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.
Constant-Diameter Curves The famous American physicist Richard P. Feynman, who disappeared in 1988, was an inexhaustible source of inspiration for anyone who lmew him personally, because of his driving enthusiasm when dealing with any kind of problem. His books are filled with acute observations and problems, often mixed with jokes to test the smartness of the reader. In one of them, [1], on pp. 167-168, Feynman tells us that when investigating the causes of the accident of the Challenger space shuttle, it occurred to him to consider the properties of what can be called constant-diameter curves. He even shows a sketch, drawn by his hands, of a curve of this kind.
In the mathematical literature these curves are referred as equichordal [2] curves. A historical example is the lima9on studied by Etienne Pascal, the father of the famous Blaise. The name "lima9on" was given by G. P. de Roberval (1602-1675), Blaise Pascal's contemporary and friend, who also proposed the concept of generalized conchoids, to which category these curves belong. Recently a long-standing problem related to constant-diameter curves was solved using techniques from dynamical systems [3].
These curves should not be confused with the constant-width [4,5] curves, but it seems that in Feynman's description they are not clearly distinguished. Talking about the roundness of the rocket booster sections of the Challenger, he writes, "NASA gave me all the numbers on how far out of round the sections can get. . . . the numbers were measurements taken along three diameters, every 60 degrees. But three matching diameters won't guarantee that things will fit; six diameters, or any other number of diameters, won't do, either."
First of all, it is a bit odd that NASA technicians would believe that three di-
4 THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
ameters could determine the circularity of a section.
Leaving aside the tricky mechanical problems, let us consider the mathematical aspects. What diameter is referred to? Is it the distance between two parallel tangents to the border, or is it the chord of the curve ofFeynman's Fig. 17 [ 1]? He doesn't explain. If you use a gauge caliber to measure the diameter you get the distance between two parallel tangent planes, but if you want to measure the chord of the curve as in his Fig. 17, you have to know the position of the equichordal point.
Actually there is an incongruity between the description Feynman gives and the sketch he draws. Furthermore, he cites the example of a Reuleaux triangle, which is a constant-width curve but not an equichordal curve! Dulcis in fundo, he tells us a story of when he was a kid and saw in a museum a mechanism with constant-diameter curves turning on shafts that wobbled but that made a gear rack move perfectly horizontally. To do this, the gears would have to be constant-width curves not equichordal curves.
This figure has all its diameters the same
length-yet it is obviously not round. (Figure
17 from R. P. Feynman, What Do You Care What Other People Think?, W.W. Norton & Co., N.Y., 1988, page 168.)
Was he joking or simply confused?
Feynman was famous for jokes related
to physics (e.g., B. F. Chao [6], A. Ru
ina [7], and M. Kuzik [7]), so the one
cited here may be an example con
cerning mathematics.
Anyway, as further proof that these
curves are doomed to generate confu
sion, one notes the article by B. Kawohl
[8], where the author, in connection
with constant-width curves, cites (at p.
21) the wrong Feynman book for the
wrong reason!
REFERENCES
1 . R. P. Feynman, What do you care what
other people think?, W.W. Norton & Co. ,
N.Y., London, 1 988.
2. M. Rychlik, "The Equichordal Point Prob
lem," Elec. Res. Announcements Amer.
Math. Soc. 2, no. 3 (1 996), 1 08-1 23.
3 . M . Rychlik, "A complete solution to the
Equichordal Problem of Fujiwara, Blaschke,
Rothe, and Weitzenbock," lnventiones Math
ematicae 129, issue 1 (1 997), 1 4 1 -2 1 2 .
4. D. Hilbert, S. Cohn-Vossen, Geometria Jn
tuitiva, Boringhieri, Turin, reprint 1 967.
5 . M . Gardner, Giochi Matematici, vol. 4 , San
soni, Florence, 2nd reprint 1 979.
6. B . F. Chao, "Feynman's Dining Hall Dy
namics," Physics Today 42 (1 989), no. 2, p.
1 5.
7. A. Ruina, M. Kuzik, "Feynman: Wobbles,
Bottles and Ripples," Physics Today 42
( 1 989), no. 1 1 , 1 27-130.
8. B. Kawohl, "Symmetry or not?", Mathemat
ical lntelligencer 20 (1 998), no. 2, 1 6-22.
Angelo Ricotta
ISAC-CNR
Via del Fossa del Cavaliere 100 001 33 Rome
Italy
e-mail: [email protected]
Hardy's Duncan Prize Book G.H. Hardy attributed his initial inter
est in mathematics to competitive in
stincts. In his Apology he wrote
I do not remember having felt, as a boy, any passion for mathematics ... .
I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.
Figure 1.
He got his chance to beat other boys,
Wykehamist boys at that, when at age
twelve he won a scholarship to Win
chester in 1889. He was considered too
Figure 2.
young to leave home at the time and his
entry to Winchester was delayed a year.
Hardy entered Winchester College as a
Foundation Scholar in September 1890.
He wasted no time in making his mark
During his first year Hardy won the
Duncan Prize in mathematics, a book
(Fig. 1) purchased from an endowment
by Philip Bury Duncan, Wykehamist
and Keeper of the Ashmolean Museum,
who "wanted the mathematical arts to
be fostered and honored among the
sons of Winchester." The book, an Eng
lish translation of Amedee Guillemin's
The Heavens, was specially bound and
stamped front and back with the Win
chester College seal, an image based on
the coat of arms of William of Wyke
ham, founder of the college (Fig. 2).
Guillemin's popular astronomy hand
book was written "for youth and un
scientific 'children of larger growth' "
just the sort of book that would appeal
to an exceptionally clever thirteen
year-old. Pasted to the inside front
cover is a printed book plate (Fig. 3) identifying the book, in Latin, as the
Duncan Prize in mathematics. Evi
dently a supply of such books was kept,
VOLUME 25, NUMBER 4, 2003 5
�x Ctrtamint Q 'OT\.','J.' J>JWPO.'ITO,
I "TF..R C) • •
.. u;o�-.:.·oA rr. �O!l �tbross,
VI 'TOHI I'R r:MII" I,
IH:I'OilT\YIT
.. ., J>JI )(,x.ct. :w
:11.1 I. xx: �
Figure 3.
as the bookplate bears the printed date
"MDCCCLXXX" with an additional �x" added by hand. The bookplate is in
scribed in a cramped hand (perhaps
that of the Reverend George Richard
son, Mathematics Master at the time) to
Godfrey Harold Hardy and is dated 22
December 1890.
The prize book is a physical link to
what may have been Hardy's first rec
ognized mathematical success outside
of the provincial confmes of his Cran
leigh childhood. It apparently disap
peared for many decades only to be
"rediscovered" recently in-of all
places-Cincinnati. It is believed to
have come to Cincinnati with Profes
sor Archibald Macintyre, who entered
Magdalene College, Cambridge in 1926
and received his Ph.D. from Cambridge
University in 1933. Macintyre left the
University of Aberdeen in 1959 to take
up an appointment as Research Pro
fessor of Mathematics at the University
of Cincinnati. How Macintyre acquired
the book is unknown, but penciled fig
ures on the inside endpapers appear
6 THE MATHEMATICAL INTELUGENCER
to be bookseller's marks, suggesting
that the book was at one time on the
second-hand market. After Macintyre's
death Hardy's book passed to Profes
sor Donald Wright. He was unaware of
the book's history, but knowing the chronic bibliophilism which afflicts the
author of this letter, Wright presented
the book to him. Hardy's prize book
will be returned soon to its natural
home, Trinity College, Cambridge.
Acknowledgments
Thanks to Don Wright, David Ball, and
Patrick Maclure, Secretary of the Wyke
hamist Society.
Charles Groetsch
Department of Mathematical Sciences
University of Cincinnati
Cincinnati, OH 45221 -0025, U.S.A.
Not-so-magical Square My attention has recently been drawn to
an article that appeared earlier in the
Mathematical Tourist department: "The
Magic Square on Sagrada Familia," by
P. Maritz, vol. 23 (2001), no. 4, 49-53. I
would like to offer a few comments.
Maritz spends the first half of his piece telling us about Gaudi, architect
of the Sagrada Familia church in
Barcelona, and most of the second half
in a general introduction to the topic of
magic squares. Thus it is that the pur
ported subject of his article, the magic
square that is found carved in stone in
the church (Fig. 1), comes in for only
very brief treatment toward the end.
1 14 14 4
11 7 6 9
8 10 10 5
13 2 3 15
Fig. 1
The square, he tells us (Maritz seems
unaware that it is executed more than
once in the church), is due to Josep
Maria Subirachs Sitjar, the renowned
Spanish sculptor. He goes on to list
seven of its "interesting properties."
The first of these is that the constant
sum is 33, the age attained by Jesus
Christ. If I understand aright, the sculp
tor saw religious significance in this nu
merical coincidence (as perhaps might
be expected in one christened Josep
Maria), and explains the inclusion of the
square in his rendering of the betrayal
of Jesus by Judas Iscariot. Tenuous as
this justification may seem, the notion
finds support in a poster on sale in the
church, depicting the "Criptograma de
Subirachs," or magic square, in which
33 separate patterns of four numbers
adding to 33 are indicated. At the bot
tom we read (in Spanish), "33 of the 310
combinations that sum to the age of Je
sus . . . " The claim that there exist 310
sets of 4 numbers that sum to 33 in the
square is in fact wrong; there are 88. In any case, it seems clear that the concept
of 33 as a number of pious import in
virtue of the 33 revolutions of the earth
made around the sun during the lifetime
of Jesus is an idea shared also by the
church authorities, freakish as the idea
may appear to many, atheists and the
ists alike.
This brings us to Maritz's next four
points: that the four corner numbers,
the four central numbers, the four cen
tral numbers in the outer rows, and the
four central numbers in the outer
columns all have the same sum. The
trouble is that these are NOT interest
ing properties of Subirach's square,
they are necessary properties of ANY
4 X 4 magic square. Moreover, his fmal
two points-that the four numbers in
each quadrant sum to 33, and that the
four numbers in each of the two short
broken diagonals also sum to 33-are
not, as he implies, independent prop
erties, but imply each other. The above
facts are easily verified from a glance
at the general formula describing every
4 X 4 magic square shown in Figure 2.
A B+a C+b D+c
C+c+x D+b A+a B-x
D+a-x c B+c A+b+x
B+b A+c D C+a
Fig. 2
In short, Maritz's seven points yield
only one distinctive mathematical
property of the Subirachs square.
Worse yet, it seems to me, is that
Maritz fails to point out what must
strike even the lowliest magic-square
buff as the most glaring feature of
Subirachs's square, namely, that it is
TRMAL. This is a technical term
(somewhat pejorative) used in the field
to denote squares that contain repeated
numbers. Subirach's square contains
two such repetitions, 14 and 10, al
though whether this renders it doubly
uninteresting or not, I am unsure.
Just as one would expect Maritz, be
fore writing about a magic square, to
acquaint himself with the rudiments of
the subject, one would surely think
that before incising a magic square on
a public building, Subirachs would
learn enough to have an idea of the rel
ative merits of the square he presented.
Had he but taken that trouble, he could
have avoided embarrassment. Figure 3
0 2 17 14 0 2 14 17 0 5 12 16
6 15 7 5 16 13 3 1 15 11 6 1
18 3 8 4 12 8 9 4 10 3 13 7
9 13 1 10 5 10 7 11 8 14 2 9
Fig. 3
gives three examples of non-trivial 4 X 4 magic squares that he might have
used instead. The sets of 16 distinct in
tegers differ in each case, while the
common constant sum in each square
remains 33:
a monumental blunder: Subirachs has
immortalized his nescience in stone.
The true significance of the Sagrada
Familia magic square is thus that it is
Lee Sallows
Johannaweg 1 2
6523 MA Nijmegen
The Netherlands
e-mail: [email protected]
A Tragic Square
Dis ont nt Wo Hardship
Gloom Sadness Suffering
Misery Tribulation Pain
(count the letters)
-Lee Sallows
Erratum In our last issue, vol. 25, no. 3, we reproduced a mosaic illustrating the death
of Archimedes. We described it as a seventeenth-century forgery. We are in
formed by the Stadtische Galerie of Frankfurt-am-Main, who had kindly au
thorized us to reproduce the work, that it is an eighteenth-century forgery.
(The correction of the date is significant, for it means that the fraud was done
after the excavation of Pompeii had heightened interest in the ancient world.)
VOLUME 25, NUMBER 4, 2003 7
ljfi(W·\·1·1 David E. Rowe, Editor I
On Projecting the Future and Assessing the Past-the 1946 Princeton Bicentennial Conference David E. Rowe
Send submissions to David E. Rowe,
Fachbereich 1 7- Mathematik,
Johannes Gutenberg University,
055099 Mainz, Germany.
atmd rs Ma Lan n 'olomon
Lefsch tz ([Mac Lane 19 220]):
In 1 40 wh n h was writing his
ond b ok on topology, [Lef-
h tz) ent draft of on tion
up to Whitn y and , lac Lane at
Harvard. Th draft · w r inc r-
tz ran as follow :
H r ' to Lefsch tz, lomon L
lrr pr ibl as h IJ. When he' a1 last beneaU1 d1e d
H 'II th n begin to h ·kl od.
