Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters

to the editor should be sent to the

editor-in-chief, Chandler Davis.

Condorcet Splitting and

Point Criteria

Sir, At my age I don't write many Let­

ters to the Editor any more. But when

I read Crespo Cuaresma's article on

"Point Splitting and Condorcet Crite­

ria" in the Summer 2001 issue of your

esteemed journal (pp. 23-26), I sat up

straight, circumstances permitting.

First of all, I was delighted to see

the name of my distinguished col­

league Condorcet hitting the headlines

yet again [1]. Of course, the voting sys­

tem carrying his name is really mine.

In fact when, some hundreds of years

after me, my system was named after

him, this only anticipated that yet an­

other few hundreds of years later,

Stigler [2] would come up with Stigler's Law of Eponomy. This states, as you

know as well as I do, that a decent man­

ner in which to properly indicate that

a scientific result is not yours is to have

it named after you. The only catch is

that the world is not told it's mine.

That's why I am so grateful to lain

McLean and John London [3], and oth­

ers as detailed in [ 4], who recently took

pains to put the facts on record. While

missing out on my very first paper on

the subject [5], they nevertheless rec­

ognized my achievements just on the

grounds of the two later papers [6, 7].

That first paper got buried away in the

catacombs of the Vatican Library and

was excavated only in October 2000

[4]. Which, incidentally, teaches the

practical lesson that even when your

paper remains unread for over seven

hundred years, it's still not too late for

it to resurface at the tum of the next

millennium and drive home its point.

Your readers may find this comforting.

Speaking of practicality, I notice that

I should come to why I am writing this

letter. It's because I was intrigued by the

eminently practical solution that Cre­

spo Cuaresma has for his friends Alan

and Charles. As the two fellows don't

know what to do with their money, they

distribute not it, but infinitely divisible

points. I particularly appreciate the in-

genious mathematization of those mun­

dane monetary mishaps because, as a

philosopher, I am thrilled by the philo­

sophical implications. When I was

active we worried much about contem­

plating an infinitely expansible uni­

verse, but an infinitely divisible point

was unthinkable. A point was a point.

An indivisible unity. Or, as I said in [5],

unus punctus. I apologize for changing

the dialect, it's just that I don't know

what you folks would say these days, a pixel?, which makes me chuckle since,

once you are on file with as many pub­

lications as I am, close to three hundred,

you can be used as the intellectual orig­

inator of almost anything. Some people

have even turned me into one of the fa­

thers of Computer Science [8], though

simultaneously picturing me as "one of

the most inspired madmen who ever

lived" does not do me justice. All through my life one of my concerns was

communication, and if communication

is promoted not only by my combina­

torial aids but also by Computer Sci­

ence, then I would hail it loudly and in­

stantly work it into my general art. As

a first attempt I have had my three

electoral papers rapidly prototyped

atwww.uni-augsburg.de/llull/, to assist your contemporaries in the

correct attribution of my ideas.

Yours truly,

Ramon Llull (1232-1316)

Left Choir Chapel

San Francisco Cathedral

Palma de Mallorca

Catalonian Kingdom

REFERENCES

[1 ] H. Lehning: "The birth of Galois and the

death of Condorcet." Mathematical lntelli­

gencer 1 3, no. 2 ( 1 99 1 ), 66-67 .

[2] S .M. Stigler: "Stigler's law of eponomy. "

Transactions of the New York Academy of

Sciences, Series 11 39 ( 1 980), 1 47-1 57.

(3] I . Mclean and J. London: "The Borda and

Condorcet principles: Three medieval ap­

plications." Social Choice and Welfare 7

( 1 990), 99-1 08.

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 3

[4] G. Hagele and F. Pukelsheim: "Liull 's writ­

ings on electoral systems." Studia Llulliana

41 (2001) , 3-38.

[5] R. Llull (before 1 283): "Artifitium electionis

personarum." Codex Vaticanus Latinus

9332, 1 1 r- 1 2v.

[6] R. Llull (about 1 283) : "En qual manera

Natanne fo eleta a abadessa. " Codex His­

panicus 67, 32v-34r.

[7] R. Llull (1 299): "De arte eleccionis." Codex

Cusanus 83, 47v-48r.

[8] M. Gardner: Logic Machines and Diagrams,

Second Edition. Harvester Press, Brighton,

1 983.

by the good offices of

Friedrich Pukelsheim

lnst. fOr Mathematik, Univ. Augsburg

D-86135 Augsburg, Germany

e-mail: Pukelsheim@Math.Uni-Augsburg.De

Errata: The Surfaces Capable of

Division into Infinitesimal

Squares by Their Curves of

Curvature

Consider the following example: let M := (eCl+i)tlt E IRI) c C � IRI2 be the logarithmic spiral. This is a perfectly good C1-submanifold (well, it is even C"') of !R2: it carries an atlas of C1-sub­manifold charts, i.e., to every point p E M there is a neighbourhood U C !RI2 of p and a diffeomorphism <p : U �

<p( U) c !RI2 such that <p( U n M) = <p(U) n (IR x {0)). On the other hand, taking p E M, p = 0 E IRI2, the orthog­onal projection 'Trp : M � Tp cannot be an infinitesimal bijection as, in any neighbourhood about 0 E IRI2, M spirals around 0 infinitely often. (Note that such p is not near standard in M: since p = 0 $ M there is no standard point

Po E M with P = Po·) Consequently, the definition I give

in [1] for a C1-submanifold (and, in consequence, also the one for a smooth submanifold) is "wrong": it cannot be shown equivalent to the usual defini­tion. In fact, it is the (standard differ­ential geometry) argument I give just before the definition that is wrong­the argument only applies to choices of standard coordinate systems. Thus, re­quiring (a)-( c) of the definition to hold only for all near standard (in M) points

p EM (as given in [3]), one can indeed show the equivalence to the usual (standard) definition [2], i.e., to the ex­istence of a C1-submanifold chart around each point. Accordingly, the following constructions and assertions remain valid, but should be restricted to near standard points.

Also, I hereby apologise to K.

Stroyan for erroneously citing him for the equivalence proof: in [3], he shows that a C1-submanifold in the sense of the (corrected!) nonclassical definition is an abstract C1-manifold1, i.e., that it carries a C1-atlas of charts (not sub­manifold charts). However, some ad­ditional remarks he gives in his paper strongly suggest the validity of the the­orem that I have just announced-and that, surprisingly, I was not able to fmd in the literature.

Finally: I should have mentioned that the surface graphics in [ 1] were produced using the computer algebra system Mathematica.

REFERENCES

1 . U. Hertrich-Jeromin: The surfaces capable

of division into infinitesimal squares by their

curves of curvature: A nonstandard analysis

approach to classical differential geometry;

Math. lntelligencer 22 (2000), no. 2, 54-61 .

2. U. Hertrich-Jeromin: A nonstandard analy­

sis characterization of standard submani­

folds in Euclidean space; Balkan J. Geom.

App/. 6 (2001 ), 1 5-22.

3. K. Stroyan: Infinitesimal analysis of curves

and surfaces; in J. Barwise, Handbook of

Mathematical Logic, North-Holland, Am­

sterdam 1 977.

Udo Hertrich-Jeromin

Department of Mathematics

TU Berlin

D-1 0623 Berlin

Germany

e-mail: udo@sfb288.math.tu-berlin.de

Indemnification

The Author represents and warrants ... that, to the best of the Author's knowledge, no formula, procedure, or prescription contained in the Ar­ticle would cause injury if used or

1And not the converse, which is certainly wrong, as the example of a a-shaped curve shows.

4 THE MATHEMATICAL INTELLIGENCER

followed in accordance with th� in­structions and/or warnings con­tained in the Article. The Author will indemnify Springer-Verlag against any costs, expenses or damages that Springer-Verlag may incur or for which Springer-Verlag may become liable as a result of any breach of these warranties. These representa­tions and warranties may be ex­tended to third parties by Springer­Verlag. -Copyright Transfer Statement

Claims continue to mount in the case of the Haas "Cross-number puzzle," published without adequate warning notices in the Mathematical Intelli­gencer (vol. 24, no. 2, p. 76). In ap­pearance just a crossword puzzle with numbers, this puzzle has turned out to be an extreme mental and physical health hazard. Hundreds of complaints have arrived from around the world, of headache, neck strain, back strain, blurred vision, dizziness, insomnia, nightmares, and inability to concen­trate, following an attempt to solve it. Several injuries were reported from readers distracted by thinking about it while driving or operating heavy equip­ment. Numerous others ask unemploy­ment compensation after being fired for doing it on the job.

Springer-Verlag is forwarding all claims directly to the author Robert Haas, whose signed copyright transfer leaves him liable for all costs.

The most tragic case to date is that of Thomas Chadbury, a promising young mathematician whom the puzzle may have permanently deranged. He is now confined to an institution. "My God, you can't argue around him, and his new ideas never stop, he's a mathemati­cian," said his psychiatrist Shrinkovsky, who himself has filed a third-party claim, citing lost clientele and the counseling costs for himself as he struggles to treat his patient.

Robert Haas

1 081 Carver Road

Cleveland Heights, OH 441 1 2

USA

[4] G. Hagele and F. Pukelsheim: "Liull 's writ­

ings on electoral systems." Studia Llulliana

41 (2001) , 3-38.

[5] R. Llull (before 1 283): "Artifitium electionis

personarum." Codex Vaticanus Latinus

9332, 1 1 r- 1 2v.

[6] R. Llull (about 1 283) : "En qual manera

Natanne fo eleta a abadessa. " Codex His­

panicus 67, 32v-34r.

[7] R. Llull (1 299): "De arte eleccionis." Codex

Cusanus 83, 47v-48r.

[8] M. Gardner: Logic Machines and Diagrams,

Second Edition. Harvester Press, Brighton,

1 983.

by the good offices of

Friedrich Pukelsheim

lnst. fOr Mathematik, Univ. Augsburg

D-86135 Augsburg, Germany

e-mail: Pukelsheim@Math.Uni-Augsburg.De

Errata: The Surfaces Capable of

Division into Infinitesimal

Squares by Their Curves of

Curvature

Consider the following example: let M := (eCl+i)tlt E IRI) c C � IRI2 be the logarithmic spiral. This is a perfectly good C1-submanifold (well, it is even C"') of !R2: it carries an atlas of C1-sub­manifold charts, i.e., to every point p E M there is a neighbourhood U C !RI2 of p and a diffeomorphism <p : U �

<p( U) c !RI2 such that <p( U n M) = <p(U) n (IR x {0)). On the other hand, taking p E M, p = 0 E IRI2, the orthog­onal projection 'Trp : M � Tp cannot be an infinitesimal bijection as, in any neighbourhood about 0 E IRI2, M spirals around 0 infinitely often. (Note that such p is not near standard in M: since p = 0 $ M there is no standard point

Po E M with P = Po·) Consequently, the definition I give

in [1] for a C1-submanifold (and, in consequence, also the one for a smooth submanifold) is "wrong": it cannot be shown equivalent to the usual defini­tion. In fact, it is the (standard differ­ential geometry) argument I give just before the definition that is wrong­the argument only applies to choices of standard coordinate systems. Thus, re­quiring (a)-( c) of the definition to hold only for all near standard (in M) points

p EM (as given in [3]), one can indeed show the equivalence to the usual (standard) definition [2], i.e., to the ex­istence of a C1-submanifold chart around each point. Accordingly, the following constructions and assertions remain valid, but should be restricted to near standard points.

Also, I hereby apologise to K.

Stroyan for erroneously citing him for the equivalence proof: in [3], he shows that a C1-submanifold in the sense of the (corrected!) nonclassical definition is an abstract C1-manifold1, i.e., that it carries a C1-atlas of charts (not sub­manifold charts). However, some ad­ditional remarks he gives in his paper strongly suggest the validity of the the­orem that I have just announced-and that, surprisingly, I was not able to fmd in the literature.

Finally: I should have mentioned that the surface graphics in [ 1] were produced using the computer algebra system Mathematica.

REFERENCES

1 . U. Hertrich-Jeromin: The surfaces capable

of division into infinitesimal squares by their

curves of curvature: A nonstandard analysis

approach to classical differential geometry;

Math. lntelligencer 22 (2000), no. 2, 54-61 .

2. U. Hertrich-Jeromin: A nonstandard analy­

sis characterization of standard submani­

folds in Euclidean space; Balkan J. Geom.

App/. 6 (2001 ), 1 5-22.

3. K. Stroyan: Infinitesimal analysis of curves

and surfaces; in J. Barwise, Handbook of

Mathematical Logic, North-Holland, Am­

sterdam 1 977.

Udo Hertrich-Jeromin

Department of Mathematics

TU Berlin

D-1 0623 Berlin

Germany

e-mail: udo@sfb288.math.tu-berlin.de

Indemnification

The Author represents and warrants ... that, to the best of the Author's knowledge, no formula, procedure, or prescription contained in the Ar­ticle would cause injury if used or

1And not the converse, which is certainly wrong, as the example of a a-shaped curve shows.

4 THE MATHEMATICAL INTELLIGENCER

followed in accordance with th� in­structions and/or warnings con­tained in the Article. The Author will indemnify Springer-Verlag against any costs, expenses or damages that Springer-Verlag may incur or for which Springer-Verlag may become liable as a result of any breach of these warranties. These representa­tions and warranties may be ex­tended to third parties by Springer­Verlag. -Copyright Transfer Statement

Claims continue to mount in the case of the Haas "Cross-number puzzle," published without adequate warning notices in the Mathematical Intelli­gencer (vol. 24, no. 2, p. 76). In ap­pearance just a crossword puzzle with numbers, this puzzle has turned out to be an extreme mental and physical health hazard. Hundreds of complaints have arrived from around the world, of headache, neck strain, back strain, blurred vision, dizziness, insomnia, nightmares, and inability to concen­trate, following an attempt to solve it. Several injuries were reported from readers distracted by thinking about it while driving or operating heavy equip­ment. Numerous others ask unemploy­ment compensation after being fired for doing it on the job.

Springer-Verlag is forwarding all claims directly to the author Robert Haas, whose signed copyright transfer leaves him liable for all costs.

The most tragic case to date is that of Thomas Chadbury, a promising young mathematician whom the puzzle may have permanently deranged. He is now confined to an institution. "My God, you can't argue around him, and his new ideas never stop, he's a mathemati­cian," said his psychiatrist Shrinkovsky, who himself has filed a third-party claim, citing lost clientele and the counseling costs for himself as he struggles to treat his patient.

Robert Haas

1 081 Carver Road

Cleveland Heights, OH 441 1 2

USA

«·)·"I"·' I I

Pub l ishing Report Henry Helson

The (}pinion column offers

mathematicians the opportunity to

write about any issue of interest to

the international mathematical

community. Disagreement and

controversy are welcome. The views

and opinions expressed here, however,

are exclusively those of the author,

and neither the publisher nor the

editor-in-chief endorses or accepts

responsibility for them. An (}pinion

should be submitted to the editor-in­

chief, Chandler Davis.

I always had a hankering to be an en­trepreneur. It was suppressed all the

years of academic work, but came out about ten years ago, before I retired. I had written three books and they were published, but the publishers didn't seem as excited about them as I was, and the books were undoubtedly in their last stage of life (as I was also!). I didn't want them to go out of print. Furthermore, I had a new manuscript and only a half-hearted, unenthusiastic offer from a publisher.

Then I took matters into my own hands. This is the history, so far, of my enterprise. I offer it as information about the economics of textbooks, something which concerns all of us in the teaching profession. I feel this has interest, because textbook publishing is an opaque industry. The real pub­lishers don't tell us much, even though we are their only customers.

I decided to publish my new book myself. I had a not-very-modem com­puter, and a 300-dot laser printer, which was obsolete even then, but which gave beautiful pages (and still does). Also I was proficient in EXP, the wysiwyg program that, unaccountably and unfortunately, seems to have lost out to TEX. I've always done my own typing, due largely to terrible hand­writing. First I got in touch with Gilbert Strang of MIT, who was already a pub­lisher. I got good advice, and I trea­sured his encouragement.

Next I produced a clean printout of my book Honors Calculus, and looked for "Printers" in the Yellow Pages. In a short while I had a big stack of books in my garage.

Holden-Day had published my Lin­ear Algebra but was going out of busi­ness. The president, Fred Murphy, had been a friend since the days when he traveled on behalf of Addison-Wesley. He gave me back my rights to the book, the old copies at his cost of production, and many rolls of transparent tape, which I still use for mailing. That was a big push; now I had two titles. In the

course of time, I brought out new edi­tions of my Harmonic Analysis

(which I got back from Brooks/Cole), Honors Calculus, and Linear Algebra. I published Notes on Complex Func­

tion Theory by Don Sarason, and An

Invitation to General Algebra and Universal Constructions by George Bergman, both colleagues at Berkeley. My last book, Calculus and Probabil­

ity, has sold some copies but has not yet been adopted anywhere. Mean­while I arranged with the Hindustan Book Agency of New Delhi to sell their book Basic Ergodic Theory by M. G. Nadkami outside of Asia; and Hindus­tan has reprinted some of my books for sale in Asia. I am writing a monograph that I hope to publish later this year.

The teaching world didn't beat a path to my garage, but I've shown a profit to the IRS every year since the beginning in 1992. I think I am a pub­lisher. I understand better than I did how the business works, and why it doesn't work better. The comments that follow are my serious opinions, but I emphasize that they are based on my own experience and not on statis­tical evidence.

I don't understand how bookstores can stay in business. I set a "list price" and bill resellers with a discount of 200/o. I think this is normal. That means the bookstore has a markup of 25% ( un­less it charges more than list price, which is difficult because that price is quoted in public databases, such as Amazon. com's catalogue). Out of that markup, the store pays for delivery, and transportation back again if the book isn't sold. Unless the order is very large, UPS will get around 100/o of the price of the book each way. Reorders (if the first order wasn't large enough) are costlier, because the number of books is small. The order and the re­order come by telephone, which isn't free. Books get damaged, stolen, lost; invoices are misplaced. Somebody has to work on every snafu. I don't know how the bookstores manage. Please

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 5

don't entertain unkind thoughts about yours, unless they don't pay their pub­lishers.

Actually (unlike other publishers) I include the shipping cost in my list price. I use the US Postal Service, which is much cheaper than alterna­tives. Bookstores prefer UPS because the shipment is tracked. That avoids the problem of accountability if the shipment doesn't arrive. But my expe­rience with the postal service is excel­lent. The only problem, for me, is wait­ing in line for service. The postal service claims to have modernized its handling of mail, and I believe it, but the local PO is still terribly obsolete. I think Congress is to blame. There are hundreds of postal rates, for the bene­fit of various political interests, with no relation to the service rendered. If rates depended not on the content of packages but only on the service re­quired, then we could go to the PO, weigh the package on the scale, enter the type of service, buy postage from a machine, and leave without seeing a clerk As it is, I cannot even put a stamped package in a mailbox, be­cause people are still fearful of a for­mer Berkeley mathematician-turned­terrorist who is not dangerous any more; instead I have to waste gas and time going to the PO. That is no way to run the postal business.

How do used books get recircu­lated? An individual store buys the used book back, but that store is un­likely to sell it again. There is a busi­ness of getting used books to the places where they are currently adopted; I don't know how it works, but I am surprised if anyone can make a profit. The book will have to be stored, to wait for the course that needs it. If nobody does, it is a total loss. If somebody does, there seem to be two UPS charges to cover. Assum­ing the student who sold it got back half the list price of the new book, and it is then resold for three-quarters of list price, I don't see how there is any worthwhile margin of profit.

A university used my Linear Alge­bra. I got back some unsold copies with a price sticker that was higher than my list price. I complained to the bookstore and was told, sorry, but they

6 THE MATHEMATICAL INTELLIGENCER

had bought some used copies for more than my list price, and marked the rest accordingly! They were decent and ac­tually gave students refunds.

I like selling one or two books at a time to libraries and individuals. The price is full list, they always send a check promptly, and sometimes people tell me they like my books. My ad­vanced books mostly go out this way. But that business is too small to be re­ally profitable. I would like my local in­dependent bookstore to stock my ti­tles, but they won't; they want a 400/o discount on list. Nevertheless they buy and stock used copies of my books, which they offer more expensively than I do new ones, and which are sold quickly. This tempts me to leave a note with my price and telephone number in my books, but I haven't done so yet.

I have had only three orders that

Instructors do not

take the choice of

texts seriously. were never paid for. One bookstore went out of business, after telling lies over the telephone for months. Two European distributors have just never paid, in spite of repeated requests. One other, in Paris, did fmally pay, after I came by in person. I think they were surprised by my visit!

For foreigners, paying is a problem. In spite of what we hear about inter­national capital flows, my bank won't accept payment from anyone except another American bank, or else a wire transfer that is too expensive. So a for­eign bookseller has to maintain an ac­count in the United States. Within Eu­rope it is messy too, although that does not affect me. In spite of the allegedly common currency, if you deposit a check in euros in one euro country drawn on a bank in another, it is still a foreign-exchange transaction. For some reason, banks in all countries see no reason to simplify things.

As in other parts of our economy, the cost of production in the publish­ing business is determined by the (high) cost of human time and the (low) cost of manufacturing by rna-

chine. Printing two thousand copi�s of a book costs surprisingly little more than a hundred. (The process used is different.) The work of preparing a book for publication is the same re­gardless of the number of copies to be printed; this makes more advanced texts expensive. The editor who comes to a booth at a meeting (and his hotel bill) costs the company a bundle. The representative who calls at our offices, with no purpose except to be nice and offer complimentary copies of relevant texts, does too (but I haven't seen one in recent years). All the complimentary copies are expensive to mail. Thus the overhead is high, but if a book does sell a lot of copies, it is very profitable indeed.

I don't have these expenses, and I do everything myself except the actual printing, so I can set my prices much

lower than a commercial publisher could, and I do. But my small scale makes advertising a problem for me. I can't afford to advertise in print ( al­though I have tried); a single small in­sertion in the American Mathemati­cal Monthly costs hundreds of

dollars, and I can't even think of the Notices of the AMS. Mailing individual letters doesn't work well; I don't know why. Libraries will only buy a title if it is in a series by a big publisher, or if someone asks for it, so there is no point in writing to librarians. I send out dozens of "examination copies," and probably that is doing some good, be­cause I get orders out of the blue from people who must have seen the book somewhere.

Are textbooks too expensive? Yes, if we assume you would rather not pay for the excess length of the modem cal­culus book, or its too generous mar­gins, or pretty colors. I paid $2.90 for Osgood's calculus (still worth reading) in 1943. If you apply an inflation factor of 10, it should cost about $30 now. It would cost more than that, but not much more, and the quality of printing is very much better now, even leaving colors and margins aside. But a mod­em calculus text costs another $40 more yet, and the added cost is largely waste. The publisher wants to be sure no topic is omitted that any potential user could want, and therefore the text

is twice as long as Osgood was, and

most of the book will never be read.

Then there is the froth: the colors, the

wide margins, that are supposed to

please students. I don't think students

are pleased, but their instructors seem

to be, because they choose these mon­

strosities all the time. The blame lies

with us, the faculty who adopt text­

books and don't give a thought to what

the book will cost. The publishers just

give us what we want, and a little more.

I sunnise that publishers lose

money on many of the elementary

books they publish. They are expensive

to print and expensive to transport,

and a lot will have few adoptions. They

will be gone in a couple of years. More

of the same continue to appear be­

cause every publisher is looking for the

new Thomas. Few find him.

This is reminiscent of the automo­

bile business a few decades ago. Every

American producer wanted to hit the

center of the market with a product

that everybody would like. The result

was products that did not fit the needs

of a lot of people. Then foreigners in­

vaded the American market with cars

each addressed to some particular seg­

ment of the market. There were small

cheap cars of different kinds, and big

expensive ones, and each was appre­

ciated by the people for whom it was

intended. Pretty soon there was not

much left in the middle for the mass

marketers.

The textbook field is ripe for a sim­

ilar development, although it is not

likely to be brought about by competi­

tion from abroad. Our educational in­

stitutions and the students in them are

varied, but our publishers continue to

churn out cloned copies of old calcu­

lus texts. They are not that different

from the first Granville that I learned

from. Certainly they do not serve the

diverse student bodies that buy them.

The situation can't improve until pub­

lishers give up the idea of the all-pur­

pose text, and try to serve well the sev­

eral smaller markets that exist now.

They will not do that until instructors

ask seriously for texts that are appro­

priate for their students.

I have been disappointed to see how

instructors at my university and else­

where do not take the choice of texts

seriously. The ones they choose have

for their only virtue that they will be

easy to teach from, because they will

not arouse anxiety in their students.

We complain about how little respect

students have for our subject, but we

require them to study texts that con­

descend to them and offer them noth­

ing meriting respect.

Actually my publishing venture is

not entirely the result of a passion for

entrepreneurial activity. If it were, I

would be a good deal richer. Like many

others who think that universities are

for teaching (as well as research), I felt

challenged by the crisis in the teaching

of mathematics and wanted to try to do

something about it. The best way I

could think of was to write texts that

incorporate my ideas for teaching in

one of those segments just mentioned.

Since there is no present market for

those ideas, I had to publish the books

myself.

Naturally, students should get all

the help we can give them: competent

lecturing to begin with, and then office

hours, review sessions, math clubs,

and especially other students to talk to.

After that, the student has some re­

sponsibility. There is no way to elimi­

nate the lonely job of making sense out

of lecture and text. Finally a student

has to come to terms with the subject.

Then all the reassuring, chatty digres­

sions that pad these thick books are

just confusing. At the moment of actu­

ally learning something, it is important

to have a text that tells it like it is, with­

out pretending that learning is easy,

without a mass of irrelevant story­

telling, without fake applications, and

above all without assuming that the

student-reader is an idiot.

To be clear, a text should be as sim­

ple and brief as possible. It simply is

not true, for the students I have in

mind, that an idea is easier to under-

stand told imprecisely in six para­

graphs than told carefully in one. Fur­

thermore, not all true statements are of

the same importance. The text should

direct the student to what is most im­

portant, and leave inessential details to

be filled in by the lecturer, or presented

in problems.

If we want good textbooks, first we

have to write them. The calculus man­

uscripts I get to review suggest that

writers, like publishers, want to hit that

jackpot, and are not trying to write

carefully to a narrower target. If we ask

for good books, publishers will do their

part in providing them. Then we fac­

ulty need the courage to choose ones

that are right for our students, and

learn to teach from them. Students

won't like it; and with our promotions

dependent on student evaluation

forms, we've got a problem. Should we

face it, or just keep on moaning about

how hard it is to teach mathematics?

HENRY HELSON

15 The Crescent

Berkeley, CA 94708 USA

e-mail: helson@math.berkeley.edu

Henry Helson, beginning with his stu­

dent years at Harvard, has had a long

career in harmonic analysis, from the

classical to the functional-analytic.

Most of it has been spent at the Uni­

versity of California Berkeley, where

he is now Emeritus Professor.

VOLUME 24. NUMBER 3, 2002 7

MANUEL RITORE AND ANTONIO ROS

Some Updates on lsoperimetric Prob ems

lready in ancient times Greek mathematicians treated the isoperimetric

properties of the circle and the sphere, the latter of which can be formu­

lated in two equivalent ways: (i) among all bodies of the same volume,

the round ball has the least boundary area, (ii) among all surfaces of the

same area, the round sphere encloses the largest volume.

The first proof of the isoperimetric property of the cir­

cle is due to Zenodorus, who wrote a lost treatise on

isoperimetric figures, known through the fifth book of the

Mathematical CoUection by Pappus of Alexandria [13].

Zenodorus proved that among polygons enclosing a given

area, the regular ones have the least possible length. This

implies the isoperimetric property of the circle by a stan­

dard approximation argument. Since then many proofs and

partial proofs have been given. Among the many mathe­

maticians who have considered these problems are Euler,

the Bemoullis, Gauss, Steiner, Weierstrass, Schwarz, Levy,

and Schmidt, among others.

Nowadays by an isoperimetric problem we mean one in

which we try to find a perimeter-minimizing surface (or hy­

persurface) under one or more volume constraints and with

possibly additional boundary and symmetry conditions.

Thanks to the development of Geometric Measure Theory

in the past century (see, for instance, the text [15] and the

references therein) we have existence and regularity re­

sults for most of the "natural" isoperimetric problems we

can think about. By regularity we mean that the solution

of the problem either is a smooth surface, or has well­

understood singularities, as in the double-bubble problem,

which we spotlight below.

We will describe how to seek the solutions of some

isoperimetric problems in the Euclidean space �3, including

the double-bubble problem. For other ambient manifolds

such as n-dimensional spheres or hyperbolic spaces, we re­

fer to the reader to Burago and Zalgaller' s treatise [ 4] on geo­

metric inequalities, where an extensive bibliography can be

found. We will not treat either some recent interesting ad­

vances in the study of isoperimetric domains in surfaces.

The Classical lsoperimetric Problem in IR3 We wish to fmd, among the surfaces in �3 enclosing a fixed

volume V > 0, the ones with the least area. From general

results of Geometric Measure Theory [ 15], this problem has

at least a smooth compact solution. Moreover, from varia­

tion formulae for area and volume, the mean curvature of

such a surface must be constant. The mean curvature at a

point of the surface is the arithmetic mean of the principal

curvatures, which indicates how the surface is bent in

space. It is not difficult to show, from the second variation

formula for the area, keeping constant the volume en­

closed, that the solution surface (and hence the enclosed

domain) has to be connected.

There are several ways to prove that the sphere is the

only solution to this problem. Perhaps the most geometri­

cal ones are the various symmetrization methods due to

Steiner and Schwarz [4] and Hsiang [14]. Let us explain

briefly their arguments. Consider an isoperimetric body 0. Steiner's method applies to the family of lines ortho­

gonal to a given plane P, and, for every line L in this fam­

ily, replaces L n 0 by the segment in L centered at P n L of the same length. This procedure yields another body 0' with the same volume as n, and strictly less boundary area

unless the original body n was symmetric about a plane

parallel to P. This implies that 0 must have been symmet­

ric about a plane parallel to P.

Schwarz considers a given line L. For every plane P or-

© 2002 SPRINGER� VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 9

thogonal to L, the intersection P n 11 is replaced by the disc in P centered at P n L of the same area. Again a new body 11' is obtained with the same volume as 11, and a smaller boundary area unless 11 was rotationally symmetric about a line parallel to L. In a similar way, one can use a family of concentric spheres instead of parallel planes to obtain a new symmetrization known as spherical symmetrization.

A third symmetrization was used by Hsiang. He consid­ered a plane p dividing n into two equal volume parts n+ and n-. Assuming that area(O+) :::s area(O-), he took the domains n = n+ u n- and 11' = n+ u reo+), where r is the reflection in P. Then 11' is also an isoperimetric domain, from which we conclude that area(O+) = area(O-). We also have by regularity that an and an' are constant mean curvature surfaces, and, by construction, they coincide in an+. By general properties of constant mean curvature sur­faces we conclude an = an', and son= 11', which means that 11 was symmetric with respect to P.

By applying Steiner or Hsiang symmetrization, it follows that n is symmetric with respect to a plane parallel to any given one; by Schwarz symmetrization, that 11 is symmetric with respect to a line parallel to an arbitrary one. It is not dif­

ficult to see from these properties (and the compactness and connectedness of aO) that an must be a sphere.

Hence a symmetrization method suffices to character­ize the isoperimetric domains in Euclidean space �3. This is due to the large group of isometries of this space. We will see other situations where this is not enough to char­acterize the isoperimetric domains.

There is also a symmetrization method for embedded constant mean curvature surfaces, known as the Alexan­drov reflection method [22], which shows that such a sur­face embedded in �3 is symmetric with respect to a plane parallel to a given one, and hence has to be a sphere.

Some Other lsoperimetric Problems In

Euclidean Space

We consider in this section a modified version of the clas­sical isoperimetric problem in �3. For a regular region R C �3 and for V :::s vol R we want to find a surface of least area � C R separating a region 11 C R of volume V. The surfaces admitted can have boundary, which is contained in the boundary of R. That is, region 11 is bounded by � and per­haps by a piece of aR.

This is often referred to as ajree boundary problem with a volume constraint. We emphasize that the area of ann aR is not considered in this problem (Fig. 1 ).

Geometric Measure Theory [15] ensures the existence of the solution � at least for compact R, and its regularity, at least in low dimensions. Moreover, any solution � has constant mean curvature and meets the boundary of R

R

Figure 1 . lsoperimetric domains in a region R.

given R it is certainly difficult to characterize the isoperi­metric solutions, but the following coJ\iecture is plausible.

CONJECTURE. Any solution to the isoperimetric problem in a strictly convex region is homeomorphic to a disc.

Let us now consider some other choices of the region R.

The lsoperimetric problem In a halfspace Let us assume that R is the halfspace z 2: 0. We will find the surfaces � that separate a region 11 c R of fixed vol­ume with the least perimeter. Because R is noncompact, the existence of isoperimetric domains requires proof, as a minimizing sequence could diverge, but this is solved by using translations. So we have existence and also regular­ity, which is a local matter. In this case we have

THEOREM. Isoperimetric domains in the halfspace z 2: 0 are haljballs centered on the plane z = 0 (Fig. 2).

For the proof of this theorem we first observe that the isoperimetric region 11 must touch the plane z = 0. Other­wise, moving n until it becomes tangent to the plane z = 0 we get an isoperimetric region such that � = an touches z = 0, but neither at a� nor orthogonally.

Also 11 is connected: otherwise we could move two components of n until they touch, producing a singularity in the boundary. We now apply Hsiang symmetrization, but only for planes orthogonal to z = 0, to conclude that n is rotationally symmetric about a line L orthogonal to z = 0.

Hence � is obtained by rotating a plane curve to get a constant mean curvature surface. It turns out that there are only a few types of curves that produce, when rotated, a constant mean curvature surface. They were studied by Ch. Delaunay in 1841 [6], and they are depicted in Figure 3.

Since our curve touches the line of revolution (it has a max­imum of the z-coordinate ), looking at the list, we conclude that it is part of a circle, and so � is a halfsphere.

The lsoperimetric problem In a ball Let us now assume that R is a ball. Spherical sym-

at a� orthogonally. m e t r i z a t i o n When R is strictly convex Q'· proves that an

the surface � is connecteLd, · · ·-IJ �-------/.- ---, isoperimetric sur-

and bounds on its genus · · �: .. t _ :, � face � is a surface of :� · 8 and on the number of ---'--- revolution around some components of a� "--------------------------'· line L containing the cen-

are known [21]. For a Figure 2. lsoperimetric domains in a halfspace. ter of the ball. As

10 THE MATHEMATICAL INTELLIGENCER

Figure 3. Generating curves of surfaces of revolution with constant mean curvature. The horizontal line is the axis of revolution. From left to right and above to below, the generated constant mean curvature surfaces are unduloids, cylinders, nodoids, spheres, catenoids, and planes orthogonal to the axis of revolution.

shown in Figure 4, there are surfaces of this kind which

are not spheres. What we know is that I is a piece of a

sphere or a flat disc if I touches L.

We will sketch the proof of

THEOREM ([21]). Isoperimetric domains in a ball are those bounded by a flat disc passing through the center of the ball or by spherical caps meeting orthogonally the bound­ary of the ball.

To prove the theorem, assume that I is neither a piece

of a sphere nor a flat disc, so that I does not touch L.

Choose p E I at minimum distance from L. Consider the

Killing field X of rotations around the axis L' orthogonal to

L passing through p. The set C of points of I where X is

tangent to I can be shown to consist of a finite set of closed

curves. This set includes ai and the intersection of the

plane (L, L'), generated by Land L ', with I. By the special

properties of the field X, there is another curve in C pass­

ing throughp apart from ( L, L') n I. We conclude that I -C has at least four connected components.

