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.
p
structions
(standard) definition [2], i.e., to the ex
tained in the Article.The Author will
istence
Cusanus 83, 47v-48r.
of
a
C1-submanifold
and/or
warnings
con
indemnify Springer-Verlag against
chart
any costs, expenses or damages that
around each point. Accordingly, the following constructions and assertions
Springer-Verlag may incur or for
remain valid, but should be restricted
which Springer-Verlag may become
to near standard points.
liable as a result of any breach of
K.
these warranties. These representa
Stroyan for erroneously citing him for
tions and warranties may be ex
Also,
[7] R. Llull (1 299): "De arte eleccionis." Codex
followed in accordance with th� in
EM (as given in [3]), one can indeed
show the equivalence to the usual
I
hereby
apologise
to
tended to third parties by Springer
the equivalence proof: in [3], he shows
[8] M. Gardner: Logic Machines and Diagrams,
that a C1-submanifold in the sense of
Verlag.
Second Edition. Harvester Press, Brighton,
the (corrected!) nonclassical definition
-Copyright Transfer Statement
1 983.
is an
abstract C1-manifold1, i.e., that it
carries a C1-atlas of charts (not sub
by the good offices of
manifold charts). However, some ad
Friedrich Pukelsheim
ditional remarks he gives in his paper
lnst. fOr Mathematik, Univ. Augsburg
strongly suggest the validity of the the
D-86135 Augsburg, Germany
orem that I have just announced-and
e-mail:
[email protected]
that, surprisingly, I was not able to fmd in the literature.
Errata: The Surfaces Capable of
Finally: I should have mentioned
Claims continue to mount in the case of the Haas "Cross-number puzzle," published without adequate warning notices in the Mathematical Intelli gencer (vol. 24, no. 2, p. 76). In ap pearance just a crossword puzzle with numbers, this puzzle has turned out to be an extreme mental and physical
Division into Infinitesimal
that the surface graphics in [1] were
health hazard. Hundreds of complaints
Squares by Their Curves of
produced using the computer algebra
have arrived from around the world, of
Curvature
system Mathematica.
Consider the following example: let
M
:=
(eC l +i)tl t E IRI) c C
�
IRI2 be the
headache, blurred
neck strain, back strain,
vision,
dizziness,
insomnia,
REFERENCES
nightmares, and inability to concen
logarithmic spiral. This is a perfectly
1 . U. Hertrich-Jeromin: The surfaces capable
trate, following an attempt to solve it.
good C1-submanifold (well, it is even
of division into infinitesimal squares by their
Several injuries were reported from
curves of curvature: A nonstandard analysis
readers distracted by thinking about it
approach to classical differential geometry;
while driving or operating heavy equip
Math. lntelligencer 22 (2000), no. 2, 54-61 .
ment.Numerous others ask unemploy
!R2: it carries an atlas of C1-sub manifold charts, i.e., to every point p E M there is a neighbourhood U C !RI2 of p and a diffeomorphism
=
(IR
x {0)). On the other hand,
E IRI2, the orthog onal projection 'Trp : M � Tp cannot be
taking p
E M,
p
=
0
2. U. Hertrich-Jeromin: A nonstandard analy sis characterization of standard submani
3. K. Stroyan: Infinitesimal analysis of curves and surfaces; in J. Barwise, Handbook of
neighbourhood about 0
Mathematical Logic, North-Holland, Am
around 0 infinitely often. (Note that
sterdam 1 977.
such p is not near standard in M: since
0 $ M there is no standard point Po E M with P = Po·)
p
=
Consequently, the definition I give
Springer-Verlag is
folds in Euclidean space; Balkan J. Geom. App/. 6 (2001 ), 1 5-22.
an infinitesimal bijection as, in any
E IRI2, M spirals
ment compensation after being fired for doing it on the job. 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
Udo Hertrich-Jeromin
may have permanently deranged. He is
Department of Mathematics
now confined to an institution. "My God,
TU Berlin
in [1] for a C1-submanifold (and, in
D-1 0623 Berlin
consequence, also the one for a smooth
Germany
submanifold) is "wrong": it cannot be
e-mail:
[email protected]
you can't argue around him, and his new
mathemati cian," said his psychiatrist Shrinkovsky,
ideas never stop, he's a
who himself has filed a third-party
shown equivalent to the usual defini
claim,
tion. In fact, it is the (standard differ
counseling costs for himself as he
ential geometry) argument I give just before the definition that is wrong
Indemnification
citing lost clientele and the
struggles to treat his patient.
The Author represents and warrants
the argument only applies to choices of
... that, to the best of the Author's
standard coordinate systems. Thus, re
knowledge, no formula, procedure,
1 081 Carver Road
quiring (a)-(c) of the definition to hold
or prescription contained in the Ar
Cleveland Heights, OH 441 1 2
only for all near standard (in M) points
ticle would cause injury if used or
USA
1And not the converse, which is certainly wrong, as the example of a a-shaped curve shows.
4
THE MATHEMATICAL INTELLIGENCER
Robert Haas
[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
p
EM (as given in [3]), one can indeed
followed in accordance with th� in
show the equivalence to the usual
structions
(standard) definition [2], i.e., to the ex
tained in the Article.The Author will
istence
of
a
C1-submanifold
chart
and/or
warnings
con
indemnify Springer-Verlag against any costs, expenses or damages that
personarum." Codex Vaticanus Latinus
around each point. Accordingly, the
9332, 1 1 r- 1 2v.
following constructions and assertions
Springer-Verlag may incur or for
remain valid, but should be restricted
which Springer-Verlag may become
to near standard points.
liable as a result of any breach of
[6] R. Llull (about 1 283): "En qual manera Natanne fo eleta a abadessa. " Codex His panicus 67, 32v-34r.
K.
these warranties. These representa
Stroyan for erroneously citing him for
tions and warranties may be ex
Also,
[7] R. Llull (1 299): "De arte eleccionis." Codex Cusanus 83, 47v-48r.
I
hereby
apologise
to
the equivalence proof: in [3], he shows
tended to third parties by Springer
[8] M. Gardner: Logic Machines and Diagrams,
that a C1-submanifold in the sense of
Verlag.
Second Edition. Harvester Press, Brighton,
the (corrected!) nonclassical definition
-Copyright Transfer Statement
1 983.
is an abstract C1-manifold1, i.e., that it carries a C1-atlas of charts (not sub
by the good offices of
manifold charts). However, some ad
Friedrich Pukelsheim
ditional remarks he gives in his paper
lnst. fOr Mathematik, Univ. Augsburg
strongly suggest the validity of the the
D-86135 Augsburg, Germany
orem that I have just announced-and
e-mail:
[email protected]
that, surprisingly, I was not able to fmd in the literature.
Errata: The Surfaces Capable of
Finally: I should have mentioned
Claims continue to mount in the case of the Haas "Cross-number puzzle," published without adequate warning notices in the Mathematical Intelli gencer (vol. 24, no. 2, p. 76). In ap pearance just a crossword puzzle with numbers, this puzzle has turned out to be an extreme mental and physical
Division into Infinitesimal
that the surface graphics in [ 1] were
health hazard. Hundreds of complaints
Squares by Their Curves of
produced using the computer algebra
have arrived from around the world, of
Curvature
system Mathematica.
Consider the following example: let
M
:=
(eC l +i)tl t E IRI) c C
�
IRI2 be the
headache, neck strain, back strain, blurred vision,
dizziness,
insomnia,
REFERENCES
nightmares, and inability to concen
logarithmic spiral. This is a perfectly
1 . U. Hertrich-Jeromin: The surfaces capable
trate, following an attempt to solve it.
good C1-submanifold (well, it is even
of division into infinitesimal squares by their
Several injuries were reported from
curves of curvature: A nonstandard analysis
readers distracted by thinking about it
C"') of !R2: it carries an atlas of C1-sub
manifold charts, i.e., to every point E M there is a neighbourhood U C !RI2 of p and a diffeomorphism
p
=
(IR
x {0)). On the other hand,
taking p E M, p onal projection
=
0 E IRI2, the orthog
'Trp : M
�
Tp cannot be
an infinitesimal bijection as, in any
approach to classical differential geometry;
while driving or operating heavy equip
Math. lntelligencer 22 (2000), no. 2, 54-61 .
ment.Numerous others ask unemploy
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
ment 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.
neighbourhood about 0 E IRI2, M spirals
Mathematical Logic, North-Holland, Am
around 0 infinitely often. (Note that
sterdam 1 977.
of Thomas Chadbury, a promising
Udo Hertrich-Jeromin
may have permanently deranged. He is
Department of Mathematics
now confined to an institution. "My God,
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
young mathematician whom the puzzle
TU Berlin D-1 0623 Berlin
consequence, also the one for a smooth
Germany
submanifold) is "wrong": it cannot be
e-mail:
[email protected]
shown equivalent to the usual defini tion. In fact, it is the (standard differ ential geometry) argument I give just before the definition that is wrong
The most tragic case to date is that
you can't argue around him, and his new
ideas never stop, he's a mathemati
cian," said his psychiatrist Shrinkovsky,
who himself has filed a third-party claim, citing lost clientele and the
Indemnification
counseling costs for himself as he struggles to treat his patient.
The Author represents and warrants
the argument only applies to choices of
... that, to the best of the Author's
standard coordinate systems. Thus, re
knowledge, no formula, procedure,
1 081 Carver Road
quiring (a)-(c) of the definition to hold
or prescription contained in the Ar
Cleveland Heights, OH 441 1 2
only for all near standard (in M) points
ticle would cause injury if used or
USA
1And not the converse, which is certainly wrong, as the example of a a-shaped curve shows.
4
THE MATHEMATICAL INTELLIGENCER
Robert Haas
«·)·"I"·' I I
Publishing Report
I
always had a hankering to be an en trepreneur. It was suppressed all the
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
tions
of
my
Harmonic
Analysis
years of academic work, but came out
(which I got back from Brooks/Cole),
about ten years ago, before I retired. I
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
had written three books and they were published, but the publishers didn't seem as excited about them as I was,
Henry Helson
course of time, I brought out new edi
and the books were undoubtedly in their last stage of life (as I was also!).
Bergman, both colleagues at Berkeley.
I didn't want them to go out of print.
My last book, Calculus and Probabil
Furthermore, I had a new manuscript
ity, has sold some copies but has not
and only a half-hearted, unenthusiastic
yet been adopted anywhere. Mean
offer from a publisher.
while I arranged with the Hindustan
Then I took matters into my own
Book Agency of New Delhi to sell their
hands. This is the history, so far, of my
book Basic Ergodic Theory by M. G.
enterprise. I offer it as information
Nadkami outside of Asia; and Hindus
about the economics of textbooks,
tan has reprinted some of my books for
something which concerns all of us in
sale in Asia. I am writing a monograph
the teaching profession. I feel this has
that I hope to publish later this year.
interest, because textbook publishing
The teaching world didn't beat a
is an opaque industry. The real pub
path to my garage, but I've shown a
lishers don't tell us much, even though
profit to the IRS every year since the
we are their only customers.
beginning in 1992. I think I am a pub
I decided to publish my new book myself. I had a not-very-modem com
lisher. I understand better than I did
how the business works, and why it
puter, and a 300-dot laser printer,
doesn't work better. The comments
which was obsolete even then, but
that follow are my serious opinions,
are exclusively those of the author,
which gave beautiful pages (and still
but I emphasize that they are based on
and neither the publisher nor the
does). Also I was proficient in EXP, the
my own experience and not on statis
editor-in-chief endorses or accepts
wysiwyg program that, unaccountably
tical evidence.
and opinions expressed here, however,
responsibility for them. An (}pinion
and unfortunately, seems to have lost
I don't understand how bookstores
out to TEX. I've always done my own
can stay in business. I set a "list price"
should be submitted to the editor-in
typing, due largely to terrible hand
and bill resellers with a discount of
chief, Chandler Davis.
writing. First I got in touch with Gilbert
200/o. I think this is normal. That means
Strang of MIT, who was already a pub
the bookstore has a markup of 25% (un
lisher. I got good advice, and I trea
less it charges more than list price,
sured his encouragement.
which is difficult because that price is
Next I produced a clean printout of
quoted in public databases, such as
my book Honors Calculus, and looked
Amazon.com's catalogue). Out of that
for "Printers" in the Yellow Pages. In a
markup, the store pays for delivery,
short while I had a big stack of books
and transportation back again if the
in my garage.
book isn't sold. Unless the order is very
Holden-Day had published my Lin ear Algebra but was going out of busi
large, UPS will get around 100/o of the
ness. The president, Fred Murphy, had
(if the first order wasn't large enough)
price of the book each way. Reorders
been a friend since the days when he
are costlier, because the number of
traveled on behalf of Addison-Wesley.
books is small. The order and the re
He gave me back my rights to the book,
order come by telephone, which isn't
the old copies at his cost of production,
free. Books get damaged, stolen, lost;
and many rolls of transparent tape,
invoices are misplaced. Somebody has
which I still use for mailing. That was
to work on every snafu. I don't know
a big push; now I had two titles. In the
how the bookstores manage. Please
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002
5
don't entertain unkind thoughts about
had bought some used copies for more
chine. Printing two thousand copi�s of
yours, unless they don't pay their pub
than my list price, and marked the rest
a book costs surprisingly little more
accordingly! They were decent and ac
than a hundred. (The process used is
lishers. Actually (unlike other publishers) I
tually gave students refunds.
different. ) The work of preparing a
include the shipping cost in my list
I like selling one or two books at a
book for publication is the same re
price. I use the US Postal Service,
time to libraries and individuals. The
gardless of the number of copies to be
which is much cheaper than alterna
price is full list, they always send a
printed; this makes more advanced
tives. Bookstores prefer UPS because
check promptly, and sometimes people
texts expensive. The editor who comes
the shipment is tracked. That avoids
tell me they like my books. My ad
to a booth at a meeting (and his hotel
the problem of accountability if the
vanced books mostly go out this way.
bill) costs the company a bundle. The
shipment doesn't arrive. But my expe
But that business is too small to be re
representative who calls at our offices,
rience with the postal service is excel
ally profitable. I would like my local in
with no purpose except to be nice and
lent. The only problem, for me, is wait
dependent bookstore to stock my ti
offer complimentary copies of relevant
The postal
tles, but they won't; they want a 400/o
texts, does too (but I haven't seen one
service claims to have modernized its
discount on list. Nevertheless they buy
in recent years). All the complimentary
handling of mail, and I believe it, but
and stock used copies of my books,
copies are expensive to mail. Thus the
the local PO is still terribly obsolete. I
which they offer more expensively
overhead is high, but if a book does sell
think Congress is to blame. There are
than I do new ones, and which are sold
a lot of copies, it is very profitable indeed.
ing in line for service.
hundreds of postal rates, for the bene
quickly. This tempts me to leave a note
fit of various political interests, with no
with my price and telephone number
I don't have these expenses, and I
relation to the service rendered. If
in my books, but I haven't done so yet.
do everything myself except the actual
rates depended not on the content of
I have had only three orders that
printing, so I can set my prices much
packages but only on the service re quired, 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, be
Instructors do not take the choice of texts seriously.
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 (al though I have tried); a single small in sertion in the
cal
Monthly
American Mathemati costs
hundreds
of
cause people are still fearful of a for
were never paid for. One bookstore
dollars, and I can't even think of the
mer Berkeley mathematician-turned
went out of business, after telling lies
Notices of the AMS.
terrorist who is not dangerous any
over the telephone for months. Two
letters doesn't work well; I don't know why. Libraries will only buy a title if it
Mailing individual
more; instead I have to waste gas and
European distributors have just never
time going to the PO. That is no way to
paid, in spite of repeated requests. One
is in a series by a big publisher, or if
run the postal business.
other, in Paris, did fmally pay, after I
someone asks for it, so there is no
How do used books get recircu
lated?
An
individual store buys the
came by in person. I think they were
point in writing to librarians. I send out
surprised by my visit!
dozens of "examination copies," and
used book back, but that store is un
For foreigners, paying is a problem.
probably that is doing some good, be
likely to sell it again. There is a busi
In spite of what we hear about inter
cause I get orders out of the blue from
ness of getting used books to the
national capital flows, my bank won't
people who must have seen the book
places
currently
accept payment from anyone except
somewhere.
adopted; I don't know how it works,
another American bank, or else a wire
Are textbooks too expensive? Yes,
but I am surprised if anyone can make
transfer that is too expensive. So a for eign bookseller has to maintain an ac
if we assume you would rather not pay
for the excess length of the modem cal
stored, to wait for the course that
count in the United States. Within Eu
culus book, or its too generous mar
needs it. If nobody does, it is a total
rope it is messy too, although that does
gins, or pretty colors. I paid $2.90 for
where
they
are
a profit. The book will have to be
loss. If somebody does, there seem to
not affect me. In spite of the allegedly
Osgood's calculus (still worth reading)
be two UPS charges to cover. Assum
common currency, if you deposit a
in 1943. If you apply an inflation factor
ing the student who sold it got back
check in euros in one euro country
of 10, it should cost about $30 now. It
half the list price of the new book, and
drawn on a bank in another, it is still a
would cost more than that, but not
it is then resold for three-quarters of
foreign-exchange transaction. For some
much more, and the quality of printing
list price, I don't see how there is any
reason, banks in all countries see no
is very much better now, even leaving
worthwhile margin of profit.
reason to simplify things.
colors and margins aside. But a mod
Linear Alge
As in other parts of our economy,
em calculus text costs another $40
I got back some unsold copies
the cost of production in the publish
more yet, and the added cost is largely
with a price sticker that was higher
ing business is determined by the
waste. The publisher wants to be sure
than my list price. I complained to the
(high) cost of human time and the
no topic is omitted that any potential
bookstore and was told, sorry, but they
(low) cost of manufacturing by rna-
user could want, and therefore the text
A university used my
bra.