IAfhile working on this essay, I
WW found myself thinking about
some general questions raised by some
of the discussions that took place in
Princeton several decades ago. For ex
ample, does it make sense to talk about
"progress" in mathematics in a global
sense, and, if so, what are its hallmarks
and how do mathematicians recognize
such improvements? Or does mathe
matics merely progress at the local
level through conceptual innovations
and technical refinements made and
appreciated only by the practitioners
of specialized subdisciplines? Special
ists in modem mathematical commu
nities are, of course, regularly called
upon to assess the quality of work un
dertaken in their chosen field. But
what criteria do mathematicians apply
when they express opinions about the
depth and importance of contempo
rary research fairly far removed from
their own expertise? Presumably those
in leadership positions expect their
general opinions to carry real weight
and sometimes even to have significant
practical consequences. So how do
opinion leaders justify their views
8 THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
when trying to assess the importance
of past research or guide it into the fu
ture? How do they determine the rela
tive merits of work undertaken in dis
tinct disciplines, and on what basis do
they reach their conclusions?
Clearly various kinds of external
forces-money comes to mind-influ
ence mathematical research and chan
nel the talent and energy in a commu
nity. Yet as every researcher knows,
even under optimal working condi
tions and without external constraints,
success can be highly elusive. Small re
search groups are often more effective
than isolated individuals, but projects
undertaken on a larger scale can also
pose unforeseeable difficulties. So to
what extent can mathematicians really
direct the course of future investiga
tions? How important are clearly con
ceived research programs, or do such
preconceived ideas tend to hamper
rather than promote creative work?
And if "true progress" can only be as
sessed in retrospect or within the con
text of specialized fields of research,
shouldn't opinionated mathematicians
think twice before making sweeping
pronouncements about the signifi
cance of contemporary developments?
These kinds of questions are, of
course, by no means new; their rele
vance has long been recognized, even
if mathematicians have usually tried to
sweep them under their collective
rugs. More recently, historians and so
ciologists have cast their eyes on such
questions just as mathematicians be
came increasingly sensitized to the
contingent nature of most mathemati
cal activity (see [Rowe 2003a]). Until
recent decades, however, conven
tional wisdom regarded mathematical
knowledge as not just highly stable,
but akin to a stockpile of eternal
truths. If since The Mathematical Experience [Davis & Hersh 1981] this
classical Platonic image of mathemat
ics has begun to look tired and anti
quated, we might begin to wonder how
this could have happened. Those who
eventually turned their backs on con
ventional Platonism surely realized
that doing so carried normative impli
cations for mathematical research (as
well as for historians of mathematics,
see [Rowe 1996]). So long as doing
mathematics was equated with finding
eternal truths, practitioners could ply
their craft as a high art and appeal to
the ideology of "art for art's sake," like
the fictive expert on "Riemannian hy
persquares" in The Mathematical Ex
perience. But once deprived of this tra
ditional Platonist crutch, many
mathematicians had difficulty finding
a substitute prop to support their
work When P. J. Davis and Reuben
Hersh poked fun at the inept re
sponses of their expert on Rieman
nian hypersquares who was unable to
explain what he did (never mind why),
this didn't mean that these questions
are easy to answer. How, after all, do leading authorities form judgments
about the quality or promise of a fel
low mathematician's work? What cri
teria are used to assess the relative
importance of work undertaken in two
different, but related fields? Mathe
matics may well be likened to a high
art, but then artists are normally ex
posed to public criticism by non
artists, such as professional critics.
Clearly, mathematicians seldom find
themselves in a similar position; their
work is too esoteric to elicit comment
other than in the form of peer review.
So what is good mathematics and who
decides whether it is really good or
merely "fashionable"? If research in
terests shift with the fashions of the
day, to what extent do fashionable
ideas reflect ongoing developments in
other fields? And who, then, are the
fashion moguls of a given mathemati
cal era or culture, and how do they
make their influence felt? Can anyone really predict the future course of
mathematical events or at least sense
which areas are likely to catch fire?
Hilbert's Inspirations No doubt plenty of people have tried,
most famously David Hilbert, the lead
ing trendsetter of the early twentieth
century. In 1900 he captured the atten
tion of a generation of mathematicians
who subsequently took up the challenge
of solving what came to be known as the
twenty-three "Hilbert problems" [Brow
der 1976], [Gray 2000]. Some of these
had been kicking around long before
Hilbert stepped to the podium at the
Paris ICM in 1900 to speak about
"Mathematische Probleme" [Hilbert
1935, 290-329]. Moreover, a few of the
fabled twenty-three (numbers 6 and 23
come readily to mind) were not really
problems at all, but rather broadly con
ceived research programs.
The idea behind Hilbert's address
was to suggest fertile territory for the
researchers of the early twentieth cen
tury rather than merely enumerate a
list of enticing problems. Indeed, his
main message emphatically asserted
that mathematical progress-signified
by the solution of difficult problems
leads to simplification and unification
rather than baroque complexity.
"Most alluring,"
Minkowski wrote,
"would be the
attempt to look into
the future "
"Every real advance," he concluded,
"goes hand in hand with the invention
of sharper tools and simpler methods
which at the same time assist in un
derstanding earlier theories and cast
aside older mathematical develop
ments . ... The organic unity of math
ematics is inherent in the nature of this
science, for mathematics is the foun
dation of all exact knowledge of nat
ural phenomena." (Quoted from
"Hilbert's Lecture at the International
Congress of Mathematicians," in [Gray
2000, 282].)
Hilbert badly wanted to make a
splash at the Paris ICM. Initially he
thought he could do so by challenging
the views of the era's leading figure,
Henri Poincare, who stressed that the
vitality of mathematical thought was
derived from physical theories. This vi
sion rubbed against Hilbert's deeply
engrained purism, so he sought the ad
vice of his friend, Hermann Minkowski.
The latter dowsed cold water on
Hilbert's plans to counter Poincare's
physicalism, but then gave him an en
ticing idea for a different kind of lec
ture. "Most alluring," Minkowski
wrote, "would be the attempt to look
into the future, in other words, a char
acterization of the problems to which
the mathematicians should turn in the
future. With this, you might conceiv
ably have people talking about your
speech even decades from now. Of
course, prophecy is indeed a difficult
thing" (Minkowski to Hilbert, 5 Janu
ary 1900, [Minkowski 1973, 119-120]).
Hilbert, who now stood at the height
of his powers, rose to Minkowski's
challenge. He never doubted his vision
for mathematics, and his success
story-indeed, the whole Hilbert leg
end-took off with the publication of
the Paris lecture with its full list of 23
problems (at the Paris ICM he pre
sented only ten of them). Now that
more than a century has elapsed, we
realize that Hilbert's views on founda
tions, as adumbrated in his 1900
speech, were hopelessly naive and far
too optimistic. Even his younger con
temporaries-most notably Brouwer
and Hermann W eyl-sensed they were
inadequate, though Hilbert continued
to fight for them bravely as an old man.
When Kurt Gi:idel dealt the formalist
program a mortal blow in 1930,
Hilbert's vision of a simple, harmo
nious Cantorian paradise died with it.
Still, his reputation as the "Gi:ittingen
sage" lived on, making Minkowski's
prediction-that mathematicians might
still be "talking about your speech even
decades from now"-the most prophetic
insight of all.
Hilbert died on a bleak day in mid
February 1943, just after the German
army surrendered at Stalingrad, setting
the stage for the final phase of the Nazi
regime. As the SS and Gestapo intensi
fied their efforts to round up and ex
terminate European Jews, Hilbert's
first student, Otto Blumenthal, got
caught in their web; he died a year later
in the concentration camp in There
sienstadt. In the meantime, several of
his most illustrious students had found
their way to safer havens (see [Sieg
mund-Schultze 1998] for a detailed ac
count of the exodus). Two of them,
VOLUME 25, NUMBER 4, 2003 9
The Problems of Mathematics I. Morse. M.. Institute for Ad
vanced Study 2. Ancocbea. G .. University of
Salamanaca. Spain 3. Borsuk. K .• University of
Warsaw, Poland 4. Cramer. H .• University of
Stockholm. Sweden S. Hlavaty, V .. University of
Praaue. Czechoslovakia 6. Whitehead. J. H. C.. University
of Oxford. En&)and 7. Gardina, L. J., Princeton 8. Riesz, M., University of
Lund, Sweden 9. Lefachetz, S .. Princeton
10. Veblen, 0.. Institute for Advanced S1udy
II. Hopf, H .. Federal Technical School, Switzerland
12. Newman. M. H. A., University of Manchester, EnaJand
13. Hodae. W. V. D .. Cambridge. En&land
14. Dirac. P. A. M .. Cambridge University, En&)and
IS. Hua. L. K., Tsing Hua Univer-sity, China
16. Tukey, J. W .. Princeton 17. Harrold. 0. G .. Princeton 18. Mayer, W., Institute for Ad
vanced Study 19. Mautner, F. 1., Institute for Ad
vanced Study 20. GOdel, K., Institute for Advanced
Study
21. Levinson, N., Massachusens Institute of Technology
21. Cohen, I. S., University of Pennsylvania
23. Seidenberg. A., University of California
24. Kline, J. R., University of Penn-sylvania
15. Ellenbei'J, S., Indiana University
26. Fox, R. H .. Princeton 17. Wiener. N .. Massachusetts Insti
tute of Technology 18. Rademacher, H .. University of
Pennnsylvania 19. Salem. R., Massachuseus Insti
tute of Technology
10 THE MATHEMATICAL INTELLIGENCER
30. Tarski, A., University of California
31. Bargmann, V., Princeton
32. Jacobson, N .. The Johns Hopkins University
33. Kac, M., Cornrll University
34. Stonr. M. H .. University of Chicago
35. von Neumann, J., lnstitutr for Advanced Study
36. Hedlund, G. A., University of Virginia 37. Zariski, 0., University of Illinois
38. Whyburn, G. T .. Univrrsity of Virginia. 39. McShane, E. J.. University of
Vir&inia -40. Quine, W. V., Harvard 41. Wilder, R. L., University of
Michipn
41. Kaplansky, 1 .. lnstitutr for Advanced Study
43. Bochner, S .. Princeton
44. Leibler. R. A .. Institute for Advanced Study
45. Hildebrandt, T. H .. University of Michipn
46. Evans. G. C., University of California
47. Widder, D. V .. Harvard
48. Hotelling, H., University of Nonh Carolina
49. Peck, L. G., Institute for Advanced Study
50. Synge. J. L., Carnegie Institute of Technology
Sl. Rosser, J. B .. Cornell
52. Murnaghan, F. D., The Johns Hopkins University
53. Mac lanr, S., Harvard
54. Cairns, S. S., Syracuse Univrrsity
SS. Brauer. R .. University of Toronto, Canada
56. Schoenbei'J, I. J .. University of Pennsylvania
57. Shiffman, M., Now York Univrrsity
58. Milgram. A. N .. Institute for Advanced Study
59. Walker. R. J .. Cornell
60. Hurewicz, W., Massachusetts Institute of Technology
61. McKinsey, J. C. C .. Oklahoma Agricultural and Mechanical
62. Church, A., Princeton 63. Robenson. H. D., Princeton 64. Bullin. W. M., BuUin and Mid
dleton, Louisville, Ky. 65. Hille, E .• Yale University 66. Alben, A. A., University of
Chicaao 67. Rado, T., The Ohio State Uni-
versity 68. Whitney, H., Harvard 69. Ahlfors, L. V., Harvard 70. Thomas, T. Y., Indiana Univer
sity 71. Crosby, D. R .• Princrton 72. Weyl, H., lnstitutr for Advanced
Study 73. Walsh, J. L .. Harvard 74. Dunford, N., Yale 75. Spenser, D. C., Stanford Univer-
sity 76. Montgomery, D., Yale 77. Birkhoff, G., Harvard 78. Kleene, S. C .. University of Wis..
consin 79. Smith, P. A .. Columbia Univer
sity 80. Youngs. J. W. T .. Indiana Uni
versity 81. Steenrod, N. E .. University of
Michipn
81. Wilks, S. S., Princeton 83. Boas, R. P., Mathematical Re
views, Brown Univenity 84. Doob, J. L., University of Illi
nois 85. Feller. W., Cornell University 86. Zygmund, A., University of
Pennsylvania 87. Anin, E., Princrton 88. Bohnenblust. H. F.. California
Institute of Trchnolosy 89. Allendoerfer, C. B.. Haverford
College 90. Robinson, R. M .. Princeton 91. Jkllman, R .. Princeton 92. Beglr, E. G .. Yale 93. Tucker, A. W .. Princeton
Hermann Weyl and Richard Courant,
met again nearly four years after
Hilbert's death in Princeton to take
part in an event that brought to mind
their former mentor's famous Paris lec
ture. There, on the morning of 17 De
cember 1946, Luther Eisenhart opened
Princeton's Bicentennial Conference
on "Problems of Mathematics," a three
day event that brought together some
one hundred distinguished mathemati
cians.
Princeton Agendas The stated purpose of this event was
"to help mathematics to swing again
for a time toward unification" after a
long period during which a "unified
viewpoint in mathematics" had been
neglected. Its program was both broad
and ambitious, but as a practical con
sideration the Conference Committee
decided to omit applied mathematics,
even though significant connections
between pure mathematics and its ap
plications were discussed. The larger
vision set forth by its organizers also
carried distinctly Hilbertian overtones:
The forward march of science has been marked by the repeated openingup of new fields and by increasing specialization. This has been balanced by interludes of common activity among related fields and the development in common of broad general ideas. Just as for science as a whole, so in mathematics. As many historical instances show, the balanced development of mathematics requires both specialization and gener
alization, each in its proper measure. Some schools of mathematics have prided themselves on digging deep wells, others on excavation over a broad area. Progress comes most easily by doing both. The increasing tempo of modern research makes these interludes of common concern and assessment come more and more frequently, yet it has been nearly fifty years since much thought has been broadly given to a unified viewpoint in mathematics. It has seemed to us that our conference offered a unique opportunity to help mathematics to swing again for a time toward unification [Lefschetz 1947, p. 309}.