But this is enough to show that I cannot be an isoperi­

metric surface by using Courant's Nodal Domain Theorem

[5] . The intuitive idea is that we can rotate (infmitesimally)

two of these components to get a nonsmooth surface which

encloses the same volume and have the same area as I; the

new surface should be also isoperimetric, which is a con­

tradiction because it is not regular.

Observe that the isoperimetric domains in a ball are

never symmetric with respect to the center of the ball. We

L'

L Figure 4. There are candidates to be isoperimetric domains in a ball which are not spheres nor flat discs.

may complicate the problem by imposing this symmetry.

The following problem is still open.

PROBLEM. Among surfaces in a ball which are symmetric with respect to the center of the ball, find those of least area separating a fixed volume.

The lsoperimetric problem in a box The convex region R given by [a, a'] X [b, b'] X [c, c ' ] will be called a box. For this region no symmetrization can be

applied to the isoperimetric domains. The most reasonable

conjecture for such a region is

CONJECTURE. The surfaces bounding an isoperimetric do­main in a box R are

(i) an octant of a sphere centered at one vertex of R, or

(ii) a quarter of a cylinder whose axis is one of the edges of R, or

(iii) a piece of a plane parallel to some of the faces of R. The type of solution depends on the shape of the box R and on the value of the enclosed volume.

What is known at this moment? Some partial results. We

know that the conjecture is true when one edge is much larger

than a second one, which is huge compared with the third

one [20], [18]. Also that the candidates are constant mean cur­

vature surfaces which are graphs over the three faces of the

box (Fig. 5). Apart from the ones stated in the above con­

jecture, we have two families of constant mean curvature sur­

faces which might be isoperimetric solutions [19]. They are

depicted in Figure 6. The right-hand family is a three-para­

meter one and includes a part of the classical Schwarz '!P­

minimal surface. This surface has been shown to be stable

Figure 5. Probable solutions of the isoperimetric problem in a box.

VOLUME 24. NUMBER 3, 2002 11

Figure 6. Candidates to be solutions of the isoperimetric problem in a box.

(n01megative second variation of area enclosing a fixed vol­

ume) by M. Ross, although it cannot be a solution of the

isoperimetric problem by results of Hadwiger [8]; see also

Barthe-Maurey [3]. The left-hand family is a two-parameter

one. It is also known that the isoperimetric solution for half

of the volume is a plane in the case of the cube.

The lsoperlmetric problem in a slab Let us assume now that R is a slab bounded by two paral­

lel planes P1 and P2 in !R3. Existence in this noncompact

region is ensured by applying translations parallel to the

planes Pi to any minimizing sequence. One can also apply

symmetrization (with respect to planes orthogonal to Pi) to conclude that an isoperimetric solution is symmetric

with respect to some line L orthogonal to Pi. Possible so­

lutions in this case are halfspheres centered at some of the

planes Pi, tubes, and unduloids (see Figure 3). A careful

analysis of the stability of the generating curves is required to discard unduloids, getting (Fig. 7)

THEOREM ([2], [24), [16]). The surfaces bounding an isoperi­metric domain in a slab in !R3 are

(i) haifspheres centered on one of the boundary planes, or

(ii) tubes around a line orthogonal to the boundary planes.

This result remains true in jRn+ 1, for n ::::; 7, but not for n 2: 9 (the case n = 8 remains open). In high dimensions one can

prove the existence of unduloids which are solutions to the

isoperimetric problem [16]. The argument is a simple com­

parison: for n 2: 9, a halfsphere with center on one of the

Figure 7. lsoperimetric problems in a slab. The one on the right is an unduloid, which appears in large dimensions.

12 THE MATHEMATICAL INTELLIGENCER

Figure 8. The standard double bubble.

boundary planes and tangent to the other cannot be an

isoperimetric solution by regularity. But it has less perimeter

than a tube of the same volume. We conclude that there is

an isoperimetric solution that is neither a sphere nor a tube.

The only remaining possibility is an unduloid.

Multiple Bubbles

The standard double bubble is seen in nature when two

spherical soap bubbles come together. It is composed of

three spherical caps (one of which may degenerate to a flat

disc) spanning the same circle. The caps meet along the

circle in an equiangular way. The whole configuration is ro­

tationally invariant around a line. Standard bubbles are can­

didates to be solutions of the following isoperimetric prob­

lem, known as "the double-bubble problem" (Fig. 8).

PROBLEM. Among surfaces enclosing and separating two given volumes, find the ones with the least possible total area.

For existence we refer to Almgren's work [1]; for regu­

larity, to Taylor [23], who showed that any solution consists

of constant mean curvature sheets in such a way that either

(i) three sheets meet along a curve at equal angles of 120 de­

grees, or (ii) in addition, four such curves and six sheets meet

at some point like the segments joining the barycenter of a

regular tetrahedron with the vertexes (sheets go out to the

edges of the tetrahedron). Natural candidates to be solutions

of this isoperimetric problem are the standard double bub­

bles (Fig. 9) (there is precisely one for every pair of volumes),

and it turns out they are the best:

THEOREM. The standard double bubble is the least-perim­eter way to enclose and separate two given volumes in !R3.

Figure 9. Double bubbles. The one on the right was shown to be un­stable, and hence it does not appear in nature. Pictures by John Sul­livan, University of Illinois (http://www.math.uiuc.edu/-jms)

Figure 10. The horizontal line is the axis of revolution. When rotated the curves give the whole bubble. Each curve is a piece of a Delaunay curve. The ones touching the axis are circles. When three curves meet, they meet at 1 20° angles. For equal volumes just the first configura­tion has to be considered. In the second configuration one of the regions is disconnected.

This result was first proved by Hass and Schlafly [9) for

the case of two equal volumes. The general case was solved

by Hutchings, Morgan, Ritore, and Ros [12] (announced in

[11)). As in the previous examples, one tries to find some kind

of symmetry in the problem. This was done by Foisy [7) and Hutchings [10) following an idea of Brian White: for up to

three volumes in IR3, Borsuk-Ulam's theorem (more precisely,

one of its corollaries known as "the ham sandwich theorem")

shows that we can find a plane P1 dividing each region of a

solution � of the double-bubble problem in two equal volume

parts. Hutchings [10] proved that such a plane is a symmetry

plane. A second application of Borsuk-Ulam shows that there

is another plane P2, orthogonal to Pt. which divides each re­

gion again in two equal volume parts, and it is again a sym­metry plane. But now it is easy to conclude that any plane

which contains the line L = P1 n P2 divides each region of

the bubble in two equal volume parts, and so it is a plane of

symmetry. We conclude that � is a surface of revolution

around the line L. So in fact we have some curves that, ro­

tated around a certain axis, give us the whole bubble. Be­

cause these curves generate constant mean curvature sur­faces, they are among the Delaunay curves in Figure 3.

As in the previously discussed isoperimetric problems,

symmetrization is not enough to classify the isoperimetric

solutions. Using again Hutching's results and stability tech­

niques, we are able to reduce the candidates different from

the standard double bubble to the possibilities depicted in

Figure 10. The final argument is again a stability one. By using ro­

tations orthogonal to the axis of revolution of the double

bubble, we prove

PROPOSITION. Consider a configuration of curves that gen­erates a solution of the double-bubble problem by rotation. Assume there are points {p1, . . . , Pnl in the regular part

'·

b ·""' ..... ....

p Figure 1 1 . The partition method.

of the curves so that the normal lines meet at some point p, possibly ao, in the axis of revolution. Then (p1, . . . , Pnl

cannot separate the configuration.

We illustrate the power of this Proposition by easily dis­

carding the first type of candidates. Pick the line L equidis­

tant from intersection points a and b. Assume that this line

meets the axis of revolution at point p. In each one of the

curves joining a and b there is at least one point p1 at max­

imum distance from p and at least one p2 at minimum dis­

tance from p. Then p 1 and P2 separate the configuration,

so that the generated bubble cannot be a solution of the

double-bubble problem.

In order to discard the second type of candidates some

more work is needed, but it has been done in [11).

Of course we can ask about the surfaces of least area which enclose and separate n regions in IR3. Existence and

regularity follow from the Almgren and Taylor results. For n = 3, 4 there are two natural candidates (see Fig. 12),

which we shall call again standard bubbles. For these vol­umes we also have the following

CONJECTURE. The standard n-bubble, n � 4, is the least­perimeter way to enclose and separate n given volumes in IR3.

However, the situation is extremely complicated when

we consider n > 4 regions, and in this case we even don't

have an applicant to solve the problem.

Symmetrization works for double bubbles in Euclidean

spaces of any dimension. It seems natural to hope that the

standard double bubble be the least-perimeter way to en­close and separate two given regions in IRn, for any n 2:: 3.

Figure 12. A standard triple bubble. Picture by John Sullivan, Uni· versity of Illinois (http://math.uiuc.edu/-jms)

VOLUME 24, NUMBER 3, 2002 1 3

In case n = 4 this has been proved, by using the arguments

of [11], in [17].

REFERENCES

1 . F. J. Almgren, Jr. , Existence and regularity almost everywhere of

solutions to elliptic variational problems with constraints, Mem.

Amer. Math. Soc. 4 (1 976), no. 1 65.

2. Maria Athanassenas, A variational problem for constant mean cur­

vature surfaces with free boundary, J. Reine Angew. Math. 377

(1 987), 97-1 07.

3. F. Barthe and B. Maurey, Some remarks on isoperimetry of Gauss­

ian type, Preprint ESI 721 , 1 999.

4 . Yu. D. Burago and V. A Zalgaller, Geometric inequalities, Springer­

Verlag, Berlin, 1 988, Translated from the Russian by A B. Sosin­

ski!, Springer Series in Soviet Mathematics.

5. R. Courant and D. Hilbert, Methods of mathematical physics. Vol.

I, lnterscience Publishers, Inc., New York, N.Y. , 1 953.

6 . C. Delaunay, Sur Ia surface de revolution dont Ia courbure moyenne

est constante, J. Math. Pure et App. 16 (1 841 ), 309-321 .

7 . Joel Foisy, Soap Bubble Clusters in IR2 and in IR3, Undergraduate

thesis, Williams College, 1 991 .

8. H. Hadwiger, Gitterperiodische Punktmengen und lsoperimetrie,

Monatsh. Math. 76 (1 972), 41 0-418.

9. Joel Hass and Roger Schlafly, Double bubbles minimize, Ann. of

Math. (2) 151 (2000), no. 2, 459-51 5.

1 0. Michael Hutchings, The structure of area-minimizing double bub­

bles, J. Geom. Anal. 7 (1 997), no. 2, 285-304.

1 1 . Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros,

Proof of the double bubble conjecture, Electron. Res. Announc.

Amer. Math. Soc. 6 (2000), 45-49 (electronic).

1 2 . Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros,

Proof of the double bubble conjecture, Annnals of Math. (2) 155

(2002), no. 2, 459-489.

A U T H O R S

MANUEL RITORE

Departmento de Geometria y Topologia

Universidad de Granada

18071 Granada

Spain

e-mail: ritore@ugr.es

Manuel Ritore, born in 1 966, studied at the Universidad de

Extremadura. He got his doctorate at the Universidad de

Granada in 1 994 under the supervision of Antonio Ros. He

continues to work on minimal surfaces, surfaces of constant

mean curvature, and isoperimetric problems.

14 THE MATHEMATICAL INTELLIGENCER

1 3. Wilbur R. Knorr, The ancient tradition of geometric problems, Dover

Publications, Inc., New York, 1 993.

1 4 . Blaine Lawson and Keti Tenenblat (eds.), Differential geometry, A

Symposium in Honor of Manfredo do Carma. Longman Scientific

& Technical, Harlow, 1 991 .

1 5 . Frank Morgan, Geometric measure theory, A beginner's guide.

Third ed. , Academic Press Inc. , San Diego, CA, 2000.

1 6 . Renato H. L. Pedrosa and Manuel Ritore, lsoperimetric domains in

the Riemannian product of a circle with a simply connected space

form and applications to free boundary problems, Indiana Univ.

Math. J. 48 (1 999), no. 4 , 1 357-1394.

1 7 . Ben Reichardt, Cory Heilmann, Yuan Y. Lai, and Anita Spielman,

Proof of the double bubble conjecture in IR4 and certain higher di­

mensions, Pacific J. Math. (to appear), 2000.

1 8 . Manuel Ritore, Applications of compactness results for harmonic

maps to stable constant mean curvature surfaces, Math. Z. 226

(1 997), no. 3, 465-481 .

1 9. --, Examples of constant mean curvature surfaces obtained from

harmonic maps to the two sphere, Math. Z 226 (1 997), no. 1 , 1 27-1 46.

20. Manuel Ritore and Antonio Ros, The spaces of index one minimal

surfaces and stable constant mean curvature surfaces embedded

in flat three manifolds, Trans. Amer. Math. Soc. 348 (1 996), no. 1 ,

391 -410.

21 . Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of

constant mean curvature with free boundary, Geom. Dedicata 56

(1 995), no. 1 , 1 9-33.

22. Michael Spivak, A comprehensive introduction to differential geom­

etry, vol. 4, Publish or Perish, Berkeley, 1 979.

23. Jean E. Taylor, The structure of singularities in soap-bubble-like

and soap-fi lm-like minimal surfaces, Ann. of Math. (2) 103 (1 976),

no. 3, 489-539.

24. Thomas I. Vogel, Stability of a liquid drop trapped between two

parallel planes, SIAM J. Appl. Math. 47 (1 987), no. 3, 5 1 6-525.

ANTONIO ROS

Departamento de Geometria y Topologia

Universidad de Granada

1 8071 Granada

Spain

e-mail: aros@ugr.es

Antonio Ros was bom in 1 957. He discovered Classical Dif­

ferential Geometry in the textbooks Differential Geometry of

Curves and Surfaces by M.P. do Carmo and A Survey of Min­

imal Surfaces by R. Osserman. His research interests include

variational problems for surfaces in Euclidean three-space .

M athe rnatica l l y Bent

The proof is in the pudding.

Opening a copy of The Mathematical

lntelligencer you may ask y ourself

uneasily , "lf'hat is this anyway-a

mathematical journal, or what?" Or you may ask, "lf'here am /?" Or even

"lf'ho am /?" This sense of disorienta­

tion is at its most acute when y ou

open to Colin Adams's column.

Rela:c. 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 01 267 USA

e-mail: Colin.C.Adams@williams.edu

Col i n Adam s , Editor

Dr. Yecke l and M r. H ide Colin Adams

"oh, what a shame, what a

shame." Inspector Armand

looked down at the now still form of

Dr. Yeckel. "Such a waste." "But Inspector," said Sargeant Lani­

gan with horror. "That man lying there

looks different than when he collapsed

just now. He has transformed into

someone else while lying there. I am

sure of it."

"Ah, Lonigan, in a sense he is the

same man and in a sense he is not."

"Yer speaking nonsense there, In­spector."

"Well, then sit down here, Lanigan,

and I will tell you a tale. A tale that will make your blood run colder than the

Thames in January."

"It's not one of those math stories of

yours, is it, Inspector?"

"In fact, it is, indeed. A story that

will make your teeth chatter like a

squirrel in heat."

"Fire away, Inspector. My teeth

need a good chatter."

"This story began with a young in­structor of mathematics, name of Dr.

Yeckel. A new Ph.D., he was bright,

friendly, and well scrubbed. Students

loved him. Had a job at the university

there in town. Taught calculus mostly

and sometimes linear algebra."

"Oh, I've heard linear algebra is

quite the course."

"Yes, Lanigan, it is, it certainly is.

Now this Dr. Yeckel, he el\ioyed teach­

ing. He liked the sound of chalk on a

board, the rustle of students in their

seats, the smell of Lysol in the bath­

rooms. He especially liked that smell.

"And the students knew he liked it.

Teaching, that is. And they appreciated

the attention he showered on them. His

willingness to meet with them at odd

hours, to answer their e-mails, and to

help them with the problems. He liked

them and they liked him. Yes, he was

happy as a mongoose in a snake pit, he

was. But you see, teaching was only part of his job there at the institution

of higher learning. Because, you see,

that institution was what is called a "re­

search university." Do you know what

that means, Lanigan?"

"Errr, does that mean they do some

kind of nasty experiments, Inspector?"

"Not exactly, Lanigan. It means they

search for new truths. Sometimes with

nasty experiments, and sometimes

without.

"Now this Dr. Yeckel had special­

ized in an area called 'number theory.' That is the study of numbers, like 2, 3,

5, and 7. When he was focused on his

teaching he was fine. But then he

would get involved in his research. And

suddenly a transformation would over­

come him."

"You mean he would become a crea­ture."

"Exactly, Lanigan. His hair would become unkempt. His fingernails

would become dirty. His eyes would

get bloodshot, and his shirt would be­

come unacceptable in its odor."

"Oh, my goodness, Inspector, a crea­

ture. Was he dangerous?"

"You have no idea, Lonigan. He was

in a deranged state of mind. The world

as we know it meant nothing to him.

He could easily step in front of a mov­

ing car without thinking to look if it

was safe. His mind would be off on Dio­

phantine approximations, a very ab­

stract area of mathematical considera­

tion indeed."

"Sounds fancy."

"Oh, believe me, it is. And he was

hooked on the Stillwell conjecture." "Is

that some kind of hard math problem?"

"Only the greatest open conjecture in all of Diophantine Approximation is

all. He became obsessed with it. Sud­

denly, his students weren't so impor­

tant to him. He would forget to meet

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3. 2002 1 5

his classes. The students who had

loved him so much would sit waiting

for him hour after hour, but rarely did

he come. When he did come, it was

even worse. They hardly recognized

him. They would ask him a question,

with their bright eyes and inquisitive

minds, and he would say, 'Hmmm?' and

lean against the wall lost in thought.

They would purposely make extra-loud

rustling sounds, but he could not hear

them. Sometimes, he would drop the

chalk in mid-lecture and wander out of

the room. The poor students, their lit­

tle hearts were broken."

"A sad tale indeed, Inspector."

"Sometimes we don't know what we

have until it is gone, Lonigan. Such is

how it was with him. Eventually, his

enrollments dropped. His wife left him,

his dog ran away, and the university

threatened to fire him if he didn't

meet his classes."

"As it should be."

"Perhaps so. But then, Lonigan, as

often happens in life, fate provided a

sudden and unexpected twist."

"How so, Inspector?"

"Y eckel showed that the Stillwell

conjecture was equivalent to Q equals

NQ."

"Yer talking gibberish, Inspector. I'm

no mathematician. I just walk a beat."

"Leave it to say, Lonigan, that he had

made a major step toward the solution

of the Stillwell conjecture. Suddenly he

was a mathematical celebrity. He was

invited to speak at colleges and uni­

versities all over the country. Recep­

tions were thrown in his honor, with

sparkling cider and little stuffed mush­

rooms. He received large federal grants

to continue his research and the uni-

16 THE MATHEMATICAL INTELLIGENCER

versity received the overhead on the

grants. Of course, all was forgiven, and

he was given tenure."

"Doesn't that mean they cannot fire

him for as long as he lives?''

"It does, indeed, Lonigan, it does in­

deed. And now the problem became

worse. His hair and beard grew longer

and more tangled. The t-shirt he wore

began to come apart at the arm pits.

His pants were frayed and stained up

and down with coffee."

"Did the university take action?"

"Oh, yes, they did. They made him

chair of the Mathematics Department."

"Now why did they do that?"

"Well, he was the most successful

mathematician in the department."

"And what does that have to do with

running a department?"

"A sad tale

indeed,

Inspector . "

"Ah, Lonigan, now you have wacked

the nail on its tiny top. It needn't have

anything to do with running a depart­

ment. And in this case, it did not.

Yeckel continued to work on his re­

search to the exclusion of all his other

duties. Appointments were missed.

Staffing reports were not submitted.

Hires were not made. Within a year's

time, the Mathematics Department was

in a shambles."

"Well, Inspector, then the university

must have realized its mistake."

"But Lonigan, you must remember,

universities are not like individuals

with common sense and the ability to

act on it. No, they are more like a thou­

sand toads trapped in a Bentley, all

hoping this way and that, slapping

against the windows and muddying the

plush leather interior."

"Oh, I see."

"So the university left him in charge.

And the department finally revolted.

Whole subdisciplines jumped ship. By

the end of his term, there was no one

left but the lifers who couldn't get work

elsewhere."

"And is that what did him in, In­

spector?"

"No, Lonigan, no. He couldn't have

cared less what happened to the rest

of the department. All that mattered to

him was his research. But then he

opened the paper one morning to find

that Q = NQ had been proved by a

graduate student from Southampton.

This immediately implied the Stillwell

conjecture. It was too much for him

to bear. His mathematical heart

broke."

"Ah. I see."

"He collapsed on the spot. This spot

right before us. And since his dream of

proving the Stillwell conjecture had

been destroyed, his body reverted to its

former state. He was no longer the

driven intellectual who derived all his

meaning from the pursuit of knowledge,

ignoring the real world around him. Now he reverted to the man he had

been, the nurturing, caring teacher

whom the students had loved so much."

"Ah, but it was too late, Inspector."

"Hardly. Nothing a good bath, a hair­

cut, and a breath mint can't fix. Help

him up, Lonigan. He should be fme in

an hour or so."

SCOTT W. WILLIAMS

M i ion - Buck Prob ems

pon publication of Apostolos Doxiadis 's new novel, Uncle Petros & Gold­

bach's Conjecture in 2000, the publishers, Faber and Faber in Britain and

Bloomsbury Publishing USA, offered $1,000,000 to individual(s) who solve

Goldbach's Conjecture. On May 10, The Clay Mathematical Sciences Insti-

tute inaugurated a $7,000,000 Millennium Prize, a million­

dollar award for the solution of each of seven famous prob­

lems. Contrary to belief, this publicity stunt has precedence

in Mathematics. This article is a result of my personal re­

view of the history of a few famous unsolved problems

whose statements can be understood by a person with an

undergraduate mathematics degree or less.

When I was a student, the Burnside Problem, the Sim­

ple Odd Group Conjecture (1963), and the Continuum Hy­

pothesis had just been resolved but the Riemann Hypoth­

esis, the Four-Color Map Problem, Fermat's Last Theorem,

the Bieberbach Conjecture, the Poincare Conjecture, and

the Goldbach Conjecture were all famous open problems.

Ten years later, the Four-Color Problem and the Alexan­

drov Conjecture were solved. In twenty years the Bieber­

bach Conjecture was proved. Thirty years later Fermat's

Last Theorem is gone and just a few of the aforementioned

problems remain, although others have surfaced. A solu­

tion to any of these problems brings "fame" and occasion­

ally one of the major mathematical prizes such as the

$145,000 Steele Prize, the $50,000 Wolf Prize, a special gold

medal (along with $15,000) called The Fields Medal, infor­

mally known as the "Nobel Prize of Mathematics," or what

I call the real "Nobel Prize" for mathematicians, the Royal

Swedish Academy of Sciences' $500,000 Crafoord Prize.

The seven problems whose solutions will bring

$1,000,000 each from the Clay Mathematical Sciences In-

An earlier version of this article appeared in the NAM Newsletter XXX1 (2)(2000).

stitute (http://www.claymath.org) are the Poincare Con­

jecture and the Riemann Hypothesis, both discussed below,

and the P versus NP problem, the Hodge Conjecture, the

Yang-Mills Existence and Mass Gap, the Navier-Stokes Ex­

istence and Smoothness; and the Birch and Swinnerton­

Dyer Conjecture. The problems are accompanied by arti­

cles written by Stephen Cook, Pierre Deligne, Enrico

Bombieri, Charles Fefferman, and Andrew Wiles.

Attaching monetary value to mathematics questions is

not new. In 1908 German industrialist Paul Wolfskehl es­

tablished a prize of 10,000 DM (approximately $1,000,000

at the time) for a proof of Fermat's Last Theorem. (See No­

tices A.M.S. 44 no. 10 (1997), 1294-1302.) Unfortunately in­

flation diminished the prize value so that in 1997 Wiles col­

lected just $50,000; however, the Royal Swedish Academy

of Sciences also awarded Wiles the Schock Prize, and he

received the Prix Fermat from the Universite Paul Sabatier.

DeBranges was awarded the Ostrowski Prize for proving a

much stronger conjecture than the Bieberbach Conjecture.

"The Prince of Problem-Solvers and the Monarch of Prob­

lem-Posers," the late Paul Erdos, who won the $50,000 Wolf

Mathematics Prize, was famous for offering cash prizes to

those mathematicians who solved certain of his problems.

These prizes ranged from $10,000 for what he called "a hope­

less problem" in number theory to $25 for something that

he considered not particularly difficult but still tricky, pro­

posed in the middle of a lecture. Since Erdos's 1996 death,

© 2002 SPRINGER· VERLAG NEW YORK, VOLUME 24. NUMBER 3, 2002 1 7

other mathematicians have continued this practice. Now a corporation offers one million dollars and an institute of­fers more.

Fields Medals have not been awarded to persons over the age of forty. Concerning solutions of famous problems, some Fields Medals were awarded to:

Selberg (1950) for his work on the Riemann Hypothesis; Cohen (1966) for his resolution of the Continuum Hypoth­esis; Smale (1966) for his work on the Generalized Poin­care Conjecture for n > 4; Thompson (1970) for his part in the solution of the Odd Simple Group Conjecture; Bombieri (1974) for his work on the local Bieberbach Conjecture; Faltings (1986) for his solution of Mordell's Conjecture; Freedman (1986) for his work on the Generalized Poincare Conjecture for n = 4; Borcherds (1998) for his solution of the Monstrous Moonshine Conjecture.

Perhaps via "fame" a solution will bring to some a mod­est fortune. The unsolved problems below (Goldbach's Conjecture, The Kolakoski sequence, The 3x + 1 Problem, Schanuel's Conjectures, Box-Product Problem, Odd Perfect Number Problem, Riemann Hypothesis, Twin Primes Con­jecture, Lost-in-a-Forest Problem, Palindrome Problem, The Poincare Conjecture) all have simple statements. Some of these problems (the Riemann Hypothesis and the Poin­care Conjecture) are usually taken to have more value to the field than others. However, there have been lesser prob­lems which were not resolved by simply pushing the ex­isting techniques further than others had done, but rather by introducing highly original ideas which were to lead to many developments. I, therefore, call them all million­buck problems because I believe (the techniques involved in) their resolution will be worth at least $1 million to Mathematics.

1 . Goldbach's Conjecture

On June 7, 1742, Christian Goldbach wrote a letter to L. Euler suggesting every even integer is the sum of two primes, and this is unproved still, although it is known to be true for all numbers up to 4 · 1013. The closest approx­imation to a solution to Goldbach's Conjecture is Chen­Jing Run's recent result that every "sufficiently large" even number is of the form p + qr, where p, q, r are primes. For the $1,000,000 prize, Faber and Faber in Britain, and Bloomsbury Publishing USA, issued a stringent set of re­quirements, which included publishing the solution to Goldbach's Conjecture. Contestants had until March 2002 to submit their applications and March 2004 to publish the solution. If there is a winner, the prize will be awarded by the end of 2004.

A still-unsolved consequence of Goldbach's Conjecture is the odd Goldbach Conjecture, "every odd integer greater than five is the sum of three primes." This has been shown to be true for odd integers greater than 107000000 and will probably fall when proper computing power is devoted to it.

2. Beat's Conjecture

This is a generalization of Fermat's Last Theorem. If Ax +

J3Y = CZ, where A, B, C, x, y, and z are positive integers and

1 8 THE MATHEMATICAL INTELLIGENCER

x, y, and z are all greater than 2, then A, B, and C must have a common factor. Andrew Beal is a banker and an am­ateur mathematician, yet he offers $75,000 for the resolu­tion of this conjecture, which was first announced in 1997. The prize committee consists of Charles Fefferman, Ron Graham, and R. Daniel Mauldin, and the funds are held in trust by the American Mathematical Society.

3. Schanuel's Two Conjectures

(not to be confused with the Schanuel Lemma

or the Ax-Schanuel Theorem)

In the early 1960s, Stephen Schanuel made two conjectures about the algebraic behavior of the complex exponential function. Schanuel offers $2,000, $1,000 each, for the pub­lished resolution of the conjectures in his lifetime. The Schanuel Conjecture is the following independence prop­erty of (C,eZ): If Z1, z2, . . . , Zn in C are complex numbers linearly independent over the rationals, then some n of the 2n numbers Z1, z2, . . . , Zn, e01, e-<2, . . . ezn are algebraical1y independent. The Converse Schanuel Conjecture says that there is nothing more to be said. Explicitly, let F be a count­able field of characteristic zero and E : F � F a homomor­

phism from the additive group to the multiplicative group whose kernel is cyclic. The conjecture is that if (F,E) has the independence property, then there is a homomorphism of fields h : F � C such that h(E(x)) = r/'(x). Either of the two conjectures would imply, for example, algebraic inde­pendence of e and 7r. [For the first take z1 = 1, z2 = 7Ti; for the second, one must construct (F,E) with an element p such that E(ip) = - 1 and so that E(l), p are algebraically independent. ] At present, we don't even know that e + 7r

is irrational.

4. The Kolakoski Sequence

Consider the sequence of ones and twos

if = ( 1221 12122122 1 121 12212 1 121221 121121 221221 12122121121 122122 1 12).

A block of if is a maximal constant subsequence. We con­sider the blocks and their lengths. For example, beginning from the left, the first block (I) has length 1. The second block (22) has length 2. The third block (11) has length 2. Continue in this fashion and notice that the sequence A = (1221 12122 1 . . . ) of block lengths is an initial segment of if. The Kolakoski Sequence is the (unique) infinite sequence if of ones and twos, beginning with 1, for which the se­quence A of block lengths satisfies A = if. Chris Kimberling (see http://cedar.evansville.edu/-ck6/index.html) promises a prize of $200 to the first person to publish a solution of all five problems below (he says chances are if you solve one, you'll see how to solve the others). Considering the last 4 questions as one makes the Kolakoski Sequence questions interesting:

i. Is there a formula for the nth term of if? ii. If a string (e.g., 2122 1 1) occurs in if, must it occur again? iii. If a string occurs in if, must its reversal also occur? (1 122 12 occurs)

iv. If a string occurs in cr, and all its 1s and 2s are swapped,

must the new string occur? (121 122 occurs)

v. Does the limiting frequency of 1s in cr exist and is it 1/2?

5. The Box-Product Problem

Given countably infinitely many copies of the interval [0, 1] ,

the typical (Tychonov) product topology on their product is

topologically a copy of the Hilbert Cube. Give it Urysohn's

1923 box-product topology instead (so open sets are unions

of products of arbitrary open intervals). The Box-Product

Problem asks, "Is the box-product topology on the product

of countably infinitely many copies of the real line normal?"

In other words, can disjoint closed sets be separated by dis­

joint open sets? In 1972 Mary Ellen Rudin showed that the

continuum hypothesis implies YES, but in 1994 L. Brian

Lawrence proved the answer is NO to the corresponding

problem for uncountably many copies. What is known

about the problem is no different whether the real line is

replaced by such related spaces as the closed interval [0, 1 )

o r the convergent sequence and its limit (the space X = { 2 -n: n E N} U {0} C R) and is related to combinatorial

questions in Set Theory. Scott Williams offers (with appeal

to A Hitch-Hiker's Guide to the Galaxy) a $42 prize to the

person who settles the box-product problem in his lifetime.

6. The Collatz 3x + 1 Conjecture

Because it is easy to program your computer to look for

solutions, many youngsters (and adults) have played with

the 3x + 1 problem: On the positive integers define the

function F(x) = 3x + 1 if x is odd and F(x) = x/2 if x is

even. Iterations of F lead to the sequences (1 , 4, 2, 1), (3,

10, 5, 16, . . . , 1), and (7, 22, 1 1 , 34, 17, 52, 26, 13, 40, 20, 10,

. . . , 1). The 3x + 1 conjecture, stated in 1937 by Lothar

Collatz, is, "For each integer x, applying successive itera­

tions of F, eventually yields 1." During Thanksgiving vaca­

tion in 1989 I programmed my desktop computer to verify

the conjecture by testing integers in their usual order. M­

ter 3 days it verified that the first 500,000 integers satisfied

the 3x + 1 conjecture. Currently, the conjecture has been

verified for all numbers up to 5.6 · 1013, but not by me.

For fun, consider the different conclusions to three

slightly different versions of this problem obtained by ex­

changing 3x + 1 for one of 3x - 1, 3x + 3, or 5x + 1.

7. Odd Perfect Number Problem

Does there exist a number that is perfect and odd? A num­

ber is perfect if it is equal to the sum of all its proper divi­

sors. This question was first posed by Euclid and is still

open. Euler proved that if N is an odd perfect number, then

in the prime power decomposition of N, exactly one expo­

nent is congruent to 1 mod 4 and all the other exponents

are even. Using computers, it has been shown that there

are no odd perfect numbers < 10300.

8. Riemann Hypothesis

This is the most famous open problem in mathematics. In

his 1859 paper On the Number of Primes Less Than a

Given Magnitude, Bernhard Riemann (1826--1866) ex­

tended the zeta function, defined by Euler as

00 1 ?(s) = L - for s > 1,

n� l ns

to be defined for every complex number. Riemann noted

that his zeta function trivially had zeros at s = -2, -4, - 6,

. . . , and that any remaining, nontrivial zeros were symmet­

ric about the line Re( s) = 1/2. The Riemann Hypothesis says

all nontrivial zeros are on this line; i.e., they have real part 1/2.

9. Twin Primes Conjecture

A twin prime is an integer p such that both p + 1 and p -

1 are prime numbers. The first five twin primes are 4, 6, 12,

18, and 30. The Twin Primes Conjecture states there are in­

finitely many twin primes. It is known there are 27,412,679

twin primes <1010• The largest known twin prime is

2,409,1 10, 779,845 · 260000, which has 18,072 digits. However,

the sum of the reciprocals of the twin primes is finite.

1 0. The Poincare Conjecture

Henri Poincare said, "Geometry is the art of applying good

reasoning to bad drawings." For a positive integer n, an n­manifold is a Hausdorff topological space with the prop­

erty that each point has a neighborhood homeomorphic to

n-space Rn. The manifold is simply connected if each loop

in it can be deformed to a point (not possible if it, like a

doughnut, has a hole). The Generalized Poincare Conjec­

ture says that each simply connected compact n-manifold

is homeomorphic to the n-sphere. Near the end of the 19th

century, Poincare conjectured this for n = 3, and the Gen­

eralized Poincare Conjecture has been solved in all cases

except n = 3.

1 1 . Palindrome Problem

A palindrome is a phrase or word which is the same if you

reverse the position of all the letters. A integer palindrome

has the same property; e.g., 121. Here is an algorithm which

one might think leads to a palindrome: Given an integer x,

let x* be the reverse of n's digits, and F(x) = x + x*. Now

iterate the process. Considering sequences of iterations of

F, we have (29, 29 + 92 = 121) and (176, 176 + 671 = 847,

1595, 7546, 14003, 44044). The examples show that itera­

tions of 29 and 176 lead, respectively, to palindromes 121

and 44044. The Palindrome Problem is "Given any integer

x, do iterations of F lead to a palindrome?" This is unsolved

even in the case x = 196.

1 2. Lost-in-a-Forest Problem

In 1956 R. Bellman asked the following question: Suppose

that I am lost without a compass in a forest whose shape

and dimensions are precisely known to me. How can I es­

cape in the shortest possible time? Limit answers to this

question for certain two-dimensional forests: planar re­

gions. For a given region, choose a shape of path to follow

and determine the initial point and direction which require

the maximum time to reach the outside. Then minimize the

maximum time over all paths. For many plane regions the

answer is known: circular disks, regular even-sided poly-

VOLUME 24, NUMBER 3, 2002 19

gonal regions, half-plane regions (with known initial dis­

tance), equilateral triangular regions. However, for some

regions-for regular odd-sided polygonal regions in general

and triangular regions in particular-only approximations

to the answer are known.

