ttingen im S.S. 1908, lecture notes

Mathematics Department University of Louisiana at Lafayette 217 Maxim Doucet Hall, P.O. Box 41010, Lafayette, LA 70504-1010, USA e-mail: [email protected]


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Number Theory Through Inquiry by David C. Marshall, Edward Odell, and Michael Starbird WASHINGTON, DC, THE MATHEMATICAL ASSOCIATION OF AMERICA, 2007, HARDCOVER, IX + 140 PP., US$51.00, ISBN: 978-0-88385-751-9 REVIEWED BY JOHN J. WATKINS

R

. L. Moore strode like a giant over the mathematical landscape of America during the first half of the twentieth century. Born in Dallas, Texas, he spent 49 of his illustrious 64-year career at the University of Texas. A key member of the American school of point-set topology, Moore served as president of the American Mathematical Society, produced 50 doctoral students, six of whom eventually became either president or vicepresident of the AMS, while five served as president of the Mathematical Association of America, and now these 50 students have themselves produced 2239 doctoral descendants. In 1999, at the end of the millennium, Keith Devlin in his MAA Online column claimed that R. L. Moore was the ‘‘greatest university mathematics teacher ever.’’ It is hard to dispute that claim, though sadly it should be added that Moore’s reputation carries with it an often ignored blemish: Moore held deeply racist attitudes toward black students. While still a student at the University of Chicago from 1903–1905, Robert Lee Moore conceived of a radical new style of teaching. He would soon mold this idea into a highly successful technique, a new method of teaching that would eventually bear his name: the Moore Method. It is very likely that, in the end, it is this innovative teaching method that will be Moore’s most lasting contribution to mathematics. Even now, more than a hundred years later, people enthusiastically follow the example set by Moore. An extremely ambitious and attractive new textbook, Number Theory Through Inquiry, by David C. Marshall, Edward Odell, and Michael Starbird, has been written specifically to use the teaching 82

technique that has now become known as the Modified Moore Method. Moore developed his method to produce research mathematicians. He hand picked students for his graduate course in topology, and as far as he was concerned, the less they knew, the better. He once told a young woman who had written to him asking for advice about preparing for his course, ‘‘whatever else you read about this summer, do not read any point-set theory if you can help it.’’ He would begin on the first day of his course by giving several definitions and stating a few theorems. The students were then left on their own to prove the theorems and were not allowed to collaborate or do any reading whatsoever. Gradually, students would work out proofs of the theorems for themselves and present them to the rest of the class. The key as Moore saw it was not to feed information to students by lecturing or giving them a text to read, but to have them gain for themselves the power of being able to do mathematics by making mistakes, getting things wrong, and yet eventually discovering their own arguments for settling questions correctly. His guiding principle was: ‘‘That student is taught the best who is told the least.’’ The book Number Theory Through Inquiry has been designed to be used in the spirit of Moore, but not at all in the rigid way that Moore treated his own graduate students. These days, the Modified Moore Method is a far gentler and much more patient style of teaching that allows for a wide range of students—both in terms of abilities and levels, including undergraduates—to experience through a process of guided discovery the genuine benefits of learning to think independently, to depend on their own resources rather than those of an authority, and to discover that they have within themselves the power to create truly important ideas. The authors believe, and I certainly agree, that these benefits extend well beyond the mathematics classroom. As Paul Halmos put it, the Moore approach of trying to instill in the student an ‘‘attitude of questioning everything and wanting to learn answers actively’’ is ‘‘a good thing in every human endeavor, not only in mathematical research.’’


This is why Number Theory Through Inquiry is an ambitious book, because it is not merely trying to teach number theory, it is trying to change student attitudes. The goal is to liberate students for a lifetime of learning, discovery, and exploration. It could be said, however, that there is nothing particularly new about all this as a goal in teaching. The Socratic method, after all, has been around for a very long time. All great teachers—Mr. Chips is one notable example—are far less interested in the specific subject matter at hand than in the overall growth of their students (one of R. L. Moore’s students in later years even referred to Moore as ‘‘Mr. Chips with attitude’’). Jaime Escalante, memorably portrayed by Edward James Olmos in the 1987 film Stand and Deliver, borrowed from Nike their inspirational trademark ‘‘Just do it’’ to invoke for students the spirit of active versus passive learning that was at the core of his own remarkable teaching style. Number Theory Through Inquiry is an extremely thin book. A typical page contains a definition or two, and then several questions, exercises, and theorems connected by the barest minimum of prose. An instructor choosing this book for a course needs to be fully committed to the intended method of instruction. An instructor also needs to truly believe the course is about empowering students and not about covering material. Students could not possibly prove anywhere near all the theorems in the book in a typical semester course. An instructor would have many decisions to make about how to teach such a course. Do you want the students to collaborate? How much do you guide the students? One of the most difficult things when using this method is learning to resist the natural impulse to jump in and correct students when they make mistakes. Allowing students the luxury of making mistakes, finding their mistakes, or having other students find the mistakes is at the very heart of the method. What do you do if no one in the class can seem to get started on a proof? This list of pedagogical questions could go on and on. An instructor choosing this book for a course not only needs to be an excellent teacher, and probably