These pronouncements make clear
that the Princeton Conference on
"Problems of Mathematics" was no or
dinary meeting of mathematical minds.
As the editors of A Century of Mathematics in America duly noted:
The world war had just ended, mathematicians had returned to their university positions, and large numbers of veterans were beginning or resuming graduate work. It was a good time to take stock of open problems and to try to chart the future course of research [Duren 1989, p. ix}.
The Conference Committee, chaired
by Solomon Lefschetz, reflected the
pool of talent that had been drawn to
Princeton as a result of the flight from
European fascism, listing such stellar
names as Emil Artin, Valentin Barg
mann, Salomon Bochner, Claude
Chevalley, and Eugene Wigner. Thus
the Princeton Bicentennial came at a
propitious time for such a meeting,
though the scars of the Second World
War were still fresh and the threat of
nuclear holocaust a looming new dan
ger. The tensions of this political at
mosphere, but above all the Princeton
mathematicians' hopes for the future
were echoed in their conference re
port:
Owing to the spiritual and intellectual ravage caused by the war years, it seemed exceedingly desirable to have as many participants from abroad as possible. As the list of members shows, considerable success was
attained in this. Our conference became, as it were, the first international gathering of mathematicians in a long and terrible decade. The manifold contacts and friendships renewed on this occasion will, we all hope, in the words of the Bicentennial announcement, "contribute to the advancement of the comity of all nations and to the building of a free and peaceful world" [Lefschetz 1947, p. 310}.
Just over a decade had passed since
the last International Congress of
Mathematicians was held in Oslo, and
several who were present at that 1936
event also attended the Princeton con-
ference, including Oswald Veblen, Nor
bert Wiener, Hermann Weyl, Garrett
Birkhoff, Lars Ahlfors, and Marcel
Riesz. Among the distinguished math
ematicians who attended the Princeton
Bicentennial were Paul Dirac and
William Hodge from England, Zurich's
Heinz Hopf, and China's L. K. Hua. Of
the 93 mathematicians-all of them
men-pictured in the group photo,
eleven (all seated in the front row)
came from overseas. A large percent
age of the others, however, were Eu
ropean emigres, many of whom had
come to North America during the pre
vious ten years.
Nativism vs. Internationalism in American Mathematics Yet if internationalism had a nice ring,
this theme played a secondary role at
the Princeton Bicentennial, which had
little in common with the ICMs of the
past. On the contrary, as the frrst large
scale gathering of America's mathe
matical elite at the onset of the post
war era, this meeting was strongly
colored by domestic conflicts. Intent
on laying the groundwork for their own
vision of a "new mathematical world
order," the Princetonians seized on
their university's bicentennial as an op
portunity to place themselves at the
fulcrum of a now dynamic, highly Eu
ropeanized American mathematical
community. Princeton's Veblen, unlike
Harvard's G. D. Birkhoff, had played a
major part in helping displaced Euro
pean mathematicians find jobs in the
United States. Given these circum
stances, Princeton could legitimately
host an intellectual event with the
explicitly stated moral agenda of aim
ing to promote harmonious rela
tions among the world's mathemati
cians. But the Princeton community
was, in this respect, almost singular in
the United States.
Harvard's reputation as a bastion of
conservatism placed it in natural op
position to Princeton, thereby height
ening tensions within the American
mathematical community. G. D. Birk
hoff had long despised Lefschetz dur
ing an era when anti-Semitism at Ivy
League universities was pervasive
[Reingold 1981, 182-184]. As the first
native-trained American to compete
VOLUME 25, NUMBER 4, 2003 1 1
Solomon Lefschetz was impulsive, frank and
opinionated; enough so that many found him
obnoxious. He loved to argue and never openly
admitted his mistakes, however glaring. But
his student Albert W. Tucker was convinced
that Lefschetz's bark was worse than his bite.
On a train ride from Princeton to New York he
overheard a conversation between Lefschetz
and Oscar Zariski, who were both discussing
an important new paper in algebraic geome
try. Lefschetz wasn't sure whether to classify
the author's techniques as topological or al
gebraic, which led Zariski to ask: "How do you
draw the line between algebra and topology?"
Lefschetz answered in a flash: "Well, if it's just
turning the crank, it's algebra, but if it's got an
idea in it, it's topology!" (Mathematical People.
Profiles and Interviews, ed. Donald J. Albers
and G. L. Alexandeson. Boston: Birkhauser,
1985, p. 350.)
on equal terms with Europe's elite
mathematicians, Birkhoff sought to
bring the United States to the forefront
of the world scene. Coming from E. H.
Moore's ambitious Chicago school, he
embodied the Midwestern ideals of
Americans determined to demonstrate
their own capabilities and talent
through incessant hard work During
the 1920s, he molded Harvard into the
strongest department in the U.S., par
ticularly in his own field, analysis and
dynamical systems. Like other Harvard
departments, it was not a model of eth
nic diversity, a fact appreciated by
M.I. T. 's Norbert Wiener and, somewhat
later, New York University's Richard
Courant (see [Siegmund-Schultze 1998,
181-185]). Five years after the Nazi
takeover, Birkhoff offered a survey of
the first fifty years of American math
ematics as part of the AMS Semicen
tennial celebrations. This lecture
caused a major stir because of certain
oft-repeated remarks about the influx
of first-class foreign mathematicians to
the United States. The latter, Birkhoff
felt, threatened to reduce the chances
of native Americans, who could be
come "hewers of wood and drawers of
water" within their own community.
He then added: "I believe we have
reached the point of saturation. We
Princeton's . .
organ1z1ng
committee clearly
set its sights high in
preparing for this
memorable event.
must definitely avoid the danger"
[Birkhoff 1938, 276-277).
During the final years of Birkhoffs
career-he died in 1944-he tangled
with Princeton's Hermann Weyl in a
dispute over gravitational theory. Birk
hoff had set forth an alternative to Ein
stein's general theory of relativity
which dispensed with the equivalence
principle, the very cornerstone of Ein
stein's theory. After some rather petty
exchanges, Birkhoff and Weyl broke off
their debate, agreeing that they should
disagree. Veblen, who er1ioyed having
both Einstein and W eyl as colleagues,
took a rather dismissive view of Birk-
hoffs ideas about gravitational theory.
He also distanced himself from the Har
vard mathematician's rather provincial
views about the "dangers" posed by for
eigners within the American mathemat
ical community. In his necrology of
Birkhoff he wrote that
. . . a sort of religious devotion to
American mathematics as a "cause"
was characteristic of a good many of
[Birkhoffs} predecessors and contem
poraries. It undoubtedly helped the
growth of the science during this pe
riod. By now [ 1944 j mathematics is
perhaps strong enough to be less na
tionalistic. The American mathemat
ical community has at least been
healthy enough to absorb a pretty sub
stantial number of European mathe
maticians without serious complaint.
[Veblen 1944}
After the war, the senior Birkhoff
having passed from the scene, Lef
schetz no longer had to contend with
his former nemesis. During the Prince
ton Bicentennial Conference, he
emerged in his full glory as the new gray
eminence of American mathematics.
Princeton's 12-man organizing com
mittee clearly set its sights high in
preparing for this memorable event.
The conference dealt with develop
ments in nine fields, some venerable
(algebra, algebraic geometry, and
analysis), others more modem (math
ematical logic, topology), and a few of
even more recent vintage (analysis in
the large, and "new fields"). Each of the
nine sessions was chaired by a distin
guished figure in the field, whose open
ing remarks were followed by more ex
tensive discussion led by one or more
experts. 1 This format was designed to
promote informal exchanges, rather
than forcing the participants to spend
most of their time listening to a series
of formal presentations. The results
were carefully recorded by specially
chosen reporters, who summarized the
main points discussed.
1The nine sessions were (1) algebra (chair (C): E. Artin, discussion leader(s) (D): G. Birkhoff, R. Brauer, N. Jacobson; (2) algebraic geometry (S. Lefschetz (C), W. V. D.
Hodge, 0. Zariski (D)); (3) differential geometry (0. Veblen (C), V. Hlavaty, T. Y. Thomas (D)); (4) mathematical logic (A Church (C), A Tarski (D)); (5) topology (A W.
Tucker (C), H. Hopi, D. Montgomery, N. E. Steenrod, J. H. C. Whitehead (D)); (6) new fields (J. von Neumann (C), G. C. Evans, F. D. Murnaghan, J. L. Synge, N. Wiener
(D)); (7) mathematical probability (S. S. Wilks (C), H. Cramer, J. L. Doob, W. Feller (D)); (8) analysis (S. Bochner (C), L. V. Ahlfors, E. Hille, M. Riesz, A Zygmund (D));
(9) analysis in the large (H. P. Robertson (C), S. Mac Lane, M. H. Stone, H. Weyl (D)).
12 THE MATHEMATICAL INTELLIGENCER
Garrett Birkhoff had numerous opportunities
to witness the traditional rivalry between Har
vard and Princeton during G. D. Birkhoff's
heyday. He later recalled this incident: one
day Lefschetz came to Harvard-this must
have been around 1942-to give a colloquium
talk. After the talk my father asked him,
"What's new down at Princeton?" Lefschetz
gave him a mischievous smile and replied,
"Well, one of our visitors solved the four-color
problem the other day." My father said: "I
doubt it, but if it's true I'll go on my hands
and knees from the railroad station to Fine
Hall." He never had to do this; the number of
fallacious proofs of the four-color problem is,
of course, legion. (Mathematical People. Pro
files and Interviews, ed. Donald J. Albers and
G. L. Alexanderson. Boston: Birkhiiuser, 1985,
pp. 12-13.)
A Rivalry Lives On Judging from these conference reports,
which the organizers characterized as
giving "much of the flavor and spirit of
the conference," they must have found
many of the sessions disappointing (as
suming they took the stated agenda of
the conference seriously). Still, the
Russian-born Lefschetz surely felt a
deep satisfaction in hosting an event
which demonstrated the dominance of
Princeton's Europeanized community
over its traditional rival, the Harvard
department once led by G. D. Birkhoff.
This rivalry lived on and was manifest
throughout the meeting. In the opening
sessions on algebra, chaired by Emil
Artin, Harvard's Garrett Birkhoff began
by noting the contrast between the dis
cussion format chosen for the Prince
ton meeting and the more conventional
one adopted at the Harvard Tercente
nary meeting ten years earlier (though
he apparently did not state any prefer
ence). The younger Birkhoff then pro
ceeded to make some rather pompous
pronouncements about the state of his
discipline. He characterized algebra as
"dealing only with operations involving
a finite number of elements," noting
that this led to three distinct types of
algebraic research: (1) "trivial" results;
(2) those which also employ the axiom
of choice, which he felt were "becom
ing trivial"; and (3) general results, like
his own work relating to the Jordan
Holder theorem.
Artin had only recently arrived from
Indiana, so he had not yet fully
emerged as the "cult figure" of the
Princeton department described by
Gian-Carlo Rota [Rota 1989]. Still, he
had known Garrett Birkhoff for some
time, as the latter had twice stopped
off in Hamburg during the mid-1930s to
visit him on the way to European con
ferences [Birkhoff 1989, p. 46]. Pre
dictably, he brushed aside Birkhoffs
definition of algebra based on systems
to which finitely many operations are
applied. "What about limits," he fired
back, noting that these are indispens
able for valuation theory? Birkhoff
merely replied that he didn't consider
this part of algebra, but added "this
doesn't mean that algebraists can't do
it." Mter this, a number of others
chimed in-Mac Lane, Dunford, Stone,
Rad6, and Albert-mainly adding re
marks that seem to have contributed
little toward clarifying major trends in
algebraic research. One senses a num
ber of different competing agendas
here, particularly in the exchange be
tween Artin and Birkhoff. As Rota later
recalled, at Princeton Artin made no
secret of his loathing for the whole An
glo-American algebraic tradition "asso
ciated with the names Boole, C. S.
Peirce, Dickson, the later British in
variant-theorists, ... and Garrett Birk
hoffs universal algebra (the word 'lat
tice' was strictly forbidden, as were
several other words)." Birkhoff pre
sumably had more than an inkling of
this attitude, which must have grated
on him, since Artin's arrogance was al
most in a class by itself.
This particular rivalry may be seen as
part of the ongoing conflict between "na
tivists" and "internationalists" within the
American mathematical community,
Garrett Birkhoff having been a leading
representative of the former group.
The year 1936 was undoubtedly still
very vivid in Birkhoffs mind when he
attended the Princeton conference a
decade later. Many years later he re
called how he was "dazzled by the
depth and erudition of the invited
speakers" at the 1936 ICM in Oslo. He
was pleased that the two Fields medal
ists-Lars Ahlfors and Jesse Douglas
"were both from Cambridge, Massa
chusetts, and delighted that the 1940
International Congress was scheduled
to be held at Harvard, with my father
as Honorary President!" [Birkhoff
1989, 46]. He remembered the "serene
atmosphere of Harvard's Tercentenary
celebration," which took place the fol
lowing September in conjunction with
the summer meeting of the American
Mathematical Society. The event at
tracted more than one thousand per
sons, including 443 members of the
AMS. He admitted that the invited lec
tures were over his head, but he knew
that only very few in the large audi
ences that attended could follow the
presentations.
Ten years later, much had changed,
as Birkhoff had become a leading fig
ure in the American mathematical
community. With his famous name and
rising reputation, he clearly saw him
self as carrying Harvard's banner into
the rival Princeton camp, and he prob
ably missed the kind of serene plea
sures he associated with his alma mater. He may well have been unhappy
about the format of the conference,
given that all nine sessions were
chaired by Princeton mathematicians.