This article is dedicated to John Isbell. Concerning this

article, I had personal correspondence with William

Massey, Mohan Ramachandran, Samuel Schack, and

Stephen Schanuel. All errors, however, are mine.

REFERENCES

General References

J Korevaar, Ludwig Bieberbach's conjecture and its proof by Louis de

Branges, Amer. Math. Monthly 93 (1 986), 505-514 .

For a wealth of information on some of the unsolved problems above,

also see the MathSoft web page: http://www.mathsoft.com/asolve/

index.html

The extraordinary story of Fermat's Last Theorem: http://www.cs.uleth.

ca/�kaminski/esferm03.html

Erdos References

http://vega.fmf.uni-lj.si!�mohar/Erdos.html

http://www. maa.org/features/erdos.html

http://www-groups.dcs.st-and.ac.ukl�history/

1 . Goldbach's Conjecture References

Chen, Jing Run: On the representation of a large even integer as the

sum of a prime and the product of at most two primes. Sci. Sinica

1 6 (1 973), 1 57-1 76.

http://www.utm.edu/researchlprimes/glossary/GoldbachConjecture.html

2. Beal's Conjecture References

R. Daniel Mauldin, A Generalization of Fermat's Last Theorem: The Beal

Conjecture and Prize Problem, Notices of the AMS, December 1 997,

p. 1 437.

3. Schanuel's Conjecture References

Chow, T. Y. , What is a Closed-Form Number? Amer. Math. Monthly

1 06 (1 999), 44()--448.

Macintyre, A. , Schanuel's Conjecture and Free Exponential Rings, Ann.

Pure Appl. Logic 51 ( 1991) , 241 -246.

John Shackell , Zero-equivalence in function fields defined by algebraic

differential equations, Transactions of the Amer. Math. Soc. 336

(1 993), 1 5 1 -1 71 .

Jacob Katzenelson, Shlomit S. Pinter, Eugen Schenfeld, Type match­

ing, type-graphs, and the Schanuel Conjecture. ACM Transactions

on Programming Languages and Systems 1 4 (1 992), 574-588.

4. Kolakoski Sequence References

W. Kolakoski, Problem 5304, Amer. Math. Monthly, 73 (1 966),

681 -682.

I. Vardi , Computational Recreations in Mathematics, Addison-Wesley,

1 991 ' p. 233.

5. Box-Product Problem References

L. Brian Lawrence, Failure of normality in the box product of uncount­

ably many real lines. Trans. Amer. Math. Soc. 348 (1 996), 1 87-203.

S.W. Williams, Box products. Handbook of Set-Theoretic Topology (K.

Kunen and J.E. Vaughan eds.), North-Holland (1 984), 1 69-200.

Web reference: http:/ /www.math.buffalo.edu/ �sww/Opapers/Box.Product.

Problem.html

20 THE MATHEMATICAL INTELLIGENCER

6. Collatz 3x + 1 Conjecture References

http://www.cs.unb.ca/�alopez-o/math-faq/node61 .html

Richard K. Guy, Unsolved problems in number theory Springer, Prob­

lem E16 .

G.T. Leavens and M. Vermeulen. 3x + 1 search programs. Journal

Comput. Math. Appl. 24 (1 992), 79-99.

7. Odd Perfect Number References

http://www.cs.unb.ca/�alopez-o/math-faq/node55.html

8. Riemann Hypothesis References

http://www. utm .edu/research/primes/notes/rh . html

http://www.math.ubc.ca/�pugh/RiemannZeta/RiemannZetaLong.html

http://match.stanford.edu/rh/

9. Twin Primes References

http://www.utm.edu/research/primes/lists/top20/twin.html

1 0. Poincare Conjecture References

http:/ /mathworld . wolfram .com/PoincareConjecture.html

1 1 . Palindrome Problem References

http://www .seanet.com/ �ksbrown/kmath004. htm

1 2. References for Bellman's Lost-in-a-Forest

R. Bellman, Minimization problem. Bull. Amer. Math. Soc. 62 (1 956), 270.

J .R. Isbell, An optimal search pattern , Naval Res. Logist. Quart. 4 (1 957),

357-359.

Web survey and reference article: http://www.mathsoft.com/asolve/

forest/forest. html

A U T H O R

SCOTT W. WILLIAMS

Department of Mathematics State University at Buffalo

Buffalo, NY 14260-2900

USA

e-mail: sww@buffalo.edu

web: http://www.math.buffalo.edu/-sww/

Scott Williams was raised in Baltimore and got his doctorate

at Lehigh University in 1 969. His thesis and most of his pub­

lications are in general topology, with the usual tie-ins such as

set theory and spaces of continuous functions. He has been

at SUNY Buffalo with only occasional wandering since 1 971 .

Married, with three daughters.

He has been a professional musician, and has many ex­

hibited works in the visual arts. He is seriously interested in

the African-American heritage, including his own. At present

he is editing the newsletter of the National Association of Math­

ematicians.

ROBERT FINN

Eight Remarkab e Propert ies of Cap i ary Surfaces

hysically, a capillary surface is an interface separating two fluids that are adja­

cent to each other and do not mix. Examples are the interface separating air and

water in a "capillary tube" (Figure 1), the outer surface of the "sessile liquid drop"

resting on a horizontal plate, and that of the "pendent drop" supported in stable

equilibrium by such a plate (Figure 2). The seeming con­flict in these three examples with the intuitive dictum that "water seeks its own level" certainly was of historical sig­nificance in drawing attention to the problems and devel­oping a general theory.

In general, one considers a connected volume "V of liq­uid resting on a rigid support surface 'W (Figure 3). One notes that the shape of the free interface � depends strongly on the shape of 'W (and orientation in a gravity or other field g, if any); it may be less immediately evident that the form of � is also strongly dependent on the mate­rial composition of 'W.

The underlying mathematical modeling on which most modem theory is based was initiated by Young [ 1 ) and by Laplace [2) in the early nineteenth century. The theory was put onto a unified conceptual footing by Gauss [3] , who used the Principle of Virtual Work formulated by Johann Bernoulli in 1717 to characterize such surfaces as equilib­ria for the mechanical energy of the system. I adopt that formulation in what follows, although one should note some uncertainties about it that are pointed out in [4]. In modem notation, the position vector x on the free surface � satisfies

llx = 2HN. (1a)

Here H is the scalar mean curvature of � (the average of two sectional curvatures by orthogonal planes contain­ing a common normal to �), and N a unit vector normal to �. The variational condition leads to an expression for H in terms of position. The operator ll denotes the intrinsic Laplacian on � (the Laplacian in the metric of �, obtained by evaluating the traditional Laplacian in conformal coor­dinates and multiplying by the local area ratio). For exam­ple, on a sphere of radius R one finds H = 1/R, and the Laplacian of a function on the sphere at a point p can be obtained as the Laplacian on the tangent plane at p, con­sidered as the stereographic projection from the diametri­cally opposite point on the sphere. In general, the Lapla­cian in (1a) is a highly nonlinear operator.

The free surface � meets the rigid surface 'W in a con­tact angle y that depends only on the physical character­istics of the materials, and not on the shape of 'W or of �, nor on the thickness of'W, nor on the presence or absence of external (gravity) fields. Thus, if all materials are ho­mogeneous, then

y == canst. (lb)

Differing materials give rise to widely differing values of y. From a mathematical point of view, y is prescribed; we may normalize 0 :::::; y :::::; 7T.

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 21

Figure 1 . Capillary tube; 'Y < 1r /2.

The position vector of every smooth surface satisfies (la). Capillary surfaces are distinguished by the particular form of H, arising from the physical conditions. In a verti­cal gravity field g (which may vanish or be negative) we find

pg H(x) = - z + canst.

(T (2)

where p is the density change across ';1, u the surface ten­sion, and z the height above a reference level. The constant in (2) is to be determined by an eventual volume constraint. We are then faced with the problem of finding a surface whose mean curvature is a prescribed function of position, and which meets prescribed bounding walls in a prescribed angle y. In the following sections, I examine the behavior of solutions to this problem in varying contexts.

Property 1 . Discontinuous Disappearance

We consider a capillary tube with general section n in the absence of gravity. That would not be appropriate for the con­figuration of Figure 1, as in that case all fluid would flow ei­ther out to infinity if y < Tr/2, or to the bottom of the tube if

y > Trl2. I will therefore assume that the tube has been re­moved from the bath and closed at the bottom, and that a prescribed (finite) volume of fluid covering the base has been inserted at the bottom. It can be shown [5] that every solu­tion surface for (la) bounded by a simple closed curve en­circling the side walls projects simply onto the base, and thus admits a representation z = u(x, y). We then find from (la)

div Tu = 2H = const., 'Vu Tu = Yl + 1Vul2 (3a)

b

Figure 2. a) Sessile drop; b) Pendent drop.

22 THE MATHEMATICAL INTELLIGENCER

Figure 3. General configuration.

in n, with

v · Tu = cos y (3b)

on an; here , is the unit exterior-directed normal vec­tor. Note that H is determined by these conditions. In fact, the di;ergen�e theorem together with (3a,b) yields 2Hjfll = lan,cos y, mdependent of volume.

In the special case in which n is a circular disk, (3a,b) can be solved explicitly by a spherical cap; Figure 4 illus­trates the case 0 ::s y < Trl2. To some extent, this same so­lution applies when n is a regular polygon, because the ver­tical planes through the sides cut any incident sphere in constant angles. For example, if in Figure 5 we choose the circumscribing circle to be the equatorial circle of a lower hemisphere ';1: v(x, y), then v(x, y) solves the problem for the value y such that a + y = Tr/2.

Values of y yielding a + y > Tr/2 are obtained simply by increasing the radius of the circle. However, a + y < Trl2 cannot be achieved this way, as the equatorial disk would no longer cover the entire domain fl. The difficulty that appears is not an accident of the procedure; it reflects rather a gen­eral characteristic of the local behavior of solutions of (3a, b) near comer points. The following result is proved in (6].

Figure 4. Circular section; surface interface.

Figure 5. Hexagonal section; equatorial circle of lower hemisphere.

THEOREM 1 .1 : If a + y < n/2 at any corner point P of open­ing angle 2a, then there is no neighborhood of p in n, in which there is a solution of (3a) that assumes the data (3b) at boundary points in a deleted neighborhood of P on '2:.

In Theorem 1.1 no growth condition is imposed at P. It can be shown that if a + y 2: n/2, then any solution defined in a neighborhood of P is bounded at P. Further, in partic­ular cases (such as the regular polygon discussed above) solutions exist whenever a + y 2: n/2. Thus, there can be a discontinuous change of behavior, as y decreases across the dividing mark a + y = n/2, in which a family of uni­formly smooth bounded solutions disappears without dis­cernible trace.

Figure 6. Water in wedges formed by acrylic plastic plates; g > 0. a) a + 'Y > 7r/2; b) a + 'Y < 7r/2.

This striking and seemingly strange behavior was put to experimental test by W. Masica in the 132-meter drop tower at the NASA Glenn Laboratory in Cleveland, Ohio. This drop tower provides about five seconds of free fall in vac­uum, in effective absence of gravity. Figure 6a,b shows two identical cylindrical containers, having hexagonal sections, after about one second of free fall; the configuration did not noticeably change during the remaining period of fall. The containers were partially filled, with alcohol/water mixtures of different concentrations, leading to data on both sides of critical. In Figure 7a, a + y > 1r/2, and the spherical cap solution is observed. In Figure 7b, a + y <

1r/2. The fluid climbs up in the edges and partly wets the top of the container, yielding a surface interface 9' that folds back over itself and doubly covers a portion of n, while

{h

Figure 7. Different fluids in identical hexagonal cylinders during free-fall. a) a + 'Y > 7r/2; b) a + 'Y < 7r/2.

VOLUME 24, NUMBER 3, 2002 23

'Y

Figure 8. Behavior of interface in corner; a + 'Y < 7TI2.

leaving neighborhoods of the vertices of n uncovered (see Figure 8). Thus a physical surlace exists under the given con­ditions as it must, but it cannot be obtained as solution of (3) over n. The seeming "non-existence" paradox appeared be­cause we were looking for the surlace in the wrong place.

I emphasize again that the change in behavior is dis­continuous in terms of the parameter y. Were the top of the container to be removed when a + y < 7T/2, the fluid would presumably flow out the corners until it disappeared entirely to infmity. For any larger y, the fluid height stays bounded, independent of y.

In the presence of a downward-directed gravity field, equation (3a,b) must be replaced by

divTu = KU + const. (4a)

in n,

v · Tu = cosy (4b)

on an, with K = pg/u. There is again a discontinuous change at the same critical y; although in this case a solu­tion continues to exist as y decreases across the critical value. The discontinuous behavior is evidenced in the sense that every solution with a + y < 7T/2 is necessarily un­bounded at P, whereas if a + y ::::: 7T/2 then all solutions in a fixed neighborhood of P are bounded, independent of y in that range; see the discussion in [7], Chapter 5.

This result was tested experimentally by T. Coburn, who formed an angle with two acrylic plastic plates meeting on a vertical line, and placed a drop of distilled water between

24 THE MATHEMATICAL INTELLIGENCER

them at the base. Figure 6 shows the result of closing down the angle about two degrees across the critical opening. On the left, a + y 2: 7T/2; the maximum height is slightly below the predicted upper bound. On the right, a + y < 7T/2. The liquid rises to over ten times that value. The experiment of Coburn establishes the contact angle between water and acrylic plastic to be 80° ± 2°.

There is not universal agreement on the physical def­inition of contact angle. In view of "hysteresis" phenomena leading to difficulties in its measurement, the concept has been put into some question, and the notions of "advanc­ing" and "receding" angles were introduced. Also these quantities are not always easily reproducible experimen­tally. The procedure just described gives a very reliable and reproducible measurement for the "advancing" angle, when y is close to 7T/2; but if y is small, the region at the vertex over which the rise height is large also becomes very small, which can lead to experimental error. This difficulty was in large part overcome by the introduction of the "canoni­cal proboscis" [8, 9, 10], in which the linear boundary seg­ments are replaced by precisely curved arcs, leading to large rise heights over domains whose measure can be made as large as desired. The procedure has the drawback that it can require a zero-gravity space experiment over a large time period. Nevertheless, its accuracy has been suc­cessfully demonstrated [ 11 ] , and it can yield precise an­swers in situations for which conventional methods fail.

Property 2. Uniqueness and Non-uniqueness

Let us consider a fixed volume V of liquid in a vertical cap­illary tube closed at the base n, as in Figure 9a. Let I = an be piecewise smooth, that is, I is to consist of a fmite number of smooth curved segments that join with each other in well-defined angles at their end points and do not otherwise intersect. One can prove ([7], Chapter 5):

THEOREM 2.1: Let Io be any subset of I, of linear Hausdorff measure zero. Then if K ::::: 0, any solution of ( 4a) in n, such that ( 4b) holds at aU smooth points ofi'-2.0, is uniquely de­termined by the volume V and the data on I'-2.0•

Note that if K > 0, then a solution always exists; see, e.g., [ 12, 13]. If K = 0 then further conditions must be imposed to ensure existence; see Property 5 below.

In Theorem 2.1, no growth conditions are imposed; nev­ertheless, the data on any boundary subset of Hausdorff measure zero can be neglected in determining the solution. This property distinguishes the behavior of solutions of (3) or of ( 4) sharply from that of harmonic functions, for which failure to impose the boundary condition at even a single point in the absence of a growth condition leads to non­uniqueness.

Uniqueness has also been established for the sessile drop of Figure 9b. The known proof [14] proceeds in this case in a very different way. But in view of the uniqueness property in these particular cases, it seemed at first natural to expect that the property would persist during a contin­uous convex deformation of the plane into the cylinder, as indicated in Figure 9c.

a c

� g ""' ""' ""' ""'

""' ""' ""' ""' ""' " ""' ""' ""' ""'

""' ""' � ""' ""' b

QJ Figure 9. Support configurations: (a) capillary tube, general section; (b) horizontal plate; (c) convex surface.

Efforts to complete such a program turned out to be fruitless, for good reason. Consider, as a possible interme­diate configuration in such a process, a vertical circular cylinder closed at the bottom by a 45° right circular cone (Figure 10). If one fills the cone almost to the joining cir­cle, with a fluid whose contact angle with the bounding walls is 45°, a horizontal surface provides a particular so­lution of (1) with that contact angle. That is the case in any vertical (or vanishing) gravity field. On the other hand, if a large enough amount of fluid is added, the fluid will cover the cone and the contact curve will lie on the vertical cylin­der. In this case, the fluid cannot be horizontal at the bound­ing walls in view of the 45° contact angle, and a curved in­terface will result, as in the figure. It is known that if g :=:::: 0, there is a symmetric solution interface whose contact line is a horizontal circle, and that the interface lies entirely below that circle. Adding or removing fluid does not change the shape of the interface, as long as the contact line lies above the joining circle with the cone. It is thus clear that

Figure 1 0. Non-uniqueness.

one can remove fluid until the prescribed volume is at­tained, and obtain a second solution in the container, as in­dicated in the figure.

The construction indicated can be extended in a re­markable way [15, 16] :

THEOREM 2.2: There exist rotationally symmetric contain­ers admitting entire continua of rotationally symmetric equilibrium interfaces ':!, all with the same mechanical energy and bounding the same fluid volume.

This result holds for any vertical gravitational field g. The case g = 0 is illustrated in Figure 1 1. Some physical con­cerns about the construction are indicated in [4]; neverthe­less, it is strictly in accord with the Gauss formulation.

The question immediately arises, which of the family of interfaces will be observed if the container is actually filled with the prescribed volume V. An answer is suggested by the following further result [ 16, 17, 18]:

THEOREM 2.3: All of the interfaces described in Theorem 2.2 are mechanically unstable, in the sense that there ex­ist interfaces arbitrarily close to members of the family, bounding the same volume and satisfying the same boundary conditions, but yielding smaller mechanical energy.

These other interfaces are necessarily asymmetric. Be­cause it is known [19] that a surface of minimizing energy exists, the construction provides an example of "symmetry breaking," in which symmetric conditions lead to asym­metric solutions.

This prediction was tested computationally by M. Calla­han [20], who studied the case g = 0 and found a local min­imum (potato chip) and a presumed absolute minimum (spoon); see Figure 12. It was then tested experimentally in a drop tower by M. Weislogel [21] , who observed the "spoon" surface within the five-second limit of free fall. In

VOLUME 24, NUMBER 3, 2002 25

, ' - - - - - - - -- - - ... ... _

- - - - - - - - - - - - -

' , ... _ _ _ _ _ _ _ _ _ '

... _ _ _ _ _ _ _

, , , ,

Figure 1 1 . Continuum of interfaces in exotic container; g = 0. All in­terfaces yield the same sum of surface and interfacial energy, bound the same volume, and meet the container in the same angle y = 80°.

a more extensive experiment on the Mir Space Station, S. Lucid produced both the potato chip and the spoon [22]. Her obsetvation is compared with the computed surfaces in Figure 12.

Property 3. Liquid Bridge Instabilities, Zero g; Fixed Parallel Plates

In recent years, a significant literature has appeared on sta­bility questions for liquid bridges joining parallel plates with prescribed angles in the absence of gravity, as in Figure 13.

The bulk of this work assumes rigid plates and exaniines the effects of free surface perturbation; see, e.g., [23-29) . In general terms, it has been shown that stable bridges in this sense are uniquely determined rotationally symmetric surfaces, known as catenoids, nodoids, unduloids, or, as particular cases, cylinders or spheres. There is evidence to suggest that corresponding to the two contact angles 'Yb /'2, and separation distance h of the plates, there is a criti­cal volume VcrC y1, y2; h) such that the configuration will be unstable if V < Vcr and stable if V > Vcr· That assertion has not been completely proved.

Because stability criteria are invariant under homothety, the above assertion would imply that if the plate separa­tion is decreased without changing the volume or contact angles, then an initially stable configuration will remain sta­ble. In [29], Finn and Vogel raised the question: suppose that a bridge is initially stable; will every configuration with the same liquid profile, but with plates closer together, also be stable? One would guess a positive answer, because wi:th plates closer together there is less freedom for fluid per­turbation. But we note that we will have to change the con­tact angles, resulting in changed energy expressions, and the requirement of zero volume change for admissible per­turbations has differing consequences for the energy changes resulting from perturbations.

In fact, Zhou in [26] showed that the answer can go ei­ther way, and even can move back and forth several times during a monotonic change in separation h, so that the sta­bility set will be disconnected in terms of the parameter h. Zhou considered bridges whose bounding free surfaces are catenoids, which are the rotationally symmetric minimal

Figure 12. Symmetry breaking in exotic container, g = 0. Below: calculated presumed global minimizer (spoon) and local minimizer (potato chip). Above: experiment on Mir: symmetric insertion of fluid (center); spoon (left); potato chip (right).

26 THE MATHEMAnCAL INTELLIGENCER

Figure 13. Liquid bridge joining parallel plates; g = 0.

surlaces. She proved that if the contact angles on both plates are equal, and if the plates are moved closer to­gether equal distances without changing the profile, then an initial stability will be preserved. However, that need not be so if only one of the plates is moved. Let y1 be the contact angle with the lower plate, and hold this constant; Zhou showed that there are critical contact angles y' = 14.38°, Yo = 14.97°, such that if y' < Y1 < Yo then if the up­per plate is sufficiently distant in the range Y1 < Y2 < 7T - y�, the corifiguration will be unstable. On moving that plate downward, it will enter a stability interval; on con­tinued downward motion, the configuration will again become unstable, and finally when the plates are close enough, stability will once more ensue.

Property 4. Liquid Bridge Instabilities, Zero g; Tilting of Plates

In the discussion just above, motion of the plates was ex­cluded from the class of perturbations introduced in the stability analysis. More recently, the effect of varying the inclination of the plates was examined, with some unex­pected results [30].

THEOREM 4.1: Unless the initial configuration is spheri­cal, every bridge is unstable with respect to tilting of ei­ther plate, in the sense that its shape must change dis­continuously on infinitesimal tilting.

It should be noted that a spherical bridge joining paral­lel plates is a rare event, occurring only under special cir­cumstances. A necessary condition is y1 + Y2 > 7T ; for each such choice of contact angles, there is exactly one volume that yields a spherical bridge.

A spherical bridge can change continuously on plate tilt­ing; however, for general tubular bridges, instability must be expected, in the sense of discontinuous jump to another configuration. With regard to what actually occurs, one has

THEOREM 4.2: If Y1 + Y2 > 7T, a discontinuous jump from a non-spherical bridge to a spherical one is feasible. If y1 + 'Y2 s; 7T, no embedded tubular bridge can result from in­finitesimal tilting; further, barring pathological behavior, no drop in the wedge formed by the plates can be formed.

In the latter case, presumably the liquid disappears dis­continuously to infinity. By a "drop in the wedge" is meant a connected mass of fluid containing a segment of the in­tersection line :£ of the planes as well as open subsets of

each of the planes on its boundary, and whose outer sur­face ';! is topologically a disk.

A spherical bridge with tubular topology can exist in a wedge of opening 2a if and only if y1 + y2 > 7T + 2a. In contrast to the case of parallel plates, whenever this con­dition holds, spherical bridges of arbitrary volume and the same contact angles can be found. McCuan proved [31] that if YI + Y2 s; 7T + 2a, then no embedded tubular bridge ex­ists. Wente [32] gave an example of an immersed tubular bridge, with 'YI = Y2 = 7T/2.

The unit normal N on the surlace ';! of a drop in a wedge of opening 2a can be continuous to :£ only if ( 'Yb y2) lies in the closed rectangle m of Figure 14. It is proved in [33] that

if ( Yl, 'Y2) is interior to m then the interface ';! of every such drop is metrically spherical. It is col\iectured in that refer­ence that there exist no drops with unit normal to ';! dis­continuous at :£. In [30] it is shown that the col\iecture can­not be settled by local considerations at the "juncture" of the surface with :£; in fact, there exist surfaces ';! that exhibit such discontinuous behavior locally. The col\iecture asserts that no such surfaces are drops in the sense indicated above.

Property 5. C-singular Solutions

As noted in the discussion of Property 1 above, for capil­lary tubes of general piecewise smooth section 0, solutions of (3a,b) do not always exist. Failure of existence is not oc­casioned specifically by the occurrence of sharp comers; existence can fail even for convex analytic domains. The following general existence criterion appears in [7]:

Referring to Figure 15, consider all possible subdomains 0* * 0,0 of 0 that are bounded on k by subarcs k* C k and within 0 by subarcs f* of semicircles of radius IOVCik lcos y), with the properties

i) the curvature vector of each f* is directed exterior to 0*, and

1t

Figure 14. Domain of data for continuous normal vector to drop in wedge.

VOLUME 24, NUMBER 3, 2002 27

I. *

Figure 15. Extremal configuration for the functional <1>.

ii) each f* meets �. either at smooth points of � in the angle y measured within fl* or else at re-entrant cor­ner points of � at an angle not less than y.

We then have

THEOREM 5. 1: A solution u(x) of (3a,b) exists in fl if and only if for every such configuration there holds

<l>(fl*;y) = lf* l - l�*lcos y + 2H cos y > 0 (5)

with

_ m 2H -lfll

cos y.

Every such solution is smooth interior to fl, and uniquely determined up to an additive constant.

In this result, the circulars arcs f* appear as extremals for the functional <t>, in the sense that they are the bound­aries in fl of extremal domains fl* arising from the "sub­sidiary variational problem" of minimizing <t>.

The following result is proved in [34]:

THEOREM 5.2: Whenever a smooth solution of (3a,b) fails to exist, there will always exist a solution U(x,y) over a subdomain flo bounded within fl by circular subarcs r 0 of semicircles of radius 112Ho, for some positive H0 :::::; H. The arcs meet � in the angle y or else at re-entrant cor­ner points of� in angles not less than y, as in Figure 15. As the arcs fo are approached from within flo, U(x, y) is asymptotic at infinity to the vertical cylinders over those arcs.

We refer to such surfaces U(x,y) as cylindrically sin­gular solutions, or "C-singular solutions". The subarcs are the extremals for the <t> functional, corresponding to H = Ho in (10). Figure 16 illustrates the behavior. Such solu­tions have been observed experimentally in low gravity as surfaces going to the top of the container instead of to the vertical bounding walls.

THEOREM 5.3: C-singular solutions may be unique or not unique, depending on the geometry. They can co-exist with regular solutions, but can fail to exist in cases for which regular solutions do exist.

Figure 17 indicates a case in which a C-singular solution appears for any y < (7T/2) - a. In this case uniqueness can

28 THE MATHEMATICAL INTELLIGENCER

"'

Q:

u=oo

Figure 16. C-singular surface interface.

be shown, and no smooth solution exists. If we consider two such domains with different opening angles, reflect one of them in a vertical axis, expand it homothetically so that the vertical heights of the extremals are the same for both domains, and then superimpose the domains at their tips and discard what is interior to the outer boundary, we ob­tain the configuration of Figure 18. In this case two distinct C-singular solutions can appear, for the same y, with re­gions of regularity, respectively, to the left of one of the in­dicated extremals or to the right of the other one. It has not been determined whether a regular solution exists in this case; however, in the "double bubble" configuration of Figure 19, if the two radii are equal and the opening is small enough, then both regular and C-singular solutions will oc­cur, for any prescribed y. Finally it can be shown that in the disk domain of Figure 20, a regular solution exists for every y, but there can be no C-singular solution.

Figure 17. If a + 1' < Trl2, there exists exactly one C-singular solu­tion, up to an additive constant; no regular solution exists.

Figure 18. At least two C-singular solutions exist.

Property 6. Discontinuous Reversal of

Comparison Relations

Consider surface interfaces :J' in a capillary tube as in Fig­ure 1, in a downward gravity field g and without volume constraint. The governing relations become

divTu = KU in n, K > 0; v . Tu = cos 'Y on �- (6)

Here u is the height above the asymptotic surface level at infinity in the reservoir. About 25 years ago, M. Miranda raised informally the question whether a tube with section n0 always raises liquid to a higher level over that section than does a tube with section nl :J :J no (Figure 21). An al­most immediate response, indicating a particular configu­ration for which the answer is negative, appears in [35]. A number of conditions for a positive answer were obtained; see [36] and [7], Sec. 5.3. A further particular condition for a negative answer is given in [7], Sec. 5.4.

Very recently [37] it was found that negative answers must be expected in many seemingly ordinary situations; further, these negative answers can even occur with height differences that are arbitrarily large. Beyond that, the an­swer can change in a discontinuous way from positive to negative, under infmitesimal change of domain. What is perhaps most remarkable is that such discontinuous change in behavior occurs for the circular cylinder, which is the section for which one normally would expect the smoothest and most stable behavior.

r ��/,. I I \ '

'

Figure 19. Double-bubble domain. For a small enough opening, both a regular and a C-singular solution exist, given any 'Y·

I illustrate the possible behavior with a specific exam­ple. Denote by n1 a square of side 2, and by n(t) = nt the domain obtained by smoothing the comers of n1 by circu­lar arcs of radius (1 - t), 0 :::s t ::::; 1. Thus, no becomes the inscribed disk (Figure 22).

For y � 7T/4, it can be shown that there exists a solution of (6) in any of the nt. Denote these solutions by ut(x; K). One can prove:

Figure 20. In a disk, a regular solution exists for any 'Y; but no C­singular solution exists.

Figure 21 . Does Oo raise fluid higher over its section than does 01 over that same section?

VOLUME 24, NUMBER 3, 2002 29

Q(t)

Figure 22. Configuration for example.

THEOREM 6.1 : There exists Co > 0 with the property that for each t in 0 < t < 1, there exists C(t) > 0 such that

u1(x; K) - ut(x; K) > (C(t)IK) - C0 (7)

uniformly over Ot. On the other hand, we have

THEOREM 6.2: For aU K > 0,

u0(x; K) > u1(x; K) (8)

in 00. Thus, no matter how closely one approximates the inscribed disk by making t small, the solution in the square will dominate (by an arbitrarily large amount) the solution in Ot if K is small enough. However, the solution in the disk itself dominates the one in the square, regardless of K. The limiting behavior of u1(x; K) - ut(x; K) as K ---" 0 is thus dis­continuous at the value t = 0, and in fact with an infinite jump.

Paul Concus and Victor Brady tested this unexpected re­sult independently by computer calculations. Figure 23 shows u1 - Ut for 'Y = 7T/3, evaluated at the symmetry point x = (0,0), as function of t for four different values of the (non-dimensional) Bond number B = Ka2, with a being a representative length. In the present case, a was chosen to be the radius of the inscribed disk, so that B = K. One sees that u1 - u0 is always negative, as predicted, while for any e > 0, u1 - U13 becomes arbitrarily large positive with de­creasing K. Note that the vertical scale in Figure 23 is log­arithmic, so that each unit height change corresponds to a factor of ten.

Property 7. An Unusual Consequence of

Boundary Smoothing

The discussion under Property 6 above indicates that the specific cause of failure of existence for solutions of (3a,b)

1� �-r----,-----�----r-----r---��--�----�-----r----,-----��

1cl

"": 101 0 •

q_ :I

8 -;-10°

0 0.1 0.2 0.3 0.4 0.5 1

0.6

D ... 0 ...

0.7

B= .oo.J 1 B= .0 1 B= 1 B= 100

0.8 0.9

Figure 23. u1(0; B) - ut(O; B) as function of t; 'Y = 7TI3. Note negative values that minimize when t = 0, and large slopes at end points when B is small.

30 THE MATHEMATICAL INTELLIGENCER

p

Figure 24. Solution exists when 'Y = 59°. Smoothing at P leads to non­existence.

in a domain n is not the occurrence of corner points, but

rather the presence of boundary segments of locally large

inward-directed curvature, relative to averaged curvature

properties of the domain. For a circular domain, solutions

exist for any contact angle y; however, given a value 'Y in

0 ::; 'Y < 1r/2, existence will fail if the domain is modified to

a sufficiently eccentric ellipse. Existence will also fail if a

protruding corner of opening 2a appears with a + 'Y < 1r/2.

(We define the curvature at a protruding corner to be +oo.)

In view of such observations, one might expect that if a so­

lution exists in a domain n whose boundary contains a

point P of strict maximal inward boundary curvature, then

if an is smoothed at that point by a circular arc of smaller

curvature that is tangent to an on either side of P, so as to

make the domain closer to circular in form, then a solution

will again exist in the smoothed domain. Figure 24 indi­

cates a configuration in which that assertion fails, see [38].

The angle at P is 90°, the angle formed by the extended seg­

ments at Q is 60°, and the domain is smooth except at the

single corner point P. A solution exists when 'Y = 59°, but

smoothing at P would lead to non-existence at that contact

angle. The parameters of the construction must be chosen

carefully for that behavior to occur.

Property 8. Isolated Singularities

It is known [39] that if K 2:: 0, then every isolated singu­larity of a solution of ( 4a,b) is removable. Thus, also in

this respect, the solutions behave strikingly differently from

those of linear equations.

If K < 0, the situation cannot be described so simply, and

in fact Concus and Finn [40] proved the existence, in an in­

terval 0 < r < 8, of a rotationally symmetric solution U(r) of the equation (obtained from (4a) by a normalization)

divTu = -u (9)

with an isolated singularity at the origin, and admitting a

(divergent) asymptotic expansion

U(r) = _ _!_ + � r2 -567

r7 r 2 8

+ 123149 11 21246673 1 15

+ (10) 16

r -128

r · · ·

They conjectured (a) that U(r) is the unique symmetric so­

lution of (9) with a non-removable isolated singularity at

the origin, and (b) that 8 = oo. The latter conjecture was

proved by Bidaut-Veron [41], who then later showed [42]

that any singular solution satisfying the specific estimate

I 1 I 10 ur(r) l - r2 < 3 ( 1 1)

is uniquely determined.

The singular solution U(r) is related in a striking way to

the pendent liquid drop, illustrated in Figure 2. Concus and

Finn showed [43] that if one allows the vertex height u0 of

the drop to decrease toward negative infmity, one obtains

a family of globally defmed solutions of the related para­

metric equations, exhibiting a succession of shapes that are

initially bubble-like along the vertical axis near the vertex,

and then smooth out, cross the horizontal axis, and con­

tinue to infinity; see Figure 25, where the computed shapes

are compared with a computed U(r). On the basis of these

observations, those authors conjectured that the bubble­

like solutions converge as the vertex point tends to nega­

tive infinity, uniformly in any compact set, to the singular

solution of Conjecture (a) above. This conjecture was par­

tially proved in [ 44], where it was shown that there is a

subsequence converging to a singular solution with the

asymptotic properties of U(r). Most recently Nickolov [45] showed that every rota­

tionally symmetric solution of (9) with a non-removable iso­

lated singularity satisfies ( 1 1); in view of the results of [42],

this work completes the proof of the Conjecture (a). As a

corollary, note that in the succession of bubble-solutions

considered just above, there is no need to choose a subse­

quence; every sequence converges, and to the same limit.

Summarizing,

THEOREM 8. 1: If K 2:: 0, then every isolated singularity of a solution of divTu = KU is removable. If K < 0, then there exists a unique rotationally symmetric solution with a non-removable isolated singularity; this solution admits (after normalization) the divergent asymptotic expan­sion (10). Additionally when K < 0, there exists, for any negative vertex height u0, a global "bubble-solution" as de­scribed above, having the general character of a pendent liquid drop. As u0 � -oo, these surfaces converge uni­formly to the singular solution.