6
THE MATHEMATICAL INTELLIGENCER
is twice as long as Osgood was, and
ask seriously for texts that are appro
stand told imprecisely in six para
most of the book will never be read.
priate for their students.
graphs than told carefully in one. Fur
Then there is the froth: the colors, the
I have been disappointed to see how
thermore, not all true statements are of
wide margins, that are supposed to
instructors at my university and else
the same importance. The text should
please students. I don't think students
where do not take the choice of texts
direct the student to what is most im
are pleased, but their instructors seem
seriously. The ones they choose have
portant, and leave inessential details to
to be, because they choose these mon
for their only virtue that they will be
be filled in by the lecturer, or presented in problems.
strosities all the time. The blame lies
easy to teach from, because they will
with us, the faculty who adopt text
not arouse anxiety in their students.
If we want good textbooks, first we
books and don't give a thought to what
We complain about how little respect
have to write them. The calculus man
the book will cost. The publishers just
students have for our subject, but we
give us what we want, and a little more.
require them to study texts that con
uscripts I get to review suggest that
writers, like publishers, want to hit that
descend to them and offer them noth
jackpot, and are not trying to write
I
sunnise
that
publishers
lose
money on many of the elementary
ing meriting respect.
carefully to a narrower target. If we ask
books they publish. They are expensive
Actually my publishing venture is
for good books, publishers will do their part in providing them. Then we fac
to print and expensive to transport,
not entirely the result of a passion for
and a lot will have few adoptions. They
entrepreneurial activity. If it were, I
ulty need the courage to choose ones
will be gone in a couple of years. More
would be a good deal richer. Like many
that are right for our students, and
of the same continue to appear be
others who think that universities are
learn to teach from them. Students
cause every publisher is looking for the
for teaching (as well as research), I felt
won't like it; and with our promotions
new Thomas. Few find him.
challenged by the crisis in the teaching
dependent
This is reminiscent of the automo
of mathematics and wanted to try to do
forms, we've got a problem. Should we
on
student
evaluation
bile business a few decades ago. Every
something about it. The best way I
face it, or just keep on moaning about
American producer wanted to hit the
could think of was to write texts that
how hard it is to teach mathematics?
center of the market with a product
incorporate my ideas for teaching in
that everybody would like. The result
one of those segments just mentioned.
was products that did not fit the needs
Since there is no present market for
of a lot of people. Then foreigners in
those ideas, I had to publish the books
vaded the American market with cars
myself.
each addressed to some particular seg
Naturally, students should get all
ment of the market. There were small
the help we can give them: competent
cheap cars of different kinds, and big
lecturing to begin with, and then office
expensive ones, and each was appre
hours, review sessions, math clubs,
ciated by the people for whom it was
and especially other students to talk to.
intended. Pretty soon there was not
After that, the student has some re
much left in the middle for the mass
sponsibility. There is no way to elimi
marketers.
nate the lonely job of making sense out
The textbook field is ripe for a sim
of lecture and text. Finally a student
ilar development, although it is not
has to come to terms with the subject.
likely to be brought about by competi
Then all the reassuring, chatty digres
tion from abroad. Our educational in
sions that pad these thick books are
15 The Crescent
stitutions and the students in them are
just confusing. At the moment of actu
Berkeley, CA 94708
varied, but our publishers continue to
ally learning something, it is important
churn out cloned copies of old calcu
to have a text that tells it like it is, with
HENRY HELSON
USA
e-mail:
[email protected]
lus texts. They are not that different
out pretending that learning is easy,
from the first Granville that I learned
without a mass of irrelevant story
Henry Helson, beginning with his stu
from. Certainly they do not serve the
telling, without fake applications, and
dent years at Harvard, has had a long
diverse student bodies that buy them.
above all without assuming that the
career in harmonic analysis, from the
The situation can't improve until pub
student-reader is an idiot.
classical to the
lishers give up the idea of the all-pur
To be clear, a text should be as sim
functional-analytic.
Most of it has been spent at the Uni
pose text, and try to serve well the sev
ple and brief as possible. It simply is
versity of California Berkeley, where
eral smaller markets that exist now.
not true, for the students I have in
he is now Emeritus Professor.
They will not do that until instructors
mind, that an idea is easier to under-
VOLUME 24. NUMBER 3, 2002
7
MANUEL RITORE AN D 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
found. We will not treat either some recent interesting ad
vances in the study of isoperimetric domains in surfaces.
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
The Classical lsoperimetric Problem in IR3
�3 enclosing a fixed V > 0, the ones with the least area. From general
We wish to fmd, among the surfaces in volume
implies the isoperimetric property of the circle by a stan
results of Geometric Measure Theory [15], this problem has
dard approximation argument. Since then many proofs and
at least a smooth compact solution. Moreover, from varia
partial proofs have been given. Among the many mathe
tion formulae for area and volume, the mean curvature of
maticians who have considered these problems are Euler,
such a surface must be constant. The mean curvature at a
the Bemoullis, Gauss, Steiner, Weierstrass, Schwarz, Levy,
point of the surface is the arithmetic mean of the principal
and Schmidt, among others.
curvatures, which indicates how the surface is bent in
Nowadays by an isoperimetric problem we mean one in
space. It is not difficult to show, from the second variation
which we try to find a perimeter-minimizing surface (or hy
formula for the area, keeping constant the volume en
persurface) under one or more volume constraints and with
closed, that the solution surface (and hence the enclosed
possibly additional boundary and symmetry conditions.
domain) has to be connected.
Thanks to the development of Geometric Measure Theory
There are several ways to prove that the sphere is the
in the past century (see, for instance, the text [15] and the
only solution to this problem. Perhaps the most geometri
references therein) we have existence and regularity re
cal ones are the various symmetrization methods due to
sults for most of the "natural" isoperimetric problems we
Steiner and Schwarz [4] and Hsiang [14]. Let us explain
can think about. By
regularity we mean that the solution
of the problem either is a smooth surface, or has well
briefly their arguments. Consider an isoperimetric body
P, and, for every line L in this fam L n 0 by the segment in L centered at P n L of the same length. This procedure yields another body 0'
understood singularities, as in the double-bubble problem,
gonal to a given plane
which we spotlight below.
ily, replaces
We will describe how to seek the solutions of some
isoperimetric problems in the Euclidean space
�3, including
with the same volume as
n, and strictly less boundary area n was symmetric about a plane
the double-bubble problem. For other ambient manifolds
unless the original body
such as n-dimensional spheres or hyperbolic spaces, we re
parallel to
fer to the reader to Burago and Zalgaller' s treatise [4] on geo
ric about a plane parallel to
metric inequalities, where an extensive bibliography can be
0.
Steiner's method applies to the family of lines ortho
P. This implies that 0 must have been symmet P. Schwarz considers a given line L. For every plane P or-
© 2002 SPRINGER�VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002
9
Pn 11 is replaced by the disc P centered at Pn Lof the same area. Again a new body
thogonal to L, the intersection in
R
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 consid
n into two equal volume parts n+ 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 ered a plane p dividing and
curvature surfaces, and, by construction, they coincide in
an+. By general properties of constant mean curvature sur faces we conclude an = an', and son= 11', which means that 11 was symmetric with respect to P.
Figure 1 . lsoperimetric domains in a region R.
given
R it is certainly difficult to characterize the isoperi
metric 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
By applying Steiner or Hsiang symmetrization, it follows
that
n is symmetric with respect to a plane parallel to any 11 is symmetric
given one; by Schwarz symmetrization, that
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.
R.
The lsoperimetric problem In a halfspace
R is the halfspace z 2: 0. We will find � that separate a region 11 c R of fixed vol ume with the least perimeter. Because R is noncompact,
Let us assume that the surfaces
the existence of isoperimetric domains requires proof, as
Hence a symmetrization method suffices to character
a minimizing sequence could diverge, but this is solved by
�3. This
using translations.So we have existence and also regular
ize the isoperimetric domains in Euclidean space
is due to the large group of isometries of this space. We
will see other situations where this is not enough to char acterize the isoperimetric domains. There is also a symmetrization method for embedded
ity, 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).
constant mean curvature surfaces, known as the Alexan
For the proof of this theorem we first observe that the
drov reflection method [22], which shows that such a sur
11 must touch the plane z = 0. Other 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
face 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 clas sical 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
R. That is, region 11 is bounded by � and per 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 ).
isoperimetric region
wise, moving
=
in the boundary. We now apply Hsiang symmetrization, but only for planes orthogonal to z =
0, to conclude that n is Lorthogonal to z = 0.
rotationally symmetric about a line Hence
� is obtained by rotating a plane curve to get a
constant mean curvature surface. It turns out that there are
boundary of
only a few types of curves that produce, when rotated, a
haps by a piece of
constant mean curvature surface. They were studied by Ch.
Geometric Measure Theory [15] ensures the existence
� 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
Delaunay in 1841 [6], and they are depicted in Figure 3. Since our curve touches the line of revolution (it has a max imum of the z-coordinate), looking at the list, we conclude that it is part of a circle, and so
of the solution
a� orthogonally. When R is strictly convex the surface � is connected,
at
and bounds on its genus and on the number of components of a� are known [21]. For a
10
L
The lsoperimetric problem In a ball
---, �--------/. Q '· Let us now assume that
··
R is a ball. Spherical sym-
·-IJ
�:..t _ :, � :� ---'---· 8 ··
"-------------------------' ·
Figure 2. lsoperimetric domains in a halfspace.
THE MATHEMATICAL INTELLIGENCER
� is a halfsphere.
metrization proves
that
isoperimetric face
an sur-
� is a surface of
revolution around some
line
L containing the center of the ball. As
L'
L Figure 4. There are candidates to be isoperimetric domains in a ball Figure 3. Generating curves of surfaces of revolution with constant
which are not spheres nor flat discs.
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.
may complicate the problem by imposing this symmetry. The following problem is still open. PROBLEM.
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 bound ary 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 -
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 do main 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 thefaces 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
Figure 8. The standard double bubble.
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 isoperi metric 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
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-perim eter way to enclose and separate two given volumes in !R3.
Figure 9. Double bubbles. The one on the right was shown to be un Figure 7. lsoperimetric problems in a slab. The one on the right is
stable, and hence it does not appear in nature. Pictures by John Sul
an unduloid, which appears in large dimensions.
livan, University of Illinois (http://www.math.uiuc.edu/-jms)
12
THE MATHEMATICAL INTELLIGENCER
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 configura tion 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 sym metry 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 sur faces, 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
of the curves so that the normal lines meet at some point p, possibly ao, in the axis of revolution. Then (p 1 , . . . , Pn l 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 vol umes we also have the following =
CONJECTURE. The standard n-bubble, n � 4, is the least perimeter 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 en close and separate two given regions in IRn, for any n 2:: 3.
PROPOSITION. Consider a configuration of curves that gen erates a solution of the double-bubble problem by rotation. Assume there are points {p 1 , . . . , Pnl in the regular part
'·
b
·""'
Figure 1 1 . The partition method.
p
. ....
....
Figure 12. A standard triple bubble. Picture by John Sullivan, Uni· versity of Illinois (http://math.uiuc.edu/-jms)
VOLUME 24, NUMBER 3 , 2002
13
In case n = 4 this has been proved, by using the arguments of [11], in [17].
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
REFERENCES
Symposium in Honor of Manfredo do Carma. Longman Scientific
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.
Third ed. , Academic Press Inc. , San Diego, CA, 2000.
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.
ian type, Preprin t ESI 721 , 1 999. 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.
Math. J. 48 (1 999), no. 4, 1 357-1394.
16
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.
(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.
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 -4 1 0.
9. Joel Hass and Roger Schlafly, Double bubbles minimize, Ann. of Math. (2) 151 (2000), no. 2, 459-51 5.
2 1 . Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56
1 0. Michael Hutchings, The structure of area-minimizing double bub Geom. Anal. 7 (1 997), no. 2, 285-304.
(1 995), no. 1 , 1 9-33. 22. Michael Spivak, A comprehensive introduction to differential geom
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).
etry, vol. 4, Publish or Perish, Berkeley, 1 979.
23. Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2)
1 2. Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros, Proof of the double bubble conjecture, Annnals of Math. (2)
155
(1 976),
103
no. 3, 489-539. 24. Thomas I. Vogel, Stability of a liquid drop trapped between two parallel planes, SIAM J. Appl. Math.
(2002), no. 2, 459-489.
47
(1 987), no. 3, 5 1 6-525.
A U T H O R S
ANTONIO ROS
MANUEL RITORE
Departamento de Geometria
Departmento de Geometria y Topologia
Universidad de Granada
1 8071 Granada
1 8071 Granada
e-mail:
[email protected]
e-mail:
[email protected]
Spain
Spai n
Manuel Ritore, born in 1 966,
mean
14
to
studied at the
his doctorate
Granada in 1 994 under the continues
Unive rsidad de
at
the Universidad de
supervision
of Antonio Ros. He
work on minimal surfaces, surfaces
curvature,
y Topologia
Universidad de Granada
Extremadura. He got
and isoperimetric problems.
THE MATHEMATICAL INTELLIGENCER
226
(1 997), no. 3, 465-481 .
6. C. Delaunay, Sur Ia surface de revolution dont Ia courbure moyenne
J.
the Riemannian product of a circle with a simply connected space
1 7. Ben Reichardt, Cory Heilmann, Yuan Y. Lai, and Anita Spielman,
4. Yu. D. Burago and V. A Zalgaller, Geometric inequalities, Springer
bles,
1 6. Renato H. L. Pedrosa and Manuel Ritore, lsoperimetric domains in form and applications to free boundary problems, Indiana Univ.
3. F. Barthe and B. Maurey, Some remarks on isoperimetry of Gauss
est constante, J. Math. Pure et App.
& Technical, Harlow, 1 99 1 .
1 5. Frank Morgan, Geometric measure theory, A beginner's guide.
of co nstan t
Antonio Ros was
bom in
1 957.
He discovered
ferential Geometry in the textbooks
Classical Dif
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 rn ati c a l l y Bent
Col i n Adam s , E d itor
Dr. Yeckel and M r. Hide Colin Adams 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 y ou 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]
"
o
h, 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, In spector." "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 in structor 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 crea ture." "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
15
The students who had
versity received the overhead on the
with common sense and the ability to
loved him so much would sit waiting
grants. Of course, all was forgiven, and
act on it. No, they are more like a thou
for him hour after hour, but rarely did
he was given tenure."
sand toads trapped in a Bentley, all
even worse. They hardly recognized
him for as long as he lives?''
his classes.
he come. When he did come, it was
"Doesn't that mean they cannot fire
hoping this way and that, slapping against the windows and muddying the
him. They would ask him a question,
"It does, indeed, Lonigan, it does in
with their bright eyes and inquisitive
deed. And now the problem became
"Oh, I see."
minds, and he would say, 'Hmmm?' and
worse. His hair and beard grew longer
"So the university left him in charge.
lean against the wall lost in thought.
and more tangled. The t-shirt he wore
And the department finally revolted.
They would purposely make extra-loud
began to come apart at the
pits.
Whole subdisciplines jumped ship. By
rustling sounds, but he could not hear
His pants were frayed and stained up
the end of his term, there was no one
them. Sometimes, he would drop the
and down with coffee."
left but the lifers who couldn't get work
arm
chalk in mid-lecture and wander out of
"Did the university take action?"
the room. The poor students, their lit
"Oh, yes, they did. They made him
tle hearts were broken."
chair of the Mathematics Department."
"A sad tale indeed, Inspector. "
"Now why did they do that?"
"Sometimes we don't know what we
"Well, he was the most successful
have until it is gone, Lonigan. Such is how it was with him Eventually, his .
enrollments dropped. His wife left him,
mathematician in the department." "And what does that have to do with running a department?"
plush leather interior."
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
Q = NQ had been proved by a
his dog ran away, and the university
that
threatened to fire him if he didn't
graduate student from Southampton.
"A sad tale
meet his classes."
"As it should be."
This immediately implied the Stillwell conjecture. It was too much for him
indeed,
"Perhaps so. But then, Lonigan, as often happens in life, fate provided a
to
Inspector . "
sudden and unexpected twist. "
His
mathematical
heart
"Ah. I see."
"How so, Inspector?"
"He collapsed on the spot. This spot
"Yeckel showed that the Stillwell conjecture was equivalent to
bear.
broke."
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
"Ah, Lonigan, now you have wacked
the nail on its tiny top. It needn't have
anything to do with running a depart
right before us. And since his dream of
proving the Stillwell conjecture had
been destroyed, his body reverted to its
ment. And in this case, it did not.
former state. He was no longer the
Yeckel continued to work on his re
driven intellectual who derived all his
search to the exclusion of all his other
meaning from the pursuit of knowledge,
duties.
ignoring the real world around him.
Appointments
were
missed.
of the Stillwell conjecture. Suddenly he
Staffing reports were not submitted.
was a mathematical celebrity. He was
Hires were not made. Within a year's
been,
invited to speak at colleges and uni
time, the Mathematics Department was
whom the students had loved so much."
versities all over the country. Recep
in a shambles."
tions were thrown in his honor, with sparkling cider and little stuffed mush
"Well, Inspector, then the university must have realized its mistake. "
rooms. He received large federal grants
"But Lonigan, you must remember,
to continue his research and the uni-
universities are not like individuals
16
THE MATHEMATICAL INTELLIGENCER
Now he reverted to the man he had the nurturing,
caring
teacher
"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 Pro b 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 1 0, The Clay Mathematical Sciences Insti$7,000,000 Millennium Prize, a million
stitute (http://www.claymath.org) are the Poincare Con
dollar award for the solution of each of seven famous prob
jecture and the Riemann Hypothesis, both discussed below,
lems. Contrary to belief, this publicity stunt has precedence
and the P versus NP problem, the Hodge Conjecture, the
tute inaugurated a
in Mathematics. This article is a result of my personal re
Yang-Mills Existence and Mass Gap, the Navier-Stokes Ex
view of the history of a few famous unsolved problems
istence and Smoothness; and the Birch and Swinnerton
whose statements can be understood by a person with an
Dyer Conjecture. The problems are accompanied by arti
undergraduate mathematics degree or less.
cles written by Stephen Cook, Pierre Deligne, Enrico
When I was a student, the Burnside Problem, the Sim ple Odd Group Conjecture
(1963), and the Continuum Hy
Bombieri, Charles Fefferman, and Andrew Wiles. Attaching monetary value to mathematics questions is
pothesis had just been resolved but the Riemann Hypoth
not new. In
esis, the Four-Color Map Problem, Fermat's Last Theorem,
tablished a prize of
the Bieberbach Conjecture, the Poincare Conjecture, and
at the time) for a proof of Fermat's Last Theorem. (See No
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
1908 German industrialist Paul Wolfskehl es 10,000 DM (approximately $1,000,000
tices A.M.S.