Just a glance at their names would
have been enough to bring home that
Cambridge, even with the combined re
sources of both Harvard and M.I.T.,
was no match for the mathematical
community in Princeton with Artin,
Lefschetz, Veblen, Alonzo Church,
A. W. Tucker, John von Neumann, S. S.
Wilks, Bochner, Marston Morse, and
VOLUME 25, NUMBER 4, 2003 13
H. P. Robertson. With its university and the Institute for Advanced Study, Princeton had drawn together an unprecedented pool of mathematical talent, which was on full display at this celebratory meeting. Even Einstein was in the audience, at least briefly.
More sparks flew in the session on algebraic geometry, chaired by Lefschetz, in which William Hodge and Oscar Zariski served as discussion leaders. The latter two made illuminating remarks on the Hodge conjecture, one of the Clay Prize problems for the twenty-first century, and on minimal models in birational geometry. Lefschetz, commenting on Zariski's presentation, remarked: "To me algebraic geometry is algebra with a kick All too often algebra seems to lack direction to specific problems." To this, Birkhoff countered: "If the algebraic geometers are so ambitious, why don't they do something about the real field?" Lefschetz answered by suggesting that the geometry of real curves was analogous to number theory before the utilization of analytic methods, when one had only scattered results without a unifying theory. He pointed to Hilbert's still unsolved sixteenth problem on the nesting configurations for the components of real curves as an illustration of the lack of suitable general methods.
In the session on mathematical logic, chaired by Church, most of the discussions centered on decision problems. Oddly enough, Hilbert's tenth Paris problem, the decision problem for Diophantine equations (proved unsolvable by Yuri Matijacevic in 1970) was not even mentioned, though it was only in the 1930s that the notion of a computable algorithm became tractable. Church called attention to the recent theorem of Emil Post, who proved that the word problem for semi-groups is unsolvable. This prompted him to suggest that the word problem for groups and the problem of giving a complete set of knot invariants ought to be tackled next. J. H. C. Whitehead expressed a different opinion about these problems in the topology session, where he mentioned the word problem
in the same breath as the Poincare conjecture, noting that "our knowledge of these matters is practically nil."
Further discussions on mathematical logic were led by Alfred Tarski, who conducted a survey of the decision problem in various logical domains. An interesting argument ensued when Kurt Godel proposed an expansion of the countable formalized systems that he had investigated on the way to his famous incompleteness theorem of 1931. Church apparently took issue with Godel's claim that "the set of all things of which we can think" is probably denumerable. A philosophical debate then ensued over what it meant to have a "proof' and when a purported proof could be "reasonably" doubted. However, these reflections appear to have enriched rather than deflected the general discussions in this session, which were both focused and informative. Unlike some of the participants in the algebra and algebraic geometry sessions, the logicians avoided the temptation to grandstand or make sweeping pronouncements about the status of a particular area of research. The contrast was reflected by the organizers' characterizations of the logic session, which showed "the liveliness of mathematical logic and its insistent pressing on toward the problems of the general mathematician," as opposed to the discussion about general algebra. Should limits and topological methods, which were required for many vital results, be defined out of algebra? Clearly, Artin and Lefschetz didn't think they should, as otherwise "algebra would lose much power."
Taking Stock Having touched upon the overall at
mosphere at this meeting shortly before Christmas 1946 as well as some of the specific exchanges during these three days of discussions, let's now jump ahead to the year 1988 when the AMS was celebrating its own centenary. The following year saw the publication of the second volume of A Century of Mathematics in America in which the proceedings of the Prince-
ton meeting were reprinted [Duren 1989, 309-334]. The editors also asked several experts in relevant fields to comment on the discussions that had taken place in 1946 as recorded for these proceedings. 2 In view of the purpose of the Princeton meeting-which aimed to cast its eye on what the future held---one might have thought that such a retrospective analysis would have proven useful in order to take stock of the progress made during the intervening period. If so, the editors were forced to conclude that these commentaries underscored "how different mathematics was in 1946."
Almost all of the experts noted the immense gap that separated state of the art research in their field ca. 1988 and the interests of leading practitioners forty years earlier. Several noted that some highly significant work already published before 1946 received no attention at the Princeton conference. Thus, Robert Osserman was astonished that names like Cartan, Chern, and Weyl did not appear in the report on recent work on differential geometry. Chern's intrinsic proof of the generalized Gauss-Bonnet theorem had been presented "in Princeton's own backyard at the Institute for Advanced Study" in 1943! J. L. Doob's comments on the probability session are particularly illuminating, given that he had participated in it as a discussant:
The basic difference between the roles of mathematical probability in 1946
and 1988 is that the subject is now accepted as mathematics whereas in
1946 to most mathematicians mathematical probability was to mathe
matics as black marketing to market
ing; that is, probability was a source
of interesting mathematics but exam
ination of the background context was
undesirable. And the fact that proba
bility was intrinsically related to sta
tistics did not improve either subject's
standing in the eyes of pure mathe
maticians. In fact the relationship be
tween the two subjects inspired heated fruitless discussions of "What is probability?" and thereby encouraged the
2The commentaries covered eight of the nine sessions: algebra (J. Tate and B. Gross), algebraic geometry (H. Clemens), differential geometry (R. Osserman), mathe
matical logic (Y. N. Moschovakis), topology 0/'J. Browder), mathematical probability (J. L. Doob), analysis (E. M. Stein), and analysis in the large (K. Uhlenbeck).
14 THE MATHEMATICAL INTELLIGENCER
confusion between probability and the
phenomena to which it is applied
[ Doob 1989, 353}.
Doob went on to note that Kol
mogorov's program for the founda
tions of probability had been set forth
in 1933. It nevertheless took several
decades before the idea of treating ran
dom variables as measurable functions
gained acceptance. As Doob put it,
"some mathematicians sneered that
probability should not bury its spice in
the bland soup of measure theory, that
perhaps probability needed rigor, but
surely not rigor mortis."
Two commentators, William Brow
der and Karen Uhlenbeck, were struck
by some general remarks that Her
mann Weyl made in his after-dinner
speech at the close of the 1946 meet
ing. As one of the last great represen
tatives of the Gottingen mathematical
tradition, it was surely fitting that Weyl
was asked to speak at the closing cer
emonies. And it was equally fitting that
Weyl mentioned Minkowski's 1905
speech honoring Dirichlet, in which
Weyl's former teacher stated that the
"true Dirichlet principle" was to solve
mathematical problems "with a mini
mum of blind calculation and a maxi
mum of seeing thought." Hilbert had
been a leading advocate of this philos
ophy, but even in his youth Weyl had
deep reservations about this whole ap
proach to mathematical knowledge
(see [Rowe 2003b ]). These misgivings
had evidently not lessened during the
twilight of his career, and in Princeton
he went so far as to formulate a
counter-principle: "I find the present
state of mathematics, that has arisen
by going full steam ahead under this
slogan (the "true Dirichlet principle"),
so alarming that I propose another
principle: Whenever you can settle a
question by explicit construction, be
not satisfied with purely existential
arguments."
Although he had long since parted
company with Brouwer's brand of in
tuitionism, Weyl continued to believe
that pure mathematics can only thrive
when its tendency toward abstraction
is sustained by ideas of a non-formal
nature. He made the point in Princeton
by quoting himself in 1931 when he of
fered these remarks at a conference in
Bern:
Before one can generalize, formalize,
or axiomatize, there must be a mathematical substance. I am afraid that
the mathematical substance in the
formalization of which we have exer
cised our powers in the last two
decades shows signs of exhaustion.
Thus I foresee that the coming gener
ation will have a hard lot in mathe
matics.
Despite the tumultuous political events
that had intervened, Weyl's views in
1946 reflected much the same opinion:
The challenge, I am afraid, has only
partially been met in the intervening
fifteen years. There were plenty of en
couraging signs in this conference.
But the deeper one drives the spade the
harder the digging gets; maybe it has
become too hard for us unless we are
given some outside help, be it even by
such devilish devices as high-speed
computing machines.
No doubt John von Neumann, who had
chaired the session entitled simply
"New Fields," was smiling in approval.
Neither he nor Weyl knew what the fu
ture held, but they probably sensed
that they were standing on the brink of
a new era.
REFERENCES
[Birkhoff 1 938] G. D. Birkhoff, "Fifty Years of
American Mathematics, " in Semicentennial
Addresses of the American Mathematical
Society, vol. 2, Providence, R. 1 . : American
Mathematical Society, 1 938, pp. 27Q-3 1 5 .
[Birkhoff 1 989] Garrett Birkhoff, "Mathematics
at Harvard, 1 836-1 944," in [Duren 1 989,
3-58].
[Browder 1 976] Felix Browder, ed. , Mathemat
ical Developments Arising from Hilbert's
Problems, Symposia in Pure Mathematics,
vol. 28, Providence, R. 1 . : American Mathe
matical Society, 1 976.
[Davis & Hersh 1 981] P. J . Davis and Reuben
Hersh, The Mathematical Experience,
Boston: Birkhauser, 1 981 .
[Doob 1 989] J. L. Doob, "Commentary on
Probability," in [Duren 1 989, 353-354].
[Duren 1 988] Peter Duren et at. , eds. , A Cen
tury of Mathematics in America, vol. 1 , Prov
idence, R. I . : American Mathematical Soci
ety, 1 988.
[Duren 1 989] Peter Duren et a/. , eds. , A Cen
tury of Mathematics in America, vol. 2, Prov
idence, R. 1 . : American Mathematical Soci
ety, 1 989.
[Gray 2000] Jeremy Gray, The Hilbert Chal
lenge. Oxford: Oxford University Press,
2000.
[Hilbert 1 935] David Hilbert, Gesammelte Ab
handlungen, vol. 3, Berlin: Springer, 1 935.
[Lefschetz 1 94 7] Solomon Lefschetz et at. , eds.,
Problems of Mathematics, Princeton Univer
sity Bicentennial Conferences, Series 2, Con
ference 2, reprinted in [Duren 1 989, 309-334].
[Mac Lane 1 989] Saunders Mac Lane, "Topol
ogy and Logic at Princeton," in [Duren 1 989,
21 7-221 ] .
[Minkowski 1 973] Hermann Minkowski, Briefe an
David Hilbert, Hg. L. Rudenberg und H.
Zassenhaus, New York: Springer-Verlag, 1 973.
[Nye 2003] Mary Jo Nye, ed. , The Cambridge
History of Science. Volume 5: The Modern
Physical and Mathematical Sciences, Cam
bridge: Cambridge University Press, 2003.
[Reid 1 970] Constance Reid, Hilbert. New York:
Springer-Verlag, 1 970.
[Reingold 1 981 ] Nathan Reingold, "Refugee
Mathematicians in the United States of
America 1 933-1 941 , " Annals of Science 38
(1 981) : 31 3-338; reprinted in [Duren 1 988,
pp, 1 75-200] .
[Rota 1 989] Gian-Carlo Rota, "Fine Hall in its
Golden Age: Remembrances of Princeton in
the Early Fifties , " in [Duren 1 989, 223-236] .
[Rowe 1 996] David E. Rowe, "New T rends and
Old Images in the History of Mathematics,"
in Vita Mathernatica. Historical Research and
Integration with Teaching, ed. Ronald
Calinger, MAA Notes Series, vol. 40, Wash
ington, D.C.: Mathematical Association of
America, 1 996, pp. 3-1 6.
[Rowe 2003a] -- , "Mathematical Schools,
Communities, and Networks," in [Nye 2003,
pp, 1 1 3-1 32].
[Rowe 2003b] -- , "Hermann Weyl, the Re
luctant Revolutionary," Mathematical lntelli
gencer, 25(1 ) (2003), 61 -70.
[Siegmund-Schultze 1 998] Reinhard Siegmund
Schultze, Mathernatiker auf der Flucht vor
Hitler, Dokumente zur Geschichte der Mathe
matik, Bd. 1 0, Braunschweig: Vieweg, 1 998.
[Weyl 1 944] Hermann Weyl, "David Hilbert and
his Mathematical Work," Bulletin of the Amer
ican Mathematical Society, 50, 61 2-654.
VOLUME 25, NUMBER 4, 2003 15
DONALD G. SAARI AND STEVEN BARNEY
Conseq uences of Revers i ng Preferences
ther than standard election disruptions involving shenanigans, strategic vot-
ing, and so forth, it is reasonable to expect that elections are free from diffi-
culties. But this is far from being true; even sincere election outcomes admit all
sorts of counterintuitive conclusions.
For instances, suppose after the winner of an important
departmental election was announced, it was discovered
that everyone misunderstood the chair's instructions.
When ranking the three candidates, everyone listed his top,
middle, and bottom-ranked candidate in the natural order
first, second, and third. For reasons only the chair under
stood, he expected the voters to vote in the opposite way.
As such, when tallying the ballots, he treated a first and
last listed candidate, respectively, as the voter's last and
first choice. 1
Imagine the outcry if after retallying the ballots the chair
reported that the election ranking remained unchanged; in
particular, the same person won. Skepticism might be the
kindest reaction to greet an announcement that the elec
tion ranking for a profile-a listing which specifies the
number of voters whose preferences are given by each
(complete, transitive) ranking of the candidates-is the
same for the profile where each voter's preference order
ing is reversed. Surprisingly, this seemingly perverse be
havior can sincerely occur with most standard election pro
cedures. It is intriguing that this phenomenon can be
explained in terms of simple mathematical symmetries. Of
particular interest, the same arguments explain all of the
election paradoxes which have perplexed this area for the
last two centuries.
This issue appears to have been first introduced in [Saari
1995] where a section of this book showed that some pro
cedures allow the same election ranking to occur with a
profile and with its reversal. There is no interest in this phe
nomenon when the common ranking is a complete tie, but
when the common ranking is not a tie, this effect is called
a "reversal bias." The word "bias" is intended to foreshadow
how this anomaly affects election outcomes.