The question of stability for the pendent drop surfaces

has been addressed by many authors. Notably H. Wente

[46] showed that the portion of such a surface below a pre­

scribed height-considered as the height of a supporting

plane-is stable when the height is at or even somewhat

higher than that of the initial inflection point; but the con­

figuration is unstable if two inflections are present below

the plane. The "bubble-profiles" described above yield in

general physically unstable interfaces when considered

globally; nevertheless, all profiles exist globally as analytic

curves extending smoothly to infinity asymptotic to the r­axis, and uniquely determined by u0. The initial tip region

of such surfaces, below the instability point between the

VOLUME 24, NUMBER 3, 2002 31

Figure 25. Bubble solutions and singular solution for divTu = -u. (a) u0 = -4; (b) u0 = -8; (c) u0 = - 1 6.

first and second inflections, is realizable physically as a sta­

ble drop pendent from a horizontal plate; such drops may

be observed on the ceiling of the living room of my home

during rainstorms. When the volume increases to a critical

value above what is needed to produce an inflection but

less than a value producing two inflections, they become

unstable and fall to the floor, or occasionally to an inter­

vening bald spot.

A U T H O R

ROBERT FINN

Department of Mathematics

Stanford University

Stanford. CA 94305

USA

e-mail: finn@math.stanford.edu

Robert Finn was Professor of Mathematics at Stanford from

1 959 until his normal retirement in 1 992. He continues re­

search in hydrodynamics and other fields; collaboration with

undergraduates; and supervision of doctoral students (among

whom until now 27 Ph.D.'s have appeared).

He wishes he knew how to stop the monstrous military

preparations which ravage the earth and prepare for the an­

nihilation of everything of human value. He does not. Still he

is grateful that he can enjoy his family, the music of Haydn,

and the continued pursuit of theorems.

32 THE MATHEMATICAL INTELLIGENCER

Comments

The above exposition overlaps my two other recent survey

articles [47] and [48]. Some of the material is common to all

three articles, however the presentations take differing points

of view, and each of the articles contains items not found in

the others. The topics discussed above were chosen with ac­

cessibility to non-specialists in mind, and with a view to call­ing attention to the large body of fertile and largely unex­

plored mathematical territory that is concealed behind the

deceptively simple appearance of the governing equations (la,b ). The topics chosen do not in any sense exhaust the

range of discoveries that have appeared during the past thirty

years. There has been great and unabating activity during that period, expressing the rejuvenation of a major field of study

that flourished during most of the nineteenth century and

somewhat beyond, and then suffered a half-century hiatus.

The reference list below should offer leads to some of the

principal new directions that have been developing.

Acknowledgments

This work was supported in part by the National Science

Foundation. I wish to thank the Max-Planck-Institut ftir

Mathematik in den Naturwissenschaften, in Leipzig, for its

hospitality during preparation of the manuscript. I am in­

debted to many students, colleagues and co-workers for

conversations extending over many years, that have deep­

ened my comprehension and insight.

REFERENCES

1 . T. Young: An essay on the cohesion of fluids. Phi/as. Trans. Roy.

Soc. Land. 95 (1 805), 65-87.

2. P.S. Laplace: Traite de mecanique celeste: supplements au Livre

X, 1 805 and 1 806, resp. in CEuvres Completes Vol. 4. Gauthier­

Villars, Paris. See also the annotated English translation by N.

Bowditch (1 839), reprinted by Chelsea, New York (1 966).

3. C.F. Gauss: Principia Generalia Theoriae Figurae Fluidorum. Com­

ment. Soc. Regiae Scient. Gottingensis Rec. 7 (1 830). Reprinted

as Grundlagen einer Theorie der Gestalt von Flilssigkeiten im Zus­

tand des Gleichgewichtes in Ostwald's Klassiker der Exakten Wis­

senschaften, vol. 1 35, W. Engelmann, Leipzig, (1 903).

4. R. Finn: On the equations of capillarity. J. Mathem Fluid Mech, 3

(2001 ) , 1 39-1 5 1 .

5 . T. l . Vogel: Uniqueness for certain surfaces of prescribed mean cur­

vature. Pacific J. Math. 1 34 (1 988), 1 97-207.

6. P. Concus, R. Finn: On capil lary free surfaces in the absence of

gravity. Acta Math. 1 32 (1 974), 1 77-1 98.

7. R. Finn: Equilibrium Capillary Surfaces. Springer-Verlag , New York

( 1 986). Russian translation (with Appendix by H.C. Wente) Mir Pub­

lishers (1 988).

8. B. Fischer, R. Finn: Non-existence theorems and measurement of

capillary contact angle. Z. Anal. Anwend. 12 (1 993), 405-423.

9. R. Finn, T.L. Leise: On the canonical proboscis. Z. Anal. Anwend.

1 3 (1 994), 443-462.

1 0. R. Finn, J . Marek: The modified canonical proboscis. Z. Anal. An­

wend. 1 5 (1 996), 95-1 08.

1 1 . P. Concus, R. Finn, M. Weislogel : Measurement of critical contact

angle in a microgravity space experiment. Exps. Fluids 28 (2000),

1 97-205.

1 2. M. Emmer: Esistenza, unicita e regolarita nelle superfici de equilib­

ria nei capillari. Ann. Univ. Ferrara 18 (1 973), 79-94.

1 3. R. Finn, C. Gerhardt: The internal sphere condition and the capil­

lary problem. Ann. Mat. Pura Appl. (4) 1 1 2 (1 977), 1 3-31 .

1 4. R. Finn: The sessile liquid drop 1: symmetric case. Pacific J. Math.

88 (1 980), 541 -587.

1 5 . R. Gulliver, S. Hildebrandt: Boundary configurations spanning

continua of minimal surfaces. Manuscr. Math. 54 (1 986), 323-

347.

1 6. R. Finn: Nonuniqueness and uniqueness of capillary surfaces. Man­

user. Math. 61 (1 988), 347-372.

1 7. P. Concus, R. Finn: Instability of certain capillary surfaces. Manu­

scr. Math. 63 (1 989), 209-21 3.

1 8. H.C. Wente: Stability analysis for exotic containers. Dynam. Con­

tin. Discrete lmpuls. Systems (Waterloo, ON, 1 997). Dynam. Con­

tin. Discrete lmpuls. Systems 5 (1 999), 1 5 1 -1 58.

1 9. J.E. Taylor: Boundary regularity for solutions to various capil lar­

ity and free boundary problems. Comm. P.D. E. 2 (1 977), 323-

357.

20. M. Callahan, P. Concus, R. Finn: Energy minimizing capillary sur­

faces for exotic containers (with accompanying videotape), in Com­

puting Optimal Geometries, J.E . Taylor, ed. , AMS Selected Lec­

tures in Mathematics, American Mathematical Society, Providence,

Rl (1 991 ) , 1 3-1 5.

21 . P . Concus, R. Finn, M. Weislogel : Drop tower experiments for cap­

illary surfaces in an exotic container. AIM J. 30 (1 992), 1 34-1 37.

22. P. Concus, R. Finn, M. Weislogel: Capillary surfaces in an exotic

container; results from space experiments. J. Fluid Mech. 394

(1 999), 1 1 9-1 35.

,23. M. Athanassenas: A variational problem for constant mean curva­

ture surfaces with free boundary. J. Reine Angew. Math. 377

(1 987), 97-107.

24. T. l . Vogel: Stability of a liquid drop trapped between two parallel

planes I I : general contact angles. SIAM J. Appl. Math. 49 ( 1 989),

1 009-1 028.

25. D. Langbein: Stability of liquid bridges between parallel plates. Mi­

crogr. Sci. Techn. 1 (1 992), 2-1 1 .

26. L.-M. Zhou: On stability of a catenoidal liquid bridge. Pacific J.

Math. 1 78 (1 997), 1 85-198.

27. L.-M. Zhou: The stability of liquid bridges. Dissertation, Stanford

Univ , 1 996.

28. T.l. Vogel: Local energy minimality of capil lary surfaces in the pres­

ence of symmetry. Pacific J. Math. , to appear.

29. R. Finn, T. l . Vogel : On the volume infimum for liquid bridges. Z. Anal. Anwend. 1 1 (1 992), 3-23.

30. P . Concus, R. Finn, J . McCuan: Liquid bridges, edge blobs, and

Scherk-type capillary surfaces. Indiana Univ. Math. J. 50 (2001 )

41 1 -441 .

31 . J. McCuan: Symmetry via spherical reflection and spanning drops

in a wedge. Pacific J. Math. 1 80 ( 1 997), 291-324.

32. H.C. Wente: Tubular capillary surfaces in a convex body. Advances

in geometric analysis and continuum mechanics (Stanford, CA,

1 993), 288-298. Internal. Press, Cambridge, MA (1 995).

33. R. Finn, J. McCuan: Vertex theorems for capillary drops on sup­

port planes. Math. Nachr. 209 (2000), 1 1 5-1 35.

34. R. Finn, R.W. Neel: C-singular solutions of the capillary problem.

J. Reine Angew. Math. 512 ( 1 999), 1 -25.

35. P. Concus, R. Finn: On the height of a capillary surface. Math. Z. 1 47 (1 976), 93-95.

36. D. Siegel: Height estimates for capillary surfaces. Pacific J. Math.

88 (1 980), 471 -51 6.

37. R . Finn, A.A. Kosmodem'yanskii, Jr.: Some unusual comparison

theorems for capillary surfaces. Pacific J. Math. , to appear.

38. R. Finn: A curious property of capillary surfaces. Ann. Univ. Fer­

rarra, to appear.

39. R. Finn: On partial differential equations whose solutions admit no

isolated singularities. Scripta Math. 26 (1 961 ) , 1 07-1 1 5.

40. P. Concus, R. Finn: A singular solution of the capillary equation 1 : existence. Invent. Math. 29 (1 975), 1 43-1 48.

41 . M.-F. Bidaut-Veron: Global existence and uniqueness results for

singular solutions of the capillarity equation. Pacific J. Math. 1 25

( 1 986), 31 7-334.

42. M.-F. Bidaut-Veron: Rotationally symmetric hypersurfaces with pre­

scribed mean curvature. Pacific J. Math. 1 73 (1 996), 29-67.

43. P. Concus, R. Finn: The shape of a pendent liquid drop. Phil. Trans.

Roy. Soc. London. Ser. A 292 ( 1 979), 307-340.

44. R. Finn: Green's identities and pendent liquid drops: I. Inti. Syrnp.

Univ. Ferrara; in Developments in Partial Differential Equations and

Applications in Mathematical Physics, ed. G. Buttazzo, G.P. Galdi,

L. Zanghirati. Plenum Press, London (1 992), 39-58.

45. R. Nickolov: Uniqueness of the singular solution to the capillary

equation. Indiana Univ. Math. J. 51 (2002) to appear.

46. H .C. Wente: The stability of the axially symmetric pendent drop.

Pacific J. Math. 88 (1 980), 421 -470.

47. R. Finn: Capillary surface interfaces. Notices Amer. Math. Soc. 46 ( 1 999), 770-781 .

48. R. Finn: Some properties of capillary surfaces. Rend. Sem. Mat.

Milano (in press).

VOLUME 24, NUMBER 3, 2002 33

l@ffli•i§\:6ih£11£ii§4@11,j,i§.id M ichael Kleber and Ravi Vaki l , Editors

The Keg Index and a Mathematica l Theory of Drunkenness Christopher Tuffley

This column is a place for those bits of

contagious mathematics that travel

from person to person in the

community, because they are so

elegant, suprising, or appealing that

one has an urge to pass them on.

Contributors are most welcome.

Consider the following problem:

Problem 0. There are n people sitting in a ring, one of whom takes a swig from a keg of beer and then passes it right or left with a 50% probability in­dependently of what has happened be­fore. The process repeats until every­

one has had at least one swig, then

stops. Show that the probability that the keg stops at a particular (non­starting) person is independent of that person's position.

I'm not going to discuss this problem

until the end, because to my mind there's

a much bigger issue here, and that is: just

how drunk are these people??

After some thought, we might sup­

pose that completely sober people

would realise that the most efficient

(not to mention fairest!) method of

having everyone get a swig from the

keg is to have the keg travel round the

circle in either a clockwise or anti­

clockwise direction, and that they

would pass the keg accordingly: on re­

ceiving the keg from their left they

would pass it to their right, and vice

versa. Completely drunk people, on the

other hand, would be capable of little

more than shoving the keg back where

it came from; while someone some­

where between these two states might

pass it back where it came from with

some probability p (increasing with

drunkenness) or pass it on with prob­

ability 1 - p. This leads us to defme the

keg index:

Someone is p-drunk (or a p-drunk) if he or she passes the keg back with probability p, and passes it on with probability 1 - p.

Please send all submissions to the

Mathematical Entertainments Editor,

Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380,

Stanford, CA 94305-21 25, USA

e-mail : vakil@math.stanford.edu

Recapping, "sober" corresponds top = 0, "drunk" corresponds to p = 1. So ac­

cording to the keg index, the people in

Problem 0 are half drunk! Well! That

answers the question we set out to an-

34 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

swer, but now that we have a method

of measuring drunkenness (and one so

readily estimated by anyone, any­

where, without any need for fancy

equipment-simply use the observed

frequencies with which the subject

passes the keg on, and back), we may

have a new-found predictive power.

Maybe now we can answer some of

those questions you've always wanted

to ask but never knew how, such as:

just how drunk is the combination of a

p-drunk person and a q-drunk person?

Suppose a p-drunk person P is

standing with a q-drunk person Q on

the left and is passed the keg from the

right (see Figure 1). P might pass it

straight back with probability p, or

pass it to Q with probability 1 - p, who

might pass it on with probability 1 - q,

or back to P with probability q. P in

turn may pass it out (with probability

1 - p) or back (probability p ). In this

way it might shuttle back and forth be­

tween P and Q for some time before

eventually emerging from one side or

the other. Each round trip from Q to P

and back again (or from P to Q and

back, after the first) occurs with prob­

ability pq, so we get a geometric series.

The probability that P passes it out is

00 p + c1 _ p)2q I pkqk

k�O

(1 - p)2q = p + 1 - pq

= p - 2pq + q

1 - pq

while the probability that Q passes it

out is

00 o - P)C1 - q) I pkqk

k�O

= 1 - p + pq - q

1 - pq

I make two observations about these

two probabilities: first, that they sum

to 1, and second, that they're symmet­

ric in p and q. That they sum to 1 is un-

-- ----..._ ,/'" (

� .'\ ( \

�� 4� \ ) "' -- ? /

R ---9 • / '----- ....___

--

Figure 1 . Just how drunk is a p-drunk person with a q-drunk person?

surprising, but nevertheless comforting if you want a swig from the keg too, while the fact they're symmetric in p and q is quite unexpected. Unexpected, and fortuitous, saving us from having to introduce such concepts as left- and right­drunkenness: together these two facts imply that the (p,q)­drunk combination behaves as a single person with the net keg index

Since

P[the keg is passed back] = P � 2pq + q.

- pq

p - 2pq + q - - p(l - q)2 > 0 q - - ' 1 - pq 1 - pq

(1)

it follows that the (p,q)-drunk combination is drunker than max(p,q }-at last a mathematical proof of something read­ily confirmed experimentally.

Before going on, let's take another look at equation (1). At first sight there seems to be a problem here when p = q = 1, because the denominator is zero. However, a closer inspection shows that everything's okay after all, as the numerator is zero too and the limit as p,q � 1- is 1, which is what we expect. In spite of this, to make things nicer later on, I will assume that complete drunks don't exist. This is mathematics: we're allowed to do this kind of thing. And if sober people start being a problem, I just might as­sume them out of existence too.

Some people may object to the calculation leading to equation (1) on the grounds that the effect of the swigs on the individual keg indices is not taken into account when calculating the net keg index. This is clearly an issue that will have to be addressed, but it turns out to be not as se­rious a flaw as it may at first appear. The effect of the swigs will depend on the number taken-the fewer swigs, the bet­ter the swig-free approximation will be. We therefore cal­culate the expected number of swigs before P and Q pass out the keg, obtaining

00 E[swigs] = p + (1 - p)(1 - q) I 2k(pq)k-l

k�l 00

+ c1 - p)2q I c2k - 1)(pq)k-2. c2) k�2

Using the series

1 = � k k- 1

2 L X ' (1 - x) k�o

we obtain the expressions

2x - _!_ ( 1 - 1 ) = I 2kx2k-l (1 - x2)2 - 2 (1 - x)2 (1 + x)2 k�o

and

1 + x2 _ _!_ ( 1 + 1 ) (1 - x2)2 - 2 (1 - x)2 (1 + x)2

00 = I (2k - 1)x2Ck- l).

k�l

Applying these to (2) with x = (pq)112, we find

E[swigs] = p + (1 - p)(1 - q) (1 _2pq)2

3 - pq q - p + (1 - p)2 q (1 - pq)2 = 2 + 1 - pq '

and because pq ::;::; p, q < 1, we have

lq - p

l < 1 1 - pq

on [0, 1) X [0, 1). Therefore the expected number of swigs lies strictly between 1 and 3, and we may sleep peacefully in the knowledge that in the low swig-alcohol-content limit, the correction to the swig-free approximation isn't too large.

Now suppose we have three people: a p-drunk, a q-drunk, and an r-drunk, named P, Q, and R. To calculate their net keg index we may simply use the above result to treat P and Q as a single person S, and then apply the re­sult again to S and R. We get

P - 2pq + q - 2 P - 2pq + q r + r

1 - pq 1 - pq

1 - P - 2pq + q r 1 - pq

p + q + r - 2qr - 2pr - 2pq + 3pqr 1 - qr - pr - pq + 2pqr

The result is symmetric in p, q, and r, as we now expect in light of the p,q-symmetry of equation (1): interchanging two neighbours doesn't change the net keg index, and such transpositions generate the symmetric group. It follows that we may use equation (1) to assign a well-defmed net keg index to any group of people, without first having to line them up in a row, and we may thereby extend the keg index to less orderly arrangements. As an application, it's easy to show inductively that n half-drunks are equivalent to a single nl(n + 1)-drunk. The rapid convergence of this figure to 1 as n � oo may go a long way towards explain­ing crowd behaviour in pubs.

VOLUME 24, NUMBER 3, 2002 35

In fact, if we define ++ : [0, 1) X [0, 1) � [0, 1) (pro­

nounced double-vision-plus) by

p - 2pq + q p++q = 1 ' - pq

then ([0, 1), ++) forms an abelian, associative sernigroup

with identity: the keg semigroup. So the keg index forms a

structure with a number of nice properties-in fact, about

the only nice property we'd like to have but don't is the ex­

istence of inverses. What's more, inverses would seem to

have a natural interpretation in "sober-up" pills: pills or po­

tions that, when taken, sober you up, or at the very least,

make you less drunk. Can the keg sernigroup be embedded

in a group?

Yes! Indeed, there is a natural candidate for the target

group: as an addition law defined on a half-open interval

and satisfyingp++q 2::: max{p,q}, the keg semigroup is rem­

iniscent of the non-negative real numbers under addition.

Furthermore, the fact that n half-drunks are equivalent to

a single n/(n + I)-drunk suggests an explicit map: the func­

tion if; : R;,:0 - [0, 1) given by

t lj;(t) = -.

t + 1

A simple check shows that ljJ is indeed an isomorphism,

with inverse

f/J(p) = _P_, 1 - p

so mathematically speaking, sober-up pills exist! To con­

struct them simply extend ljJ to all of R. The fact that

if;( - 1) = oo appears to present a problem, but this is re­

solved by defming

++oo = lim p - 2pq + q

= 2P - 1

= 1/J(f/l(p) - 1) p q->oo 1 - pq p

for p i= oo, and (taking a limit once more) oo + + oo = 2 = ljJ( -2). We then have lj;(t + u) = lj;(t) + + lj;(u) for all t, u E R (in fact, we may regard p = 1 as ljJ( oo ), in which case this

holds for all t, u E R U (oo}), so ljJ maps (R, + ) isomorphi­

cally onto (R U (oo} \ { 1 }, + + ). There! Sober-up pills exist,

and correspond to things that pass the keg back with prob­

abilities outside [0, 1]-so perhaps "pills" is not the correct

word, but in any case, as mathematicians our part is done:

we've proven the theoretical existence of sober-up some­things and we may leave the details of physically realising

them to disciplines better suited to the task.

More Sober Reflections

Now that the keg index has been developed to such a sat­

isfactory conclusion, let's turn our attention back to the

original problem. How do you solve Problem 0 , and what

36 THE MATHEMATICAL INTELLIGENCER

leads to such a surprising result? Our intuition suggests

that the keg should be more likely to finish further from

rather than closer to the starting point. Where is it going

wrong?

To answer the first question, suppose you're sitting

somewhere in the circle and did not start with the keg.

The keg will stop at you if and only if both your neigh­

bours have swigs before you do; and for this to happen,

the keg, having visited one of them, must make it all the

way around the circle to the other without ever being

passed to you. The probabilities of the two events "your

left neighbour gets the keg before your right" and "your

right neighbour gets the keg before your left" do depend

on where you're sitting, but their sum does not and equals

1 . The probability of the keg stopping at you is then the

probability of it getting from one of your neighbours to

the other without ever being passed to you. But this is just

the probability of it stopping at you given that it started

at your neighbour, and so does not depend on where you

are in the circle.

More generally, consider a random walk on a connected

graph G that begins at some vertex x, moves at each step

with equal probability to any neighbour of the current ver­

tex, and stops as soon as it has visited every vertex. Such

a walk is called a cover tour, and in these terms, the result

of Problem 0 is that a cover tour from any vertex on a cy­

cle is equally likely to end at any other vertex. This is true

of complete graphs too, by symmetry. In a paper with ac­

knowledgments "to Ron Graham for extra incentive, and to

the Hunan Palace, Atlanta GA, for providing the napkins,"

Lovasz and Winkler [1] show that complete graphs and cy­

cles are the only graphs with this property. In doing so they

show that our intuition is correct in general, and give in­

sight into where it is failing us for the cycle; namely, let­

ting L(x,y) be the event that a cover tour beginning at x ends at y, they show the following:

THEOREM (Lovasz and Winkler [ 1 ]) Let u and v be nonad­jacent vertices of a connected graph G. Then there is a neighbour x of u such that P[L(x,v)] :s P[L(u,v)]; further, the inequality can be taken to be strict if the subgraph in­duced by V(G) \ {u,v} is connected.

The theorem is proved by showing that P[L(u,v)] is equal

to the average of P[L(x,v)] at its neighbours, plus a non­

negative correction term that is positive if G \ { u, v l is con­

nected. This implies that for a fixed finishing vertex y the

minimum of P[L(x,y)] occurs at a neighbour of y, as we

expect; but it is not a strict minimum for the cycle, a cycle

being disconnected by the removal of any two nonadjacent

vertices. However, there is a surprise: they give an exam­

ple to show that for fixed initial vertex x the minimum of

P [L(x,y)] need not occur at a neighbour of x. The example

is a complete graph Kn with an extra path u, x, y, z, v join-

Continued on page 67

P. SCHOLL, A. SCHURMANN, J. M. WILLS

Po yhedral Models of Fe ix K ein 's Group

''I have the polyhedron on my desk. I love it!" John H. Conway, Aug. 19, 1993

felix Klein's group (which comes accompanied by Klein's curve, Klein's regular map,

and Klein's quartic) is one of the most famous mathematical objects; in A. M.

Macbeath's words ([L], p. 104), ''It is a truly central piece of mathematics. "

Felix Klein discovered this finite group PSL(2, 7) of or­der 168 in 1879 [K], and since then its properties have been investigated, generalized, applied, and discussed in hun­dreds of papers.

The recent book The Eightfold Way [L] contains several survey articles by prominent experts, which collect and dis­cuss the essentials of Klein's group from various aspects. This book was issued on the occasion of the installation at the Berkeley campus of a nice geometric model of Klein's group made of Carrara marble by the artist H. Ferguson.

The idea of visualizing Klein's group by geometric mod­els is not new. Felix Klein himself gave a planar and a 3-di­mensional model. The planar one is the unsurpassable Poincare model (Figure 2), well known from classical com­plex analysis. Klein's 3-dimensional model consists of three hyperboloids whose axes meet at right angles.

In this paper we consider 3-dimensional models which are as close as possible to the Platonic solids, built up of planar polygons with or without self-intersections and with maximal possible symmetry. Polyhedral realizations of groups or regular maps can be considered as contributions to H.S.M. Coxeter's general concept of "groups and geom­etry." We will show polyhedral realizations of Klein's group, two of them "old" and two new. For this we need to review some basic properties of Klein's group. For more details we refer to [C], [CM], [K], [L], [MS] or [SW1].

Maps, Flags, and Symmetries

First we consider the icosahedral group and its polyhedral realizations, the regular icosahedron and dodecahedron

(Figure 1). The 60 elements of the icosahedral group can be represented by the 60 black (or white) triangles of the pattern on the sphere in Figure 1. Such a pattern is called a "regular map," and the 60 black (or white) triangles are indistinguishable under rotations of the sphere. A reflec­tion transposes the black triangles into the white ones and vice versa, giving the extension to the full icosahedral group of order 120.

Now the 120 triangles of this regular map on the sphere can be collected in two dual ways to build up a convex reg­ular polyhedron. Either one collects the 3 white and 3 black triangles around each 6-valent vertex of the map, which yields the icosahedron with 20 triangles and 12 5-valent ver­tices; or the 5 white and 5 black triangles are collected around the 10-valent vertices, which yields the dodecahe­dron with 12 pentagons and 20 3-valent vertices. Each black or white triangle of the map corresponds to an ordered triplet of a vertex, an edge, and a face of the icosahedron or of the dodecahedron; these triplets are called "flags." So the flags (or the black and white triangles) correspond to the elements of the group; i.e., they represent the elements of the icosahedral group.

In the same way, the 168 black (hatched) and 168 white triangles in the Poincare model (Figure 2) represent the el­ements of Klein's group. Again the triangles of the map can be collected in two dual ways as for the icosahedral group. If black triangles may be interchanged with white, we have a group of 336 elements.

If one considers the 6-valent vertices in Figure 2, then again 3 black and 3 white triangles fit together to form one

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 37

Figue 1 . Icosahedron and dodecahedron and their "regular map."

38 ll-IE MAll-IEMATICAL INTEWGENCER

9

t3

t't Figure 2. Klein's group as a regular map.

&

5

large triangle, and one obtains 56 ( = 336/6) triangles, but

seven of them meet at a common vertex, 24 altogether.

From the 14-valent vertices in Figure 2, one obtains 24 hep­

tagons, which meet at 56 3-valent vertices.

The only problem is to make a polyhedron with flat

faces. There are some differences between the icosahedral

group and Klein's group, which cause difficulties.

Hidden Symmetries and Petrie-Polygons

The main difference is that the icosahedral group is one of

the rotation groups in Euclidean 3-space, and so the rela­

tion between the map on the sphere and the polyhedra is

natural and obvious. In fact the geometric objects-the Pla­

tonic solids-existed long before the mathematical back­

ground (groups, maps) was understood. One had a vague

idea of their deeper importance.

Klein's group of order 168 (or 336 for the full group) is

much larger than the icosahedral group of order 60 (full:

120) and is not a symmetry group in Euclidean 3-space. We

can create a polyhedron and consider a group of 168 trans­

formations of it, but they can't all be congruences!

Klein's group does contain a geometric subgroup,

namely the "octahedral rotation group" of order 24, which

was of course known to Felix Klein. We seek, then, a model

of Klein's group with octahedral symmetry, the elements

(or flags) fall into seven orbits of 24 elements each as 168 = 7 · 24 (or 14 orbits for the full

group). Hence not all automor­

phisms of the group can be seen,

and those which do not occur as

geometric symmetries are called

"hidden symmetries." These hid­

den symmetries, though not given

by Euclidean motions, are combi­

natorial and geometric automor­

phisms of the polyhedron.

For example, the hidden sym­

metries are shown by the fact that

all faces are of the same type (tri­

angles or heptagons), and so are

the vertices (7-valent or 3-valent).

Another tool to discover hidden

symmetries are Petrie-polygons.

A Petrie-polygon is a skew

polygon (or zigzag line) where

every two but no three consecu­

tive edges belong to the same face

of the polyhedron. On a regular

figure all possible Petrie-polygons

have the same length, and for the

icosahedral map and hence for the

regular icosahedron and dodeca­

hedron, this is 10.

The length r of the Petrie-poly­

gons, together with p, the number

of sides of a face, and q, the va­

lence of the vertices, character-

So {3, 5}10 denotes the regular icosahedron and {5, 3}10 the

regular dodecahedron.

For Klein's map the Petrie-polygons have length 8, and

so the two dual representations are {3, 7}8 and {7, 3}8. This

explains the title of the book The Eightfold Way. Finally we sketch that a polyhedral realization of Klein's

group has genus 3, i.e., it is topologically equivalent to a

sphere with three handles. For regular maps of genus g � 2 with p-gons and q-valent vertices, there is the famous Rie­

mann-Hurwitz identity, which relates all relevant numbers,

in particular the genus g and the order A of the automor­

phism group:

A = 2(g - 1) - + - - -( 1 1 1 )-1

p q 2

From p = 3, q = 7 (or vice versa) and A = 168 follows g = 3. As a consequence, such groups have maximal order

84(g - 1), and Klein's group is the first one of these rare

"Hurwitz groups."

Polyhedral Models with Tetrahedral Symmetry

It can easily be shown that any 3-dimensional . model of

Klein's group with maximal-i.e., octahedral-symmetry

has self-intersections, so in order to avoid self-intersec­

tions, polyhedral embeddings have at most the next lower

symmetry, tetrahedral rotation symmetry of order 12.

izes a regular polyhedron {p, q)r. Figure 3. Polyhedral embedding of {3, 7Ja with tetrahedral symmetry.

VOLUME 24, NUMBER 3, 2002 39

In 1985 E. Schulte and J. M. Wills gave such a polyhe­dral embedding of (3, 7}8 in [SW1], built up of 56 triangles, which meet at 7-valent vertices, 24 altogether (Figure 3).

Each of the four holes of the model has a strong twist, and it is not clear a priori that this can be done without self-intersections. The 24 vertices split into two orbits of 12 vertices under the tetrahedral rotation group. The outer orbit of 12 vertices can be realized by the vertices of an Archimedean solid, namely the truncated tetrahedron. Sev­eral cardboard and metal models and computer films were made of this realization. (See also [BW], and Conway's com­ment at the head of this article). In its symmetry and em­bedding properties, it corresponds to Ferguson's model, but it is 8 years older. H.S.M. Coxeter's comment (Dec. 3, 1984) on this model: " . . . a wonderful result." The con­structions and incidences can be found in detail in [SW1] and [SW2]. For more details see [SSW], where one can fmd also models with integer coordinates.

We now come to the dual map { 7, 3}s of Klein's group, built up of heptagons. Ferguson's model is the realization of {7, 3}8 on the standard model of an oriented smooth sur­face of genus 3 with tetrahedral symmetry. It shows the 24 heptagons, and, hence it corresponds to the regular do­decahedron (5, 3}IO· It is a help in understanding Klein's group. Ferguson's model is curved, so the heptagons are nonplanar and the model is not a polyhedron.

The construction of a polyhedral model of (7, 3}8 is �ore difficult, but it can be done with modem computer pro­grams. The result is shown in Figure 4 (for details of con­struction see [SSW]). The bizarre model is complicated, and is of no help in understanding Klein's group. This under­lines the simplicity of its dual polyhedral embedding of ( 3, 7}8. In the next section we explain why dual polyhedral realizations of the same group can differ so much.

Polyhedral Models with Octahedral Symmetry

As already mentioned, any 3-dimensional model of Klein's group with maximal (octahedral) symmetry has self-inter­sections; in particular this is true of Klein's curved model of three intersecting hyperboloids.

So it is a bit surprising that the simplest polyhedral model of Klein's group is a polyhedral immersion with oc­tahedral symmetry. It was found by E. Schulte and J.M. Wills in 1987 [SW 2] and is shown in Figure 5.

Its octahedral symmetry implies that the symmetry group acts transitively on its 24 vertices: the vertices are all alike. The vertices can be chosen so that their convex hull is the snub cube, hence one of the 13 Archimedean solids. As a consequence, 32 of the 56 triangles are even regular. The three intersecting tunnels of this model cor­respond to Klein's three intersecting hyperboloids, and the Petrie polygons can easily be seen. Altogether this polyhe-

dral model provides the easiest way to understand the structure of Klein's group PSL (2, 7).

In sharp contrast to this simple model, its dual (7, 3}8 is extremely bizarre (see Figure 6). Although its octahedral symmetry group acts transitively on its 24 congruent heptagons, the model is compli­cated. Again, the model was con­structed by computer; for more details, refer to [SSW].

The heptagons have self-inter­sections, so the model is related to the classical Kepler-Poinsot poly­hedra and to Coxeter's regular complex polyhedra.

Figure 4. Kepler-Poinsot-type realization of {7, 3}s with tetrahedral symmetry.

It might be surprising that the realizations of a pair of dual maps of the same group can be so dif­ferent. But the reason is quite sim­ple: In the triangulations the facets are, by definition, triangles, so they are convex and free of self­intersections. All topological com­plications, twists, and curvature are hidden in the vertices, whose shape is flexible. In the dual, with 3-valent vertices, all complica­tions have to be stored in the hep­tagons, which makes the models star-shaped and bizarre. So this

40 THE MATHEMATICAL INTELLIGENCER

Figure 6. Kepler-Poinsot-type realiza­tion of {7, 3)s with octahedral symme­try.

Figure 5. Polyhedral immersion of {3, 7la with octahedral symmetry.

A U T H O R S

PETER SCHOLL

Fachbereich Mathematik

Universitat Siegen

D-57068 Siegen

Germany

ACHILL SCH0RMANN

School of Mathematical Science

Peking University

J. M. WILLS

Fachbereich Mathematik

Universitat Siegen

D-57068 Siegen

Germany

Beijing 1 00871

Ch ina

e-mail: Peter.Scholl@unix-ag.org e-mail: wil ls@mathematik. uni-siegen.de

Peter Scholl, a native of Siegen, receives

his doctorate at the University there in 2002,

with a thesis on "Sphere-Packings and Mi­

croclusters." His favorite hobby is chess.

Achill SchOrmann completed his doctoral

work at Siegen with a prize-winning thesis

on "Sphere-Packings." He is now on a

postdoctoral research visit with Professor

Chuanming Zong. His hobbies are football

(soccer) and cycling .

Jorg Wills has worked on extremal prob­

lems and convexity, and also on symme­

try and combinatorial geometry. He has re­

ported to The lntelligencer on both sides

of his work, as in vol. 20 (1 998), no. 1 ,

1 6-21 . His main hobbies are music and

art.

model is not a conceptual tool to understand Klein's group, in sharp contrast to its dual.

But all these realizations of Klein's group may qualify as contributions to "Art and Mathematics" -and as contribu­tions to Felix Klein's and H.S.M. Coxeter's general idea of bringing algebra and geometry closer together.

REFERENCiiS

[BW] J. Bokowski and J.M. Wills, Regular polyhedra with hidden sym­

metries. Mathematical lntelligencer 1 0 (1 988), no. 4, 27-32.

[C] H.S.M. Coxeter, Regular Complex Polytopes, Cambridge University

Press, Cambridge, 2nd edit. 1 991 .

[CM] H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for

Discrete Groups, Springer, Berlin 1 980 (4th edit.)

[G) J . Gray, From the History of a Simple Group, The Mathematical ln­

telligencer 4 (1 982), no. 2 , 59-67 (reprint in [L)).

42 THE MATHEMATICAL INTEWGENCEA

[GS] B. Grunbaum, G. Shephard, Duality of polyhedra, in: Shaping

Space, eds. G . Fleck and M. Senechal, Birkha.user, Boston

1 988.

[K) F. Klein, Ueber die Transformation siebenter Ordnung der elliptis­

chen Funktionen, Math. Ann. 1 4 (1 879), 428-471 (English transl. by

S. Levy in [L)).

[L) S. Levy (edit.), The Eightfold Way, MSRI Pub!., Cambridge Univ.

Press, New York 1 999.

