The Mathematical Intelligencer Vol 31 No 1 Januray 2009

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 BestMathematical Books of theTwentieth Century

IIn The Mathematical Intelligencer,vol. 29 (2007), no. 1, I askedfor suggestions for the best mathe-

matical books of the twentiethcentury. I am very grateful to theseven readers who responded. Therewas widespread disagreement aboutwhich of my two categories—essen-tially ‘‘academic’’ mathematics andwhat I have now learned to call‘‘paramathematics’’—the books fellinto, so I have abandoned thedistinction.

The table below shows the booksthat received more than one vote.Despite the small sample size, we havean interestingly varied list and aclear winner. If any readers (whetheror not they are members of the van derWaerden fan club) feel sufficientlyinspired or irritated by the results toe-mail me some more suggestions, Iwill happily incorporate their ideasand 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, ‘‘Elements de Mathematique’’

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

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

2 A. Grothendieck, ‘‘Elements de Geometrie Algebrique’’

2 D. R. Hofstadter, ‘‘Godel, 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 Morphogenese’’

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

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

Mathematically Bent Colin Adams, Editor

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 disorienta-

tion is at its most acute when you

open to Colin Adams’s column.

Relax. Breathe regularly. It’s

mathematical, it’s a humor column,

and it may even be harmless.

Column editor’s address: Colin Adams,

Department of Mathematics,

Bronfman Science Center, Williams College,

Williamstown, MA 01267, USA


Riot at theCalc ExamCOLIN ADAMS

TThere had been a lot of unrest inthe classroom all semester. To acertain extent, I was to blame. I

decided right at the beginning of thecourse not to waste any time. So thefirst day, I introduced the triple inte-gral. It was quite a shock for studentswho had yet to see a single integral.But that’s what I wanted. Shock ther-apy. Shock calculus. Embed an idea sodeeply in their brains it would neverget out. Brand their brains with a hotbranding iron that said $$$. That firstday, the smell of burning brain matterwas overwhelming.

Slack-jawed students with bulgingeyes gaped in disbelief and horror.Several flipped frantically through theirclass schedules to check if they were inthe right room.

I hit them with a Fourier series toget their attention. By the end of thatfirst day, they looked like the morningafter an all night dorm party of the‘‘Oktoberfest Meets Mardi Gras’’ vari-ety; the bowed heads, the bloodshoteyes, the looks of nausea andanguish.

‘‘It’s no use sobbing,’’ I told onewoman as she staggered out of theclassroom. ‘‘Either you can do it or youcan’t. Toughen up or get stomped allover.’’

Then she really started bawling.But it’s the truth. Knowledge isn’t

for the faint of heart. There is somenasty knowledge out there. You haveto be able to take it. The citadel oflearning isn’t for the lily-livered.

By the next class at least half of thelivers were gone. It was a bigimprovement; we were down to amuch more workable group. Now Icould practice the one-on-one intimi-dation at which I excelled.

We covered about 75 years ofmathematics that day, from about 1837to 1912. But I still wanted three morestudents to drop. Any more than thatand I would fall below the 18 studentminimum for the course to run.

So I gave them a pop quiz oncomplex analytic functors over qua-ternionic tangent bundles. At the endof the 15 minutes, 13 students rushedup with drop slips for me to sign. Isigned three.

The first few weeks of the semester,I didn’t assign any homework. Whyshould I collect assignments that I hadno intention of grading? It would justclutter up my office. But the studentsstarted to get nervous. Homeworkbegan to appear in my mailbox,problems from sections in the text thatwe were supposed to be covering.

It infuriated me, this unsolicitedwork. I would return it the next classwithout having looked at it, only tofind it in my mailbox again severalhours later.

So, finally, I was forced to assignand collect homework. I would slashover it with red magic markers andcrayons at random. Then I would putbig red Fs at the top of each and writethings like, ‘‘Have you considered a jobat McDonald’s?’’

And so, we settled into the routineof the semester. Every few weeks, Iwould call the registrar to insist that theclassroom was inadequate and had tobe changed. It was a quick way toensure that the number of studentspresent was kept to a reasonable size. Iknew that those few who really wan-ted to learn would find out where theclass had gone. Optimal teachingconditions occur when the number ofstudents per faculty is kept to theminimum possible. Logically, then, mygoal was to lose them all.

But students in pursuit of passinggrades are not easily shaken. Everytime the class moved, at least a fewwould show up. Usually, though, thesearch for the new classroom madethem late, and if there is one thing Iwill not abide, it is a lack of respect for

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

time. At exactly 10:00, I would lock thedoor to the classroom. No matter howdesperately the late students bangedon the door, whimpering and plead-ing, I wouldn’t let them in. You have tolearn discipline before you can learnmathematics.

There was one student namedWattle who was consistently late. Hedidn’t even bother coming to the door.Each class meeting, at about 10:05, hishead would appear at the window ofwhatever classroom we were in. Hewould stand outside and take notes forthe entire hour. I considered havingthe class moved to the second floor,but after a while I became almost fondof that head bobbing outside thewindow.

As the semester progressed, stu-dents began to hound me. I could nolonger go to my office during my officehours for fear of running into them. Itgot to where I would find them millingaround my door at all hours of the dayand night, wearing those sad hangdoglooks, asking me where the class hadmoved to and what were the assign-ments. The students, they just want tosuck you dry.

About the eighth week, I lost con-trol of the class. The balance of powerin a classroom is always tenuous, andkeeping the respect and awe of thestudents requires a delicate hand. Irealized I had lost that control whenthey began to catch the blackboarderasers I hurled at them and fling themback. The student–teacher relationshipwas breaking down.

Things deteriorated quickly. When Istrolled across campus, I was hit by toomany Frisbees.� Of course, anyonestrolling across a college campus has toexpect to be hit by one or two Fris-bees. It’s part of being a pedestrian in ayouth-encumbered environment. But Iwas averaging nearer to 10 or 12

Frisbees, a couple of footballs, and abaseball or two. Those baseballs canreally sting.

And then, one day, I was late toclass. I had been debating the correctdefinition of the word ‘‘tangential’’with a colleague and, in my excite-ment, I lost track of the time. I arrivedat the classroom at 10:05 and the doorwas locked. I could hear snickeringinside. That was when I decided I hadhad enough.

Wattle never saw me coming. Igrabbed him in a headlock and saidthrough the window, ‘‘Unlock it orWattle arrives in hell earlier thanexpected.’’ There was some debate, asWattle was not particularly popular oncampus. But as he began to gurgle andwave his arms dramatically, theyopened the door.

From then on, the students left mealone. They kept their distance. Thefew who continued to come to classusually sat in the back and seemedinordinately skittish.

On the day of the final, I passed outthe exam. I was pretty pleased withmyself. Instead of having to make uptwo exams, one for the graduatecourse I was teaching and one for thecalculus course, I used the graduateexam for both. Perhaps that was unfair,but there are a lot of constraints on afaculty member’s time. One has toweigh all one’s responsibilities. In thiscase, making up the calc exam wasoutweighed by the Barbara Mandrellspecial on TV.

Unfortunately, Wattle noticed thatthe exam was for the graduate course.I had forgotten to change the coursenumber in the upper right hand cornerof the front page. He stood up.

‘‘That’s it. We don’t have to take thisanymore.’’ He picked up the examand crumpled it in his fist. A cheerwent up.

‘‘Sit down, Wattle. Sit down or youflunk,’’ I yelled. He took the balled-upexam and hurled it at me. Suddenly, allthe students were up, screaming andyelling. Crumpled exams flew every-where. A chair sailed across the room. Iran for the door.

When the police arrived, two stu-dents were wrapping the cord from theoverhead projector around my neck.Quite frankly, I am lucky to be alive. Ihave only the students’ penchant foralcohol and their resultant slowedmotor skills to thank for my survival.

At the trial, I explained how thestudents, disgruntled with the down-ward spiral in their grades, haddecided to take action. But rather thanhitting the books and working dili-gently to improve their minds, theychose instead to murder the professor,and, thereby, prevent the distributionof grades. It was an unsuccessfulattempt at cold-blooded premeditatedfirst-degree murder. I was particularlyeloquent. The judge sentenced themall to 10 to 12 years in one of America’soldest educational facilities, the Fed-eral Penitentiary in Leavenworth,Kansas.

It was difficult for me to decide onthe grades that they deserved. I con-sidered petitioning the college tointroduce a new grade, Z, whichwould essentially mean, ‘‘This studentdid so abysmally in the course thatthey deserve to die, but we did thenext best thing. We had them lockedaway.’’ In the end I gave them all Fsexcept for Wattle. In a moment ofweakness, I gave him a D-. I canimagine exactly how his head mustlook, bobbing behind the barred win-dow at Leavenworth.

My next semester starts in a week. Itis clear that I will have to run myclasses more strictly this time. No moreMr. Nice Guy.


Mathematics Is Nota Game But...ROBERT THOMAS

AAs a mathematician I began to take an interest in phi-losophy of mathematics on account of my resent-ment 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]. Variouselements of this notion are false and some are alsooffensive.

Mathematical effort, especially in recent decades—andthe funding of it—indicate as clearly and concretely as ispossible that mathematics is a serious scientific-type activitypursued by tens of thousands of persons at a professionallevel. While a few games may be pursued seriously bymany and lucratively by a professional few, no one claimsspectator sports are like mathematics. At the other end ofthe notion, paper is inessential, merely helpful to thememory. Communication (which is what the paper mighthint at) is essential; our grip on the objectivity of mathe-matics depends on our being able to communicate ourideas effectively.

Turning to the more offensive aspects of the notion, wethink often of competition when we think of games, and inmathematics one has no opponent. Such competitors asthere are are not opponents. Worst of all is the meaning-lessness attributed to the paradigm of clear meaning; whatcould be clearer than 2 + 2 = 4? Is this game idea notirredeemably outrageous?

Yes, it is outrageous, but there is within it a kernel ofuseful insight that is often obscured by outrage at the mainnotion, which is not often advocated presumably for thatreason. I know of no one that claims that mathematics is agame or bunch of games. The main advocate of the ideathat doing mathematics is like playing a game is DavidWells [3]. It is the purpose of this essay to point to theobscured kernel of insight.

Mathematics Is Not a GameMathematics is not a collection of games, but perhaps it issomehow like games, as written mathematics is somehowlike narrative. I became persuaded of the merit of somecomparison with games in two stages, during one of whichI noticed a further fault with the notion itself: there are nomeaningless games. Meaningless activities such as tics andobsessions are not games, and no one mistakes them forgames. Meanings in games are internal, not having to dowith reference to things outside the game (as electrons, forexample, in physics are supposed to refer to electrons inthe world). The kings and queens of chess would notbecome outdated if all nations were republics.

The ‘meaningless’ aspect of the mathematics-as-gamenotion is self-contradictory; it might be interesting to knowhow it got into it and why it stayed so long.

Taking it as given then that games are meaningful totheir players and often to spectators, how are mathematicalactivities like game-playing activities? The first stage ofwinning me over to a toleration of this comparison came inmy study of the comparison with narrative [4].

One makes sense of narrative, whether fictional or fac-tual, by a mental construction that is sometimes called theworld of the story. Keeping in mind that the world of thestory may be the real world at some other time or right nowin some other place, one sees that this imaginative effort isa standard way of understanding things that people say; itneed have nothing at all to do with an intentionally creativeimagining like writing fiction. In order to understand con-nected speech about concrete things, one imagines them.This is as obvious as it is unclear how we do it. We oftensay that we pretend that we are in the world of the story.This pretence is one way—and a very effective way—ofindicating how we imagine what one of the persons we arehearing about can see or hear under the circumstances ofthe story. If I want to have some idea what a person in


certain circumstances can see, for example, I imaginemyself in those circumstances and ask myself what I cansee [5]. Pretending to be in those circumstances does notconflict with my certain knowledge that on the contrary Iam listening to the news on my radio at home. This maymake it a weak sort of pretence, but it is no less useful forthat. The capacity to do this is of some importance. Itencourages empathy, but it also allows one to do mathe-matics. One can pretend what one likes and consider theconsequences 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, inchanging one’s response to ‘what is 2 + 2?’ from ‘2 + 2what?’ to the less concrete ‘four’ [6].

This ludic aspect of mathematics is emphasized by BrianRotman in his semiotic analysis of mathematics [7] andacknowledged by David Wells in his comparison ofmathematics and games. Admitting this was the first stageof my coming to terms with games. The ludic aspect issomething that undergraduates, many of whom havedecided that mathematics is either a guessing game (a badcomparison of mathematics and games) or the execution ofrigidly defined procedures, need to be encouraged to dowhen they are learning new ideas. They need to foolaround with them to become familiar with them. Changingthe parameters and seeing what a function looks like withthat variety of parameter values is a good way to learn howthe function behaves. And it is by no means only studentsthat need to fool around with ideas in order to becomefamiliar with them. Mathematical research involves a gooddeal of fooling around, which is part of why it is a plea-surable activity. This sort of play is the kind of play thatKendall Walton illustrates with the example of boys inwoods not recently logged pretending that stumps arebears [8]. This is not competitive, just imaginative foolingaround.

I do not think that this real and fairly widely acknowl-edged—at least never denied—aspect of mathematics hasmuch to do with the canard with which I began. Thecanard is a reductionistic attack on mathematics, for it saysit is ‘nothing but’ something it is not: the standard

reductionist tactic. In my opinion, mathematics is anobjective science, but a slightly strange one on account ofits subtle subject matter; in some hands it is also an art [9].Having discussed this recently at some length [10], I do notpropose to say anything about what mathematics is here,but to continue with what mathematics is like; becausesuch comparisons, like that with narrative, are instructiveand sometimes philosophically interesting.

The serious comparison of mathematics with games isdue in my experience to David Wells, who has summed upwhat he has been saying on the matter for twenty years in astrange document, draft zero of a book or two calledMathematics and Abstract Games: An Intimate Connection[3]. Wells is no reductionist and does not think that math-ematics is any sort of a game, meaningless or otherwise. Heconfines himself to the comparison (‘like a collection ofabstract games’—p. 7, a section on differences—pp. 45-51),and I found this helpful in the second stage of my seekinginsight in the comparison. But I did not find Wells’s directcomparison as helpful as I hope to make my own, whichbuilds on his with the intent of making it more compre-hensible and attractive (cf. my opening sentence).

Doing Mathematics Is Not Like Playing a GameDepending on when one thinks the activities of our intel-lectual ancestors began to include what we acknowledge asmathematics, one may or may not include as mathematicsthe thoughts lost forever of those persons with the cunei-form tablets on which they solved equations. The tabletsthemselves indicate procedures for solving those particularequations. Just keeping track of quantities of all sorts ofthings obviously extended still farther back, to somethingwe would not recognize as mathematics but which gaverise to arithmetic. Keeping track of some of the many thingsthat one cannot count presumably gave rise to geometricalideas. It does seem undeniable that such procedural ele-ments are the historical if not the logical basis ofmathematics, and not only in the Near East but also in Indiaand China. I do not see how mathematics could arisewithout such pre-existing procedures and reflections onthem— probably written down, for it is so much easier toreflect on what is written down.

This consideration of procedures, and of course theirraw material and results, is of great importance to mycomparison of mathematics and games because my com-parison is not between playing games and doingmathematics. I am taking mathematics to be the sophisti-cated activity that is the subject matter of philosophy ofmathematics and research in mathematics. I do not meanactions such as adding up columns of figures. Mathematicsis not even those more complicated actions that we arehappy to transfer to computers. Mathematics is what wewant to keep for ourselves. When playing games, we stickto the rules (or we are changing the game being played),but when doing serious mathematics (not executing algo-rithms) we make up the rules—definitions, axioms, andsome of us even logics. As Wells points out in the section ofhis book on differences between games and mathematics,in arithmetic we find prime numbers, which are a wholenew ‘game’ in themselves (metaphorically speaking).

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

AU

TH

OR ROBERT THOMAS was an undergraduate

at the University of Toronto when Chan-

dler Davis arrived there and obliged him to

get acquainted with quantifiers among

other things. He later studied also atWaterloo 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


� 2008 Springer Science+Business Media, LLC., Voulme 31, Number 1, 2009 5

http://www.philmat.oxfordjournals.org

While mathematics requires reflection on pre-existingprocedures, reflection on procedures does not becomerecognizable as mathematics until the reflection hasbecome sufficiently communicable to be convincing.Conviction of something is a feeling, and so it can occurwithout communication and without verbalizing or sym-bolizing. But to convince someone else of something, weneed to communicate, and that does seem to be anessential feature of mathematics, whether anything iswritten down or not— a fortiori whether anything is sym-bolic. And of course convincing argument is proof.

The analogy with games that I accept is based on thepossibility of convincing argument about abstract games.Anyone knowing the rules of chess can be convinced that amove has certain consequences. Such argument does notfollow the rules of chess or any other rules, but it is basedon the rules of chess in a way different from the way it isbased on the rules of logic that it might obey. To discuss theanalogy of this with mathematics, I think it may be useful tocall upon two ways of talking about mathematics, those ofPhilip Kitcher and of Brian Rotman.

Ideal AgentsIn his book The Nature of Mathematical Knowledge [11],Kitcher introduced a theoretical device he called the idealagent. ‘We can conceive of the principles of [empirical]Arithmetic as implicit definitions of an ideal agent. An idealagent is a being whose physical operations of segregationdo satisfy the principles [that allow the deduction inphysical terms of the theorems of elementary arithmetic].’(p. 117). No ontological commitment is given to the idealagent; in this it is likened to an ideal gas. And for this reasonwe are able to ‘specify the capacities of the ideal agent byabstracting from the incidental limitations on our owncollective practice’ (ibid.).

The agent can do what we can do but can do it forcollections however large, as we cannot. Thus modality isintroduced without regard to human physical limitations.‘Our geometrical statements can finally be understood asdescribing the performances of an ideal agent on idealobjects in an ideal space.’ (p. 124). Kitcher also alludes to the‘double functioning of mathematical language—its use as avehicle for the performance of mathematical operations aswell as its reporting on those operations’ (p. 130). ‘To solve aproblem is to discover a truth about mathematical opera-tions, and to fiddle with the notation or to discern analogiesin it is, on my account, to engage in those mathematicaloperations which one is attempting to characterize.’ (p. 131).

Rotman is at pains to distinguish what he says from whatKitcher had written some years before the 1993 publicationof The Ghost in Turing’s Machine [7] because he developedhis theory independently and with different aims, butwe readers can regard his apparatus as a refinement ofKitcher’s, for Rotman’s cast of characters includes an Agentto do the bidding of the character called the Subject. TheSubject is Rotman’s idealization of the person that readsand writes mathematical text, and also the person thatcarries out some of the commands of the text. For example,it is the reader that obeys the command, ‘Consider triangle

ABC.’ But it is the Agent (p. 73) that carries out suchcommands as, ‘Drop a perpendicular from vertex A to theline BC,’ provided that the command is within the Agent’scapacities. We humans are well aware that we cannot drawstraight lines; that is the work of the agents, Kitcher’s andRotman’s. We reflect on the potential actions of theseagents and address our reflections to other thinking Sub-jects. Rotman’s discussion of this is rich with details like thetenselessness of the commands to the Agent, indeed thecomplete lack of all indexicality in such texts. The tense-lessness is an indication of how the Subject is anidealization as the Agent is, despite not being blessed withthe supernatural powers of the Agent. The Agent, Rotmansays, is like the person in a dream, the Subject like theperson dreaming the dream, whereas in our normal statewe real folk are more like the dreamer awake, what Rot-man 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 willresult when certain operations are performed on signs(p. 77). We need not follow him there to appreciate theserviceability of his semiotic distinctions.

The need for superhuman capacities was noted long agoin Frege’s ridicule [2] of the thought that mathematics isabout empty symbols:

([...] we would need an infinitely long blackboard, aninfinite 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 canbe theorems in a theory of chess’ (p. 168, § 93, myemphasis). According to Frege,

The distinction between the game itself and its theory,not drawn by Thomae, makes an essential contributiontowards our understanding of the matter. [...] in thetheory of chess it is not the chess pieces which areactually investigated; it is a question of the rules andtheir consequences. (pp. 168-169, § 93)

The Analogies Between Mathematics and GamesHaving at our disposal the superhuman agents of Kitcherand Rotman, we are in a position to see what is analogousbetween mathematics and games.

It is not playing the game that is analogous to mathe-matics, but our reflection in the role of subject on theplaying of the game, which is done by the agent. When acolumn of figures is added up, we do it, and sometimeswhen the product of two elements of a group is required,we calculate it; but mathematics in the sense I am usinghere is not such mechanical processes at all, but theinvestigation of their possibility, impossibility, and results.For that highly sophisticated reflective mathematical activ-ity, the agent does the work because the agent can drawstraight lines. Whether points are collinear depends onwhether they are on the agent’s straight lines, not onwhether they appear on the line in our sketch. We can putthem on or off the line at will; the agent’s results are con-strained by the rules of the system in which the agent is

6 THE MATHEMATICAL INTELLIGENCER

working. Typically we have to deduce whether the agent’sline is through a point or not. The agent, ‘playing the game’according to the rules, gets the line through the point ornot, but we have to figure it out. We can figure it out; theagent just does it. The analogy to games is two-fold.

1. The agent’s mathematical activity (not playing a game) isanalogous to the activity of playing a game like chesswhere it is clear what is possible and what is impossi-ble—the same for every player—often superhuman butbound by rules. (Games like tennis depend for what ispossible on physical skill, which has no pertinencehere.)

2. Our mathematical activity is analogous to (a) gameinvention and development, (b) the reflection on theplaying of a game like chess that distinguishes expertplay from novice play, or (c) consideration of matters ofplay for their intrinsic interest apart from playing anyparticular 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. Anal-ogous to the hypotheses of our theorems are chesspositions, about which it is possible to reason as depend-ably as in mathematics because the structure is sufficientlyprecisely set out that everyone who knows the rules cansee what statements about chess positions are legitimateand what are not. Chains of reasoning can be as long as welike without degenerating into the vagueness that plagueschains of reasoning about the real world. The ability tomake and depend on such chains of reasoning in chess andother games is the ability that we need to make such chainsin mathematics, as David Wells points out.

To obtain a useful analogy here, it is necessary to riseabove the agent in the mathematics and the mere physicalplayer in the game, but the useful analogy is dependent onthe positions in the game and the relations in the mathe-matics. The reflection in the game is about positions morethan the play, and the mathematics is about relations andtheir possibility more than drawing circles or taking com-pact closures. Certainly the physical pieces used in chessand the symbols on paper are some distance below what isimportantly going on.

I hope that the previous discussion makes clear whysome rules are necessary to the analogy despite the fact thatwe are not bound by those rules. The rules are essentialbecause we could not do what we do without them, but itis the agent that is bound by them. We are talking about, asit were, what a particular choice of them does and does notallow. But our own activity is not bound by rules; we cansay anything that conveys our meaning, anything that isconvincing to others.

Here is objectivity without objects. Chess reasoning isnot dependent upon chess boards and chess men; it isdependent on the relations of positions mandated by therules of the game of chess. Mathematics is not dependenton symbols (although they are as handy as chess sets) buton the relations of whatever we imagine the agent to workon, specified and reasoned about. Our conclusions areright or wrong as plainly as if we were ideal agents loose in

Plato’s heaven, but they are right or wrong dependent onwhat the axioms, conventions, or procedures we havechosen dictate.

Outside mathematics, we reason routinely about whatdoes not exist, most particularly about the future. As thenovelist Jim Crace was quoted on page R10 of the 2007 6 2Toronto Globe and Mail, ‘As a good Darwinist, I know thatwhat doesn’t confer an advantage dies out. One advantage[of narrative (Globe and Mail addition)] is that it enables usto play out the bad things that might happen to us and torehearse what we might do.’ In order to tell our own sto-ries, it is essential to project them hypothetically into thefuture based on observations and assumptions about thepresent. At its simplest and most certain, the skill involvedis what allows one to note that if one moves this pawnforward one square the opponent’s pawn can take it. It’sabout possibilities and of course impossibilities, all of themhypothetical. It is this fundamental skill that is used both inreflection on games and in mathematics to see what isnecessary in their respective worlds.

I must make clear that David Wells thinks that entities inmaths and abstract games have the same epistemologicalstatus but that doing mathematics is like (an expert’s) play-ing a game in several crucial respects, no more; he disagreeswith the usefulness of bringing in ideal agents, indeedopposes doing so, apparently not seeing the advantage ofsplitting the analogy into the two numbered aspects above.This section is my attempt to outline a different but accept-able game analogy—a game-analysis analogy.

ConclusionGames such as bridge and backgammon, which certainlyinvolve strategy, have a stochastic element that preventslong chains of reasoning from being as useful as they are inchess. Such chains are, after all, an important part of howcomputers play chess. The probabilistic mathematicsadvocated by Doron Zeilberger [12] is analogous to theanalysis of such a stochastic game, and will be shunned bythose uninterested in such analysis of something in whichthey see nothing stochastic. Classical (von Neumann) gametheory, on the other hand, actually is the analysis of situa-tions that are called games and do involve strategy. Thegame theory of that current Princeton genius, John Con-way, is likewise the actual analysis of game situations [13].Does the existence of such mathematical analysis count foror against the general analogy between mathematics andgame analysis?

On my version of the analogy, to identify mathematicswith games would be one of those part-for-whole mistakes(like ‘all geometry is projective geometry’ or ‘arithmetic isjust logic’ from the nineteenth century); but identification isnot the issue. It seems to me that my separation of gameanalysis from playing games tells in favour of the analogyof mathematics to analysis of games played by other—notnecessarily superhuman—agents, and against the analogyof mathematics to the expert play of the game itself. This isnot a question David Wells has discussed. For Wells, him-self an expert at abstract games such as chess and go, playis expert play based unavoidably on analysis; analysis is just


part of playing the game. Many are able to distinguish theseactivities, and not just hypothetically.

One occasionally hears the question, is mathematicsinvented or discovered?—or an answer. As David Wellspoints out, even his game analogy shows why bothanswers and the answer ‘both’ are appropriate. Once agame is invented, the consequences are discovered—gen-uinely discovered, as it would require a divine intelligenceto know just from the rules how a complex game couldbest be played. When in practice rules are changed, onemakes adjustments that will not alter the consequences toodrastically. Analogously, axioms are usually only adjustedand the altered consequences discovered.

What use can one make of this analogy? One use thatone cannot make of it is as a stick to beat philosophers intoadmitting that mathematics is not problematic. Like math-ematicians, philosophers thrive on problems. Problems arethe business of both mathematics and philosophy. Solvingproblems is the business of mathematics. If a philosophercame to regard the analogy as of some validity, then shewould import into the hitherto unexamined territory ofabstract games all of the philosophical problems concern-ing mathematics. Are chess positions real? How do weknow about them? And so on; a new branch of philosophywould be invented.

What use then can mathematicians make of the analogy?We can use it as comparatively unproblematic material indiscussing mathematics with those nonphilosophers desir-ing to understand mathematics better. I have tried toindicate above some of the ways in which the analogy isboth apt and of sufficient complexity to be interesting; it isno simple metaphor but can stand some exploration. Someof this exploration has been carried out by David Wells, towhose work I need to refer the reader.

REFERENCES

[1] Thomae, Johannes. Elementare Theorie der analytischen Func-

tionen einer complexen Veranderlichen. 2nd ed. Halle: L. Nebert,

1898 (1st ed. 1880), ridiculed by Frege [2].

[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 Uni-

versity 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 Mathe-

matical 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, Washing-

ton, 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-rig-

orous 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,

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, andVarchenko onRational GeneratingFunctions for ConesMATTHIAS BECK, CHRISTIAN HAASE, AND FRANK SOTTILE

OOur aim is to illustrate two gems of discrete geome-try, namely formulas of Michel Brion [7] and ofJames 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 aslight feeling of mystery in us. Let us start with someexamples.

Suppose we would like to list all positive integers.Although there are many, we may list them compactly inthe form of a generating function:

x1 þ x2 þ x3 þ � � � ¼X

k [ 0

xk ¼ x

1� x: ð1Þ

Let us list, in a similar way, all integers less than or equalto 5:

� � � þ x�1 þ x0 þ x1 þ x2 þ x3 þ x4 þ x5 ¼X

k� 5

xk

¼ x5

1� x�1:

ð2Þ

Adding the two rational function right-hand sides leads to amiraculous cancellation

x

1� xþ x5

1� x�1¼ x

1� xþ x6

x � 1¼ x � x6

1� x

¼ x þ x2 þ x3 þ x4 þ x5:

ð3Þ

This sum of rational functions representing two infiniteseries collapses into a polynomial representing a finiteseries. This is a one-dimensional instance of a theorem dueto Michel Brion. We can think of (1) as a function listing theinteger points in the ray [1,?) and of (2) as a functionlisting the integer points in the ray (-?,5]. The respectiverational generating functions add up to the polynomial (3)that lists the integer points in the interval [1, 5]. Here is apicture of this arithmetic.

+

=1 2 3 4 5


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

Analogous to the generating functions (1) and (2) are thegenerating functions of the cones at each vertex generatedby the edges at that vertex. For example, the two edgestouching the origin generate the nonnegative quadrant,which has the generating function

X

m;n� 0

xmyn ¼X

m� 0

xm �X

n� 0

yn ¼ 1

ð1� xÞ �1

ð1� yÞ :

The two edges incident to (0, 2) generate the cone ð0; 2Þ þR� 0ð0;�2Þ þ R� 0ð4; 0Þ; with the generating function

X

m� 0;n� 2

xmyn ¼ y2

ð1� xÞð1� y�1Þ :

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

x4y2

ð1� x�1Þð1� x�1y�1Þ :

Finally, the fourth vertex cone is ð2; 0Þ þ R� 0ð2; 2Þ þR� 0ð�2; 0Þ; with the generating function

x2

ð1� xyÞð1� x�1Þ :

Inspired by our one-dimensional example above, we addthose four rational functions:

1

ð1� xÞð1� yÞ þy2

ð1� xÞð1� y�1Þ

þ x4y2

ð1� x�1Þð1� x�1y�1Þ þx2

ð1� xyÞð1� x�1Þ¼ y2 þ xy2 þ x2y2 þ x3y2 þ x4y2

þ y þ xy þ x2y þ x3y

þ 1þ x þ x2:

The sum of rational functions again collapses to a poly-nomial, which encodes precisely those integer points thatare contained in the quadrilateral Q:

Brion’s Theorem says that this magic happens for anypolytope P in any dimension d; provided that P hasrational vertices. (More precisely, the edges of P haverational directions.) The vertex cone Kv at vertex v is thecone with apex v and generators the edge directionsemanating from v. The generating function

rKvðxÞ :¼

X

m2Kv\Zd

xm

for such a cone is a rational function (again, provided thatP has rational vertices). Here we abbreviate xm forxm1

1 xm22 � � � xmd

d : Brion’s Formula says that the rationalfunctions representing the integer points in each vertexcone sum up to the polynomial rPðxÞ encoding the integerpoints in P:

rPðxÞ ¼X

v a vertex of PrKvðxÞ:

A second theorem, which shows a similar collapse ofgenerating functions of cones, is due (independently) toJames Lawrence and to Alexander Varchenko. We illustrateit with the example of the quadrilateral Q: Choose adirection vector n that is not perpendicular to any edge ofQ; for example we could take n = (2, 1). Now at eachvertex v of Q; we form a (not necessarily closed) conegenerated by the edge directions m as follows. If w � n[ 0,

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

AU

TH

OR

S MATTHIAS BECK was an undergraduateat Wurzburg, 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 Uni-

versity, he held postdoctoral positions at

SUNY Binghamton, MSRI, and the Max

Planck Institute. His research is in discrete

and combinatorial geometry and numbertheory—and more particularly in enumer-

ating integer points in polyhedra.