Rather than an election ranking, voters more typically care
only about who wins, or who is elected for, say, the depart
mental budget committee. This raises the question whether
an election procedure would allow the same winner, or the
same two candidates, .. . , or the same k candidates to be
top-ranked with a profile and its reversal. Call this situation
a "k-winner reversal bias." Common sense suggests that we
should question the reliability of an election procedure if it elects the same committee with a profile and with the pro
file of reversed preferences-i.e., if the procedure allows a
k-winner reversal bias. As one of us (Barney) discovered, an
Internet discussion group worrying about election methods
is particularly concerned about the case k = 1, which we call
the "top-winner reversal bias." It should be a concern be
cause, as shown here, rather than a rare and obscure phe
nomenon, we can expect some sort of reversal behavior
about 25% of the time with the standard plurality vote.
Our thanks to Hannu Nurmi, Tom Ratliff, and two referees for their comments on an earlier version.
1This is not a hypothetical story, but actually occurred in an academic department to which one of us (Saari) belonged. The chair was promoted to a higher adminis
trative position.
© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 4, 2003 1 7
Positional Methods Among the widely used election methods are what William
Riker [ 1982] calls positional methods. Riker, who was a pi
oneer in using mathematics to address problems from po
litical science, coined the word "positional" to refer to a
method where a ballot for the n 2: 2 candidates is tallied
by assigning specified weights, WI, w2, . . . , Wn, respec
tively, to a voter's first, second, . . . , and nth ranked can
didates. The candidates are then ranked according to the
sum of weights from all ballots. Since the election ranking
remains unchanged after adjusting the weights so that Wn = 0, assume that this is the case. The plurality vote is the
commonly used "vote for one" system where WI = 1 and
w2 = · · · = Wn = 0. The weights for the Borda Count
(named after Jean Charles de Borda, an eighteenth-century
French mathematician, inventor, explorer, warrior in the
American Revolution, and one of the founders of the met
ric system) specify the number of candidates ranked below
a specified candidate, so WI = n - 1, w2 = n - 2, . . . , Wn = n - n = 0. Actually, any choice of weights defines a posi
tional method as long as WI > Wn = 0 and Wj 2: Wj+ I for
j = 1, . . . , n - 1. (A positive fixed multiple of the weights
scales the tally and yields the same election ranking.)
To demonstrate, we compute each candidate's tally for
all positional methods for the profile
Number Prefer A B c
4 A>C>B 4w1 0 4w2 3 A>B>C 3w1 3w2 0 (1 ) 4 B>C>A 0 4w1 4w2 3 C>B>A 0 3w2 3w,
Total 7w1 4w1 + 6w2 3w1 + 8w2
Thus the plurality vote, where WI = 1, w2 = w3 = 0, results
in the ranking A>B>C with a 7:4:3 tally. With the antiplu
rality vote defined by WI = w2 = 1, W3 = 0 (called "antiplu
rality" because by voting for all but one candidate, each voter
is effectively voting against a candidate; the method is a "neg
ative plurality vote"), the election ranking is C>B>A with a
11: 10:7 tally. Notice the conflict with the plurality outcome.
Now reverse each voter's ranking to obtain the reversed
profile
Number Prefer A B c
4 B>C>A 0 4w1 4w2 3 C>B>A 0 3w2 3w1 4 A>C>B 4w, 0 4w2 (2)
3 A>B>C 3w1 3w2 0
Total 7w1 4w1 + 6w2 3w1 + 8w2
The point to notice is that each candidate's tally for each
positional procedure is the same with the table 1 profile as
with its table 2 reversal. Thus, unless the outcome is a com
plete tie, the procedure exhibits a reversal bias. A complete
tie requires 7WI = 4wi + 6w2 = 3wi + 8w2, or WI = 2w2,
the Borda Count. Consequently, with the sole exception of
the Borda Count, all other positional methods experience
a reversal bias with this profile. The source of this phenomenon is the considerable sym-
1 8 THE MATHEMATICAL INTELLIGENCER
metry embedded in the profile's two pairs of two rankings.
The flrstpairis the rankingA>C>Bwith its reversalB>C>A;
each is preferred by four voters. Likewise, with the pair of
A> B>C and its reversal C> B> A, each is supported by three
voters. As positional methods respect anonymity (i.e., we do
not know who has what preferences), the profile and its re
versal are the same. Being indistinguishable, the profile and
its reversal must give the same outcome.
To describe this symmetry, first let ffi,j be the permu
tation of candidates that interchanges i andfs names. So,
if p is a profile then ui, j (p) interchanges each voter's rank
ing of i andj. It is easy to show that all positional methods
f satisfy what is called neutrality; namely,
./(ui,j(p)) = ffi,jCf(p)). (3)
In words, if 211 voters confused Sue with Mary when mark
ing the ballot (instead of the correct profile p, they used
<rs,M(p)), then the correct outcome is found by exchanging
Mary's and Sue's tallies (namely, use <rs,M(./(us,M(p))), as
it equalsf(p)).
Our example involves the symmetry Ilk, which reverses
rankings. More precisely, if l!k(p) reverses each voter's
ranking of the candidates, we want to identify all proce
dures where
f(l!k(p)) = l!k(j(p)). (4)
As reversing a reversal returns to the initial ranking, Eq. 4
means that llk(.f(l!k(p))) = j(p). Using the introductory ex
ample where all marked their ballots in the reversed manner
(rather than p, the ballots are marked as l!k(p)), if the elec
tion procedure satisfied Eq. 4, then a way to find the correct
f(p) outcome is to reverse the j(l!k(p)) ranking. This seem
ingly natural property can fail with most procedures.
A way to spot the methods susceptible to these prob
lems is to mimic the table 1 example by using profiles of
the l!k(p) = p type; i.e., those profiles where each ranking
in p is accompanied by the same number of voters prefer
ring its reversal. With these profiles, .f(l!k(p)) = j(p ). So, if
Eq. 4 is true, we have that l!k(f(p)) = .f(p). But l!k(f(p)) = j(p) holds only ifj(p) is a complete tie. Thus, we just need
to identify those procedures which fail to deliver a com
plete tie for these special l!k(p) = p profiles.
THEOREM 1. For three-candidate elections, only the Borda
Count never exhibits the reversal or k-winner reversal bias,
k ::::; 2. AU other positional methods suffer the reversal, top
winner, and 2 winner-reversal bias. For a procedure to ex
hibit these effects, a profile must have a su.fficienay large
component of rankings with their reversal.
The Borda Count always satisfies Eq. 4 for any n ;::: 3,
so it never has a reversal or a k-winner reversal bias. Al
most all positional methods fail to satisfy the equalities
WI = W2 + Wn-I = W3 + Wn-2 = · · · = Wn-I + W2 = Wii (5)
methods jailing Eq. 5 allow reversal and k-winner rever
sal biases for any k ::5 n - 1 . Indeed, if Eq. 5 jails, then select any ranking; a profile can be constructed where the
profile and its reversal support the specified ranking with the same tally.
While simple, Eq. 5 has surprisingly strong conse
quences. It means, for example, that all commonly used
methods, such as "vote for one," or "vote for two," or meth
ods based on almost any choices of weights are suscepti
ble to the full array of reversal problems. Moreover, since
it is arguable that profiles of this p = ffi(p) type should end
in a tie, it follows that procedures which fail Eq. 5 bias the
outcome; a measure of this bias is the difference in value
between the smallest and largest Eq. 5 terms. For three al
ternatives, then, the bias that a positional method intro
duces into the election outcome is captured by the non
zero difference w1 - 2w2• We will return to this comment
when discussing general election paradoxes.
Once we understand the origin of Eq. 5 and how to con
struct examples, a formal proof is immediate. To explain
Eq. 5, the election tallies for the profile consisting of
Ct>Cz> · · · > Cn and its reversal Cn> · · · > c2>c1 are, re
spectively, Wt + Wn, Wz + Wn- b Ws + Wn-2, · · · , Wt + Wn· If these tallies fail to agree, they violate Eq. 5 (remember,
Wn = 0) and the non-tied outcome means that the proce
dure suffers a reversal bias.
To construct profiles asserted by Theorem 1 with the elec
tion ranking c1>c2>c3>c4, we exploit the bias caused when
Eq. 5 is not satisfied. So if w1 + w4 < w2 + w3, exploit the
larger w2 + w3 sum by putting into the profile voters for
whom c1 and c2 are, respectively, second and third ranked.
The two remaining candidates, c3 and c4, can be ordered in
two ways. Use both orderings to define the two rankings
cs>c1>cz>c4 and c4>c1>c2>cs. Include the reversal for each
ranking to obtain what we call the "c2 unit" of (c3>c1>c2>c4, c4>c2>c1>csl and (c4>c1>c2>cs, cs>c2>c1>c4}. For this c2 unit, c1 and c2 each receive 2( w2 + w3) points, while c3 and
c4 each receive the smaller 2(w1 + w4) = 2w1 value.
Replace c2 with c1 to create a c3 and a c4 unit. The num
ber of the c1 units needed to design a profile depends on
the desired outcome; e.g., one choice of a p which gener
ates the specified election ranking consists of two c2, one
cs, and no C4 units. The c1>c2>cs>c4 election outcome has
the tally 2[3(w2 + ws))] > 2[2(w2 + w3) + wt] > 2[(w2 + ws) + 2wt ] > 2[3w1]. By construction p = ffi(p ), so the
conclusion of Theorem 1 is satisfied. The formal proof just
verifies that this approach extends to any n. Notice a conspicuous gap: of all positional methods sat
isfying Eq. 5, Theorem 1 only excuses the Borda Count from
these reversal effects for n 2:: 4 alternatives. For instance,
the weights (2,1, 1,0), or (2,2 - y,y,O), 0 :s: y :s: 1, or (4,4 - z,
2,z,O), 0 :s: z :s: 2, define positional methods which satisfy
Eq. 5, but Theorem 1 does not state whether they suffer the
reversal bias. They do not (the technical proof is omitted),
but they have other problems.
These conclusions extend to a much wider class of vot
ing procedures. For instance, Saari and Van Newenhizen
[ 1989] define a multiple voting procedure as one which is
equivalent to having the voter mark the ballot and then se
lect the positional procedure to tally this particular ballot.
"Approval Voting" is the multiple voting procedure where
a voter can vote for as many candidates as he or she wishes;
by voting for one, or two, or say three candidates, the voter
is effectively selecting, respectively, the positional methods
(1,0, . . . ,0), or (1 , 1 ,0, . . . ,0), or (1 ,1 , 1,0, . . . ,0). (As one
might anticipate from the variability, this procedure, used
by both the MAA and AMS, has several serious flaws [Saari,
2001] .) Other multiple procedures are "truncated voting"
where a voter ignores instructions by voting only for some
candidates, and "cumulative voting" where a voter can dis
tribute a specified number of points among the candidates
in any desired manner, etc. As these methods clearly fail
Eq. 5, the following assertion is immediate.
THEOREM 2. All multiple voting procedures suffer both a reversal and a k-winner reversal bias. In particular, this includes Approval Voting, cumulative voting, and truncated voting when used with any positional method.
We leave it for the reader to determine (which is not dif
ficult) whether the method of single transferable vote (STV)
used by the AMS suffers these problems. In STV, when the
goal is to select, say, two of three candidates, as soon as a
candidate receives over a third of the vote, she is elected;
any remaining ballots that have her top-ranked are reas
signed to the second-listed candidate.
Three-Candidate Positional Elections A more important objective is to understand how reversal
effects affect election outcomes. In doing so, we verify our
earlier statement about the likelihood of these reversal ef
fects, along with the claim that these reversal behaviors ex
plain voting paradoxes. This last theme uses the observa
tion that a positional method which fails to satisfy Eq. 5
can bias the election outcome. Indeed, as we will see, all
differences among positional election outcomes reflect the
w1 - 2w2 differences. Along the way, some easily used con
ditions are developed to identify, for instance, when re
versing a profile will not reverse the ranking. In deriving
new conclusions while outlining how to find others, the
three-candidate setting is emphasized for ease of exposi
tion. Our approach uses the "procedure line" and a geo
metric representation of profiles introduced in [Saari 1994,
1995] and used in several ways by Nurmi [1999, 2002] .
Equilateral triangles, such as Fig. 1a, are useful devices to
describe three-candidate election outcomes. Assign a rank
ing to a point in the triangle according to its distance from
each vertex where "closer is better." For instance, any point
in the small triangular region of Fig. 1a with "11" is closest
to B, next closest to C, and farthest from A, so it is assigned
the B>C> A ranking. Represent a profile by listing the num
ber of voters who have each preference ranking in the ap
propriate region; e.g., Fig. 1a displays the profile given in (6).
No.
5
4
Ranking
A>B>C
A>C>B
C>A>B
No.
3
1 1
0
Ranking
C>B>A
B>C>A
B>A>C
(6)
VOLUME 25, NUMBER 4, 2003 1 9
4 + 15s C
A "'---..._--� B 9 + s 10 14 11 + 8s a. Profile
Figure 1. Representing profiles and tallies.
This geometry simplifies computing election tallies. To
see why, notice that all rankings with A>B are to the left
of the vertical line, so the 5 + 4 + 1 = 10 sum of the num
bers in these three Fig. 1a regions is A's tally in an {A,B)
pairwise election. All pairwise tallies are similarly com
puted and listed next to the appropriate triangle edge.