44 no. 10 (1997), 1294-1302.) Unfortunately in 1997 Wiles col lected just $50,000; however, the Royal Swedish Academy
flation diminished the prize value so that in
bach Conjecture was proved. Thirty years later Fermat's
of Sciences also awarded Wiles the Schock Prize, and he
Last Theorem is gone and just a few of the aforementioned
received the Prix Fermat from the Universite Paul Sabatier.
problems remain, although others have surfaced. A solu
DeBranges was awarded the Ostrowski Prize for proving a
tion to any of these problems brings "fame" and occasion
much stronger conjecture than the Bieberbach Conjecture.
ally one of the major mathematical prizes such as the
"The Prince of Problem-Solvers and the Monarch of Prob lem-Posers," the late Paul Erdos, who won the
$50,000 Wolf
$145,000 Steele Prize, the $50,000 Wolf Prize, a special gold medal (along with $15,000) called The Fields Medal, infor
Mathematics Prize, was famous for offering cash prizes to
mally known as the "Nobel Prize of Mathematics," or what
those mathematicians who solved certain of his problems.
I call the real "Nobel Prize" for mathematicians, the Royal
These prizes ranged from $10,000 for what he called "a hope
Swedish Academy of Sciences'
less problem" in number theory to
The
seven
problems
$500,000 Crafoord Prize.
whose
solutions
will
bring
$1, 000,000 each from the Clay Mathematical Sciences In-
$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,
An earlier version of this article appeared in the NAM Newsletter XXX 1 (2)(2000).
© 2002 SPRINGER· VERLAG NEW YORK, VOLUME 24. NUMBER 3, 2002
17
other mathematicians have continued this practice. Now a corporation offers one million dollars and an institute of fers more.
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 am ateur mathematician, yet he offers $75,000 for the resolu
Fields Medals have not been awarded to persons over
tion of this conjecture, which was first announced in 1997.
the age of forty. Concerning solutions of famous problems,
The prize committee consists of Charles Fefferman, Ron
some Fields Medals were awarded to:
Graham, and R. Daniel Mauldin, and the funds are held in
Selberg (1950) for his work on the Riemann Hypothesis;
trust by the American Mathematical Society.
Cohen (1966) for his resolution of the Continuum Hypoth esis; Smale (1966) for his work on the Generalized Poin care 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 mod est 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 Con jecture, Lost-in-a-Forest Problem, Palindrome Problem, The Poincare Conjecture) all have simple statements. Some of these problems (the Riemann Hypothesis and the Poin care Conjecture) are usually taken to have more value to the field than others. However, there have been lesser prob lems which were not resolved by simply pushing the ex isting 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 million buck problems because I believe (the techniques involved in) their resolution will be worth at least $1 million to Mathematics.
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 1 be true for all numbers up to 4 10 3. The closest approx ·
imation to a solution to Goldbach's Conjecture is Chen Jing Run's recent result that every "sufficiently large" even
+ 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 re quirements, 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
(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 pub lished resolution of the conjectures in his lifetime. The Schanuel Conjecture is the following independence prop erty 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, . . .
independent.
ezn are algebraical1y The Converse Schanuel Conjecture says that
there is nothing more to be said. Explicitly, let F be a count able 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 inde pendence 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
is irrational.
+
7r
4. The Kolakoski Sequence
1 . Goldbach's Conjecture
number is of the form p
3. Schanuel's Two Conjectures
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.
Consider the sequence of ones and twos if = ( 1221 12122122 1 12 1 12212 1 12122 1 121121 22122 1 12122 1 2 1 1 2 1 122 122 1 12). A
block
of if is a maximal constant subsequence. We con
sider 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 ( 1 1 ) has length 2. Continue in this fashion and notice that the sequence
A=
(1221 12 122 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 se quence
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?
2. Beat's Conjecture
ii. If a string (e.g., 2122 1 1) occurs in if, must it occur again?
This is a generalization of Fermat's Last Theorem. If Ax +
iii. If a string occurs in if, must its reversal also occur?
J3Y = CZ, where A,
(1 122 12 occurs)
18
B, C, x, y, and z are positive integers and
THE MATHEMATICAL INTELLIGENCER
iv. If a string occurs in
cr,
and all its 1s and 2s are swapped,
v. Does the limiting frequency of 1s in
cr
Given Magnitude, Bernhard Riemann (1826--1866) ex tended the zeta function, defined by Euler as
must the new string occur? (121 122 occurs) exist and is it 1/2?
?(s) 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 1 994 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 Hitch-Hiker's Guide to the Galaxy) a $42 prize to the
to A
person who settles the box-product problem in his lifetime.
1 - for ns � l n
L 00
=
s > 1,
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 <10 10 • 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 n manifold 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
6. The Collatz 3x + 1 Conjecture
Because it is easy to program your computer to look for
doughnut, has a hole). The Generalized Poincare Conjec
solutions, many youngsters (and adults) have played with
ture says that each simply connected compact n-manifold
+ 1 problem: On the positive integers define the 3x + 1 if x is odd and F(x) x/2 if x is
century, Poincare conjectured this for n = 3, and the Gen
the 3x
function F(x) =
=
is homeomorphic to the n-sphere. Near the end of the 19th
even. Iterations of F lead to the sequences ( 1 , 4, 2, 1), (3,
eralized Poincare Conjecture has been solved in all cases
10, 5, 16, . . . , 1), and (7, 22, 1 1 , 34, 17, 52, 26, 13, 40, 20, 10,
except n = 3.
. . . , 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
·
10 13, 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
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 = 1 2 1 ) and ( 1 76, 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
sors. This question was first posed by Euclid and is still
In 1956 R. Bellman asked the following question: Suppose
open. Euler proved that if N is an odd perfect number, then
that I am lost without a compass in a forest whose shape
in the prime power decomposition of N, exactly one expo
and dimensions are precisely known to me. How can I es
nent is congruent to 1 mod 4 and all the other exponents
cape in the shortest possible time? Limit answers to this
are even. Using computers, it has been shown that there are no odd perfect numbers < 10300.
gions. For a given region, choose a shape of path to follow
question for certain two-dimensional forests: planar re and determine the initial point and direction which require
8. Riemann Hypothesis
the maximum time to reach the outside. Then minimize the
This is the most famous open problem in mathematics. In
maximum time over all paths. For many plane regions the
the Number of Primes Less Than a
answer is known: circular disks, regular even-sided poly-
his 1859 paper On
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.
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 E 1 6 . G.T. Leavens and M . Vermeulen. 3x
+
1 search programs. Journal
Comput. Math. Appl. 24 (1 992), 79-99. This article is dedicated to John Isbell. Concerning this article,
I
Massey,
had
personal
Mohan
correspondence
Ramachandran,
with
Samuel
William
Schack,
and
Stephen Schanuel. All errors, however, are mine.
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
REFERENCES
http://match.stanford.edu/rh/
General References
9. Twin Primes References
J Korevaar, Ludwig Bieberbach's conjecture and its proof by Louis de Branges, Amer. Math. Monthly 93 (1 986), 505-5 1 4 . For a wealth o f information o n some o f 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
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
Erdos References
R. Bellman, Minimization problem. Bull. Amer. Math. Soc. 62 (1 956), 270.
http://vega.fmf.uni-lj.si!�mohar/Erdos.html
J.R. Isbell, An optimal search pattern , Naval Res. Logist. Quart. 4 (1 957),
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
357-359. Web survey and reference article: http://www.mathsoft.com/asolve/ forest/forest. html A U T H O R
sum of a prime and the product of at most two primes. Sci. Sinica 16
(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.
SCOTT W. WILLIAMS
Pure Appl. Logic 51 ( 1 99 1 ) , 241 -246.
Department of Mathematics
John Shackell, Zero-equivalence in function fields defined by algebraic
State University at Buffalo
differential equations, Transactions of the Amer. Math. Soc. 336
Buffalo, NY 14260-2900
(1 993), 1 5 1 -1 71 .
USA
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
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 97 1 . Married, with three daughters.
L. Brian Lawrence, Failure of normality in the box product of uncount
He has been a professional musician, and has many ex
ably many real lines. Trans. Amer. Math. Soc. 348 (1 996), 1 87-203.
hibited works in the visual arts. He is seriously interested in
S.W. Williams, Box products. Handbook of Set-Theoretic Topology (K.
the African-American heritage, including his own. At present
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
he is editing the newsletter of the National Association of Math ematicians.
ROBERT FINN
Ei g ht Remarkab e Properties of Cap i ary Su rfaces
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 con flict in these three examples with the intuitive dictum that "water seeks its own level" certainly was of historical sig nificance in drawing attention to the problems and devel oping a general theory. In general, one considers a connected volume "V of liq uid 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 mate rial 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 equilib ria 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 contain ing 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 coor dinates and multiplying by the local area ratio). For exam ple, 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, con sidered as the stereographic projection from the diametri cally opposite point on the sphere. In general, the Lapla cian in (1a) is a highly nonlinear operator. The free surface � meets the rigid surface 'W in a con tact angle y that depends only on the physical character istics 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 ho mogeneous, 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 3. General configuration.
in n, with v ·
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 verti cal gravity field g (which may vanish or be negative) we find
pg H(x) = - z + canst.
(2)
(T
where p is the density change across ';1, u the surface ten sion, 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
Tu = cos y
on an; here , is the unit exterior-directed normal vec tor. 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 illus trates the case 0 ::s y < Trl2. To some extent, this same so lution applies when n is a regular polygon, because the ver tical 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 gen eral characteristic of the local behavior of solutions of (3a, b) near comer points. The following result is proved in (6].
We consider a capillary tube with general section n in the absence of gravity. That would not be appropriate for the con figuration of Figure 1, as in that case all fluid would flow ei ther 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 re moved 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 solu tion surface for (la) bounded by a simple closed curve en circling 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.,
Tu =
Yl
'Vu + 1Vul2
(3a)
b
Figure 2. a) Sessile drop; b) Pendent drop.
22
THE MATHEMATICAL INTELLIGENCER
(3b)
Figure 4. Circular section; surface interface.
Figure 5. Hexagonal section; equatorial circle of lower hemisphere.
Figure 6. Water in wedges formed by acrylic plastic plates; g > 0.
a)
THEOREM 1 . 1 : If a + y < n/2 at any corner point P of open ing 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 partic ular 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 uni formly smooth bounded solutions disappears without dis cernible trace.
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 vac uum, 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
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 def inition 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 "advanc ing" and "receding" angles were introduced. Also these quantities are not always easily reproducible experimen tally. 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 "canoni cal proboscis" [8, 9, 10], in which the linear boundary seg ments 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 suc cessfully demonstrated [ 1 1 ] , and it can yield precise an swers in situations for which conventional methods fail.
'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 con ditions as it must, but it cannot be obtained as solution of (3) over n. The seeming "non-existence" paradox appeared be cause we were looking for the surlace in the wrong place. I emphasize again that the change in behavior is dis continuous 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)
v · Tu = cosy
(4b)
in n,
on an, with K = pg/u. There is again a discontinuous change at the same critical y; although in this case a solu tion 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 un bounded 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
Property 2. Uniqueness and Non-uniqueness
Let us consider a fixed volume V of liquid in a vertical cap illary 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 de termined 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; nev ertheless, 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 non uniqueness. 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 contin uous 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 interme diate 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 cir cle, with a fluid whose contact angle with the bounding walls is 45°, a horizontal surface provides a particular so lution 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 cylin der. In this case, the fluid cannot be horizontal at the bound ing walls in view of the 45° contact angle, and a curved in terface 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 at tained, and obtain a second solution in the container, as in dicated in the figure. The construction indicated can be extended in a re markable way [ 15, 16] : THEOREM 2.2: There exist rotationally symmetric contain ers 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 con cerns about the construction are indicated in [4]; neverthe less, 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 ex ist 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. Be cause 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 asym metric solutions. This prediction was tested computationally by M. Calla han [20], who studied the case g = 0 and found a local min imum (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 in
terfaces 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 sta bility 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) . I n 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 criti cal volume VcrCy1 , 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 separa tion is decreased without changing the volume or contact angles, then an initially stable configuration will remain sta ble. 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 per turbation. But we note that we will have to change the con tact angles, resulting in changed energy expressions, and the requirement of zero volume change for admissible per turbations has differing consequences for the energy changes resulting from perturbations. In fact, Zhou in [26] showed that the answer can go ei ther way, and even can move back and forth several times during a monotonic change in separation h, so that the sta bility 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
each of the planes on its boundary, and whose outer sur face ';! is topologically a disk.
A spherical bridge with tubular topology can exist in a
wedge of opening 2a if and only if y 1 + y2 > 7T + 2a. In contrast to the case of parallel plates, whenever this con dition holds, spherical bridges of arbitrary volume and the same contact angles can be found. McCuan proved [31] that
Figure 13. Liquid bridge joining parallel plates; g
if YI + Y2 s; 7T + 2a, then no embedded tubular bridge ex ists. Wente [32] gave an example of an immersed tubular =
0.
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 ('Yby2) lies in surlaces. She proved that if the contact angles on both
the closed rectangle m of Figure 14. It is proved in [33] that
plates are equal, and if the plates are moved closer to gether 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
if (Yl, 'Y2) is interior to m then the interface ';! of every such drop is metrically spherical. It is col\iectured in that refer
Zhou showed that there are critical contact angles y'
surface with :£; in fact, there exist surfaces ';! that exhibit
contact angle with the lower plate, and hold this constant; =
< Y1 < Yo then if the up per plate is sufficiently distant in the range Y1 < Y2 < 14.38°, Yo
=
14.97°, such that if y'
7T y�, the corifiguration will be unstable. On moving that plate downward, it will enter a stability interval; on con tinued downward motion, the configuration will again become unstable, and finally when the plates are close enough, stability will once more ensue.
ence that there exist no drops with unit normal to ';! dis
continuous at :£. In [30] it is shown that the col\iecture can
not be settled by local considerations at the "juncture" of the
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 capil lary tubes of general piecewise smooth section
0, solutions
of (3a,b) do not always exist. Failure of existence is not oc casioned specifically by the occurrence of sharp comers;
Property 4. Liquid Bridge Instabilities, Zero g;
Tilting of Plates
In the discussion just above, motion of the plates was ex cluded 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 unex pected results [30].
THEOREM 4.1: Unless the initial configuration is spheri cal, every bridge is unstable with respect to tilting of ei ther plate, in the sense that its shape must change dis continuously on infinitesimal tilting.
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 IOVCiklcos y), with the properties i) the curvature vector of each f* is directed exterior to 0*, and
It should be noted that a spherical bridge joining paral lel plates is a rare event, occurring only under special cir cumstances. A necessary condition is y1 + Y2 > 7T ; for each such choice of contact angles, there is exactly one volume
1t
that yields a spherical bridge.
A spherical bridge can change continuously on plate tilt ing; 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 in finitesimal 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 dis continuously to infinity. By a "drop in the wedge" is meant a connected mass of fluid containing a segment of the in tersection line :£ of the planes as well as open subsets of
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 cor ner points of � at an angle not less than y.
Q:
We then have THEOREM 5. 1: A solution u(x) of (3a,b) exists in fl if and only iffor every such configuration there holds
(fl*; y) = l f* l
-
l�*lcos y +
2H cos y > 0
(5)
with
_ 2H - m cos y. lfll 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 , in the sense that they are the bound aries in fl of extremal domains fl* arising from the "sub sidiary variational problem" of minimizing . 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 r0 of semicircles of radius 112Ho, for some positive H0 :::::; H. The arcs meet � in the angle y or else at re-entrant cor ner 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.
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 ob tain the configuration of Figure 18. In this case two distinct C-singular solutions can appear, for the same y, with re gions of regularity, respectively, to the left of one of the in dicated 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 oc cur, 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.
We refer to such surfaces U(x,y) as cylindrically sin gular solutions, or "C-singular solutions". The subarcs are the extremals for the functional, corresponding to H = Ho in (10). Figure 16 illustrates the behavior. Such solu tions 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
Figure 17. If
a
+ 1' < Trl2, there exists exactly one C-singular solu
tion, up to an additive constant; no regular solution exists.
Figure 18. At least two C-singular solutions exist.
Property 6. Discontinuous Reversal of
I illustrate the possible behavior with a specific exam
Comparison Relations
Consider surface interfaces :J' in a capillary tube as in Fig
ure 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
ple. Denote by n 1 a square of side 2, and by n(t) = nt the
domain obtained by smoothing the comers of n 1 by circu lar arcs of radius (1
-
t), 0 :::s t
inscribed disk (Figure 22).
::::;
1. Thus, no becomes the
For y � 7T/4, it can be shown that there exists a solution t of (6) in any of the nt. Denote these solutions by u (x; K).
One can prove:
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 n l :J :J no (Figure 21). An al most immediate response, indicating a particular configu
ration 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 an swer 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.