Department of Mathematics

San Francisco State University

San Francisco, CA 94132, USA


URL: http://math.sfsu.edu/beck

CHRISTIAN HAASE was born and raised in

Berlin. After a respectable apprenticeship as

an algebraic topologist, he ventured into

polyhedral geometry, presaging his slide intolattice-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 & InformatikFreie Universitat Berlin, 14195 Berlin,

Germany


URL: http://erhart.math.fu-berlin.de


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

For example, the edge directions at the origin are alongthe positive axes, and so this cone is again the nonnegativequadrant. At the vertex (2, 0) the edge directions are(-2, 0) and (2, 2). The first has negative dot productwith n and the second has positive dot product, and so weobtain the half-open cone ð2; 0Þ þ R\0ð�2; 0Þ þ R� 0ð2; 2Þ¼ ð2; 0Þ þ R[ 0ð2; 0Þ þ R� 0ð2; 2Þ: At the vertex (4, 2) bothedge directions have negative dot product with n and weget the open cone ð4; 2Þ þ R[ 0ð0; 4Þ þ R[ 0ð2; 2Þ; and atthe vertex (0, 2) we get the half-open cone ð0; 2Þ þR� 0ð2; 0Þ þ R[ 0ð0; 2Þ: The respective generating functionsturn out to be

1

ð1� xÞð1� yÞ ;x3

ð1� xÞð1� xyÞ ;

x6y3

ð1� xyÞð1� yÞ ; andy3

ð1� xÞð1� yÞ :

Now we add them with signs according to the parity of thenumber of negative (w � n\ 0) edge directions w at thevertex. In our example, we obtain

1

ð1� xÞð1� yÞ �x3

ð1� xÞð1� xyÞ

þ x6y3

ð1� xyÞð1� yÞ �y3

ð1� xÞð1� yÞ¼ y2 þ xy2 þ x2y2 þ x3y2 þ x4y2

þ y þ xy þ x2y þ x3y

þ 1þ x þ x2:

This sum of rational functions again collapses to thepolynomial that encodes the integer points in Q: Thisshould be clear here, for the integer points in the non-negative quadrant are counted with a sign ±, dependingupon the cone in which they lie, and these coefficients cancelexcept for the integer points in the polytopeQ:

The identity illustrated by this example works for anysimple polyope—a d-polytope where every vertex meetsexactly d edges. Given a simple polytope, choose a direc-tion vector n 2 R

d that is not perpendicular to any edgedirection. Let Eþv ðnÞ be the edge directions w at a vertex v

with w � n[ 0, and E�v ðnÞ be those with w � n\ 0. Definethe cone

Kn;v :¼ v þX

w2Eþv ðnÞR� 0w þ

X

w2E�v ðnÞR\0w:

This is the analogue of the cones in our previous example.The Lawrence–Varchenko Formula says that adding therational functions of these cones with appropriate signsgives the polynomial rPðxÞ encoding the integer points inP:

rPðxÞ ¼X

v a vertex of Pð�1ÞjE

�v ðnÞj rKn;vðxÞ:

Here, rKn;vðxÞ is the generating function encoding theinteger points in the cone Kn;v: An interesting feature of thisidentity, which also distinguishes it from Brion’s Formula, isthat the power series generating functions have a commonregion of convergence. Also, it holds without any restric-tion that the polytope be rational. In the general case, thegenerating functions of the cones are holomorphic func-tions, which we can add, as they have a common domain(the common region of convergence).

ProofsBrion’s original proof of his formula [7] used theLefschetz–Riemann–Roch theorem in equivariant K-theory[3] applied to a singular toric variety. Fortunately for us, theremarkable formulas of Brion and of Lawrence–Varchenkonow have easy proofs, based on counting.

Let us first consider an example based on the cone K ¼R� 0ð0; 1Þ þ R� 0ð2; 1Þ: The open circles in the picture onthe left in Figure 1 represent the semigroup Nð0; 1Þ þNð2; 1Þ; which is a proper subsemigroup of the integerpoints K \ Z

2 in K: The picture on the right shows howtranslates of the fundamental half-open parallelepiped Pby this subsemigroup cover K: This gives the formula

rKðxÞ ¼ rPðxÞ �X

m;n� 0

xmðx2yÞn ¼ 1þ xy

ð1� xÞð1� x2yÞ ;

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

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, andelsewhere, 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 MathematicsTexas A&M University, College Station

TX 77843, USA


URL: http://www.math.tamu.edu/~sottile

Figure 1. Tiling a simple cone by translates of itsfundamental parallelepiped.


as the fundamental parallelepiped P contains two integerpoints, the origin and the point (1, 1).

A simple rational cone in Rd has the form

K :¼ vþXd

i¼1

kiwi j ki 2 R� 0

( )¼ vþ

Xd

i¼1

R� 0wi;

where w1; . . .;wd 2 Zd are linearly independent. This cone

is tiled by the ðNw1 þ � � � þ NwdÞ-translates of the half-open parallelepiped

P :¼ vþXd

i¼1

kiwi j 0� ki\1

( ):

The generating function for P is the polynomial

rPðxÞ ¼X

m2P\Zd

xm;

and so the generating function for K is

rKðxÞ ¼X

a2Nw1þ��þNwd

xa � rPðxÞ ¼rPðxÞ

ð1� xw1Þ � � � ð1� xwd Þ ;

which is a rational function. This formula and its proof donot require that the apex v be rational, but only that thegenerators wi of the cone be linearly independent vectorsin Z

d :A general rational cone K with apex v and generators

w1; . . .;wn 2 Zd has the form

K ¼ vþ R� 0w1 þ � � � þ R� 0wn:

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 decomposedinto simple cones K1; . . .;Kl having pairwise disjoint inte-riors, each with apex v and generated by d of thegenerators w1; . . .;wn of K: We would like to add thegenerating functions for each cone Ki to obtain the gen-erating function for K: However, some of the cones mayhave lattice points in common, and some device is neededto treat the subsequent overcounting.

An elegant way to do this is to avoid the overcountingaltogether by translating all the cones [5]. We explain this.There exists a short vector s 2 R

d such that

K \ Zd ¼ ðsþKÞ \ Zd ; ð4Þ

and no facet of any cone sþK1; . . .; sþKl contains anyinteger points. This gives the disjoint irrationaldecomposition

K \ Zd ¼ ðsþK1Þ \ Zd t � � � t ðsþKlÞ \ Z

d ;

and so

rKðxÞ ¼X

m2K\Zd

xm ¼Xl

i¼1

rsþKiðxÞ ð5Þ

is a rational function.For example, suppose that K is the cone in R

3 with apexthe origin and generators

w1 ¼ ð1; 0; 1Þ; w2 ¼ ð0; 1; 1Þ; w3 ¼ ð0;�1; 1Þ;and w4 ¼ ð�1; 0; 1Þ:

If we let K1 be the simple cone with generatorsw1;w2;w3; and K2 be the simple cone with generatorsw2;w3;w4; thenK1 andK2 decomposeK into simple cones.If s ¼ ð18 ; 0;� 1

3Þ; then (4) holds, and no facet of sþK1 or ofsþK2 contains any integer points. We display these cones,together with their integer points having z-coordinate 0,1, or 2.

The cone sþK1 contains the 5 magenta points shownwith positive first coordinate, whereas sþK2 containsthe other displayed points. Their integer generating func-tions are

rsþK1ðxÞ ¼ xz þ xz2

ð1� yzÞð1� y�1zÞð1� xzÞ ;

rsþK2ðxÞ ¼ 1þ z

ð1� yzÞð1� y�1zÞð1� x�1zÞ ; and

rKðxÞ ¼ð1þ zÞ2ð1� zÞ

ð1� yzÞð1� y�1zÞð1� xzÞð1� x�1zÞ :

Then rsþK1ðxÞ þ rsþK2

ðxÞ ¼ rKðxÞ, as

ðxz þ xz2Þð1� x�1zÞ þ ð1þ zÞð1� xzÞ ¼ 1þ z � z2 � z3

¼ ð1þ zÞ2ð1� zÞ:

While the cones that appear in the Lawrence–Varchenkoformula are all simple, and those in Brion’s formula arestrictly convex, we use yet more general cones in theirproof. A rational (closed) halfspace is the convex subset ofR

d 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 whoseboundary hyperplanes have some point in common. Weassume this intersection is irredundant. The apex of K is theintersection of these boundary hyperplanes, which is anaffine subspace.

The generating function for the integer points in K is theformal Laurent series

SK :¼X

m2Kxm: ð6Þ


It is a priori less clear how to interpret this formal series as arational function if K is not strictly convex, that is, if its apexis not a single point. The apex is a rational affine subspaceL, and the cone K is stable under translation by any integervector w that is parallel to L. If m 2 K \ Zd ; then the seriesSK contains the series

xm �X

n2Zxnw

as a subsum. As this can converge only for x = 0, the seriesSK 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 anotherformal series. Thus the additive group C½½x�1

1 ; . . .; x�1d �� of

formal Laurent series is a module over the ringC½x�1

1 ; . . .; x�1d � of Laurent polynomials. The space PL of

polyhedral Laurent series is the C½x�11 ; . . .; x�1

d �-submoduleof C½½x�1

1 ; . . .; x�1d �� generated by the set of formal series

fSK j K is a simple rational coneg:

Since any rational cone may be triangulated by simplecones, PL contains the integer generating series of allrational cones.

Let Cðx1; . . .; xdÞ be the field of rational functions on Cd ;

which is the quotient field of C½x�11 ; . . .; x�1

d �: According toIshida [11], the proof of the following theorem is due toBrion.

THEOREM 7 There is a unique homomorphism of

C½x�11 ; . . .; x�1

d �-modules

u : PL �! Cðx1; . . .; xdÞ;

such that uðSKÞ ¼ rK for every simple cone K in Rd :

PROOF. Given a simple rational coneK ¼ v þ hw1; . . .;wdiwith fundamental parallelepiped P; we have

Yd

i¼1

ð1� xwi Þ � SK ¼ rPðxÞ:

Hence, for each S 2 PL, there is a nonzero Laurentpolynomial g 2 C½x�1

1 ; . . .; x�1d � such that gS ¼ f 2

C½x�11 ; . . .; x�1

d �. If we define uðSÞ :¼ f =g 2 Cðx1; . . .; xdÞ;then u(S) is independent of the choice of g. This defines therequired homomorphism.

The map u takes care of the nonconvergence of thegenerating series SK when K is not strictly convex.

LEMMA 8 If a rational polyhedral cone K is not strictly

convex, then uðSKÞ ¼ 0:

PROOF. Let K be a rational polyhedral cone that is not

strictly convex. Then there is a nonzero vector w 2 Zd such

that wþK ¼ K; and so xw � SK ¼ SK: Thus xwuðSKÞ ¼uðSKÞ: Since 1� xw is not a zero-divisor in Cðx1; . . .; xdÞ;we conclude that uðSKÞ ¼ 0:

We now establish Brion’s Formula, first for a simplex,and then, using irrational decomposition, for the general

case. (A d-dimensional simplex is the intersection of d + 1halfspaces, one for each facet.)

For a face F of the simplex P; let KF be the tangent coneto F, which is the intersection of the halfspaces correspondingto the d - dim(F) facets containing F. Let ; be the emptyface of P; which has dimension -1. Its tangent cone is P:

THEOREM 9 If P is a simplex, then

0 ¼X

F

ð�1ÞdimðFÞSKF; ð10Þ

the sum over all faces of P.

PROOF. 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 becomesX

G�F

ð�1ÞdimðGÞ:

But this vanishes, as every interval in the face poset of P is aBoolean lattice.

Now we apply the evaluation map u of Theorem 7 to theformula (10). Lemma 8 implies that uðSKF

Þ ¼ 0 except whenF ¼ ; or F is a vertex, and then uðSKF

Þ ¼ rKFðxÞ: This gives

0 ¼ �rPðxÞ þX

v a vertex of PrKvðxÞ;

which is Brion’s Formula for simplices.Just as for rational cones, every polytope P may be

decomposed into simplices P1; . . .;Pl having pairwisedisjoint interiors, using only the vertices of P :

P ¼ P1 [ � � � [ Pl :

Then there exists a small real number �[ 0 and a shortvector s such that if we set

P0 :¼ sþ ð1þ �ÞP and P0i :¼ sþ ð1þ �ÞPi

for i ¼ 1; . . .; l;

then P0 \ Zd ¼ P \ Z

d ; and no hyperplane supporting anyfacet of any simplex P0i meets Zd : If we write KðQÞw for thetangent cone to a polytope Q at a vertex w; then for v avertex of P with v0 = (1 + �)v + s the corresponding ver-tex of P0; we have KðP0Þv0 \ Z

d ¼ KðPÞv \ Zd and so this is

an irrational decomposition. ThenX

v a vertex of PrKðPÞvðxÞ ¼

X

v a vertex of P0rKðP0ÞvðxÞ

¼Xl

i¼1

X

v a vertex of P0i

rKðP0iÞvðxÞ

¼Xl

i¼1

rPiðxÞ ¼ rP0 ðxÞ ¼ rPðxÞ:

The second equality holds because the vertex cones KðP0iÞvform an irrational decomposition of the vertex cone KðP0Þv;and because the same is true for the polytopes. Thiscompletes our proof of Brion’s Formula.


Consider the quadrilateral Q; which may be triangu-lated by adding an edge between the vertices (2, 0) and(0, 2). Let � ¼ 1

4 and s ¼ ð� 12 ;� 1

4Þ: Then ð1þ �ÞQ þ s hasvertices

ð� 12;� 1

4Þ; ð2;� 14Þ; ð� 1

2; 2þ 14Þ; ð4þ 1

2; 2þ 14Þ :

We display the resulting irrational decomposition.

Q

We use the map u to deduce a very general form of theLawrence–Varchenko formula. Let P be a simple polytope,and for each vertex v of P choose a vector nv that is notperpendicular to any edge direction at v. Form the coneKnv;v as before. Then we have

rPðxÞ ¼X

v a vertex of Pð�1ÞjE

�v ðnvÞj rKnv ;v

ðxÞ: ð11Þ

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 changewhen any of the vectors nv is rotated.

Pick a vertex v and vectors n, n0 that are not perpen-dicular 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 direc-tions w at v such that n � w and n0 � w have the same sign,but with apex the affine line vþ Rm: Thus we have theidentity of formal series

SKn;v � SK ¼ �SKn0 ;v :

Applying the evaluation map u gives

rKn;vðxÞ ¼ �rKn0 ;vðxÞ;

which proves the claim, and the generalized Lawrence–Varchenko formula (11).

ValuationsValuations provide a conceptual approach to these ideas.Once the theory is set up, both Brion’s Formula and theLawrence–Varchenko Formula are easy corollaries ofduality being a valuation. We are indebted to Sasha Barv-inok who pointed out this correspondence to the secondauthor during a coffee break at the 2005 Park City Mathe-matical Institute. Let us explain.

Consider the vector space of all functions Rd ! R. Let Vbe the subspace that is generated by indicator functions ofpolyhedra:

½P�: x 7! 1 if x 2 P;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

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

Already this simple example shows that our generators donot form a basis: they are linearly dependent. For P0 ¼½0; 3� and Q0 ¼ ½1; 2�; we get the same sum.

But this is the only thing that can happen.

THEOREM 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 somevector space. Some standard examples are

That rPðxÞ is a valuation is a deep result of Khovanskii–Pukhlikov [12] and of Lawrence [14]. The last example iscalled the Euler characteristic. This valuation is surprisinglyuseful. For example, it can be used to prove Theorem 13below.

The most interesting valuation for us comes from thepolar construction. The polar P_ of a polyhedron P is thepolyhedron 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.

V mðPÞR volðPÞ

PL SPðxÞ

Cðx1; . . .; xdÞ rPðxÞ

R 1


(2)

The polar of a cone K … is the cone K_ :¼fx j hx; yi� 1 for all y 2 Kg:

(3) Suppose that P is a polytope whose interior containsthe origin and F is a face of P:

Then the polar of the tangent cone KF … is the convexhull of the origin together with the dual face F_ :¼fx 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_;Fi ¼ 1: Conversely, if x 2 K_F ; thenhx; :i is maximized over KF at F by example (2), and itis at most 1 there.

In these examples, the polar of the polar is the originalpolyhedron. This happens if and only if the original poly-hedron 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.

THEOREM 13 (Lawrence [14]) The assignment ½P� 7! ½P_�defines a valuation.

This innocent-looking result has powerful conse-quences. Suppose that P is a polytope whose interiorcontains the origin. Then we can cover P_ by pyramidsconvð0;F_Þ over the codimension-one faces F_ of P_. Theindicator functions of P and the cover differ by indicatorfunctions of pyramids of smaller dimension.

½P_� ¼X

F_½convð0; F_Þ� � lower dimensional pyramids:

ð14Þ

The Euler–Poincare formula for general polytopesorganizes this inclusion-exclusion, giving the exactexpression

½P_� ¼Xð�1Þcodim F_þ1½convð0; F_Þ�:

We illustrate this when P is the square.

=+ − +− + −+ − +

= + + +

− − − − +

If we apply polarity to (14), we get the Brianchon–GramTheorem [6, 9].

½P� ¼X

v vertex

½Kv�

� tangent cones of faces of positive dimension:

ð15Þ

This is essentially the indicator-function version of Theo-rem 9, but for general polytopes. If we now apply thevaluation r, and recall that r evaluates to zero on cones thatare not strictly convex, we obtain Brion’s Formula.

Next, suppose that we are given a generic directionvector n. On a face F of P; the dot product with nachieves its maximum at a vertex vnðFÞ: For a vertex v ofP; we set

F_n ðvÞ :¼[

F :vnðFÞ¼v

relint F_:

(The relative interior, relintðPÞ; of a polyhedron P is thetopological interior when considered as a subspace of itsaffine hull.) In words, we attach the relative interior of alow-dimensional pyramid convð0;F_Þ to the full-dimen-sional pyramid convð0; v_Þ that we see when we look inthe n-direction from convð0;F_Þ: In this way, we obtainan honest decomposition

½P_� ¼X

v

½convð0;F_n ðvÞÞ�: ð16Þ

For the polar of the square, this is

To compute the polar of the half-open polyhedronconvð0;F_n ðvÞÞ; we have to write its indicator function


½convð0;F_n ðvÞÞ� as a linear combination of indicator func-tions of (closed) polyhedra. If P is a simple polytope, thenall the dual faces F_ are simplices. It turns out that thepolar of convð0;F_n ðvÞÞ is precisely the forward tangentcone Kn;v at the vertex v. So the Lawrence–Varchenkoformula is just the polar of (16).

This gives a fairly general principle for constructingBrion-type formulas: Choose a decomposition of (theindicator function of) P_; and then polarize. We invite thereader to set up his or her own equations this way.

An ApplicationBrion’s Formula shows that certain data of a polytope—thelist of its integer points encoded in a generating function—can be reduced to cones. We have already seen how toconstruct the generating function rKðxÞ for a simple coneK: General cones can be composed from simple ones viatriangulation and either irrational decomposition or inclu-sion-exclusion. Given a rational polytope P; Brion’sFormula allows us to write the possibly huge polynomialrPðxÞ as a sum of rational functions, which stem from(triangulations of) the vertex cones. A priori it is not clearthat this rational-function representation of rPðxÞ is anyshorter than the original polynomial. That this is indeedpossible is due to the signed decomposition theorem ofBarvinok [1].

To state Barvinok’s Theorem, we call a rational d-coneK ¼ vþ

Pdi¼1 R� 0wi unimodular if w1; . . .;wd 2 Z

d gen-erate the integer lattice Z

d : The significance of aunimodular cone K for us is that its fundamental (half-open) parallelepiped contains precisely one integer pointp, and so the generating function of K has a very simpleand short form

rKðxÞ ¼xp

1� xw1ð Þ � � � 1� xwdð Þ :

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

THEOREM 17 (Barvinok) For fixed dimension d, the

generating function rK for any rational cone K in Rd

can be decomposed into generating functions of unimod-

ular cones in polynomial time; that is, there is a

polynomial-time algorithm and (polynomially many) uni-

modular cones Kj such that rKðxÞ ¼P

j �jrKjðxÞ; where

�j 2 f�1g:

Here polynomial time refers to the input data of K; thatis, the algorithm runs in time polynomial in the input lengthof, say, the halfspace description of K:

Brion’s Formula implies that an identical complexitystatement can be made about the generating function rPðxÞfor any rational polytope P: From here it is a short step(which nevertheless needs some justification) to see thatone can count integer points in a rational polytope inpolynomial time.

We illustrate Barvinok’s short signed decomposition forthe cone K :¼ ð0; 0Þ þ R� 0ð1; 0Þ þ R� 0ð1; 4Þ; ignoringcones of smaller dimension.

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

In general the cone ð0; 0Þ þ R� 0ð1; 0Þ þ R� 0ð1;nÞ is thedifference of two unimodular cones, but it has a uniquedecomposition into n unimodular cones.

Arguably the most famous consequence of Barvinok’sTheorem applies to Ehrhart quasipolynomials—thecounting functions LPðtÞ :¼ # tP \ Zd

� �in the positve-

integer variable t for a given rational polytope [4] P: Onecan show that the generating function

Pt� 1 LPðtÞ xt is a

rational function, and Barvinok’s Theorem implies that thisrational function can be computed in polynomial time.Barvinok’s algorithm has been implemented in the softwarepackages barvinok [17] and LattE [8]. The method ofirrational decomposition has also been implemented inLattE; considerably improving its performance [13].

ACKNOWLEDGMENTS

Research of Beck supported in part by NSF grant DMS-

0810105. Research of Haase supported in part by NSF

grant DMS-0200740 and a DFG Emmy Noether fellow-

ship. 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 Mathe-

matics, 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, Theoreme nouveau sur les polyedres, J. Ecole

(Royale) Polytechnique 15 (1837), 317–319.


7. M. Brion, Points entiers dans les polyedres convexes, Ann. Sci.

Ecole Norm. Sup. 21 (1988), no. 4, 653–663.

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

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

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.

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

convex sets, Pacific J. Math. 75 (1978), no. 2, 397–410.

11. M.-N. Ishida, Polyhedral Laurent series and Brion’s equalities,

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

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

for integrals and sums of quasipolynomials on virtual polytopes,

Algebra i Analiz 4 (1992), 188–216.

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

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

236.

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

(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

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

lozhen. 21 (1987), no. 1, 11–22.

17. S. Verdoolaege, software package barvinok (2004), electroni-

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

18. W. Volland, Ein Fortsetzungssatz fur additive Eipolyhederfunk-

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

149.


Four Poems from When She Was Kissedby the MathematicianSandra 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, sometimeseven, placid and perfectly

divisible, the way she imaginescertain numbers are,

sometimes stopped by oddirregular murmurs, listens and counts

his breaths, his unintelligible wordsas if she wrapped a rosary of integers

around her wrists, his wrists, linking them bothin the smaller infinity, the kind you can count

and maybe comprehend,the one that Zeno scorned.

His chest with its mane of grayrises and falls as she counts, crawls nearer,

wishing he’d explain again or elseembrace her, silence this abacus

of prayer that ticks in her head:O God, whatever you are, let this one

be —and the bed swellsin the heat, in the dark, and the single

sheet that holds them closewinds round and round like the great

enfolding spaces through which arrowsfly and breaths and prayers

on their eccentric routetoward Zeno’s black incalculable target.


They Debate Triangles and Medians

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

necessity, each a moment brimmingwith 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, unlikelyas the medians that cling together

at the center of each triangle,knotting altitudes and perpendicularsinto 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 mediansof you and me and themlast and glow till one by one

the fastenings unclaspand that which must be linearsheds the comforts of shape,

each line going its lonely distanceto the non-Euclidean placewhere parallels diverge in darkness.

She Grapples With Operations Research

... the by now famous problem of the jeep... concernsa jeep which is able to carry enough fuel to travel adistance d, but is required to cross a desert whosedistance is greater than d (for example 2d). It is to dothis by carrying fuel from its home base and estab-lishing fuel depots at various points along its route sothat it can refuel as it moves farther out... [But] ingeneral, the more jeeps one sends across, the lowerthe fuel consumption per jeep.

—David Gale, ‘‘The Jeep Once More or Jeeper by theDozen’’

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 kindlierMojave 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 templesclenched with calculations

and desire.

At stated stations

palms, dates, springs of comfortwill 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. Long-

time 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]

� 2008 Sandra M. Gilbert, Volume 31, Number 1, 2009 19

http://mathsite.math.berkeley.edu/main.html

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 herselfinto a crystal dish of peaches.

The mathematician is crossing the desert,crossing, journeying past Zeno, past the infinite.

She wants to be the firstoasis that he reaches.

He Explains the Book Proof

The shadowy clatter of the cafeframes the glittering doorway.

A white cup and a blue bowlinscribe pure shapes on the table.

The mathematician says, Let’s turn the pagesand 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 allbeen written down on sheer

sheets of the invisible,and held quite still,

so that to think hard enoughis 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.



Years Ago David E. Rowe, Editor

JakobSteiner’sSystematischeEntwickelung:The Culmina-tion ofClassicalGeometryVIKTOR BLASJO

Send submissions to David E. Rowe,

Fachbereich 08—Institut fur Mathematik,

Johannes Gutenberg University,

D55099 Mainz, Germany.


Introduction

SSteiner’s Systematische Ent-wickelung of 1832 was amonumental unification of

classical geometry based on a newconception of projective geometry anda new approach to conic sections. Weshall study this work in some detail. Ishall also claim that Steiner was com-mitted to the unification and reverenceof classical geometry, and that hiswork was a remarkable success bythese standards, but that later genera-tions imposed different standards ofsuccess, downplaying historical conti-nuity and favouring intrinsicallymotivated programmatic agendas—inthe case of projective geometry epito-mised by von Staudt (1847)—whichcaused a lasting and undeserveddepreciation of Steiner’s work.

It ought to be uncontroversial thatthe Systematische Entwickelung wasintended as a unification of classicalgeometry since, first, it is packed withclassical theorems and references toclassical geometers, and, second, Stei-ner says so. In his preface, Steinerexplains that the goal of his work is to‘‘extract a thread of continuity and acommon root’’ from classical geometryby uncovering ‘‘fundamental proper-ties that contain the germ of alltheorems, porisms, and problems ofgeometry, so generously made avail-able to us in older and modern times.’’Classical geometry has thus far pro-duced ‘‘a collection of separate tricks,however clever, but no organicallyconnected whole. This work tries touncover the organism by which themost varied features of the spatialworld are connected. A small numberof very simple fundamental relation-ships make up the schema by whichthe remaining mass of theorems can bedeveloped consistently and withoutdifficulty.’’ Elsewhere he said: ‘‘As ateacher I tried whenever possible totreat each subject as consisting of asingle idea, and to see the individual

theorems as mere consequences left asfootprints in the development of thisone idea. Almost unconsciously, thisled me to the actual genetic viewpoint,as it must have appeared to the ancientgeometers, although I approached it inthe opposite way. Since I had a wealthof solved problems and theoremsavailable to me, my task was to focusnot on individual theorems but ratherthe general principles of syntheticconstruction from which all theseinventions follow, to present them inthis capacity and treat them exhaus-tively according to these principles.’’(Graf (1897, p. 12–13); for more ofSteiner’s own words on these matters,see Lange (1899, pp. 19–21, 23–24)).Finally, I cannot help but perceive assymbolic an observation made byJacobi (1843) in a letter to his wifewritten when Jacobi and Steiner wereboth in Rome: ‘‘Steiner has an aptitudefor finding walled-in old Doric col-umns in old stables and worn-downbuildings . . . through which one isclearly reminded that one is walkingon classical soil.’’

Nevertheless, many commentatorsmiss this point and consequently fail toappreciate Steiner’s work. Back in theold days the Systematische Entwicke-lung used to be called ‘‘epoch making’’(Kotter (1901, p. 252)), a ‘‘masterpiece’’(Zacharias (1912, p. 41)), ‘‘a model of acomplete method and execution for allother branches of mathematics’’(Jacobi (1845), quoted in Burckhardt(1976, p. 18)), and so on. Not so today,however. As a framework for this dis-cussion, let us state a principle whichshould be a truism but which is in factviolated in many commentaries onSteiner. Suppose a mathematician Xwrites a work aiming to achieve Y.Anyone wishing to criticise this workcould reasonably be expected to arguethat either

(i) X does not achieve the objective Y;or

� 2008 SPRINGER SCIENCE+BUSINESS MEDIA, LLC., Volume 31, Number 1, 2009 21

(ii) the objective Y is not worthpursuing.

A charge of the type ‘‘X does not dealwith Z’’ is plainly irrelevant. Still, this isprecisely the type of criticism offeredby many commentators.

Leading the crusade is Klein (1979,p. 118), who writes in his discussion ofthe Systematische Entwickelung: ‘‘inretreating from the ground won byMobius and rejecting the principle ofsigns from synthetic geometry, [Steiner]deprived himself of the possibility ofmore general formulations. Thus, whendealing with cross-ratios he was forcedalways to fix the order of the elements;but, above all, he lost the occasion ofmastering the imaginary. He neverreally understood it and fell into the useof such terms as ‘the ghost’ or ‘the sha-dow land of geometry.’ And of coursehis system had to suffer from this self-imposed restriction. Thus, though thereare two conics x2

1 þ x22 � x2

3 ¼ 0 andx2

1 þ x22 þ x2

3 ¼ 0 from a projectivepoint of view, in Steiner’s system there isno room for the second. Von Staudt wasthe first to liberate synthetic geometryfrom these and other imperfections.’’