already quite experienced, but also needs to know number theory cold. Even a pro like Paul Halmos talked about needing a couple of months preparing to teach a course using this method and said, ‘‘As the course goes along, I must keep preparing for each meeting: to stay on top of what goes on in class. I myself must be able to prove everything. In class, I must stay on my toes every second.’’ Here are just a few examples of how the authors approach specific material. To have students prove there are infinitely many primes, they first ask them to think of a natural number that is divisible by 2, 3, 4, and 5. Next, they ask them to think of a natural number that has a remainder 1 when divided by 2, 3, 4, and 5. Then they get them to generalize this idea by proving that for any natural number k, there is another (much larger) natural number not divisible by any natural number less than k except 1. Of course, by now the students have been handed the answer (k - 1)! + 1 on a silver platter. The next theorem they have the students prove is that for any natural number k, there is a prime larger than k, which in turn leads immediately to the infinitude of primes theorem. So, it is somewhat debatable as to whether at this point the students have actually discovered a proof of the infinitude of the primes. Perhaps they have just been led by the hand to what is effectively Euclid’s proof. Still, this is a good, active way to present this proof. The authors follow this up in an excellent way by having students prove what they call the infinitude of 4k + 3 primes theorem. Then they ask whether there are other theorems like this that can be proved. Here, I suspect many students will fall into the trap of believing they can prove an infinitude of 4k + 1 primes theorem in a similar way, but then won’t have time in their course to reach page 90, where they might learn that in order to prove this important theorem you really need to know that –1 is a quadratic residue modulo primes of the form 4k + 1. A difficulty with the technique of guided discovery—all too familiar to anyone who has ever used it—is that students often don’t go where we think we are leading them. This happened to me (playing the role of the student) in

a section of the book where Euler’s theorem is to be used to solve congruences, and the authors asked whether I could think of an appropriate operation to apply in each case to both sides of the congruences x5 : 2 (mod 7) and x3 : 7 (mod 10). Not realizing where they were hoping to lead me, I multiplied both sides of the first congruence by x, and then could easily solve 1 : 2x (mod 7) by inspection to get x = 4; for the second congruence, since / (10) = 4, I again multiplied both sides of the congruence by x and could solve 1 : 7x (mod 10) by inspection to get x = 3. But, as I discovered in the next paragraph, they had really intended for me in each case to raise both sides of the congruence to some appropriate exponent (5 works for the first congruence, and 3 works for the second), because this is the approach that generalizes to the theorem they were trying to lead us toward. Occasionally, the notion of discovery or inquiry seemed to go completely out of the window: For example, when students are simply told that half the numbers less than an odd prime p are quadratic residues and half are not. Why not let them discover this on their own? Or when the authors simply list for the students the primes among the first 30 primes for which 2 is a quadratic residue, and those for which it is not. There also are places in the text where the authors may not be giving students enough guidance. I’d be surprised if students could prove that Euler’s /-function is multiplicative, having been told only to circle numbers relatively prime to numbers such as, say, 36 written down in a 4 by 9 array; or if they could prove the case n = 4 of Fermat’s last theorem even having been told to prove the stronger statement that there are no nontrivial integer solutions to x4 + y4 = z2. I would like to try using Number Theory Through Inquiry for a course, though perhaps not exactly in the way intended by the authors. Rather than have the students use it as a text, I think I’d prefer to take a somewhat less guided approach in order to remain truer to the Modified Moore Method by providing students with selected definitions and theorems taken from the book. I might, however, have them start