Figure 20. In a disk, a regular solution exists for any 'Y; but no C singular solution exists.
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·
Figure 21 . Does Oo raise fluid higher over its section than does 01
over that same section?
VOLUME 24, NUMBER 3, 2002
29
THEOREM 6.2: For aU K > 0,
0 u (x; K) > u1(x; K)
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
u 1 (x; K) - ut(x; K) > (C(t)IK) - C0
(7)
(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 u 1 (x; K) - u t(x; K) as K ---" 0 is thus dis continuous at the value t = 0, and in fact with an infinite jump. Paul Concus and Victor Brady tested this unexpected re sult independently by computer calculations. Figure 23 shows u 1 - 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 u 1 - u0 is always negative, as predicted, while for any e > 0, u 1 - U13 becomes arbitrarily large positive with de creasing K. Note that the vertical scale in Figure 23 is log arithmic, 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)
uniformly over Ot . On the other hand, we have
1� �-r----,-----�--r---r---��--�-�
1cl D ... 0 ... "": 101
B= .oo.J 1 B= .0 1 B= 1 B= 1 00
0 •
q_ :I
8 -;-10°
0
0.1
Figure 23. u1 (0; B) - ut(O; B) as function of t; 'Y
is small.
30
THE MATHEMATICAL INTELLIGENCER
0.2
=
7TI3.
0.3
0.4
0 .5 1
0.6
0.7
Note negative values that minimize when t
0.8
=
0.9
0, and large slopes at end points when B
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
p
ur(r) l -
1
r2
is uniquely determined. The singular solution
I<3 10
( 1 1)
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 Figure 24. Solution exists when 'Y
=
59°. Smoothing
at P leads to non
existence.
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,
n is not the occurrence of corner points, but
and then smooth out, cross the horizontal axis, and con
rather the presence of boundary segments of locally large
tinue to infinity; see Figure 25, where the computed shapes
inward-directed curvature, relative to averaged curvature
are compared with a computed
properties of the domain. For a circular domain, solutions
observations, those authors conjectured that the bubble
in a domain
y; however, given a value 'Y in
exist for any contact angle 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
U(r). On the basis of these
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).
n whose boundary contains a
Most recently Nickolov [45] showed that every rota
point P of strict maximal inward boundary curvature, then
tionally symmetric solution of (9) with a non-removable iso
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
lated singularity satisfies ( 1 1); in view of the results of [42], this work completes the proof of the Conjecture (a).
make the domain closer to circular in form, then a solution
corollary, note that in the succession of bubble-solutions
will again exist in the smoothed domain. Figure 24 indi
considered just above, there is no need to choose a subse
lution exists in a domain if
As a
cates a configuration in which that assertion fails, see [38].
quence; every sequence converges, and to the same limit.
The angle at P is 90°, the angle formed by the extended seg
Summarizing,
ments at Q is 60°, and the domain is smooth except at the
A solution exists when 'Y = 59°, but
single corner point P.
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
if K 2:: 0, then every isolated singu larity of a solution of (4a,b) is removable. Thus, also in It is known [39] that
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) div Tu
=
-u
(9)
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 expan sion (10). Additionally when K < 0, there exists, for any negative vertex height u0, a global "bubble-solution" as de scribed above, having the general character of a pendent liquid drop. As u0 � -oo, these surfaces converge uni formly 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
with an isolated singularity at the origin, and admitting a
figuration is unstable if two inflections are present below
(divergent) asymptotic expansion
the plane. The "bubble-profiles" described above yield in
U(r)
=
_
_!_ + � r2 r
567
2
+
8
123149 16
general physically unstable interfaces when considered
r7
globally; nevertheless, all profiles exist globally as analytic
r u0. The initial tip region
curves extending smoothly to infinity asymptotic to the
r
11
-
2 1246673 1 128
15
r
+
· · ·
(10)
axis, and uniquely determined by
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
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
=
-
8;
(c) u0
=
- 1 6.
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 call ing 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
ROBERT FINN Department of Mathematics Stanford University Stanford. CA 94305 USA
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.
e-mail: [email protected]
REFERENCES
Robert Finn was Professor of Mathematics at Stanford from 1 959 until his normal retirement in 1 992. He continues re
1 . T. Young: An essay on the cohesion of fluids. Phi/as. Trans. Roy. Soc. Land. 95 (1 805), 65-87.
hydrodynamics and other fields; collaboration with
2. P.S. Laplace: Traite de mecanique celeste: supplements au Livre
undergraduates; and supervision of doctoral students (among
X, 1 805 and 1 806, resp. in CEuvres Completes Vol. 4. Gauthier
search
in
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.
Villars, Paris. See also the annotated English translation by N. Bowditch (1 839), reprinted by Chelsea, New York (1 966).
3. C.F. Gauss: Principia Generalia Theoriae Figurae Fluidorum. Com
ment. Soc. Regiae Scient. Gottingensis Rec. 7 (1 830). Reprinted as Grundlagen einer Theorie der Gestalt von Flilssigkeiten im Zus tand des Gleichgewichtes in Ostwald's Klassiker der Exakten Wis senschaften, vol. 1 35, W. Engelmann, Leipzig, (1 903).
32
THE MATHEMATICAL INTELLIGENCER
4. R. Finn: On the equations of capillarity. J. Mathem Fluid Mech, 3
25. D. Langbein: Stability of liquid bridges between parallel plates. Mi
5. T.l. Vogel: Uniqueness for certain surfaces of prescribed mean cur
26. L.-M. Zhou: On stability of a catenoidal liquid bridge. Pacific J.
(200 1 ) , 1 39-1 5 1 .
vature. Pacific J. Math. 1 34 (1 988), 1 97-207. 6. P. Concus, R. Finn: On capillary free surfaces in the absence of gravity. Acta Math. 1 32 (1 974), 1 77-1 98. 7. R. Finn: Equilibrium Capillary Surfaces. Springer-Verlag , New York (1 986). Russian translation (with Appendix by H.C. Wente) Mir Pub lishers (1 988). 8. B. Fischer, R. Finn: Non-existence theorems and measurement of capillary contact angle. Z. Anal. Anwend. 12 (1 993), 405-423.
9. R. Finn, T.L. Leise: On the canonical proboscis. Z. Anal. Anwend. 13
(1 994), 443-462.
1 0. R. Finn, J. Marek: The modified canonical proboscis. Z. Anal. An wend. 1 5 (1 996), 95-1 08.
crogr. Sci. Techn. 1 (1 992), 2-1 1 . Math. 1 78 (1 997), 1 85-198. 27. L.-M. Zhou: The stability of liquid bridges. Dissertation, Stanford Univ , 1 996. 28. T.l. Vogel: Local energy minimality of capillary surfaces in the pres ence of symmetry. Pacific J. Math., to appear.
29. R. Finn, T. l . Vogel : On the volume infimum for liquid bridges. Z. Anal. Anwend. 1 1 (1 992), 3-23.
30. P. Concus, R. Finn, J. McCuan: Liquid bridges, edge blobs, and Scherk-type capillary surfaces. Indiana Univ. Math. J. 50 (200 1 ) 41 1 -441 . 31 . J. McCuan: Symmetry via spherical reflection and spanning drops in a wedge. Pacific J. Math. 1 80 (1 997), 291-324.
1 1 . P. Concus, R. Finn, M. Weislogel : Measurement of critical contact
32. H.C. Wente: Tubular capillary surfaces in a convex body. Advances
angle in a microgravity space experiment. Exps. Fluids 28 (2000),
in geometric analysis and continuum mechanics (Stanford, CA,
1 97-205. 1 2. M. Emmer: Esistenza, unicita e regolarita nelle superfici de equilib ria nei capillari. Ann. Univ. Ferrara 18 (1 973), 79-94. 1 3. R. Finn, C. Gerhardt: The internal sphere condition and the capil lary problem. Ann. Mat. Pura Appl. (4) 1 1 2 (1 977), 1 3-31 . 1 4. R. Finn: The sessile liquid drop 1: symmetric case. Pacific J. Math. 88
(1 980), 541 -587.
1 5. R. Gulliver, S. Hildebrandt: Boundary configurations spanning continua of minimal surfaces. Manuscr. Math. 54 (1 986), 323347. 1 6. R. Finn: Nonuniqueness and uniqueness of capillary surfaces. Man user. Math. 61 (1 988), 347-372. 1 7. P. Concus, R. Finn: Instability of certain capillary surfaces. Manu scr. Math. 63 (1 989), 209-21 3. 1 8. H.C. Wente: Stability analysis for exotic containers. Dynam. Con tin. Discrete lmpuls. Systems (Waterloo, ON, 1 997). Dynam. Con tin. Discrete lmpuls. Systems 5 (1 999), 1 5 1 -1 58.
1 993), 288-298. Internal. Press, Cambridge, MA (1 995). 33. R. Finn, J. McCuan: Vertex theorems for capillary drops on sup port planes. Math. Nachr. 209 (2000), 1 1 5-1 35. 34. R. Finn, R.W. Neel: C-singular solutions of the capillary problem. J.
Reine Angew. Math. 5 1 2 (1 999), 1 -25.
35. P. Concus, R. Finn: On the height of a capillary surface. Math. Z. 1 47
(1 976), 93-95.
36. D. Siegel: Height estimates for capillary surfaces. Pacific J. Math. 88
(1 980), 471 -51 6.
37. R. Finn, A.A. Kosmodem'yanskii, Jr.: Some unusual comparison theorems for capillary surfaces. Pacific J. Math., to appear. 38. R. Finn: A curious property of capillary surfaces. Ann. Univ. Fer rarra, to appear. 39. R. Finn: On partial differential equations whose solutions admit no isolated singularities. Scripta Math. 26 (1 961 ), 1 07-1 1 5. 40. P. Concus, R. Finn: A singular solution of the capillary equation 1 : existence. Invent. Math. 2 9 (1 975), 1 43-1 48.
1 9. J.E. Taylor: Boundary regularity for solutions to various capillar
41 . M.-F. Bidaut-Veron: Global existence and uniqueness results for
ity and free boundary problems. Comm. P.D. E. 2 (1 977), 323-
singular solutions of the capillarity equation. Pacific J. Math. 1 25
357. 20. M. Callahan, P. Concus, R. Finn: Energy minimizing capillary sur
(1 986), 31 7-334. 42. M.-F. Bidaut-Veron: Rotationally symmetric hypersurfaces with pre
faces for exotic containers (with accompanying videotape), in Com
scribed mean curvature. Pacific J. Math. 1 73 (1 996), 29-67.
puting Optimal Geometries, J.E. Taylor, ed. , AMS Selected Lec
43. P. Concus, R. Finn: The shape of a pendent liquid drop. Phil. Trans.
tures in Mathematics, American Mathematical Society, Providence, Rl (1 99 1 ) , 1 3-1 5.
Roy. Soc. London. Ser. A 292 (1 979), 307-340. 44. R. Finn: Green ' s identities and pendent liquid drops: I. Inti. Syrnp.
2 1 . P. Concus, R. Finn, M. Weislogel : Drop tower experiments for cap
Univ. Ferrara; in Developments in Partial Differential Equations and
illary surfaces in an exotic container. AIM J. 30 (1 992), 1 34-1 37.
Applications in Mathematical Physics, ed. G. Buttazzo, G.P. Galdi,
22. P. Concus, R. Finn, M. Weislogel: Capillary surfaces in an exotic container; results from space experiments. J. Fluid Mech. 394 (1 999), 1 1 9-1 35. ,23. M. Athanassenas: A variational problem for constant mean curva ture surfaces with free boundary. J. Reine Angew. Math. 377 (1 987), 97-107. 24. T.l. Vogel: Stability of a liquid drop trapped between two parallel planes II: general contact angles. SIAM J. Appl. Math. 49 (1 989), 1 009-1 028.
L. Zanghirati. Plenum Press, London (1 992), 39-58. 45. R. Nickolov: Uniqueness of the singular solution to the capillary equation. Indiana Univ. Math. J. 51 (2002) to appear. 46. H .C. Wente: The stability of the axially symmetric pendent drop. Pacific J. Math. 88 (1 980), 421 -470. 47. R. Finn: Capillary surface interfaces. Notices Amer. Math. Soc. 46 (1 999), 770-781 . 48. R. Finn: Some properties of capillary surfaces. Rend. Sem. Mat. Milano (in press).
VOLUME 24, NUMBER 3, 2002
33
l@ffli•i§\:6ih£11£ii§ 4@11,j,i§.id
Consider the following problem:
The Keg Index and a Mathematical Theory of Drunkenness
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 in dependently of what has happened be fore. 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 (non starting) person is independent of that person's position.
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.
M i chael K l e b e r and Ravi Vaki l , Editors
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
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil,
Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 1 25, USA e-mail : [email protected]
34
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. 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-
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
p)2q I pkqk 00
p + c1
_
k�O
=p +
=
(1 - p)2q 1 - pq
p - 2pq + q 1 - pq
while the probability that Q passes it out is
I pkqk 00
o - P)C 1 - q)
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-
( (
\
,/'"
Using the series
----..._
.'\ \
�
�� 4�
"' -- ?
R ---9 --
--
•
'-----
....___
/
/
)
1 (1 - x)2
=
� kX k - 1 '
L k�o
we obtain the expressions 2x
(1 - x2)2 and
-
_!_
2
(
)
1 1 = I 2kx2k - l - x)2 (1 + x)2 k�o
(1
_!_ (
1 + x2 _ 1 + (1 - x2)2 2 (1 - x)2
(1
1 + x)2
= I (2k - 1)x2Ck - l). 00
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 right drunkenness: together these two facts imply that the (p,q) drunk combination behaves as a single person with the net keg index P[the keg is passed back] =
+q P � 2pq . - pq
k�l
Applying these to
(2) with x = (pq) 112, we find
E[swigs] = p
+ (1 - p)(1 - q)
+ (1 - p)2 q and because pq ::;::; p, q
(1)
p - 2pq + q p(l - q)2 > 0 q1 - pq - ' 1 - pq it follows that the (p,q)-drunk combination is drunker than max(p,q }-at last a mathematical proof of something read ily 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 as sume 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 se rious a flaw as it may at first appear. The effect of the swigs will depend on the number taken-the fewer swigs, the bet ter the swig-free approximation will be. We therefore cal culate the expected number of swigs before P and Q pass out the keg, obtaining
+ (1 - p)(1 - q) I 2k(pq)k - l k�l 00
+ c1 - p )2q I c2k - 1)(pq)k - 2. c2) k� 2 00
3 - pq
(1 - pq)2
2 _ (1 pq)2 =
2+
q -p 1 - pq '
< 1, we have
q -p l 1 - pq l < 1
Since
E [swigs] = p
)
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 re sult again to S and R. We get
+ q - 2 P - 2pq + q r + r P -1 2pq 1 - pq - pq 2pq + q r 1-P1 - 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 explain ing crowd behaviour in pubs.
VOLUME 24, NUMBER 3, 2002
35
In fact, if we define ++ : [0, 1) X [0, 1) nounced double-vision-plus) by
p++q =
�
[0, 1) (pro
p - 2pq + 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 satisfying p + + 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
lj;(t) =
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
p + +oo = lim
q->oo
2 1 p - 2pq + q = P = 1/J(f/l(p) - 1) 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 some things 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 nonad jacent 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 in duced 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 joinContinued on page 6 7
P. SCHOLL, A. SCHURMANN, J. M. WILLS
Po yhedral M odels of Fe ix K ein 's Group ''I
have the polyhedron on my desk. I love it!" John H. Conway, Aug. 19, 1993
f
elix 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. 1 04), ''It is a truly central piece of mathematics. "
Felix Klein discovered this finite group PSL(2, 7) of or der 168 in 1879 [K], and since then its properties have been investigated, generalized, applied, and discussed in hun dreds of papers. The recent book The Eightfold Way [L] contains several survey articles by prominent experts, which collect and dis cuss 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 mod els is not new. Felix Klein himself gave a planar and a 3-di mensional model. The planar one is the unsurpassable Poincare model (Figure 2), well known from classical com plex 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 geom etry." 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 reflec tion 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 reg ular 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 ver tices; or the 5 white and 5 black triangles are collected around the 10-valent vertices, which yields the dodecahe dron 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 el ements 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."
&
9
5
t3
t't Figure 2. Klein's group as a regular map.
38
ll-IE MAll-IEMATICAL INTEWGENCER
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
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:
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 1 1 1 -1 A = 2( g - 1) +tonic solids-existed long before the mathematical back p q 2 ground (groups, maps) was understood. One had a vague From p = 3, q = 7 (or vice versa) and A = 168 follows g = idea of their deeper importance. 3. As a consequence, such groups have maximal order Klein's group of order 168 (or 336 for the full group) is 84(g - 1), and Klein's group is the first one of these rare much larger than the icosahedral group of order 60 (full: "Hurwitz groups." 120) and is not a symmetry group in Euclidean 3-space. We can create a polyhedron and consider a group of 168 trans Polyhedral Models with Tetrahedral Symmetry formations of it, but they can't all be congruences! It can easily be shown that any 3-dimensional . model of Klein's group does contain a geometric subgroup, Klein's group with maximal-i.e., octahedral-symmetry namely the "octahedral rotation group" of order 24, which has self-intersections, so in order to avoid self-intersec was of course known to Felix Klein. We seek, then, a model tions, polyhedral embeddings have at most the next lower of Klein's group with octahedral symmetry, the elements symmetry, tetrahedral rotation symmetry of order 12. (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, characterizes 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 polyhe dral 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. Sev eral cardboard and metal models and computer films were made of this realization. (See also [BW], and Conway's com ment at the head of this article). In its symmetry and em bedding 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 con structions 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 sur face of genus 3 with tetrahedral symmetry. It shows the 24 heptagons, and, hence it corresponds to the regular do decahedron ( 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 pro grams. The result is shown in Figure 4 (for details of con struction see [SSW]). The bizarre model is complicated, and is of no help in understanding Klein's group. This under lines 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-inter sections; 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 oc tahedral 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 cor respond to Klein's three intersecting hyperboloids, and the Petrie polygons can easily be seen. Altogether this polyhedral 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 compli cated. Again, the model was con structed by computer; for more details, refer to [SSW]. The heptagons have self-inter sections, so the model is related to the classical Kepler-Poinsot poly hedra and to Coxeter's regular complex polyhedra. It might be surprising that the realizations of a pair of dual maps of the same group can be so dif ferent. But the reason is quite sim ple: In the triangulations the facets are, by definition, triangles, so they are convex and free of self intersections. All topological com plications, twists, and curvature are hidden in the vertices, whose shape is flexible. In the dual, with 3-valent vertices, all complica tions have to be stored in the hep tagons, which makes the models Figure 4. Kepler-Poinsot-type realization of {7, 3}s with tetrahedral symmetry. star-shaped and bizarre. So this
40
THE MATHEMATICAL INTELLIGENCER
Figure 5. Polyhedral immersion of {3, 7la with octahedral symmetry.