The ‘‘Steiner: bad—von Staudt:good’’ dichotomy is very prevalent. Forexample, Coolidge’s discussion (1934,pp. 222–223) of the SystematischeEntwickelung is mostly concerned withfinding ‘‘slips’’ and is immediately fol-lowed by an ecstatic discussion of vonStaudt, ‘‘this deep thinker,’’ who ‘‘per-ceived two essential weaknesses in thesynthetic geometry of his predeces-sors. (a) The basis of projectiverelations was the cross-ratio. This isprojectively invariant but, as previ-ously given, was based on distancesand angles which are not in themselvesunalterable. (b) What are imaginarypoints anyway? What can be said aboutthem, except that they are imaginary?’’

Although Klein, as a proud discipleof Plucker, is certainly biased in thisissue, his point of view has neverthelesstaken hold quite widely. Laptev & Ro-zenfel’d (1996, p. 37), for example, donot hesitate to state that Steiner’s dis-regard for negative and complexnumbers is ‘‘an important defect.’’ Eventhe ‘‘Geometry’’ article in the Encyc-lopædia Britannica (Heilbron (2007)),supposedly an objective source, takes aKleinian stance: ‘‘Poncelet’s followers

realized that they were hamperingthemselves, and disguising the truefundamentality of projective geometry,by retaining the concept of length andcongruence in their formulations, sinceprojections do not usually preservethem. . . . Efforts were well under wayby the middle of the 19th century, by . . .von Staudt . . . among others, to purgeprojective geometry of the last super-fluous relics from its Euclidean past.’’

These commentators miss the point.Steiner achieves exactlywhat he sets outto achieve—a systematic unification ofclassical geometry—whereas von Sta-udt of course never comes close toanything of the sort. The intrinsic theorypoint of view naturally makes Steiner’swork look flawed, but the real issue iswhether the ideas of von Staudt et al.would have helped Steiner further hisobjective. I say that the answer is almostalways no. Steiner is not ‘‘forced’’ to fixthe order of the elements in the cross-ratio; he chooses to use this classicalnotion of the cross-ratio and it serveshim well. It is not that there is ‘‘no room’’for the conic x2

1 þ x22 þ x2

3 ¼ 0; there isin fact no point in it since Steiner’s onlyinterest is the classical theory. His workdoes not ‘‘suffer’’ from exclusion ofimaginary elements, because they arenot needed, and therefore Steiner couldnot care less ‘‘what can be said aboutthem.’’ As for using metric notionswhere purely projective ones are pos-sible—so what? If our goal is to unifyclassical geometry, metric notions areby no means an ‘‘essential weakness.’’To insist onpurelyprojectivemethods isa ‘‘self-imposed restriction’’ if there everwas one. What was ‘‘purged’’ by vonStaudt et al. was not ‘‘superfluous rel-ics’’ but historical continuity.

Klein (1979, p. 118) also claims that‘‘further imperfections affect the basicdefinitions of Steiner’s system, so thatmany more theorems have exceptionsthan Steiner was aware of.’’ To supportthis claim, Klein refers to Baldus(1923). But this seems to me to bebased on a misunderstanding due to atoo modern reading of Steiner. Baldusdoes not so much attack Steiner’swork, per se, but rather contemporaryauthors using his definition of projec-tivity, his argument being that it cannotproperly handle pencils centred atinfinity (p. 87). But Steiner himself

does not use pencils centred at infinity;indeed, doing so would have causedhim great difficulty, since, as the abovegentlemen so eagerly remarked, hisdefinition of the cross-ratio is metrical.

Projective Geometry in theSystematische EntwickelungI shall now provide an overview of themathematical content of the System-atische Entwickelung. First, we studythe cross-ratio, which is the foundationof the entire theory. This immediatelyyields swift proofs of the classical the-orems of Pappus and Desargues. Thereal triumph, however, is Steiner’stheory of conic sections, which weshall study subsequently.

The Cross-Ratio

We shall now see how Steiner arrivedat the cross-ratio. Consider a line A anda pencil at B, and let a; b; c; . . . be thepoints where the line is intersected bythe lines a; b; c; . . . of the pencil(Figure 1). The line and the pencil aresaid to be in perspective. If we movethe line or the pencil, then a; b; c; . . .will no longer correspond to a; b; c; . . .,but there will be some definite relationbetween the two entities (i.e., knowingthe lengths ab, bc, etc., means knowingthe angles \ab, \bc, etc.). We find thisrelation as follows. Let p be the line ofthe pencil perpendicular to A, anddraw the perpendicular d1a to d. ThenBpd and ad1d are similar, giving

Bp

Bd¼ ad1

ad;

or, since ad1 ¼ Ba � sinð\adÞ,ad

sinð\adÞ ¼Ba �Bd

Bp:

By the same argument,

ac

sinð\acÞ ¼Ba �Bc

Bp;

bc

sinð\bcÞ ¼Bb �Bc

Bp;

bd

sinð\bdÞ ¼Bb �Bd

Bp;

which combine to give

ad

sinð\adÞ

. bd

sinð\bdÞ¼ ac

sinð\acÞ. bc

sinð\bcÞ


or

ad

bd

. ac

bc¼ sinð\adÞ

sinð\bdÞ. sinð\acÞ

sinð\bcÞ ;

which no longer depends on thepositions of the line and the pencil, sowe have found the relation we werelooking for. This is the cross-ratio. Animmediate consequence ðxx5; 10Þ is thetheorem that for any line cutting apencil, the points a; b; c; d of the linecorresponding to the lines a, b, c, d ofthe pencil always have the same cross-ratio (since the right-hand side in theabove expression stays the same),which is a more conventional state-ment of the projective invariance of thecross-ratio. In particular, since thecross-ratio is preserved when a line isprojected onto another line, such aprojection is determined by its effecton any three points ðxx6; 10Þ.

Pappus’ Theorem (§23)We are given a hexagonB1BB2a1eaB1 with vertices alternatelyon two lines B1eB2 and a1Ba

(Figure 30; I shall reproduce Steiner’sfigures with their original numbering).We wish to show Pappus’s theorem:The three points of intersection ofopposite sides are on a line. Draw thelines ae (A) and BB2, meeting in f, anddraw the lines a1e (A1) and BB1,meeting in l1. The points f and l1 arethe first two Pappus points; we need toshow that the third point, the inter-section of a1B2 and aB1, is on thesame line (A2). To do this, we note that

the triangle efl1 is inscribed in the tri-angle BB1B2. This means that e, f, andl (the intersection of A and Bl1) allcome back onto themselves when sentthroughout the series of projectionsA! A1 ! A2 ! A defined by theprojection points B;B2;B1. So, by thethree-point determinacy, this cycle ofprojections is the identity. Followingthe course of a through these projec-tions traces out a triangle aa1a2, with

a2, the third Pappus point, located onA2, as was to be shown.

Desargues’ Theorem (§21)

Steiner’s proof of Desargues’s theorembegins with the following lemma(Figure 27). Let A, A1, A2 be threeprojectively related lines, i.e., there areprojections

A ! A1 : a;b;c;... 7! a1;b1;c1;...;A ! A2 : a;b;c;... 7! a2;b2;c2;...;A1 ! A2 : a1;b1;c1;... 7! a2;b2;c2;...;

and let the lines meet in a point e;e1;e2.The lemma says that the three points ofprojection B;B1;B2 are on a line.Proof: The line BB1 is a projection lineof A!A1 as well as A!A2, so it goesthrough three corresponding pointsd;d1;d2. But d1d2 must be a projectionline of the third projection A1!A2, soB2 must also be on this line. ThusB;B1;B2 are collinear, and the lemmais proved. Desargues’s theorem maynow be proved as follows. Let aa1a2

and bb1b2 be two triangles in per-spective, i.e., the lines A, A1, A2

connecting corresponding verticesmeet in a point e. Let B;B1;B2 be theintersections of the extensions of cor-responding sides of the triangles, and

Figure 1.

Figure 30.


use these points to project A!A1,A!A2, and A1!A2. Under theseprojections, a;b;e, and a1;b1;e, anda2;b2;e correspond by construction, soby the three-point determinacy ofprojections, the three lines are projec-tively related, so, by the result justproved B;B1;B2 must be on a line,which is Desargues’s theorem.

Conic Sections in theSystematische EntwickelungAs promised above, we shall now seehow a remarkably unified and simpleapproach to conic sections is madepossible by the basic theory ofprojections.

The Fundamental Theorem on

Conic Sections (§§37–39)

Steiner’s fundamental theorem onconic sections is, according to himself,‘‘more important than all the previ-ously known theorems about them, forit is the true fundamental theorem,since it is so comprehensive thatalmost all other properties of thesefigures follow from it in the simplestand clearest way, and the method bywhich they will be deduced surpassesany known point of view in terms ofsimplicity and convenience’’ ðx39Þ.

We shall discuss the theorem first interms of circles. It extends to generalconics by projection, of course. Con-sider a circle with center M and two ofits tangents A, A1 (Figure 38). Anyother tangent a pairs a point of A witha point of A1. We shall prove that thiscorrespondence A! A1 is a projec-tion. First, let q,r be the two tangents

parallel to A, A1; thus if we let q be thepoint at infinity of A, then q1 will be theintersection 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 andr. Consider now two other tangentsa, b. To show that the correspondenceis a projection we need to show thatthe cross-ratio of a; r; b; q is the same asthe cross-ratio of a1; r1; b1; q1, i.e.,

ar

br

. aq

bq¼ a1r1

b1r1

. a1q1

b1q1

:

Since q and r1 are the points at infinity,this simplifies to

ar

br

.1 ¼ 1

. a1q1

b1q1

or

ar � a1q1 ¼ br � b1q1:

Therefore, to show that the corre-spondence is a projection, we need toshow that the quantity ar � a1q1 isindependent of the choice of a. To dothis, we note first that rq1 cuts the circle

in half, making an equilateral triangledrq1 with base angles a = a1. Connect-ing a to M obviously bisects the angleat a, and, similarly, a1M bisects theangle at a1. Also, by comparing theangle sums of the triangle aMa1 andthe quadrilateral arq1a1, we see thatthe angle \aMa1 is equal to a. There-fore, the triangles aMa1, arM, Mq1a1

are similar, giving ar=rM ¼Mq1=q1a1,or ar � a1q1 ¼Mr �Mq1, so ar � a1q1

is indeed independent of a, as weneeded to show.

Thus we have proved the followingtheorem ðx38:IVÞ: Any two projectivelyrelated lines define a conic section towhich they and all their projectionlines are tangents (Figure A(i)). Or,dually, any two projectively relatedpencils define a conic section throughthe centers of the pencils as the locusof intersections of corresponding lines(Figure A(ii)).

The two dual forms of the funda-mental theorem subsume, as we shallprove below, two of the most promi-nent earlier attempts at systematicapproaches to conic sections, namelythose of Pascal and Newton.

Pascal (1640) envisioned a unifica-tion of the theory of conic sectionsbased on his ‘‘hexagrammum mysti-cum.’’ Immediately after havingintroduced his theorem he says: ‘‘[W]epropose to give a complete text on theelements of conics, that is to say, all theproperties of diameters and otherstraight lines, of tangents, etc., to con-struction of the cone from substantiallythese data, the description of conicsections by points, etc.’’ (Pascal (1640),quoted from Struik (1969, p. 165)).This program saw a revival with thediscovery of Brianchon’s theorem.

Figure 27.

Figure 38.


Newton (1667/68) unified muchconic section theory by the followingconstruction (Figure B): Given tworules HFG and RKS with fixed angles\HFG and \RKS, and fixed points Fand K, if we make one intersection, S,move along a line, then the otherintersection, R, traces out a conic. Thisis a very efficient tool, not least forsolving construction problems involv-ing conics. Newton later put some ofthis theory in the Principia (1687,Book I, Section V), although it does

not do much good there, appearing, asit does, in a section having ‘‘but littleconnection with the rest of the Prin-cipia’’ (Ball (1893, p. 81)). A moresystematic account was later providedby Maclaurin (1720).

The fundamental theorem also hasseveral interesting immediate corollar-ies ðxx40�41Þ. If, for instance, the twogenerating lines are similar, then theline at infinity is a projection line andthus a tangent to the conic, so there-fore the conic must be a parabola. The

fundamental theorem also shows that aconic is determined by five tangents(or, dually, five points): Two tangentsare taken as the generating lines andthe other three determine the projec-tive relation between them, by thethree-point determinacy of projectivetransformations.

Furthermore, the fundamental con-struction of conic sections extended tospace becomes a construction of one-sheeted hyperboloids: For any twoprojectively related lines in space,their projection lines generate a one-sheeted hyperboloid ðx51:IVÞ.

I should also mention that Steiner’sfundamental theorem is commonlycalled ‘‘Steiner’s definition’’ of conicsections, and is sometimes criticised assuch; e.g., ‘‘Steiner’s definition assignsa special role to two points on theconic, obscuring its essential symme-try’’ (Coxeter (1993, p. ix)). Also, Kline(1972, pp. 847–848), in his discussionof the Systematische Entwickelung,speaks of Steiner’s ‘‘now standardprojective method of defining theconic sections’’ and claims that ‘‘he didnot identify his conics with sections ofa cone,’’ which is false (they are pro-jective images of circles, as we haveseen), and, I might add, fundamentallyinconsistent with Steiner’s commitmentto classical geometry. The freedom todefine the objects of study as onepleases comes only when a theorymatures into the stage of intrinsicmotivation.

Pascal’s and Brianchon’s

Theorems (§42)

We shall prove Brianchon’s theorem.Pascal’s theorem, of course, followsdually. Consider a hexagon B1B2kaa1

l1B1 circumscribing a conic (Figure 31;the conic is not shown). Take the twosides ak (A) and a1l1 (A1) as the twogenerating tangents of the fundamentaltheorem. Then the fundamental theo-rem says that these two sides A and A1

are projectively related and that theother four sides are projection linesconnecting corresponding points of Aand A1. Now project A onto kl1 (A2)from B1, and project A1 onto A2 fromB2. These two projections agree onthree points—the images of b and b1, k

and k1, and l and l1—so, by the three-point determinacy, they must alsoagree on the image of a and a1, i.e.,

Figure A. The two dual ways of generating conic sections

by the fundamental theorem (From Courant & Robbins

(1941, pp. 208, 205)).

Figure B. Newton’s organic construction of conic

sections. (From Newton (1967–1981, Vol. 2, p. 118).)


B1a and B2a1 intersect A2 at the samepoint. So the three diagonalsB1a;B2a1;A2 meet at a point (a2),which is Brianchon’s theorem.

Newton’s Organic Construction

of Conic Sections (§46)

Newton’s organic construction of con-ics follows immediately from thefundamental theorem. Recall fromSection 3.1 that we have two rules withfixed angle, and we wish to show thatas one intersection moves along a linethe other traces out a conic. The factthat the first intersection moves along aline means in our language that wehave two projective pencils, B and B1

(Figure 50). The second intersection isthe intersection of two other pencils,B2 and B3, which, since the angles ofthe rules are fixed, are in fact justrotated copies of the previous pencils.So, by the fundamental theorem, thesetwo projectively related pencils definea conic section.

Pole and Polar Theory (§44–45)

In this section, we shall see how thetheory of poles and polars may beapproached through Steiner’s funda-mental theorem on conic sections. Thisis particularly interesting since thistheory was a precursor of the modernnotion of duality, which, although ithad been recognised before, Steinerwas the first to give a prominent placeat the very foundation of projectivegeometry ðx1Þ.

Poles and Polars Before SteinerPolar reciprocation with respect to acircle associates a line with every pointand a point with every line, as follows.Consider a line that cuts through thecircle. It meets the circle in two points.Draw the tangents to the circle throughthese points. The two tangents meet ina point. This point is the pole of theline. Conversely, the line is the polar ofthe point. Thus, we can deal with linesthat cut through the circle and, equiv-alently, points outside the circle. Butwhat about a line outside the circle (or,equivalently, a point inside the circle)?

Let l be such a line. For every point onl, there is a polar line through the cir-cle, as above. We claim that all thesepolar lines have one point in common,so that this point is the natural pole ofl. Monge ð1799; x39Þ proves this bycleverly bringing in the third dimen-sion. Imagine a sphere that has thecircle as its equator. Every point on l isthe vertex of a tangent cone to thissphere, where the two tangents to theequator are part of this cone and thepolar line is the perpendicular projec-tion of the circle of intersection of thesphere and the cone. Now consider aplane through l tangent to the sphere.It touches the sphere at one pointP. Every cone contains this point(because the line from any point on lto P is a tangent to the sphere and so ispart of the tangent cone). Thus, forevery cone, the perpendicular projec-tion of the intersection with the spheregoes through the point perpendicu-larly below P, and this is the pole of l,and l is the polar of this point.

Polar Theory and DualityPolar reciprocation thus suggests aduality between points and lines. Bri-anchon (1806) was perhaps the first touse this idea in his proof of ‘‘Brianchon’stheorem,’’ the dual of Pascal’s theorem.The polar reciprocation approach toduality was systematised by Poncelet(1822). In this tradition, Brianchon’stheorem can be derived as follows. Westart with Pascal’s theorem, i.e., we havea hexagon inscribed in a conic. Whenwe apply polar reciprocation, the conicgoes to a conic, because through anypoint off the conic there are two tan-gents to it, and they go to two collinearpoints on the new curve, so the newcurve has degree two, so it is a conic.And the vertices of the hexagon, beingpoints of the conic, go to tangents to thenew conic, and thus the hexagon goesto a circumscribed hexagon, and thesides of the original hexagon go to thevertices of the new, and the line whereextensions of opposite sides meet goesto the point where lines connectingopposite vertices meet.

It is, however, not necessary toapproach duality through polar recip-rocation. Gergonne (1825/26) arguedthat the same effect is achieved ‘‘bymerely exchanging the two wordspoints and lines with one another.’’

Figure 31.

Figure 50.


This suggests itself from an extensionof the above ideas beyond the domainof conics: Any curve can be assigned adual curve, namely the curve envel-oped by the polars of all its points.

These two contrasting views led toa dispute between Poncelet andGergonne; see, e.g., Gray (2007,Chapter 5). Steiner notes in his prefacethat this dispute is put in perspective byhis work, concluding that while ‘‘Ger-gonne’s principle proves, in retrospect,to be more primitive, closer to thesource, Poncelet has made an equallyvaluable contribution, in his develop-ment and furthering of syntheticgeometry, so that this field may nolonger be disregarded, as it has been alltoo often and all too frivolously inmodern times.’’

Steiner’s Approach to PolarReciprocationSteiner’s approach to polar reciproca-tion may be illustrated by his proof ofthe theorem of Monge studied above.The proof uses the harmonic propertyof the complete quadrilateral, so weshall prove that first. A harmonic setof points is four points with cross-ratio 1,

ad

bd

. ac

bc¼ 1

(or -1 according to many otherauthors, since, in the typical case, bd is‘‘negative’’). A complete quadrilateralis a figure determined by four lines.Figure 26 shows a complete quadrilat-eral with sides a,b,a1,b1 and diagonalsAE; a1E; aC. We wish to show thateach diagonal is divided harmonicallyby the other two, i.e., aDb1C anda1DbE and ACBE are all harmonicsets. Consider the three lines a,b,cthrough a, and consider the fourth lined that makes a, d, b, c harmonic. In thesame way, consider the line d1 thatmakes a1, d1, b1, c harmonic. Whenthe four lines a, d, b, c or the four linesa1, d1, b1, c are intersected by thediagonal aC, the result is then a har-monic set of points. It must be the sameset in both cases, by the three-pointdeterminacy, since the two sets havethree points in common: a and a1 meetat a, and b and b1 meet at b1, and c isshared. Therefore, d and d1 must meetat some point D0 that makes aD0b1C

harmonic. Applying the same argument

with a1C in place of aC shows that dand d1 must also meet at some point D00

that makes a1D00bE harmonic. But by

construction, D0 must be on the diag-onal aC, and D00 must be on thediagonal a1E, so both D0 and D00 mustin fact be D, the intersection of thesetwo diagonals. Thus aDb1C and a1DbE

are harmonic sets, as was to be shown.Now we are ready for Steiner’s proof

of Monge’s theorem above. ConsiderFigure 43. We wish to show that as theintersection f of the two tangents A1, A3

at the points a1; a3 moves in a line y,then the line a1a3 turns around the pointy. Draw two other tangents A, A2 tomake a complete quadrilateral. By theharmonic property of the completequadrilateral, e; y; d; z is a harmonic setof points. Projecting from f, we see thata1; y; a3; v is a harmonic set of points.Projecting from z, we see that a; y; a2; uis 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 beshown.

ConclusionLet me now revisit my thesis that Stei-ner was committed to historicalcontinuity. We have seen that there isgood reason to interpret the System-atische Entwickelung in this way. Wehave also seen that some remarks ofSteiner himself point in this direction.This is the sum of my evidence.Biographical accounts of Steiner—of

which there are very few, and none offull length—do not offer direct supportfor the thesis. On the contrary, thethesis appears to contradict a standardcharacterisation of Steiner.

Criticism of Steiner goes hand inhand with biographical accounts ofhim as intuitively gifted but unschol-arly and somewhat flimsy. Lampe(1900, p. 138), for example, says that‘‘it is very probable that he neverstudied the writings of other mathe-maticians, but merely looked throughthem to compare his results with thoseof his predecessors.’’ How, then, is oneto explain that the Systematische Ent-wickelung reads like a monumentalunification of classical geometry? Thebiographers propose an easy solution:They simply maintain that the wholework was more or less ghostwritten byJacobi, ‘‘who, unlike Steiner, read anextraordinary amount and was wellversed in the mathematical literature’’(Graf (1897, p. 13–14)). Despite neverreading any books, Steiner still pro-vides frequent historical references inthe Systematische Entwickelung,including, on one occasion, referencesto 15 mathematicians on a single page(Werke, I, p. 340). Geiser (1874, p. 249)thinks that ‘‘one may assume’’ that itwas Jacobi who ‘‘made the careful lit-erary references possible,’’ an opinionshared by Obenrauch (1897, p. 253).Lampe (1900, pp. 138–139) andBiermann (1963, p. 40) also emphasise

Figure 26.


Jacobi’s role in teaching the ignorantSteiner about the literature.

I believe Jacobi’s role is overstatedbythese authors. Surely Steiner wouldhave discussed such matters withJacobi, and indeed there are such indi-cations in the the excerpts from theircorrespondence published by Jahnke(1903b, e.g., p. 271). But in the sameletters Steiner also brings up many ref-erences himself. Also, the references inthe Systematische Entwickelung are notvery careful at all (not up to Jacobianstandards); most of the time, Steiner justgives a name with an occasional ‘‘bek-anntlich’’ thrown in here and there. Andif the references were primarily sup-plied by Jacobi, then it would perhapsseem strange for Steiner to write

explicitly in his preface that ‘‘all impor-tant theorems already discovered byothers I have, to the extent of myknowledge, credited to their originaldiscoverers.’’

The fact that Steiner gives more fre-quent references in the SystematischeEntwickelung than elsewhere couldeasily be accounted for without assum-ing the interference of Jacobi by the factthat the Systematische Entwickelung isfundamentally a unification of classicalgeometry, so references are highly rele-vant, whereas many of his other worksare, in essence, self-contained, so refer-ences would not contribute to thepurpose of the work. This is also con-sistent with the common claim thatSteiner did not always give proper

references when he borrowed ideasfrom 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 lentcredibility to his work. For example, as Ihave noted elsewhere (Blasjo (2005,x1)), in his work on the isoperimetricproblem, Steiner (1842b, xx13�14)solves a particular subproblem treatedunsuccessfully by the Greeks, where-upon he promptly refers to Pappus andothers; see also the preface to Steiner(1842a).

Furthermore, although some of thereferences may be considered cosmetic(e.g., pointing out origins of terminol-ogy) and could easily have been addedby Jacobi, almost all of them are verynaturally linked to the mathematicalcontent. Since the standard view is thatSteiner ‘‘did not care much for literaturestudy [and] book knowledge’’ and that‘‘he created ‘his’ geometry from himself,from his exceptional intuition’’ (Bier-mann (1963, p. 31)), it must be aremarkable coincidence then that hiswork happens to contain a slew ofclassical theorems in every section, allset for Jacobi to go over and pen in thereferences.

To sum up, the caricature of Steineras unscholarly is often propagated butrarely, if ever, backed up by evidence.In fact, this view is so plainly incon-sistent with Steiner’s work that itsproponents need an elaborate andunsubstantiated ghostwriter theory toprotect it. Thus, I feel justified in notregarding this biographical material asdisproof of my thesis.

Finally, if I may venture a generalisa-tion, I think the factor of adherence tohistorical continuity is important forunderstanding 19th-century mathemat-ics beyond Steiner. I feel that the notionof what constitutes legitimate researchunderwentaquite radical transformationin the late 19th-century, to some extentmotivated mathematically but perhapsmost of all institutionally, promptedby a great increase in the number ofmathematicians, doctoral students, andpublication quantity and pace, a processinwhich historical continuitywas largelysacrificed. Indeed, nowadays popularmathematicians such as Riemann andCantor are celebrated for the revolu-tionary character of their work, whereasthe likes of Steiner are very much

Figure 43.


depreciated. Perhaps this is due to afailure to recognise the proper historicalsetting for these works and, in par-ticular, a lack of appreciation of theirvision of mathematics. I, for one, do notbelieve that the demise of Steiner andthat of historical continuity coincide byaccident.

REFERENCES

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

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

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der Projektivitat. Math. Ann., 90(1–2), 86–

102.

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

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

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biographische Skizze. Nova acta Leopol-

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Blasjo, 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’ecole Polytechnique, 17, 297–302. Eng-

lish excerpt in Smith (1959, pp. 331–336).

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12–22.

Coolidge, J. L. (1934). The Rise and Fall of

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Monthly, 41(4), 217–228.

Courant, R. & Robbins, H. (1941). What Is

Mathematics?. Oxford University Press,

New York.

Coxeter, H. S. M. (1993). The Real Projective

Plane. Springer-Verlag, New York, third

edition.

Geiser, C. F. (1874). Zur Erinnerung an Jakob

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chen Naturforschenden Gesellschaft, 56,

215–251.

Gergonne, J. (1825/26). Considerations philo-

sophiques sur les elements de la science

de l’etendue. Annales de mathematiques

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Graf, J. H. (1897). Der Mathematiker Jakob

Steiner von Utzenstorf. K. J. Wyss, Bern.

Gray, J. (2007). Worlds Out of Nothing: A

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(1907, p. 107).

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v. Eichhorn, 8 July 1845. In Jahnke

(1903a).

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Geometrie. B. G. Teubner, Leipzig.

Department of Philosophy,

Logic and Scientific Method

London School of Economics

London

United Kingdom



Mathematical Entertainments Michael Kleber and Ravi Vakil, Editors

The World’sTallestCrypticKEVIN WALD

This column is a place for those bits

of contagious mathematics that travel

from person to person in the

community, because they are so

elegant, surprising, or appealing that

one has an urge to pass them on.

Contributions are most welcome.

Please send all submissions to the

Mathematical Entertainments Editor,

Ravi Vakil, Stanford University,

Department of Mathematics, Bldg, 380,

Stanford, CA 94305-2125, USA


TTall, isn’t it?’’

‘‘It tries to be. Choose your entriesright, 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.’’

Across

2:

Lord Vader comprehends horrible woe,

not the correct way to read the final

answer to this puzzle (4, 2, 5)

11:

Emotionless jerk comes after mug andring (7)

13:

Note about comedienne Margaret’sacademic organization (6)

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73

89 90 91 92 93

94 95 96 97 98 99

100 101 102

103 104

105 106

107 108 109

110 111 112 113 114

115 116 117 118 119 120

121 122

123

124 125 126 127

128 129 130

131 132

133 134 135 136 137 138

139 140 141 142 143 144

0

1

2

...

...

...

...

...

...

...

...

...

...

...

...

...

...


14:

Our lady, in shackles, reveals the exact

length of the final answer to this puzzle

(4, 7)

16:

Revolutionary obtains submachinegun (4)

18:

Celebrated getting the German areaunited into a larger political entity (9)

19:

At last, that weirdo barber Sweeney (4)

21:

Ripe to tangle with a supernaturalbeing (4)

22:

A bunch of stitches look sound (4)

25 + 45n:

Semitic language lacks masculine wordmeaning ‘‘beer’’ (4)Look and sound of resentment (4)Noisily drops horseriding equipment(4)

26 + 45n:

The French gutlessly honor the Cow-ardly Lion’s portrayer (4)Fifty-one-foot elevator (4)Knocks box over (4)

28 + 45n:

Change caused by an effect of themoon spinning (4)Norse God in love with noise (4)Behead cod; otherwise, it stinks (4)

31 + 45n:

Network showing Cavemen, a cartoonabout 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 con-fusion (6, var. spelling)Druggie starting to snort printer ink(6)

33 + 45n:

Fired after getting old, misguided Danbombed (10)Hauled around a nob that’s tossedback bubbly (10)Vehicles containing Mr. Weasley, fruitdrink, 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 expres-sion 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 LoneRanger’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’sanimals (5)Eye part of over-trimmed veal (5)Perform numbers with one actressnamed Reed (5)

43 + 45n:

Humongous nude actor named McK-ellen (3)Attention: This is a serving of corn (3)Californian airport is negligent (3)

45 + 45n:

A mass of ice secondarily absorbs aten-millionth of a joule (4)Is in the wrong English train, son (4)Shamuses mentioned in issues of anMIT paper (4)

47 + 45n:

One unbelievably long time (3)Stimpy’s pal is almost torn apart (3)Where you permanently store data fora 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 ascripting language (4)Exhibit surprise by staring at the tail-biting snake (4)

56 + 45n:

Heard about Damon’s pad (3)Needlefish found in hangars (3)

Entertainment Editors’ Note

To the right sort of puzzle fan, January means one thing: the MITMystery Hunt. This annual event challenges teams of MIT students andenthusiasts from around the world to solve a succession of puzzles of allshapes, sizes, and styles, some never before seen. Answers to individualpuzzles somehow fit together to reveal the higher-order metapuzzles ateam must solve to win—earning them the right to run the hunt thefollowing year.

This puzzle is a cryptic crossword which appeared in the 2008 MysteryHunt, reproduced here with the kind permission of its constructor, KevinWald. The ‘‘final answer’’ to this puzzle is a single word, which may notbe 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’sonline 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 thesets of Across and Up clues. Even with these answers in hand, filling in thegrid and discovering the final answer poses a challenge. If all else fails, thepuzzles, solutions, and explanations for the entire 2008 MIT MysteryHunt can be found at http://web.mit.edu/puzzle/www/08/.