Instead of using ( wb w2, 0) for positional method elec
tions, an easier way to compare procedures is to normalize
the weights by dividing by w1; this defines (1, s, 0) where the
fixed s = w'2fw1 value, 0 ::::; s ::::; 1, is assigned to a second
ranked candidate. In this manner the Borda Count (2,1,0) be
comes (1,t,Q), and the (7,5,0) method becomes (l,t,Q). To
tally positional-method ballots, notice from Fig. 1a that A is top ranked in the two regions with A as a vertex, so add these
numbers. Next, A is the second ranked in the two adjacent
regions; in Figure 1a these are the two regions containing 1
and 0. Thus, add s times this sum to compute A's final tally
of (5 + 4) + s(1 + 0) = 9 + s; this value is placed near the A
vertex. The similarly computed tallies for the other two can
didates are listed next to the appropriate vertex.
In the three-dimensional space of election tallies, R3, the
A, B, C tallies of (9 + s, 11 + 8s, 4 + 15s) describe a line
connecting the plurality tally (where s = 0) with the an
tiplurality outcome (where s = 1 ); this is the procedure line [Saari 1992, 1994, 1995]. This line identifies all positional
method tallies; the tally for (1, s, 0) is s of the way from the
plurality to the antiplurality tally. Since the Borda Count is
given by s = t• the Borda tally is at the midpoint. Proce
dure lines have proved to be a convenient tool. For in
stance, by using the procedure line, Tabarrok [2001] dis
covered surprising conclusions about the 1992 presidential
election involving Clinton. History buffs will enjoy the
Tabarrok and Spector [1999] paper using a natural exten
sion of the procedure line [Saari 1992] to characterize
everything that could have happened with the 1860 elec
tion involving Abraham Lincoln.
An advantage of the procedure line for theoretical pur
poses is that it identifies all positional-method outcomes.
This suggests that a way to find all consequences of re
versing a profile p is to compare the p and ?Jt(p) procedure
lines. But first we need to represent a reversed profile.
Finding the reversed profile
To find the reversed profile, place the number from each
triangular region of the original profile in the diametrically
20 THE MATHEMATICAL INTELLIGENCER
C 5 + 15s
A ---..._--� B 14 + s 14 10 5 + 8s b. Reversed profile
opposite region (relative to the center of the triangle); e.g.,
the Fig. 1b profile is the reversal of the Fig. 1a profile. No
tice from Fig. 1b that while each candidate's plurality tal
lies (the s = 0 values) for the profile and the reversed pro
file differ, the coefficients of "s" remain the same. This
always is true. To explain, when tallying ballots as de
scribed above, the s coefficient is the sum of terms in dia
metrically opposite regions; consequently, reversing a pro
file preserves the sum.
The s coefficients are the differences between the pro
cedure line's endpoints-the antiplurality and plurality tal
lies-so they represent the tallies of the voters' secondplace vote. Call this difference vector, or line segment, the
Second Place Tallies (the SPT). For instance, the Fig. 1a
SPT is the vector (10, 19, 19) - (9, 11, 4) = (1, 8, 15).
THEOREM 3. For any three-candidate profile p, the SPT for p and ?Jt(p) agree. This forces the directions and lengths of their procedure lines to agree; the two lines are parallel.
Proof The direction and length of a line are determined by
the difference between its end points; this difference be
tween a procedure line's antiplurality and plurality tallies
is the SPT. For instance, the Fig. 1b SPT of (15, 13, 20) -
(14, 5, 5) = (1, 8, 15) is the same as for Fig. 1a. As a pro
file's SPT is defmed by the s coefficients, which always
agree for p and ?Jt(p ), the theorem is proved. 0 Compare a profile's SPT to a straight piece of wire which
behaves like a compass needle; when moved, it points in
the same direction in the three-dimensional space of tal
lies. So the tally of one positional method and the SPT com
pletely determine the procedure line. As the SPT for p and
?Jt(p) agree (Theorem 3), knowing how the tallies for a des
ignated procedure change from p to ?Jt(p) completely de
termines the p and ?Jt(p) procedure lines. Since only the
Borda Count is immune to reversal effects, it is the desig
nated procedure.
To illustrate this description with the Figure 1 example,
the (normalized) Borda Count tallies for p and ?Jt(p) are,
respectively, (9.5, 15, 11.5) and (14.5, 9, 12.5). The Borda
Count defines the midpoint of the procedure line, so p's
procedure line is found by placing the (1, 8, 15) SPT so that
its midpoint is at (9.5, 15, 11.5). Similarly, to find ?Jt(p)'s
procedure line, move the same SPT so that its midpoint
now is at (14.5, 9, 12.5).
This description attributes all reversal effects to changes
in the Borda tally. These changes involve another symme
try involving how each candidate's Borda vote differs from
the average Borda score. To illustrate, table 7 computes
these differences for the Fig. 1a profile and its Fig. 1b re
versal; the total number of Borda points is 36, so the aver
age of Borda points assigned to the candidates is 36/3 = 12.
Profile A 8 c
Original, Fig. 1a 9.5 - 1 2 = -2.5 1 5 - 1 2 = 3 1 1 .5 - 1 2 = -0.5 (7)
Reversed, Fig. 1 b 1 4.5 - 1 2 = 2.5 9 - 1 2 = - 3 1 2.5 - 1 2 = 0.5
As table 7 suggests, reversing a profile just changes the sign
of each candidate's Borda differential from the average
Borda score. The reason for this behavior is that each can
didate's Borda tally can be computed by adding the points
she receives in all pairwise elections. So to compute a can
didate's difference from the average Borda score, in each
pairwise election add how much the tally differs, either
above or below, a complete tie. (To have normalized Borda
values, divide by two.) But eil(p) reverses all pairwise tal
lies-and all differences from the average-so only the sign
changes when computing differences from the average
Borda score.
This effect indicates how to find geometrically all three
candidate properties associated with reversing a profile.
Profile p's procedure line is determined by its midpoint, the Borda tallies, and its SPT vector. For an n-voter profile p, the average Borda score is C%n)/3 = �· so point (�, �' �) on the diagonal x = y = z indicates each candidate's average Borda score. Construct a line segment where p's Borda tally is one endpoint and the segment's midpoint is (� , �· %); the segment's other endpoint is the eil(p) Borda tally, so it is the midpoint of the eit(p) procedure line. To find the eit(p) procedure line, slide p's SPT
to this eil(p) Borda tally. All consequences of reversing a profile p are found by comparing differences in the p and 0't(p) procedure lines.
In Fig. 2a, the solid line on the right represents the Fig. 1a
procedure line and the bullet designates the Borda tally in
the B>C>A region; the plurality endpoint is in the B>A>C
region. To find the eil(p) shift, construct a line (the Fig. 2a
dashed line) from p's Borda tally passing orthogonally
through the x = y = z diagonal; eil(p )'s Borda tally is equidis-
c
A a. Profile line and transfer
tant on this line on the other side of the diagonal. To find
eil(p )'s procedure line, which is the slanted line on the left,
slide p's SPT so that its midpoint is at this flipped Borda tally.
Some geometry
We can use this geometry to indicate why the B>A>C plu
rality ranking for the Fig. 1a profile p changes to A>C - B for eil(p ). First, the plurality endpoint of p's procedure line
is close to a B - A tie. The SPT remains invariant, so the
key is the flipped Borda tally; it favors A, helps C, but hurts
B (table 7). In eil(p )'s procedure line, this flip pushes the
SPT deeper into the region favoring A, somewhat helping
C, but hurting B. For readers comfortable with three-dimensional geom
etry, this description suffices to explain the new results
given below. For most of us, however, the Fig. 2a three-di
mensional geometry is difficult to envision. So replace ac
tual tallies with the fraction of the total vote each candi
date receives; e.g. , replace a (50, 140, 10) tally with (2��, ���' 2�0) . Geometrically, the normalized tally is a point in
the simplex { (x, y, z) lx + y + z = 1, x, y, z 2: 0} (Fig. 2b).
Although Fig. 2b helps to visualize election outcomes,
problems arise because the projection distorts geometric
properties; this distortion is not dissimilar to the difficulties
of observing objects through a convex mirror. Rather than
the midpoint, for instance, the normalized Borda tally is two
thirds of the way from the plurality point. As dramatically
demonstrated in Fig. 2b, the projected p and eil(p) parallel
procedure lines are skewed. The reason for this distortion is that the plurality tallies are divided by the number of voters
while the antiplurality tallies are divided by twice this num
ber. Consequently the two normalized plurality endpoints al
ways are twice as far apart as the antiplurality endpoints.
The two dotted lines in Fig. 2b-the bottom and top one
connect, respectively, the p and eil(p) plurality and an
tiplurality outcomes-are parallel; this provides an inter
esting research tool. In the figure, p's Borda tally (the bul
let) is reflected along the dashed line about the center point
to identify eil(p )'s Borda tally; this flipped Borda position
determines eil(p )'s procedure line. While the projected pro
cedure lines rarely are parallel, the dashed and dotted lines
always are. After formally stating this observation and then
indicating which lines in a triangle can be procedure lines,
these facts are combined to derive new conclusions.
c
b. Two-dimensional projection
Figure 2. Procedure lines for a profile and its reversal.
VOLUME 25, NUMBER 4, 2003 21
THEOREM 4. For any p and any two (1 , s, 0) positional methods, the lines connecting each method's (normalized) taUy for p and (J}l,(p) are paraUel in a Fig. 2b representation. The length of the line connecting the (1, s, 0) outcomes is -1- times the length of the line connecting plu-1 + s rality outcomes, or 2(1: s) times the length of the line con-necting the Borda outcomes. The (1, s, 0) outcome on a procedure line is � of the distance from the plurality 1 + s to the antiplurality endpoint.
The proof is a straightforward exercise in elementary
geometry which we leave to the reader.
Procedure lines
Before using Theorem 4, we need a simple way to fmd
all possible positional lines. The surprisingly relaxed rules
[Saari 2001] to identify which line segments in a triangle
are procedure lines are described in terms of how voters
cast their first (the plurality outcome) and second (the SPT)
place votes. For any integers selected in the following man
ner, a unique profile exists with the specified election out
comes and tallies.
• Choose a non-negative integer value for each candidate's
plurality tally, the sum determines the total number of
voters.
• Select any nonnegative integers to define the SPT where
-their sum equals the number of voters, and
-since the sum of the SPT entries for any two candi-
dates includes the plurality tally for the third candi
date, it must be at least this large.
A candidate's Borda tally is the average of her assigned plu
rality and antiplurality values.
To illustrate how easy it is to construct the unique sup
porting profile, suppose the plurality tallies for A, B, and C
are, respectively, 4, 5, and 6 as indicated in Fig. 3, and the
SPT is (10,5,0). Adding the plurality and SPT tallies deter
mines the antiplurality tallies (in parentheses) of respectively
14, 10, 6. The zero SPT value requires the diagonal terms
defining Cs s coefficient to be zero, so the 4 and 5 for A's
and B's plurality tallies must be positioned as indicated in
Fig. 3a. It is trivial to find the division of Cs six plurality
votes which allows the correct antiplurality outcomes.
For procedure lines on the equilateral triangle, it is eas
ier to describe the plurality and antiplurality endpoints. The
following rules follow from properties of the projection.
{6) , 6 c
A &...... _ ____.......__ _ _..;a B {14) , 4 a. {10) , 5 Figure 3. Creating a profile.
22 THE MATHEMATICAL INTELLIGENCER
• Any non-negative rational value can be each candidate's
normalized plurality tally as long as the values sum to
unity; this defines the plurality endpoint of the procedure
line.
• The antiplurality endpoint can be any non-negative ra
tional value which
-is at least half as large as the assigned plurality value
and all values sum to unity,
-is bounded above by one half, and
-for any two candidates, the sum of twice their an-
tiplurality value minus their plurality value is at least
as large as the third candidate's normalized plurality
value.
• The Borda outcome is two thirds of the way from the
plurality to antiplurality endpoint.
To illustrate, select (i, i• t) and (±, ±• i), respectively,
for plurality and antiplurality values. By multiplying the
first by 6 and the second by twice this value, integer tallies
of (3, 1 , 2) and (3, 3, 6) emerge. The corresponding integer
profile is in Fig. 3b; fractional values follow by dividing each
value by 6. Because fractions are dense, the line segments
which depict properties of positional methods can be
drawn in almost any way near a complete tie.
Finding new results
It now is easy to fmd new conclusions about reversal ef
fects. Just draw a line in the triangle-to represent p's pro
cedure line-and use the above structures to compute
(J}l,(p)'s procedure line. Results follow by comparing differ
ences and similarities of outcomes on the two procedure
lines. Thus, all possible results are determined by all pos
sible ways these lines can be drawn. Moreover, a sense
about the likelihood of different conclusions is associated
with the flexibility in drawing appropriate lines.
We illustrate this approach by using the horizontal pro
cedure line drawn in Fig. 4a which meets seven ranking re
gions (three regions are lines which represent tie votes).
Thus the corresponding profile p has seven different elec
tion rankings that vary with changes in the positional
method; they range from the plurality A>C>B through
A>B>C and the Borda's B>A>C to the antiplurality's
B>C>A. The Borda tally is identified by the bullet. To find
(J}l,(p)'s positional line, flip the Borda tally about the center.
(Construct a dashed line from Borda tally through the cen
ter; (J}l,(p )'s Borda tally is equidistant on this line on the other
side of the center.) Next, draw a dotted line (on the left)
b.
R(p) .
a.
Figure 4. Finding new results.
parallel to the dashed line; start it from p's plurality tally. According to Thm. 4, (lh(p )'s plurality point is on this dotted line and 3/2 as far as the distance between Borda tallies. As these two points define a straight line, they determine (lh(p)'s procedure line; (lh(p)'s antiplurality outcome is on the parallel dotted line on the right.