using the text at some point midway in the course once they had fully developed their own independence. This is a hybrid technique suggested by one of Moore’s students, F. Burton Jones, who allowed his own topology students to use Kelley for bedtime reading beginning about Christmastime. Anyone who is thinking of adopting Number Theory Through Inquiry as a text should be aware that there is an instructor’s manual available from the publisher where the authors discuss the philosophy of the book and provide tips for the first five chapters—the likely content for a first course—based on their own experience with students. There is much to be gained teaching a course using a well conceived and well executed text such as Number Theory Through Inquiry—one might even use this book as a template for designing one’s own course in virtually any subject. However, something valuable could be lost, too. Number theory developed not in the inevitable way that a subject such as calculus did, but in an undeniably quirky human way, and this treatment of number theory strips away so much of its rich history that it is left a bit too bare and lifeless for my taste. REFERENCES

[1] D. W. Cohen, A Modified Moore Method for Teaching Undergraduate Mathematics, The American Mathematical Monthly, 89 (1982) 473–474; 487–490. [2] K. Devlin, The Greatest Math Teacher Ever, Devlin’s Angle, MAA Online (May 1999) http://www.maa.org/devlin/devlin_5_99. html. [3] K. Devlin, The Greatest Math Teacher Ever, Part 2, Devlin’s Angle, MAA Online (June 1999) http://www.maa.org/devlin/devlin_ 6_99.html. [4] F. B. Jones, The Moore Method, The American

Mathematical

Monthly,

84

(1977) 273–278 . [5] J. Parker, R. L. Moore: Mathematician & Teacher, Washington, DC, The Mathematical Association of America, 2005.

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


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King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts WALKER & COMPANY, 2006, 386 PP, US$27.95 ISBN 13 978-0-8027-1499-2 REVIEWED BY ULF PERSSON

M

odern mathematics is a forbidding subject: Highly technical, involving a formidable conceptual apparatus, necessitating years of study before it even starts making sense in the way many other sciences immediately make sense to the public. Is there a royal way to mathematics, a way of getting to the heart of the subject without extended preliminaries? There famously is no royal way to geometry, but maybe geometry itself is the royal way to mathematics. If so, who would be more fitting to be the king than the subject of the book under review— H. S. M. Coxeter? A man who showed that, even with elementary tools, it is possible to penetrate deeply into mathematics, giving heart to the hope that the subject can be enjoyed directly without the alienation that comes with high technology: In short, that it is still possible to retain this sense of innocent wonder which initially seduced most of us into the subject. Harold Scott MacDonald Coxeter (known as Donald) was born in 1907, the only child of a mismatched couple. His father, Harold Coxeter, an amateur sculptor and chain-smoking baritone singer, made a living in the family business of purveying surgical instruments; his mother was a painter of some renown, specializing in portraits and landscapes. When in spite of shared cultural interests they later divorced, his father remarried a woman only six years older than his teenage son, thereby exacerbating an emotional trauma Donald would never fully overcome.

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As a bright solitary child, Donald found pleasures in lonely pursuits such as inventing imaginary languages, composing music, and engaging in mathematical investigations. His parents were both proud of and concerned for their gifted but delicate child. They sought professional guidance how to best care for his talents. It was decided early on that music was really not his forte. His prospects in mathematics were far more promising. At the age of 16, on the advice of Bertrand Russell, they contacted the mathematician E. H. Neville, who recommended that Donald drop all subjects except mathematics and German. Neville delegated him to Alan Robson, a senior mathematician at Marlborough College. Donald sat for the Cambridge University entrance exam in 1925 and qualified for King’s College, but Robson urged him to try for Trinity. Another year of study did the trick. He submitted his first mathematical paper when he was about to enter University. He had done some work on spherical tetrahedra, obtaining definite integrals he challenged readers to evaluate directly. The paper appeared in the Mathematical Gazette and intrigued G. H. Hardy, who could never resist the temptation of a definite integral. At Trinity, Coxeter came under the tutelage of Littlewood, devoted himself single-mindedly to his studies, and tried to resolve his recurrent problems of digestion by turning himself into a lifelong vegetarian, which caused him to lose weight and render him his characteristic taut and timeless appearance, so fitting for a geometer. Predictably he did very well on the Tripos, earning himself the rank of Wrangler. On a visit to Austria in the summer of 1928, Coxeter discovered the work of Schla¨fli in the Vienna University library. Schla¨fli, a Swiss schoolteacher ignored during his lifetime, had anticipated many of Coxeter’s later discoveries (in fact, the standard notation introduced by Coxeter for regular polytopes is an adaptation of Schla¨fli’s), most notably the classification of regular solids in four dimensions. Coxeter would champion him from then on. Coxeter’s association with the luminaries at Cambridge was somewhat marginal. He started his doctoral studies under the aged geometer H. F. Baker, a