The Entertainments Editors welcome submissions of crosswords andother puzzles with similar appeal at all levels of accessibility.

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

cryptics1

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

http://web.mit.edu/puzzle/www/08/

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

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

57 + 45n:

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

58 + 45n:

Tail wild female singers (4)Legendary king of the Huns, at LongIsland (4)Each hurt! (4)Shaft put MDMA into beer (4)

59 + 45n:

Prefix on ‘‘form,’’ ‘‘phyll,’’ or ‘‘loch,’’oddly (6)Dance in a medical facility, wearing astring tie (6)

60 + 45n:

Variant lyre modified to serve thepurposes of a bard (11)Disturbed rest isn’t his motivation fordrinking (11)That senior’s remarkably husky vocalquality (11)

63 + 45n:

Made certain act involving $5 pastaperverse (11)Petey and Gertie Minuit will get drunkin 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 thinksthey run things (4, abbr.)Pat’s failing a high school exam (4, abbr.)

68 + 45n:

Weapon in a room (3)Seldom ignoring every odd characterwith 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)

6:

Organic compound a biblical bookdiscussed (5)

7:

Hears about flightless birds (5)

8:

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

9:

Socks a biblical prophet (5)

10:

Lump of uranium extracted from massof condensed vapor (4)

12:

As announced, bishopric had to yield (4)

15:

Three or four centers from the localfootball team (4)

17, 62 + 45n:

Italian wines damage dateless veggiedishes (8)Surveillance agents capturing onecriminal with dead body parts (8)Lab I rave about is not always the same(8)

18, 64 + 45n:

Spilled dirt from large books (6)Former Philippine president’s manu-script about an oil company (6)Mistakenly put Roman ‘‘I’’ in JohnWayne’s first name (6)

20, 66 + 45n:

Naiad frolicking with goddess (5)Weeping leaves Rory disheartened (5)Cooked tater and fish (5)

21, 67 + 45n:

Men’s sad, pathetic, irrationality (7)Ms. Zadora left fashionable streetmusician (7)Shamus turned down pierced Caryaglabra 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, tawdryCaltha palustris (5, 8)This part has no misdirected arrowsfired while retreating (8, 5)

25 + 45n:

Wild rice I got can be created by pro-karyotes (13)Cause excitment in Paris’s summerwith 999 (about a thousand) spacecraftcomponents (6, 7)

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 proteinsused in photosynthesis (13)

29 + 45n:

Legendary queen of Carthage finallyuttered wedding vow (4)A bird rendered extinct by party afterparty (4)Deer takes small amount of drugs (4)

30 + 45n:

Embarrassed after mostly unnecessaryvehicle is frequently punctured (6–7)Philosopher and seer can’t see Dr.Awkward (4, 9)Crusading group and mad doctorreunite (8, 5)

32 + 45n:

Allows a bit of discussion of this school(6)Powerful ditty about Roosevelt (6)

35 + 45n:

Mentioned eatery in an Indian city (5)Discourage retrospective about Sena-tor Kennedy (5)One who catches long fish U.S. Grant’sopponent tossed back (5)

36 + 45n:

Poorly written ‘‘if’’ ran further downthe page (5)Ruled by German and British Queen(5)

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) somer-saulting thesaurus writer (5)

48 + 45n:

No, I’m (gevalt!) the guy who playedSpock (5)


Bloom and Minderbinder’s Greekisland (5)Caroming truck follows MonsieurMarcel Marceau et al. (5)

50 + 45n:

Prize is a reversible tie (5)‘‘Flying Ur’’ is a former name for aflying 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 notrequiring a prescription (3, abbr.)Sphere a male sib spun (3)

53 + 45n:

Assemble like soldiers entirelyenthralled by Flipper (4, 2)Leaf is folded up to form a bra pad (6)

57 + 45n:

Reportedly obtain water from theFrench Quarter of a city (6)

Leave unfinished number with Faust’sauthor (6)Band featuring Matt Johnson andwildly 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)


Tallest CrypticClue Answers andBrief Explanations

In the explanations, ‘‘ana’’ = anagram,‘‘rev’’ = reversal, ‘‘hom’’ = homophone,and ‘‘double def’’ = double denfinition.

Across

2

DOWN TO EARTH (DARTH ‘‘compre-hends’’ 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 (PIQUEhom), 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 (INREAL ana), STONER (S[nort] +

TONER)

33 + 45n

CANNONADED (CANNED ‘‘after get-ting’’ O + DAN rev), CARBONATED(CARTED ‘‘around’’ A + NOB rev),CARRONADES (CARS ‘‘containing’’RON + ADE)

34 + 45n

ADS ([l]ADS), YEN (double def)

37 + 45n

EDH (HE’D ana), FEH (HEF rev), NTH(‘‘in’’ hyaciNTHs)

38 + 45n

EASERS (ERASES ana), ELDEST (LETSED 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 (OFrev + 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)

57 + 45n

TAM (TAM[e]), GAM (MAG rev), GAL(LAG rev)

58 + 45n

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

59 + 45n

CHLORO (OR LOCH ana), BOLERO(ER ‘‘wearing’’ BOLO)

60 + 45n

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

63 + 45n

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

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])

17, 62 + 45n

MARSALAS (MAR + SALADS - D),TOENAILS (TAILS ‘‘capturing’’ ONEana), VARIABLE (LAB I RAVE ana)

18, 64 + 45n

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

20, 66 + 45n

DIANA (NAIAD ana), TEARY (TEA +

R[or]Y), TETRA (TATER ana)

21, 67 + 45n

MADNESS (MEN’S SAD ana), PIANIST(PIA + IN rev + ST), PIGNUTS (PI +

STUNG rev)

23, 68 + 45n

A-ONE (‘‘from’’ mAO NEvertheless),EIRE (IRE hom)


24, 69 + 45n

MARSH MARIGOLD (MARS + HOLD‘‘gripping’’ A GRIM ana), PARTHIANSHOTS (THIS PART HAS NO ana)

25 + 45n

BACTERIOGENIC (RICE I GOT CANBE ana), ROCKET ENGINES (ROCK +

ETE + NINES ‘‘about’’ G), POCKETLIGHTER (PLIGHTER ‘‘has’’ COKE ana+ T)

27 + 45n

FEAST OR FAMINE (FE + ASTOR +

FAME ‘‘has’’ IN), HEARTBREAKING(THE RIGA BANKER ana), PLASTO-CYANINS (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’’NEE-DLES[s] + CAR),RENE DESCARTES(SEER CAN’T SEEDR ana), TEUTONIC ORDER (DOC-TOR REUNITE ana)

32 + 45n

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

35 + 45n

DELHI (DELI hom), DETER (RE TEDrev), EELER (R. E. LEE rev)

36 + 45n

INFRA (IF RAN ana), UNDER (UND +

ER)

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(doubledef), MIMES (SEMI rev ‘‘follows’’ M)

50 + 45n

AWARD (A + DRAW rev), USAIR (URIS A ana)

51 + 45n

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

52 + 45n

ARC (‘‘within’’ fARCe), OTC (TO rev +

C[odeine]), ORB (BRO rev)

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



The Searchfor Quasi-Periodicityin Islamic 5-foldOrnamentPETER R. CROMWELL

Introduction

TThe Penrose tilings are remarkable in that they arenon-periodic (have no translational symmetry) butare clearly organised. Their structure, called quasi-

periodicity, can be described in several ways, includingvia self-similar subdivision, tiles with matching rules, andprojection of a slice of a cubic lattice in R

5. The tilings arealso unusual for their many centres of local 5-fold and 10-fold rotational symmetry, features shared by some Islamicgeometric patterns. This resemblance has promptedcomparison, and has led some to see precursors of thePenrose tilings and even evidence of quasi-periodicity intraditional Islamic designs. Bonner [2] identified threestyles of self-similarity; Makovicky [20] was inspired todevelop new variants of the Penrose tiles and later, withcolleagues [24], overlaid Penrose-type tilings on traditionalMoorish designs; more recently, Lu and Steinhardt [17]observed the use of subdivision in traditional Islamicdesign systems and overlaid Penrose kites and darts onIranian designs. The latter article received widespreadexposure in the world’s press, although some of thecoverage overstated and misrepresented the actualfindings.

The desire to search for examples of quasi-periodicity intraditional Islamic patterns is understandable, but we musttake care not to project modern motivations and abstrac-tions into the past. An intuitive knowledge of group theoryis sometimes attributed to any culture that has producedrepeating patterns displaying a wide range of symmetrytypes, even though they had no abstract notion of a group.There are two fallacies to avoid:

• abstraction: P knew about X and X is an example of Ytherefore P knew Y.

• deduction: P knew X and X implies Y therefore P knewY.

In both cases, it is likely that P never thought of Y at all, andeven if he had, he need not have connected it with X.

In this article I shall describe a tiling-based method forconstructing Islamic geometric designs. With skill andingenuity, the basic technique can be varied and elaboratedin many ways, leading to a wide variety of complex andintricate designs. I shall also examine some traditionaldesigns that exhibit features comparable with quasi-periodictilings, use the underlying geometry to highlight similaritiesand differences, and assess the evidence for the presence ofquasi-periodicity in Islamic art.

A few comments on terminology. Many of the con-structions are based on tilings of the plane. A patch is asubset of a tiling that contains a finite number of tiles and ishomeomorphic to a disc. I use repeat unit as a generic termfor a template that is repeated using isometries to create apattern; it is not so specific as period parallelogram orfundamental domain. A design or tiling with radial sym-metry has a single centre of finite rotational symmetry. Theother terminology follows [8] for tilings, supplemented by[33] for substitution tilings.

Islamic Methods of ConstructionAlthough the principles of Islamic geometric design are notcomplicated, they are not well-known. Trying to recoverthe principles from finished artwork is difficult, as the most


conspicuous elements in a design are often not thecompositional elements used by the designer. Fortunatelymedieval documents that reveal some of the trade secretshave survived. The best of these documents is manuscriptscroll MS.H.1956 in the library of the Topkapi Palace,Istanbul. The scroll itself is a series of geometric figuresdrawn on individual pages, glued end to end to form acontinuous 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 morethan a pattern book as it shows construction lines. A half-size colour reproduction can be found in [25], which alsoincludes annotations to show the construction lines andmarks scored into the paper with a stylus, which are notvisible in the photographs. References in this article tonumbered panels of the Topkapi Scroll use the numberingin [25].

Islamic designs often include star motifs. These come ina variety of forms but, in this article, we need only a fewsimple shapes that correspond to the regular star polygonsof plane geometry. Taking n points equally spaced arounda circle and connecting points d intervals apart by straightlines produces the star polygon denoted by {n/d}. This,however, is the star of the mathematician; it is rare for anartist to use the whole figure as an ornamental motif. Moreoften, the middle segments of the sides are discarded.

Many of the early Islamic designs are created byarranging 6-, 8- or 12-point stars at the vertices of thestandard grids of squares or equilateral triangles. The moregeneral rhombic lattice allows other stars to be used. Anexample based on {10/3} is shown in Figure 1(a). Theangles in the rhombus are 72� and 108�, both being mul-tiples of 36�—the angle between adjacent spikes of the star.Draw a set of circles of equal radius centred on the verticesof the lattice and of maximal size so that there are points oftangency. Place copies of the star motif in the circles so thatspikes fall on the edges of the lattice. This controls thespacing and orientation of the principal motifs, but the

design is not yet complete. There are some spikes of eachmotif that are not connected to a neighbouring motif butare free and point into the residual spaces between thecircles. The lines bounding these free spikes are extendedbeyond the circumcircle until they meet similarly producedlines from nearby stars. This simple procedure bridges theresidual spaces and increases the connectivity of the starmotifs. The same pattern of interstitial filling should beapplied uniformly to all the residual spaces and the sym-metry of the design as a whole should be preserved as faras possible. The result is shown in Figure 1(c). In this casethe kites in the interstitial filling are congruent to those inthe star. This pattern is one of the most common decagonaldesigns, and we shall name it the ‘stars and kites’ pattern forreference.

This basic approach produces a limited range of peri-odic designs with small repeat units and it only works forstars with an even number of points. A more generalmethod that can be used with all stars, and also enablescombinations of different stars to be used in a single design,is based on edge-to-edge tilings containing regular convexpolygons with more than four sides. Figure 1(b) shows atiling formed by packing decagons together, leaving non-convex hexagonal tiles between them. After placing {10/3}stars in each decagon tile, we use the same kind of inter-stitial filling procedure as before to develop the pattern inthe hexagons.

This change from circle to polygon may seem minor, butit gives rise to a range of generalisations. We are no longerrestricted to a lattice arrangement of the stars—any tilingwill suffice. The tiling may contain regular polygons ofdifferent kinds allowing different star motifs to be com-bined in the same design; the tiling naturally determines therelative sizes of the different stars. We can even discard theregular star motifs that initiate the interstitial filling and seedthe pattern generation process from the tiling itself. In thislast case, we place a pair of short lines in an X configurationat the midpoint of each edge, then extend them until theyencounter other such lines—this is similar to applyinginterstitial filling to every tile. The angle that the lines makewith the edges of the tiling, the incidence angle, is aparameter to be set by the artist and it usually takes thesame value at all edges. There is no requirement to termi-nate the line extensions at the first point of intersection; ifthere are still large empty regions in the design, or it isotherwise unattractive, the lines can be continued until newintersections arise.

This technique, known as ‘polygons in contact’ (PIC),was first described in the West by Hankin [9–13], whoobserved the polygonal networks scratched into the plasterof some designs, while working in India. Many panels inthe Topkapi Scroll also show a design superimposed on itsunderlying polygonal network. Although the purpose ofthe networks is not documented, it does not seem unrea-sonable to interpret them as construction lines. Bonner [2,3] argues that PIC is the only system for which there isevidence of historical use by designers throughout theIslamic world. The method is versatile and can account fora wide range of traditional patterns, but it is not universallyapplicable. An alternative approach is used by Castera [5],

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

AU

TH

OR PETER CROMWELL 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 algo-

rithmic problems. He was recently awarded

a research fellowship by the Leverhulme

Trust to work on the mathematical analysis

of interlaced patterns.

Pure Mathematics Division,

Mathematical Sciences Building

University of Liverpool, Peach Street

Liverpool L69 7ZL

England



who arranges the shapes seen in the final design withoutusing a hidden grid.

The PIC method is illustrated in the next four figures.Figure 2 shows two designs produced from a tiling byregular decagons, regular pentagons, and irregular convexhexagons. In part (b), a star motif based on {10/4} is placedin the decagon tiles, which gives an incidence angle of 72�for the other edges; the completed design is one of themost widespread and frequently used of all star patterns.Part (c) shows a design that is common in Central Asia andbased on {10/3} with an incidence angle of 54�. A {10/2} starand an incidence angle of 36� reproduces the stars andkites pattern. The design in Figure 3 is from [14] and con-tains star motifs based on {7/3}; in the tiling the 7-gons areregular but the pentagons are not. Figure 4 is based on a

tiling containing regular 9-gons and 12-gons. I have chosenan incidence angle of 55� to make the convex 12-gon ele-ments in the design into regular polygons and some linesegments inside the non-convex hexagonal tiles join upwithout creating a corner, but, as a consequence, neitherstar motif is geometrically regular. Plates 120–122 in [4] aretraditional designs based on the same tiling. Figure 5 showsa design with 10-fold rotational symmetry based on panel90a of the Topkapi Scroll, which Necipoglu labels as adesign for a dome [25]. The original panel shows a templatefor the figure containing one-tenth of the pattern with thedesign in solid black lines superimposed on the tilingdrawn in red dotted lines. Notice that some of the tiles aretwo-tenths and three-tenths sectors of a decagon. Domeswere also decorated by applying PIC to polyhedral

(a) (b)

(c)

Figure 1. The ‘stars and kites’ pattern.


networks. Patterns with a lower concentration of stars wereproduced by applying PIC to k-uniform tilings composed ofregular 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 commonIslamic motif. In each design, a set of hexagons sur-rounding a star has been highlighted in grey. Theenlarged star motif is called a rose and the additionalhexagons are its petals. In this case, the rose arisesbecause the decagon in the underlying tiling is sur-rounded by equilateral polygons, but they can also beconstructed using a set of tangent circles around the cir-cumcircle of the star [16] and used as compositionalelements in their own right.

You can see the PIC method in action and design yourown star patterns using Kaplan’s online Java applet [34]—you select a tiling and the incidence angles of the starmotifs, then inference logic supplies the interstitial pattern.

The tilings used as the underlying networks for the PICmethod of construction often have a high degree of sym-metry, and they induce orderly designs. Islamic artists alsoproduced designs that appear to have a more chaoticarrangement of elements with local order on a small scalebut 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 fromcopies of a small set of equilateral tiles (see Figure 6)whose angles are multiples of 36�:

(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 motifin each pattern are highlighted.


• 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 methodwith an incidence angle of 54�. The barrel hexagon and the

decagon have two forms of decoration. One decagon motifis just the star {10/3} and its constituent kites are congruentto those on the bow-tie; the other decagon motif is morecomplex and the symmetry is reduced from 10-fold to5-fold rotation.

The shapes of the tiles arise naturally when one tries totile with decagons and pentagons. The bow-tie and barrelhexagons are familiar from the previous figures. Theoctagon and the remaining hexagon are shapes that can beobtained as the intersection of two overlapping decagons.The motif on the hexagon resembles a spindle or bobbinwound with yarn. This distinctive motif is easy to locate in a

(a) (b)

Figure 3. Design containing regular 7-point stars.

(a) (b)

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


design, and its presence is a good indication that the designcould be constructed from these tiles.

The promotion of irregular tiles from supplementaryshapes to compositional elements in their own rightmarked a significant development in Islamic design.Regarding the tiles as the pieces of a jigsaw allows a lessformal approach to composition. A design can be grownorganically in an unplanned manner by continuallyattaching tiles to the boundary of a patch with a free choiceamong the possible extensions at each step. This newapproach gave artists freedom and flexibility to assemblethe tiles in novel ways and led to a new category of designs.It seems to have been a Seljuk innovation as examplesstarted to appear in Turkey and Iran in the 12th–13thcenturies. The widespread and consistent use of thesedecorated tiles as a design system was recognised by Luand 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 oftenmultiple solutions to fill a given area. Even in the simple

combination of a bobbin with a bow-tie shown in part (a),the positions of the tiles can be reflected in a vertical line sothat the bow-tie sits top-right instead of top-left. The patchin part (d) can replace any decagonal tile with a conse-quent loss of symmetry, as the bow-tie can point in any often directions. Patches (b) and (c) are another pair ofinterchangeable fillings with a difference in symmetry.

Figure 8 shows some traditional designs made from thetiles. Parts (a) and (b) are from panels 50 and 62 of theTopkapi Scroll, respectively; in both cases the originalpanel shows a template with the design in solid black linessuperimposed on the tiling drawn in red dots. The designsin 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 toshow the underlying structure of the designs, but in thefinished product these construction lines would be erasedto leave only the interlaced ribbons. This conceals theunderlying framework and helps to protect the artist’smethod. The viewer sees the polygons of the backgroundoutlined by the ribbons, but these are artifacts of the con-struction, not the principal motifs used for composition.

The internal angles in the corners of the tiles are allmultiples of 36� so all the edges in a tiling will point in oneof five directions—they will all lie parallel to the sides of apentagon. Fitting the tiles together spontaneously producesregular pentagons in the background of the interlacing, andcentres of local 5-fold or 10-fold rotational symmetry in thedesign. This symmetry can be seen in some of the config-urations of Figure 7. However, in patterns generated bytranslation of a template, this symmetry must break downand cannot hold for the design as a whole. This is a con-sequence of the crystallographic restriction: the rotationcentres in a periodic pattern can only be 2-, 3-, 4- or 6-fold.This was not proved rigorously until the 19th century but itmust surely have been understood on an intuitive level bythe Islamic pattern makers. Perhaps these tilings wereappealing precisely because they contain so many forbid-den centres; they give the illusion that one can break freefrom this law of nature. Unfortunately, when a largeenough section of a tiling is shown for the periodicity to beapparent, any (global) rotation centres are only 2-fold, and

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

Rhombus Pentagon Barrel (1) Octagon

Bow-tie Decagon (1)

Barrel (2)

Bobbin Decagon (2)

Figure 6. An Islamic set of prototiles.


the symmetry type of the (undecorated) tiling is usually oneof pgg, pmm or, more commonly, cmm.

Figure 9(b) shows the design on one wall of theGunbad-i Kabud (Blue Tower) in Maragha, north-westIran; similar designs decorate the other sides of the tower.At first sight the design appears to lack an overallorganising principle but it fits easily into the frameworkshown in Figure 9(a). Centred at the bottom-right cornerof the panel is the patch of Figure 7(g) surrounded by aring of decagons. A similar arrangement placed at the top-left corner abuts the first, leaving star-shaped gaps. Therings of decagons are filled with the patch of Figure 7(d)with the bow-ties facing outwards, except for the one onthe bottom edge of the panel, which is filled with a

decagonal tile. The star-shaped gaps are filled with thefive rhombi of Figure 7(b). The design does containirregularities and deviations from this basic plan, particu-larly in the bottom-left corner of the panel. Also thedecagon 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 thedesign shown in Figure 10. The centres of the rose motifs inthe centre of the figure and in the top-left corner are diag-onally opposite corners of a rectangle that is a repeat unit forthe design. The underlying framework in this rectangle isthe same as that of the Maragha panel. The full design isgenerated from this cell by reflection in the sides of therectangle. Note that it is the arrangement of the tiles that is

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 7. Small patches of tiles.


(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 8. Periodic designs.

(a) (b)

Figure 9. Design from the Gunbad-i Kabud, Maragha, Iran.


reflected, not the tiles with their decorative motifs; theinterlacing of the full design remains alternating. Theboundaries of the unit rectangle are mostly covered by thesides of tiles or mirror lines of tiles, both of which ensurecontinuity of the tiling across the joins. However, in the top-right and bottom-left corners (the cell has 2-fold rotationalsymmetry about its centre), the tiles do not fit in the rect-angle but overhang the edges. This is not a problem withthis method of generating designs: the overhanging tiles aresimply cut to fit and the reflections take care of the conti-nuity of the ribbons. In Figure 10 this is most obvious in themiddle near the bottom where pairs of bow-ties and bob-bins merge. The centre of the tiling can be filled with the

patch shown in Figure 7(g) but this has been discarded infavour of a large rose motif. A different construction for thispattern is presented by Rigby in [26].

When experimenting with the tiles of Figure 6, one soonlearns that those in the top row are more awkward to usethan the others—the 108� angles must occur in pairsaround a vertex and this limits the options. Indeed manydesigns avoid these tiles altogether and are based solelyon the three shapes in the bottom row. The design inFigure 11 is unusual in that it is largely composed ofawkward tiles (rhombi, pentagons, and octagons) togetherwith a few bobbins. The large star-shaped regions in thetiling can be filled with the patch shown in Figure 7(f),

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


continuing the use of the same set of tiles, but instead thismotif is replaced by the star {10/4}.

Once a design has been constructed, it can be finishedin different ways according to context and the materialsused. In some of the accompanying figures, the regionshave been given a proper 2-colouring (chessboard shad-ing), in others the lines have been made into interlacedribbons. The basic line drawing can also be used by itself aswhen it is inscribed in plaster.

What is Quasi-Periodicity?The discovery of crystalline metal alloys with 5-fold sym-metry in their diffraction patterns caused great excitementin the 1980s. Sharp spots in a diffraction pattern are evi-dence of long-range order which, at that time, wassynonymous with periodicity, but 5-fold rotations areincompatible with the crystallographic restriction so a newkind of phenomenon had been observed. The novel solids

became known as quasi-crystals and the underlying orderas quasi-periodicity. For crystallographers, the productionof sharply defined points in a diffraction pattern is adefining characteristic of quasi-periodicity. In the study ofthe decorative arts, however, the term ‘quasi-periodic’ isused somewhat informally and does not have an agreeddefinition. Readers should be aware of this potential sourceof confusion when comparing papers. For the tilings andthe related geometric designs discussed in this article, oneoption is to impose a homogeneity condition on the dis-tribution of local configurations of tiles (this is weaker thanthe crystallographic definition). This and other propertieswill be illustrated through the following example.

The example is constructed from the patches shown inFigure 12. The patches are chosen only to demonstrate thetechnique and not for any artistic merit—the unbalanceddistribution of bow-ties leads to poor designs. Any patchtiled by bow-ties, bobbins, and decagons can be converted

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

(a) (b) (c)

Figure 12. Subdivisions of three tiles into smaller copies of the same three tiles.The scale factor is 1

2 7þffiffiffi5p� �

� 4:618.


into a larger such patch by subdividing each tile as shownin the figure and then scaling the result to enlarge the smalltiles to the size of the originals. This process of ‘subdivideand enlarge’ is called inflation. Each side of each com-posite tile is formed from two sides of small tiles and themajor diagonal of a small bobbin; in the inflated tiling thehalf-bobbins pair up to form complete tiles.

Let P0 be a single decagon and let Pi+1 be the patchobtained by inflating Pi for all i 2 N . Figure 12(b) shows P1

and Figure 13 shows P2. We can iterate the inflation processto tile arbitrarily large regions of the plane. Furthermore,because P1 contains a decagon in the centre, Pi+1 contains acopy 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 atiling, P?, of the whole plane. Notice that the symmetry ofthe initial patch is preserved during inflation so P? will havea global centre of 10-fold symmetry and hence cannot beperiodic.

In general, inflation only provides the ability to createarbitrarily large patches that need not be concentric, sosome work is required to show that the limit exists and it isa tiling of the plane [19]. Two tilings are said to be locallyindistinguishable if a copy of any patch from one tilingoccurs in the other tiling, and vice versa. The family ofsubstitution tilings defined by the prototiles and subdivi-sions shown in Figure 12 is the set of all tilings that arelocally indistinguishable from P?. There are, in fact, an

uncountable number of tilings in the family but any patchin any one of them will be contained in some Pn.

The basic combinatorial properties of a substitution til-ing based on a finite set of n prototiles T1,...,Tn can beencoded in an n 9 n matrix: the entry in column j of row iis the number of small Ti in a composite Tj. For the examplehere with the tiles in the order bow-tie, bobbin, decagon,this substitution matrix is

10 5 207 11 250 2 11

!:

A matrix is said to be primitive if some power of it has onlypositive non-zero entries. If a substitution matrix is primi-tive then the patch of tiles produced by repeated inflationof any tile will eventually contain copies of all the proto-tiles. Properties of the tiling can be derived from thealgebraic properties of a primitive matrix. For example,the largest eigenvalue is the square of the scale factor of theinflation and the corresponding eigenvector containsthe relative frequencies of the prototiles in a full tiling of theplane; the corresponding eigenvector of the transposedmatrix contains the relative areas of the three proto-tiles. In our example the frequency eigenvector is5þ 5

ffiffiffi5p

; 5þ 7ffiffiffi5p

; 4� �

. Since some of the ratios betweenthe entries are irrational, any substitution tiling made fromthese subdivisions is non-periodic [30, 31].

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


Although our substitution tilings have no translationalsymmetry, they do share some properties with periodictilings. First each tiling is edge-to-edge; it is constructedfrom a finite number of shapes of tile, each of which occursin a finite number of orientations; there are finitely manyways to surround a vertex. The tiling is said to have finitelocal complexity. For primitive substitution tilings this has animportant consequence: given any patch X in the tiling thereis some number R such that a disc of radius R placed any-where on the tiling will contain a copy of X. A tiling with thisproperty is called repetitive. This means that copies of anyfinite portion of the tiling can be found evenly distributedthroughout the tiling. You cannot determine which part ofthe tiling is shown in any finite diagram of it.

For the purposes of this article, a tiling is called quasi-periodic if it is non-periodic, has finite local complexity,and is repetitive. By extension we can call an Islamic designconstructed using the PIC method quasi-periodic if itsunderlying polygonal network is a quasi-periodic tiling.Unfortunately, it is impossible to tell from any finite subsetof a tiling whether it is quasi-periodic or not. So, in order toassert that a tiling could be quasi-periodic, we need toidentify a process such as inflation that could have beenused to generate the piece shown and can also be used togenerate a complete quasi-periodic tiling.

Multi-level DesignsSome panels of the Topkapi Scroll show designs of differ-ent scales superimposed on one another. This interplay ofdesigns on multiple scales is a feature of some large Islamicdesigns found on buildings where viewers experience asuccession of patterns as they approach. From a distance,

large-scale forms with high contrast dominate but, closer in,these become too large to perceive and smaller forms takeover. Early methods to achieve this transition from big andbold through medium range to fine and delicate weresimple, often just a matter of progressively filling voids inthe background to leave a design with no vacant spaces.(There is a secondary pattern of this form on the Gunbad-iKabud.) Differences in size and level of detail wereexpressed using variation in density, depth of carving,colour and texture. Later designs are more ambitious anduse the same style on more than one scale. It is evenpossible to re-use the same pattern.

Designs that can be read on several scales are oftenreferred to as self-similar but this term itself has multiplelevels of meaning. In its strictest sense it means scaleinvariant: there is a similarity transformation (an isometryfollowed by an enlargement) that maps the design ontoitself. The transformation can be weakened to a topologicalequivalence—for example the homeomorphisms in iter-ated function systems leading to fractals. In a weaker sensestill, it means only that motifs of different scales resembleeach other in style or composition but are not replicas. Weshall use the term hierarchical for multi-level designs ofthis latter form.

In panel 28 of the Topkapi Scroll three drawings aresuperimposed on the same figure: a small-scale polygonalnetwork is drawn in red dots, the corresponding small-scale design is drawn in a solid black line, and a large-scaledesign is added in a solid red line. The polygonal networkcorresponding to the large-scale design is not shown butcan be deduced—the two polygonal networks are shownsuperimposed in Figure 14(a). The other parts of the figure

(a) Panel 28 (b) Panel 31

(c) Panel 32 (d) Panel 34

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


show the polygonal networks underlying three more 2-level designs from the scroll, but neither of the networks isshown in these panels, only the finished 2-level designs inblack and red.

Superimposing the large- and small-scale polygonalnetworks of these panels reveals subdivisions of some ofthe tiles: a rhombus in panel 28, two pentagons in panel 32,and a bobbin in panel 34. In all cases, the side of a com-posite tile is formed from the sides of two small tiles andthe diagonal of a small decagon. We can also identify thefragments of the large-scale polygons cropped by theboundaries of the panels. These panels are not arbitrarilychosen parts of a design—they are templates to be repe-ated by reflection in the sides of the boundary rectangle.Although a superficial glance at Figure 14(d) might suggestthat the large-scale network is a bobbin surrounded by sixpentagons, a configuration that can be seen in the small-scale network, reflection in the sides generates rhombi,pentagons, and barrels. The large-scale design generatedby panel 31 is shown in Figure 8(g). Panel 28 appears to betruncated on the right and is perhaps limited by the avail-able space. If it had 2-fold rotational symmetry about thecentre of the large rhombus, the large-scale design wouldbe that of Figure 8(h). A consistent choice of subdivisionemerges in all four panels and the subdivisions of the five

tiles used are shown in Figure 15. I believe this has notbeen reported before.