[SSW] P. Scholl, A. Schurmann and J .M. Wills, Polyhedral models of

Klein's quartic, http://www.math.uni-siegen.de/wills/klein/

[SW1 ) E. Schulte, J .M. Wills, A polyhedral realization of Felix Klein's

map {3, 7)8 on a Riemann surface of genus 3, J. London Math. Soc.

32 (1 985), 539-547.

[SW2] E. Schulte, J.M. Wills, Kepler-Poinsot-type realization of regular

maps of Klein, Fricke, Gordon and Sherk, Canad. Math. Bull. 30

(1 987), 1 55-164.

l$@jj:J§.&h1¥119.1,1 .. pt,iih¥J M arjorie Senechal , Ed itor I

Mathemat ics and Narrative by R . S. D. Thomas

This column is a forum for discussion

of mathematical communities

throughout the world, and through all

time. Our definition of "mathematical

community" is the broadest. We include

"schools" of mathematics, circles of

correspondence, mathematical societies,

student organizations, and informal

communities of cardinality greater

than one. What we say about the

communities is just as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the

Mathematical Communities Editor,

Marjorie Senechal, Department

of Mathematics, Smith College,

Northampton, MA 01 063 USA

e-mail: senechal@minkowski .smith.edu

It is now forty-five years since C. P.

Snow published "The Two Cultures"

(New Statesman, 1956 10 6), drawing

the intellectual world's attention to the

unhealthy division between scientists

and their less easily described com­

plement [1 ] . While the community of

mathematicians has a complex relation

to these cultures-some of us belong

to one, some to the other, and some,

like the scientist-novelist Snow him­

self, to both-for many purposes

mathematics is still considered, espe­

cially by us, as the queen of the sci­

ences. We expend much effort ex­

plaining to those that have not been

attracted to mathematics at school

what mathematics is like, despairing

that the audience will ever learn it from

personal experience. Comparisons with

poetry and music are often used to this

end [2], [3].

It occurred to me several years ago

that a comparison with narrative might

be at least as illuminating [ 4]. Most of

what follows is that comparison. I will

emphasize mainly points of similarity

because they are less obvious than the

many and large differences [5]. (I will

not discuss appearances of mathemat­

ics in literature, an interesting subject

that begins (at least) with Plato and

continues in our time in the quite dif­

ferent ways of Samuel Beckett, the

largely French Oulipo group, and re­

cent English-language plays such as

Arcadia and Proof That is another

subject entirely.)

The comparison that follows is lim­

ited in intent. It is not sociology; I have

no contribution to our video-game ver­

sion of the Science Wars. Nor is it philo­

sophical; I explore its minor philo­

sophical significance elsewhere [6]. My

modest goal is to enlarge the scope of

our analogies. The mathematical genre

of theorem and proof is in some ways

like the genre of the story. (An algo­

rithm, on the other hand, is a story, but

prescribed rather than reported.) I

claim no originality: the similarities

between proofs and stories that I will

note have been noted by others; all I

have done is to bring them together.

I compare theorems and proofs with

narratives, both fictional and histori­

cal. I stay clear, for instance, of the dis­

tinction between fiction and history, a

distinction that does not matter for my

purposes, however important it is to

philosophy of history. I am interested

in the kind of history that tells a story

and in the simpler kinds of fiction: fairy

tales, fables, mythical tales, and much

in the genres of romance, murders, ad­

venture stories, science fiction, and

fantasy. Factual basis and literary qual­

ity are both irrelevant to my case.

Stories

Let me begin by pointing out what it is

about stories, what features I take the

appropriate stories to have, that makes

the comparison worth anything at all.

The fundamental one is the postula­

tion, at the start, of characters and per­

haps props in some sort of relation to

one another that is worked out in the

telling of the story. A story has a be­

ginning that is signalled in some way,

for example by the proscenium arch or

by "once upon a time." The end of the

story is, like the beginning, a situation

involving the characters, whether it is

"they lived happily ever after" or only

that they have stopped, as in Hamlet.

While the characters and/or events in

a story may be historical, there is no

need for this, just as mathematicians

prefer to avoid questions about the re­

ality or otherwise of so-called mathe­

matical objects. I contend that a story,

if it is fictional, is about significant re­

lations among the characters (and per­

haps props). If the story is about his­

torical persons, then it is about them

as well as the relations discussed.

Stories engage the attention and fire

the imagination of a reader in a way

that other sorts of description of rela­

tions would not. A story about a father

and a son is intrinsically more engag­

ing than an essay on fatherhood and

sonship without examples; the rela-

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 43

tions in a story are seen relating rather than abstracted and objectified. The world of the story is a creation of the reader in co-operation with the teller of the story; it is imaginary yet filled with authenticating detail. Despite be­ing invented, the relations described in a story, unlike the invented characters, are almost exclusively ordinary rela­tions drawn from the real world; little new vocabulary is needed and no new semantics. Also, despite the readers' contributions, different readers will tell recognizably the same story after­wards. Non-historical characters have only those characteristics that are given to them by the storyteller, and the important characteristics are mainly relational (we need not be told that Hamlet is human). A lot of pre­suppositional baggage (Lear is a king, Othello is a Moor, Macbeth is a Scot) is conveyed in telling a story by stag­ing a play with actors. The shift from this relational limitation to the compli­cated development of character and seeing inside characters' heads makes much fiction since 1800 unsuitable for purposes of this comparison.

The writer and reader, or teller and listener, are complicit in a shared act of make-believe, a kind of waking dream. The story is not an arbitrary succession of descriptions; rather the actions of the story follow in accor­dance with physical causality and char­acters' reasons and intentions. A story has structure; it is not just a list of who was where when; a list of facts about the basket, the grandmother, the little girl, and the wolf do not convey "Little Red Riding Hood" even to people who already know the story. The logical ca­pacity as well as the imagination of the reader is engaged; each new remark, character, and event needs to be fitted into the picture already drawn in men­tal space. This engagement is the source of much of the pleasure of a story. Plots and subplots, however complex, have to be presented in a lin­ear way with devices like flashbacks to fill in out-of-order details. Not every­thing is given equal weight; the more dramatic episodes are given emphasis, tension builds, conflict is resolved. If we want to know what happened in the story, in the world of the story, we

44 THE MATHEMATICAL INTELLIGENCER

must return to the text, which is the touchstone of the story's objectivity.

Although the reader imagines the story, the reader does not make it up. There is some evidence that the oppo­site is true: one is, to a degree, the stock of one's imagination, the product of the stories that one knows, includ­ing of course one's own. The stories we know tell us how things can be in the world; through them, we see the world, though not necessarily as it is. The world can be more than one way; we happily know more than one story about the same relations, sometimes even about the same historical charac­ters.

Stories can be applied to the real world, for example when we say his­tory is repeating itself, or that someone

Discovery is real

despite its basis

in invention.

is a Scrooge or Uriah Heep; fictional as well as historical behaviours become paradigmatic. The truths of fiction are not those of accurate reportage but the revelation of significance (the moral of the story.)

In addition to the telling of stories, there can be serious discourse about the subject matter of a fictional story. One can talk, indeed argue, about Sher­lock Holmes without telling a Holmes story, but our discussion depends on those stories. We may ask, in serious discussions, how fictional characters are created, or we may ask "factual" questions ("how long is Hamlet's nose?"). Such "fictional" questions may not be answerable if the text is silent and deduction is impossible. Finally, stories are important and pervasive; some fictional and many historical characters are better known than al­most all of our contemporaries.

Transition

The preceding list of characteristics of stories was unengaging in part because it had neither narrative nor logical structure. I did not know how to give it either, and anyway, it may be in-

structive to have a small sample of prose lacking both. What follows is structured by following that list. The attractiveness of narrative is, I think, one of the reasons that mathematics teachers introduce history in their courses. It is as though students learn best when using both sides of their brains. Yet although the capacity to prove mathematical theorems presum­ably arises later than the capacity to tell stories, it depends mostly on the same imaginative and reasoning ca­pacities.

Mathematics

When we prove something in mathe­matics, we postulate some things to talk about and the relations that they are to have among themselves. These are the dramatis personae; everyone recognizes this move as the beginning of a piece of mathematics by the time they have learned to let x be the num­ber of coins in Johnny's pocket. We frequently do this by calling upon a standard set provided by an axiomatic system. This is analogous to telling a new story about characters already known from their appearance in the lo­cal mythology: Hera is the wife and sis­ter of Zeus, etc. The working out of such relations in mathematics-actu­alizing initial potential, as it is called in stories-is primarily deductive; the time passing is the reader's time, but the presentation is linear (up to a point-see below) as in a story. The end of a proof is another set of rela­tions among the characters; it is a mat­ter of choice where to end a deductive chain, as most situations have further consequences. Where we choose to leave off defmes the conclusion of the theorem. It used to be thought that the objects of geometry had to be ab­stracted from the physical world, just as some early novels like Gulliver's Travels and Robinson Crusoe mas­queraded as memoirs (we don't know why), but mathematics like fiction has become more frankly invention.

Fiction and history concentrate on the significant relations among char­acters, but accidents, chance encoun­ters, and other external events may en­ter into the narrative. Mathematics is supposed to depend only upon the

stated relations among the mathemati­cal objects; to impute any other rela­tions is an error, however helpful intu­ition may be in deciding what to prove and how. Nevertheless, one feature of fiction that is not obvious to readers but is well known to writers is a fic­tional analogue of mathematical de­duction: their story must work itself out despite their having invented everything in it but human and physi­cal nature. This fact sheds penetrating light on the reasoning that, because mathematical deduction is objective, the things of which it is true must be present somewhere. In both cases dis­covery is real despite its basis in in­vention. The importance of intuition as a source of such discoveries is com­mon to mathematics and fiction, and perhaps history.

Logical consequence is the gripping analogue in mathematics of narrative consequence in fiction; all physical causes, personal intentions, and logical consequences in stories are mapped to implication in mathematics, still often represented in the old time-sequence locution, if . . . , then . . . . All are an­swers to the implicit question, "how will it turn out?". Mathematical facts, without some understanding of why they are the way they are, are almost impossible to learn and too boring to keep awake for.

The relation of imagination and de­duction in stories and mathematics is interestingly different, almost oppo­site. In mathematics one imagines in order to see why what is implied is im­plied; whereas in stories one deduces locally to know what to imagine, how to see the story unfolding. One does not deduce on a large scale in reading because in stories, as in life, there are too many imponderables and border­line cases for deduction to be depend­able. But on a larger scale, the function of much mathematics learning is to stock the imagination, not wholly dif­ferently from the way learning stories stocks the imagination. One sees what may be and-unlike stories-what cannot be. As well as learning to see in mathematical (relational) terms, one also learns that some relations are not possible in the presence of others. Fic­tion is more purely permissive.

The engaging feature of mathemati­cal discourse is that the relations dis­cussed actually relate: they are not ab­stract. Geometry has nothing to say about collinearity as an abstraction; all study of collinearity and non-collinear­ity is about points that are or are not collinear. As soon as we have three non-collinear points we have a triangle and can reason about that triangle; that is, among other relations, about their

non-collinearity. Shakespeare, too, did not write essays on jealousy, ambi­tious treachery, and procrastinating re­venge, but put persons into those rela­tions in order to engage our attention.

Mathematical facts,

without some

understanding of

why they are the

way they are, are

almost i mpossi ble

to learn.

In speaking of staging a play, I men­tioned the presuppositional baggage that mathematical objects are by defi­nition free of and have historically shed gradually. Even the most abstract mathematics lets entities enter into the relations that are being discussed; we draw general conclusions from con­sideration of cases that we take care are not special cases (for example, by using the language of set theory, in which the entities are completely char­acterless and the relations that we imagine are all specified in terms of a few simple relations like set member­ship and set inclusion.)

Narrative, in contrast, treats what are frankly and ultimately special cases. Mathematicians are usually in­terested in special cases for the pat­terns they reveal. The best of narra­tive's special cases have a similar but implicit purpose (that's why Scrooge and Heep became paradigmatic.)

We use simple relations and only those that we need, but we talk about

them as though they were real, just as storytellers talk as though their events were real. Despite the imaginative ef­fort that is required to learn a mathe­matical proof, the reader's contribu­tion being considerable, if the proof is then repeated, most of what was imag­ined will be ignored and the proof given will be substantially the same. We somehow grasp a proof as a whole, as we somehow grasp a story as a whole. Some aspects of presentation help with the grasping. Because stories and proofs are linear, we have lemmas like flashbacks that allow us to prove things out of order. We define certain results to be of greater importance and specify them as theorems or lemmas for that reason. There is an analogue in proof of dramatic tension, but I need not spell out how it works: its elegant release is instantly recognizable.

Mathematicians often find it simpler to discuss mathematical systems that are isomorphic in different ways in dif­ferent circumstances. H. B. Griffiths has pointed out to me that specializing an algebraic structure, distinguishing one specific example of a class of iso­morphic structures, "is like casting a play, and the flavour of the special mathematics corresponds to that of a particular production: all such produc­tions have the same abstract struc­ture." The dependence on the text of the story, which in fiction is absolute, is much less in mathematics. Never­theless, the importance of the text for the objectivity of the mathematics has led to the philosophical position­extreme formalism-that the text of the proof is all that there is. Just as we can entertain more than one story about some mythological characters, we can welcome different formaliza­tions of the positive integers within set theory and different proofs of the same result.

Finally I turn to application and truth. Mathematics can be applied in the same way as stories. A triangle, like many other mathematical phenomena, is something that can be recognized in the real world (even in stories.) The more interesting application of mathe­matics is the application of a whole theory, like Euclidean space, to a whole scientific theory, like Newtonian

VOLUME 24, NUMBER 3, 2002 45

mechanics. Then the words of the

mathematical theory are made to refer

not to the mathematical objects of the­

orems but to real or idealized physical

things. The characterlessne� of math­

ematical objects, which the analogy

with fictional characters brings out,

makes plausible my contention that it

is the relations among them that are

compared with the relations among the

physical objects. For instance, the Sun

and the Earth are not compared with

Euclidean points, but rather it is the

distances between two Euclidean

points, the focus and moving point on

an ellipse, that are compared to the dis­

tances between Sun and Earth. It is

when so applied that mathematics be­

comes true in the sense in which "the

sky is blue" is true: not by deduction

from premises but by correspondence.

The usual sense of "true" in mathe­

matics is the more attenuated one that

usually coincides with "validly de­

ducible" -analogous to "true in the

story." Besides the value represented

by validity there is another value re­

vealed by proof, significance. A really

good idea in mathematics, like

Descartes's representation of loci by

equations, is not cashed out by proving

it but by proving things with it; it has a

revelatory power that the best stories

have in their different way. Likewise,

there are questions that can be asked

in mathematics to which no answer

can be given because there is no text

in which to look them up (Erdos's

Book) and we cannot deduce them ei­

ther. The logical systems where ques­

tions always have answers are too sim­

ple for mathematics.

Conclusion

I think that the comparison of mathe­

matics with narrative is deeper and

more far-reaching than analogies with

music and poetry. Though I have not

seen the comparison stated as exten­

sively elsewhere (Paulos's book ex­

cepted), I realize that there is much

more to be said. That proof and narra­

tive are different ways of working out

the consequences of relational hy-

46 THE MATHEMATICAL INTELLIGENCER

potheses is illustrated by A K. Dewd­

ney's article in a recent issue of The Mathematical InteUigencer (22, no. 1,

46-51), "The Planiverse Project: Then

and now." Instead of boring deduction

of the consequences of his two-dimen­

sional imaginary world ("a dry read"­

Dewdney, p. 48), he says, "It would

have to be work of fiction, set in the

planiverse itself." As I wrote above, sto­

ries engage the attention and fire the

imagination of a reader.

But enough. I hope that these re­

marks will stimulate debate and dis­

cussion in the mathematical commu-

A real ly good idea

in mathematics,

. is not cashed

out by proving it

but by proving

things with it

nity. I would be grateful for additional

aspects that I have not mentioned, re­

actions to what I have said, and in par­

ticular, news of the usefulness of this

comparison.

NOTES

[1 ] He completed this task in his Rede Lecture

at Cambridge in 1 959, "The Two Cultures

and the Scientific Revolution."

[2] Scott Buchanan, Poetry and Mathematics,

second edition (first edition, 1 929).

Chicago: Chicago University Press, 1 962.

[3] Edward Rothstein, Emblems of Mind, New

York: Times Books, 1 995.

[4] John Allen Paulos had much the same idea

and wrote Once Upon a Number (New

York: Basic Books, 1 998), which I recom­

mend to anyone who finds my discussion

interesting.

[5] The philosopher Mario Bunge has pub­

lished two slightly different lists of gross

ways in which mathematics and fiction in

particular differ; some apply to history and

some do not. All can be debated. Treatise

on Basic Philosophy. Volume 7 , Epistemol­

ogy and Methodology Ill: Philosophy of Sci­

ence and Technology. Part I : Formal and

Physical Sciences. Dordrecht: Reidel,

1 985. "Moderate mathematical fictionism"

in Philosophy of Mathematics Today. E.

Agazzi and G. Darvas, eds. Dordrecht:

Kluwer, 1 997; pp. 51-71 .

[6] I have done this in two papers, "Mathe­

matics and Fiction 1: Identification," and

"Mathematics and Fiction I I : Analogy," to

appear in Logique et Analyse.

' A U T H O R

ROBERT S. D. THOMAS

St John's College and

Department of Mathematics

University of Manitoba

Winnipeg, Manitoba R3T 2N2 Canada

e-mail: thomas@cc.umanitoba.ca

Robert Thomas studied at the uni­

versities of Toronto, Waterloo, and

Southampton. His non-professional in­

terests have always extended into the

humanities. He has been at the Uni­

versity of Manitoba since 1 970, suc­

cessively in Computer Science, where

he studied braids algorithmically; Ap­

plied Mathematics, where he studied

elastic waves in shells; and Mathemat­

ics. He is editor of Philosophia Mathe­

matica (www.umanitoba.ca/pm) and

treasurer of the Canadian Society for

History and Philosophy of Mathemat­

ics (www.cshpm.org). His wife, now

in children's literature though once

trained and employed as a chemist,

regales him with stories as they jog.

More on the ROJAS Magic Square Aldo Domenicano and

Istvan Hargittai

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cafe

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

.lf so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follow in your tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium

e-mail : dirk. huylebrouck@ping. be

R 0 T A s

0 p E R A

T E N E T

A R E p 0

s A T 0 R

Almost 2000 years after its appearance

as a graffito on the plaster of a column

of a Pompeii palaestra, this alphabetic

magic square (the ROTAS square) is still

attracting interest and curiosity. Our

note in The Mathematical InteUigencer [1] stimulated two Letters to the Editor

[2,3]. This prompted us to provide addi­

tional information about the sites where

this magic square has been found and

the efforts made to unravel its meaning.

The ROT AS square is often found in

medieval buildings and manuscripts in

France, Germany, and Italy. In a more

or less modified form it can be traced

in an area extending from Britain to

Ethiopia and Asia Minor. A compre­

hensive account of its occurrence from

ancient to modem times was published

by De Jerphanion in 1935 [4].

A second ROTAS square from

Abruzzi (Italy), in addition to that de­

scribed in our previous note [1 ] , is

found in a medieval bas-relief in the

church of Santa Lucia at Magliano dei

Marsi, a town about 40 km south of

L' Aquila. The bas-relief is the first from

the left in a set of four, inserted in the

upper portion of the facade. The five

words are arranged in five oblique lines

to fit the space under the abdomen of

a griffon; they can hardly be read with­

out a pair of binoculars.

Until 1868 no examples of the

ROT AS square were known earlier than the 6th century A.D. In that year an ex­

ample dating from Roman times was

found scratched on the wall-plaster of

a Roman house near Cirencester, Eng­

land [5]. Four other examples were

found in 1932 by American archaeolo­

gists on the walls of a Roman military

barracks at Dura Europos, Syria [6].

The Romans left Dura Europos soon

after A.D. 256, which sets an upper limit

to the age of this fmding. The example

from Pompeii (which cannot be later

than A.D. 79, when Pompeii was de­

stroyed by an eruption ofMt. Vesuvius)

was discovered in 1936 [7].

In 1954 still another example dating

from Roman times was found on a brick

of the Governor's palace at Aquincum

in Budapest, Hungary [8]. The letters

were written on the clay before firing

(see Figure 1). The text reads:

R 0 M A T l B I s u B

T A R 0 · T A s

0 p E R A

T E N E T

A R E p 0

s A T 0 R

The first three words are the beginning

of the well known versus recurrens (palindromic verse) ROMA TIBI SUB ITO

MOTIBUS IBIT AMOR; the word ITA

means "in this way." Archaeological ev­

idence dates this fmding to the begin­

ning of the 2nd century A.D. The brick

is exhibited in the Aquincum Museum

at the site of the excavations, which is easily reached from downtown Bu­

dapest by suburban railway.

All known examples of the square

dating from Roman times begin with

the word ROTAS. Those dating from

medieval times generally begin with

SA TOR (i.e., the sower, often identified

with God in the Christian tradition). In some of the medieval squares, however,

including the two from Abruzzi, the first

word is ROT AS. If the different letters

are regarded as points of different col­

ors, then the SATOR square is the mir­

ror image of the ROT AS square.

Among the dozens of interpreta­

tions of the ROTAS square we will mention just a few. The difficulty is, of

course, in the unknown word AREPO.

The oldest interpretation is found in a

Greek bible of the 14th century [9]. The

magic square is reported there in Latin

(using Greek characters) and is fol­

lowed by a Greek translation. The

word AREPO is translated as lXporpov (plough). This led Carcopino [10] to re­

late AREPO to the celtic word arepen­nis, an ancient unit of land, and to pos-

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 47

Figure 1. The brick with the magic square from the Aquincum excavations (Budapest, Hungary). Photograph courtesy of Aquincum Museum.

tulate a latinized word arepus meaning

plough. This would give the following

meaning to the five words of the

square: the sower with his plough

looks after the wheels. To other schol-

48 THE MATHEMATICAL INTELLIGENCER

ars [5, 1 1] , AREPO is the personal name

of the sower. The latest interpretation,

whereby the word AREPO is split into

three latin words, a re po, has been pre­

sented in The Intelligencer [3].

Whether the ROTAS square has a

Christian or pre-Christian origin is a

much debated question [4,7,8,10-14];

the arguments put forward by those

who favor a pre-Christian origin [8,1 1-

13] seem to be prevailing. Anyhow, it

is likely that the five crosses appearing

in the square (i.e., the four letters T and

the two words TENET crossing at the

center) and the presence of the word

SATOR have favored its diffusion

through Christendom from an early

time. In Nubia and Ethiopia the five

words of the square were curiously as­

sociated to the nails used to crucify Je­

sus Christ [4].

The use of the ROTAS (or SA TOR)

square as an amulet is well docu­

mented [4]. It was used against fever,

to prevent rabies from dog bites, and

as an aid to parturient women. It is of­

ten found surrounded by the four

beasts of the Apocalypse, symbolizing

the Evangelists, or inserted in a penta­

cle or a Solomon's seal.

In 1926 Grosser [ 14] noted that the

Latin words PATER NOSTER (Our Fa­

ther, the beginning of Lord's Prayer),

repeated twice and crossing at the cen­

tral, unique letter N, could be con­

structed using 21 out of the 25 letters

of the square (the reader is referred to

the illustration in [2]). The remaining

four letters, two A and two 0, could be

taken as standing for alpha (the begin­

ning) and omega (the end). This led

Grosser to enunciate the view that the

square was Christian in origin, and in­

vented during a time of persecution as

a secret sign by which the Christians

could recognize each other.

On the other hand, a dozen Latin

anagrams have been constructed out of

the 25 letters of the ROT AS square [4].

Some of them may qualify as Christian,

e.g., oro te Pater, oro te Pater, sanas (I pray you Father, I pray you Father,

heal), while others are evocative of

black magic, e.g., Satan, ter oro te, opera praesto (Satan, thrice I pray you,

act soon). As the letters of the 5 X 5

square (one N, two S and P, four A, E,

0, R, and T) can be combined in

487,000,580,566,500,000 different ways

to construct a string of 25 letters, it is

by no means surprising that some of

the combinations make sense!

But do we really need to parse the

ROTAS square? Its association with

a palindromic verse in the Aquincum

brick, and the presence of many mean­

ingless words and palindromic verses

among the Pompeii graffiti [12] makes

it unlikely that a precise meaning should

be associated to the five words ROTAS

OPERA TENET AREPO SA TOR. Is there

any satisfactory interpretation for an­

other magic square

R 0 M A

0 L I M

M I L 0

A M 0 R

which was found as a graffito in Ostia,

the ancient port of Rome, and in Pom­

peii as well [12]? We think that having

an obscure text adds to the magic. The

25 letters of the ROT AS square are

arranged in a highly symmetrical way;

this makes it possible to read them

from various directions yielding al­

ways the same intriguing, mysterious

text. All this has fascinated the rich in­

habitants of Pompeii, the soldiers de­

fending the remote borders of the Ro­

man Empire, the pious Christians of

the Middle Ages, and the archaeolo­

gists of the 20th century. It apparently

still entices the sophisticated Readers

of The Mathematical Intelligencer.

Acknowledgment. We thank the

associates of the Aquincum Museum

(Budapest, Hungary) for their kind

assistance.

REFERENCES

1 . A. Domenicano and I . Hargittai, "Alpha­

betic Magic Square in a Medieval Church,"

The Mathematical lntelligencer 22 (2000),

no. 1 , 52-53.

2. B. Artmann, "Conceptual Magic Square,"

The Mathematical lntelligencer 22 (2000),

3, 4 .

3 . N . Gauthier, "Parsing a Magic Square,"

The Mathematical lntelligencer 22 (2000),

no. 4, 4 .

4. G. De Jerphanion, "La Formule Magique

Sator Arepo ou Rotas Opera: Vieilles

Theories et Faits Nouveaux," Recherches

de Science Religieuse 25 (1 935), 1 88-225.

5. F. Haverfield, The Archaeological Journal

56 (1 899), 31 9-323; R. G . Collingwood,

The Archaeology of Roman Britain, Lon­

don, 1 930, pp. 1 7 4-1 76 (quoted in Refs.

4 and 1 2) .

6. M. I . Rostovtzeff, The Excavations at Oura­

Europos: Preliminary Report of Fifth Sea­

son of Work, New Haven 1 934, pp.

1 59-1 61 .

7. M. Della Corte, " I I Crittogramma del Pater

Noster Rinvenuto a Pompei ," Rendiconti

della Pontificia Accademia Romana di

Archeologia 1 2 (1 936), 397-400.

8. J. Szilagyi, "Ein Ziegelstein mit Zauber­

formel aus dem Palast des Statthalters in

Aquincum," Acta Antiqua Academiae Sci­

entiarum Hungaricae 2 (1 954), 305-310 .

9. C . Wescher, Bulletin de Ia Societe des An­

tiquaires de France ( 187 4), 1 52-1 54

(quoted in Refs. 4 and 1 0).

1 0. J. Carcopino, Etudes d'Histoire Chreti­

enne: le Christianisme Secret du Carre

Magique; les Fouilles de Saint-Pierre et Ia

Tradition, Albin Michel, Paris, 1 953.

1 1 . G. De Jerphanion, "Osservazioni suii'Orig­

ine del Ouadrato Magico Sator Arepo,"

Rendiconti della Pontificia Accademia Ro­

mana di Archeologia 1 2 (1 936), 401 -404;

G. De Jerphanion, "A Propos des Nou­

veaux Exemplaires, Trouves a Pompei, du

Carre Magique Sator," Comptes Rendus

des Seances de I'Academie des Inscrip­

tions & Belles-Lettres (1 937), 84-93.

1 2 . M. Guarducci, "II Misterioso Quadrato

Magico: l ' lnterpretazione di Jerome Car­

copino, e Documenti Nuovi, " Archeologia

Classica 1 7 (1 965), 21 9-270.

1 3. A. Frugoni, "Sator Arepo Tenet Opera Ro­

tas," Rivista di Storia e Letteratura Re/i­

giosa 1 (1 965), 433-439.

1 4. F. Grosser, "Ein Neuer Vorschlag zur Deu­

tung der Sator-Formel ," Archiv fOr Reli­

gionswissenschaft 24 (1 926), 1 65-1 69.

Aldo Domenicano

Department of Chemistry, Chemical Engineer­

ing and Materials

University of L'Aquila

1-671 00 L'Aquila, Italy

e-mail: domenica@univaq.it

Istvan Hargittai

Budapest University of Technology and Eco­

nomics

H-1 521 Budapest, Hungary

e-mail: hargittai@tki.aak.bme.hu

VOLUME 24, NUMBER 3, 2002 49

Joannes Keplerus Leomontanus: Kep ler's Ch i ldhood in Wei l der Stadt and Leon berg 1 571 - 1 584 by Hans-Joachim Albinus

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium

e-mail: dirk.huylebrouck@ping.be

From a modem point of view Jo­hannes Kepler (27 December

1571-15 November 1630) is described as an astronomer, a physicist, a car­tographer, a calendar expert, and not the least, a mathematician. In astron­omy his name is associated with Ke­pler's laws of planetary motion, 1 in mathematics with Kepler's rule2 (Ke­plersche Fassregel), an approximation he developed to determine the volume of barrels. 3 He was instrumental in the beginnings of astronomy as a science, for passing from mere description of the observed phenomena to search for their inner connections.

Here, I want to take a closer look at the place where he was born and those where he grew up and went to school. All of these places are located in a rel­atively small area in and around the an­cient duchy of Wtirttemberg. They are easily accessible by car or train, for ex­ample when visiting the universities of Stuttgart, Tiibingen, or Karlsruhe.

Birth and Early Childhood

in Weil der Stadt

Staufer Emperor Friedrich II founded the town of Weil der Stadt sometime before 1241.

There Johannes Kepler was born on 27 December 1571, 2:30 P.M. He was probably named Johannes because it was the saint's day of John the Evange­list (in German Johannes). His parents, Heinrich Kepler and Katharina, (nee Guldenmann), had married in May 1571, and Johannes was falsely declared a premature baby (seven months): his parents probable feared the stigma of an illegitimate conception.

Heinrich Kepler also came from W eil der Stadt, a small town located in the valley of the river Wiirm. At that time, it had about 1000 inhabitants and was situated on a trade route leading from Switzerland to France. The elder Kepler was the fourth son of the mer­chant and innkeeper, Sebald Kepler, who was the mayor of the town, and his wife Katharina (nee Miiller), who came from nearby Marbach am Neckar.4 Kepler's mother Katharina came from Eltingen, located some 14 kilometers east of W eil der Stadt, where her parents Melchior and Mar­garetha Guldenmann lived. A farmer and innkeeper, Melchior Guldenmann was the mayor of Eltingen.

The house where Kepler's paternal grandparents lived is still standing on the right side of the marketplace beside the Weil der Stadt town hall. A plaque on the house carries the inscription, "Marktplatz 5/16. Jh-1902 Gasthof 'Zum Engel'/Wohnhaus des Biirger­meisters Sebald Kepler t 1596/1986 Sanierung + Restaurierung" (Market­place no. 5, from 16th century to 1902 inn Zum Engel, home of mayor Sebald Kepler, who died in 1596; restored in 1986).

The house where Johannes was born and where the Kepler family lived still exists, too. The half-timbered house with a stone base is situated, somewhat hidden, next to the market­place left of the town hall at the be­ginning of the passage toward the main church. It was damaged in 1648 by French troops during the Thirty Years' War, but was later repaired. In 1938 the Kepler Association bought the house

1 These are the basics of celestial mechanics: ( 1 ) The orbit of each planet is an ellipse that has the sun at one

focus. (2) The radius vector from the sun to each planet sweeps out equal areas in equal times. (3) The ratio

of the squares of the revolution periods of two planets equals the ratio of the cubes of their mean orbital axes.

2This is, in modern notation, a formula for numerical integration:

Jb b - a b - a a f(x) d:x: == -6 - (f(a) + 4j (-2 -) + f(b) ),

where the approximation is exact for polynomials of degree :5 3. This is a special case of Simpson's formula.

3The circumstances of his second marriage in Eferding, near Linz, which led Kepler to an intensive occupation

with stereometry, are described in [1 8] .

4Marbach is known as the birthplace of the poet Friedrich Schiller and the mathematician and astronomer Jo­

hann Tobias Mayer.

50 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

The house where Kepler was born, now Ke­pler-Museum in Weil der Stadt.

and renovated it. On the initiative of

the historian and Kepler expert Max

Casper, the Kepler-Museum was

opened here in 1940. In seven rooms,

the most important phases of Kepler's

life are portrayed: his childhood in

Swabia, his youth and student days, his

stays in Graz, Prague, Linz, and Re­

gensburg; there are also exhibits on

Kepler's role in modem natural sciences.

The exhibition includes first editions of

some of his works, some of his mathe­

matical and astronomical instruments,

computer simulations as well as audio

and video presentations about Kepler's

discoveries. Near the entrance, there is

also a large bust of Kepler created

sometime after 1930 by Gustav Adolf

Bredow, a sculptor from Stuttgart.

5The others were Schorndorf and Waiblingen.

Johannes Kepler's early childhood

was not very harmonious. His father

Heinrich was a violent-tempered man,

often involved in fights, and he had

consequently been admonished and

punished several times. In 1574, at the

age of 27, he went to Belgium as a mer­

cenary and, although a Protestant, en­

tered the Catholic Spanish army, which

was at war with the Calvinist Nether­

lands. Katharina and the children (Jo­

hannes and his younger brother Hein­

rich, who was the second of altogether

seven children) stayed in Weil der

Stadt.

Soon after that Katharina fell ill with

the plague, but she recovered well. In

July 1575 she followed her husband to

Belgium, meaning to bring him back

home. The two sons stayed with their

grandparents in W eil der Stadt. During

his parents' absence, Johannes fell ill

with smallpox. He suffered from poor

health ever after and also suffered last­

ing damage to his eyes, which later pre­

vented him from making exact astro­

nomical observations; all the more

reason to admire his astronomical

achievements. In September 1575 the

parents returned to their children.

In 1870 a memorial was erected in

Weil der Stadt's marketplace in honor

of Johannes Kepler. It still stands to­

day, although the pedestal was altered

in 1940. The design-Kepler, looking

up to the sky, wearing a Spanish cos­

tume and holding his celestial globe,

compasses, and a scroll showing a

drawing of the elliptic planetary or­

bits-was conceived by August von

Kreling, who was the director of the Art School of Nuremberg at that time.

On each of the four comers of the

pedestal stands a different statue:

• the astronomer Nicolaus Copernicus

(1473-1543), whose heliocentric con­

cept of the world formed the basis

for Kepler's research

• the mathematician and astronomer

Michael Mastlin (1550-1631), Ke­

pler's teacher at the University of

Tubingen, who acquainted him with

Copernicus's theory, and remained his lifelong friend

• the Danish astronomer Tycho Brahe

(1546-1601), whose extensive and

accurate astronomical database en­

abled Kepler to discover and mathe­

matically verify the laws of planetary

motion

• the Swiss mathematician Jobst Burgi

(1552-1632), with whom Kepler

worked in Prague at the court of Em­

peror Rudolph II.

On the sides of the pedestal are four

reliefs, one called Astronomia, show­

ing Urania the muse of astronomy, and

the others showing scenes from Ke­

pler's life:

• Mathematica shows Mastlin as he

explains the Copernican system to

Kepler. (In the background are busts

of Hipparchus and Ptolemy.)

• Physica shows a scientific dispute

between Brahe and Kepler, whose

important works Astronomia Nova of 1609 and Tabulae Rudolphinae of

1627 lie on the table. At Brahe's feet

is a plan of the wall quadrant of his

observatory Uraniborg on the island

of Hven. (In the background are the

Emperor Rudolph II and Primo

W allestein, at whose courts Kepler

worked, and some printers at work)

• Optica shows Biirgi in his workshop

in Prague, joined by Kepler. Burgi

watches the planet Jupiter through a

telescope designed by Kepler. It has

two convergent lenses, which is still

the basic form of the modem re­

fractive telescope.