British devotee of the Italian school of algebraic geometry. Baker had, among other things, instigated a tradition of Saturday tea parties for his students and associates, mixing mathematics with biscuits. Once, Coxeter invited the aged lady Alicia Boole Stott (a daughter of the well-known Boole and a niece of the surveyor Everest), a mathematical autodidact almost 50 years his senior, who had rediscovered Schla¨fli’s results by spatial intuition. Coxeter’s thesis on polytopes earned him a prize and, in 1932, a Rockefeller Foundation fellowship to Princeton. There, Solomon Lefschetz nicknamed him Mr. Polytope and remarked, after one of Donald’s seminars, that it was sometimes good to hear about trivial things. Oswald Veblen was a bit more supportive, if distantly so. Yet the mathematical environment at Princeton exposed him to a great variety of intellectual stimulation, such as that provided by John von Neumann. During this time he developed his well-known notation for reflection groups. He returned for a second stint at the Institute of Advanced Study in 1934–1935. The second visit turned out to be even more fruitful, as his study of discrete reflection groups tied in with Hermann Weyl’s investigation of continuous group representations and root lattices, and he was invited to contribute an appendix to Weyl’s seminar notes, which were widely distributed. On Hardy’s recommendation, he was invited to edit Ball’s Mathematical Recreations and Essays (an ambivalent appreciation coming from Hardy, who in A Mathematician’s Apology was rather dismissive of Ball). The task delighted Coxeter, and he certainly was the perfect man for it, removing outdated material and replacing it with chapters on polytopes and other geometrical gems. The summer of 1936 turned out to be crucial to Coxeter’s personal life. He met his first girlfriend, a Dutch au pair, who agreed to be his wife after a rather short courtship. Just before the scheduled marriage later that summer, his father unexpectedly suffered a heart attack and drowned as he was teaching his younger daughters to swim. The wedding went through anyway, but without celebration. The young couple took off

for Toronto, where Coxeter would remain until his death almost 70 years later. As one would expect, Coxeter preferred to stay aloof of administrative duties, which may be why he became a full-fledged tenured professor only in 1948, 12 years after his initial appointment. The same year saw the publication of Regular Polytopes, which in some sense he had been working on for the preceding 24 years. The book made his reputation. During the Second World War, Coxeter espoused pacifism, which was not entirely comme-il-faut in provincial Toronto. During the McCarthy era, he took a public stand for civil liberties and was instrumental in finding a sanctuary for Chandler Davis, a victim of the witch hunt. He championed nuclear disarmament and was a vocal foe of American involvement in Vietnam. In his later years, he signed a petition to protest the bestowal of an honorary degree on former President George Bush, Sr. Coxeter claimed that there could be nothing worse than the first President Bush—until the second one came along. Over the years, Coxeter won his share of prizes and distinctions. In 1950, he was elected a Fellow of the Royal Society and later became an honorary fellow of both the Edinburgh and London mathematical societies. In 1997, he was appointed a ‘‘Companion of the Order of Canada,’’ the highest of three levels of honor that Canada bestows. The biography contains an apparently complete list of Coxeter’s 250-odd publications (including subsequent editions and translations of his books) spanning almost 80 years, starting with his entry in 1926 in the Mathematical Gazette to his posthumous 2005 inclusion in the memorial volume dedicated to Bolyai. His articles appear both in technical journals and popular magazines such as

the American Mathematical Monthly and the Mathematics Teacher, in addition to the Gazette; but regardless of where they are, all are accessible to the general mathematician with geometrical combinatorial leanings. The various titles reveal the breadth of his interest, ranging from polytopes and group theory to physics, viral macromolecules, the art of M. C. Escher, and music. He even contributed to a philosophical anthology with the article ‘‘Cases of Hyperdimensional Awareness.’’ As to his books, in addition to the one on polytopes, his Introduction to Geometry was widely acclaimed, and his Generators and Relations for Discrete Groups (coauthored with W. O. J. Moser) presents his main contribution to professional mathematics. Coxeter was blessed with a long life and, more importantly, with a mind that remained lucid to the very end. His last public lecture was given in Budapest in July 2002. This is the event with which the author introduces her account of his life. Just days before he died, nine months later, he was busy readying the lecture for publication. After his death, his brain joined a ‘‘brain bank’’ at McMaster University in Hamilton, Ontario, where a neuroscientist already had acquired the brain of Einstein. In presenting the life of a mathematician, it is the work that matters. This poses a serious challenge to any biographer writing for the lay reader. Coxeter was for most of his career definitely out of fashion. The mathematics he was doing was seen by most mathematicians of the time as a quaint vestige of Victorian mathematics. The moral lesson that Coxeter provides is to disregard fashion: Eventually you will be vindicated (not necessarily within your lifetime). But one should not, as Roberts is somewhat prone to do, pit Coxeter as a valiant David against the Goliath of modern mathematics,