Figure 16 shows my 2-level design based on panel 32.The composite tiles generate the large-scale design (shownin grey) and the small tiles generate a small-scale design(black and white) that fills its background regions. Thebarrel tile has two forms of decoration: I have used thesimple motif for the large-scale design and the other motifon the small-scale design. Completing the small-scaledesign in the centre of a composite pentagon is problem-atic. For a pentagon of this scale, only a partial subdivisionis possible: once the half-decagons have been placed,one is forced to put pentagons at the corners; only apentagon or a barrel can be adjacent to the corner penta-gons, and both cases lead to small areas that cannot betiled. The grey area in Figure 15(b) indicates one suchessential hole. I have chosen a slightly different filling fromthe one in the Topkapi Scroll. The large-scale design is thatof Figure 2(c).

Figure 17 gives a similar treatment to panel 34. It con-tains four copies of the template rectangle shown inFigure 14(d), two direct and two mirror images. In thiscase, the large-scale pattern is expressed using shading ofthe regions. Examples of both styles can be found onbuildings in Isfahan, Iran.

(a)

(b)

(c)

(d)

(e)

Figure 15. Subdivisions derived from the Topkapi Scroll. The scale factor is3þ

ffiffiffi5p� 5:236.


The bow-tie is notable by its absence from Figure 15. Itsuffers the same fate as the pentagon: the tiles at its twoends are forced and its waist cannot be tiled. (The large-scale polygonal network underlying panel 29 of the scrollhas a quarter of a bow-tie in the top right corner sur-rounded by pieces of decagons, but it is not based onsubdivision in the same way as the others.)

In Figure 16 the visible section of the large-scale designcan also be found as a configuration in the small-scaledesign. However, larger sections reveal that the pattern isnot scale invariant. This is a general limitation of thesesubdivisions. It is not possible to use the subdivisions ofFigure 15 as the basis of a substitution tiling because,

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

A Design from the AlhambraThe design illustrated in Figure 18 forms the major part of alarge panel in the Museum of the Alhambra—see [24] for aphotograph. The panel has been assembled from fragmentsuncovered in 1958, but the original would have been fromthe 14th century. The lower part of the figure shows thefinished design and the upper part shows a polygonalnetwork that I propose as the underlying framework. Theprincipal compositional element of the framework is adecagon surrounded by ten pentagons, which gives rise to

Figure 16. A 2-level design based on panel 32 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 thetop left of the figure—an inner ring of ten and an outer ringof twenty; adjacent elements share two pentagons. Theconnections between the inner and outer rings are of twokinds. The shaded rhombi contain the translation unit fromthe familiar periodic design of Figure 2(b). The construc-tion of the design in the remaining spaces is shown inFigure 19: in part (b) the design is seen to be a subset of theconfiguration of pentagonal motifs of part (a), whereas (c)shows the same design over a network that includes half-barrels and one-tenth decagons—the polygons used inFigure 18. The edges in the resulting polygonal networkare of two lengths, which are related as the side anddiagonal of a pentagonal tile. The final design can begenerated from this network using a generalisation of thePIC method: the short edges have incidence angle 72� andthe long edges have incidence angle 36�. A 20-fold rose isplaced in the centre; the tips of alternate petals meet 10-fold roses, and lines forming the tips of the intermediatepetals are extended until they meet other lines in the pat-tern. The reconstructed rectangular panel also hasquadrants of 20-fold roses placed in the four corners, acommon feature of such panels that reflects the fact thatmost are subsets of periodic patterns. However, the quad-rants are misaligned and are also the most heavily restoredareas of the panel. I have omitted them from the figure.

This design is unusual in the large number of straightlines it contains that run across the figure almost uninter-rupted. The marks in the bottom right corner of Figure 18indicate the heights of horizontal lines; there are five fam-ilies of parallels separated by angles of 36�. In some quasi-periodic tilings it is possible to decorate the prototiles withline segments that join up across the edges of the tiling toproduce a grid of continuous straight lines that extend overthe whole plane. These lines are called Ammann bars. Theintervals between consecutive parallel Ammann bars comein two sizes, traditionally denoted by S and L (short andlong). They form an irregular sequence that does notrepeat itself and never contains two adjacent Ss or threeadjacent Ls.

The lines in Figure 18 are not genuine Amman bars.Those marked with an asterisk do not align properly acrossthe full width of the piece shown but deviate so that the Sand L intervals switch sides. (Structural defects of this kindhave been observed in quasicrystals, where they areknown as phasons). The periodic design in Figure 2(b) hassimilar lines but its sequences repeat: the vertical ‘Ammannbars’ give sequence SLSL, the lines 36� from verticalgive SLLSLL, and those 72� from vertical are not properlyaligned.

Makovicky et al [24] propose Figure 18 as an exampleof a quasi-periodic design. They try to find a structuralconnection between it and the cartwheel element of the

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


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

[20] as he studied the Maragha pattern shown inFigure 9(b). Their boldest assertion is Conclusion 6[24, p. 125]:

The non-periodic cartwheel decagonal pattern from theexcavations in the Alhambra and from the Moroccanlocalities is based on a modified Penrose non-periodictiling derived recently as ‘PM1 tiling’ by Makovicky…We conclude that a symmetrized PM1-like variety ofPenrose tiling must have been known to the Merinid andNasrid artesans (mathematicians) and was undoubtedlycontained in their more advanced pattern collections.

Elsewhere in the paper, the authors are more cautious andrealistic about the nature of their speculation. They offer analternative construction based on an underlying radiallysymmetric network of rhombi whose vertices lie in thecentres of the decagonal tiles [24, Fig. 23].

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

(a) (b) (c)

Figure 19.


In order to classify a pattern as periodic or radiallysymmetric, we must have a large enough sample to be ableto identify a template and the rules for its repetition. Sim-ilarly, to classify a pattern as quasi-periodic, we mustdescribe a constructive process that allows us to see thegiven patch as part of a quasi-periodic structure coveringthe whole plane. It is not sufficient that geometric featuresof a design, such as rotation centres, can be shown to alignwith those of a familiar quasi-periodic tiling within a finitefragment. We need to find a procedure built on elements ofthe design. The set of tiles underlying Figure 18 and the setP2 shown in Figure 13 are both large patches with 10-foldsymmetry, but only in the second case do we know how toextend it quasi-periodically.

In my opinion, the design strategy underlying theAlhambra pattern does not require an understanding ofPenrose-type tilings, and is based on little more than thedesire to place large symmetric motifs (roses) in a radiallysymmetric pattern and fill the gaps. The construction out-lined at the start of this section produces the completedesign using methods and motifs believed to have beenused by Islamic artists. The general structure has the samefeel as Figure 5. The ‘Ammann bars’ are an artifact of theconstruction, although the structure of the design may haveevolved and been selected to enhance their effect. Theywould also have helped to maintain accurate alignment ofelements during its construction.

Designs from IsfahanFigure 20 shows a 2-level design that, like the TopkapiScroll examples above, is based on subdivision. The large-scale design is the stars and kites pattern derived from thebow-tie and decagon tiling of Figure 1(b). The subdivisionsof the bow-tie and decagon used to generate the small-scale design are shown in Figures 21(a) and (c) with

the large-scale pattern added in grey. The side of acomposite tile is formed from the diagonals of two bob-bins and one decagon. The pattern cannot be scaleinvariant: the polygonal network for the large-scale designcontains a bow-tie surrounded by four decagons but thislocal arrangement does not occur in the small-scalenetwork.

These subdivisions were derived by Lu and Steinhardt[17] from three hierarchical designs found on buildings inIsfahan. The grey areas in Figure 22 mark out the sectionsof the large-scale polygonal network underlying thesedesigns: the rectangular strip runs around the inside of aportal in the Friday Mosque, the triangular section is one ofa pair of mirror-image spandrels from the Darb-i Imam(shrine of the Imams), and the arch is a tympanum from aportal, also from the Darb-i Imam—see [17, 35] for photo-graphs. Bonner [2] gives an alternative subdivision schemefor the Darb-i Imam arch using the tiling of Figure 2(a) asthe basis for the large-scale design.

The mosaic in the Darb-i Imam tympanum differs fromthe symmetrically perfect construction of Figure 20 in sev-eral places. For example a bow-tie/bobbin combinationlike Figure 7(a) in the top right corner of the central com-posite bow-tie is flipped; bow-tie/bobbin combinations inthe corners of the upper composite decagon are also flip-ped; a decagon at the lower end of the curved section ofthe boundary on each side is replaced by Figure 7(d). Themodifications to the composite decagon appear to bedeliberate as the same change is applied uniformly in allcorners. Replacing the small decagons may make it easierto fit the mosaic into its alcove. The bow-tie anomaly ispossibly a mistake by the craftsman.

If we want to use the Isfahan subdivisions as the basis ofa substitution tiling, we need to construct a companionsubdivision of the bobbin tile. In doing so we should

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


emulate the characteristics of the two samples—propertiessuch as the mirror symmetry of the subdivisions, and thepositions of the tiles in relation to the grey lines. Notice thatfocal points such as corners or intersections of the greylines are always located in the centres of decagons, and theinterconnecting paths pass lengthwise through bow-ties.Figure 21(b) shows my solution: it satisfies some of thesecriteria, but it is spoilt by the fact that some of the corners ofthe grey lines are so close together that decagons centredon them overlap, and there is a conflict between runningthe path through a bow-tie and achieving mirror symmetry

at the two extremes. This extra subdivision enables theinflation process to be performed, but the resulting tilingsare probably of mathematical interest only. The large scalefactor for the subdivisions yields a correspondingly largegrowth rate for the inflation. After two inflations of adecagon the patch would contain about 15000 tiles; forcomparison, the patch shown in Figure 13 contains about1500 tiles.

Lu and Steinhardt use the Isfahan patterns in their dis-cussion of quasi-periodicity. Commenting on the spandrel,they say [17, p. 1108]:

(a)

(b)

(c)

Figure 21. Subdivisions (a) and (c) are derived from designs on buildings inIsfahan [17]. The scale factor is 4þ 2

ffiffiffi5p� 8:472.


The Darb-i Imam tessellation is not embedded in aperiodic framework and can, in principle, be extendedinto an infinite quasiperiodic pattern.By this they mean that the visible fragment of the large-

scale design is small enough that no translational symmetryis immediately apparent and so the patch could be part of anon-periodic tiling. If we only have access to a finite pieceof any tiling, it is impossible to decide whether it is periodicwithout further information on its local or global structure.Although the lack of conspicuous periodicity in the Darb-iImam design could be interpreted as a calculated display ofambiguity on the part of the artist, to me it seems morelikely to be the result of choices influenced by aestheticqualities of the design, and the relative sizes of the tesseraein the small-scale pattern and the area to be filled. The factthat the same periodic tiling is a basis for all three Isfahandesigns makes it a good candidate for the underlyingorganising principle. Translation in one direction is visiblein the Friday Mosque pattern.

Lu and Steinhardt also observe that the medieval artistsdid not subdivide a single large tile but instead used a patchcontaining a few large tiles arranged in a configuration thatdoes not appear in the small-scale network. They thenremark [17, p. 1108]:

This arbitrary and unnecessary choice means that,strictly speaking, the tiling is not self-similar, althoughrepeated application of the subdivision rule wouldnonetheless lead to [a non-periodic tiling].

This gives the impression that, if the medieval craftsmenhad wanted to, they could have started with a single tileand inflated it until it covered the available space. But wemust beware of seeing modern abstractions in earlier work.There is no evidence that medieval craftsmen understoodthe process of inflation. The mosaics require only one levelof subdivision, and they do not contain a subdivision of thebobbin that would be needed to iterate the inflation.

In my opinion the Isfahan patterns, like the 2-leveldesigns in the Topkapi Scroll, are best explained as anapplication of subdivision to generate a small-scale fillingof a periodic large-scale design. Furthermore, the choice ofthe large-scale design seems far from arbitrary: it is one ofthe oldest and most ubiquitous decagonal star patterns, and

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

Connections with Penrose TilingsThe use of subdivision and inflation to produce quasi-periodic tilings with forbidden rotation centres came toprominence in the 1970s with investigations following thediscovery of small aperiodic sets of tiles, the Penrose kiteand dart being the most famous example. Penrose tilingshave local 5-fold and 10-fold rotation centres and the factthat some Islamic designs share these unusual symmetryproperties has prompted several people to explore theconnections between the two [1, 17, 20, 24, 27].

Figure 23 shows subdivisions of the kite and dart intothe bow-tie, bobbin, and decagon tiles. As in earlierexamples, the sides of the kite and dart lie on mirror linesof the tiles. Using this substitution, any Penrose tiling canbe converted into a design in the Islamic style [27]. Fur-thermore, 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 intokites and darts. Two of the patches are familiar to studentsof Penrose tilings: (a) is the long bow-tie component ofConway worms and (b) is the hub of the cartwheel tiling.Notice also that (b) is assembled from (a) and (c) in themanner of Figure 7(d).

Kites and darts come with matching rules to prohibit theconstruction of periodic tilings when the tiles are assem-bled like a jigsaw. In Figure 24 the two corners at the‘wings’ of each dart and the two corners on the mirror lineof each kite are decorated with grey sectors; the matchingrule is that grey corners may only be placed next to othergrey corners. This prevents, for example, the bow-tie andthe decagon in the figure from being assembled in the starsand kites pattern: it is not possible to place two bow-ties onopposite corners of a decagon.

The markings on the kites and darts in Figure 24 endowthe composite tiles with a matching rule of their own. Eachside of a composite tile has a single grey spot that dividesits length in the golden ratio; we decorate each side with anarrow pointing towards the short section. Instead ofdefining the matching rule at the vertices of the tiling, as

Figure 22. Sections of the bow-tie and decagon tilingused in the Isfahan patterns.

Figure 23. Subdivisions of the Penrose kite and dart.


with the Penrose example previously described, we placeconstraints on the edges of the tiling: the arrows on the twosides forming an edge of the tiling must point in the samedirection. With these markings and matching rule, the bow-tie and bobbin are an aperiodic set. To prove this note thatthe subdivisions in Figure 25 show that we can tile theplane by inflation, and that any periodic tiling by bow-tiesand bobbins could be converted into a periodic tiling bykites and darts but this is impossible. The substitutionmatrix for these marked tiles is associated with the Fibo-nacci sequence and the ratio of bobbins to bow-ties in asubstitution tiling is the golden ratio. Notice that a hori-zontal line running through the centre of a compositebow-tie passes lengthwise through the small bow-ties andshort-ways across the small bobbins. Inflation produces alonger line with the same properties, and a substitutiontiling will contain arbitrarily long such lines. Any infinitelines must be parallel as they cannot cross each other.These lines inherit their own 1-dimensional substitutionrule.

ConclusionsIn the preceding sections I have described methods forconstructing Islamic geometric patterns, given a briefintroduction to the modern mathematics of substitutiontilings, and analysed some traditional Islamic designs. Theconclusions I reached during the course of the discussionare isolated and summarised here:

1. It is possible to construct quasi-periodic tilings from theset of prototiles used by Islamic artists (Figure 6).Examples can be generated as substitution tilings based

on inflation or using a matching rule with markedversions of the tiles.

2. Islamic artists did use subdivision to produce hierarchi-cal designs. There are examples illustrating the methodin the Topkapi Scroll, and three designs on buildings inIsfahan can be explained using this technique. Indeed,their prototiles are remarkable in their capacity to formsubdivisions of themselves in so many ways.

3. There is no evidence that the Islamic artists iterated thesubdivision process—all the designs I am aware of haveonly two levels. This is to some degree a practicalissue: the scale factor between the small-scale andlarge-scale designs is usually large and the area of thedesign comparatively small. With the subdivisions usedin the Topkapi Scroll, iteration is impossible as compositeversions of the pentagon and bow-tie do not exist.

4. There is no evidence that the Islamic artists used matchingrules. Ammann bars are the nearest thing to a form ofdecoration that could have been used to enforce non-periodicity. Similar lines that appear on some designs area by-product of the construction, not an input to thedesign process, although the designs may have beenselected because this feature was found attractive.

5. The designs analysed in this article do not provideevidence that Islamic artists were aware of a process thatcan produce quasi-periodic designs. They are periodic,generated by reflections in the sides of a rectangle, orare large designs with radial symmetry. The multi-leveldesigns are hierarchical, not scale invariant.

In this article I have concentrated on designs with local5-fold symmetry. In Spain and Morocco there are analo-gous designs with local 8-fold symmetry, including somefine 2-level designs in the Patio de las Doncellas in theAlcazar, Seville—see [22] for photographs. The geometry ofthe polygonal networks underlying these designs isgrounded on the

ffiffiffi2p

system of proportions rather than thegolden ratio. Plans of muqarnas (corbelled ceilings built bystacking units in tiers and progressively reducing the size ofthe central hole to produce a stalactite-like dome) some-times display similar features. These networks have astrong resemblance to the Ammann–Beenker quasi-periodic tiling composed of squares and 45�–135� rhombi[33]. This tiling is another substitution tiling that can begenerated by subdivision and inflation; the tiles can also bedecorated with line segments to produce Ammann bars.Similar claims to those assessed in this article have beenmade for some of the Islamic 8-fold designs [2, 6, 22, 23].

To me, it seems most likely that the Islamic interest insubdivision was for the production of multi-level designs.Islamic artists were certainly familiar with generatingdesigns by applying reflection, rotation and translation torepeat a template. They probably had an intuitive under-standing of the crystallographic restriction and a feelingthat global 5-fold and 10-fold rotation centres are somehowincompatible with periodicity. They did have the toolsavailable to construct quasi-periodic designs but not thetheoretical framework to appreciate the possibility or sig-nificance of doing so.

(a) (b) (c)

Figure 24. Patches of Penrose kites and darts.

(a) (b)

Figure 25. Subdivisions of marked tiles that preservethe markings. The scale factor is 1

2 3þffiffiffi5p� �

� 2:618.


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 Castera, Dirk Frettloh,

Chaim Goodman-Strauss, Emil Makovicky, John Rigby,

Joshua Socolar, and John Sullivan.

BIBLIOGRAPHY

1. M. Arik and M. Sancak, ‘Turkish–Islamic art and Penrose tilings’,

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teenth 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.

3. J. Bonner, Islamic Geometric Patterns: Their Historical Develop-

ment and Traditional Methods of Derivation, unpublished

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4. J. Bourgoin, Les Elements de l’Art Arabe: Le Trait des Entrelacs,

Firmin-Didot, 1879. Plates reprinted in Arabic Geometric Pattern

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5. J.-M. Castera, Arabesques: Art Decoratif au Maroc, ACR Edition,

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6. J.-M. Castera, ‘Zellijs, muqarnas and quasicrystals’, Proc. ISAMA,

(San Sebastian, 1999), eds. N. Friedman and J. Barrallo, 1999,

pp. 99–104.

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

Fivefold Symmetry, ed. I. Hargittai, World Scientific, 1992,

pp. 263–281.

8. B. Grunbaum and G. C. Shephard, Tilings and Patterns,

W. H. Freeman, 1987.

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,

Memoirs of the Archaeological Society of India, no 15, Govern-

ment of India, 1925.

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

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

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 Pro-

ceeding 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.

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 ‘‘Dec-

agonal 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 Appliques 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.

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’, Boletın

Sociedad Espanola Mineralogıa 19 (1996) 1–26.

23. E. Makovicky and P. Fenoll Hach-Alı, ‘The stalactite dome of the

Sala de Dos Hermanas—an octagonal tiling?’, Boletın Sociedad

Espanola Mineralogıa 24 (2001) 1–21.

24. E. Makovicky, F. Rull Perez and P. Fenoll Hach-Alı, ‘Decagonal

patterns in the Islamic ornamental art of Spain and Morocco’,

Boletın Sociedad Espanola Mineralogıa 21 (1998) 107–127.

25. G. Necipoglu, The Topkapi Scroll: Geometry and Ornament in

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’,

Proc. Bridges: Mathematical Connections in Art, Music and

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

pp. 41–48.

28. F. Rull Perez, ‘La nocion de cuasi-cristal a traves de los mosaicos

arabes’, Boletın Sociedad Espanola Mineralogıa 10 (1987) 291–

298.

29. P. W. Saltzman, ‘Quasi-periodicity in Islamic ornamental design’,

Nexus VII: Architecture and Mathematics, ed. K. Williams, 2008,

pp. 153–168.

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Press, 1995.

31. M. Senechal and J. Taylor, ‘Quasicrystals: The view from Les

Houches’, Math. Intelligencer 12 no 2 (1990) 54–64.

INTERNET RESOURCES

32. ArchNet. Library of digital images of Islamic architecture,

http://archnet.org/library/images/

33. E. Harriss and D. Frettloh, 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/


http://archnet.org/library/images/

http://tilings.math.uni-bielefeld.de/

http://www.cgl.uwaterloo.ca/$\sim$csk/washington/taprats/

http://www.sciencemag.org/cgi/content/full/315/5815/1106/DC1

http://www.patterninislamicart.com/

GeometricConstructions withEllipsesALISKA GIBBINS AND LAWRENCE SMOLINSKY

he geometric problems of trisecting a general angleand doubling the cube cannot be solved by the useof a straightedge and compass alone. These

beautiful results were a triumph of modern algebra, firstpublished by Pierre Laurent Wantzel [1]. The constructiblepoints are those that are in iterated quadratic extensions ofthe base field. A consequence, the Gauss-Wantzel Theo-rem, states that a regular n-gon is classically constructible ifand only if /(n) is a power of 2; here / is the Euler /function, which counts how many integers are less thanand relatively prime to n.

To the modern reader the question of what are thepossible numerical magnitudes (lengths) is answered bythe process of the topological completion of the rationalnumbers, i.e., the real numbers. The ancient Greeksshowed the existence of numerical magnitudes by affir-mative use of the axioms—proving existence is a plausiblemotive for why the Ancients engaged in constructions [2,3]. Straightedge-and-compass constructions are a rigorousapplication of Euclid’s first three postulates. However,there are other procedures the Greeks used for con-structions. For example, a procedure not grounded in thepostulates is the rotation of a plane figure to producea solid, or the intersection of a plane with a solid figure(p. 29 [4]).

Pappus of Alexandria described a classification ofmethodology for geometric problems—one which heattributed to those he called Ancients. A construction iscalled planar if it is done with straightedge and compassalone, solid if it uses conic sections, and linear if it useshigher order curves [2]. Solid solutions do exist to theclassical problems. Pappus gave two trisection construc-tions with hyperbolas that are possibly due to Apollonius.

Menaechmus, the discoverer of conic sections, is supposedto have made his discovery while working on the problemof doubling the cube, and he gave a construction usingparabolas. Those constructions have been described in TheIntelligencer [5].

In 1895 James Pierpont essentially gives the analysis ofwhich numbers are constructible using conic sections. Intwo pages at the end of a Bulletin paper, Pierpontremarks that ‘‘Greek geometers frequently allowed the useof the conic sections in a geometric construction,’’ and hedetermines that a regular n-gon allows a solid construc-tion if and only if /(n) has only factors of 2 and 3 [6, 7].Recent work by Carlos R. Videla explores solid construc-tions, and Videla gives a complete and more modernversion of Pierpont’s result [5]. Videla allows the con-struction of a conic when the focus, directrix, andeccentricity are constructible. The conically-constructiblenumbers may be obtained by circles, parabolas, andhyperbolas alone.

We will consider construction with ellipses. Theelliptic constructions in this paper are from an under-graduate project by the first author and directed bythe second. Patrick Hummel also gives a treatment ofelliptic constructions in a paper from his undergraduateproject [8]. While the abstract theory is similar, the con-structions are different. The authors are grateful to thereferee for directing them to Hummel’s paper and othersources.

We will primarily be concerned with solid constructions,but it is worthwhile to note that ancient Greek construc-tions were extremely rich and varied. A beautifuldemonstration of this variety is a construction to double thecube by Archytas of Taras, who came out of Plato’s

TT


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

x2 þ y2 ¼ x: ð1Þ

Take a point B on the circle and let b be the length of thesegment OB. Let A = (1, 0), so OA is a diameter. SeeFigure 1. Archytas can find the cube root of b.

The construction requires three dimensions. Start withthe right circular cylinder with axis parallel to the z-axiscontaining our circle, which is given by equation (1).Take the cone with vertex O obtained by rotating theline containing OB about the x-axis. This cone is givenby

b2ðx2 þ y2 þ z2Þ ¼ x2: ð2Þ

The third and surprising construction is to take the circlein the xz-plane of radius 1/2 and diameter OA and rotate itabout the z-axis. The result is a degenerate torus—a circlerotated about a tangent line. See Figure 2. The equation ofthe degenerate torus is

ðx2 þ y2 þ z2Þ2 ¼ x2 þ y2: ð3Þ

If P is a point of intersection of the cylinder, cone, anddegenerate torus, then the distance of P to the origin is

ffiffiffib3p

:

ðx2 þ y2 þ z2Þ2 ¼ x2 þ y2 by equation ð3Þ¼ x by equation ð1Þ¼ b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

pby equation ð2Þ:

The planar and solid constructions can be built from alimited number of well-defined constructions with lines,circles, and other conics, and so can be analyzed bymodern algebra a la Gauss, Wantzel, and Pierpont. Ingeneral, the rich variety of constructions introduced by theancient Greeks requires more analysis before allowingalgebra 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 lineare discussed by Arthur Baragar [10].

Wilbur Richard Knorr, in his discussion of angle-trisec-tion methods by ancient geometers in [11] (page 216),comes up with about a dozen solutions and writes, ‘‘In viewof the massive extinction of documentation from antiquity,we can hardly presume that this list would exhaust theentire range of ancient solutions.’’ There are a lot of inter-esting constructions and questions to explore.

In the modern treatment of construction problems, onefirst translates a geometric question into an algebraicquestion by use of the Cartesian or Gaussian plane, andthen analyzes the question using the power of algebra andnumber theory. This first step was already started—as itwere—in the beginning with Descartes’s analytic geometry.Rene Descartes gave a trisection construction in his 1637 LaGeometrie, the monograph in which he introduced analyticgeometry. 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 isshown in Figure 3. The construction uses the paraboladefined by y = 2x2 and the circle through the originwith center (1/2 cos (h),1). The x-coordinates of the pointsof 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

ALISKA GIBBINS is a graduate student at

Ohio State, studying geometric group the-

ory 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.


Ohio State University,

Columbus, OH 43210, USA


LAWRENCE SMOLINSKY started in topol-

ogy, getting his doctorate at Brandeis underthe late Jerry Levine. More recently he has

worked in integrable systems and represen-

tation 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!


Louisiana State University,

Baton Rouge, LA 70803, USAe-mail: [email protected]

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

O

B

A1

4

1

2

3

4

1

2

1

2

Figure 1. Example with b = 1/2.

AU

TH

OR

S


cos (3h) = 4 cos3(h) - 3cos (h). In Figure 3, there are fourdistinct points of intersection although two are very closetogether.

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

may include additional points, e.g., to form a generalangle). The set of points one may derive using only astraightedge and compass will be called classically con-structible points derived from P. The set of all numbersthat arise as the ordinate or abscissa of classically con-structible points is the set of classically constructiblenumbers.

We recall some of the facts about classically constructiblenumbers, taking some of the background notions fromHungerford [13]. It is one of the founding observations thatstarting with an initial set of points P the classically con-structible numbers form a field, i.e., using straightedge andcompass constructions one may start with two numbers andconstruct the sum, difference, product, and quotient. Fur-thermore, if (x, y) is a classically constructible point, it is asimple exercise to show that (y, x) is a classically construc-tible 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 Fis the subset F � F � R

2: Suppose P and Q are distinctpoints in the plane of F. Then the line determined by P andQ is a line in F. Similarly, the circle with center P andcontaining Q is a circle in F. It is a straightforward calcu-lation that the intersection points of two lines in F arepoints in the plane of F. Furthermore, if a circle in F isintersected with either a line in F or a circle in F, then theintersection points are in the plane of F(r), where r is the

1 1

2

1

21

1

2

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

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


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

Elliptic ConstructionsWe mention three approaches for constructing ellipses thatwere known to the Ancients and can be used to extendstraightedge-and-compass constructions to constructionswith ellipses.

The first approach is to use the fact that an ellipse is thelocus of points whose distance from the two foci is aconstant sum. This property was known at least as far backas Apollonius [2].

The second approach is to use the interpretation of anellipse as the motion of a point on a line segment whoseendpoints slide along perpendicular lines. This construc-tion was reported by Proclus, who was a head of theAcademy [2]. This construction could be accomplished withknowledge of the line containing the foci and the lengths ofthe major and semimajor axis.

The third approach is to allow the construction of anellipse given its directrix, focus, and eccentricity.

Each of the ellipse construction techniques ostensivelywould determine a different set of constructible points inthe plane, but these three sets all turn out to be the same.This fact follows from part (2) of Proposition 3 below. Weuse the first method, which may be accomplished with pinsand string. Our fundamental constructions are:

(C1) Given three points, one may insert pins in the threepoints, tighten the string around the pins, and removeone pin. Keep the string taut, and use a pen to draw theellipse around the two pins as foci and passing through thethird.

(C2) Given two points, one may insert pins in the twopoints, tighten the string around the pins, remove one pin,and use a pen to draw the circle with center one point andthe other on the circle.

(C3) Given two points, one may draw the line throughthe two points.

Which points can be reached by constructions with astraightedge and pins and string? Start with a set of points P,which include (0, 0) and (1, 0). The elliptically constructiblepoints derived from P are all the points of intersectionobtained from the ellipses and lines constructed by theoperations above. We call the obtained coordinates the fieldof elliptically constructible numbers.

Analogous to the previous definitions for classical con-structions is the following definition. Suppose F is asubfield of the real numbers R: If O, P, and Q are distinctpoints in the plane of F, then the ellipse containing O withfoci P and Q is an ellipse in F.

REMARK 1 If an ellipse is in standard position then its

equation can be given as

Ax2 þ Cy2 � F ¼ 0;

where C [ A [ 0. Let a ¼ffiffiffiFA

q; b ¼

ffiffiffiFC

qand c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � b2:p

.

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

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

x ¼ a2

c and the eccentricity is e ¼ ca : The ellipse is

constructible by the focus-and-directrix approach if and

only if a2

c and ca are constructible.