Figure 4b illustrates this construction with a p choice that admits three different election rankings. Notice how the orientation of p's procedure line affects the orientation for (lh(p )'s line. An amusing example is to choose p's procedure line to be a point. (This profile requires the proportion of the tally assigned to each candidate to be the same for all positional procedures.) The corresponding (lh(p) procedure line is a segment (on the line from the point through the center) where all rankings reverse the common p ranking.
Conclusions now are apparent. For instance, just by varying the length of the SPT and the location of the Borda tally, the procedure line could allow one, or two, or, . . . , or seven different election rankings. An interesting feature of both Fig. 4 diagrams is that the number of rankings allowed by p and (lh(p) agree; this always is the case. Also notice that the closer the Borda tally is to the center-a complete tie-the smaller the changes allowed in election outcomes when reversing the profile.
THEOREM 5. The following statements hold for three-candidate positional-method elections.
1. For any integer k, 1 ::s k ::s 7, a profile p can be found with precisely k different positional-method outcomes as the value of s varies; (!h(p) also has precisely k different outcomes. (For k > 1, some outcomes involve ties).
2. All non-Borda positional methods experience the toptwo reversal bias; that is, the same two candidates are top-ranked with p and (!h(p ), but ranked differently. For instance, p's plurality tally could be A>B>C while (lh(p) 's could be B>A>C.
3. All non-Borda positional methods experience a topwinner reversal bias. That is, the same candidate can win with p and (!h(p) but otherwise the rankings differ; e.g., p 's antiplurality ranking could be A>B>C while (!h(p) 's could be A>C>B.
4. A necessary and sufficient condition for all positional methods to have the same ranking and tally for a profile p and (!h(p) is that the Borda ranking be a complete tie.
b.
The only non-obvious fact (which follows from Theo
rem 6) is that p and (lh(p) always admit the same number of rankings. The rest of these results can be verified just by drawing lines on the triangle. For instance, no matter how a straight line is drawn, it cannot cross more than seven regions, so the upper bound of part 1 is obvious. Similarly, to create a p with three, or four, or any other number of outcomes, just draw a line meeting the specified number of regions. To verify the second part, which asserts there is a profile p with an A>B>C plurality outcome while (lh(p)'s plurality outcome is B>A > C, place the plurality endpoint of p's procedure line in the A> B> C region near a A � B tie; the plurality tip of the (lh(p) line will be in the B> A> C region if you place the Borda point in the C>A>B region. By using the approach described in the previous section, actual profiles are easy to construct.
The last assertion is the easiest to explain. If the Borda Count is a complete tie, then the p and (lh(p) procedure lines coincide. But when the Borda Count is not a complete
tie, the flip which determines the (lh(p) Borda tally changes the outcomes for all positional procedures which are sufficiently close to the Borda Count.
These results seem to suggest that almost any outcomes can occur, but this is false. As in Fig. 4, a positional procedure's p and (lh(p) outcomes must be on the same side
of the dashed line connecting Borda tallies. This geometry restricts the procedure's allowed outcomes.
Does anything reverse?
Intuition suggests that something must be reversed when a profile is reversed. This is correct; Theorem 6, which slightly generalizes a result in [Saari, 1995] describes a reversal effect which combines reversals of election rankings, profiles, and the choice of a positional method. To explain the notation, let f(p,(1, s, 0)) be the (1, s, 0) tally for profile p, and let fN(p,(1, s, 0)) be the normalized tally. Recall, the antiplurality vote is the reversal of the plurality vote, as it is equivalent to plurality voting against somebody; similarly the (1, 1 - s, 0) voting method can be viewed as
the reversal of (1, s, 0).
THEOREM 6. For any p involving n voters and for any s, 0 ::s s ::s 1, the tallies satisfy
f(p, (1, s, 0)) + f((!h(p), (1, 1 - s, 0)) = (n, n, n). (8)
VOLUME 25, NUMBER 4, 2003 23
For normalized tallies, the relationship is
(1 + s)fN (p, (1, s, 0)) + (1 + (1 - s))fN (\Yt(p), (1, 1 - s, 0)) = (1, 1, 1). (9)
The f(p, (1, s, 0)) ranking always is the reversal of the
f(\Yt(p ), (1, 1 - s, 0)) ranking.
Proof Candidate A's tally is the number of voters who have her top ranked plus s times the number who have her second-ranked. Tallying \Yt(p) with (1, 1 - s, 0) is equivalent to the number of voters with A bottom ranked plus (1 - s)
times the number who have her second ranked. As the sum is n, Eq. 8 follows. To derive Eq. 9, normalize the tallies.
The same argument generalizes Eq. 8 from three to c � 3 candidates. After normalizing the positional weights to (s1 = 1 , s2, s3, . . . , Sc- 1• Sc = 0), Eq. 8 extends to
f(p,(l, S2, . . . , Sc- 1 , 0)) + f(\Yt(p), (1, 1 - Sc- 1 , . . . , 1 - Sz, 0)) = (n, n, . . . , n).
This expression allows the above results to be extended to
any number of candidates. D To illustrate Eq. 8, the 24-voter Fig. 1a proflle has the
plurality tallies of (9, 11 , 4), while the antiplurality tallies for \Yt(p) are (15, 13, 20). It is immediate that
(9, 11 , 4) + (15, 13, 20) = (24, 24, 24).
As required by Theorem 6, p's plurality ranking of B>A>C
reverses the \Yt(p) antiplurality ranking of C>A>B. More generally, the (1, s, 0) tallies for Fig. 1a, and the (1, 1 - s,
0) for Fig. 1b, are, respectively, (9 + s, 1 1 + 8s, 4 + 15s) and (14 + (1 - s), 5 + 8(1 - s), 5 + 15(1 - s)). We find, as required by Eq. 8,
(9 + s, 1 1 + 8s, 4 + 15s) + (15 - s, 13 - 8s, 20 - 15s)
= (24, 24, 24).
As another example, suppose a 30-voter profile p is constructed to have a plurality ranking of A>C>B with tallies of (16, 4, 10). It immediately follows that the antiplurality ranking of \Yt(p) is B>C>A with tallies (30, 30, 30) - (16, 4, 10) = (14, 26, 20).
The surprising regularity of positional election rankings offered by Theorem 6 makes it easier to determine all relationships between p and \Yt(p) outcomes. For instance, to analyze the top-winner reversal bias for the plurality vote, we need to determine all ways to position the procedure line so that the plurality winner is the same for p and \Yt(p ). But according to Theorem 6, this situation holds if and only if p's antiplurality ranking has this same candidate bottom
ranked. So, rather than needing to construct \Yt(p) to determine whether this behavior occurs, we can concentrate
on properties of procedure lines for p. For instance, one such p with a top-winner bias has a procedure line with a plurality ranking A>B>C and an antiplurality ranking B>C>A; this line is easy to draw.
Armed with Theorem 6 we can identify all reversal be
havior just from the p election rankings. To illustrate with a plurality A >B>C ranking, the following lists all possible antiplurality endpoints. The \Yt(p) plurality ranking, the reversal effects, and the number of positional method outcomes (the number of regions the positional line crosses)
are also specified.
Number of
outcomes p Antlplurality 01(p} Plurality Reversal biases
A>B>C C>B>A no reversal effects
3 A>C>B B>C>A no reversal effects
5 C>A>B B>A>C two-winner reversal
7 C>B>A A>B>C ranking reversal
5 B>C>A A>C>B top-winner reversal
3 B>A>C C>A>B no reversal effects
As the above demonstrates, some sort of reversal bias
occurs for the plurality vote if and only if the procedure line permits five or more rankings. By using results from
[Saari and Tataru 1999], which compute the probabilities that positional methods have specified numbers of outcomes2, we obtain the likelihoods of different reversal behaviors. Incidentally, it also follows from Theorem 6 that if a condition permits one of these reversal phenomena to occur with the plurality method, the same behavior occurs with the antiplurality method.
THEOREM 7. For three candidates, the following probabil
ity statements hold for any probability distribution of
voter profiles where, as the number of voters grows, the
distribution is asymptotically independent with a common variance, and the mean has an equal number of vot
ers of each type.
1. A necessary and sufficient condition for all positional
method outcomes of a profile p to be reversed when the
profile is reversed is for p 's plurality and antiplural
ity outcomes to agree. The likelihood of such a behavior is 0.31.
2. A necessary and sufficient condition for a reversal ef
fect to occur for the plurality outcome is that a profile's antiplurality outcome reverse the plurality out
come. This behavior occurs with probability 0.06. 3. A necessary and sufficient condition for a plurality
(or antiplurality) top-reversal, or a two-winner rever
sal effect is for the profile to allow five different elec
tion rankings as the positional methods change (and
for the plurality outcome to be a strict ranking). This
occurs with probability 0. 19.
According to this theorem, reversal effects are surpris
ingly likely. Similar results hold for all (1, s, 0) and (1, 1 - s,
0) rules, s -=/=- �· but with larger likelihoods for the first assertion and smaller likelihoods for the other two. To extend the second statement, notice that a necessary and suf-
2The Saari-Tataru approach using procedure lines and differential geometry has subsequently been used by others, primarily various combinations of M. Tataru, V. Mer
lin, and F. Valgones, to obtain several fascinating results; e.g., see [Tataru, Merlin 1 999], [Merlin, Tataru, Valgones 2000]).
24 THE MATHEMATICAL INTELLIGENCER
ficient condition for a (1, s, 0) reversal effect to occur is
for a profile's (1, 1 - s, 0) outcome to reverse the (1, s, 0) outcome; this likelihood diminishes to zero as s � t· Constructing examples
When p = 0l(p) components of a profile cause reversal effects, it is reasonable to anticipate that the more these
Three-candidate Profile Subapacea Tr at an n-voter profile for a thr e-candidate election as a v tor in R' ,.,;th non-negative int ger c mpon nt. . Pr fil ha n d mp · d int U1 · com n nt
parts wh.i h an� t th din r nt kinds of el ·tion m I h
od [ 'aari 1999. 2000] by disco,· 'ting the appr priat c .
ordinat ·y m which <livid Jr.i int row· rnut ually or
th gonal 'Ub ·pa • . Th M utral" profil p · - (.!!, . . . i) ·p cifi th number of voters; it is a point on tlt • di
agonal. Th other three ub ·pac r ide in 01e in1ple 6
i,(3!) = {p = CPt, . • . P ) E /(> - Pi = 1t, PJ � 0], J I
through P . . Th n gativ and p iti\"
tor in i,.(3!) d rib how to m v v 1 r
to on rt the starting p · int a d ir d pro-
The first h o-dimen ional ubspace th Basic Pro
files. i panned by B,1 and Ba.
- 1 + Os C c - 1 + 0
A .a.::;...--.L.....-.....;::,jO B A .a.::;... __ .L....._.....;::,jO B 2 Os 2 - 1 + 0 - 1 + Os 2 Os
a. A-Basic, BA b. B-Bs.sic, Bs
Th B. 1 fom1 ugg t pr fer n which pr · usly had
A b ttom-rank d ar mov d t no,., ha\" .4. I p-mnk d.
(\ ith U1 obvious choic for Be, B,t + Bs B = 0.) s demo trn d by th talli r t d by th triangle • we
that pairwise and po it ional out om for Basic proalway. ar consi "t nt-no voting conflic · an oc
ur h r .
onfli t . tart wi1h lhc two-dim nsional Re1•r.rsal ub. pact> pann d b R., and RH.
1 + 2 c
A B 2 4 1 + 2s
c. A-Rever81ll, R"
C I 2s
A B 1 + 2 2 - 4s
d. B-Revcrsal, Rs
components dominate a profile, the more dramatic the reversal effects. Not only is this true, but all possible three
candidate differences among positional outcomes-the so
called "election paradoxes" -are completely determined by these reversal terms. In other words, we now know that the huge literature characterizing differences among these procedures merely describes consequences of how these
O + Os C C 1 1 - 30s
A ""'----:-L......:-.....,. B A �---'L---� B 0 + Os 0 + Os - 1 3 + 36s - 1 1 2 - 6
e. Condorcet, C r. An example
-l .1·( 1 2s) y( 1 + 2. ) > - l + .r(l + 2s)
+ y(2 - 4s) > 5 x(2 - 4s) + !J( l + 2s)
plurality vot ) ; .r = - I I . y = -4 ar the
choic
Z• '} . -
VOLUME 25, NUMBER 4, 2003 25
procedures are affected by reversal effects. This assertion
follows from a convenient decomposition of profiles which
allows us to analyze all possible positional and pairwise
elections [Saari 1999].
This decomposition expresses any profile as a union of
profiles of four types. To start, a "Neutral" configuration
has the same number of voters assigned to each of the six
rankings. Second, a "Condorcet" profile configuration af
fects only pairwise rankings; it is given by Condorcet
triplets such as A >B>C, B>C>A, C>A>B, or its reversal.
While such a triplet (a Z3 orbit of a ranking) has no effect
on positional rankings (because each candidate is ranked
first, second, and third once), it causes pairwise cycles; in
part these cycles arise because z3 does not admit symme
tries (that is, a subgroup) of order two. More is said about
this below. The "Reversal" configuration is created by us
ing pairs consisting of a ranking and its reversal; these pro
file components (the Z2 orbit of a ranking) do not affect
pairwise rankings but, as demonstrated above, they affect
positional outcomes. The remaining "Basic" configuration
requires all positional and pairwise rankings and tallies to
agree.
Using a vector approach, all three-candidate profiles can
be decomposed into components of these four types where
the coefficients may be fractions. As demonstrated by Eq.
5, only the Reversal and Basic directions affect positional
outcomes; only the Basic directions affect the Borda Count.
Consequently all differences between the Borda and any
other positional ranking are strictly due to Reversal terms
[Saari 1999].
