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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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: [email protected]
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.
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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 : [email protected]
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: [email protected] 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: [email protected]
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: [email protected]
Istvan Hargittai
Budapest University of Technology and Eco
nomics
H-1 521 Budapest, Hungary
e-mail: [email protected]
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: [email protected]
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
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• 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
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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: [email protected]
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
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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: [email protected] .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: [email protected]
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: [email protected]
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