LEMMA 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.

PROOF. 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.

PROPOSITION 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 equationax2 + bxy + y2 + dx + ey + f = 0. The ellipse E is inF 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 andonly if the lengths of its semi-major and semi-minoraxes 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 thepoints of intersection of E1 and E2 are in a field F(R),where R is the set of real roots of a quartic polynomialwith coefficients in F. If E is an ellipse in F and L is aline in F, then the coordinates of the points ofintersection of E and L are in the field F.

PROOF. 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 con-

verse is similar, but rotate E into standard position using h,

where cot ðhÞ ¼ a�cb 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. Anytwo constructible ellipses E1 and E2 have equations of theform:


a1x2 þ b1xy þ y2 þ d1x þ e1y þ f1 ¼ 0 ð4Þ

a2x2 þ b2xy þ y2 þ d2x þ e2y þ f2 ¼ 0: ð5Þ

Solving for y,

y ¼ � f1 � f2 þ ðd1 � d2Þx þ ða1 � a2Þx2

ðb1 � b2Þx þ ðe1 � e2Þ: ð6Þ

Putting this expression back into equation (4) we get

Ax4 þ Bx3 þ Cx2 þ Dx þ G ¼ 0; ð7Þ

where the coefficients are

A ¼ �e1f2 þ f 22 þ e2f1 � 2f2f1 þ f 2

1

B ¼ d1e2 � d2e1 � b1f2 þ 2d2f2 � 2d1f2 þ b2f1 � 2d2f1 þ 2d1f1

C ¼ �b1d2 þ d22 þ b2d1 � 2d2d1 þ d2

1 þ a1e2 � a2e1 þ 2a2f2

� 2a1f2 � 2a2f1 þ 2a1f1

D ¼ a1b2 � a2b1 þ 2a2d2 � 2a1d2 � 2a2d1 þ 2a1d1

G ¼ a22 � 2a2a1 þ a2

1:

Let R be the real roots of Equation (7). The proof of theother claim is similar.

Part 2 is similar in spirit to Parts 1 and 3.Note that Part 2 implies that the three methods of con-

struction given in the beginning of this section yield thesame constructible numbers.

Trisection of the General AngleSuppose we start with an angle of measure h. Translate it tothe congruent central angle \ABC with A = (cos (h),sin (h)), B = (0, 0), and C = (1, 0). To trisect this angle, wemust construct the point A0 = (cos (h/3), sin (h/3)), and\A0BC trisects the angle. Constructing A0 is equivalent toconstructing the number cos (h/3), because sin (h/3) canthen be produced as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� cos2ðh=3Þ

p:

Let q = cos (h). By a triple-angle formula, cos (h/3) is areal solution to the equation

4x3 � 3x � q ¼ 0:

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

THEOREM 4 The general angle can be trisected.

PROOF. Let F be the field of constructible numbers

derived from (0, 0), (1, 0), and (cos (h), 0). Let q = cos (h)

and p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi17� 4qp

: Consider the following ellipses with

coefficients in F:

2x2 þ 4y2 � qx þ 2py þ 2 ¼ 0

6x2 þ 4y2 þ ð2p� 4Þy � ð2þ 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 x-coordinates of the points of intersection, we obtain thatthey are the real roots of the equation

4x4 � 4x3 � 3x2 þ ð3� qÞx þ q ¼ 0;

which factors as (4x3 - 3x - q)(x - 1) = 0. One of thesolutions is cos (h/3).

The example of the trisection of h ¼ p3

is shown inFigure 4. In equations (8), q ¼ 1

2: There are four distinct

points of intersection, although two are very close together.

Cube roots and Doubling the CubeThe ability to construct a cube whose volume is double agiven cube is the same as the ability to multiply a sidelength by

ffiffiffi23p

.

THEOREM 5 If a is an elliptically constructible number,

then so isffiffiffia3p

.

PROOF. Let F be the field of constructible numbers

derived from (0, 0), (1, 0), and (a, 0). Consider the fol-

lowing equations of ellipses with coefficients in F:

2x2 þ y2 � ax þ 2ffiffiffi2p

y þ 1 ¼ 0

3x2 þ y2 � ax þ ð1þ 2ffiffiffi2pÞy þ

ffiffiffi2p¼ 0:

ð9Þ

These ellipses are in the plane of F by Proposition 3 part(1). Solving the system of equations (9) for the x-coordinates of the points of intersection, we obtain thatthey are the real roots of the equation

x4 � ax ¼ 0:

The real roots are 0 andffiffiffia3p

:

1 1

2

1

1

0.95 1

1.3

1.25

Figure 4. Trisection of h ¼ p3.


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

Concluding RemarksWe can determine which numbers are elliptically con-structible. Suppose P is a set of points and F � R is the fieldof elliptically constructible numbers determined by P. LetF ¼ F þ iF : Then F is a subfield of the complex numbersand the constructible points in the Gaussian plane. Let F0

be the field generated by the rationals and the coordinatesof the points in P.

THEOREM 6

(1) F is the smallest field containing F0, i, and the squareroots, cube roots, and conjugate of each element.

(2) F is the smallest field which contains F0 and the realroots of every fourth-degree polynomial with coefficientsin F.

PROOF. Part (1) is shown in [5]. To see part (2), note that

F ¼ ReF: By Cardano’s and Ferrari’s formulas, F contains

the real roots of fourth-degree polynomials with coeffi-

cients 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) isthat taking a cube root of a complex number is trisectingan angle and taking the cube root of a real number, i.e.,ffiffiffi

R3p

eh3i is a cube root of Rehi. Combining this observation

with Cardano’s and Ferrari’s formulas shows that

trisections and real cube roots are exactly the construc-tions required to obtain all of F:

The fields of elliptically constructible numbers and con-ically constructible numbers are the same. So the regular n-gons that are elliptically constructible were determined byPierpont [5, 6]. A regular n-gon is elliptically constructible ifand only if /(n) = 2s3t for some s and t. For example, the 7-and 9-sided regular polygons are elliptically constructiblebut not classically constructible.

Note also that one only has to allow the construction oftranslations of ellipses in standard position to do ellipticconstructions. Oblique ellipses are not required to obtainthe field of elliptically constructible numbers, for theyare not required for constructions in the proofs ofTheorems 4 and 5. The type of ellipses may be furtherrestricted: the ratio of the lengths of the major to the minoraxis may be restricted to 1;

ffiffiffi2p

;ffiffiffi3p; and

ffiffiffiffiffiffiffi3=2:

p. Can this be

improved?

REFERENCES

[1] Wantzel, L. ‘‘Recherches sur les moyens de reconnaıtre si un

probleme de Geometrie peut se resoudre avec la regle et le

compas,’’ Journal de mathematiques pures et appliquees Ser. I 2

(1837), 366–372. Available free online through the gallica library

(Bibliotheque Nationale de France).

[2] Knorr, Wilbur Richard, The Ancient Tradition of Geometric Prob-

lems, Dover Publications, New York, 1993.

[3] Zeuthen, H.G., ‘‘Die geometrische Construction als ‘Existenz-

beweis’ 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.

[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 Dis-

quisitiones 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

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 Twice-

Notched Straightedge,’’ Amer. Mathematical Monthy 109 (2002),

151–164.

[11] Knorr, Wilbur Richard, Textual Studies in Ancient and Medieval

Geometry, Birkhauser, Boston, Inc., Boston, 1989.

[12] Yates, Robert C., ‘‘The Trisection Problem’’ National Mathemat-

ics Magazine (continued as Mathematics Magazine) 15 (1941),

191–202.

[13] Hungerford, Thomas W., Algebra, New York: Springer 1997.

1

3

2

1

Figure 5. Constructingffiffiffi23p

.


The Mathematical Tourist Dirk Huylebrouck, Editor

AbstractNeo-Plasticityand ItsArchitecturalManifestationin the LuisBarraganHouse/Studioof 1947JIN-HO PARK,HONG-KYU LEE,YOUNG-HO CHO, AND

KYUNG-SUN LEE

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.

LLuis Barragan (1902–1988), bornand raised in Guadalajara, Mex-ico, was a modern architect

whose works have influenced con-temporary building designs in hisnative country and beyond. His archi-tecture responds to the contextual andnatural inheritance of Mexico, signify-ing a new residential dwellingpredicated on modernity and indige-nously rooted in the symbol ofMexican living. The manner in whichhis buildings are integrated within theirgiven ‘‘place’’ is perhaps the key factorin his significance and renown. Whiledrawing from cultural and regionalreferences of Mexico, Barragan offereda utopian vision of the unification ofthe vernacular Mexican style witharchitectural purity and simplicity.Stucco walls with bricks, intense satu-rated colors, and natural illumination

possessing a spiritual quality definedBarragan’s designs. Barragan continuesto exert a profound influence on con-temporary architecture. (See [4], [6], [7]and [8].) His vision has inspired someof the best-known contemporaryMexican architects including RicardoLegorreta, Andrea Casillas, and Enri-que Norton of TEN (Taller EnriqueNorton) Arquitectos, among others.

1

Ricardo Legorreta is among the disci-ples of Barragan who make use of hissense of color, spatial composition,and design vocabulary.

Among Barragan’s work, his ownhouse and studio stands out for itsinterplay of abstract planes and boldmasses. Its colorful walls provideinternal rooms and patios with pleas-ant filtered light. Barragan writes, ‘‘Ihave left large plane walls withoutwindow openings, both for plasticbeauty… . By the use of large wallsurfaces one can also obtain spaceswith varying luminosity, which createsan ambience more comfortable andintimate.’’2 Barragan’s architecture isassociated with two primary connec-tions. The abstract neo-plasticity of DeStijl and Bauhaus strongly inspired thegeometry of the house, whereas Bar-ragan’s association with avant gardeartistic circles, which included DiegoRivera, Frida Kahlo, and Jos ClementeOrozco, infused him with indigenousculture and regional principles.

Mathematical Intelligencer readersmay wonder why this house is thesubject of an article. Although muchhas been written about the Luis Barr-ragan house/studio, most studies ofthe house are descriptive presentationslacking formal and mathematical

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium


1A 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 Barragan, London: Phaidon Press; Eggener, K, (2001) Luis Barragan’s Gardens of El Pedregal, New York: Princeton

Architectural Press; Federica Zanco, F. (2001) Luis Barragan: 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: Birkhauser.2‘‘Luis Barragan,’’ Arts and Architecture, August [2], pp.24–25.


study. The present article reviews hisarchitectural thought in an effort tograsp the abstract nature of Barragan’sachievement, and also explores theunderlying logic applied to the housethrough an in-depth analysis andinterpretation of the house. Amongother aspects, Barragan’s worksprominently feature the transformationof rigorous combinations of simple buttimeless geometry into unique spatialcompositions. The simple geometrynot only serves as a key principle forthe consistent and systematic qualityunderlying his work, but also providesan order for the formal elements thatencompass the spatial composition. Itexpresses the clarity of his thought andinsight, and achieves overly formal andunique architectonic space forms, thusproviding a source of Barragan’sexpression. Therefore, central to thisarticle is the notion that Barragan’shouse is characterized by a powerfulcommitment to the spirit of abstractioncoupled with a strong geometricalbasis. Above all, we would encourageinterested Mathematical Intelligencerreaders to visit the house to experienceits unique formal quality and gain newarchitectonic insights.

The Barragan HouseThe house and studio of Luis Barraganis located within Mexico City. Adjacentto an earlier home he designed, itwas completed in 1947.3 The lot forthe house is 100 feet across its front(about 30 m) and 140 feet (about42 m) deep.4 The house was the pointof departure for his subsequentworks. Barragan’s work from this per-iod focuses on colors and forms andthe light that defines his buildings. Thefocus may also be on the emotionalquality of the form and light in theirabstract manifestations. Barragan livedand worked in the house alone untilhis death in 1988 from Parkinson’sdisease.

The Barragan house is known for itsunique characteristics and the sereneform of both the house and garden. Theproperty is completely hidden from the

outside neighborhood by a plain facadeon a small narrow street.5 The entirehouse is screened and turned inwards toallow for greater privacy, creating col-orful interior spaces and shadedcourtyards. An exterior view is pre-sented in Figures 1–5.

Upon entering the house, a darkentrance hall with indirect lighting andvolcanic lava floors extends to a vesti-bule facing a pink wall. In thevestibule, stairs link the volcanic lavafloor stairway to the mezzanineabove.6 To the right, one is led to themain living area. The main roomincludes the living room and a librarythat is a double-height space with darkexposed wooden girders. These stylegirders are typical in Barragan’shouses, one example being in the Lo-pez house of 1948. Here one finds aserene enclosed garden. High wallssurround the garden with bastions setat intervals. These walls serve as aprotective barrier to the outside world,bringing tranquility and comfort. Waterand lush vegetation are utilized toregulate the temperature of both thegarden and the building.

The entire layout of the processionto the interior space and the gardenvividly reflects the elaborate imageof the Lahambra Moorish garden:‘‘… [W]hile walking along the lava cre-vices, under the shadow of imposingramparts of living stocks, I suddenly

discovered, to my astonishment, a smallsecret green valley—the shepherds call them ‘‘jewels’’ —surrounded and enclosed by themost fantastic, capricious rock forma-tions … .’’7

The living room is partitioned byfolded screens and lowered walls,which are movable according to func-tional needs. These elements create anuninterrupted flow of rooms. Theexposed pine ceiling structure visiblyextends beyond the boundaries ofthe individual partitioned rooms, sothat, while remaining private, theyare not completely isolated fromone another. Reinforcing the dynamicquality of the high open space isits element, the stairs. The woodenstairs of the library lead to the mezza-nine. The stairway without a handrailbecomes a dynamic element throughits expression of flowing movement.The doorway found at the top of thestairs seems a part of the stairway,because it is the same width and ismade out of the same material.

Barragan’s studio is located next tohis house with one wall in common.The studio, with various offices, hasdirect access from the street through avestibule. A typical patio is located onthe west side of the studio, yet originallyit overlooked an enclosed patio with alarge window. The patio was enclosedwith high walls on three sides, offering a

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

Francisco Ramirez Street.

3This house is currently known as the Ortega house realized in 1940.4‘‘Luis Barragan,’’ Arts and Architecture, August [2], pp.24–25.5Clive Bamford Smith, Five Mexican Architects, Architectural Book Publishing Co., Inc. New York, [16], p.74.6Barragan Foundation, Casa Luis Barragan Guide [3], Mexico.7In his official address, 1980 Pritzker Architecture Prize, see Paul Rispa, ed., Barragan, the Complete Work, New York: Princeton Architectural Press, [15], pp. 204–207.


visual barrier to lend the serene qualityof an enclosed space, decorated withtraditional ceramic vases. Between theoffices and the studio lies an outdoorspace for cleaning.

The second-floor plan includes twobedrooms, a guest room, a dressingroom, a mezzanine for the house, and

two offices above the studio. Thehouse and the studio are not linked onthe second floor. Interestingly, theserooms rely on natural light.

The high enclosed walls on the roofterrace provide the space with a senseof privacy and serenity. The terracedgarden is blocked off from the street.

The walls help focus attention on thediscontinuity between the roof terraceand the outside environment. The ter-race becomes a totally isolated part ofthe house and offers no vista. Exposedonly to the sky, the bold roof terracebrings to mind the light sculptures ofJames Turrell.8

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.’’

8Refer 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.


Sources of Barragan’s Abstract

Neo-Plasticity

From the construction of this houseonward, Barragan begins to resolve hisplanar surfaces. Horizontal and verticalplanes begin to link, trapping rectan-gular planes within. Barragan’srectilinear designs fuse abstract neo-plasticity with the Mexican landscapetradition. He promoted a rich vocabu-lary of local materials and a wide rangeof colors within formal plasticity inno-vations. Early on, while traveling inEurope in the 1920s, Barragan wasinspired by the work of architects suchas Gropius, Mies van der Rohe, and LeCorbusier. When he started his practicein 1927, his early designs reflected theSpanish-Mexican vernacular tradition.The year 1947 was generally regardedas the beginning of Barragan’s system-atic development, a period thatcontinued until his death in 1988.Unlike his early work, his later designsexhibit simple geometric forms.

Barragan was particularly associatedwith European immigrants within theUnited States and Mexico. Among oth-ers, his association with MathiasGoeritz was of primary influence on hisabstract and plastic work. Goeritz car-ried pure plastic forms to their mostextreme limits in his designs. Filteringthrough Goeritz’s abstraction andinfluences of minimal art, Barraganincorporated Euro-American Modernist

design into the Mexican landscape andhis color schemes, creating a uniqueand exhilarating new design style.Torres de Satelite, designed by MathiasGoeritz and Luis Barragan in 1957 andbuilt in 1958, is an example of theircollaboration. It is located in CiudadSatelite, a middle-class zone, in thenorthern part of Naucalpan, Mexico.

Josef Albers’s paintings also payattention particularly to simple com-positions and contrasting color.Barragan collected a few of JosefAlbers’s paintings, and the seriesHomage to the Square was displayedon the walls of the Barragan house. Aformal analogy between Albers’spaintings and Barragan’s architecturecan be readily drawn.

Albers’s series Homage to the Squareare based on a grid system, drawn onboth horizontal and vertical divisions of20 units each. The first series of Albers’spaintings consists of four squares withfour shades of one color. The squareswithin each painting are nested pro-portionally, according to their sizes. Inbasic composition (Figure 6a), theunits to either side of the nestedsquares are twice as large as the unitson the top and bottom. The propor-tional relationship between the squaresis based on simple whole numberssuch as 1, 2, 3, etc. Accordingly, theirarrangement is bilaterally symmetricalalong a vertical axis, but not strictly

concentric, providing a dynamic result.Three more basic types are added lateron in his series. Their compositionrelies on the same divisional techniquebut an individual square is removedfrom the four-square composition(Figures 6b–d).

With this mathematically plottedframework, Albers experimented withthe retinal effects of color within aseries of nested squares.9 The squaresare used to investigate color interactionwith the adjoining colors where theycontrast, recede, or pop out. The con-secutive squares of color turn out to beperfectly harmonious and purelyabstract, unlike anything in nature.

There is no evidence that Barraganheld a particular regard for mathemat-ics. However, through the use ofsimple geometry associated withwhole-number ratios and the chro-matic colors of the paintings, it isevident that Barragan was influencedby Albers’s approaches. Perhaps Bar-ragan takes Albers’s system to achieveharmony and proportion within hisworks. Barragan clearly appreciatedAlbers’s approach to exploring thepotential of abstract values, shape,color, and texture. At the Bauhaus,Albers dealt principally with abstract,formal issues. He also stressed com-mon materials and their inherentproperties. For Albers, a deep knowl-edge of abstract composition enhanced

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

authors).

9See Josef Albers’s 1963 book, Interaction of Color, New Haven: Yale University Press.


the comprehension of materialistic qual-ity and social suitability. Barragan’sapproach was also along these lines.

Barragan’s form, defined by planarwalls of different heights and colors,involves explorations of three-dimen-sional depth using light and space.10

His work has been praised as havingattained a degree of mystical andspiritual abstraction. An example isthe color palette used within theBarragan house, which includes yel-low, purple, pink, red, and an earthy-brown, coupled with neutral graysand whites. Barragan did not followany systematic color theory such asJohannes Itten’s color circle. Also,Barragan’s use of color is not basedon material properties; instead, hesought to articulate color to influ-ence and reinforce desired spatialeffects.

Within the Barragan house, mostwalls are colored white, which acts asa foreground element that defines thespatial extension outwards. Key walls

in certain rooms are meanwhile col-ored. The vestibule best reveals theuse of pink to reinforce the spatialintent of the house. The reflected lightlanding on the pink surface leadsvisitors from the entrance hall to thevestibule, thereby reinforcing linearmovement between spaces via transi-tional zones. In the living room,Barragan applied yellow to the floor.Counterbalancing the yellow floor is adark brown wooden ceiling that con-tinues to the adjacent rooms. In thedining room, Barragan applies red tothe walls. Three-dimensional spacecombined with color is reminiscent ofGerrit Thomas Rietveld’s Schroderhouse, as well as the color drawingsof Cornelis van Eesteren and Theovan Doesburg (see Figure 7).

Typically, abstract expressionistpainters intended to move beyondrepresentation to pure form. In reality,these painters were inspired to createfrom patterns, shapes, and colors theyfound within the natural landscape.

Van Doesburg’s early window designillustrates a series of abstract pro-cesses that begin with a naturalisticimage that is transformed, step-by-step, into an abstract composition ofgeometric shapes. This is a classicexample of abstract expressionism.11

The abstract expressionist movementis described as being inclined heavilytowards conceptualization, surpassingall that is to be perceived in materialreality. This exponent of conceptual-ized abstraction influenced Barraganin terms of his abstraction of natureand his feelings about Mexican archi-tecture. Denouncing the traditionalimage, Barragan searched for a newvision of Mexican architecture throughabstraction. Barragan conceptualizedthe traditional image of the Mexicanhouse and manifested it in a newplastic volumetric morphology thatsurpassed the traditional model,shedding all formal connotations andstructural organization to trace innerforce.

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

authors).

Figure 7. Left: Cornelis van Eesteren and Theo van Doesburg, ‘‘Contra-Construction’’ of 1923;

middle: Cornelis van Eesteren and Theo van Doesburg, ‘‘Maison Particuliere’’ of 1922; right:

Barragan house color scheme (computer reconstructed).

10Barragan 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.11See Allan Doig, Theo Van Doesburg, London: Cambridge University Press, [5].


Geometrical Analysis

of the House

While Barragan was respectful of localand regional conditions, he aggres-sively pursued the abstraction of form,as in the abstract expressionists’paintings. Barragan abstracted thenaturalistic image of the Mexicanlandscape and then translated it intoabstract geometric space and form,thus introducing a new sense of order,space, and form: Not that of organicnature but of simple geometry.

Barragan’s entire house appears tobe an experiment in simple geometry;there is no evidence that Barraganused a formal system. Nevertheless,when the ground plan or facade isexamined, one first recognizes thatspatial division is based on a rectangle.

The floor plan is analyzed in anattempt to find a basic grid that willestablish an underlying geometry forthe design as a whole.12 Upon analysis,a 4.25 square meter unit module (M)best explains the house plan. Figure 8shows the plan overlaid on the deter-mined grid. This grid is essential indetermining the proportional layout ofthe house. The spatial division mightbe explained by more functional rea-sons, however, it may also be due tothe influence of abstract paintings,which go beyond functional form.

For Barragan, the relation betweenthe house and the garden is integral.He wrote, ‘‘In designing and planningthese functional gardens it is of pri-mary importance to invest effort incharacter and atmosphere, as well as inplastic beauty.’’ He continued, ‘‘Wefound that… if we were to createbeautiful architectural forms that werein harmony with [the landscape], wewould have to opt for extreme sim-plicity: Abstract quality, preferablystraight lines, flat surfaces and primarygeometric shapes.’’13 These ideas areclearly reflected within the house.When Barragan planned the house, heconsidered the garden as an emptyvolume related to the house in terms ofits shape and design.

The house is composed of threemajor zones: The residence, studio, andgarden. When these zones are dia-grammed with the primary geometric

shape, ignoring some minor areas, eachzone forms a square and double squarerepresenting a spatial territory (Fig-ure 9a). While the parti of theresidential area and the garden relies onthe square, the studio is on the doublesquare. 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 is2.5 M 9 2.5 M. When the three squaresare rearranged to the proportions ofone of Albers’s paintings, the composi-tions turn out to be similar, as shown in

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.

12The analysis of the floor plan and elevation is based on the drawings from the book Casa Luis Barragan Guide, by the Barragan Foundation.13Paul Rispa, ed. Barragan, the Complete Work, New York: Princeton Architectural Press, [15], pp. 34–35.


Figure 9b. Is this too speculative oraccidental? What we can presume is thatBarragan intuitively planned threemajor spaces according to their sizerelationships using the idea of Albers’spainting to manifest plastic form.

Facade Analysis

The Spanish-Mexican traditional houseface cannot be found in the Barraganhouse. In contrast with the plasteredplanar surfaces of the street and gardenfacade are various windows thatappear to be randomly arranged andare far from being symmetrical. Thealignment of the windows has little orno virtue on first impression.

An examination of the windowframes is significant, because theirsize, profile, and proportion arestrongly related to the character andappearance of the Barragan house.Although collectively the windows aredisorganized and of different sizes,each window is ordered using similarproportions with regard to a square.Upon closer examination, it is seenthat this square element also domi-nates the street and garden facade. Allwindow gratings and framings areformed according to the addition andsubdivision of the square. Neverthe-less, this square unit is not related tothat of the floor plans. That is, thesquare unit of the plan (4.6 m) is notcarried through to the elevations.Barragan apparently sought freedomfrom a single unit constraint.

The imposing facade of the housefaces General Francisco Ramirez Street.In the facade, the window openingsare all different and are not alignedrepeatedly, as shown in Figure 10.Various shapes of window openings aswell as gratings are created. Theseshapes are formed according to asquare and a rectangle. That is, win-dows for the guestroom, dressingroom, library, ventilation openings,garage opening, and bathroom arebased on a square, but the other twowindows for the studio offices are in arectangular form: Approximately, onewindow is 7:6 and the other is 6:5 inproportion (Figure 11).

In the garden facade, two separateplanar walls are formed according to asquare. One creates the living roomand the other the sleeping and kitchenareas. When extended to chimneyheight, the dotted line of the livingroom plane forms a square (abce inFigure 12). The other planar wallforms a square as well (defg) as shownin Figure 12.

Window shapes also appear on thegarden facade in two forms: Squareand rectangular. Square windows arefurther subdivided with simple ratiossuch as 1:1 and 2:1. The living roomwindow looking out onto the gardenrelies on a half division (Figure 13a),and the window of Barragan’s ownbedroom is divided into a tripartiteform (Figure 13b).

Rectangular windows form threetypes of ratios: A square and a third for

4:3, a square and a fifth for 6:5, and asquare and a seventh for 8:7. They maybe generated by either a square that isdeducted from the rectangle, or amodule square that is added to formrectangular windows (Figures 13c–f).For example, the 4:3 rectangle iscomposed by either assembling 108square modules or subtracting a squarefrom a rectangle, where the remainderis an undivided rectangle. Thisremainder can be further subdividedinto square modules.

It is remarkable that this procedureof making a window is very much likethe classical ‘‘anthyphairesis.’’14 Fol-lowing Fowler (1987), Lionel March[10, 11] provides a pictorial approachto the anthyphairetic procedures ofPlatonic mathematics. Fowlerexplained the approach as ‘‘a processof repeated and reciprocal subtractionwhich is then to generate a definitionof ratio as a sequence of repetitionnumbers.’’ Here, March elaborated thenotion by depicting a repetitive sub-tractive and additive composition. Forexample, an 11 9 4 rectangle is sub-tracted, thus leaving a 1 9 1 square asa unit remainder (Figure 14a). Also,based on a 1 9 1 square unit, variousmodules are added and concatenated(Figure 14b). This generates ‘‘a defini-tion of ratio as a sequence ofrepetitions numbers, namely, theanthyphairesis.’’

Most of the windows are approxi-mately measured in whole numbers inproportion. Unlike the complexity of

Figure 10. Window articulation of the street facade.

14Lionel March, Architectonics of proportion: a shape grammatical depiction of classical theory Environmental and Planning B: Planning and Design, 1999, Vol. 26, pp.

91–100.


the space forms, there involves sur-prisingly few room dimensions andcorresponding ratios. In the house, sixdifferent ratios, 1:1, 2:1, 4:3, 6:5, 7:6,and 8:7, are collected. The noticeablecharacter of the ratio is that most of thefractions advance by adding one toboth the numerator and denominator.The ratio is equivalent to the classicalsequence of superparticular numbers(March, [9]). All window proportionsused in the Barragan house areobtained using this procedure. Inaddition, related window designs withdifferent ratios can be further gener-ated with the same method as a familygroup (Figure 15).

The subdivision of the windowframe is a unique practice of Barragan.In Barragan’s other houses, there lie a

variety of subdivisions. The windowacts as a picture frame that shows agiven view of the natural landscapewithout. Albers’s and Piet Mondrian’spaintings come to mind in terms oftheir simple proportional divisions(Figure 16).

In addition, each interior windowshutter in the Barragan house isunique. The white wood panel shut-ters are carefully designed according totheir purpose. Several different typesof panel configurations are observedthroughout the house. They include: Asingle panel system, a three panelsystem with one side containing asingle panel and the other side twopanels, and four panel systems whereeach is divided in different propor-tions. These panels are double hung

using hinges. The shutters in the pri-vate rooms operate in a unique way.For example, when the shutter is divi-ded into four panels, the upper portionof the shutters is meant to be openedfirst; only then can the lower shuttersbe opened. Central to this idea is pri-vacy: The upper portion is meant onlyto allow daylight without losing pri-vacy. In other instances, two shutterpanels are hinged together. Like win-dows, each shutter is proportionallydivided (Figure 17). These divisionsare derived from simple whole numberratios such as 1:1, 2:1, 3:2, 4:3, 7:6, etc.

Barragan did not leave us detaileddescriptions of how he designed, nordid he outline his strategy for control-ling building geometry. In addition,specific geometric practices of thehouse do not appear among Barragan’ssketches or original working drawings.Due to the relative lack of documentaryevidence, one can hardly delineate anexact geometrical principle within theBarragan house. Nevertheless, throughstudying and modeling of plans, ele-vations, sections and constructiondetails, the author believes that Barra-gan, consciously and subconsciously,

Figure 13. Window design derived from the addition and

subdivision of a square.

Figure 11. Proportional design of window framing and grating.

Figure 12. The geometric design of the garden facade.


strives towards an abstract neo-plastic-ity incorporating the use of the squareto design the proportions of the facadesand floor plans. Furthermore, various

interwoven themes that influence thedevelopment of the house clearlyreflect Barragan’s approach. There islittle argument that the house is part of

the beginning of the Barragan style thatculminated in the highly interpretativework reflecting abstract artists such asMathias Goeritz and Josef Albers.

It appears that Barragan did not usea systematic computational method indesigning the house but very much anintuitive procedure of addition andsubtraction of the square. Superficially,the window layout of the street facadelooks disorganized, devoid of anyregularity. Closer observation, how-ever, reveals that the manner in whichthey are designed is similar to theclassical ‘‘anthyphairesis,’’ whichinvolves a repetitive subtractive andadditive composition. Therefore, it canbe speculated that instead of mathe-matically computed harmony, thehouse is a manifestation of abstractneo-plasticity, where the design is asearch towards a glimpsed sub-conscious conception.