In the town museum, which gives

the history of W eil der Stadt, there is

information about the erection of the

Kepler Memorial, from the first sketches

of 1840 to the fund-raising efforts of

1851 and 1860, the memorial commit­

tees and their members, photographs

and drawings of the Kepler-House, and

the memorial itself.

The Move to Leonberg: Its Con­

sequences for Kepler's Career

Leon berg, to the east of W eil der Stadt,

was founded in 1248/1249 by Count Ul­

rich I mit dem Daumen ofWiirttemberg

(i.e. , the one with the abnormal [right]

thumb) as the first of three new towns5

in the county.

VOLUME 24, NUMBER 3, 2002 51

Kepler Memorial in Weil der Stadt.

One may ask why Kepler's parents moved to Leonberg, giving up the ad­vantages of inhabitants of a free im­perial town and becoming subjects of the Duke of Wiirttemberg instead. Ac­cording to accounts Kepler wrote later, the housing conditions in his parents' house in Weil der Stadt were inadequate. Apart from his parents, his brother, and himself, younger brothers and sisters of his father were living there, too. On the ground floor there was a shop for herbs. Both Hein­rich and Katharina being very quar-

relsome, it must have been unbearable for these people to live together in such a confined space. Religious con­flict may also have played a part. In 1522 a Lutheran community came into being in Weil der Stadt, but in 1573 the Counter-Reformation6 began. Kepler's grandfather Sebald had been the leader of the Protestants, therefore the family justifiably feared financial disadvantages.

Leonberg was and remained Re­formed, and with more than 1200 in­habitants it was somewhat larger than

6The Counter-Reformation prevailed in Weil der Stadt for the first time in 1 597 and once and for all in 1 628.

W eil der Stadt. At that time, Leonberg enjoyed an economic upswing, because between 1560 and 1565 Duke Christoph of Wiirttemberg had the Leonberg cas­tle built as one of his seats of govern­ment. This, in conjunction with the rights that living in a town in general and living under the Tiibingen Contract in particular brought (e.g., the existence of law courts, the right to choose one's domicile freely), may have been the de­cisive factor for the Kepler family's choice of a new residence.

Compared to the free imperial town of Weil der Stadt, Wiirttemberg had other advantages, which were to be­come of importance later in Johannes Kepler's life. Firstly, there was com­pulsory school attendance since the great ecclesiastical constitution 7 ( Grofie Kirchenordnung) enacted 1559/1582

(as an end to the Reformation in Wiirt­temberg). All children were to learn to read and write, at least well enough to read the catechism, the Bible, and the hymnbook. Wiirttemberg was the first duchy in Germany that met this de­mand by Luther. Secondly, there was a state-run university in Tiibingen.8 It of­fered a type of higher education, the Schwabische Laufbahn, which started with attendance at a German basic school, then a Latin grammar school, and led to the university, either via the nine lower-9 and four higher10 monastic schools1 1 or via the two colleges (paed­agogium) in Stuttgart and Ttibingen. A degree in theology then enabled the graduate to become a priest, a public servant, or a teacher. The Tiibinger Stift, a seminary, was affiliated with the university as a place of living and training for the theology students. Par­ticularly talented male natives of Wiirt­temberg were given a ducal scholar­ship which included free stay at the Stift and at the university. Kepler was to benefit from this later.

7The so-called GroBe Kirchenordnung had been written by the Wurttemberg Reformer Johannes Brenz (1 499-1570). The house where he was bom in Weil der Stadt­

bought and renovated by the Protestant Church in 1 887-is still standing at Brenzgasse 2. The town museum exhibits an extensive collection about Brenz and the

Reformation in Weil der Stadt.

BThe University of Tubingen was founded in 1477 by the Wurttemberg Duke Eberhard im Barte (i.e., bearded). This was relatively late, because around Wurttemberg

the universities of Heidelberg (founded 1386), Freiburg (1 456), Basel (1 460), lngolstadt {1 472) already existed-the oldest German university was in Prague (1 348)­

and therefore rt was a great risk; but attempto {"I will risk it") was Eberhard 's motto, and the subsequent success proved him right.

9Adelberg, Alpirsbach, Anhausen, Blaubeuren, Denkendorf, Sank! Georgen, Kbnigsbronn. Lorch, Murrhardt.

10Bebenhausen, Herrenalb, Hirsau, Maulbronn.

1 1 The monasteries themselves had been closed during the Reformation.

52 THE MATHEMATICAL INTELLIGENCER

Two residential houses on Leonberg's market place, Kepler's on the left.

Childhood in Leonberg

In 1575 Heinrich Kepler bought a house

on Leonberg's marketplace and the

family moved there in December of the

same year, shortly after Johannes's

fourth birthday. The half-timbered

Kepler house still stands in Leonberg

at Marktplatz 11. A plaque on the house

bears the inscription "Elternhaus I des

Astronomen I Joh. Kepler I damaligen

Schillers / 1576--1579." (Parental home

of the astronomer Joh. Kepler 1576--

1579, then a schoolboy.)

Strangely, Marktplatz 13, just to the

right, also has a plaque, with the inscrip­

tion "Hier wohnte I Astronom I Johannes

Keppler lvon l 1572-1585" (Here lived

astronomer Johannes Kepler from 1572

to 1585). This erroneous sandstone

plaque dates to around 1870, when peo­

ple, euphoric about the founding of the

German Empire, tried to claim Leon­

berg as Kepler's place of birth.

The following year Heinrich again

went to Belgium as a mercenary and

returned to Leonberg in 1577. After

paying a citizen's fee (Burgergeld), the

Keplers became legal citizens of Leon­

berg. In the same year, Johannes en­

tered the elementary school, the Ger­

man school. It was also in 1577 that his mother showed him the appearance of

a comet. 12 The following year, he en­

tered the Latin grammar school, where

the students were taught to read and

write using Latin exercise books and

where teaching, conversations, and

recitations were exclusively in Latin.

The two grammar schools that Jo­

hannes Kepler attended were housed

in the former Beguine-House. 13 This building still stands at Pfarrstra.Be 1 ,

right next to the church steeple. 14 To­

day it is a museum with exhibits on the

prehistory and early history of the

Leonberg area, as well on everyday life

in 19th-century Leonberg. A special

memorial room in the museum is de­

voted to Schelling.

Kepler's Sojourn in Leonberg

In December 1579, beset by fmancial

problems, his father sold the house and

the family moved to Ellmendingen,

which today is a part of the town of

Keltern. It is located close to

Pforzheim, just under 40 kilometers

west of Leonberg. There Heinrich

Kepler leased the village inn Zur Sonne. That building was torn down some hun­

dred years ago, and today the Cafe Ke­pler stands at its former location at

Durlacher Stra.Be 24. There is a plaque

with the following inscription: "BIS

ZUR JAHRHUNDERTWENDE STAND I

AN DIESER STELLE DAS GASTHAUS

ZUR I SONNE. HIER WOHNTE VON

1579 BIS 1584 I HEINRICH KEPLER,

AUS WElL DER STADT, I MIT SEINER

12Back then comets were considered to be harbingers of misery and therefore attracted much attention. The astronomer Tycho Brahe watched this comet at the same

time and recorded the data.

13'fhe Beguines were houses where unmarried women and widows joined in a community similar to a convent, but without taking a binding ecclesiastical vow. Their

main duty was nursing. This movement reached its peak during the 1 3th and 1 4th centuries in France, Germany, and the Netherlands.

14Aiso in the PfarrstraBe lies the house of the parish priest. This 1 7th-century residential building-in Kepler's times it was still owned by Leonberg 's provost Lutherus

Einhorn - is the house where the philosopher Friedrich Wilhelm Schelling (1 775-1854), the theologian Heinrich Eberhard Gottlob Paulus (1 761-1851 )-first a close

friend. later a fierce critic of Schelling-and the doctor Karl Wilhelm Hochstatter ( 1 781-1 81 1 ) were born. These three men. Johannes Kepler, and the Leonberger dog.

a new breed in the 1 9th century, made Leonberg famous. A very readable description of the lives of Schelling and Paulus is [1 5].

VOLUME 24, NUMBER 3, 2002 53

Two inscriptions, the correct one on the left.

Old school in Leonberg.

54 THE MATHEMATICAL INTELLIGENCER

FRAU UND SEINEM SOHNE I DEM NACHMALS I WELTBERUHMTEN AS­TRONOMEN I JOHANNES KEPLER / ELLMENDINGEN AM HEIMATTAG 5. JULI 1959." (Until the tum of the [twen­tieth] century the inn Zur Sonne stood in this place; from 1579 to 1584 Hein­rich Kepler from Weil der Stadt lived here with his wife and son, the subse­quently world-famous astronomer Jo­hannes Kepler; Ellmendingen, 5 July 1959, Local History Day.) Of course Jo­hannes's younger brother Heinrich also lived there! In the village of Ell­mendingen there was no school. There­fore Kepler was, according to his own words, "burdened by farm work" from 1580 to 1582. At some point during that time his father showed him a lunar eclipse while they were standing in the vineyard behind the house. This and the comet of 1577 may have aroused the child's interest in astronomy.

During the winter of 1582/1583 he went back to attend the Latin grammar school in Leonberg. He probably stayed in Eltingen with his maternal grandparents, the Guldenmanns. To­day Eltingen is a district of Leonberg; in the mid-16th century the two places were two kilometers apart.

Kepler completed his elementary education at the Latin grammar school in 1583 and was a good student. His teacher recommended him and he was admitted for the examination that would qualify him for secondary school. In May 1583 he passed the Landexamen, which was given once a year in Stuttgart. This entitled him to the ducal scholarship. The house of Katharina Kepler's parents still stands at Carl­Schminke-Stra.Be 54, and is marked by a plaque with the following inscription: "Geburtshaus I der I Mutter des Astrono­men Kepler: I Katharina Guldenmann I geb. 8. 11 . 1547." (In this house the mother of the astronomer Kepler was born: Katharina Guldenmann born 8 November 1547.)

On the corner of Carl-Schminke­Stra.Be and Hindenburgstra.Be in Eltin­gen there is also a fountain, a memorial

House where Kepler's mother was born in Leonberg (Eitingen).

to Kepler's mother. The accompanying inscription "Zur Erinnerung an I Katha­rina Kepler, geb. Guldenmann I geb. zu Eltingen am 8. 11 . 1547/Errichtet von der Gemeinde I Eltingen im Jahre 1937" (In memory of Katharina Kepler, nee Guldenmann, born in Eltingen 8 No­vember 1547; erected by the munici­pality of Eltingen in 1937) is located somewhat hidden on a wall behind a lime tree.

After passing his exam in 1583 Kepler returned to Ellmendingen and again did farm work while he waited for a vacancy in the monastic school.

In the spring of 1584 the Kepler fam­ily moved back to Leonberg, this time into a house near the lower town gate. The building no longer exists. In May 1584 Johannes's sister Margaretha was born there.

Kepler's Further Life in the

Duchy of Wiirttemberg

In October 1584 Kepler attended the lower monastic school in Adelberg, the so-called Grammatistenkloster. Adel­berg lies between Schorndorf and Gop­pingen, a little more than 50 kilometers

east of Leon berg. The basic structure of the monastery, the walls, and some of the buildings are still standing and open to the public-among other things the Pralatur (prelate's house). Next to the gate of the building there is a plaque in memory of Kepler, carrying the in­scription "Hier weilte von 1584-86/der Astronom I JOHANNES KEPLER I als Klosterschiller" (Here stayed from 1584 to 1586 the astronomer Johannes Kepler as a pupil at the monastic school). On one of the corners of the building another plaque gives the history of the Pralatur including a reference to Kepler. Today, some of the remaining buildings of the monastery serve as res­idential and business buildings.

As early as October 1586, Kepler changed schools and attended the higher monastic school in Maulbronn. The former Cistercian monastery of Maulbronn lies north of Pforzheim, about 40 kilometers from Leonberg. Many of the old buildings still stand, among them some that Kepler fre­quented. However, most of them, in­cluding the school, 15 have been rebuilt several times in the course of the years.

1 5The famous pupils after Kepler to have studied in Maulbronn include the poets Friedrich Holderlin and Hermann Hesse, the theologian David Friedrich StrauB, the

poet and philosopher Friedrich Theodor Vischer, the poet and revolutionary Georg Herwegh, the poet and journalist Hermann Kurz, the philosopher and theologian Ed­

uard Zeller, the diplomat and French minister of foreign affairs Karl Friedrich Reinhard (see [22]). In Maulbronn is buried Schelling's first wife, Caroline, former wife of

the writer and phi lologian August Wilhelm Schlegel.

VOLUME 24. NUMBER 3, 2002 55

Katharina Kepler Memorial in Eltingen.

The monastery is open to the public and is part of UNESCO's world cultural heritage. Kepler stayed in Maulbronn until 1589. In October 1587 he briefly traveled to Tiibingen for matriculation at the university. But he had to delay the beginning of his studies until there was a vacancy in the Stijt. Therefore he took the Bachelor's exam in Maul­bronn in 1588.

In September 1589 Kepler entered the Tiibinger Stift, a former Augustin-

ian monastery, and began his studies in the philosophy faculty of the university (Artistenjakultat), as was common practice at that time. He finished his studies in August 1591, earning his Mas­ter's degree. Because Kepler wanted to become a priest, he went on to study in the theological faculty. He stayed on at the Stijt that housed many important theologians, scientists, and writers from Wiirttemberg. 16 Even today, after being renovated several times, it serves as a dormitory for theology studentsY There is a plaque in memory of Jo­hannes Kepler, showing a portrait and the inscription "Johannes Kepler 1571-1630 I Gestiftet von der Universitat Tiibingen zur 400 jahrigen Griindung des ev. theol. Stifts." (Johannes Kepler 1571-1630, donated by the University of Tiibingen on the occasion of the 400th anniversary of the foundation of the Protestant theological seminary.) To get to it, one has to pass the gateway connecting outer court and inner court, then take a right into the cloister, climb the stairs to the first floor, and then step out onto the balcony. 18 Two display cases located in a side corridor behind the door next to the plaque in memory of Schelling are worth seeing as well. One of them contains copies of docu­ments verifying Kepler's connections to the Stijt and the university, among other things a receipt of the reception containing his handwritten name, 19 the ducal order to admit him to the univer­sity,20 and a certificate of discharge. The other case contains documents concerning Holderlin, Schelling, and Hegel.21

There is still another place in Tiibin­gen which reminds us of Johannes Kepler. Following the usual way from the Stijt or the marketplace to the cas­tle, one has to pass the house Burgsteige 7, which has a plaque with inscription

"Hier wohnte!Prof. Michael Mastlin{aus Goppingen, /der Lehrer des Astronomen/ Johannes Kepler." (Here lived Prof. Michael Mastlin from Goppingen, teacher of the astronomer Johannes Kepler.) Certainly Kepler must have been in Mastlin's house frequently.

In March 1594 Kepler left Tiibingen before finishing his theology studies and went to Graz in Styria (today a part of Austria). In the course of spreading the Reformation, he had been proposed for a chair in mathe­matics at a corporative Protestant school. Actually Kepler considered his move to Graz as a brief interrup­tion in his theological studies, and asked the Duke for permission to postpone the completion of his stu9-­ies in Tiibingen. The Duke agreed, but fate had other plans for Kepler. Nev­ertheless in his later life he returned to Wiirttemberg several more times.

In 1596 Kepler left Graz for several months and went to Stuttgart and Tiibin­gen, among other things to prepare the printing of his first book Mysterium Cos­mographicum, a heliocentric descrip­tion of the world, but still in a traditional incorrect style. He took the opportunity to offer Duke Friedrich I to build a sil­ver miniature of the planetary model which he later repudiated. It was to cost 100 Florins, but the Duke declined. In 1609 he went on another journey to Wiirttemberg that included short trips to Stuttgart and Tiibingen, in order to pre­pare the printing of his book Astrorw­mia Nova, one of his main scientific works. Kepler used the occasion to pre­sent himself to the new Duke Johann Friedrich and ask him for a chair at the University ofTiibingen, without success.

Beginning in 1615, Kepler's mother Katharina had been suspected of be­ing a witch. That is why Kepler, who had moved to Linz in the meantime,

1 6Aiong with Kepler, these include, for example, all of the forementioned famous students in Maulbronn-with the exception of Hesse-and the poets Gustav Schwab, Wilhelm Hauff, and E!:duard Morike, the philosophers Georg Friedrich Wilhelm Hegel and Friedrich Wilhelm Schelling, the theologians Johann Albrecht Bengel and Friedrich Christian Baur, the poet and doctor Justinus Kemer, and the poet and humanist Nicodemus Frischlin (see [12) and [1 7)). From King Wilhelm II of WOrttemberg comes the aphorism, "Anybody who wants to be successful in this country, must have attended the Stitt. Anybody who wants to be successful outside of this country, must have been expelled from the Stiff. Tertium non datur."

1 7Since 1 969 women are also admitted.

1 8There are similar plaques in memory of Hegel, Holderlin, Schelling, Morike, and StrauB. 19" Joannis Keplerus Leomontanus Natus anno 7 1 . 27. Dembris." 20" . . . from Maulbronn . . . Johannes Kappeler of Leonberg". Unlike in Latin, the German spelling of names was not standardized at that time, as we can see here: Kappeler instead of Kepler. 2 1 1n 1 790 the three of them shared the same room in the Stiff!

56 THE MATHEMATICAL INTELUGENCER

working as Upper Austria's mathe­matician (Landschajtsmathematiker) and a professor at the district's school, came to Leonberg in 1617. He also went to Tiibingen and Niirtingen, visiting among others Wilhelm Schick­hardt22 (1592-1635), who later be­came a professor at the University of Tiibingen, and who invented the four­species calculating machine before Blaise Pascal. In 1623 and 1624 Schickhardt had built two of these ma­chines23; one of them got lost in the chaos of the Thirty Years' War, the other, originally meant for Kepler to facilitate his astronomical calcula­tions, was destroyed in a fire at Schickhardt's workshop. In 1937 Franz Hammer found plans by Schick­hardt,24 from which Bruno Baron von Freytag LOringhoff, then professor of philosophy at the University of TUbin­g en, was able to build a reconstruc­tion of the machine in 1957-1960. A functioning model is on display on the upper floor of the Kepler-Museum in Weil der Stadt, another in the Stadt­museum Tiibingen (town museum) as part of the exhibition on the town's history.25

From 1620 to 162 1 Kepler stayed in Wiirttemberg, in Giiglingen in fact, be­cause his mother had been arrested in August 1620 and brought there for trial. Giiglingen is about 45 kilometers north of Leonberg near Heilbronn. After a civil case to compensate the damage created by her alleged witchcraft, and another civil case brought by her against her accusers for slander, both of which had been suspended tem­porarily, there now began a witchcraft trial lasting fourteen months. Katha­rina was threatened with the death penalty. Thanks to Johannes Kepler's great commitment, making use of all

0

* lr'nperial Free Ci1y

• Abbey 0 Oi�tricl Town S W I T Z E R L A D

T H E D U C H Y O F W U R TT E M B E R G C O U TY OF MOMPE L G A R D 0 1 0 1 5

km

The Duchy of Wiirttemberg in Kepler's times, showing Weil der Stadt (a), Leonberg (b), Ell­mendingen (c), Adelberg {d), Maulbronn (e), Tubingen {f), Stuttgart {g), Guglingen (h) (accord­ing to [1 1] and [20])

juridical means, his mother was ac­quitted in October 1621, after being tor­tured in the first degree: that is, she was shown the instruments of torture, they were explained to her, and she was threatened with their immediate use. 26 The tower in which she was in­carcerated no longer exists.

Katharina Kepler died at age 7 4 in April 1622, most likely in RoBwa.Iden, lo-

cated between Kirchheimffeck and Goppingen, where her son-in-law was the local prelate. Her burial place is un­known. A plaque on the ancient ceme­tery of Leonberg next to the old part of the town, carrying the inscrip­tion "Gewidmet I dem Andenken I der auf I diesem Friedhof ruhenden I Mutter I des I Astronomen Kepler/Katharina geb. Guldenmann I gestorben hier 13. April

220ften written as Schickard; other spellings are known too.

23For more information on how the machines worked, using the concept of Napier's calculating rods, see [5], [1 0] , and [4].

24The drawings are part of the Kepler estate at the Pulkovo observatory (Saint Petersburg/Leningrad); the estate was discovered at Frankfurt in 1 765 and bought by

the Russian Tsarina Catherine II, on the advice of Leonhard Euler.

25The Stadtmuseum also is in possession of one of the rare portraits of Kepler, an original copperplate engraving from 1 621 . The odd circumstances of its production

are described in [5] (illustration E4).

26There were witch·hunts in Europe from the middle of the 1 5th until the middle of the 1 8th century, but with great regional variations. Reformers as well as Catholics

believed in magic and witchcraft. In WUrttemberg criminal cases had to be brought before a court of law ever since 1 551 , on the basis of the code of criminal proce­

dure of Emperor Karl V (Carolina). Therefore, there had been many fewer trials and fewer death sentences than elsewhere.

In all, in the administrative district of Leonberg 34 accusations of witchcraft were investigated; in 24 cases (among them 23 women) charges were brought against

the suspects; and 1 1 of these (all women) were sentenced to death. 8 of these death sentences came during the period of office of provost Lutherus Einhorn, who

was the prosecutor in the case of Katharina Kepler (see [ 1 3]). Thus, the life of Kepler's mother had been in great danger.

It is unknown if Kepler himself believed in the existence of witches. However, it was very wise of him not to raise this theological question in court.

VOLUME 24, NUMBER 3. 2002 57

1622" (Dedicated to the memory of the mother of the astronomer Kepler, Katha­rina nee Guldenmann, who died here on 13 April 1622 and who lies here in this cemetery), but its information is proba­bly incorrect. The plaque is easily found: starting at the cemetery chapel (near the entrance Seestra.Be 7 -9), one follows the main path and finds it after approxi­mately 50 meters to the left, where it is

set into the cemetery wall.

Summing Up

In the area of the old Duchy of Wiirt­temberg there are no other direct ref­erences to Kepler in buildings, memo­rials, etc. than those in Weil der Stadt, Leonberg (including Eltingen), Tiibin­gen, and Adelberg. These places27 are also connected to four major trends of contemporaneous history influencing Kepler's life: Reformation and Counter­Reformation; Ptolemaic and heliocen­tric conceptions of the world; the Thirty Years' War (1618-1648); and the witch-hunts.

Detailed information about Kepler's time in Leonberg can be found in [21] . The standard work on Kepler's life and work still is [3]; a concise text for ex­ample is [8]. Autobiographical facts about Kepler are mainly to be found in his letters. Published on the occasion of the 400th anniversary of Kepler's birthday, [7] offers a nice selection of them, along with German translation of Latin texts.

REFERENCES

[1 ] Adelberg-eine Bilddokumentation. Ge­

meinde Adelberg, 1 977.

[2] Borst, Otto; Feist, Joachim: Wei/ der Stadt.

Theiss, Stuttgart, 1 977.

[3] Casper, Max: Johannes Kepler. Kohlham­

mer, Stuttgart, 1 958 (3rd ed.). There is a 4th

edition by the Kepler Association (Verlag fUr

Geschichte der Naturwissenschaften und

der Technik, Stuttgart, 1 995), improved by

references to Kepler's original writings.

There exists an English translation of the 3rd

edition (Dover, New York, 1 993).

[4] Freytag Liiringhoff, Bruno Baron von:

"Prof. Schickards Tubinger Rechenmas­

chine von 1 623." Kleine TObinger

Schriften, 4. Stadt Tubingen, 1 981 .

[5] Gaulke, Karsten; Weber, Ricarda: Oas

Kepler-Museum in Wei/ der Stadt. Kepler­

Gesellschaft, Weil der Stadt, 1 999.

[6] Gramm, Bernadette; Walz, Eberhard: His­

torischer Altstadttohrer Leonberg. Stadt­

archiv Leonberg, 1 991 .

[7] Hammer, Franz; Hammer, Esther; Seck,

Friedrich: Johannes Kepler-Selbstzeug­

nisse. Frommann-Holzboog, Stuttgart,

1 971 '

[8] Hoppe, Johannes: "Johannes Kepler." Bi­

ographien hervorragender Naturwissen­

schaftler, Techniker und Mediziner, 1 7.

Teubner, Leipzig, 1 987 (5th ed.).

[9] Kirschner, Karl ; Stroh, Martin; Rosier, Her­

mann: Chronik von Adelberg, Hundsholz

und Nassach. Gemeindeverwaltung Adel­

berg, 1 964.

[ 1 0] Kistermann, Friedrich W. : "How to use the

Schickard calculator. Types of recon­

structed Schickard calculators." Annals of

the History of Computing, 23 (2001) , no.

1 ' 80-85.

[ 1 1 ] Methuen, Charlotte: Kepler's TObingen.

Ashgate Publishing, Aldershot, 1 998.

[1 2] Muller, Ernst; Haering, Theodor; Haering,

Hermann: Stiftskopfe. Schwabische Ah­

nen des deutschen Geistes aus dem

TObinger Stitt. Salzer, Heilbronn, 1 938.

[1 3] Raith, Anita: "Das Hexenbrennen in Leon­

berg." In : Durr, Renate (ed.): Nonne, Magd

oder Ratsfrau. Frauenleben in Leonberg

aus vier Jahrhunderten. Beitrage zur

Stadtgeschichte, 6. Stadtarchiv Leon berg,

1 998, p. 53-73.

[1 4] Schutz, Wolfgang: Die historische Alptadt

von Wei/ der Stadt. Geiger, Horb, 1 996.

[1 5] Schiinwitz, Ute: Er ist mein Gegner von je­

her. Friedrich Wilhelm Joseph Schelling

und Heinrich Eberhard Gottlob Paulus.

Keicher, Leonberg, 2001 .

[1 6] Setzler, Wilfried, et al. : Leonberg. Eine

altwurttembergische Stadt und ihre

Gemeinden im Wandel der Geschichte.

Wegrahistorik, Stuttgart, 1 992. There was

an index published seperately by the

Stadtarchiv Leonberg in 2001 ; the author

is Karl-Heinz Fischiitter.

[1 7] Setzler, Wilfried: TObingen. Auf a/ten We­

gen Neues entdecken. Ein Stadtfuhrer.

Schwabisches Tagblatt, Tubingen, 1 997.

[1 8] Sigmund, Karl: "Kepler in Eferding." The

Mathematical lntelligencer 23 (2001 ), no.

2 , 47-51 '

[1 9] Sutter, Berthold: Der Hexenprozef3 gegen

Katharina Kepler. Kepler-Gesellschaft and

Heimatverein, Weil der Stadt, 1 984 (2nd

ed.).

[20] Vann, James A: The making of a state.

Wurttemberg 1593-1 793. Cornell Univer­

sity Press, Ithaca, 1 984. German transla­

tion: Wurttemberg auf dem Weg zum

modernen Staat 1593-1 793 (Deutsche

Verlags-Anstalt, Stuttgart, 1 986).

[21 ] Walz, Eberhard: Johannes Kepler

Leomontanus. Gehorsamer Underthan und

Burgerssohn von Lowenberg, Beitrage zur

Stadtgeschichte, 3. Stadtarchiv Leonberg,

1 994.

[22] Ziegler, Hansjorg; Mahal, Gunther;

Luipold, Hans-A : Maulbronner K6pfe. Ge­

fundenes und Bekanntes zu ehemaligen

Seminaristen. Melchior, Vaihingen an der

Enz, 1 987.

lnnenministerium Baden-Wurttemberg

DorotheenstraBe 6

D-701 73 Stuttgart

Germany

e-mail: hans-joachim .albinus@im. bwl.de

27Some more connections with these places: Brenz, who also served as chancellor of the University of Tubingen, and Schelling have already been mentioned. In

1 831/1 832 Mbrike worked as substitute (Pfarrverweser) for the parish in Eltingen; the church and the priest's house are just a few steps from the house where Katha­

rine Kepler was born. From 1 796 to 1 801 Schiller's mother Elisabetha Dorothea lived in Leonberg 's castle as a widow, together with Schiller's sister Luise. Hblderlin's

friend from his time in Maulbronn, Immanuel Nast, was the son of Benjamin Nast, a baker from Leonberg, whose house is located across from Kepler's on Leonberg's

marketplace and where Hblderlin visited him in 1 788, also meeting the sweetheart of his youth, Luise Nast. Wilhelm Schickhardt's uncle Heinrich was the master builder

(Hofbaumeister) of the Duchy of Wurttemberg and laid out the so-called Pomeranzengarten (Bitter-Orange-Garden) behind Leonberg's castle, which is one of the few

existing terraced gardens from the Renaissance. Some of his buildings can also be found in Tubingen, for example, the collegium illustre, later Wilhelmsstift, the Catholic

equivalent to the Protestant Stitt.

58 THE MATHEMATICAL INTELLIGENCER

lj¥1(¥·\·[.1 David E. Rowe, Ed itor !

I s (Was) Mathemat ics an Art or a Sciencet David E. Rowe

Send submissions to David E. Rowe,

Fachbereich 1 7 - Mathematik,

Johannes Gutenberg University,

055099 Mainz, Germany.

If you teach in a department like mine, the answer to this timeless question

may actually bear on the resources your program will have available to teach mathematics in the future. In our de­partment, knowing whether we should be counted as belonging to the Geistes­or to the Naturwissenschajten (hu­manities or natural sciences) could well have serious budgetary implica­tions. Of course most mathematics de­partments are now facing a more press­ing issue, one that can perhaps be boiled down to a related question: Is mathematics closer to (a) an art form or (b) a form of computer science? If your students think the answer is cer­tainly (b), then you can probably pro­ceed on to the next round of more con­crete questions (can mathematics always be run using Windows?, etc.). But this column is concerned with his­torical matters; let me devote this one to the loftier issue raised by the (par­enthetical) question in the title above.

Looking into the recent past, we might wonder to what degree leading mathematicians saw their work as rooted in the exact sciences, as op­posed to the purist ideology espoused by G. H. Hardy in A Mathematician's Apology. Not surprisingly, then as now, opinions about what mathematics is (or what it ought to be) differed. For every Hardy, so it would seem, there was a Poincare, advocating a realist ap­proach, and vice-versa. About a cen­tury ago, when the prolific number-the­orist Edmund Landau learned that young Arnold Sommerfeld was ex­pending his mathematical talents on an analysis of machine lubricants, he summed up what he thought about this dirty business with a single sneering word (pronounced with a disdainful Berlin accent): Schmierol. What could have been more distasteful to a "real" mathematician like Landau than this stuff-even the word itself sounded schmutzig. And so SchmierOl became standard Gottingen jargon, a term of derision that summed up what many

felt: applied mathematics was inferior mathematics; or maybe not even wor­thy of the name. Sommerfeld himself may have grown tired of hearing about "monkey grease." In 1906 he left the field of engineering mathematics to be­come a theoretical physicist, one of the most successful career transitions ever made.

Even within pure mathematics there was plenty of room for hefty disputes about what mathematics ought to be. Foundational issues that had been smoldering throughout the nineteenth century became brush-fire debates af­ter 1900. By the 1920s the foundations of mathematics were all ablaze; David Hilbert battled Brouwer in the center of the inferno. Their power struggle culminated with Hilbert's triumphal speech at the ICM in Bologna in 1928, followed shortly thereafter by his uni­lateral decision to dismiss Brouwer from the editorial board of Mathema­tische Annalen. [Dal] To some in Got­tingen circles, it looked as though Hilbert had defeated the mystic from Amsterdam, but their victory celebra­tion was unearned. Formalism never faced intuitionism on the playing field of the foundations debates. Rather, the Dutchman had merely been shown the door, ostracized from the Gottingen community that had once offered him Felix Klein's former chair. By the time Kurt Godel pinpointed central weak­nesses in Hilbert's program in 1930, the personal animosities that had fueled these fires ceased to play a major role. The foundations crisis proclaimed by Hermann Weyl in 1921 was thus al­ready over by the time Godel proved his incompleteness theorem. The fire had just blown out, enabling the foun­dations experts to go on with their work in a far more peaceful atmos­phere (for an overview, see the essays in [H-P-J]).

Herbert Mehrtens suggests that fun­damental differences regarding mathe­matical existence reflected a broader cultural conflict that divided mod-

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 59

Figure 1 . This group photo was taken at the 1 920 Bad Nauheim Naturforscher meeting, which took place at nearly the same time as the con­troversial Congress of Mathematics in Strasbourg. Since German mathematicians were excluded from participation at the latter meeting, sev­eral came to Bad Nauheim. Hermann Weyl spoke at a special joint session of the German mathematical and physical societies devoted to Einstein's general theory of relativity. This brought forth the famous debate between Einstein and Philipp Lenard on the foundations of gen­eral relativity. L. E. J. Brouwer, second from left in the middle row, also delivered a lecture that also caused quite a stir, at least among math­ematicians. It was entitled: "Does every real number have a decimal representation?" Standing in the back are lsaai Schur, George P61ya, and Erich Bessel-Hagen. Seated are Bela von Kerekiart6, Brouwer, Ott6 Szasz, and Edmund Landau; on the ground is Hans Hamburger. (From George P61ya, The POiya Picture Album: Encounters of a Mathematician, ed. G. L. Alexanderson, Boston: Birkhiiuser, 1987, p. 42.)

ernists and anti-modernists [Meh-2] . No doubt philosophical disputes over existential difficulties cut deeply, but Mehrtens emphasizes that the intense foundational debates during the early twentieth century took place against a background of rapid modernization, and this had a major impact on math­ematical research. The impact of modernity on higher education in gen­eral, and on mathematics in particular, is easy enough-to discern, and yet the effects on mathematical practice de­pended heavily on how higher mathe­matics was already situated in various countries. Thus, Hardy's purism can best be appreciated by remembering that nineteenth-century Cambridge had long upheld applied mathematics in the grand tradition of its famous

60 THE MATHEMATICAL INTIELLIGENCER

Wranglers and physicists. During this same time in Germany just the oppo­site prevailed. There, mathematical purism held sway, reaching a high­water mark in Berlin in the 1870s and 80s, the heyday of Kummer, Weier­strass, and Kronecker (see [Row-2]).

Modernization at the German uni­versities elevated the status of the nat­ural sciences, which had long been overshadowed by traditional humanis­tic disciplines. As part of this trend, mathematicians began to pay closer at­tention to scientific and technological problems. Felix Klein took this as his principal agenda in building a new kind of mathematical research community in Gottingen, where Karl Schwarzschild, Ludwig Prandtl, and Carl Runge pro­moted various facets of applied re-

search. Ironically, this community has often come to be remembered as "Hilbert's Gottingen," whereas Hilbert himself has often been seen through the lens of his later "philosophical" work, the formalist program of the 1920s (for three recent reassessments of his ap­proach to foundations, see [Cor], [Row-3], and [Sieg]). Clearly, Richard Courant had· a very different image of Hilbert in mind when he wrote the first volume of Courant-Hilbert, Mathematische Meth­oden der Physik in 1924. Just as clearly, Hilbert himself saw mathematics in very broad terms, a vision sustained by strong views about the nature of math­ematical thought. The same can be said of his leading rival, Henri Poincare, whose ideas had a lasting impact on philosophers of science.

Poincare's work often drew its in­spiration from physical problems, and he made numerous important contri­butions to celestial mechanics and electrodynamics (see [B-GJ and [Dar]). In Science and Hypothesis, Poincare examined the role played by hypothe­ses in both physical and mathematical research, arguing against many of the views about mathematical knowledge that had prevailed a century earlier. In particular, he sought to demonstrate that it was fallacious to believe "math­ematical truths are derived from a few self-evident propositions, by a chain of flawless reasonings," that they are "im­posed not only on us, but on Nature it­self." ([Poi], p. xxi). Poincare's alter­native view, a doctrine that came to be known as "conventionalism," was sup­ported by a trenchant analysis of the geometry of physical space, then a mat­ter of considerable controversy (for a recent analysis of Poincare's views, see [B-MJ) .