Figure 6. Kepler-Poinsot-type realiza tion of {7, 3)s with octahedral symme try.
A U T H O R S
J. M. WILLS
PETER SCHOLL
ACHILL SCH0RMANN
Fachbereich Mathematik
School of Mathematical Science
Fachbereich Mathematik
Universitat Siegen
Peking University
Universitat Siegen
D-57068 Siegen
Beijing
1 00871
D-57068 Si egen
Ch i na
Germany
Germany
e-mail: [email protected]
e-mail: wills@mathematik. uni-siegen.de
Achill SchOrmann completed his doctoral Peter Scholl, a native of Siegen, receives
work at Siegen with a prize-winning thesis
his doctorate at the University there in 2002,
on "Sphere-Packings." He is now on a
lems and convexity, and also on symme
with a thesis on "Sphere-Packings and Mi
postdoctoral research visit with Professor
try and
croclusters." His favorite hobby is chess.
Chuanming Zong. His hobbies are football
ported to The lntelligencer on both sides
(soccer) and cycling
of his work, as in vol. 20 (1 998), no. 1 ,
.
Jorg Wills has worked on extremal prob combinatorial geometry. He has re
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 contribu tions to Felix Klein's and H.S.M. Coxeter's general idea of bringing algebra and geometry closer together.
[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.
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
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 ..
Mathematics 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: [email protected]
I
M arjorie Senechal , Ed itor
t 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
I
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
must return to the text, which is the
structive to have a small sample of
than abstracted and objectified. The
touchstone of the story's objectivity.
prose lacking both. What follows is
world of the story is a creation of the
Although the reader imagines the
structured by following that list. The
reader in co-operation with the teller
story, the reader does not make it up.
attractiveness of narrative is, I think,
of the story; it is imaginary yet filled
There is some evidence that the oppo
one of the reasons that mathematics
with authenticating detail. Despite be
site is true: one is, to a degree, the
teachers
ing invented, the relations described in
stock of one's imagination, the product
courses. It is as though students learn
introduce history in
their
a story, unlike the invented characters,
of the stories that one knows, includ
are almost exclusively ordinary rela
ing of course one's own. The stories we
brains. Yet although the capacity to
tions drawn from the real world; little
know tell us how things can be in the
prove mathematical theorems presum
new vocabulary is needed and no new
world; through them, we see the world,
ably arises later than the capacity to
semantics. Also, despite the readers'
though not necessarily as it is. The
tell stories, it depends mostly on the
contributions, different readers will
world can be more than one way; we
same imaginative and reasoning ca
tell recognizably the same story after
happily know more than one story
pacities.
wards. Non-historical characters have
about the same relations, sometimes
only
those
characteristics
that
are
given to them by the storyteller, and
best when using both sides of their
even about the same historical charac
Mathematics
ters.
When we prove something in mathe
are
Stories can be applied to the real
matics, we postulate some things to
mainly relational (we need not be told
world, for example when we say his
talk about and the relations that they
that Hamlet is human).
tory is repeating itself, or that someone
are to have among themselves. These
the
important
characteristics
A lot of pre
suppositional baggage (Lear is a king,
are the
Othello is a Moor, Macbeth is a Scot)
recognizes this move as the beginning
is conveyed in telling a story by stag ing a play with actors. The shift from this relational limitation to the compli
Discovery is real despite its basis
cated development of character and much fiction since
everyone
of a piece of mathematics by the time they have learned to let x be the num ber of coins in Johnny's pocket. We
frequently do this by calling upon a
in invention.
standard set provided by an axiomatic
is a Scrooge or Uriah Heep; fictional as
new story about characters already known from their appearance in the lo
seeing inside characters' heads makes
1800 unsuitable for
purposes of this comparison.
dramatis personae;
system. This is analogous to telling a
The writer and reader, or teller and
well as historical behaviours become
listener, are complicit in a shared act
paradigmatic. The truths of fiction are
cal mythology: Hera is the wife and sis
of make-believe,
not those of accurate reportage but the
ter of Zeus, etc. The working out of
dream. The story is not an arbitrary
revelation of significance (the moral of
such relations in mathematics-actu
succession of descriptions; rather the
the story.)
alizing initial potential, as it is called in
a kind of waking
actions of the story follow in accor
In addition to the telling of stories,
dance with physical causality and char
there can be serious discourse about
time passing is the reader's time, but
acters' reasons and intentions.
stories-is
primarily
deductive;
the
A story
the subject matter of a fictional story.
the presentation is linear (up to a
has structure; it is not just a list of who
One can talk, indeed argue, about Sher
point-see below) as in a story. The
was where when; a list of facts about
lock Holmes without telling a Holmes
end of a proof is another set of rela
the basket, the grandmother, the little
story, but our discussion depends on
tions among the characters; it is a mat
girl, and the wolf do not convey "Little
those stories. We may ask, in serious
ter of choice where to end a deductive
Red Riding Hood" even to people who
discussions, how fictional characters
chain, as most situations have further
already know the story. The logical ca
are created, or we may ask "factual"
consequences. Where we choose to
pacity as well as the imagination of the
questions
leave off defmes the conclusion of the
reader is engaged; each new remark,
nose?"). Such "fictional" questions may
theorem. It used to be thought that the
character, and event needs to be fitted
not be answerable if the text is silent
objects of geometry had to be ab
into the picture already drawn in men
and deduction is impossible. Finally,
stracted from the physical world, just
tal
stories are important and pervasive;
as some early novels like
source of much of the pleasure of a
some
Travels
story. Plots and subplots, however
characters are better known than al
complex, have to be presented in a lin
most all of our contemporaries.
space.
This
engagement
is the
("how
long
is
Hamlet's
fictional and many historical
ear way with devices like flashbacks to
and
Gulliver's Robinson Crusoe mas
queraded as memoirs (we don't know why), but mathematics like fiction has become more frankly invention.
fill in out-of-order details. Not every
Transition
thing is given equal weight; the more
The preceding list of characteristics of
Fiction and history
concentrate
on
the significant relations among char
dramatic episodes are given emphasis,
stories was unengaging in part because
acters, but accidents, chance encoun
tension builds, conflict is resolved. If
it had neither narrative nor logical
ters, and other external events may en
we want to know what happened in the
structure. I did not know how to give
ter into the narrative. Mathematics is
story, in the world of the story, we
it either, and anyway, it may be in-
supposed to depend
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THE MATHEMATICAL INTELLIGENCER
only
upon the
stated relations among the mathemati cal objects; to impute any other rela tions is an error, however helpful intu ition 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 fic tional analogue of mathematical de duction: their story must work itself out despite their having invented everything in it but human and physi cal 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 dis covery is real despite its basis in in vention. The importance of intuition as a source of such discoveries is com mon 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 an swers 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 de duction in stories and mathematics is interestingly different, almost oppo site. In mathematics one imagines in order to see why what is implied is im plied; 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 border line cases for deduction to be depend able. But on a larger scale, the function of much mathematics learning is to stock the imagination, not wholly dif ferently 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. Fic tion is more purely permissive.
The engaging feature of mathemati cal discourse is that the relations dis cussed actually relate: they are not ab stract. Geometry has nothing to say about collinearity as an abstraction; all study of collinearity and non-collinear ity 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, ambi tious treachery, and procrastinating re venge, but put persons into those rela tions in order to engage our attention.
Mathematical facts, without some understanding of why they are the way they are , are almost i m possi ble to learn. In speaking of staging a play, I men tioned the presuppositional baggage that mathematical objects are by defi nition 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 con sideration 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 char acterless and the relations that we imagine are all specified in terms of a few simple relations like set member ship and set inclusion.) Narrative, in contrast, treats what are frankly and ultimately special cases. Mathematicians are usually in terested in special cases for the pat terns they reveal. The best of narra tive'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 ef fort that is required to learn a mathe matical proof, the reader's contribu tion being considerable, if the proof is then repeated, most of what was imag ined 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 dif ferent circumstances. H. B. Griffiths has pointed out to me that specializing an algebraic structure, distinguishing one specific example of a class of iso morphic structures, "is like casting a play, and the flavour of the special mathematics corresponds to that of a particular production: all such produc tions have the same abstract struc ture." The dependence on the text of the story, which in fiction is absolute, is much less in mathematics. Never theless, the importance of the text for the objectivity of the mathematics has led to the philosophical position extreme 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 formaliza tions 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 mathe matics 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.
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
Conclusion
46
THE MATHEMATICAL INTELLIGENCER
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. 51-7 1 . [6] I have done this in two papers, "Mathe matics and Fiction 1: Identification," and
"Mathematics and Fiction II: Analogy," to
appear in Logique et Analyse.
'
A U T H O R
in mathematics , . is not cashed out by proving it but by proving things with it
ROBERT S. D. THOMAS St John's College and
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.
Department of Mathematics University of Manitoba Winnipeg, Manitoba
R3T 2N2
Canada e-mail: [email protected]
NOTES
Robert Thomas studied at the uni
[1 ] He completed this task in his Rede Lecture
versities of Toronto, Waterloo, and
at Cambridge in 1 959, "The Two Cultures
Southampton. His non-professional in
and the Scientific Revolution."
terests have always extended into the
[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
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-
on Basic Philosophy. Volume 7 , Epistemol ogy and Methodology Ill: Philosophy of Sci
York: Times Books, 1 995. [4] John Allen Paulos had much the same idea
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
and wrote Once Upon a Number (New
ics. He is editor of Philosophia Mathe
York: Basic Books, 1 998), which I recom
matica (www .umanitoba.ca/pm) and
mend to anyone who finds my discussion
treasurer of the Canadian Society for
interesting.
History and Philosophy of Mathemat
[5] The philosopher Mario Bunge has pub
ics (www.cshpm.org). His wife, now
lished two slightly different lists of gross
in children's literature though once
ways in which mathematics and fiction in
trained and employed as a chemist,
particular differ; some apply to history and
regales him with stories as they jog.
some do not. All can be debated. Treatise
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 thefamous 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 ROTAS 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 ROTAS 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
T
A
R
s u
B
0 · T
A
s A
0
p
E
R
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 SUBITO 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 SATOR (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 ROTAS. If the different letters are regarded as points of different col ors, then the SATOR square is the mir ror image of the ROTAS 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 arepen nis, 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 SATOR) 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 ROTAS 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 SATOR. 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 ROTAS 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.
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, "II 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), 305-31 0 .
9. C . Wescher, Bulletin de Ia Societe des An tiquaires de France
( 1 87 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
Acknowledgment. We thank the associates of the Aquincum Museum (Budapest, Hungary) for their kind assistance.
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
REFERENCES
1 . A. Domenicano and I . Hargittai, "Alpha
giosa 1 (1 965), 433-439.
1 4. F. Grosser, "Ein Neuer Vorschlag zur Deu
betic Magic Square in a Medieval Church,"
tung der Sator-Formel," Archiv fOr Reli
The Mathematical lntelligencer 22 (2000),
gionswissenschaft 24 (1 926), 1 65-1 69.
no. 1 , 52-53. 2. B. Artmann, "Conceptual Magic Square," The Mathematical lntelligencer 22 (2000),
3, 4. 3 . N . Gauthier, "Parsing a Magic Square,"
Aldo Domenicano Department of Chemistry, Chemical Engineer ing and Materials University of L'Aquila
The Mathematical lntelligencer 22 (2000),
1-671 00 L'Aquila, Italy
no. 4, 4.
e-mail: [email protected]
4. G. De Jerphanion, "La Formule Magique Sator Arepo ou Rotas Opera: Vieilles
Istvan Hargittai
Theories et Faits Nouveaux," Recherches
Budapest University of Technology and Eco
de Science Religieuse 25 (1 935), 1 88-225.
nomics
5. F. Haverfield, The Archaeological Journal 56 (1 899), 31 9-323; R. G . Collingwood,
H-1 521 Budapest, Hungary e-mail: [email protected]
VOLUME 24, NUMBER 3, 2002
49
Joannes Keplerus Leomontanus: Kepler's Childhood in Weil der Stadt and Leon berg 1 571 - 1 584 by Hans-Joachim Albinus
F
rom a modem point of view Jo hannes Kepler (27 December 1571-15 November 1630) is described as an astronomer, a physicist, a car tographer, a calendar expert, and not the least, a mathematician. In astron omy his name is associated with Ke pler's laws of planetary motion, 1 in mathematics with Kepler's rule2 (Ke plersche 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 rel atively small area in and around the an cient duchy of Wtirttemberg. They are easily accessible by car or train, for ex ample 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 Evange list (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 Weil 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 mer chant 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 Weil der Stadt, where her parents Melchior and Mar garetha 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 Biirger meisters Sebald Kepler t 1596/1986 Sanierung + Restaurierung" (Market place 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 market place left of the town hall at the be ginning 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:
b
J
a f(x) d:x: ==
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck,
Aartshertogstraat 42,
(
b-a f(a) 6
--
( ) ) b-a
b , + 4j 2 - + f( )
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] .
8400 Oostende, Belgium
4Marbach i s known a s the birthplace o f the poet Friedrich Schiller and the mathematician and astronomer Jo
e-mail: [email protected]
hann Tobias Mayer.
50
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
The house where Kepler was born, now Ke pler-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.
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 Weil 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 Wallestein, 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 Weil 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
Leonberg, to the east of Weil der Stadt, was founded in 1248/1249 by Count Ul rich I mit dem Daumen of Wiirttemberg (i.e., the one with the abnormal [right] thumb) as the first of three new towns5 in the county.
5The others were Schorndorf and Waiblingen.
VOLUME 24, NUMBER 3, 2002
51
Weil der Stadt. At that time, Leonberg enjoyed an economic upswing, because between
1560 and 1565 Duke Christoph
of Wiirttemberg had the Leonberg cas tle built as one of his seats of govern ment. 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 de cisive 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 be come of importance later in Johannes Kepler's life. Firstly, there was com pulsory school attendance since the great ecclesiastical constitution7 (Grofie
Kirchenordnung)
enacted
1559/1582
(as an end to the Reformation in Wiirt temberg). 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 de mand by Luther. Secondly, there was a state-run university in Tiibingen.8 It of fered a type of higher education, the Schwabische Laufbahn, which started with attendance at a German basic
Kepler Memorial in Weil der Stadt.
school, then a Latin grammar school, and led to the university, either via the
One may ask why Kepler's parents
relsome, it must have been unbearable
moved to Leonberg, giving up the ad
for these people to live together in
schools 1 1 or via the two colleges (paed
vantages of inhabitants of a free im
such a confined space. Religious con
agogium) in Stuttgart and Ttibingen. A
perial town and becoming subjects of
flict may also have played a part. In
degree in theology then enabled the
the Duke of Wiirttemberg instead. Ac
graduate to become a priest, a public
Kepler wrote
1522 a Lutheran community came into being in Weil der Stadt, but in 1573 the
later, the housing conditions in his
Counter-Reformation6 began. Kepler's
Stift, a seminary, was affiliated with
cording to
accounts
grandfather
the
the university as a place of living and
inadequate. Apart from his parents,
leader of the Protestants, therefore
training for the theology students. Par
his
brother,
and
himself,
younger
living there, too. On the ground floor
had
been
servant, or a teacher. The Tiibinger
parents' house in Weil der Stadt were
brothers and sisters of his father were
Sebald
nine lower-9 and four higher10 monastic
the family justifiably feared financial
ticularly talented male natives of Wiirt
disadvantages.
temberg were given a ducal scholar
Leonberg was and remained Re
there was a shop for herbs. Both Hein
formed, and with more than
rich and Katharina being very quar-
habitants it was somewhat larger than
1200 in
ship which included free stay at the Stift and at the university. Kepler was to benefit from this later.