let alone resort to hyperbole comparing Regular Polytopes to Darwin’s The Origin of Species. To dramatize the conflict between Coxeter and the mathematical establishment, the author sets up as the ‘‘villain’’ Jean Dieudonne´, whose rallying cry ‘‘Down with Euclid, death to triangles’’ heads one of the chapters in her book. The living geometry of Coxeter is contrasted with the strict and formal mathematics epitomized by the Bourbakists in their relentlessly logical expositions with no resource to visual imagery and intuitive reasoning. Like all cliche´s, this contains a significant element of truth, but the author’s presentation is a misleading oversimplification. And, as she admits, the supposed conflict between so-called Bourbakism and Coxeter had an ironic and happy ending, as Coxeter’s main insights of combining geometry with the symmetries combinatorially articulated through group theory became the subject of the concluding volumes of Bourbaki—some would say their most successful ones. Unlike Buckminster Fuller and M. C. Escher, the association to whom is treated at length in the biography, Coxeter was not an alternative mathematician: He was a professional whose insights were not entirely the result of some kind of transcendental meditation, but which also rested on nonmagical algebraic manipulations, without which mathematical contemplation would not rise above the level of insipid speculation. Today, Coxeter’s mathematics has come into its own, advancing from the pages of recreational mathematics to being an inescapable component of cutting-edge mathematics. Department of Mathematics Chalmers University of Technology Go¨teborg, Sweden e-mail: [email protected]


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Stamp Corner

Robin Wilson

The Philamath’s Alphabet: TUV Terrestrial globe During the 16th century, with the new interest in exploration and navigation, terrestrial globes became increasingly in demand. This terrestrial globe of 1568, made of brass, was constructed by Johannes Praetorius of Nuremberg. The map depicts the continents of Europe, Africa, Asia, and America, with America shown joined to Asia.

Thales of Miletus The earliest known Greek mathematician is Thales of Miletus (c. 624-547 BC) who, according to legend, brought geometry to Greece from Egypt. He

investigated the congruence of triangles, applying it to navigation at sea, and predicted a solar eclipse. He is also credited with proving that the base angles of an isosceles triangle are equal and that a circle is bisected by any diameter.

Tic-tac-toe The position game of tic-tac-toe, or noughts and crosses, developed in the 19th century from earlier ‘‘three-in-arow’’ games such as Three Men’s Morris. Two players take turns to place their symbol (o or 9) in a square and try to get three in a row. Variations involve larger boards (4 9 4 or 5 9 5) and three or more dimensions (4 9 4 9 4 or 3 9 3 9 3 9 3).

hat changed the face of the earth,’’ in a set of Nicaraguan stamps issued in 1971.

Ulugh Beg By the 15th century, Samarkand in central Asia had become one of the greatest centres of civilisation, especially in mathematics and astronomy. The observatory of the Turkish astronomer Ulugh Beg (1394–1449) contained a special sextant, the largest of its type in the world. Ulugh Beg constructed extensive tables for the sine and tangent of every angle for each minute of arc, to five sexagesimal places.

Vega Tsiolkovsky Konstantin Tsiolkovsky (1857–1935), a pioneer of rocket flight, invented multistage rockets and produced a celebrated mathematical law that relates the velocity and mass of a rocket in flight. Tsiolkovsky’s law was one of the ‘‘ten mathematical formulas

Logarithms were invented in the early 17th century. In the 1790s, the Slovenian Jurij Vega (1754–1802) published a celebrated compendium of logarithms, as well as seven-figure and 10figure logarithm tables that ran to several hundred editions. He also calculated p to 140 decimal places.

t-

Thales of Miletus Tic-tac-toe Tsiolkovsky

Terrestrial globe 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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