Hermann von Helmholtz had at­tacked the Kantian doctrine according

to which our intuitions of space and time have the status of synthetic a pri­ori knowledge. This position had be­come central to Neo-Kantian philoso­phers who insisted that Euclidean geometry alone was compatible with human cognition. Helmholtz argued, on the contrary, that the roots of our space perception are empirical, so that in prin­ciple a person could learn to perceive spatial relations in a different geome­try, either spherical or hyperbolic. Like Helmholtz, Poincare rejected the Kant­ian claim that the structure of space was necessarily Euclidean, but he stopped short of adopting the empiri­cist view, which implied that the issue of which space we actually live in could be put to a direct test. Poincare noted that any such test would first re­quire fmding a physical criterion to dis­tinguish between the candidate geome­tries. This, however, amounted to laying down conventions for the affine and metric structures of physical space in advance, which effectively under­mined any attempt to determine the

geometrical structure of space without the aid of physical principles.

Poincare's conventionalism reflected his refusal to separate geometry from its roots in the natural sciences, a position diametrically opposed to Hilbert's ap­proach in Grundlagen tier Geometrie (1899). Hilbert would have been the last to deny the empirical roots of geo­metrical knowledge, but these ceased to be relevant the moment the subject became formalized in a system of ax­ioms. By packaging his axioms into five groups (axioms for incidence, order, congruence, parallelism, and continu­ity), Hilbert revealed that these intu­itive notions from classical geometry continued to play a central role in structuring the system of axioms em­ployed by the modern geometer. Nev­ertheless, these groupings played no di­rect role within the body of knowledge, since they never appeared in the proofs of individual theorems. Thus, for Hilbert, the form and content of geom­etry could be strictly separated. In con­trast with Poincare's position, he re-

Figure 2. Hilbert surrounded by members of the Swiss Mathematical Society, Zurich, 1917. During this meeting he delivered his lecture on "Axiomatisches Denken," which signaled his retum to the arena of foundations research. The four gentlemen holding hats in the front row were Constantin Caratheodory, Marcel Grossmann, Hilbert, and K. F. Geiser, followed by Hermann Weyf. Grossmann befriended Einstein dur­ing their student days at the ETH, where both attended Geiser's lectures on differential geometry. Later, as colleagues at the ETH, Gross­mann familiarized Einstein with Ricci's absolute differential calculus. The short man standing in the middle of the back row is Paul Bemays, who would become Hilbert's principal collaborator in the years ahead. (From George P61ya, The P6/ya Picture Album: Encounters of a Math­

ematician, ed. G. L. Alexanderson, Boston: Birkhiiuser, 1 987, p. 40.)

VOLUME 24, NUMBER 3, 2002 61

garded the foundations of geometry as

constituting a pure science whose ar­

guments retain their validity without

any reliance on intuition or empirical

support.

Hilbert originally conceived his fa­

mous lecture on "Mathematical Prob­

lems" as a counter to Poincare's lecture

at the inaugural ICM in Zurich (see

[Gray] , pp. 80-88). Stressing founda­

tions, axiomatics, and number theory,

he set forth a vision of mathematics

that was at once universal and purist.

Aside from the sixth of his twenty­

three Paris problems, he gave only

faint hints of links with other fields.

Hilbert's address, in fact, was based on

the claim that mathematics, as a purely

rigorous science, was fundamentally

different from astronomy, physics, and

all other exact sciences. Taking up a

theme popularized by the physiologist

Emil du Bois-Reymond, who main­

tained that some of mankind's most

perplexing questions could never be

answered by science, Hilbert turned

the tables. For him, this was the quint­

essential difference between mathe­

matics and the natural sciences: in

mathematics alone there could be no ignorabimus because every well­

posed mathematical question had an

answer, and with enough effort that an­

swer could be found. But Hilbert went

further. This seemingly bold claim, he

maintained, was actually an article of

faith that every mathematician shared.

Of course this was Hilbert in 1900; he

hadn't yet met Brouwer!

For much of the twentieth century,

young North American mathemati­

cians were taught to believe that doing

mathematics meant proving theorems

(rigorously). This ethos gained a great

deal of legitimacy from the explosion

of interest after 1900 in foundations,

axiomatics, and mathematical logic,

fields which emerged along with the

first generation of home-grown re­

search mathematicians in the United

States. During the 1880s and 90s, a

number of young Americans came to

Gottingen to study under Felix Klein,

who gladly took on the role of training

those who became mentors to that first

generation. But by the mid-1890s,

Hilbert gradually took over this formi­

dable task [Par-Row, pp. 189-234,

62 THE MATHEMATICAL INTELLIGENCER

439-445] . Modernity was sweeping

through the German universities, and

throughout the two decades preceding

the outbreak of World War I enroll­

ments in science and mathematics

courses in Gottingen grew dramati­

cally, as did the number of foreigners

attending them.

Hilbert's ideas exerted a major im­

pact on American mathematics, not

just on those who studied under him in

Gottingen. Among those who re­

sponded to his message, none did so

with more enthusiasm than Eliakim

Hastings Moore, who helped launch re­

search mathematics at the University

of Chicago during the 1890s. Moore's

school owed much to Hilbert's re­

search agenda, particularly the ax­

iomatic approach to the foundations of

geometry. Oswald Veblen pursued this

program, first as a doctoral student at

Chicago and later at Princeton, but the

leading proponent of this style was an­

other Chicago product, the Texan

Robert Lee Moore.

Like his namesake and mentor, R. L.

Moore served as a "founding father" for

a distinctively American style of math­

ematics [Wil]. He and his followers

acted on their belief in a fundamentally

egalitarian approach to their subject

based on the (unspoken) principle that

"all theorems are created equal" (so

long as you can prove them!). Moore's

students at the University of Texas

spread this gospel, making point-set

topology one of the most popular sub­

jects in the mathematics programs of

American graduate schools. True, this

ethos in its purer form remained

largely confined to colleges and uni­

versities in the heart of the country.

General topology made only modest in­

roads at the older elite institutions on

the East Coast as the Princeton school

of J. W. Alexander, Solomon Lefschetz,

and Norman Steenrod emerged as the

nation's leading center for algebraic

topology.

R. L. Moore's Socratic teaching

style, the so-called "Moore method,"

played an integral part in his philoso­

phy of mathematics, which evinced the

rugged individualism typical for math­

ematicians from the prairie. Book­

learning had little appeal for them: this

was mathematics for the self-made

man who didn't need to rely on anyone

except perhaps a friendly neighbor.

Over on the West Coast, Stanford's

George P6lya gave mathematical ped­

agogy a new Hungarian twist aimed at

fostering mathematical creativity.

Whereas advocates of the "Moore

method" taught that doing mathemat­

ics was synonymous with proving the­

orems and finding counterexamples,

P6lya stressed the importance of in­

ductive thinking in solving mathemati­

cal problems. His How to Solve It sold

over a million copies and was trans­

lated into at least 1 7 languages [Alex,

p. 13] . Not to be outdone, Courant en­

listed Herbert Robbins to help him

write another popular text: What is Mathematics? Presumably Courant

thought he had the answer, but then so

did P6lya, R. L. Moore, and Bourbaki!

Back in Gottingen during the Great

War, physics and mathematics had be­

come ever more closely intertwined.

Einstein's general theory of relativity

captivated the attention of Hilbert and

his circle, and this wave of interest in

the subtleties of gravitation soon trav­

eled across the Atlantic. Columbia's Ed­

ward Kasner was the first American

mathematician to take up the challenge,

but he was soon followed by two of

E. H. Moore's star students, G. D. Birk­

hoff and 0. Veblen. Harvard's Birkhoff

had already begun to depart from the

abstract style of his Chicago mentor. In­

spired by the achievements of Poincare,

he tackled some of the toughest prob­

lems that physics had cast upon the

mathematicians' shore. His monograph

Relativity and Modern Physics ap­

peared in 1923; although nearly forgot­

ten today, it contains a result of major

significance for modem cosmology:

Birkhoffs Theorem: Any spherically

symmetric solution of Einstein's

empty space field equations is

equivalent to the Schwarzschild so­

lution, i.e., the static gravitational

field determined by a homogeneous

spherical mass (see [Haw-Ell] . Ap­

pendix B, for a modem statement

and proof of this theorem).

Both Birkhoff and Veblen got to know

Einstein in 192 1, when he delivered a

series of lectures in Princeton. Einstein

Figure 3. Gosta Mittag-Leffler, flanked by Henri Poincare and Edmund Landau, talking with his friend Carl Runge, back to camera. The occasion was probably the Second ICM held in Paris in 1900. Mittag-Leffler and Runge were perhaps reminiscing about Karl Weierstrass's famous lectures on function theory, which both heard during the 1870s. Landau, himself a gifted analyst, later joined Runge in Gottingen, where they stood at opposite ends of the pure/applied spectrum. (From George P61ya, The P6/ya Picture Album: Encounters of a Math­

ematician, ed. G. L. Alexanderson, Boston: Birkhauser, 1987, p. 26.)

afterward adapted these into book

form, and they were published the fol­

lowing year under the title The Mean­ing of Relativity. Around this time,

Veblen took up differential geometry,

joining his colleague Luther Eisen­

hart's quest to build new tools adapted

to the needs of general relativity. This

research explored the virgin territory

of spaces with semi-Riemannian met­

rics, non-degenerate quadratic differ­

ential forms that need not be positive

definite. Following Weyl's lead, the

Princeton trio of Eisenhart, Veblen,

and Tracy Thomas spearheaded re­

search on the projective space of

paths, which led to a new foundation

for general relativity closely connected

to the theory of Lorentzian manifolds

(for a survey of their work, see [Tho]).

General relativity and cosmology re­

mained major playing fields for mathe­

maticians throughout the 1930s. By the

time Einstein joined the faculty at

Princeton's new Institute for Advanced

Study in 1933, however, quantum me­

chanics had long since emerged as the

dominant field of interest among theo­

retical physicists. Led by John von Neu­

mann, a new wave of activity took place

aimed at developing operator theory

and other mathematical methods that

became the central tools for quantum

theorists. In the meantime, after fifteen

years of intense efforts to formulate a

field theory that could unite gravity and

electromagnetism, a lull set in (for an

overview, see [Gol-Rit]). Einstein, of

course, remained in the arena until his

death in 1955, surrounded by a small

group of younger men.

Back in Berlin, the first of Einstein's

many assistants had been Jakob Gram­

mer, whom he apparently met in Got­

tingen through Hilbert in the summer

of 1915. An orthodox Jew from Brest­

Litovsk, Grammer had gravitated to

Gottingen, where he was "discovered"

in a seminar run by Otto Toeplitz. Be­

ginning in 1917, he worked off and on

as Einstein's assistant for some ten

years, longer than anyone else (see

[Pais] , pp. 483-501). Thereafter, Ein­

stein was never without similar tech­

nical assistance in his quest for a uni­

fied field theory, an effort that took on

a more purely mathematical character

the longer he pursued this goal. Just as

Einstein's theory of gravitation trans­

formed differential geometry, so he

hoped that mathematics would some

day return the favor to physics, if only

by showing the physicists the kind of

theory they needed in order to explore

the outermost and innermost regions

of the universe. In Princeton, most of

Einstein's assistants were recent Eu­

ropean emigres who had managed to

flee before the full force of Nazi racial

policies took hold.

Due to his seniority, Edmund Lan­

dau was not among those who lost

their jobs in the Nazis' initial effort to

purify the German civil service [Sch].

His exodus from the scene was more

poignant and chilling, especially in

light of recent discussions of how "or­

dinary Germans" behaved during the

events leading up to the Holocaust

(Landau escaped its jaws when he died

in Berlin in 1938). Landau's lectures on

number theory and analysis at Gottin-

VOLUME 24, NUMBER 3, 2002 63

gen were delivered in the grand style

of the artiste, and his personal tastes

and idiosyncrasies at the blackboard

came to be known as the "Landau

style." Some found his passionate

dedication to rigor overly pedantic,

while others resented his ostentatious

lifestyle. By November of 1933, the

Gottingen student body was convinced

that Landau's mathematical art could

no longer be tolerated. Posting brown­

shifted SA troopers at the doors of his

lecture hall, they organized a success­

ful boycott of his classes. This effort,

however, was not led by the usual Nazi

rabble but by one of Germany's most

talented young mathematicians, Os­

wald Teichmuller (see [Sch-Sch]). Af­

terward, Ludwig Bieberbach, the new

spokesman for Aryan mathematics,

praised the Gottingen students for

their "manly actions," which showed

their refusal to be taught in such an

"un-German spirit" (see [Meh-1]). For

NS ideologues, Landau's work, like

that of the famous Berlin portraitist

Max Liebermann, was just "decadent

art" ( entartete Kunst) and treated as

such. Symptomatic of what was to fol­

low throughout Germany, nearly all of

the more talented Gottingen mathe­

maticians were gone by the mid-1930s,

their services no longer needed or de­

sired (for an overview, see [S-S]).

In the earlier era of Klein and

Hilbert, "art for art's sake" had always

played a prominent part in the Gottin­

gen milieu [Row-1] . Tastes differed, but

style mattered, and mathematical cre­

ativity found various forms of personal

expression. To Hermann Weyl,

Hilbert's most gifted student and him­

self a masterful writer, the preface to

his mentor's Zahlbericht was a literary

masterpiece. Still, in many Gottingen

circles, the spoken word, uttered in lec­

ture halls and seminar rooms, carried

an even higher premium. Some pre­

ferred Klein's sweeping overviews,

coupled with vivid illustrations, while

others favored Hilbert's systematic ap­

proach, aimed at reducing a problem

to its bare essentials.

The European emigres realized, of

course, that the Hilbertian legacy com­

prised far more than just axiomatics;

nor was Hilbert's style exclusively de­

signed for the pure end of the math-

64 THE MATHEMATICAL INTELLIGENCER

ematical spectrum. After Richard

Courant arrived at New York Uni­

versity, he continued to work in the tra­

dition of his 1924 classic, Courant­

Hilbert, eventually producing its long­

awaited second volume, with the help

of K. 0. Friedrichs. Courant, who went

on to become one of the foremost ad­

vocates of applied mathematics in the

United States, always imagined that the

spirit of "Hilbert's Gottingen" lived on at

NYU's Courant Institute. Meanwhile, in

the quieter environs of Princeton's In­

stitute for Advanced Study, Einstein,

Godel, and W eyl cultivated their re­

spective arts while contemplating the

significance of mathematics for science,

philosophy, and the human condition.

REFERENCES

[Alex] G. L. Alexanderson, George P61ya: A

Biographical Sketch, in The P61ya Picture

Album: Encounters of a Mathematician,

Boston: Birkhauser, 1 g87.

[8-G] June E. Barrow-Green, Poincare and the

Three-Body Problem, Providence, R . I . :

American and London Mathematical Soci­

eties, 1 997.

[B-M] Yemima Ben-Menahem, Convention:

Poincare and Some of His Critics, Bntish

Journal for the Philosophy of Science 52

(2001 ) :471-51 3.

[Cor] Leo Corry, The Empiricist Roots of

Hilbert's Axiomatic Approach, in [H-P-J], pp.

35-54.

[Dal] Dirk van Dalen. The War of the Mice and

Frogs, or the Crisis of the Mathematische An­

nalen, Mathematical lntelligencer, 12(4) (1 990):

1 7-31 .

[Dar] Olivier Darrigo!, Henri Poincare's Criticism

of fin de siecle Electrodynamics, Studies in

the History of Modern Physics, 26( 1 ) (1 995):

1-44.

[Goi-Rit] Catherine Goldstein and Jim Ritter,

The Varieties of Unity: Sounding Unified The­

ories, 1 920-1 930, Preprint Series of the

Max-Pianck-lnstitut fUr Wissenschafts­

geschichte, Nr. 1 49 (2000).

[Gray] Jeremy J. Gray, The Hilbert Challenge,

Oxford: Oxford University Press, 2000.

[Haw-Ell] S. W. Hawking and G. F. R. Ellis, The

Large Scale Structure of Space-Time, Cam­

bridge: Cambridge University Press, 1 973.

[H-P-J] V. F. Hendricks, S. A Pederson, and

K. F. Jorgensen, eds. Proof Theory. History

and Philosophical Significance, Synthese Li­

brary, vol. 292, Dordrecht: Kluwer, 2000.

[Meh-1 ] Herbert Mehrtens, Ludwig Bieberbach

and "Deutsche Mathematik," in Studies in �he

History of Mathematics, Esther R. Phillips, ed. , Washington, D.C. : Mathematical Asso­

ciation of America, 1 987, pp. 1 95-241 .

[Meh-2] Herbert Mehrtens, Moderne-

Sprache-Mathematik. Eine Geschichte des

Streits urn die Grundlagen der Disziplin und

der Subjekts forrnaler Systeme, Frankfurt am

Main: Suhrkamp Verlag, 1990.

[Pais] Abraham Pais, 'Subtle is the Lord. :. ' The

Science and the Life of Albert Einstein, Ox­

ford: Clarendon Press, 1 982.

[Par-Row] Karen Parshall and David E. Rowe,

The Emergence of the American Mathemat­

ical Research Community, 1876- 1900. J.J.

Sylvester, Felix Klein, and E.H. Moore,

AMS/LMS History of Mathematics Series,

vol. 8, Providence, R.i . : American Mathe­

matical Society, 1 994.

[Poi) Henri Poincare, Science and Hypothesis,

London: Walter Scott, 1 905; repr. New York:

Dover, 1 952.

[Row-1 ] David E. Rowe, Felix Klein, David

Hilbert, and the Gottingen Mathematical Tra­

dition, in Science in Germany: The Intersec­

tion of Institutional and Intellectual Issues,

Kathryn Olesko, ed., Osiris, 5(1 989): 1 86-2 1 3.

[Row-2] David E. Rowe, Mathematics in Berl in,

1 81 0-1 933, in Mathematics in Berlin, ed.

H.G.W. Begehr, H . Koch, J. Kramer, N.

Schappacher, and E.-J. Thiele, Basel:

Birkhauser, 1 998, pp. 9-26.

[Row-3] David E. Rowe, The Calm before the

Storm: Hilbert's Early Views on Foundations,

in [H-P-J], pp. 55-93.

tSch] Norbert Schappacher, Edmund Landau's

Gottingen. From the Life and Death of a

Great Mathematical Center, Mathernatical ln­

telligencer 13 (4) (1 99 1 ) : 1 2-1 8.

[Sch-Sch] Norbert Schappacher and Erhard

Scholz, eds. Oswald Teichmuller. Leben und

Werk, Jahresbericht der Deutschen Mathe­

rnatiker-Vereinigung 94 (1 992):1 -39.

[Sieg] Wilfried Sieg, Towards Finitist Proof The­

ory, in [H-P-J], pp. 95-1 1 4.

[S-S] Reinhard Siegmund-Schultze, Mathe­

matiker auf der Flucht vor Hitler, Dokumente

zur Geschichte der Mathematik, Bd. 1 0,

Braunschweig!Wiesbaden: Vieweg, 1 998.

[Tho] T. Y. Thomas, Recent Trends in Geome­

try, in Semicentennial Addresses of the

American Mathematical Society, New York:

American Mathematical Society, 1 938, pp.

98-1 35.

[Will Raymond L. Wilder, The Mathematical

Work of R. L. Moore: Its Background, Na­

ture, and Influence, Archive for History of Ex­

act Sciences, 26 (1 982):73-97.

ANNA MARTELLOTTI

On the Loca Weight Theorem

Sunto-Si enuncia un principia di conservazione locale del peso e se ne derivano alcune importanti conseguenze.

It is widely known that weight (and particularly weight loss) is an important health topic, and that its economic in­fluence is of enormous importance nowadays ([8], [5], [6]). Indeed it is not presumptuous to claim that without the weight business there would be many more depressed ar­eas in the world and most Western economies would un­dergo a dramatic downturn: just to mention a few signifi­cant examples, we might cite firms like Sweet'n Low, groups like Weight Watchers, Richard Simmons's success, and the following of Rosanna Lambertucci.

Therefore, a mathematical model concerning weight loss should be labelled applied mathematics and be given full financial support.

In this short note, following an idea originally due to Lavoisier, we state a theorem of Local conservation of weight, and derive from it a few of the important conse­quences. Our Main Theorem can be interpreted as one of the infmitely many equivalent forms of the well-known Maxima Vexatio Principle [2].

I am deeply indebted to Dr. Annarita Sambucini who strongly encouraged me during this project, and for the many tea and scones conservations we had about this topic. I also warmly thank Professor W ashek Pfeffer for his cheerful editing of this paper, and for communicating to me reference [7].

Preliminaries

By !1 c 1R3 we shall denote the whole universe. The time will be modelled as the half-line T = [0, + oo];

thus we shall assume that time never goes backward. For models admitting negative time we refer to [3] and the lit­erature there.

By local weight we mean a process W : T X n __.,. ]0, + oo]; hence W( t, w) represents the weight of the point w at the instant t. We shall need in the sequel the famed Maxima Vexatio Principle due to Brandi ([2]).

This work is dedicated to myself on the occasion of my 40th birthday.

Theorem I. (Maxima Vexatio Principle) Garbage cannot be escaped.

Theorem 1 admits the following equivalent formulation.

Theorem 2 No good result is true.

Main Theorem and Consequences

Let 7 be a T2 topology on !1, lr be the relative Borel a-alge­bra on n, f.Lr : kr __.,. [0, + oo[; be the weight measure on it. From now on we shall assume that, for every t0 E T the function W(t0, -) is Ir-measurable; this is a reasonable as­sumption because, given the already mentioned weight­concern craze, it is likely that in every instance there will be a scale available to measure the local weight.

It is well known that the total weight is a constant; that is, at any time t E T

W(t) = Jn W(t, w)f.Lr(dw) = constant. (1)

We can now state a sharper result, that is, a local version of the global formula (1).

Main Theorem. For every wo E n, and every u E T( wo), there exists a constant ku such that, for each t E T,

t W(t, w)f.Lr(dw) = ku. (2)

Hence in any neighbourhood of any point the weight would remain constant in time.

Proof Let wo E !1 be fixed, U E 7( wo) be any neighbour­hood of wo, and assume, by contradiction, that in the in­terval [t1, t2l (t1 < t2) one has

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 65

Then necessarily somebody in U is losing weight during

the time interval (t1 , t2] ; if it is you or one of your friends

this is a good result, otherwise, if it is someone who de­

serves your envy because she already has it all, it is a bad

result.

But the first case is impossible, because it would con­

tradict Theorem 2. On the other hand, if the second case

happens, by symmetry, it would be a good result for your

enemy. As the Maxima Vexatio Principle is a universal

statement, this is also a contradiction, and the proof is com­

plete.

The Main Theorem above has extensive consequences.

We shall mention just a few.

COROLLARY 1. (The never diet around another dieter principle)

If someone in your neighbourhood is losing, you wiU gain.

Proof. Indeed, if someone in your neighbourhood is losing,

by the Main Theorem someone in the same neighbourhood

will be gaining. By the Maxima Vexatio Principle with prob­

ability one, it will be you.

The above Corollary is not symmetric; that is, a state­

ment of the form: If somebody in your neighbourhood is gaining, you will lose does not hold. In fact if you were

losing, this again would be a good result, contradicting

Theorem 2.

As an iterated version of the previous corollary we get

the following:

COROLLARY 2. (The never in a spa rule)

The consequence of a holiday in a spa is ballooning.

Proof. Apply the previous Corollary to each of the dieters

around you and you'll see.

This rule had already been perceived in a coarser form in

[4], where it is called the Sweating does not help rule.

The next result, which is somewhat surprising, and

might even look paradoxical to optimists who are not fa­

miliar with the Maxima Vexatio Principle, is in fact a di­

rect consequence of choice of the suitable topology.

COROLLARY 3. (The pear-shaped silhouette statement).

The result of any solo diet is a pear-shaped silhouette.

Proof. Taking into account Corollary 1 and 2, you might try

to lead a secluded life while dieting: but this is equivalent

to restricting the neighbourhood U to yourself. Hence you

will lose in some zone, and consequently, from the Main

Theorem, gain in another zone of U. Now, applying the

Statement in [7] (a different version of the Maxima Vexatio

Principle), the zone that gains is usually the buttocks.

At the end of this section we shall mention an open prob­

lem linked to this topic which, however, may be hard to

solve in this framework. It is well known that there is a

subset, the lucky set of fl, consisting of points that stay

66 THE MATHEMAnCAL INTELLIGENCER

nicely thin irrespective of what they eat. This is the ex­

tensively studied Sambucini Phenomenon. As this is true

and not false, by Theorem 2 it is a bad result. On the other

hand everybody thinks that this is a positive event. Our

coJ\iecture is that a more sophisticated model is needed,

taking into account personal preferences, that is, a differ­

ent ordering (compare [ 1]).

REFERENCES

[1 ] C. Bardaro, Order, disorder, and reorder- the messy desk, Ufficio

Vivo, Rivista di Arredo di Uffici 1 2 (1 979), 1 -3.

[2] C. Brandi, II principia di massima sfiga, Boll. Un. Mat. !tal. (v) ser.

B 34 (1 977), 1 7-1 717.

[3] R. Ceppitelli , How to always be on time: the negative reordering of

time and the backward clock, Orologi 42 (1 960), 345-356.

[4] J. Fonda, Aerobics does no good. Progress that does not last, J.

of the Repentant Exerciser 2 (1 987), 1 2-41 .

[5] R. Lambertucci, Piu sani, piu belli, piu puliti dentro, Annuario della

Cazzata XV (1 990), 46-70.

[6] F. Lodispoto, La dieta del fantino e Ia felicita nell'equino dell'era

moderna, L'asino, il cavallo e Ia zebra 6 (1 967), 89-97.

[7] K. Triska, Good food as a smart, cheap alternative to silicone, in

Essays in honour of Dolly Parton, Brigitte Nielsen, editor, Brustansatz

Verlag, Berlin, 1 983, 8-88 (in Czech).

[8] SIAM Journal of Weight Control, 1 980-1995.

ANNA MARTELLOTTI

Oipartimento di Matematica

Universita degli Studi

061 23 Perugia

Italy

e-mail: amart@dipmat.unipg.it

Anna Martellotti was born in Perugia, was educated there, and

has lived most of her l ife there: the principal interruption was

winning a national competition which sent her to a 4-year vis­

iting position in Mathematical Analysis at Ancona. She works

in measure and integration, branching out into stochastic

processes. Divorced with one grown son, she enjoys moun­

tain trekking, gym workouts, and dancing.

ORA E. PERCUS AND JEROME K. PEROUS

Can Two Wrongs Make a Right? Coin Tossing Games and Parrondo' s Paradox

number of natural and man-made activities can be cast in the form of vari-

ous one-person games, and many of these appear as sequences of transitions

without memory, or Markov chains. It has been observed, initially with sur-

prise, that losing games can often be combined by selection, or even randomly,

to result in winning games. Here, we present the analysis

of such questions in concise mathematical form (exempli­

fied by one nearly trivial case and one which has received

a fair amount of prior study), showing that two wrongs can

indeed make a right-but also that two rights can make a

wrong!

Background

On frequent occasions, a logical oddity comes along, which

attracts a sizeable audience. One of the most recent is

known as Parrondo's paradox [5, 6]. Briefly, it is the ob­

servation that random selection (or merely alternation) of

the playing of two asymptotically losing games* can result

in a winning game.

Conceptually similar situations involving only the pro­

cessing of statistical data are not novel. What has been re­

ferred to as Simpson's paradox [8] is typified by this sce­

nario: Quite different items, say type 1 and type 2, cost

dealers the same $10 per unit. Suppose that, during a given

period, dealer A sells 20 and 80 of these two types, charg­

ing $13 and $15, respectively, per item. Dealer B, on the

other hand, who charges $14 and $16 per item, sells 80 and

20 of the two types. Then the average cost per item to dealer

A's customers is (1/5)13 + (4/5)15 = $14.60, while B's on

the average only pay (4/5)14 + (1/5)16 = $14.40, a net re­

sult that B is delighted to advertise. This despite the fact

that A sells both items more cheaply than B does! No sur­

prise, since A sells mainly the more expensively marked

"Such a game consists of repeated moves where the expected net gain per move is negative.

68 THE MATHEMATICAL INTELUGENCER © 2002 SPRINGER· VERLAG NEW YORK

item and B the cheaper one; this kind of cheating with sta­tistics must be commonplace.

In fact, it has been pointed out by Saari [7] that aggre­gation often yields statistical results qualitatively different from those apparent at a lower level, and that this is re­lated to well-lalOwn problems in game theory and economic theory.

Still, turning two losing games into a winning one (now we are playing solitaire) seems more than a bit counterin­tuitive. To demystify it a little, consider a really extreme case of the Parrondo phenomenon in which in game A, a player can only move from white to black, or black to black The outcome of a single move is a gain of $3 if the player moves from white to black, and a loss of $1 each time he moves from black to black. Because the player becomes trapped on the losing color, his expected gain per move is -$1. In game B, the role of black and white are reversed, but the expected gain per move is the same -$1. Now a random selection of game A or game B results in an ex­pected gain of $1 per move, no matter what color one is moving from (because half the time, whether A or B is played, you are moving from a winning color and half the time from a losing color). What is happening is that each game is rescuing the other.

Examples that are given need not be so obvious (we will quote a prototype later), and it is worthwhile having a math­

This corresponds to the eigenvalue Ao = 1 of the matrix T, all other eigenvalues being simple and having smaller ab­solute values.

Now let us combine the matrix T and the set of gains {wij} to form the matrix T(x), defined by

(3.3)

i.e., introduce a weight for the j � i transition of x raised to the Wij power. The reason for doing so is that if

we consider any sequence of transitions Jo, Jt, . . . JN from an initial j0, then this sequence has a probability PrCJo, . . . JN) = 1jNjN_1 • • • 7}2j1 1Jdo• and an associated gain WN(Jo, . . . JN) = WjNjN_1 + · · · + wh io• so that

N Pr( . .

) WN(J, • . . . jN) - n r c ) Jo, · · · JN X - inin-1 X · n � l

(3.4)

By summing over all N-step sequences, we produce the powerful moment-generating function of W N, given by the expectation

(3.5)

The moment-generating function is a wonderful tool for finding expectation values, and we'll use it right away. To do so, we first have to get a handle on T(x�. Suppose that A(x) is the maximum eigenvalue of T(x); if x is real and

ematical structure to orga­nize their analysis. If the "game" concept is re­stricted sufficiently to al­low a clear interpretation of the averaging strategy mentioned above, this is readily accomplished.

Turning two losing games

into a winn ing one

seems counterintuitive.

close to 1 , A(x) will still be real, close to 1, and largest in absolute value. Further­more, if we normalize the maximal right eigenvector 4>o(x) of T(x) by 1t4>o(x) =

The Expected Gain

Let's get technical! By a (one-person) game, we will mean a set of transitions from state j (among a finite set of states S of size s) to state i, with transition probability Tij; in ad­dition, to the move j � i in this Markov chain [9] we must associate a gain wii• which can be positive or negative. Of course, Tij 2: 0, and �iES Tii = 1 for any j E S, which can be written in vector-matrix form as

(3.1)

where 1 is the column vector of all 1's, and superscript t indicates transpose. The properties of such stochastic ma­trices are an old story, and in particular, we will confme our attention to the large class of irreducible stochastic ma­trices, where if one starts with a probability vector Poj =

Pr(start in stateJ) for the possible states, then iteration of the process

Po, 1Po, T2po, . . . results asymptotically in the unique mix of state probabil­ities cf>o,j for state j, regarded as components of the prob­ability vector { 4>o,i} satisfying

T4>o = 4Jo. (3.2)

1, and the corresponding left eigenvector 1/Jo(x) of T(x) by 1/Jo(x)€ 4>o(x) = 1, then T(x)NIA(x� approaches the corresponding projection:

lim T(x)N/A(x)N = 4Jo(x) 1/Jb(X). N-+oo (3.6)

Hence (3.5) implies that

(3.7)

There is a lot of information in (3. 7), but we will con­centrate on the asymptotic gain per move,

(3.8)

To find it, just differentiate (3. 7) with respect to x and set x = 1, assuming commutativity of the limiting opera­tions. Because A(l) = 1, 4Jo(1) = 4>o, lj!0(1) = 1, we have limN-+oo (E(WN) - NA'(1)) = lj!�(1)po, which is finite. Hence limN_,oo iCE(WN) - NA'(1)) = 0, or according to (3.8)

w = A'(1). (3.9)

An even more transparent alternative representation is ob­tained by differentiating T(x)4>o(x) = A(x)cf>o(x) with re-

VOLUME 24, NUMBER 3, 2002 69

spect to x and setting x = 1: T'(1)4>o + Tcf>b(l) = A'(l)cf>o +

4>6(1). Taking the scalar product with 1:

1tT'(1)</>o + 11cf>o(1) = A'(1) + 114>6(1),

so that A'(1) = 1tT'(1)cf>0. Thus,

w = 1tT'(1)cf>o, (3.10)

whose inteipretation is obvious: 4>o is the asymptotic state vec­tor whose components are cf>o,k> k = 1, . . . s; T/i1) = Wij Tij is the gain per move weighted by its probability; and 1 t adds it all up. Hence

(3. 11)

is the expected gain on making a move from state k, and we can also write (3.10) in the form

(3.12)

Game Averaging - a Simple Example

A game, in the terminology we have been using, is fully specified by the weighted transition matrix T(x), which tells us at the same time the probability Tij of a transition j � i and the gain Wij produced by that move. A random composite of games A and B can then be created by choos­ing, prior to each move, which game is to be played; A (and its associated move probability and gain per move), say, with probability a; or B, with probability 1 - a.

TA,B(x) = aTA(x) + (1 - a)TB(x). (4. 1)

What has come to be known as Parrondo's paradox ( orig­inally, a rough model of the "flashing ratchet" [ 1]), is that domain in which both wA < 0 and WB < 0, but WA,B > 0. Much of the phenomenology is already present in a variant of the simple model we have mentioned as background. Let us see how this goes:

In both games, A and B, a move is made from white or black to white or black Game A is now defined by a prob­ability p, no longer unity, of moving to black, q = 1 - p to white, with a gain of $3 on a move from white, of -$1 on a move from black Hence (with white : j = 1, black :j = 2)

TA = (! !} cf>oA = (!} TA(x) = (q:i3 qlx ); (4.2)

p;i3 pix

in game B, the roles of black and white are reversed, so that

For the composite game, we imagine equal probabilities, a = i• of choosing one game or the other, and indicate this by iA + iB, and now

70 THE MATHEMATICAL INTELLIGENCER

3

l§lijil;iiM

'iA+IB = (t n wtA+tB = G D·

It follows, most directly from (3. 10), that

(4.4)

(4.5)

Hence, in the bold region of Figure 1, for 3/4 < p ::s: 1, we in­deed have WA = WB < 0, together with wtA+tB > 0. (Note however that WA = WB > wtA+tB for p < t.) Game Averaging -Another Example

The game originally quoted in this context is as follows [2]: Each move results in a gain of + 1 or -1 in the player's cap­ital. If the current capital is not a multiple of 3, coin I is tossed, with a probability p1 of winning + 1, a probability q1 = 1 - p1 of "winning" - 1. If the capital is a multiple of 3, one instead flips coin II with corresponding p2 and q2. Hence the states can be taken as ( - 1, 0, 1) (mod 3), and the associated transition and gain matrices are

and then

( 0 - 1 1) w = 1 0 - 1 . (5. 1)

- 1 1 0

2 2 w = 1tT'(l)cf>o = 3 PlP2 - qlq2 (5.3)

2 + P1P2 + qlq2 - P1q1

Now suppose there are two games, the second specified by parameters pi, qi, pz, q2,. An averaging of the two would then define a move as: (1) choose game No. 1-call it A­

with probability a, game No. 2, B with probability 1 - a; (2) play the game chosen. Because the gain matrix w is the same for both games, this is completely equivalent to play­ing a new game with parametersfil = ap1 + (1 - a) pi, fi2 = ap2 + (1 - a)p2, etc., and so (5.3) applies as well. The "paradox" is most clearly discerned by imagining both games as fair, i.e. , p'fp2 = qyq2, or equivalently

112

0

lpldii;ifW

(5.4)

and similarly for pi, P2, creating the "operating cmve" shown

in Figure 2; winning games are above the cmve; losing games,

below. For games A and B as marked, all averaged games lie

on the dotted line between A and B, and all are winning

games. And by continuity with respect to all parameters, it is

clear that if A and B were slightly losing, most of the con­

necting dotted line would still be in the winning region. How­

ever, two slightly winning games, close to D and E, would re­

sult mainly in a losing game. So much for the paradox!