6The Counter-Reformation prevailed in Weil der Stadt for the first time in 1 597 and once and for all in 1 628. 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 1 477 by the Wurttemberg Duke Eberhard im Barte (i.e., bearded). This was relatively late, because around Wurttemberg the universities of Heidelberg (founded 1 386), 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. 1 0Bebenhausen, 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-- 1 579." (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 IAstronom I Johannes Keppler l von 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 t o 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 Ke pler 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
1 2Back 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. 1 4Aiso 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 - 1 85 1 ) -first a close friend. later a fierce critic of Schelling-and the doctor Karl Wilhelm Hochstatter ( 1 78 1 -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 AS TRONOMEN I JOHANNES KEPLER / ELLMENDINGEN AM HEIMATTAG 5. JULI 1959." (Until the tum of the [twen tieth] century the inn Zur Sonne stood in this place; from 1579 to 1584 Hein rich Kepler from Weil der Stadt lived here with his wife and son, the subse quently world-famous astronomer Jo hannes Kepler; Ellmendingen, 5 July 1959, Local History Day.) Of course Jo hannes's younger brother Heinrich also lived there! In the village of Ell mendingen there was no school. There fore 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 House where Kepler's mother was born in Leonberg (Eitingen). the child's interest in astronomy. During the winter of 1582/1583 he went back to attend the Latin grammar to Kepler's mother. The accompanying east of Leonberg. The basic structure of school in Leonberg. He probably inscription "Zur Erinnerung an I Katha the monastery, the walls, and some of stayed in Eltingen with his maternal rina Kepler, geb. Guldenmann I geb. zu the buildings are still standing and open grandparents, the Guldenmanns. To Eltingen am 8. 1 1 . 1547/Errichtet von to the public-among other things the day Eltingen is a district of Leonberg; der Gemeinde I Eltingen im Jahre 1937" Pralatur (prelate's house). Next to the in the mid-16th century the two places (In memory of Katharina Kepler, nee gate of the building there is a plaque in Guldenmann, born in Eltingen 8 No memory of Kepler, carrying the in were two kilometers apart. Kepler completed his elementary vember 1547; erected by the munici scription "Hier weilte von 1584-86/der education at the Latin grammar school pality of Eltingen in 1937) is located Astronom I JOHANNES KEPLER I als in 1583 and was a good student. His somewhat hidden on a wall behind a Klosterschiller" (Here stayed from 1584 teacher recommended him and he was lime tree. to 1586 the astronomer Johannes admitted for the examination that would After passing his exam in 1583 Kepler as a pupil at the monastic school). qualify him for secondary school. In Kepler returned to Ellmendingen and On one of the corners of the building May 1583 he passed the Landexamen, again did farm work while he waited another plaque gives the history of the which was given once a year in for a vacancy in the monastic school. Pralatur including a reference to Stuttgart. This entitled him to the ducal In the spring of 1584 the Kepler fam Kepler. Today, some of the remaining scholarship. The house of Katharina ily moved back to Leonberg, this time buildings of the monastery serve as res Kepler's parents still stands at Carl into a house near the lower town gate. idential and business buildings. As early as October 1586, Kepler Schminke-Stra.Be 54, and is marked by a The building no longer exists. In May 1584 Johannes's sister Margaretha was changed schools and attended the plaque with the following inscription: "Geburtshaus I der I Mutter des Astrono born there. higher monastic school in Maulbronn. The former Cistercian monastery of men Kepler: I Katharina Guldenmann I Maulbronn lies north of Pforzheim, geb. 8. 1 1 . 1547." (In this house the Kepler's Further Life in the about 40 kilometers from Leonberg. mother of the astronomer Kepler was Duchy of Wiirttemberg born: Katharina Guldenmann born 8 In October 1584 Kepler attended the Many of the old buildings still stand, November 1547.) lower monastic school in Adelberg, the among them some that Kepler fre On the corner of Carl-Schminke so-called Grammatistenkloster. Adel quented. However, most of them, in Stra.Be and Hindenburgstra.Be in Eltin berg lies between Schorndorf and Gop cluding the school, 15 have been rebuilt gen there is also a fountain, a memorial pingen, a little more than 50 kilometers 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 philologian 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 Maul bronn 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 Mas ter'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 Jo hannes Kepler, showing a portrait and the inscription "Johannes Kepler 15711630 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 docu ments 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 univer sity,20 and a certificate of discharge. The other case contains documents concerning Holderlin, Schelling, and Hegel.21 There is still another place in Tiibin gen which reminds us of Johannes Kepler. Following the usual way from the Stijt or the marketplace to the cas tle, 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 mathe matics at a corporative Protestant school. Actually Kepler considered his move to Graz as a brief interrup tion 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. Nev ertheless in his later life he returned to Wiirttemberg several more times. In 1596 Kepler left Graz for several months and went to Stuttgart and Tiibin gen, among other things to prepare the printing of his first book Mysterium Cos mographicum, a heliocentric descrip tion of the world, but still in a traditional incorrect style. He took the opportunity to offer Duke Friedrich I to build a sil ver 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 pre pare the printing of his book Astrorw mia Nova, one of his main scientific works. Kepler used the occasion to pre sent 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 be ing 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, have been expelled from the
Stiff. Tertium non datur."
must
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
56
THE MATHEMATICAL INTELUGENCER
Stiff!
working as Upper Austria's mathe matician (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 Schick hardt22 (1592-1635), who later be came a professor at the University of Tiibingen, and who invented the four species calculating machine before Blaise Pascal. In 1623 and 1624 Schickhardt had built two of these ma chines23; one of them got lost in the chaos of the Thirty Years' War, the other, originally meant for Kepler to facilitate his astronomical calcula tions, was destroyed in a fire at Schickhardt's workshop. In 1937 Franz Hammer found plans by Schick hardt,24 from which Bruno Baron von Freytag LOringhoff, then professor of philosophy at the University of TUbin g en, was able to build a reconstruc tion 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 Stadt museum 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, be cause 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 tem porarily, there now began a witchcraft trial lasting fourteen months. Katha rina was threatened with the death penalty. Thanks to Johannes Kepler's great commitment, making use of all
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The Duchy of Wiirttemberg in Kepler's times, showing Weil der Stadt (a), Leonberg (b), Ell mendingen (c), Adelberg {d), Maulbronn (e), Tubingen {f), Stuttgart {g), Guglingen (h) (accord ing to [1 1] and [20])
juridical means, his mother was ac quitted in October 1621, after being tor tured 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 in carcerated no longer exists. Katharina Kepler died at age 74 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 un known. A plaque on the ancient ceme tery of Leonberg next to the old part of the town, carrying the inscrip tion "Gewidmet I dem Andenken I der aufI 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 62 1 . 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, Katha rina nee Guldenmann, who died here on 13 April 1622 and who lies here in this cemetery), but its information is proba bly 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 approxi mately 50 meters to the left, where it is set into the cemetery wall. Summing Up
In the area of the old Duchy of Wiirt temberg there are no other direct ref erences to Kepler in buildings, memo rials, etc. than those in Weil der Stadt, Leonberg (including Eltingen), Tiibin gen, and Adelberg. These places27 are also connected to four major trends of contemporaneous history influencing Kepler's life: Reformation and Counter Reformation; Ptolemaic and heliocen tric 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 ex ample 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
[1 4] Schutz, Wolfgang: Die historische Alptadt von Wei/ der Stadt. Geiger, Horb, 1 996.
der Technik, Stuttgart, 1 995), improved by
[1 5] Schiinwitz, Ute: Er ist mein Gegner von je
references to Kepler's original writings.
her. Friedrich Wilhelm Joseph Schelling
There exists an English translation of the 3rd
und Heinrich Eberhard Gottlob Paulus.
edition (Dover, New York, 1 993).
Keicher, Leonberg, 2001 .
[4] Freytag Liiringhoff, Bruno Baron von:
[1 6] Setzler, Wilfried, et al. : Leonberg. Eine
"Prof. Schickards Tubinger Rechenmas
altwurttembergische
chine
Gemeinden im Wandel der Geschichte.
von
Schriften,
4.
1 623."
Kleine
TObinger
Stadt Tubingen, 1 981 .
Stadt
und
ihre
Wegrahistorik, Stuttgart, 1 992. There was
[5] Gaulke, Karsten; Weber, Ricarda: Oas
an index published seperately by the
Kepler-Museum in Wei/ der Stadt. Kepler
Stadtarchiv Leonberg in 2001 ; the author
Gesellschaft, Weil der Stadt, 1 999.
is Karl-Heinz Fischiitter.
[6] Gramm, Bernadette; Walz, Eberhard: His
[ 1 7] Setzler, Wilfried: TObingen. Auf a/ten We
torischer Altstadttohrer Leonberg. Stadt
gen Neues entdecken. Ein Stadtfuhrer.
archiv Leonberg, 1 991 .
Schwabisches Tagblatt, Tubingen, 1 997.
[7] Hammer, Franz; Hammer, Esther; Seck,
Friedrich: Johannes Kepler-Selbstzeug nisse.
Frommann-Holzboog,
Stuttgart,
1 971 '
[1 8] Sigmund, Karl: "Kepler in Eferding." The Mathematical lntelligencer 23 (2001 ), no.
2 , 47-5 1 '
[1 9] Sutter, Berthold: Der Hexenprozef3 gegen
[8] Hoppe, Johannes: "Johannes Kepler." Bi
Katharina Kepler. Kepler-Gesellschaft and
ographien hervorragender Naturwissen
Heimatverein, Weil der Stadt, 1 984 (2 nd
schaftler, Techniker und Mediziner, 1 7.
ed.).
Teubner, Leipzig, 1 987 (5th ed.). [9] Kirschner, Karl; Stroh, Martin; Rosier, Her
mann: Chronik von Adelberg, Hundsholz
[20] Vann, James A: The making of a state. Wurttemberg 1 593-1 793. Cornell Univer
sity Press, Ithaca, 1 984. German transla
und Nassach. Gemeindeverwaltung Adel
tion: Wurttemberg auf dem Weg zum
berg, 1 964.
modernen Staat 1 593- 1 793 (Deutsche
[1 0] Kistermann, Friedrich W . : "How to use the
Schickard calculator. Types of recon structed Schickard calculators." Annals of the History of Computing, 23 (200 1 ) , no. 1 ' 80-85. [1 1 ] Methuen, Charlotte: Kepler's TObingen.
Ashgate Publishing, Aldershot, 1 998.
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;
[1 2] Muller, Ernst; Haering, Theodor; Haering,
Luipold, Hans-A : Maulbronner K6pfe. Ge
Hermann: Stiftskopfe. Schwabische Ah
fundenes und Bekanntes zu ehemaligen
nen des deutschen Geistes aus dem
Seminaristen. Melchior, Vaihingen an der
TObinger Stitt. Salzer, Heilbronn, 1 938.
Enz, 1 987.
[1 3] Raith, Anita: "Das Hexenbrennen in Leon
berg." In: Durr, Renate (ed.): Nonne, Magd
lnnenministerium Baden-Wurttemberg
oder Ratsfrau. Frauenleben in Leonberg
DorotheenstraBe 6
aus
vier Jahrhunderten.
Beitrage zur
Stadtgeschichte, 6. Stadtarchiv Leon berg,
1 998, p. 53-73.
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 83 1 /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.
Is (Was) Mathematics an Art or a Sciencet David E. Rowe
Send submissions to David E. Rowe, Fachbereich 1 7 - Mathematik, Johannes Gutenberg University, 055099 Mainz, Germany.
Rowe, Ed itor !
I
f 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 de partment, knowing whether we should be counted as belonging to the Geistes or to the Naturwissenschajten (hu manities or natural sciences) could well have serious budgetary implica tions. Of course most mathematics de partments are now facing a more press ing 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 cer tainly (b), then you can probably pro ceed on to the next round of more con crete questions (can mathematics always be run using Windows?, etc.). But this column is concerned with his torical matters; let me devote this one to the loftier issue raised by the (par enthetical) 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 op posed 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 ap proach, and vice-versa. About a cen tury ago, when the prolific number-the orist Edmund Landau learned that young Arnold Sommerfeld was ex pending 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 wor thy of the name. Sommerfeld himself may have grown tired of hearing about "monkey grease." In 1906 he left the field of engineering mathematics to be come 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 af ter 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 uni lateral decision to dismiss Brouwer from the editorial board of Mathema tische Annalen. [Dal] To some in Got tingen circles, it looked as though Hilbert had defeated the mystic from Amsterdam, but their victory celebra tion 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 weak nesses 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 al ready over by the time Godel proved his incompleteness theorem. The fire had just blown out, enabling the foun dations experts to go on with their work in a far more peaceful atmos phere (for an overview, see the essays in [H-P-J]). Herbert Mehrtens suggests that fun damental differences regarding mathe matical 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 con troversial Congress of Mathematics in Strasbourg. Since German mathematicians were excluded from participation at the latter meeting, sev eral 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 gen eral 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 math ematicians. 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, 1 987, p. 42.)
[Meh-2] .
Wranglers and physicists. During this
search. Ironically, this community has
N o doubt philosophical disputes over
same time in Germany just the oppo
often
existential difficulties cut deeply, but
site
mathematical
"Hilbert's Gottingen," whereas Hilbert himself has often been seen through the
ernists and anti-modernists
prevailed.
There,
come
to
be
remembered
as
Mehrtens emphasizes that the intense
purism held sway, reaching a high
foundational debates during the early
water mark in Berlin in the 1870s and
lens of his later "philosophical" work,
twentieth century took place against a
80s, the heyday of Kummer, Weier
the formalist program of the 1920s (for
background of rapid modernization,
strass, and Kronecker (see [Row-2]).
and this had a major impact on math
Modernization at the German uni
three recent reassessments of his ap proach to foundations, see [Cor], [Row-
of
versities elevated the status of the nat
3], and [Sieg]). Clearly, Richard Courant
modernity on higher education in gen
ural sciences, which had long been
had · a very different image of Hilbert in
ematical
research.
The
impact
mind when he wrote the first volume of
eral, and on mathematics in particular,
overshadowed by traditional humanis
is easy enough -to discern, and yet the
tic disciplines.
effects on mathematical practice de
mathematicians began to pay closer at
pended heavily on how higher mathe
tention to scientific and technological
Hilbert himself saw mathematics in very
matics was already situated in various
problems. Felix Klein took this as his
broad terms,
As part of this trend,
Courant-Hilbert,
Mathematische Meth oden der Physik in 1924. Just as clearly, a vision sustained by
countries. Thus, Hardy's purism can
principal agenda in building a new kind
strong views about the nature of math
best be appreciated by remembering
of mathematical research community in
ematical thought. The same can be said
that
Cambridge
Gottingen, where Karl Schwarzschild,
had long upheld applied mathematics
Ludwig Prandtl, and Carl Runge pro
of his leading rival, Henri Poincare,
whose ideas had a lasting impact on
in the grand tradition of its famous
moted various facets of applied re-
philosophers of science.
60
nineteenth-century
THE MATHEMATICAL INTIELLIGENCER
Poincare's work often drew its in spiration from physical problems, and he made numerous important contri butions to celestial mechanics and electrodynamics (see [B-GJ and [Dar]). In Science and Hypothesis, Poincare examined the role played by hypothe ses 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 "math ematical truths are derived from a few self-evident propositions, by a chain of flawless reasonings," that they are "im posed not only on us, but on Nature it self." ([Poi], p. xxi). Poincare's alter native view, a doctrine that came to be known as "conventionalism," was sup ported by a trenchant analysis of the geometry of physical space, then a mat ter of considerable controversy (for a recent analysis of Poincare's views, see [B-MJ). Hermann von Helmholtz had at tacked the Kantian doctrine according
to which our intuitions of space and time have the status of synthetic a pri ori knowledge. This position had be come central to Neo-Kantian philoso phers 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 prin ciple a person could learn to perceive spatial relations in a different geome try, either spherical or hyperbolic. Like Helmholtz, Poincare rejected the Kant ian claim that the structure of space was necessarily Euclidean, but he stopped short of adopting the empiri cist 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 re quire fmding a physical criterion to dis tinguish between the candidate geome tries. This, however, amounted to laying down conventions for the affine and metric structures of physical space in advance, which effectively under mined 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 ap proach in Grundlagen tier Geometrie (1899). Hilbert would have been the last to deny the empirical roots of geo metrical knowledge, but these ceased to be relevant the moment the subject became formalized in a system of ax ioms. By packaging his axioms into five groups (axioms for incidence, order, congruence, parallelism, and continu ity), Hilbert revealed that these intu itive notions from classical geometry continued to play a central role in structuring the system of axioms em ployed by the modern geometer. Nev ertheless, these groupings played no di rect role within the body of knowledge, since they never appeared in the proofs of individual theorems. Thus, for Hilbert, the form and content of geom etry could be strictly separated. In con trast 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 dur
ing their student days at the ETH, where both attended Geiser's lectures on differential geometry. Later, as colleagues at the ETH, Gross mann 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
439-445] .
sweeping
man who didn't need to rely on anyone
constituting a pure science whose ar
through the German universities, and
Modernity
except perhaps a friendly neighbor.
guments retain their validity without
throughout the two decades preceding
Over on the West Coast, Stanford's
any reliance on intuition or empirical
the outbreak of World War I enroll
George P6lya gave mathematical ped
ments in science
support.
was
and
mathematics
agogy a new Hungarian twist aimed at
Hilbert originally conceived his fa
courses in Gottingen grew dramati
fostering
mous lecture on "Mathematical Prob
cally, as did the number of foreigners
Whereas
lems" as a counter to Poincare's lecture
attending them.
method" taught that doing mathemat
mathematical advocates
creativity.
of the
"Moore
at the inaugural ICM in Zurich (see
Hilbert's ideas exerted a major im
[Gray] , pp. 80-88). Stressing founda
pact on American mathematics, not
orems and finding counterexamples,
tions, axiomatics, and number theory,
just on those who studied under him in
P6lya stressed the importance of in
he set forth a vision of mathematics
Gottingen.
ductive thinking in solving mathemati
that was at once universal and purist.
sponded to his message, none did so
cal problems. His How
Aside from the sixth of his twenty
with more enthusiasm than Eliakim
over a million copies and was trans
three Paris problems, he gave only
Hastings Moore, who helped launch re
lated into at least 1 7 languages [Alex,
faint hints of links with other fields.
search mathematics at the University
p. 13]. Not to be outdone, Courant en listed Herbert Robbins to help him
Among
those
who
re
ics was synonymous with proving the
to Solve It sold
Hilbert's address, in fact, was based on
of Chicago during the 1890s. Moore's
the claim that mathematics, as a purely
school owed much to Hilbert's re
write another popular text:
What is
rigorous science, was fundamentally
search agenda, particularly the ax
Mathematics?
Courant
different from astronomy, physics, and
iomatic approach to the foundations of
thought he had the answer, but then so did P6lya, R. L. Moore, and Bourbaki!