In Barragan’s oeuvre, the house wasa turning point, establishing a newstyle in his architecture. The abstractform of geometry used in the Barraganhouse formed the groundwork of hisfuture career and established thefoundations of his developing ideas.Barragan continued to develop similarvocabularies and design elements inhis later projects, most notably the

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.

Figure 14. The anthyphairesis for an 11 9 4 rectangle: A repetitive subtractive (above) and

additive composition (below). (After Lionel March.)


Lopez house, Galvez house, and Gi-lardi house.

ACKNOWLEDGMENT

This work was supported by an INHA

University research grant.

REFERENCES

[1] Ambasz, E. Luis Barragan House and

Atelier for Barragan, Tacubaya, Mexico,

1947, GA Houses, Tokyo: ADA EDITA.

[2] Barragan, L. (1951) Luis Barragan, Arts

and Architecture, August 1951.

[3] Barragan, L. (2004) Casa Luis Barragan

Guide, Mexico: Barragan Foundation.

[4] Burri, R. (2000) Luis Barragan, London:

Phaidon Press.

[5] Doig, A. (1986) Theo Van Doesburg,

London: Cambridge University Press.

[6] Eggener, K. (2001) Luis Barragan’s Gar-

dens of El Pedregal, New York: Princeton

Architectural Press.

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

The Quiet Revolution, Milano: Skira Editore.

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

(1997) The Life and Work of Luis Barra-

gan, New York: Rizzoli.

[9] March, L. (1998) Architectonics of

Humanism, London: Academy Editions.

[10] March, L. (1999a) Architectonics of pro-

portion: a shape grammatical depiction of

classical theory, Environmental and Plan-

ning B: Planning and Design, 26: 91–100.

[11] March, L. (1999b) Architectonics of pro-

portion: historical and mathematical

grounds, Environment and Planning B:

Planning and Design, 26: 447–454.

[12] Martin, I. (1997) Luis Barragan: The

Phoenix Papers, Tempe, Arizona: Center

for Latin American Studies Press.

[13] Park, J. (1996) An Interview with Mark

Mack, POAR, Seoul: Ganhyang.

[14] Pauly, D. and Habersetzer, J. (2002)

Barragan: Space and Shadow, Walls and

Colour, Basel: Birkhauser.

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

Complete Work, New York: Princeton

Architectural Press.

[16] Smith, C.B. (1967) Five Mexican Archi-

tects, New York: Architectural Book

Publishing Co., Inc.

Department of Architecture

Inha University

253 Yonghyun-dong, Nam-gu

Incheon 402-751

Korea


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

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.


Reviews Osmo Pekonen, Editor

Feel like writing a review for The

Mathematical Intelligencer? You are

welcome to submit an unsolicited

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University of Jyvaskyla, Finland


Introduction toBayesianScientificComputing: TenLectures onSubjectiveComputingby Daniela Calvetti and Erkki

Somersalo

HEIDELBERG, SPRINGER SCIENCE + BUSINESS

MEDIA, 2007, 202 PP., EUR32.95 ISBN 978-0-

387-73393-7

REVIEWED BY URI ASCHER

TThe application of scientific com-puting as a tool for understand-ing and gaining quantitative

knowledge of physical processes typi-cally has two phases. In the first phase,a mathematical model is generated,and in the second, the model is simu-lated on a computer using appropriatenumerical methods. Now, assumingthat the mathematical model is not soincredibly complex that it must besimplified, should these two phases beindependent of each other?

There is a lot to be said for such aphase separation. Many useful numeri-cal methods for differential equations,for instance, have been derived, ana-lyzed, and programmed without aspecific application in mind. Thus, oncea researcher in biological evolution hassucceeded in formulating the propaga-tion and control of a measles epidemicas a time-dependent system of ordinarydifferential equations (ODE), there arecanned routines available that may beused for the subsequent simulation ofthe ODE system. These routines aretypically both more efficient and morereliable than what the mathematical

biologist would write for a special-pur-pose end, even though the specificapplication was not taken into accountduring the design and implementationof the general-purpose software. Prob-ably the extreme in this regard are theexcellent general packages available fornumerical linear algebra tasks such assolving a linear system of equations orfinding the eigenvalues of a matrix [2, 5].Under normal circumstances, suchpackages, and general numericalmethods, should certainly be usedrather than reproduced.

The rationale for a full-phase sepa-ration breaks down, however, when themathematical model to be simulated issignificantly incomplete or in doubt.Such is often the case with ill-posedinverse problems, where an orthodoxsolution of the mathematical modelinitially presented is neither possiblenor desirable. Note the differencebetween the objective notion of solv-ability of an ill-posed problem,considered by Hadamard more than100 years ago, and the subjective notionof desirability. For example, the prob-lem of deblurring a noisy image (say apolice snapshot of one’s license platewhen caught speeding) canbemodeledas a singular or highly ill-conditionedlinear system. The latter can be subse-quently solved approximately, usingsome form of regularization [3, 4]. Butthe desirability of such a solution maywell depend on the manner by whichthe unknown measurement noise hasbeen handled or accounted for! A nat-ural, although by no means only, way toaccount for lack of knowledge in mod-eling and for subjectivity, is theBayesian probability framework. Thisestablishes the connection betweenthree separate areas that this bookexplores.

Introduction to Bayesian ScientificComputing is a 200-page, easilyaccessible, pleasant introduction fusingBayesian approaches with numericallinear algebra methods for inverseproblems: A tutorial that one does nothave to believe in all its details toenjoy. To make it so accessible, the


authors often use informal language, alot of motivation, and an excellent setof examples. They essentially avoidformal mathematics (e.g., no theoremsand no proofs, although they do useformulae, and their underlying mathe-matics is carefully thought out). Thereis no attempt at completeness, nor ofcomprehensive referencing. Theauthors also avoid explicit mention ofmachine learning [1]. Having had somelimited previous exposure to all threecomponents of this mixture, namelyBayesian probability, numerical linearalgebra and inverse problems, thisbook’s approach has worked for me.

There are 10 chapters, each coveringa ‘‘lecture’’ in a graduate course that theauthors have given at universities inItaly, Finland, and the USA. An uniniti-ated graduate student probablywould need a week or two to absorbthe material in each of these lectures, sothere is a blueprint here for a graduatecourse of a normal trimester length.However, as the authors acknowledge,this is not a self-contained textbook,and it must be supported by a readinglist of other texts, which they supply.

The first three chapters introducethe necessary essentials for Bayesianinference. In an inverse problem, wewant to estimate an unknown quantityx from a set of indirect measurementsy. The corresponding problem of sta-tistical inference is to infer propertiesof an unknown probability densitydistribution given the data which havebeen generated from that distribution.Following essential definitions and afew basic theorems, one arrives atthe Bayes formula that says that theposterior probability density of x giveny is proportional to the likelihood,which is the density of y given x

(corresponding to the forward prob-lem in inverse problem parlance)times the density of the prior (corre-sponding to introducing a prioriinformation such as past experienceor image smoothness). A maximuma posteriori (MAP) estimator can sub-sequently be obtained for x.

Chapter 4 then introduces the thirdlink, numerical methods for linear sys-tems of algebraic equations, with an eyetowards ill-conditioned problems andthe smoothing properties of truncatedconjugate gradient-type iterations. Thisis followed in Chapter 6 with the prob-abilistic design of preconditioners,called here priorconditioners, whichallow a few iterations towards thesolution of an ill-conditioned linearsystem to capture more features of adesired solution. There is also a quicksection on designing a prior based on atraining set and on model reductionusing principal component analysis.

Chapters 7 and 8 are concerned withconditional Gaussian densities andyield some rather important formulaefor noisy linear systems of algebraicequations. The basic task is to obtain theprobability distribution of some com-ponents of a multivariate, normallydistributed random variable with thevalues of the other components fixed.Here, there is a good emphasis andexploitation through examples of theadditional information that the Bayes-ian probability framework yields,namely, not only a point solution (or asingle output), but also means forassessing its worth and trustworthinessin terms of credibility envelopes. Thereusually are, after all, other, often simplerways to incorporate prior informationinto a regularization method, if thatwere the only thing at stake.

Chapters 5 and 9 address theimportant issue of sampling from agiven distribution in order to verify thatthe distribution is what we think it is(or to approximate integrals in manydimensions). This exposition culmi-nates in the Markov Chain Monte Carlo(MCMC) sampling and the classicalMetropolis-Hastings algorithm.

Finally, Chapter 10 wraps it up byusing concepts and methods from dif-ferent previous chapters, introducinghypermodels and solving an exampleof deblurring a one-dimensional sur-face with discontinuities.

What I like most about this book isthe apparent enthusiasm of the authorsand their genuine interest in explain-ing rather than showing off. Thisenthusiasm is contagious, and theresult 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 Com-

pany, 1996.

Department of Computer Science

University of British Columbia

Vancouver, BC V6T 1Z4, Canada



A CertainAmbiguity.A MathematicalNovelby Gaurav Suri and Hartosh Sing Bal

PRINCETON, PRINCETON UNIVERSITY PRESS,

2007, US$ 27.95, 292 PP. ISBN-13: 978-0-691-

12709-3

REVIEWED BY TOM PETSINIS

WWorks of fiction containingvarying degrees of mathe-matics have burgeoned in

recent years. They include historicalfiction based on mathematicians, sur-real works exploring mathematicalideas, and speculative fiction based onfamous theorems and conjectures.Publishers see such books as demysti-fying the subject and perhaps makingit accessible to a wider readership.Conversely, the general reading publicmust be interested in the subject mat-ter, or new titles would not beemerging with such frequency.

Co-authored by Suri and Bal, ACertain Ambiguity is a weighty addi-tion to what is becoming a genre. Thenovel is narrated by Ravi, a youngIndian whose interest in mathematics isignited by his grandfather Vijay Sahni,who dies tranquilly at the beginning ofthe book. At 18, Ravi leaves India for anunnamed American university, intenton pursuing a career in finance. In hisfirst semester, he enrolls in an electiveunit called ‘‘Thinking about Infinity,’’presented by the inspiring Nico Alip-rantis, an unconventional lecturerwhose skill in making baklava suggestsa Greek background. Ravi soon learnsthat his grandfather visited America inthe early decades of the 20th century,and this sets him off on a trail ofdetection. It turns out that as a visitingscholar in the fictional town of

Morisette, New Jersey, Vijay wasimprisoned for expounding mathe-matical ideas that were construed asblasphemous. (He was prisoner num-ber 1729—an obvious reference to theIndian mathematician Ramanujan who,on his death-bed, saw this number asthe smallest number that could beexpressed as the sum of two cubes intwo different ways.) From newspapersand transcripts of the trial, Ravi dis-covers that Vijay developed a rapportwith John Taylor, the presiding judge.The two men discussed at some lengththe question of God’s existence,Euclid’s axioms, and the nature of ulti-mate truth. In the end, the judgequestioned his own religious views,secured Vijay’s early release, and thetwo men went on to maintain a lifelongfriendship.

As Ravi’s course progresses, he isforced to decide between a career infinance to repay his family’s invest-ment in his education, or one inmathematics in honour of his belovedgrandfather. The authors withhold hisdecision, though the reader feels theyoung man’s journey, his enquiringnature, and the influence of thecharismatic Nico will draw him tomathematics.

The relationship between Ravi andhis grandfather recalls another novelwith mathematical content: UnclePetros and Goldbach’s Conjecture byApostolos Doxiadis. The authors of ACertain Ambiguity could have usedDoxiadis’s economy in structuring theirnovel. As it stands, their work is overlydiscursive, with long tracts of mathe-matical exposition that interrupt thenarrative flow. At times it appears asthough the narrative is nothing morethan a vehicle for lectures on mathe-matics. This is often the problem withnovels of this type: They fail to strike theright balance between the didactic andthe dramatic. In this case, the novel’sdidactic sections are lucid and engag-ing. The history of mathematical infinityis clearly outlined, beginning withZeno’s Paradoxes through to conver-gent series. Cantor’s hierarchy of

infinity is expressed and illustrated in amanner accessible to the generalreader. Euclid’s fifth postulate is dis-cussed at some length, with interestingreferences to Gauss, Bolyai, Lobachev-sky, Riemann, and Einstein. But despitethe admirable collection of well-explained ideas, the book falls short inmany areas of literary fiction. Ravi, themain character, is thinly drawn and failsto grip the reader with his first-personvoice. There is little sense of time andplace. The mathematical interpolationsare too lengthy and come at theexpense of narrative, to the extent ofdiminishing the reader’s interest in thecharacters. Other material appears inthe novel without preparation or justi-fication. There a several gratuitous‘‘diary’’ entries from mathematiciansranging from Pythagoras to Godel. Inwhat is essentially a realistic novel,these entries are not sufficiently framedby the story. Had they been Ravi’sdreams or daydreams, there may havebeen some justification for them: As it is,they are simply further expositionwithout integration. The novel endswith a chapter-length diary record ofJudge Taylor’s experiences and trip toIndia to visit Vijay. The literary writing isat its strongest here, with the Judge,more so than Ravi, emerging as thenovel’s best-developed character.

Authors who embark on novels ofideas, especially mathematical, face thechallenge of making those ideasappear to come naturally from thecharacters; in other words, the ideasmust be shown through flesh andblood. One of the attributes of goodfiction is its power to pull readers intoits world and keep them interested inits characters. A Certain Ambiguity isquite strong on mathematical exposi-tion. Unfortunately, as a novel, itdoesn’t fully draw the reader into itsfictional landscape.

Teaching and Learning Services

Victoria University

Footscray Park, Melbourne

VIC 8001, Australia



Symmetryand the Monsterby Mark Ronan

NEW YORK, OXFORD UNIVERSITY PRESS, 2006,

255 PP. US$27.00 ISBN 978-0-19-280722-9.

REVIEWED BY KISHORE MARATHE

SSymmetry and the Monsterrecounts the story of an excep-tional result in the history of

mathematics: The classification of finitesimple groups. The existence anduniqueness of the largest sporadicgroup, dubbed the Monster, wasthe last piece in the classification. Thecomplete classification is arguably thegreatest achievement of 20th centurymathematics. In fact, it is unique in thehistory of mathematics: The result ofhundreds of mathematicians working inmany countries around the world forover a quarter century. This global ini-tiative was launched by DanielGorenstein, whose book [5] is still anexcellent general reference for thismaterial.

We now describe the highlights ofthis fascinating story. The first fourchapters introduce groups and theirapplication in Galois’s work. Recallthat a group is called simple if it hasno proper nontrivial normal sub-groups. Thus, an Abelian group issimple if and only if it is isomorphicto one of the groups Zp, for p a primenumber. This is the simplest exampleof an infinite family of finite simplegroups. Another infinite family offinite simple groups is the family ofalternating groups An, n [ 4 that westudy in the first course in algebra.These two families were known in the19th century. The last of the familiesof finite groups, called groups of Lietype, were defined by Chevalley inthe mid 20th century. Chapters 5 to 9discuss this material. By the early 20thcentury, the Killing–Cartan classifica-tion of simple Lie groups defined overthe field C of complex numbers hadproduced four infinite families andfive exceptional groups. This classifi-cation starts by classifying simple Liealgebras over C and then constructing

corresponding simple Lie groups. In1955, using this structure but replac-ing the complex numbers by a finitefield, Chevalley’s fundamental papershowed how to construct finitegroups of Lie type. This work led tothe classification of all infinite familiesof finite simple groups.

However, it was known that therewere finite simple groups, called spo-radic groups, that did not belong to anyof these families. Chapters 10 to 14 aredevoted to the discoveries of the 26sporadic groups. The first sporadicgroup was constructed by Mathieu in1861. In fact, he constructed five spo-radic groups, now called Mathieugroups. There was an interval of morethan 100 years before the sixth sporadicgroup was discovered by Janko in 1965.Two theoretical developments played acrucial role in the search for new simplegroups. The first of these appeared inBrauer’s address at the 1954 ICM inAmsterdam. It gave the definitive indi-cation of the surprising fact that generalclassification theorems would have toinclude sporadic groups as exceptionalcases. In fact, Fischer discovered andconstructed his first three sporadicgroups in the process of proving such aclassification theorem. Brauer’s workmade essential use of elements of order2. The second came in 1961, when Feitand Thompson proved that every non-Abelian simple finite group contains anelement of order 2. The proof of thisone line result occupies an entire 255-page issue of the Pacific Journal ofMathematics (Volume 13, 1963). Beforethe Feit–Thompson theorem, the clas-sification of finite simple groupsseemed to be a rather distant goal. Thistheorem and Janko’s new sporadicgroup greatly stimulated the mathe-matics community to look for newsporadic groups.

John Leech had discovered his 24-dimensional lattice while studying theproblem of sphere packing. The Leechlattice provides the tightest spherepacking in 24 dimensions. (However,the sphere packing problem in otherdimensions is still wide open.) Sym-metries of the Leech lattice containedMathieu’s largest sporadic group. Italso had a large number of symmetriesof order 2. Leech believed that thesymmetries of his lattice containedother sporadic groups as well. Leech

was not a group-theorist and he couldnot get group-theorists interested in hislattice. But he did find a young math-ematician (who was not a group-theorist) to study his work. In 1968,John Conway was a junior facultymember at Cambridge. He quicklybecame a believer in Leech’s ideas. Hetried to get Thompson, the great guruof group-theorists, interested in hiswork. Thompson told him to find thesize of the group of symmetries andthen call him. Conway later remarkedthat he did not know that he was usinga folk theorem which says: The twomain steps in finding a new sporadicgroup are (i) find the size of the groupof symmetries, and (ii) call Thompson.Conway worked very hard on thisproblem and soon came up with anumber. This work turned out to be hisbig break. It changed the course ofhis life and has made him into aworld-class mathematician. He calledThompson with his number. Thomp-son called back in 20 minutes and toldhim that half his number could be apossible size of a new sporadic groupand that there were two other newsporadic groups associated with it.These three groups are now denotedby Co1, Co2, Co3 in Conway’s honor.Further study by Conway and Thomp-son showed that the symmetries of theLeech lattice give 12 sporadic groups inall, including all five Mathieu groups. Inthe early 1970s, Conway started theATLAS project to collect all essentialinformation (mainly the charactertables) about the sporadic groups andsome others. The work continued intothe early 1980s when all the sporadicgroups were finally known.

After Conway’s work, the nextmajor advance in finding new sporadicgroups came through the work ofBerndt Fischer. Working under Baer,Fischer became interested in groupsgenerated by transpositions. Recallthat, in a permutation group, a trans-position interchanges two elements.Fischer first proved that a group Ggenerated by such transpositions fallsinto one of six types. The first type is apermutation group and the next fourlead to known families of simplegroups. It was the sixth case that led tothree new sporadic groups, each rela-ted to one of the three largest Mathieugroups. The geometry underlying the

76 THE MATHEMATICAL INTELLIGENCER � Springer Science+Business Media, LLC.

construction of G is that of a graphassociated to generators of G. Permu-tation groups and the classical groupsall have natural representations asautomorphism groups of such graphs.Fischer’s graphs give not only someknown groups, but also his three newsporadic groups. He published thiswork in 1971 as the first of a series ofpapers; no further papers in the seriesever appeared. In fact most of hiswork is not published. Fischer contin-ued studying other transpositiongroups. This led him first to a newsporadic group, now called the BabyMonster. By 1981, 20 new sporadicgroups were discovered, bringing thetotal to 25. The existence of the 26thand the largest of these groups wasconjectured independently by Fischerand Griess in 1973. Several scientistsconjectured that this exceptionalgroup must have relations with otherareas of mathematics as well as withtheoretical physics. The results thathave poured in since then seem tojustify this early assessment. Somestrange coincidences noticed first byMacKay and Thompson were investi-gated by Conway and Norton. Theycalled this group the Monster and theirunbelievable set of conjectures ‘‘Mon-strous Moonshine.’’ Their paper [2]appeared in the Bulletin of the LondonMathematical Society in 1979. Thesame issue of the Bulletin containedthree papers by Thompson discussinghis observations of some numerologybetween the Fischer–Griess Monster Mand the elliptic modular functions.Thompson stated his conjecturesabout the relation of the characters ofthe Monster and Hauptmoduls forvarious modular groups. He alsoshowed that there is at most one groupwhich possesses the propertiesexpected of M and has a complex,irreducible representation of degree196883 = 47.59.71 (47, 59 and 71 arethe three largest prime divisors of theorder of the Monster group). Conwayand Norton had conjectured earlier thatthe Monster should have a complex,irreducible representation of degree196883. Based on this conjecture,Fischer, Livingstone and Thorne (Bir-mingham notes 1978) computed theentire character table of the Monster.

The construction of the Monster wasannounced by Griess in 1981, and the

complete detailswere given in [6].Griessfirst constructed a commutative, nonas-sociative algebra A of dimension 196884and then showed that theMonster groupis its automorphism group. In the sameyear, the final step in the classification offinite simple groups was completed byNorton by establishing that the Monsterhas an irreducible complex representa-tion of degree 196883 (the proofappeared in print later). Combined withthe earlier result of Thompson, thisproved the uniqueness of the Monster.So the classification of finite simplegroups was complete. The various partsof the classification proof together fillthousands of pages. The project toorganize all this material and to preparea flow chart of the proof is expected tocontinue for years to come.

The last three chapters give a briefaccount of the construction of the Mon-ster and the Monstrous MoonshineConjectures. We now give a mathemati-cal formulation of these conjectures.

Monstrous MoonshineConjectures

1. For each g [ M there exists a Mac-Kay–Thompson series Tg(z) withnormalized Fourier series expan-sion given by

TgðzÞ¼q�1þX1

1

cgðnÞqn ;q¼ e2piz :

ð1Þ

There exists a sequence Hn of rep-resentations of M, called the headrepresentations, such that

cgðnÞ¼vnðgÞ; ð2Þ

where vn is the character of Hn.2. For each g [ M, there exists a Hau-

ptmodul Jg for some modular groupof genus zero, such that Tg = Jg. Inparticular,

(a) T1 = J1 = J, the Jacobi Hauptmo-dul for the modular group C.

(b) If g is an element of prime orderp, then Tg is a Hauptmodul forthe modular group Gp studiedby Ogg.

3. Let [g] denote the set of all elementsin 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 classfunction in the usual sense. How-ever, [g] is not the usual conjugacyclass. There are 194 conjugacy clas-ses of M but only 171 distinctMacKay–Thompson series.

Conway and Norton calculated all thefunctions Tg and compared their firstfew coefficients with the coefficients ofknown genus-zero Hauptmoduls. Sucha check turns out to be part of Borc-herds’s proof, which he outlined in hislecture at the 1998 ICM in Berlin [1].The first step was the construction ofthe Moonshine Modul. The entire book[3] by Frenkel, Lepowsky, and Meur-man is devoted to the construction ofthis module, denoted by V \: It has thestructure of an algebra called theMoonshine vertex operator algebra(also denoted by V \). They proved thatthe automorphism group of the infinitedimensional graded algebra V \ is thelargest of the finite, sporadic, simplegroups, namely, the Monster.

The second step was the construc-tion by Borcherds of the Monster Liealgebra using the Moonshine vertexoperator algebra V \: He used this alge-bra to obtain combinatorial recursionrelations between the coefficients cg(n)of the MacKay–Thompson series. It wasknown that the Hauptmoduls satisfiedthese relations and that any functionsatisfying these relations is uniquelydetermined by a finite number of coef-ficients. In fact, checking the first fivecoefficients is sufficient for each of the171 distinct series. Thus all the ‘‘Mon-strous Moonshine’’ conjectures are nowparts of what we can call the ‘‘Moon-shine Theorem.’’ Its relation to vertexoperator algebras, which arise as chiralalgebras in conformal field theory andstring theory, has been established. Inspite of the great success of these newmathematical ideas, many mysteriesabout the Monster are still unexplained.A recent update on the Moonshine maybe found in the book by Gannon [4].

We conclude this summary with acomment, a modification of the remarksmadebyOgg in [8]when the existenceofthe Monster group and its relation tomodular functions were still conjectures(strongly supported by computationalevidence). Its deep significance for the-oretical physics is still emerging. Somathematicians and physicists, young

� 2008 The Author(s). This article is published with open access at Springerlink.com Volume 31, Number 1, 2009 77

and old, should rejoice at the emergenceof a new subject, guaranteed to be richand varied and deep, with many newquestions to be asked and many of theconjectured results yet to be proved. It isindeed quite extraordinary that a newlight should be shed on the theory ofmodular functions, one of the mostbeautiful and extensively studied areasof classical mathematics, by the largestand the most exotic sporadic group, theMonster. That its interaction goesbeyond mathematics, into areas of the-oretical physics, such as conformal fieldtheory, chiral algebras and string theory,may be taken as strong evidence for anewareaof researchwhich this reviewerhas called in [7] ‘‘Physical Mathematics.’’

Symmetry and the Monster is writtenin nontechnical language and yet con-veys the excitement of a greatmathematical discovery usually accessi-ble only to professional mathema-ticians. The author knew many of thecontributors, and this brings a nice per-sonal touch to the narrative. His use ofnonstandard terminology seems quiteunnecessary, however. The term ‘‘atomof symmetry’’ is not more illuminatingthan ‘‘simple group’’ for the lay readerand is annoying to anyone who hastaken a first course in algebra. There areseveral factual errors and misstatements.The worst puts Newton and Leibnizdeveloping calculus in the 16th century(p. 87) and again in the 17th century

(p. 89). Janos Bolyai’s appendix is at theend ofhis father’s bookongeometry andnot in the book by Gauss (p. 195). Partsdealing with physics, especially the lastchapter, contain misstatements. There isnoevidence at this time that string theorycombines quantum physics and generalrelativity (p. 72) or that it provides amodel for elementary particles (p. 218).The level of material varies greatly. It isdoubtful that a reader who needs to bereminded of the quadratic formula,golden ratio or p and e will take awaymuchmathematics from thisbook.But inspite of these shortcomings, the bookgives a gooddescriptionofmanyaspectsof an important event in the history ofmathematics.

OPEN ACCESS

This article is distributed under the

terms of the Creative Commons Attri-

bution Noncommercial License which

permits any noncommercial use, dis-

tribution, 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-

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.

[3] I. Frenkel, J. Lepowsky, and A. Meurman.

Vertex Operator Algebras and the Mon-

ster. Pure and App. Math., # 134.

Academic Press, New York, 1988.

[4] T. Gannon, Moonshine Beyond the Mon-

ster. Cambridge University Press,

Cambridge, 2006.

[5] D. Gorenstein, Finite Simple Groups. Ple-

num 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 Sci-

ences. 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.


City University of New York

Brooklyn College

Brooklyn, NY 11210, USA

e-mail: [email protected];

[email protected]

Max Planck Institute for Mathematics in the

Sciences Leipzig

Dresden, Germany


Ernst Zermelo. AnApproach to HisLife and Workby Heinz-Dieter Ebbinghaus, in

cooperation with Volker Peckhaus

BERLIN, SPRINGER SCIENCE + BUSINESS MEDIA,

2007, XIV + 356 PP, €49.95, ISBN: 978-3-540-

49551-2

REVIEWED BY HENRY E. HEATHERLY

EErnst Zermelo was the most influ-ential set-theorist of the first halfof the 20th century. He is best

known now for his work on the Axiomof Choice and for being the first personto give an axiomatic treatment of settheory [1]. However, his mathematicalcareer began along quite different lineswith a doctoral dissertation on thecalculus of variations (Berlin, 1894), anassistantship with Max Planck at theInstitute for Theoretical Physics inBerlin, and several important, albeitcontroversial, papers on thermody-namics. Ebbinghaus has given us a fullbiography of Zermelo the mathemati-cian and scientist, as well as aninsightful description of Zermelo’sprofessional and personal life, and hisinteractions with colleagues andadversaries.

Zermelo left Planck’s Institute in1897 for further study in theoreticalphysics at Gottingen, which resulted inan Habilitation thesis on hydrody-namics in 1899. However, soon after hearrived in Gottingen, Zermelo began totake serious interest in set theory andlogic. He later wrote (in 1930) that thiswas due to the influence of David Hil-bert. In the winter semester of 1900/1901, Zermelo gave the first course everdevoted entirely to set theory. By 1902,he had published his first paper on thesubject, a short note on transfinite car-dinal number addition. At the ThirdInternational Congress of Mathemati-cians (Heidelberg, August 1904), JuliusKonig delivered a lecture where heclaimed that Cantor’s ContinuumHypothesis was false and that the car-dinality of the continuum is not analeph. He also claimed to have refuted

the Well Ordering Principle (WOP).This caused considerable controversyin which Zermelo was activelyinvolved. He quickly found an error inKonig’s argument, as did others. Thisaffair focused Zermelo’s attention onthe WOP.

By late September 1904, Zermelohad a proof of the WOP and hadexplicitly stated his ‘‘principle ofchoice,’’ later called the ‘‘Axiom ofChoice.’’ He communicated theseresults in a letter to Hilbert, and Hilberthad the relevant parts published [2, pp.139–141]. At the end of the paper,Zermelo states that he owed the ideaof using the principle of choice toErhard Schmidt.

Zermelo’s proof of the WOPbecame the object of intense criticismthat arose from three main sources: Anuneasiness with his new vehicle, theprinciple of choice; suspicion of anyargument that seemed similar to thosewhich had led to recently discoveredparadoxes; and an old mistrust ofCantor’s set theory. Among those whowere highly critical were Poincare,Schoenflies, Borel, and Felix Bernstein.In 1908, Zermelo gave a second proofof the WOP, again using the principleof choice [2, pp. 183–198]. Also in thispaper is a critique of the objectionsthat had been raised against the firstproof.

To solidify the foundation for hisproof of the WOP, further its compre-hensibility, and make clear the role ofthe principle of choice, Zermelo for-mulated the first axiomatic treatment ofset theory, which he published in 1908[2, pp. 198–215]. He did this usingseven axioms, with the Axiom ofChoice (AC) being number six. Henoted that he was unable to prove theconsistency of these axioms, but heshowed that several of the knownparadoxes of set theory cannot beobtained from his axioms. AlthoughZermelo’s system was immediatelyused by some, the general response toit was ambivalent (see [1, Section 3.3]).It is telling that Hausdorff did not usethe axiomatic point of view in his veryinfluential book on set theory pub-lished in 1914. The first textbook onaxiomatic set theory was by Fraenkel in1919, who used Zermelo’s system as hisbase. In the early 1920s, both Fraenkeland Skolem refined Zermelo’s system.

In 1930, Zermelo reformulated his sys-tem, adding two more axioms. Hecalled this system the ‘‘Zermelo–Fraenkel axiom system.’’ It includedAC, so now we would call it ZF + AC,or ZFC.