The example most frequently quoted is specialized in

that game B has only one coin, equivalent to two identical

coins, pi = p2 ( = 1/2 for a fair game, point C); and is mod­

ified in that A and B are systematically switched, rather

than randomly switched. Qualitatively, this is much the

same.

Asymptotic Variance

Much of the activity that we have been discussing arose

from extensive computer simulations [3, 4] , carried out to

the point of negligible fluctuations in the gain. How far does

one have to go to accomplish this? A standard criterion in­

volves looking at the variance of the gain per move as a

function of the number of moves, N, that have been made:

(6. 1)

The computation of a2( w; N) proceeds routinely from

the same starting point (3. 7) used previously to compute

w = limN_.oc E(WNIN). This time, differentiate (3.7) both

once and twice with respect to x and set x = 1, again as­

suming commutativity of limiting operations. Again using

A(1) = 1, <f>o(l) = <Po, f/io(1) = 1 , this results in

lim (E(WN) - N A' (1)) = #/(1)Po N---;oo )_!.� [E(WN(WN - 1)) - 2N E(WN)A'(1) - NA"(l) (6.2)

+N(N - 1) A'(1)2] = I/Jot(1)po,

which we combine to read

)_!.� [E(W�) - (E(WN))2 - N A"(1) - N A'(1)2 - N A'(l)]

= !fi!/(1)Po - (t/lbt(1)Po? + #/(1)po. (6.3)

We see then that

(6.4)

In other words, we have found that the standard devia­

tion is given asymptotically in N by

a(w; N) � N-112[A"(l) + A '(1)2 + A'(1)] 112, (6.5)

with a readily computable coefficient. For example, in the

"Parrondo" case of (5.1), where

A U T H O R S

ORA E. PERCUS JEROME K. PERCUS

Courant Institute of Mathematical Sciences

251 Mercer Street

New York, NY 1 001 2 USA

(6.6)

Ora E. Percus received an M .Sc. in Mathematics at Hebrew

University, Jerusalem, and a Ph.D. in Mathematical Statistics

from Columbia University in 1 965. She has been active in sev­

eral areas of mathematics, including probability, statistics, and

combinatorics.

Jerome K. Percus received a B.S. in Electrical Engineering, an

M.A. in Mathematics, and a Ph.D. in Physics, in 1 954, from

Columbia University. He has worked in numerous areas of ap­

plied mathematics, primarily in chemical physics, mathemati­

cal biology, and medical statistics.

They have had many collaborations, but the best of them can­

not be found in the scientific literature under their names; in­

stead, they are called Orin and Allon.

VOLUME 24, NUMBER 3, 2002 71

we find that A.(x) satisfies

A.(x)3 - (p1Q2 + QJP2 + PlQl)A.(x) + qyqz!x + PIP2X = 0. (6. 7)

By successive differentiation with respect to x, followed by x = 1, it follows that

A. ' (l) = (pW2 - qyq2)!D

A."(l) = C -2A.'(l) + 2PTP2 + 4qyq2)1D (6.8)

1 where D = 3 (2 + PJP2 + Q1Q2 - P1Q1),

and so we have

(6.9)

Concluding Remarks

We have shown here that Parrondo's "paradox" operates in two regions. One can win at two losing games by switching between them, but one can also lose by switching between two winning games. The precise fashion in which these oc­cur of course depends upon details of the games involved. Aside from details, the take-home message is that the pro­cedure of averaging strategies to improve the outcome-in

essence allowing each one to rescue the other-is effe,.::tive under a large variety of circumstances. It is certainly taken advantage of by nature and man, although not necessarily in the transparent form of the discussion of equation (5.4).

REFERENCES

[1 ] Doering, C.R. Randomly rattled ratchets, Nuovo Cim. 017 (1 995),

685-697.

[2] Harmer, G.P. , Abbot, D. Losing strategies can win by Parrondo's

paradox, Nature 402 (1 999), 864.

[3] Harmer, G.P. , Abbot, D. Parrondo's paradox. Statistical Science,

14 (1 999), 206-213.

[4] Harmer, G.P . , Abbot, D. , and Taylor, P.G. The paradox of Par­

rondo's games. Proc. R. Soc. Land. A 456 (2000), 247-259.

[5] Klarreich, E. Playing Both Sides, The Sciences (2001 ) , 25-29.

[6] Parrondo, J .M.R. , Harmer, G.P. , Abbot, D. New paradoxical games

based on Brownian ratchets, Phys. Rev. Lett. 85 (2000), 5226-

5229.

[7] Saari, D. Decisions and Elections. Cambridge: Cambridge Unil(er­

sity Press (2001 ) .

[8] Simpson, E .H . The Interpretation of Interaction in Contingency Ta­

bles, J. Roy. Stat. Soc. 813 (1 951 ) , 238-241 .

[9] Takacs, L. Stochastic processes. Methuen's Monographs on Ap­

plied Probability and Statistics (1 960).

Puzzle Solution for Cross-Number Puzzle (24, no. 2, p. 76)

3 1 4 1 5 9 2 6 5 3 1 0 4 0 • 3 3 3 3 • 6 1 • 1 0 3 5 4 8 7 2 0 • • 0 1 • 2 8 9 2 • 7 4 7 • 1 6 1 2 7 8 5 5 • 5 6 3 • 3 7 0 8 • 1 1 • • 0 5 6 8 2 0 0 2 9 • 2 1 • 4 4 0 0 • 1 0 0 6 2 7 1 8 2 8 1 8 2 8

72 THE MATHEMATICAL INTELLIGENCER

l;\§lh§l.lfj .Jet Wi m p , Editor I

Fee/ like writing a review for The

Mathematical Intelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or, if

you would welcome being assigned

a book to review, please write us,

telling us your expertise and your

predilections.

Column Editor's address: Department

of Mathematics, Drexel University,

Philadelphia, PA 1 9 1 04 USA.

Math Talks for Undergraduates by Serge Lang

NEW YORK: SPRINGER-VERLAG 1 999

PAPERBACK 1 1 2 pp.; US $29.95, ISBN: 0387987495

REVIEWED BY STEVEN G. KRANTZ

Ever since the time of Einstein and

Heisenberg, physicists have been

explaining what they do to the public.

Ever since Darwin and Mendel, biolo­

gists have been explaining what they

do to the public. Ever since Lavoisier

and Kelvin, chemists have been ex­

plaining what they do to the public.

Mathematicians are newcomers, vir­

tual tyros, at this game. Many of us

want to tell the layman what we are up

to; but it is not part of our culture to

do so. We do not have the skills, and

often we do not have the patience. We

frequently find ourselves falling back

on the tired old saw of, "Well, it's all

very technical. I really would have

trouble explaining these ideas to an­

other mathematician."

Phooey. Open up the Proceedings of the National Academy of Sciences,

or Physica B, or another journal from

a science other than mathematics.

These folks do not lack for jargon, nor

for technical ideas, nor for obscurity.

It would be just as easy for them to

hide behind the details and arcana of

their subject as it is for us. But they

have trained themselves to formulate

"toy" versions of their problems, to fib

a bit when necessary, and to give the

lay reader an encapsulated notion of

what is going on. Do you think that a

physicist ever really tells a journalist

what is going on with black holes, or

that a geneticist ever really discusses

the delicate issues of gene splicing and

cloning? Of course not. They speak in

vague generalities, and they cheat.

This is something that mathematicians

simply do not know how to do. Put

in other words, the mathematician's

greatest intellectual triumph is also his

Achilles' heel: Hilbert and Bourbaki,

among others, taught us to ply our

craft with precision and with rigor. We

are all trained to do so. And, as a re­

sult, we find ourselves hamstrung by

our own intellectual infrastructure.

We cannot express ourselves in any ar­

got but the most rigorous and most

technical. If we leave out a hypothesis

or a condition or a detail, then we de­

velop cold sweats and insomnia. Pass

the Prozac.

Serge Lang has broken out of this

mold. First of all, he has written a great

many books at all levels. They are

widely admired and universally read.

Second, he has made strenuous efforts

to communicate. One of the most as­

tonishing of these is a collection of

talks (Serge Lang fait des maths en public: 3 debats au Palais de la de­couverte, Paris) that he gave to a broad

and diverse audience in Paris. The

talks were about prime numbers, Dio­

phantine equations, hyperbolic geome­

try, and other advanced topics. Reports

are that he had these Parisians jump­

ing out of their chairs, making conjec­

tures, and arguing the finer points of

advanced mathematics. It is really re­

markable that anyone could do this.

But Serge Lang is a remarkable man.

The book under review is another

instance of Lang's gift. I wish that the

Preface could be written on vellum and

framed in every math department. It

says in part that the author is puzzled

over the dictum that mathematics

should be "a pump, not a filter" (the

reader may know that this is the battle

cry of the reform movement). He asks

whether p-adic L-functions are a pump

or a filter. Then he muses on whether

the Riemann hypothesis is a pump or

a filter. He heaps derision on the con­

cept of "vertical integration" and even

© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002 73

takes to task the NSF's VIGRE pro­gram. 1 I quote:

Are the talks in this book vertically integrated? Or horizontally? Or at what angle? Who's kidding whom? I wish it was a matter of kidding; ac­tually it's a matter of funding.

It is all great fun, and one cannot help but admire Lang's insight and nerve and wit.

But now let us turn to the "math talks" themselves. Were I to give a math talk to undergraduates, I might tell them about the four-color problem, or elementary aspects of Fermat's Last Theorem, or Bertrand's paradox, or ba­sic topology and the Poincare conjec­ture, or perhaps about Ramsey theory. The point is that one wants to tell them about something that they will be able to understand, but also about some­thing serious and relevant. You would never, for instance, catch me telling un­dergraduates--or any audience for that matter-about the EMIRP num­bers (these are integers such that, if you reverse the order of their digits, then you obtain a prime). Lang will have none of this nonsense. He does not stoop in order to communicate with a less sophisticated audience. He tells them about the real stuff:

Talk 1 is about prime numbers. Ob­viously this is a topic that you could ex­plain to your grandmother. But Lang is no wilting flower. On page 4 he intro­duces the 7T function for the distribu­tion of primes, on page 6 he gives a (rather unusual) formulation of the Riemann hypothesis, and on page 9 he introduces the Bateman/Hom conjec­ture. What is amazing is that this ma­terial is all quite accessible. Lang knows just when to wave his hands, just when to fib a little (although he is completely honest, and always says when he is fibbing), and just when to provide some rigorous detail. Perhaps a brief passage from his text will give a feeling for how Lang operates:

. . . The smaller the error term, the better is the approximation of 7T(x)

by the sum. To make everything pre­cise, we should have an estimate for the error term, which makes it as small as possible compared to 7T(x). We shall come to such an estimate in a moment, but first I want to point out that the sum can be rewritten another way, which will use calcu­lus. Skip it if you don't know cal­culus.

You can see that Lang makes his case with confidence, but with the utmost sensitivity to his audience. This is a real art, and one worth mastering.

Talk 2 is about the abc Conjecture. The reader may know that this is a de­lightful and accessible conjecture that implies Fermat's Last Theorem, the Mordell conjecture, and has many other important connections. Because many students will have read of the ex­ploits of Andrew Wiles, this is sure to be a crowd pleaser. Let me stress that Lang does not simply talk about the mathematics. He does mathematics. On page 3 of this discussion he is already proving the Mason-Stothers theorem. The proof is rigorous, and it is non-triv­ial, but it involves only moves that a sophomore could apprehend. Here is another passage from Lang:

Now you should have an irresistible impulse to do something to this equa­tion. What do you do to functions? You take their derivatives. So we get the new equation R' + S' = 0.

Isn't this delightful? Can't you see un­dergraduates perking up and paying at­tention?

Let me stress two interesting fea­tures of these lectures. One is that they are peppered with questions and com­ments from the students. I am sure that Lang selected some of the best ones, but they are nevertheless heartening for their serious tone, and for their insight. A second note is that each of these lec­tures has a worthwhile bibliography of­fering further readings that an under­graduate could actually attempt.

Talk 3 is about global integration of locally integrable vector fields. That is

fairly sophisticated stuff, but Lang in­troduces the topic with aplomb:

The material of this talk stems from Artin's presentation of Cauchy's the­orem, but really belongs in connec­tion with basic real analysis, in a course on calculus of several (actu­ally two) variables. No knowledge of complex analysis is assumed. Some basic knowledge about partial derivatives is taken for granted, but no more than what is in [La87] .

One can easily see the listener (an ap­prehensive undergraduate) being put at ease, and settling back for an edify­ing hour of new mathematics.

Talk 4 is about approximation t�e­orems of analysis. A rather sophisti­cated topic to be sure, and Lang does not shrink at all from the challenge. A

hack would fall prey to showing a lot of computer graphics and waving his hands over the ideas. Lang instead tells the reader of the Dirac mass, of Poisson distributions, convolutions, Fourier series, harmonic functions on the upper half plane, the heat kernel, and theta functions. A real tour de force.

Talk 5 is about Bruhat-Tits spaces. Holy cow! Imagine explaining abstract spaces to an undifferentiated audience of 21-year-olds! But Lang does it with style and grace.

Talk 6 is about harmonic and sym­metric polynomials. It explores such topics as symmetry, positive definite­ness, and eigenfunctions and charac­ters. This is a boffo finish to an inspir­ing collection of lectures. And I say this not only because the lectures will in­spire the students in the audience, but also because they ought to inspire me and my colleagues to go out and give lectures like this.

I must say that, when I first looked at this book, my reaction was, "This stuff is too sophisticated. Maybe he could give talks like this to undergrad­uates at Yale or MIT. But not at any public university, nor any math de­partment below the top ten." But I was dead wrong. First, Lang chose his top-

1 Note that VIGRE is the acronym for "Vertical Integration of Graduate Research and Education." The purpose of the program is to promote the weaving of graduate

education into the ongoing research programs of mathematics departments.

74 THE MATHEMATICAL INTELLIGENCER

ics very shrewdly. Some are quite fancy, but he saw clever passages into accessible insights. That is what he shared with the students. He strictly followed the dictum of "Tell them the truth, tell them nothing but the truth, but for God's sake don't tell them the whole truth." He always made it clear when he was giving a real proof and when he was just telling a parable. In short, what Lang does is a good model of mathematics, pitched at a level that is accessible to the uninitiated.

Lang's other efforts at communica­tion (in book form) include

• Math!: Encounters with High School Students

• Geometry: A High School Course • The Beauty of Doing Mathematics:

Three Public Dialogues

It is astonishing to me that a first-rank mathematician will take the time and ef­fort to have truck with such a wide va­riety of people. And it is not just a mat­ter of writing down the mathematics. One must figure out good topics for one's audience and then figure out how to explain the ideas to people of whose backgrounds one does not really have a clue. Most of us have been so long im­mured in abstract mathematics that we have virtually no idea of the struggles of an undergraduate to understand the most basic notions of logic and rigor. But Lang dives right in; he has no fear.

Of course we have all observed Serge's courage in other contexts. He has taken on the National Institutes of Health (in his battle over a scandal in­volving AIDS Research), the National Academy of Sciences (in a battle over the admission of Samuel P. Hunting­ton, a social scientist), the social sci­ence establishment (in the Ladd!Lipset brouhaha), various Nobel laureates (in the David Baltimore scandal), and many other institutions and sacred cows as well. Lang has no concern for his own well-being, for whether his grant might be in danger (he will not apply for them), nor for any other crea­ture comforts. As he has told us repeatedly, all he cares about is the truth, and for exposing pomposity, stonewalling, intimidation, and the ma­nipulation of power. And we are fortu­nate that he has shared his findings

with the rest of us. It also takes real courage to dive in and learn how to communicate with undergraduates. Af­

ter one person has taken the risk, it is easier for others to follow.

I believe-and this is the main point of the present review-that Lang sets a wonderful example. No matter how sublime is the mathematics that many of us do, I am afraid that we are not very good role models. We are solip­sistic, we are selfish, and we are largely oblivious to the world around us. Of course the competitive nature of our profession virtually requires self-ab­sorption and single-mindedness. But Lang is sufficiently dedicated and en­ergetic that he can transcend the boundaries that limit most of us, and he can demonstrate-by doing-that it is indeed possible to talk about math­ematics with people from all walks of life.

And it is important that we do so. If we care about the visibility of our sub­ject, if we care about the contributions we can make to society, if we care about convincing the politicians who fund us that what we do is worthwhile, then we must attend to these matters. As an example, R. R. Coifman made dozens of trips to Washington to pro­mote wavelets. He spoke to the Direc­tors of the Offices of Navy, and Army, and Airforce Research, but he also spoke to senators and congressmen and military people. He figured out how to communicate. And look how well off the wavelet people are today, what a high priority they have in the funding picture, and how prominent their work has become. Imagine how great life would be if all of mathemat­ics had the same strong profile.

G. H. Hardy reveled in the utter use­lessness of what he did. But he lived in another time, when expectations were more modest. Now mathematics is the driving force in much of science and technology. We do not want to be left on the sidelines: after all, this is our subject, and our bailiwick We should be front and center, leading the pack Therefore we must expand our ability to have a meeting of minds with a broad cross-section of the populace­from politicians to high-school stu­dents to parents to administrators.

While Lang's books may not be per­fect, they have certainly pointed the way and taken the first steps. This is fertile ground that I hope others will plough. I would hope that each of us will take up the challenge and present one of these talks to the local math club. Then we should tum to writing some of our own. Who knows? It might become a movement, and somebody might learn something. We would no longer have to worry about pumps and filters and vertical integration because the ideas behind that jargon would be built into the system. We would have a stronger infrastructure for producing new, young mathematicians at a time when they are much needed. And we could thank Serge Lang for setting the example.

Department of Mathematics

Washington University

St. Louis, MO 631 30-4899

e-mail: sk@math.wustl .edu

A Beautiful M ind: A Biography By Sylvia Nasar

NEW YORK: SIMON AND SCHUSTER. 1 998, 459 pp $25, ISBN: 0-648-81 906-6

A Beautiful M ind FILM, 2001 UNIVERSAL STUDIOS AND DREAMWORKS LLC

The Essential John Nash EDITED B Y HAROLD KUHN AND SYLVIA NASAR

PRINCETON UNIVERSITY PRESS, 2002

ISBN 0-691 -09527-2

REVIEWED BY DAVID GALE

It is now more than seven years since John Nash won the Nobel Prize in

economics, but, thanks to a successful book and now a popular movie, the Nash story seems to be bigger than ever. Indeed, the film about Nash's ca­reer, A Beautiful Mind, has won that other great emblem of Western culture, the Academy Award Oscar for best film of 2001. And it is surely not purely co­incidental that the play Proof, which also deals with mathematics and men-

VOLUME 24, NUMBER 3, 2002 75

tal illness, was the 2001 Pulitzer Prize

winner. It seems mathematics has be­

come box office. What does it all mean?

Of course, the public appeal of this

material derives in large part from the

depiction of mental illness, and, in the

case of the movie, the love story. In a

way this is too bad, because even if these

things had not occurred (and some of

the film events definitely did not), the

scientific story by itself is an interesting

chapter in mathematical history. Per­

haps one of its most unusual features

is that, unlike most important scientific

achievements, Nash's prize-winning

work is accessible. However, a very ca­

sual sampling of some of my colleagues

has led me to believe that probably

most mathematicians have only a

vague idea, if any, of what Nash actu­

ally did. Because I contend that the

whole thing can be understood by any­

one willing to spend the time to learn

a few definitions, let me try to support

this claim by describing the result as

one might present it, say, to a junior

high school class. (For a much more

thorough exposition of this material

and its significance at the level of a

working mathematician, see the first

section of John Milnor's article, "A No­

bel Prize for John Nash," volume 17,

number 3, of this magazine.)

Because he is a mathematician

rather than an economist, Nash did

what mathematicians do. He proved a

theorem. Here it is.

Nash's Nobel-Prize-Winning Theorem:

Every finite n-player game has an equilibrium point in mixed strate­gies.

What does all that mean, and why

does it matter? I will approach the

question indirectly.

First, what is a game? Here are some

examples.

Games children play: Odds and

Evens, Scissors-Paper-Rock, Tic­

Tac-Toe

Games grown-ups play: Chess,

Checkers, Poker, Tennis, Monopoly

Games teams play: Football,

Baseball, Bridge

76 THE MATHEMATICAL INTELLIGENCER

Note that among games I did not list

are hopscotch, tiddlywinks and pitch­

ing horseshoes. The reason is that

while these games require skill they

don't involve making choices, whereas

in Odds and Evens one must make a

decision, whether to throw one or two

fmgers. The key property of a game for

our purposes is that it must involve

such choices, which in game-theory

terminology are called strategies. In

poker one must decide whether to call,

raise, or fold. In baseball the pitcher

must decide whether to throw a curve

or a fastball, and the batter must de­

cide whether to swing or take. In foot­

ball the strategy session is institution­

alized in the huddle.

Formally, a finite game consists of

n players each equipped with a finite

set of strategies. In a play of the game

each player selects one of his/her

strategies and these jointly determine

an outcome. An outcome for the two­

player case might be a winner and a

loser, or for the general case a numer­

ical payoff, positive or negative, to

each of the n players, but it could in­

clude non-numerical rewards like win­

ning an election or capturing a fugitive.

With this degree of generality much of

social behavior fits into the game

framework Thus we have

Games politicians play. Impeaching a

president

Games businessmen play. The Enron

Game

Games countries play: The War on

Terrorism

Game theory as a distinct discipline

started in 1944 with the publication of

The Theory of Games and Economic Behavior by J. von Neumann and 0. Morgenstern; as its title suggests, the

premise was that the game-theory

point of view would be useful in ac­

counting for economic and perhaps

even political phenomena.

DEFINITION. An equilibrium point of a

game is an n-tuple of strategies, one

for each player, with the property that

no player can change his/her strategy

in such a way as to obtain a preferred

outcome, provided the other pleyers

leave their strategies unchanged.

Now in fact most games of any in­

terest don't have equilibrium points.

Thus, in Odds and Evens if you beat me

it's because I chose the "wrong" strat­

egy. Had I thrown two fingers instead

of one I would have beaten you. How­

ever, if we play the game repeatedly we

will both try to switch from one to two

fingers in some hopefully unpre­

dictable way. Similarly, the pitcher

mixes curves and fastballs so that the

batter won't know what to expect. This

is formalized by introducing mixed

strategies, which are simply probabil­

ity combinations of the originally given

"pure" strategies. Thus, the pitcl).er

might throw a curve a third of the time,

the poker player might bluff one time

out of five, etc.

With these definitions in hand the

meaning of Nash's theorem should

now be clear, but to appreciate its im­

plications let us look again at some ex­

amples. For Odds and Evens the mixed

strategy equilibrium is clearly for each

player to throw one or two fingers de­

pending on, say, the toss of a fair coin.

A less trivial example is the following

poker-like game. I toss a fair coin, and

the idea is that you pay me one dollar

if it falls heads, but I pay you one dol­

lar if it falls tails. However, the rules

specify that you don't get to see the

outcome of the toss: I simply report it

to you-but I need not tell the truth.

Further, in case I report heads, you

have an option: you may either pay me

a dollar ("fold"-i.e., accept my report

that you lost), or you may "call," in

which case you get to see the result of

the toss, and you then win or lose two

dollars according to whether I was ly­

ing or not. It seems intuitively clear

that honesty is probably not the best

policy for this game. If I never lie I can

do no better than break even. On the

other hand, if I always lie, you will dis­

cover this and always call, so once

again I win only half the time. I can do

better by mixing strategies. The proper

mixture is given by the equilibrium­

point theorem. (It turns out that I

should lie two-thirds of the time. The

junior high school class can verify this

by finding the equilibrium mixed

strategies for the two players.)

I should hasten to say at this point

that for the special case of win-lose

games (more generally "zero-sum"

games), like the examples above, the

Nash theorem had already been proved

20 years earlier by von Neumann. In

fact this is what motivated Nash to

make his discovery. As this involved

my own brush with history, let me give

a brief eyewitness account. (This is

also described in Sylvia Nasar's book.

The movie version is quite different, as

we shall see a little later.)

In the fall of 1949 I was a Fine In­

structor at Princeton, having just got­

ten my degree under Al Tucker, who

was also Nash's adviser. Tucker was

away on sabbatical at Stanford that se­

mester and asked me to report to him

periodically on Nash's progress. One

memorable morning Nash walked up

to me in Fine Hall and said, "I have a

generalization of von Neumann's min­

max theorem," and he described the re­

sult we have been discussing. It didn't

take long for me to realize that this was

progress with a capital P, so besides

passing the news along to Tucker I per­

suaded Nash to submit his result for

quick publication to the Proceedings of the National Academy of Sciences, which he did. This now historical doc­

ument ran only a little more than a sin­

gle page!

Sylvia Nasar's excellent and very

comprehensive book gives lively and

quite accurate general descriptions of

the prize-winning theorem, as well as

Nash's other major scientific achieve­

ments, on real algebraic manifolds,

isometric embedding of Riemannian

manifolds, and parabolic partial-differ­

ential equations. Unlike the equilib­

rium-point theorem, however, which

one could see was correct right away,

his other considerably deeper results

seem to have emerged by a sequence

of successive approximations. Nash

was eager, not to say persistent, in

managing to talk to the experts on his

problems, while at the same time re­

fusing to read up on the contributions

of others. In each case the people he

consulted first thought that the thing

he was trying to prove couldn't be true,

or later that it might be true but his ap­

proach to the problem would lead

nowhere. Concerning the parabolic

equations project, for example, Lars

H0rmander writes, "He came to see me

several times: 'what did I think of such

and such an equation?' At first his con­

jectures were obviously false. He was

inexperienced in these matters. Nash

did things from scratch without using

standard techniques, he had not the pa­

tience to [study earlier work]"; but then

"after a couple more times he'd come

up with things that were not so ob­

viously wrong," and eventually he

obtained the desired result but by com­

pletely original and non-standard tech­

niques.

The saga of the embedding theorem

is similar. This time the captive audi­

ence was Norman Levinson (for the

real algebraic manifolds it had been

Norman Steenrod). "Week after week

Nash would tum up in Levinson's of­

fice. . . . He would describe to Levin­

son what he had done and Levinson

would show him why it wouldn't

work." Nash nevertheless wrote up and

submitted his result. "The editors of

the Annals of Mathematics hardly

knew what to make of Nash's manu­

script . . . . It hardly had the look of a

mathematics paper. It was thick as a

book, printed by hand rather than

typed." The paper was sent to Herbert

Federer to referee. "The collaboration

between author and referee took

months. . . . Nash did not submit the

revised version of the paper until

nearly the end of the following sum­

mer." The published paper runs 98

pages, longer by a factor of three than

any of Nash's other works, and remains

formidably difficult even for experts in

the field. In a footnote Nash writes, "I

am profoundly indebted to H. Federer

to whom may be traced most of the im­

provement over the first chaotic for­

mulation of this work." I think this was

not uncharacteristic of Nash. Even in

the Proceedings note on the Equilib­

rium-Point Theorem he gives me credit

for suggesting the use of the Kakutani

fixed-point theorem to simplify the

proof. One of the few things that bother

me a bit about the book is the empha­

sis on Nash's arrogance. He was tact-

less, blunt, very ambitious, certainly,

but I don't think he had a swelled head.

(The film, which I will come to shortly,

makes an even bigger point of Nash's

arrogance.) For people interested

mainly in Nash's scientific accom­

plishments, I strongly recommend the

second book above, The Essential John Nash edited by Nasar and Harold

Kuhn, which reproduces his major pa­

pers along with some further bio­

graphical material, plus illuminating

commentaries by the two editors and

by Nash himself.

Of course, Nasar is not writing his­

tory of mathematics but rather, as the

title makes clear, a study of a person

with an exceptional mind, exception­

ally penetrating at first and later on

exceptionally disturbed. Finally she

describes how Nash once again does

the totally unexpected thing by re­

covering from a mental illness that

was thought to be essentially incur­

able. As a reporter, Nasar has gotten

hold of a wonderful story,. the "Phan­

tom of Fine Hall" who after 30 years

of barely hanging on, ends up winning

the Nobel Prize, and she tells it well.

The John Nash who emerges from

these pages is, in his rational mo­

ments, at once intriguing and exas­

perating. In describing the time after

his mental problems begin-two of

the five sections of the book are con­

cerned with this period-Nasar es­

sentially lets her subject tell his own

story, using quotations from some of

the many letters written to friends and

colleagues, often from abroad. These

were typically written in ink of three

or four colors, and were a bizarre

combination of art, poetry, mathe­

matics, and politics, often conveying

a strange ironic humor. Later, back at

Princeton, Nash could be found "print­

ing painstakingly on one of the nu­

merous blackboards that lined the

subterranean corridors linking Jadwin

and New Fine":

Mau Tse-Tung's Bar Mitzvah was 13

years, 13 months and 13 days after

Brezhnev's circumcision.

Can Hironaka resolve this singularity?

VOLUME 24, NUMBER 3, 2002 77

It is painful to imagine what it must

have been like to be tormented over

those many years by such delusional

aberrations.

There is of course much more to the

book than what I have mentioned. Con­

siderable space is given to Nash's rela­

tionships, with men and women, espe­

cially with his wife Alicia (this is no

doubt what recommended the story to

Hollywood). Finally, in what is perhaps

the most exciting chapter, Nasar. de­

scribes the rather wild last-minute fight

in the Swedish Academy over whether

Nash should receive the prize at all.

In giving the above sampling of the

book's contents I emphasized what I

suppose to be of most interest to pro­

fessional mathematicians, namely, the

process of mathematical creation. You

won't learn much about that sort of

thing from the film. I knew before see­

ing it that there was no point in wor­

rying about scientific or historical ac­

curacy, and I was prepared to judge the

film on its own terms, how well does

it succeed in what it's trying to do. I

can't resist, however, describing the

treatment of the "eureka" moment,

when Nash discovers the prize-winning

theorem.

The screenwriters have invented a

beer tavern near the campus patron­

ized by grad students and alluring un­

attached women. Nash, thinking out

loud to his companions, muses that if they all go after the blonde there can

be only one winner and the losers will

be rejected by the other girls who will

resent being second choice, whereas,

if they pay attention to the other girls.

. . . I wasn't clear about the exact rec­

ommended optimal strategy, but the

payoff was quite explicit: "everyone

gets laid." From there, in a leap of the

imagination, the whole thing suddenly

becomes clear, "Adam Smith was

78 THE MATHEMATICAL INTELLIGENCER

wrong!" and economics would never

be the same.

On its own terms, then, what is A Beautiful Mind trying to do? Obviously,

it is trying to grab the audience, and I

found it a pretty good audience-grabber.

A large part of the credit for this goes

to lead actor, Russell Crowe, who

makes us laugh or cry or shake with

fright in all the right places. The film contains many familiar ingredients.

There is suspense (will the baby drown

in the bathtub?), violence (the obliga­

tory car chase; since it didn't really hap­

pen Nash has to hallucinate it), and of

course romance. There is also the por­

trayal of academic life, which I suppose

audiences will find interesting, though

much of it will seem rather hilarious to

people who have been there, as for ex­

ample the professorial pep talk in the

film's opening scene, and the tedious

"ceremony of the pens"-which isn't

even good "Hollywood."

As the film drew to a close (I

watched it a second time for purposes

of writing this review), I became in­

creasingly aware of how each profes­

sion has its own rules and objectives

and outlook on the world, its own idea

of what the game is all about. Perhaps

I should extend my earlier list to include

Games mathematicians play: Proving

theorems, e.g., the Equilibrium-Point Theorem Games film makers play. Grabbing

audiences, e.g., A Beautiful Mind

For both these endeavors, the payoff

turns out to be a prize.

But the most striking aspect of the

John Nash story, to me, is not his

quirky personality, nor his Odyssey

from illness to recovery, nor his win­

ning of a prestigious prize, nor even his

mathematical achievements, impres-

sive though they were. Rather it is t,hat

in a competitive and very active pro­

fession he did things nobody else

would have attempted, using methods

no one else had ever thought of. The

recurrent words in the Nasar book are

innovative, original, unexpected.

By contrast, the film is for the most

part fairly predictable. Its one off-beat

attempt is in the handling of the hallu­

cinations. As one unhappy critic put it,

the picture "pulls a flagrant scam:

whole characters and episodes are pre­

sented as urgently authentic only to be

revealed as figments of a cracked imag­

ination." On the other hand, some view­

ers apparently didn't mind being suck­

ered, and find this to be one of the

film's most compelling features.

To conclude, let me return to my orig­

inal question: what are we to make of

this most unexpected interaction be­

tween mathematics and the entertain­

ment industry? Some people have spo­

ken scornfully about A Beautiful Mind because of its biographical inaccuracies

and mathematical misrepresentations.

In one respect, however, I think the film has done the subject a service by por­

traying mathematics not only as a seri­

ous and important enterprise, but also

as an exciting one in which new and

quite surprising discoveries are often

made. This is a refreshing break from the

usual stereotype of mathematicians as

strange characters who spend their lives

thinking about numbers. If it takes Os­

cars and Pulitzer Prizes to get this point

across, let's not complain. In this case it

seems what's good for Hollywood is

good for mathematics.

Department of Mathematics

University of California

Berkeley, CA 94720-0001

USA

e-mail: gale@math.berkeley.edu

Ki£B,j.k$·h•i§i R o b i n Wilson I

Geometry of Space

' IV CENTENARJO DE sAO PAULO 1,,.. 1954

Sculpture "Continuity"

Please send all submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes,

MK7 6M, England

e-mail: r.j.wilson@open.ac.uk

The geometry of space can take

many forms-in the symmetry of

bridges, in the design of buildings, or

in the sculptures that decorate our

towns and cities.

Mobius strip

Sculpture "Expansion"

A celebrated geometrical object is

the Mobius strip, named after the Ger­

man astronomer and mathematician

August Ferdinand Mobius in 1858 (al­

though flrst discovered a few months

earlier by Johann Benedict Listing). It has only one side and one edge.

An attractive three-dimensional

sculpture in the form of a Mobius strip,

"Continuity," by the Swiss architect Max

Bill, can be seen in front of the Deutsche

Bank in Frankfurt. It was carved from a

single piece of granite weighing 80 tonnes.

Another spectacular object, this time

an enormous helix, is the Brazilian sculp­

ture "Expansion," symbolizing progress.

A "ruled surface" is a curved surface

constructed from closely packed straight

lines; one surface that can be made this

way is the hyperbolic paraboloid. There

are several famous buildings using ruled

surfaces: here is one, the German pavil­ion for the 1967 World's Fair in Montreal.

1 0 0

German pavilion

0 z < ..J J: u

© 2002 SPRINGER· VERLAG NEW YORK. VOLUME 24. NUMBER 3, 2002 79