Presumably
all other exact sciences. Taking up a
geometry. Oswald Veblen pursued this
theme popularized by the physiologist
program, first as a doctoral student at
Back in Gottingen during the Great
Emil du Bois-Reymond, who main
Chicago and later at Princeton, but the
War, physics and mathematics had be
tained that some of mankind's most
leading proponent of this style was an
come ever more closely intertwined.
perplexing questions could never be
other
Texan
Einstein's general theory of relativity
answered by science, Hilbert turned
Robert Lee Moore. Like his namesake and mentor, R. L.
his circle, and this wave of interest in
the tables. For him, this was the quint
Chicago
product,
the
captivated the attention of Hilbert and
essential difference between mathe
Moore served as a "founding father" for
the subtleties of gravitation soon trav
matics and the natural sciences: in
a distinctively American style of math
eled across the Atlantic. Columbia's Ed
mathematics alone there
could be no
ematics [Wil]. He and his followers
ward Kasner was the first American
ignorabimus
every
well
acted on their belief in a fundamentally
mathematician to take up the challenge,
posed mathematical question had an
egalitarian approach to their subject
but he was soon followed by two of
answer, and with enough effort that an
based on the (unspoken) principle that
E. H. Moore's star students, G. D. Birk hoff and 0. Veblen. Harvard's Birkhoff
because
swer could be found. But Hilbert went
"all theorems are created equal" (so
further. This seemingly bold claim, he
long as you can prove them!). Moore's
maintained, was actually an article of
students at the University of Texas
faith that every mathematician shared.
spread this gospel, making point-set
Of course this was Hilbert in 1900; he
topology one of the most popular sub
hadn't yet met Brouwer!
jects in the mathematics programs of
lems that physics had cast upon the
American graduate schools. True, this
mathematicians' shore. His monograph
ethos
Relativity and Modern Physics
For much of the twentieth century, young
North
American
mathemati
in
its
purer
form
remained
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
ap
cians were taught to believe that doing
largely confined to colleges and uni
mathematics meant proving theorems
versities in the heart of the country.
ten today, it contains a result of major
(rigorously). This ethos gained a great
General topology made only modest in
significance for modem cosmology:
peared in 1923; although nearly forgot
deal of legitimacy from the explosion
roads at the older elite institutions on
of interest after 1900 in foundations,
the East Coast as the Princeton school
Birkhoffs Theorem: Any spherically
axiomatics, and mathematical logic,
of J. W. Alexander, Solomon Lefschetz,
symmetric
fields which emerged along with the
and Norman Steenrod emerged as the
empty
first generation of home-grown re
nation's leading center for algebraic
equivalent to the Schwarzschild so
search mathematicians in the United
topology.
lution, i.e., the static gravitational
States. During the 1880s and 90s, a
R.
L.
Moore's
Socratic
teaching
solution
space
field
of Einstein's equations
is
field determined by a homogeneous
number of young Americans came to
style, the so-called "Moore method,"
spherical mass (see [Haw-Ell]. Ap
Gottingen to study under Felix Klein,
played an integral part in his philoso
pendix B, for a modem statement
who gladly took on the role of training
phy of mathematics, which evinced the
and proof of this theorem).
those who became mentors to that first
rugged individualism typical for math
generation.
But
by
the
mid- 1890s,
ematicians from the
prairie.
Book
Both Birkhoff and Veblen got to know
Hilbert gradually took over this formi
learning had little appeal for them: this
Einstein in 192 1 , when he delivered a
dable
was mathematics for the self-made
series of lectures in Princeton. Einstein
62
task
[Par-Row,
pp.
THE MATHEMATICAL INTELLIGENCER
189-234,
Princeton's new Institute for Advanced Study in 1 933, 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 1 9 1 7, he worked off and on as Einstein's assistant for some ten years, longer than anyone else (see [Pais] , pp. 483-50 1). 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 Figure 3. Gosta Mittag-Leffler, flanked by Henri Poincare and Edmund Landau, talking with
day return the favor to physics, if only
his friend Carl Runge, back to camera. The occasion was probably the Second ICM held in
by showing the physicists the kind of
Paris in 1900. Mittag-Leffler and Runge were perhaps reminiscing about Karl Weierstrass's
theory they needed in order to explore
famous lectures on function theory, which both heard during the 1 870s. Landau, himself a
the outermost and innermost regions
gifted analyst, later joined Runge in Gottingen, where they stood at opposite ends of the
of the universe. In Princeton, most of
pure/applied spectrum. (From George P61ya, The P6/ya Picture Album: Encounters of a Math ematician,
ed. G. L. Alexanderson, Boston: Birkhauser, 1 987, p. 26.)
Einstein's assistants were recent Eu ropean emigres who had managed to flee before the full force of Nazi racial policies took hold.
afterward adapted these into
book
form, and they were published the fol lowing year under the title
ing of Relativity.
The Mean
Around this time,
Following Weyl's lead, the
Due to his seniority, Edmund Lan
Princeton trio of Eisenhart, Veblen,
definite.
dau was not among those who lost
and Tracy Thomas spearheaded re
their jobs in the Nazis' initial effort to
search on the
purify the German civil service [Sch].
projective
space
of
Veblen took up differential geometry,
paths, which led to a new foundation
His exodus from the scene was more
joining his colleague
Luther Eisen
for general relativity closely connected
poignant and chilling,
hart's quest to build new tools adapted
to the theory of Lorentzian manifolds
light of recent discussions of how "or
to the needs of general relativity. This
(for a survey of their work, see [Tho]).
dinary Germans" behaved during the
research explored the virgin territory
General relativity and cosmology re
events leading up to the Holocaust
especially in
of spaces with semi-Riemannian met
mained major playing fields for mathe
(Landau escaped its jaws when he died
rics, non-degenerate quadratic differ
maticians throughout the 1930s. By the
in Berlin in 1938). Landau's lectures on
ential forms that need not be positive
time
number theory and analysis at Gottin-
Einstein joined the faculty at
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 Weyl cultivated their re spective arts while contemplating the significance of mathematics for science, philosophy, and the human condition.
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, 1 990. [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, Sylvester,
Felix Klein,
1876- 1 900.
and
E.H.
J.J.
Moore,
AMS/LMS History of Mathematics Series, vol. 8, Providence, R.i.: American Mathe
matical Society, 1 994. REFERENCES
[Poi) Henri Poincare,
[Alex] G. L. Alexanderson, George P61ya: A Biographical Sketch, in Album:
Encounters
The P61ya Picture
of a
Mathematician,
Three-Body
Problem ,
Dover, 1 952. [Row-1 ] David E. Rowe, Felix Klein, David
Hilbert, and the Gottingen Mathematical Tra
Boston: Birkhauser, 1 g87. [8-G] June E. Barrow-Green,
Science and Hypothesis,
London: Walter Scott, 1 905; repr. New York:
Poincare and the
Providence,
R.I.:
American and London Mathematical Soci
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 Berlin,
eties, 1 997. [B-M] Yemima Ben-Menahem, Convention: Poincare and Some of His Critics,
Bntish
Journal for the Philosophy of Science
52
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.
(200 1 ):471-51 3. [Cor] Leo Corry, The Empiricist Roots of
[Row-3] David E. Rowe, The Calm before the
Hilbert's Axiomatic Approach, in [H-P-J], pp.
Storm: Hilbert's Early Views on Foundations, in [H-P-J], pp. 55-93.
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 .
Gottingen. From the Life and Death of a
Great Mathematical Center, Mathernatical ln telligencer 13
[Dar] Olivier Darrigo!, Henri Poincare's Criticism of fin de siecle Electrodynamics, the History of Modern Physics ,
Studies in
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
The Hilbert Challenge,
[Haw-Ell] S. W. Hawking and G. F. R. Ellis, Large Scale Structure of Space-Time,
The
Cam
bridge: Cambridge University Press, 1 973. [H-P-J] V. F. Hendricks, S. A Pederson, and Proof Theory. History
and Philosophical Significance,
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, matiker auf der Flucht vor Hitler,
Mathe
Dokumente
Braunschweig!Wiesbaden: Vieweg, 1 998. [Tho] T. Y. Thomas, Recent Trends in Geome
Oxford: Oxford University Press, 2000.
K. F. Jorgensen, eds.
(4) (1 99 1 ) : 1 2-1 8.
[Sch-Sch] Norbert Schappacher and Erhard
zur Geschichte der Mathematik, Bd. 1 0,
geschichte, Nr. 1 49 (2000). [Gray] Jeremy J. Gray,
tSch] Norbert Schappacher, Edmund Landau's
Synthese Li
brary, vol. 292, Dordrecht: Kluwer, 2000. [Meh-1 ] Herbert Mehrtens, Ludwig Bieberbach
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 Weig ht 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 in fluence 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 ar eas in the world and most Western economies would un dergo a dramatic downturn: just to mention a few signifi cant 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 conse quences. 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 Washek 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 lit erature 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]). =
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-alge bra 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 as sumption because, given the already mentioned weight concern 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 neighbour hood of wo, and assume, by contradiction, that in the in terval [t1 , t2 l (t 1 < t2) one has
This work is dedicated to myself on the occasion of my 40th birthday.
© 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:
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
12
(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. C eppitelli , 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.
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
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 life 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 Wron gs Make a Rig ht? Coin Tossing Games and Parrondo' s Paradox
number of natural and man-made activities can be cast in the form of various one-person games, and many of these appear as sequences of transitions without memory, or Markov chains. It has been observed, initially with surprise, that losing games can often be combined by selection, or even randomly, to result in winning games. Here, we present the analysis
Conceptually similar situations involving only the pro
of such questions in concise mathematical form (exempli
cessing of statistical data are not novel. What has been re
[8] is typified by this sce 1 and type 2, cost
fied by one nearly trivial case and one which has received
ferred to as Simpson's paradox
a fair amount of prior study), showing that two wrongs can
nario: Quite different items, say type
$10 per unit. Suppose that, during a given 20 and 80 of these two types, charg
indeed make a right-but also that two rights can make a
dealers the same
wrong!
period, dealer A sells ing
On frequent occasions, a logical oddity comes along, which attracts a sizeable audience. One of the most recent is known as Parrondo's paradox
$13 and $15, respectively, per item. Dealer B, on the $14 and $16 per item, sells 80 and
other hand, who charges
Background
[5, 6]. Briefly, it is the ob
20 of the two types. Then the average cost per item to dealer (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
A's customers is
=
servation that random selection (or merely alternation) of
sult that B is delighted to advertise. This despite the fact
the playing of two asymptotically losing games* can result
that A sells both items more cheaply than B does! No sur
in a winning game.
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 sta tistics must be commonplace. In fact, it has been pointed out by Saari [7] that aggre gation often yields statistical results qualitatively different from those apparent at a lower level, and that this is re lated 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 counterin tuitive. 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 ex pected 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 ematical structure to orga nize their analysis. If the "game" concept is re stricted sufficiently to al low a clear interpretation of the averaging strategy mentioned above, this is readily accomplished.
This corresponds to the eigenvalue Ao = 1 of the matrix T, all other eigenvalues being simple and having smaller ab solute 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 7}2j1 1Jdo• and an associated gain PrCJo, . . . JN) = 1jNjN_1 WN(Jo, . . . JN) = WjN jN_1 + + wh io• so that •
•
•
·
·
. WN(J, Pr(Jo, . · · · JN)X
·
• . . .
jN) -
N
n rin in- 1 cX) · n�l
(3.4)
By summing over all N-step sequences, we produce the powerful moment-generating function of WN, 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 close to 1 , A(x) will still be real, close to 1, and largest in absolute value. Further more, if we normalize the maximal right eigenvector 4>o(x) of T(x) by 1 t4>o(x) = 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:
Turning two losi n g games into a winn ing one
seems counterintuitive.
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 ad dition, 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 ma trices are an old story, and in particular, we will confme our attention to the large class of irreducible stochastic ma trices, where if one starts with a probability vector Poj = Pr(start in state J) for the possible states, then iteration of the process
Po, 1Po, T2po, . . . results asymptotically in the unique mix of state probabil ities cf>o,j for state j, regarded as components of the prob ability vector { 4>o i } satisfying ,
T4>o = 4Jo.
(3.2)
lim T(x)N/A(x)N = 4Jo(x) N-+oo
1/Jb(X) .
(3.6)
Hence (3.5) implies that (3.7) There is a lot of information in (3. 7), but we will con centrate 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 opera tions. 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 ob tained 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) 4>6(1). Taking the scalar product with 1: =
=
3
A ' (l)cf>o +
1tT'(1) o + 1 1 cf>o(1) = A'(1) + 1 14>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 vec tor 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
l§lijil;iiM
(3. 1 1)
'iA +IB
is the expected gain on making a move from state k, and we can also write (3.10) in the form (3.12)
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 choos ing, 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. =
aTA(x) + (1 - a)TB(x).
(4. 1)
What has come to be known as Parrondo's paradox (orig inally, 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 prob ability 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 =
(! !} ( )
q:i3 qlx TA(x) = ; p;i3 pix
cf>oA =
(!}
THE MATHEMATICAL INTELLIGENCER
(4.4)
It follows, most directly from (3. 10), that Hence, in the bold region of Figure 1, for 3/4 < p ::s: 1, we in deed 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 cap ital. 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 - p 1 of "winning" - 1. If the capital is a multiple of 3, one instead flips coin II with corresponding p 2 and q2. Hence the states can be taken as ( - 1, 0, 1) (mod 3), and the associated transition and gain matrices are =
w=
(
0 1 -1
-1 0 1
)
1 -1 . 0
(5. 1)
and then
(4.2)
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
G D·
=
in game B, the roles of black and white are reversed, so that
=
=
(t n (4.5)
Game Averaging - a Simple Example
TA,B(x)
wtA+tB
=
w = 1tT'(l)cf>o = 3
2 2 PlP2 - q lq2 2 + P1P2 + q lq2 - P 1q 1
(5.3)
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 play ing a new game with parameters fil = 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'fp 2 = qyq2, or equivalently
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
112
(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 0
+
A ' (1)] 112,
(6.5)
with a readily computable coefficient. For example, in the "Parrondo" case of
lpldii;ifW
(5.1) , where
(6.6)
(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
A U T H O R S
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
B has only one coin, equivalent to two identical = p2 ( = 1/2 for a fair game, point C); and is mod ified in that A and B are systematically switched, rather
that game coins,
pi
than randomly switched. Qualitatively, this is much the same.
ORA E. PERCUS
Asymptotic Variance
251 Mercer Street
Much of the activity that we have been discussing arose from extensive computer simulations
[3, 4],
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) a2(w; N) proceeds routinely from (3. 7) used previously to compute
the same starting point
w=
limN_.oc E(WNIN).
This time, differentiate
once and twice with respect to
x and set x =
(3. 7) both 1 , again as
suming commutativity of limiting operations. Again using
A(1)
=
1,
o(l) =
f/io (1)
=
New York, NY 1 001 2 USA
carried out to
the point of negligible fluctuations in the gain. How far does
The computation of
JEROME K. PERCUS
Courant Institute of Mathematical Sciences
1 , this results in
Nlim ---;oo (E(WN) - N A' (1)) = #/ (1)Po )_!.� [E(WN(WN - 1)) - 2N E(WN)A'(1) - NA"(l) (6.2) +N(N - 1) A'(1)2] = I/Jot (1)po,
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 - (p 1 Q2 + QJP2 + PlQl)A.(x) + qyqz!x + PIP2X = 0. (6. 7) By successive differentiation with respect to x, followed by 1, it follows that
x
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
=
A.' (l) = (pW2 - qyq2)!D
685-697.
A."(l) = C - 2A.'(l) + 2PTP2 + 4qyq2)1D where D
=
[1 ] Doering, C.R. Randomly rattled ratchets, Nuovo Cim. 017 (1 995),
(6.8)
1
3 (2 + PJP2 + Q 1 Q2 - P 1 Q1) ,
[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, 1 4 (1 999), 206-2 1 3.
and so we have
[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.
(6.9)
[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-
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 oc cur of course depends upon details of the games involved. Aside from details, the take-home message is that the pro cedure of averaging strategies to improve the outcome-in
5229.
[7] Saari, D. Decisions and Elections. Cambridge: Cambridge Unil(er sity Press (200 1 ) . [8] Simpson, E . H . The Interpretation o f Interaction i n Contingency Ta bles, J. Roy. Stat. Soc. 813 (1 95 1 ) , 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)
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THE MATHEMATICAL INTELLIGENCER
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l;\§lh§l.lfj
.Jet Wi m p , Editor
I
This is something that mathematicians
Math Talks for Undergraduates
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,
by Serge Lang
among others, taught us to ply our craft with precision and with rigor. We
NEW YORK: SPRINGER-VERLAG 1 999 PAPERBACK 1 1 2 pp.;
are all trained to do so. And, as a re
US $29.95, ISBN: 0387987495
REVIEWED
Fee/ like writing a review for The Mathematical Intelligencer? You are
E
BY
sult, we find ourselves hamstrung by our own intellectual infrastructure.