From 1902 until 1907, Zermelo was aPrivatdozent at Gottingen. His aca-demic career was progressing slowly,possibly because of several interrup-tions due to illness. Using his verysignificant influence with the PrussianMinistry of Cultural Affairs, Hilbert wasable to have Zermelo appointed to alectureship in mathematical logic atGottingen. Because of health problems,Zermelo could not begin his lectures asscheduled in the winter semester 1907/1908, but in the summer semester of1908, he gave the first course in math-ematical logic ever offered at a Germanuniversity. Zermelo’s plans to write abook on mathematical logic neverachieved fruition, however; the lecturenotes from his course on mathematicallogic are in the Zermelo Nachlass, [3].

The lectureship in logic was only atemporary expedient. Zermelo wanteda permanent position. By 1909, he hadimpressive research credentials with asubstantial list of published work incalculus of variations, thermodynam-ics, hydrodynamics, and set theory. Healso was strongly supported by Hilbert.Yet he had been passed over severaltimes for permanent positions. Thereasons were illness and controversy.

Zermelo’s early papers on thermo-dynamics contained some sharpdisagreements with earlier work byLudwig Boltzmann, which led to apublic controversy between the twolasting until just after Boltzmann’sdeath in 1906. Ebbinghaus (see p. 25)illustrates the degree of acrimony inthis affair with excerpts from a letterfrom Boltzmann to Felix Klein. Inaddition to the general controversyover Zermelo’s work on the WOP andAC, Zermelo was noted for the‘‘polemical aspect’’ of his personality,an attribute that was still being com-mented on when Zermelo was in hissixties. Helmuth Gericke, who wasZermelo’s research assistant in 1934,later commented that ‘‘sometimes heeven insulted his friends.’’ Amongother things, this caused a personalcontroversy with Felix Bernstein,which in turn became linked to


arguments within the PhilosophicalFaculty at Gottingen.

Zermelo’s ill health, extending backto his youth, frequently left him unableto satisfy his teaching commitments. By1905, it had become a serious cause ofinterruptions in his career. He sufferedfrom respiratory illnesses and wasdiagnosed in June 1906 with tubercu-losis of the lungs. To this one mustadd his occasionally erratic mentalcondition.

In 1910, Zermelo’s outstandingresearch record and the strong recom-mendations by Hilbert and othersovercame the negatives and he wasoffered a professorship at the Univer-sity of Zurich. His time in Zurich, 1910to 1916, is the only period in his life thathe held a paid university professorship.In Zurich, Zermelo continued hisresearch on set theory as well asworking on measure theory, abstractalgebra, and game theory. The latterconcerned an application of set theoryto the game of chess (he was anenthusiastic chess player). Zermelo’sonly doctoral student at Zurich wasWaldemar Alexandrow, who com-pleted a dissertation on the foundationsof measure theory in 1915. Paul Bern-ays and Ludwig Bieberbach each madetheir Habilitation under Zermelo inZurich. In light of his later renown in settheory and logic, it is somewhat sur-prising that Bernays’s thesis in Zurichwas on modular elliptic functions; helater did a second Habilitationsschrifton logic at Gottingen under the direc-tion of Hilbert. Long interruptions dueto illness hindered Zermelo in super-vising the mathematical developmentof students. His health continued todeteriorate until he was forced to retirefrom 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 athealth resorts, but he continued towork mathematically. In October 1916,he was awarded the Alfred Ackermann-Teubner Prize of Leipzig University. InMarch 1921, Abraham Fraenkel began acorrespondence with Zermelo con-cerning the independence of Zermelo’saxioms for set theory. In one such let-ter, Zermelo formulated a second-orderversion of the axiom of replacement,yet he also criticized this because of itsnondefinite character. He also took this

ambivalent position publicly, forexample at a 1921 meeting of the Ger-man Mathematical Union in Jena.

In October 1921, Zermelo moved toFreiburg in southwestern Germany,where he lived until his death in 1953. In1926, he was appointed ‘‘full honoraryprofessor’’ at the Mathematical Instituteof the University of Freiburg. This car-ried no salary, but he was given the feespaid by participants in his courses. Hishealth improved somewhat, allowinghim to undertake modest teachingactivities. He typically gave one courseeach semester, and these ranged fromapplied mathematics to foundations, aswell as courses in complex analysis, realfunctions, and differential equations.His scientific activity increased remark-ably. His work on applied mathematicsincluded a paper on ‘‘navigation in theair as a problem of the calculus of vari-ations’’ and one on evaluating theresults of chess tournaments. Of course,he continued work on the foundationsof set theory and on logic. This includedwork on infinitary languages and infi-nitary logic.

During his Freiburg period, Zermelobecame involved in several acrimoni-ous controversies with leading figuresin set theory and logic. The first waswith Fraenkel. Zermelo served as editorfor the publication of Cantor’s collectedmathematical and philosophical works.This project, which Zermelo recom-mended to the Berlin publishing houseSpringer-Verlag in 1926, led to contro-versy and hard feelings with Fraenkel.The main point of contention con-cerned the biographical essay of Cantorthat Fraenkel wrote for the collectedworks and Zermelo’s highly criticalremarks concerning this in correspon-dence with Fraenkel.

The next controversy Ebbinghauscalls ‘‘A ‘War’ Against Skolem.’’ In bothhis published comments and in per-sonal correspondence, Zermelo reactedvigorously and even harshly to ThoralfSkolem’s 1929 paper which gave a ver-sion of definiteness that essentiallycorresponds to second-order definabil-ity. Zermelo felt that axiomatic settheory was threatened by Skolem’sresults and that he had ‘‘a particularduty’’ to fight against it.

Soon after this, Zermelo becameinvolved in the controversy swirlingaround Godel’s Incompleteness Theorem

of 1930. This led to a lively personal cor-respondencewithGodel, aswell as publicor published remarks by each of them.Zermelo conducted himself in his usualtactless style and seemed to be concernedwith ‘‘plots’’ against him. During this per-iod, he suffered a nervous breakdownfrom which he quickly recovered.

Less than two years after the Nazistook power in Germany, Zermelo ranafoul of the regime. In January 1935,Eugen Schlotter, an assistant at theMathematical Institute and an ardentNazi, denounced Zermelo to the uni-versity authorities. Other accusationsquickly followed, and a formal inves-tigation was undertaken by the rector,who officially recommended that Zer-melo give up his teaching duties. InMarch 1935, Zermelo resigned from hisposition at the University of Freiburg.

The loss of his honorary professor-ship and his disappointment over thebehavior of some of his former col-leagues left Zermelo bitter. The year1935 marks the beginning of a decline inZermelo’s mental energy. He worked ona book on set theory which was nevercompleted. By the end of the 1930s, hehad withdrawn completely from thescene of foundations of mathematics.However, he continued some smallermathematical projects, pureandapplied,as evidencedbyhis Nachlass. InOctober1943, he married Gertrud Seekamp,whom he had known for some time. In1946, Zermelo was reappointed as hon-orary professor at Freiburg, but he wasunable to return to lecturing because ofincreasing blindness and infirmities ofage. He died on May 21, 1953. Gertrudlived to be over 100, outliving Ernst bymore than 50 years.

Thebiographyunder reviewcontainsnumerous photographs of Zermelo, hisfamily, and his colleagues. Someof thesegive insight into Zermelo’s personality,e.g., Zermelo at the dinner table with hisdog (p. 175) and the small three-wheeledcar that he drove in the early 1930s (p.146). Of considerable interest, as well asbeing helpful in getting an accurate,unvarnished perception of events andpersonalities, are the many excerptsfrom letters, not only toor fromZermelo,but also correspondence between othermajor figures. There is anextensive list ofreferences and a helpful chronologicalvita. The book is well edited, with only afew minor typographical errors. It is


highly recommended for universitylibraries and for those interested inthe history of mathematics of the 20thcentury.

REFERENCES

[1] Gregory H. Moore, Zermelo’s Axiom of

Choice. Its Origins, Development, and

Influence, Springer-Verlag, New York,

1982.

[2] Jean van Heijenoort, From Frege to Godel.

A Source Book in Mathematical Logic,

1879–1931, Harvard University Press,

Cambridge, 1967.

[3] Ernst Zermelo, Mathematische Logik. Vor-

lesungen gehalten von Prof. Dr. E. Zermelo

zu Gottingen im S.S. 1908, lecture notes

by Kurt Grelling. Universitatsarchiv Frei-

berg, C129/224 (Part I) and C129/215

(Part II), Freiberg.

Mathematics Department

University of Louisiana at Lafayette

217 Maxim Doucet Hall, P.O. Box 41010,

Lafayette, LA 70504-1010, USA



Number TheoryThrough Inquiryby David C. Marshall, Edward Odell,

and Michael Starbird

WASHINGTON, DC, THE MATHEMATICAL ASSOCI-

ATION OF AMERICA, 2007, HARDCOVER, IX +

140 PP., US$51.00, ISBN: 978-0-88385-751-9

REVIEWED BY JOHN J. WATKINS

RR. L. Moore strode like a giant overthe mathematical landscape ofAmerica during the first half of

the twentieth century. Born in Dallas,Texas, he spent 49 of his illustrious64-year career at the University ofTexas. A key member of the Americanschool of point-set topology, Mooreserved as president of the AmericanMathematical Society, produced 50doctoral students, six of whom even-tually became either president or vice-president of the AMS, while five servedas president of the Mathematical Asso-ciation of America, and now these 50students have themselves produced2239 doctoral descendants. In 1999,at the end of the millennium, KeithDevlin in his MAA Online columnclaimed that R. L. Moore was the‘‘greatest university mathematics tea-cher ever.’’ It is hard to dispute thatclaim, though sadly it should be addedthat Moore’s reputation carries with itan often ignored blemish: Moore helddeeply racist attitudes toward blackstudents.

While still a student at the Univer-sity of Chicago from 1903–1905, RobertLee Moore conceived of a radical newstyle of teaching. He would soon moldthis idea into a highly successful tech-nique, a new method of teaching thatwould eventually bear his name: theMoore Method. It is very likely that, inthe end, it is this innovative teachingmethod that will be Moore’s mostlasting contribution to mathematics.Even now, more than a hundred yearslater, people enthusiastically follow theexample set by Moore. An extremelyambitious and attractive new textbook,Number Theory Through Inquiry, byDavid C. Marshall, Edward Odell, andMichael Starbird, has been writtenspecifically to use the teaching

technique that has now becomeknown as the Modified Moore Method.

Moore developed his method toproduce research mathematicians. Hehand picked students for his graduatecourse in topology, and as far as he wasconcerned, the less they knew, thebetter. He once told a young womanwho had written to him asking foradvice about preparing for his course,‘‘whatever else you read about thissummer, do not read any point-set the-ory if you can help it.’’ He would beginon the first day of his course by givingseveral definitions and stating a fewtheorems. The students were then lefton their own to prove the theorems andwere not allowed to collaborate or doany reading whatsoever. Gradually,students would work out proofs of thetheorems for themselves and presentthem to the rest of the class. The key asMoore saw it was not to feed informa-tion to students by lecturing or givingthem a text to read, but to have themgain for themselves the power of beingable to do mathematics by makingmistakes, getting things wrong, andyet eventually discovering their ownarguments for settling questions cor-rectly. His guiding principle was: ‘‘Thatstudent is taught the best who is toldthe least.’’

The book Number Theory ThroughInquiry has been designed to be usedin the spirit of Moore, but not at all inthe rigid way that Moore treated hisown graduate students. These days,the Modified Moore Method is a fargentler and much more patient style ofteaching that allows for a wide rangeof students—both in terms of abilitiesand levels, including undergradu-ates—to experience through a processof guided discovery the genuinebenefits of learning to think indepen-dently, to depend on their ownresources rather than those of anauthority, and to discover that theyhave within themselves the power tocreate truly important ideas. Theauthors believe, and I certainly agree,that these benefits extend well beyondthe mathematics classroom. As PaulHalmos put it, the Moore approach oftrying to instill in the student an ‘‘atti-tude of questioning everything andwanting to learn answers actively’’ is ‘‘agood thing in every human endeavor,not only in mathematical research.’’

This is why Number TheoryThrough Inquiry is an ambitious book,because it is not merely trying to teachnumber theory, it is trying to changestudent attitudes. The goal is to liberatestudents for a lifetime of learning, dis-covery, and exploration. It could besaid, however, that there is nothingparticularly new about all this as a goalin teaching. The Socratic method, afterall, has been around for a very longtime. All great teachers—Mr. Chips isone notable example—are far lessinterested in the specific subject matterat hand than in the overall growth oftheir students (one of R. L. Moore’sstudents in later years even referred toMoore as ‘‘Mr. Chips with attitude’’).Jaime Escalante, memorably portrayedby Edward James Olmos in the 1987film Stand and Deliver, borrowed fromNike their inspirational trademark ‘‘Justdo it’’ to invoke for students the spiritof active versus passive learning thatwas at the core of his own remarkableteaching style.

Number Theory Through Inquiry isan extremely thin book. A typical pagecontains a definition or two, and thenseveral questions, exercises, and theo-rems connected by the barestminimum of prose. An instructorchoosing this book for a course needsto be fully committed to the intendedmethod of instruction. An instructoralso needs to truly believe the course isabout empowering students and notabout covering material. Studentscould not possibly prove anywherenear all the theorems in the book in atypical semester course.

An instructor would have manydecisions to make about how to teachsuch a course. Do you want the stu-dents to collaborate? How much do youguide the students? One of the mostdifficult things when using this methodis learning to resist the natural impulseto jump in and correct students whenthey make mistakes. Allowing studentsthe luxury of making mistakes, findingtheir mistakes, or having other studentsfind the mistakes is at the very heart ofthe method. What do you do if no onein the class can seem to get started on aproof? This list of pedagogical ques-tions could go on and on.

An instructor choosing this book fora course not only needs to be anexcellent teacher, and probably


already quite experienced, but alsoneeds to know number theory cold.Even a pro like Paul Halmos talkedabout needing a couple of monthspreparing to teach a course using thismethod and said, ‘‘As the course goesalong, I must keep preparing for eachmeeting: to stay on top of what goeson in class. I myself must be able toprove everything. In class, I must stayon my toes every second.’’

Here are just a few examples ofhow the authors approach specificmaterial. To have students prove thereare infinitely many primes, they firstask them to think of a natural numberthat is divisible by 2, 3, 4, and 5. Next,they ask them to think of a naturalnumber that has a remainder 1 whendivided by 2, 3, 4, and 5. Then they getthem to generalize this idea by provingthat for any natural number k, there isanother (much larger) natural numbernot divisible by any natural numberless than k except 1. Of course, by nowthe students have been handed theanswer (k - 1)! + 1 on a silver platter.The next theorem they have the stu-dents prove is that for any naturalnumber k, there is a prime larger thank, which in turn leads immediately tothe infinitude of primes theorem. So, itis somewhat debatable as to whether atthis point the students have actuallydiscovered a proof of the infinitude ofthe primes. Perhaps they have justbeen led by the hand to what iseffectively Euclid’s proof. Still, this is agood, active way to present this proof.

The authors follow this up in anexcellent way by having studentsprove what they call the infinitude of4k + 3 primes theorem. Then they askwhether there are other theorems likethis that can be proved. Here, I suspectmany students will fall into the trap ofbelieving they can prove an infinitudeof 4k + 1 primes theorem in a similarway, but then won’t have time in theircourse to reach page 90, where theymight learn that in order to provethis important theorem you really needto know that –1 is a quadratic residuemodulo primes of the form 4k + 1.

A difficulty with the technique ofguided discovery—all too familiar toanyone who has ever used it—is thatstudents often don’t go where we thinkwe are leading them. This happened tome (playing the role of the student) in

a section of the book where Euler’stheorem is to be used to solve con-gruences, and the authors askedwhether I could think of an appropri-ate operation to apply in each case toboth sides of the congruences x5 : 2(mod 7) and x3 : 7 (mod 10).

Not realizing where they werehoping to lead me, I multiplied bothsides of the first congruence by x, andthen could easily solve 1 : 2x (mod7) by inspection to get x = 4; for thesecond congruence, since / (10) = 4,I again multiplied both sides of thecongruence by x and could solve1 : 7x (mod 10) by inspection to getx = 3. But, as I discovered in the nextparagraph, they had really intended forme in each case to raise both sides ofthe congruence to some appropriateexponent (5 works for the first con-gruence, and 3 works for the second),because this is the approach that gen-eralizes to the theorem they weretrying to lead us toward.

Occasionally, the notion of discov-ery or inquiry seemed to go completelyout of the window: For example, whenstudents are simply told that half thenumbers less than an odd prime p arequadratic residues and half are not.Why not let them discover this on theirown? Or when the authors simply listfor the students the primes among thefirst 30 primes for which 2 is a quadraticresidue, and those for which it is not.

There also are places in the textwhere the authors may not be givingstudents enough guidance. I’d be sur-prised if students could prove thatEuler’s /-function is multiplicative,having been told only to circle num-bers relatively prime to numbers suchas, say, 36 written down in a 4 by 9array; or if they could prove the casen = 4 of Fermat’s last theorem evenhaving been told to prove the strongerstatement that there are no nontrivialinteger solutions to x4 + y4 = z2.

I would like to try using NumberTheory Through Inquiry for a course,though perhaps not exactly in the wayintended by the authors. Rather thanhave the students use it as a text, I thinkI’d prefer to take a somewhat less gui-ded approach in order to remain truerto the Modified Moore Method byproviding students with selected defi-nitions and theorems taken from thebook. I might, however, have them start

using the text at some point midway inthe course once they had fully devel-oped their own independence. This is ahybrid technique suggested by one ofMoore’s students, F. Burton Jones, whoallowed his own topology students touse Kelley for bedtime reading begin-ning about Christmastime. Anyone whois thinking of adopting Number TheoryThrough Inquiry as a text should beaware that there is an instructor’s man-ual available from the publisher wherethe authors discuss the philosophy ofthe book and provide tips for the firstfive chapters—the likely content for afirst course—based on their own expe-rience with students.

There is much to be gained teachinga course using a well conceived andwell executed text such as NumberTheory Through Inquiry—one mighteven use this book as a template fordesigning one’s own course in virtuallyany subject. However, somethingvaluable could be lost, too. Numbertheory developed not in the inevitableway that a subject such as calculus did,but in an undeniably quirky humanway, and this treatment of numbertheory strips away so much of its richhistory that it is left a bit too bare andlifeless 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 Mathemat-

ical Association of America, 2005.


and Computer Science

Colorado College

Colorado Springs, CO 80903, USA



http://www.maa.org/devlin/devlin_5_99.html





King of InfiniteSpace: DonaldCoxeter, the ManWho SavedGeometryby Siobhan Roberts

WALKER & COMPANY, 2006, 386 PP, US$27.95

ISBN 13 978-0-8027-1499-2

REVIEWED BY ULF PERSSON

MModern mathematics is a forbid-ding subject: Highly technical,involving a formidable con-

ceptual apparatus, necessitating yearsof study before it even starts makingsense in the way many other sciencesimmediately make sense to the public.Is there a royal way to mathematics, away of getting to the heart of the subjectwithout extended preliminaries? Therefamously is no royal way to geometry,but maybe geometry itself is the royalway to mathematics. If so, who wouldbe more fitting to be the king than thesubject of the book under review—H. S. M. Coxeter? A man who showedthat, even with elementary tools, it ispossible to penetrate deeply into math-ematics, giving heart to the hope thatthe subject can be enjoyed directlywithout the alienation that comes withhigh technology: In short, that it is stillpossible to retain this sense of innocentwonder which initially seduced most ofus 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 amateursculptor and chain-smoking baritonesinger, made a living in the family busi-ness of purveying surgical instruments;his mother was a painter of somerenown, specializing in portraits andlandscapes. When in spite of sharedcultural interests they later divorced, hisfather remarried a woman only six yearsolder than his teenage son, thereby ex-acerbating an emotional trauma Donaldwould never fully overcome.

As a bright solitary child, Donaldfound pleasures in lonely pursuits suchas inventing imaginary languages, com-posing music, and engaging in mathe-matical investigations. His parents wereboth proud of and concerned for theirgifted but delicate child. They soughtprofessional guidance how to best carefor his talents. It was decided early onthat music was really not his forte. Hisprospects in mathematics were far morepromising. At the age of 16, on theadvice of Bertrand Russell, they con-tacted the mathematician E. H. Neville,who recommended that Donald dropall subjects except mathematics andGerman. Neville delegated him to AlanRobson, a senior mathematician at Marl-borough College. Donald sat for theCambridge University entrance exam in1925 and qualified for King’s College,but Robson urged him to try for Trinity.Another year of study did the trick. Hesubmitted his first mathematical paperwhen he was about to enter University.He had done some work on sphericaltetrahedra, obtaining definite integralshe challenged readers to evaluate direc-tly. The paper appeared in the Mathe-matical Gazette and intrigued G. H.Hardy, who could never resist the temp-tation of a definite integral.

At Trinity, Coxeter came under thetutelage of Littlewood, devoted himselfsingle-mindedly to his studies, and triedto resolve his recurrent problems ofdigestion by turning himself into a life-long vegetarian, which caused him tolose weight and render him his charac-teristic taut and timeless appearance, sofitting for a geometer. Predictably he didvery well on the Tripos, earning himselfthe rank of Wrangler.

On a visit to Austria in the summer of1928, Coxeter discovered the work ofSchlafli in the Vienna University library.Schlafli, a Swiss schoolteacher ignoredduring his lifetime, had anticipatedmany of Coxeter’s later discoveries (infact, the standard notation introducedby Coxeter for regular polytopes is anadaptation of Schlafli’s), most notablythe classification of regular solids in fourdimensions. Coxeter would championhim from then on.

Coxeter’s association with the lumi-naries at Cambridge was somewhatmarginal. He started his doctoral studiesunder the aged geometer H. F. Baker, a

British devotee of the Italian school ofalgebraic geometry. Baker had, amongother things, instigated a tradition ofSaturday tea parties for his students andassociates, mixing mathematics withbiscuits. Once, Coxeter invited the agedlady Alicia Boole Stott (a daughter of thewell-known Boole and a niece of thesurveyor Everest), a mathematical auto-didact almost 50 years his senior, whohad rediscovered Schlafli’s results byspatial intuition.

Coxeter’s thesis on polytopes earnedhim a prize and, in 1932, a RockefellerFoundation fellowship to Princeton.There, Solomon Lefschetz nicknamedhim Mr. Polytope and remarked, afterone of Donald’s seminars, that it wassometimes good to hear about triv-ial things. Oswald Veblen was a bitmore supportive, if distantly so. Yet themathematical environment at Princetonexposed him to a great variety of intel-lectual stimulation, such as that pro-vided by John von Neumann. Duringthis time he developed his well-knownnotation for reflection groups. Hereturned for a second stint at the Insti-tute of Advanced Study in 1934–1935.The second visit turned out to beeven more fruitful, as his study of dis-crete reflection groups tied in withHermann Weyl’s investigation of con-tinuous group representations and rootlattices, and he was invited to contributean appendix to Weyl’s seminar notes,which were widely distributed. OnHardy’s recommendation, he was invi-ted to edit Ball’s Mathematical Rec-reations and Essays (an ambivalentappreciation coming from Hardy, whoin A Mathematician’s Apology wasrather dismissive of Ball). The taskdelighted Coxeter, and he certainlywas the perfect man for it, removingoutdated material and replacing itwith chapters on polytopes and othergeometrical gems.

The summer of 1936 turned out to becrucial toCoxeter’s personal life.Hemethis first girlfriend, a Dutch au pair, whoagreed to be his wife after a rather shortcourtship. Just before the scheduledmarriage later that summer, his fatherunexpectedly suffered a heart attackand drowned as he was teaching hisyounger daughters to swim. The wed-ding went through anyway, but withoutcelebration. The young couple took off


for Toronto, where Coxeter wouldremain until his death almost 70 yearslater.

As one would expect, Coxeter pre-ferred to stay aloof of administrativeduties, which may be why he becamea full-fledged tenured professor onlyin 1948, 12 years after his initial ap-pointment. The same year saw thepublication of Regular Polytopes, whichin some sense he had been working onfor the preceding 24 years. The bookmade his reputation.

During the Second World War,Coxeter espoused pacifism, which wasnot entirely comme-il-faut in provin-cial Toronto. During the McCarthy era,he took a public stand for civil libertiesand was instrumental in finding asanctuary for Chandler Davis, a victimof the witch hunt. He championednuclear disarmament and was a vocalfoe of American involvement in Viet-nam. In his later years, he signed apetition to protest the bestowal of anhonorary degree on former PresidentGeorge Bush, Sr. Coxeter claimed thatthere could be nothing worse than thefirst President Bush—until the secondone came along.

Over the years, Coxeter won hisshare of prizes and distinctions. In 1950,he was elected a Fellow of the RoyalSociety and later became an honoraryfellow of both the Edinburgh andLondon mathematical societies. In 1997,he was appointed a ‘‘Companion of theOrder of Canada,’’ the highest of threelevels of honor that Canada bestows.The biography contains an apparentlycomplete list of Coxeter’s 250-odd pub-lications (including subsequent editionsand translations of his books) spanningalmost 80 years, starting with his entryin 1926 in the Mathematical Gazette tohis posthumous 2005 inclusion in thememorial volume dedicated to Bolyai.His articles appear both in technicaljournals and popular magazines such as

the American Mathematical Monthlyand the Mathematics Teacher, in addi-tion to the Gazette; but regardless ofwhere they are, all are accessible to thegeneral mathematician with geometri-cal combinatorial leanings. The varioustitles reveal the breadth of his interest,ranging from polytopes and group the-ory to physics, viral macromolecules,the art of M. C. Escher, and music.He even contributed to a philosophicalanthology with the article ‘‘Cases ofHyperdimensional Awareness.’’ As tohis books, in addition to the one onpolytopes, his Introduction to Geometrywas widely acclaimed, and his Genera-tors and Relations for Discrete Groups(coauthored with W. O. J. Moser) pre-sents his main contribution to profes-sional mathematics.

Coxeter was blessed with a long lifeand, more importantly, with a mind thatremained lucid to the very end. His lastpublic lecture was given in Budapest inJuly 2002. This is the event with whichthe author introduces her account ofhis life. Just days before he died, ninemonths later, he was busy readying thelecture for publication. After his death,his brain joined a ‘‘brain bank’’ atMcMaster University in Hamilton, On-tario, where a neuroscientist already hadacquired the brain of Einstein.

In presenting the life of a mathema-tician, it is the work that matters. Thisposes a serious challenge to any biog-rapher writing for the lay reader.Coxeter was for most of his career def-initely out of fashion. The mathematicshe was doing was seen by most math-ematicians of the time as a quaintvestige of Victorian mathematics. Themoral lesson that Coxeter provides is todisregard fashion: Eventually you willbe vindicated (not necessarily withinyour lifetime). But one should not, asRoberts is somewhat prone to do, pitCoxeter as a valiant David againstthe Goliath of modern mathematics,

let alone resort to hyperbole comparingRegular Polytopes to Darwin’s The Ori-gin of Species.

To dramatize the conflict betweenCoxeter and the mathematical estab-lishment, the author sets up as the‘‘villain’’ Jean Dieudonne, whose rally-ing cry ‘‘Down with Euclid, death totriangles’’ heads one of the chaptersin her book. The living geometry ofCoxeter is contrasted with the strict andformal mathematics epitomized by theBourbakists in their relentlessly logicalexpositions with no resource to visualimagery and intuitive reasoning. Likeall cliches, this contains a significantelement of truth, but the author’spresentation is a misleading oversim-plification. And, as she admits, thesupposed conflict between so-calledBourbakism and Coxeter had an ironicand happy ending, as Coxeter’s maininsights of combining geometry withthe symmetries combinatorially articu-lated through group theory becamethe subject of the concluding volumesof Bourbaki—some would say theirmost successful ones. Unlike Buckmin-ster Fuller and M. C. Escher, the asso-ciation to whom is treated at length inthe biography, Coxeter was not analternative mathematician: He was aprofessional whose insights were notentirely the result of some kind of tran-scendental meditation, but which alsorested on nonmagical algebraic manip-ulations, without which mathematicalcontemplation would not rise above thelevel of insipid speculation.

Today, Coxeter’s mathematics hascome into its own, advancing from thepages of recreational mathematics tobeing an inescapable component ofcutting-edge mathematics.


Chalmers University of Technology

Goteborg, Sweden



Stamp Corner Robin Wilson

ThePhilamath’sAlphabet:TUV

Terrestrial globeDuring the 16th century, with the newinterest in exploration and navigation,terrestrial globes became increasinglyin demand. This terrestrial globe of1568, made of brass, was constructedby Johannes Praetorius of Nuremberg.The map depicts the continents ofEurope, Africa, Asia, and America, withAmerica shown joined to Asia.

Thales of MiletusThe earliest known Greek mathemati-cian is Thales of Miletus (c. 624-547BC) who, according to legend, broughtgeometry to Greece from Egypt. He

investigated the congruence of trian-gles, applying it to navigation at sea,and predicted a solar eclipse. He is alsocredited with proving that the baseangles of an isosceles triangle areequal and that a circle is bisected byany diameter.

Tic-tac-toeThe position game of tic-tac-toe, ornoughts and crosses, developed in the19th century from earlier ‘‘three-in-a-row’’ games such as Three Men’sMorris. Two players take turns to placetheir symbol (o or 9) in a square andtry to get three in a row. Variationsinvolve 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).

TsiolkovskyKonstantin Tsiolkovsky (1857–1935), apioneer of rocket flight, inventedmultistage rockets and produced acelebrated mathematical law thatrelates the velocity and mass of arocket in flight. Tsiolkovsky’s law wasone of the ‘‘ten mathematical formulas

t-

hat changed the face of the earth,’’ in aset of Nicaraguan stamps issued in1971.

Ulugh BegBy the 15th century, Samarkand incentral Asia had become one of thegreatest centres of civilisation, espe-cially in mathematics and astronomy.The observatory of the Turkishastronomer Ulugh Beg (1394–1449)contained a special sextant, the largestof its type in the world. Ulugh Begconstructed extensive tables for thesine and tangent of every angle foreach minute of arc, to five sexagesimalplaces.

VegaLogarithms were invented in the early17th century. In the 1790s, the Slove-nian Jurij Vega (1754–1802) publisheda celebrated compendium of loga-rithms, as well as seven-figure and 10-figure logarithm tables that ran to sev-eral hundred editions. He alsocalculated p to 140 decimal places.

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


Terrestrial globe

Thales of Miletus

Tic-tac-toe

Tsiolkovsky

Ulugh Beg Vega


The Mathematical Intelligencer Vol 31 No 1 Januray 2009

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Transcript of The Mathematical Intelligencer Vol 31 No 1 Januray 2009