STEVEN G. KRANTZ
We cannot express ourselves in any ar
ver since the time of Einstein and
got but the most rigorous and most
Heisenberg, physicists have been
technical. If we leave out a hypothesis
explaining what they do to the public.
or a condition or a detail, then we de
review of a book of your choice; or, if
Ever since Darwin and Mendel, biolo
velop cold sweats and insomnia. Pass
gists have been explaining what they
the Prozac.
you would welcome being assigned
do to the public. Ever since Lavoisier
Serge Lang has broken out of this
a book to review, please write us,
and Kelvin, chemists have been ex
mold. First of all, he has written a great
plaining what they do to the public.
many books at all levels. They are
Mathematicians are newcomers, vir
widely admired and universally read.
welcome to submit an unsolicited
telling us your expertise and your predilections.
tual tyros, at this game. Many of us
Second, he has made strenuous efforts
want to tell the layman what we are up
to communicate. One of the most as
to; but it is not part of our culture to
tonishing of these is a collection of
do so. We do not have the skills, and
talks
often we do not have the patience. We frequently find ourselves falling back on the tired old saw of, "Well, it's all
and diverse audience in Paris.
very technical. I really would have
talks were about prime numbers, Dio
The
trouble explaining these ideas to an
phantine equations, hyperbolic geome
other mathematician."
try, and other advanced topics. Reports
Phooey. Open up the
Proceedings
are that he had these Parisians jump
of the National Academy of Sciences,
ing out of their chairs, making conjec
or Physica B, or another journal from
tures, and arguing the finer points of
a science other than mathematics.
advanced mathematics. It is really re
These folks do not lack for jargon, nor
markable that anyone could do this.
for technical ideas, nor for obscurity.
But Serge Lang is a remarkable man.
It would be just as easy for them to
The book under review is another
hide behind the details and arcana of
instance of Lang's gift. I wish that the
their subject as it is for us. But they
Preface could be written on vellum and
have trained themselves to formulate
framed in every math department. It
"toy" versions of their problems, to fib
says in part that the author is puzzled
a bit when necessary, and to give the
over the
lay reader an encapsulated notion of
should be "a pump, not a filter" (the
what is going on. Do you think that a physicist ever
really
dictum that
mathematics
reader may know that this is the battle
tells a journalist
cry of the reform movement). He asks
what is going on with black holes, or
whether p-adic L-functions are a pump
that a geneticist ever
Column Editor's address: Department
(Serge Lang fait des maths en public: 3 debats au Palais de la de couverte, Paris) that he gave to a broad
really
discusses
or a filter. Then he muses on whether
the delicate issues of gene splicing and
the Riemann hypothesis is a pump or
of Mathematics, Drexel University,
cloning? Of course not. They speak in
a filter. He heaps derision on the con
Philadelphia, PA 1 9 1 04 USA.
vague generalities,
cheat.
cept of "vertical integration" and even
and they
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 3, 2002
73
takes to task the NSF's VIGRE pro gram. 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; ac tually 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 ba sic topology and the Poincare conjec ture, 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 some thing serious and relevant. You would never, for instance, catch me telling un dergraduates--or any audience for that matter-about the EMIRP num bers (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. Ob viously this is a topic that you could ex plain to your grandmother. But Lang is no wilting flower. On page 4 he intro duces the 7T function for the distribu tion of primes, on page 6 he gives a (rather unusual) formulation of the Riemann hypothesis, and on page 9 he introduces the Bateman/Hom conjec ture. What is amazing is that this ma terial 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 pre cise, 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 calcu lus. Skip it if you don't know cal culus. 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 de lightful 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 ex ploits 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-triv ial, 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 equa tion. 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 un dergraduates perking up and paying at tention? Let me stress two interesting fea tures of these lectures. One is that they are peppered with questions and com ments 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 lec tures has a worthwhile bibliography of fering further readings that an under graduate could actually attempt. Talk 3 is about global integration of locally integrable vector fields. That is
fairly sophisticated stuff, but Lang in troduces the topic with aplomb: The material of this talk stems from Artin's presentation of Cauchy's the orem, but really belongs in connec tion with basic real analysis, in a course on calculus of several (actu ally 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 ap prehensive undergraduate) being put at ease, and settling back for an edify ing hour of new mathematics. Talk 4 is about approximation t�e orems of analysis. A rather sophisti cated 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 2 1-year-olds! But Lang does it with style and grace. Talk 6 is about harmonic and sym metric polynomials. It explores such topics as symmetry, positive definite ness, and eigenfunctions and charac ters. This is a boffo finish to an inspir ing collection of lectures. And I say this not only because the lectures will in spire 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 undergrad uates at Yale or MIT. But not at any public university, nor any math de partment 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.
are quite
with the rest of us. It also takes real
While Lang's books may not be per
fancy, but he saw clever passages into
Some
courage to dive in and learn how to
fect, they have certainly pointed the
accessible insights. That is what he
communicate with undergraduates. Af
way and taken the first steps. This is
shared with the students. He strictly
ter one person has taken the risk, it is
fertile ground that I hope others will
followed the dictum of "Tell them the
easier for others to follow.
truth, tell them nothing but the truth,
plough. I would hope that each of us
I believe-and this is the main point
will take up the challenge and present
but for God's sake don't tell them the
of the present review-that Lang sets
one of these talks to the local math
whole truth." He always made it clear
a wonderful example. No matter how
club. Then we should tum to writing
when he was giving a real proof and
sublime is the mathematics that many
some of our own. Who knows? It might
when he was just telling a parable. In
of us do, I am afraid that we are not
become a movement, and somebody
short, what Lang does is a good model
very good role models. We are solip
might learn something. We would no
of mathematics, pitched at a level that
sistic, we are selfish, and we are largely
longer have to worry about pumps and
is accessible to the uninitiated.
oblivious to the world around us. Of
filters and vertical integration because
course the competitive nature of our
the ideas behind that jargon would be
profession virtually requires self-ab
built into the system. We would have a
sorption and single-mindedness. But
stronger infrastructure for producing
Lang's other efforts at communica tion (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 ef
Lang is sufficiently dedicated and en
new, young mathematicians at a time
ergetic
when they are much needed. And we
that
he
can
transcend
the
boundaries that limit most of us, and
could thank Serge Lang for setting the
he can demonstrate-by doing-that it
example.
is indeed possible to talk about math ematics with people from all walks of
Department of Mathematics
life.
Washington University
fort to have truck with such a wide va
And it is important that we do so. If
St. Louis, MO 631 30-4899
is not just a mat
we care about the visibility of our sub
e-mail: [email protected] .edu
ter of writing down the mathematics.
ject, if we care about the contributions
One must figure out good topics for
we can make to society, if we care
how to explain the ideas to people of whose backgrounds one does not really have a clue. Most of us have been so long im
about convincing the politicians who
riety of people. And it
one's audience and then figure out
mured in abstract mathematics that we
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 pro
A Beautiful Mind: A Biography By Sylvia Nasar
have virtually no idea of the struggles of
mote wavelets. He spoke to the Direc
NEW YORK: SIMON AND SCHUSTER. 1 998, 459 pp
an undergraduate to understand the
tors of the Offices of Navy, and Army,
$25, ISBN: 0-648-81 906-6
most basic notions of logic and rigor.
and Airforce Research, but he also
But Lang dives right in; he has no fear. Of course we have
spoke to senators and congressmen
all observed
and military people. He figured out
Serge's courage in other contexts. He
how to communicate. And look how
has taken on the National Institutes of
well off the wavelet people are today,
Health (in his battle over a scandal in
what a high priority they have in the
volving AIDS Research), the National
funding picture, and how prominent
Academy of Sciences (in a battle over
their work has become. Imagine how
the admission of Samuel P. Hunting
great life would be
ton, a social scientist), the social sci
ics had the same strong profile.
if all of mathemat
ence establishment (in the Ladd!Lipset
G. H. Hardy reveled in the utter use
brouhaha), various Nobel laureates (in
lessness of what he did. But he lived in
the
another time, when expectations were
David
Baltimore
scandal),
and
A Beautiful Mind 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
I
BY
DAVID GALE
t is now more than seven years since John Nash won the Nobel Prize in
many other institutions and sacred
more modest. Now mathematics
cows as well. Lang has no concern for
driving force in much of science and
his own well-being, for whether his
technology. We do not want to be left
book and now a popular movie, the
grant might be in danger (he will not
on the sidelines: after all, this is our
Nash story seems to be bigger than
apply for them), nor for any other crea
subject, and our bailiwick We should
As he has told us
be front and center, leading the pack
reer, A
repeatedly, all he cares about is the
Therefore we must expand our ability
other great emblem of Western culture,
truth, and for exposing pomposity,
to have a meeting of minds with a
stonewalling, intimidation, and the ma
broad cross-section of the populace
the Academy Award Oscar for best film
nipulation of power. And we are fortu
from politicians to high-school stu
incidental that the play
nate that he has shared his findings
dents to parents to administrators.
also deals with mathematics and men-
ture
comforts.
is the
economics, but, thanks to a successful
ever. Indeed, the film about Nash's ca
of
Beautiful Mind,
has won that
2001. And it is surely not purely co Proof, which
VOLUME 24, NUMBER 3, 2002
75
2001 Pulitzer Prize
Note that among games I did not list
outcome, provided the other pleyers
winner. It seems mathematics has be
are hopscotch, tiddlywinks and pitch
leave their strategies unchanged.
tal illness, was the
come box office. What does it all mean?
ing horseshoes.
The reason is that
Of course, the public appeal of this
while these games require skill they
Now in fact most games of any in
material derives in large part from the
don't involve making choices, whereas
terest don't have equilibrium points.
depiction of mental illness, and, in the
in Odds and Evens one must make a
Thus,
decision, whether to throw one or two
it's because I chose the "wrong" strat
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
in Odds and Evens if you beat me
fmgers. The key property of a game for
egy. Had I thrown two fingers instead
our purposes is that it must involve
of one I would have beaten you. How
the film events definitely did not), the
such choices, which in game-theory
scientific story by itself is an interesting
terminology are called
strategies.
In
ever,
if we play the game repeatedly we
will both try to switch from one to two
chapter in mathematical history. Per
poker one must decide whether to call,
fingers
haps one of its most unusual features
raise, or fold. In baseball the pitcher
dictable way.
is that, unlike most important scientific
must decide whether to throw a curve
mixes curves and fastballs so that the
achievements,
or a fastball, and the batter must de
batter won't know what to expect. This
In foot
is formalized by introducing mixed
Nash's
prize-winning
in
some
hopefully
unpre
Similarly, the pitcher
work is accessible. However, a very ca
cide whether to swing or take.
sual sampling of some of my colleagues
ball the strategy session is institution
strategies, which are simply probabil
has led me to believe that probably
alized in the huddle.
ity combinations of the originally given
most
mathematicians
have
only
Formally, a finite game consists of
a
"pure"
strategies.
Thus, the pitcl).er
vague idea, if any, of what Nash actu
n
players each equipped with a finite
might throw a curve a third of the time,
ally did. Because I contend that the
set of strategies. In a play of the game
the poker player might bluff one time
whole thing can be understood by any
each player selects
out of five, etc.
one willing to spend the time to learn
strategies and these jointly determine
a few definitions, let me try to support
an
outcome. An
one
of his/her
outcome for the two
With these definitions in hand the meaning of Nash's theorem should
this claim by describing the result as
player case might be a winner and a
now be clear, but to appreciate its im
one might present it, say, to a junior
loser, or for the general case a numer
plications let us look again at some ex
high school class. (For a much more
ical payoff, positive or negative, to
amples. For Odds and Evens the mixed
thorough exposition of this material
each of the
and its significance at the level of a
n
players, but it could in
strategy equilibrium is clearly for each
clude non-numerical rewards like win
player to throw one or two fingers de
working mathematician, see the first
ning an election or capturing a fugitive.
pending on, say, the toss of a fair coin.
section of John Milnor's article, "A No
With this degree of generality much of
A less trivial example is the following
social behavior fits
poker-like game. I toss a fair coin, and
bel Prize for John Nash," volume number
17,
3, of this magazine.)
Because
he
is
a
into the game
framework Thus we have
mathematician
rather than an economist, Nash did
Games politicians play. Impeaching a
what mathematicians do. He proved a
president
theorem. Here it is.
Every finite n-player game has an equilibrium point in mixed strate gies. 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.
Game
to you-but I need not tell the truth.
in case I report heads, you
have an option: you may either pay me a dollar ("fold"-i.e., accept my report
Terrorism
that you lost), or you may "call, " in Game theory as a distinct discipline
which case you get to see the result of
1944 with the publication of The Theory of Games and Economic Behavior by J. von Neumann and 0.
the toss, and you then win or lose two
started in
dollars according to whether I was ly ing or not. It seems intuitively clear
Morgenstern; as its title suggests, the
that honesty is probably not the best
premise
game-theory
policy for this game. If I never lie I can
point of view would be useful in ac
do no better than break even. On the
was
that
the
counting for economic and perhaps
Evens, Scissors-Paper-Rock, Tic
even political phenomena.
Tac-Toe
Checkers, Poker, Tennis, Monopoly
outcome of the toss: I simply report it Further,
Games countries play: The War on
Games children play: Odds and
Games grown-ups play: Chess,
lar if it falls tails. However, the rules specify that you don't get to see the
Games businessmen play. The Enron
Nash's Nobel-Prize-Winning Theorem:
the idea is that you pay me one dollar
if it falls heads, but I pay you one dol
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
DEFINITION. An equilibrium point of a
better by mixing strategies. The proper
game is an n-tuple of strategies, one
mixture is given by the equilibrium
for each player, with the property that
point theorem.
Games teams play: Football,
no player can change his/her strategy
should lie two-thirds of the time. The
Baseball, Bridge
in such a way as to obtain a preferred
junior high school class can verify this
76
THE MATHEMATICAL INTELLIGENCER
(It turns out that I
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
wrong!" and economics would never
have been like to be tormented over
be the same.
those many years by such delusional
in a competitive and very active pro
On its own terms, then, what is
A
Beautiful Mind trying to do? Obviously,
aberrations.
sive though they were. Rather it is t,hat fession
he
did things nobody else
would have attempted, using methods
There is of course much more to the
it is trying to grab the audience, and I
no one else had ever thought of. The
book than what I have mentioned. Con
found it a pretty good audience-grabber.
recurrent words in the Nasar book are innovative, original, unexpected.
siderable space is given to Nash's rela
A large part of the credit for this goes
tionships, with men and women, espe
to lead actor,
who
By contrast, the film is for the most
cially with his wife Alicia (this is no
makes us laugh or cry or shake with
part fairly predictable. Its one off-beat
Russell
Crowe,
fright in all the right places. The film
attempt is in the handling of the hallu
Hollywood). Finally, in what is perhaps
contains
the most exciting chapter, Nasar. de
There is suspense (will the baby drown
the picture "pulls a flagrant scam:
scribes the rather wild last-minute fight
in the bathtub?), violence (the obliga
whole characters and episodes are pre
in the Swedish Academy over whether
tory car chase; since it didn't really hap
sented as urgently authentic only to be
Nash should receive the prize at all.
pen Nash has to hallucinate it), and of
revealed as figments of a cracked imag ination." On the other hand, some view
doubt what recommended the story to
many
familiar
ingredients.
cinations.
As one unhappy critic put it,
In giving the above sampling of the
course romance. There is also the por
book's contents I emphasized what I
trayal of academic life, which I suppose
ers apparently didn't mind being suck
suppose to be of most interest to pro
audiences will find interesting, though
ered, and find this to be one of the
fessional mathematicians, namely, the
much of it will seem rather hilarious to
film's most compelling features.
process of mathematical creation. You
people who have been there, as for ex
won't learn much about that sort of
ample the professorial pep talk in the
thing from the film. I knew before see
film's opening scene, and the tedious
To conclude, let me return to my orig inal question: what are we to make of
this most unexpected interaction be
ing it that there was no point in wor
"ceremony of the pens"-which isn't
tween mathematics and the entertain
rying about scientific or historical ac
even good "Hollywood."
ment industry? Some people have spo
curacy, and I was prepared to judge the film on its own terms, how well does
As the film drew to a close (I
ken scornfully about A
Beautiful Mind
watched it a second time for purposes
because of its biographical inaccuracies
it succeed in what it's trying to do. I
of writing this review), I became in
and mathematical misrepresentations.
can't resist, however, describing the
creasingly aware of how each profes
treatment of the "eureka" moment,
sion has its own rules and objectives
has done the subject a service by por
when Nash discovers the prize-winning
and outlook on the world, its own idea
traying mathematics not only as a seri
theorem.
of what the
game is all
In one respect, however, I think the film
about. Perhaps
ous and important enterprise, but also
I should extend my earlier list to include
as an exciting one in which new and
ized by grad students and alluring un
Games mathematicians play: Proving
made. This is a refreshing break from the
attached women. Nash, thinking out
theorems, e.g., the
The screenwriters have invented a beer tavern near the campus patron
loud to his companions, muses that
if
quite surprising discoveries are often
Equilibrium-Point
Theorem
usual stereotype of mathematicians as strange characters who spend their lives
they all go after the blonde there can
Games film makers play. Grabbing
be only one winner and the losers will
audiences, e.g.,
A Beautiful Mind
be rejected by the other girls who will
thinking about numbers. If it takes Os
cars and Pulitzer Prizes to get this point across, let's not complain.
In this
case it
resent being second choice, whereas,
For both these endeavors, the payoff
seems what's good for Hollywood
if they pay attention to the other girls.
turns out to be a prize.
good for mathematics.
. . . I wasn't clear about the exact rec
But the most striking aspect of the
ommended optimal strategy, but the
John Nash story, to me, is not his
payoff was quite explicit: "everyone
quirky personality, nor his Odyssey
University of California
gets laid. " From there, in a leap of the
from illness to recovery, nor his win
Berkeley, CA 94720-0001
imagination, the whole thing suddenly
ning of a prestigious prize, nor even his
USA
becomes
mathematical achievements,
e-mail: [email protected]
78
clear,
"Adam
THE MATHEMATICAL INTELLIGENCER
Smith
was
impres-
Department of Mathematics
is
Ki£B,j.k$·h•i§i
Robin Wilson
Geometry of Space
I
T
he 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
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 pavil ion for the 1967 World's Fair in Montreal.
1 00
0 z < ..J
J: u
'
IV CENTENARJO DE sAO PAULO 1,,.. 1954 Sculpture "Continuity"
Sculpture "Expansion"
German pavilion
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]
© 2002 SPRINGER· VERLAG NEW YORK. VOLUME 24. NUMBER 3, 2002
79