The new twist added here is a description how these
components geometrically affect the positioning of the pro
cedure line. (Details can be verified by using [Saari 1999].)
The procedure line (in a three-dimensional space) gener
ated by the Basic portion of a profile is parallel to the x = y = z diagonal. So, for all positional methods the difference
of each candidate's Basic tally from the average number of
assigned points is the same. All differences in election out
comes, then, are introduced by Reversal terms; they pivot
the procedure line about the Borda outcome. If a strong
Reversal component creates a B>C>A plurality outcome,
for instance, then the pivoting of the procedure line cre
ates a tendency for the other endpoint-the antiplurality
outcome-to define the opposite A >C>B outcome.
To illustrate by creating examples, start with the Fig. 5a
O + Os C
A &....-----""---.......;::a B 6 + 3s 6 3 3 + 6s a. Starting profile
Figure 5. Creating examples.
26 THE MATHEMATICAL INTELLIGENCER
profile. The average number of assigned points per candi
date for (1, s, 0) is [(6 + 3s) + (3 + 6s)]/3 = 3 + 3s, so each
candidate's tally minus this average is (3, 3s, -3 - 3s). Be
cause these differences change with the s value, it means
that the profile has a Reversal component. (This is not ob
vious; it illustrates a case with a fractional coefficient.) By
comparing the differences for the antiplurality ( s = 1) with
the Borda (s = t) pivot point, the (3, 3, - 6) - (3, 1.5,
-4.5) = (1, 1 .5, - 1 .5) components show that the effect of
this hidden Reversal term is to create a bias for the an
tiplurality outcome favoring B at the expense of A and C.
Indeed, this distortion causes the antiplurality ranking of
A - B>C to conflict with the A >B>C conclusion for all
other positional procedures.
To make these statements more concrete, we modify the
Fig. 5a profile to create a p with a plurality top-winner re
versal bias; p's plurality ranking will be B> A>C and 0Jt(p )'s
plurality ranking will be B>C>A. To achieve this goal, we
need to add reversal terms. That is, select y and z values
from Fig. 5b so that adjoining it to the profile of Fig.
5a gives the plurality B>A>C and antiplurality A >C>B
rankings.
Adding each candidate's tallies from the Figs. 5a, b tri
angles, the desired B> A>C plurality outcome and A>C> B
antiplurality outcomes occur, respectively, if and only if
z + y + 3 > y + 6 > z,
or
6 + y > z > 3, (10)
and
9 + y + 2z > 2y > 9 + y + z,
or
9 + z > y > 9. ( 1 1)
The simplest choice of y = 10, z = 4 defines the p in Fig.
6a; 0Jt(p) is in Fig. 6b. While the Fig. 6a plurality ranking
changes from that of Fig. 5a, the Borda ranking remains the
same, reflecting Borda's immunity to reversal terms.
This construction can also be used to demonstrate how
the "size" of the Reversal term affects the 0Jt(p) outcomes.
While all Eq. 10 choices define a p with a plurality B> A>C
ranking, we know from the properties of the procedure line
that different Reversal components generate different 0Jt(p)
C z + 2ys
A &....-----""---.......;::a B y + 2zs y + z b. Reversal terms
4 + 20s C
A c....--......... --� B 16 + 21s 20 17 17 + 6s
a. Profile p Figure 6. Final example.
plurality rankings. To illustrate with the smallest value of
z = 4, observe in Eq. 12 how the \Jt(p) plurality ranking
changes as the y value increases; the \Jt(p) outcome moves
through five different rankings until the y � 14 values re
quire the plurality ranking to be the same for p and \Jt(p ).
As p's Borda ranking is immune to Reversal terms, it re
mains A>B>C; the \Jt(p) Borda ranking reverses p's Borda
outcome to become C> B> A.
y value,
z = 4
0 :S y :S 8
y = 9
1 0 :S y :S 1 2
y = 1 3
y :0: 1 4
!'lt(p) plurality
ranking
C>B>A
C - B>A
B>C>A
B>C - A
B>A>C
Reversal behavior
Top-winner
Top-winner
Reversal
(1 2)
These Reversal terms provide a tool which now makes
it trivial to create paradoxical examples. Of interest for our
earlier claim, because these terms are fully responsible for
all possible differences among three-candidate positional
method election outcomes, they explain this two-century
mathematical mystery about election procedures.
Using Parts Now consider those election methods which are based on
pairwise majority votes. As illustrated by the pairwise tal
lies in Figs. 1 and 6, \Jt(p) always reverses p's pairwise rank
ings and tallies; this suggests that maybe pairwise proce
dures never suffer reversal problems. After all, should the
pairwise rankings form a transitive ranking, then any rea
sonable procedure will select the top-ranked candidate. But
\Jt(p) reverses the rankings, so p's bottom-ranked candi
date becomes \Jt(p )'s top-ranked candidate; reversal biases
cannot occur.
The reason difficulties arise is that there are 2(�) ways
to rank the (�) pairs of the n candidates. Consequently, re
versal problems may be created by the way these proce
dures handle the 2(�) - n! non-transitive pairwise out
comes. For n = 3, there are only 23 - 6 = 2 possibilities,
but for n = 4 there are 26 - 24 = 40 such situations. Since
the non-transitive settings significantly outnumber the tran
sitive ones once n � 4, plenty of opportunities exist for un
expected behavior. What helps in our analysis is that we
now know [Saari 1999, 2000] that all non-transitive settings
C13 + 20s
A B 10 + lls 17 20 14 + 6s
b. Profile 'R(p)
for n candidates-hence all possible reversal problems
are caused by the "Zn cyclic symmetry orbits" of the n al
ternatives; these are natural generalizations of the Con
dorcet triplets. To construct such a profile component, start
with any n-candidate ranking such as A> B>C · · · > Z. For
the second ranking, move the top-ranked candidate to the
bottom to have B>C> · · · >Z>A. Continue until there are
n rankings. With n candidates, this Condorcet n-tuple cre
ates the cyclic outcomes A>B, B>C, . . . Z>A, each with
n - 1 : 1 tallies.
To see how reversal problems can occur, consider an
agenda. This is a form of tournament where candidates are
compared with a pairwise vote in a specified manner; af
ter each comparison the winner is advanced to be com
pared with the next specified candidate. One example,
then, is where the winner of an A and B pairwise vote is
compared with C. With a Condorcet triplet [A>B>C,
B>C>A, C>A>B), A beats B to advance to a vote with C;
C wins by a 2: 1 vote. \Jt(p) is the reversed Condorcet triplet
[C>B>A, B>A>C, A>C>B) with the opposite pairwise cy
cle ofA>C, C>B, and B>A with the 2: 1 tallies. With \Jt(p),
B beats A in the first comparison, but loses to C in the sec
ond election. Since C is the winner with both p and \Jt(p ),
an agenda admits the top-winner reversal bias.
This cyclic effect for the agenda example suggests that
aU positional-method runoff procedures-where the top
two candidates in a positional election are advanced to a
majority-vote runoff-allow a top-winner reversal bias. To
explain, if profile p satisfies Eq. 4 with an A>B>C out
come, then A and B are advanced to the runoff while the
\Jt(p) outcome of C> B> A advances B and C to the runoff.
To have a top-winner reversal bias, the profile needs to
have a cyclic effect where B beats A with p and B beats C
with \Jt(p ). Again, what simplifies the construction is that
such an example requires C to beat B with p.
To construct illustrating examples, notice that a Con
dorcet n-tuple does not affect positional-method election
rankings (as each candidate is ranked in each position
once). So, as illustrated in Fig. 7, create a p = PI + pz
where (according to Theorem 7) PI has the same plurality
and antiplurality ranking and p2 defines an appropriate cy
cle. Profile PI is given in Fig. 7a for x = 0; the positional
method outcomes are A> B>C with 4 + s : 2 + 2s : 3s tallies.
The p2 portion is the Condorcet triplet given by the x's
in Fig. 7a. As these terms add the same x + xs value to each
VOLUME 25, NUMBER 4, 2003 27
3s C
A �----'---� B 4 + s 2 + 2s
a. Profile Pl is where x = 0 Figure 7. Adding cycles.
candidate's positional method tally, they do not affect the positional method rankings. But as indicated in the figure, the x terms can change in the pairwise rankings. In particular, for B to beat A, and C to beat B with P 1 + P2, select x where 2 + 2x > 4 + x; i.e., x :=:: 3. The x = 3 choice in Fig. 7b defines a p with the top-winner bias for any positional method runoff.
A word of caution; not all elimination procedures suffer these reversal problems. An example is Nanson's method [Nanson 1882] which, at each stage, drops all candidates who fail to receive more than the average Borda score; the remaining candidates are reranked with the Borda method and the process continues until a single candidate remains. Since the Nanson winner survives the first cut with p, when the average Borda score is subtracted from her Borda tally, it must be positive. But, as demonstrated earlier, with ffi(p) this difference is negative. Because the Nanson winner with p is dropped at the first stage with ffi(p ), there is no reversal problem. So while a Borda runoff can suffer a topreversal bias, Nanson's approach never does.
To obtain a general result for the n :=:: 3 alternatives (ab a2, . . . , anJ, represent the tallies for the C:D pairs with a point in Rm. To do so assign an axis for each (aj, ak) pair. The value used for a (aj, ak} tally is the difference between aJs and ak's votes divided by the number of voters. Thus, the outcomes are on the [1 , - 1] interval of this axis, where 1 means that aj wins unanimously, 0 means a tie, and - 1 means that ak wins unanimously. All pairwise outcomes, then, are in a cube of Rm centered at the origin 0 called the representation cube. 3 The coordinate planes define 2 (2) orthants in the presentation cube; each orthant contains all pairwise tallies supporting a specific choice of pairwise rankings.
To illustrate with n = 3 and Fig. 8, let the x, y, z coordinates represent, respectively, the rankings A>B, B>C, C>A. While the cube [ - 1, 1 ]3 has eight vertices, only six of them can be identified with the six transitive rankings. It is not difficult to show that the labeled vertices in Fig. 8 correspond to transitive rankings; for instance V1 corresponds to A>B, B>C, A>C or A>B>C. The two remaining vertices, (1, 1, 1) and ( - 1, - 1, - 1 ), correspond to cyclic rankings.
C 3 + 6s
A �--�--� B 7 + 4s 7 8 5 + 5s b. Profile p = Pl + P2
The representation cube is the convex hull of the six labeled vertices; it turns out [Saari 1995] that the rational points in this hull represent all possible pairwise election outcomes. Notice that this hull meets the positive and negative orthants; the points in these two orthants are the pairwise cyclic outcomes that can cause problems. Indeed, the 15-voter Fig. 7b choice of p defines the point C � 8 , 7 � 8 , 4 �511 ) ; as all components are negative the election rankings form the cycle B>A, C>B, A>C. The ffi(p) point reverses each sign; it is (8 - 7 , 8 - 7 ,
1 1 - 4). For any n, the 15 15 15 . p and ffi(p) tallies differ only oy the s1gn of each com-ponent, so they are endpoints of a line segment with 0 as the midpoint. This statement introduces a geometric test for a top-winner reversal bias. (Procedures mentioned in this theorem which have not been introduced are described below.)
THEOREM 8. Suppose a specified election method using pairwise votes is given. For each candidate, find all points in the representation cube which elect that candidate. If a line segment of positive length centered at 0 with the p and ffi(p) pairwise outcome as endpoints has both endpoints in the same candidate's region, then the procedure has a top-winner Reversal bias. Thus, for instance, agendas and Dodgson's method (for n :=:: 4) have the top-winner Reversal bias. If all such line segments have the endpoints in regions for different candidates, then the method never has a top-winner Reversal bias. As examples, Copeland's, Borda's and Kemeny's methods never experience a top-winner Reversal bias.
To Illustrate Theorem 8, consider the agenda where the winner of an A and B pairwise vote is compared with C. Each orthant of the representation cube in RC�) = R3 determines a specific agenda winner. But 3! = 6 of these eight orthants represent transitive rankings where the topranked candidate is the agenda winner. Since reversing a transitive ranking makes the previously bottom-ranked candidate top-ranked, none of these six regions passes the line segment test. It remains to examine the two remaining orthants where the pairwise rankings define cycles. Both orthants elect C and they are diametrically opposite one
"The set of all admissible pairwise tallies is a subset which can be determined; this is done in [Saari 1 995] for n = 3, and a similar approach holds for all n.
28 THE MATHEMATICAL INTELLIGENCER
another, so a top-winner reversal bias must occur with any profile which allows a cycle.
An intriguing election approach was introduced by the mathematician Charles Dodgson, who is better known as Lewis Carroll of Alice in Wonderland fame. Dodgson's method selects the Condorcet winner-the candidate who beats all others in pairwise comparisons. If a Condorcet winner does not exist, replace the actual rankings with the "closest" set of rankings which have a Condorcet winner. For Dodgson, "closest" is the minimum number of adjacent changes in individual rankings which create a new profile with a Condorcet winner. Ratliff [2001, 2002, 2003) has discovered a surprising array of unexpected behaviors allowed by this procedure.
Dodgson's method selects the top-ranked candidate from a transitive ranking, so ignore the n! orthants with transitive outcomes. Similarly, suppose the non-transitive rankings for p define a Condorcet winner and a Condorcet loser (a candidate who loses to each of the other candidates). Since the reversal converts p's Condorcet loser into ffi(p )'s Condorcet winner, no reversal bias occurs. More generally, any p which defines a Condorcet loser which differs from the Dodgson winner cannot have the top-winner reversal bias.
Next consider profiles with Condorcet winner A but no Condorcet loser. Because A is the ffi(p) Condorcet loser, it is reasonable to suspect that nothing can go wrong. What makes the actual story more complicated is that ffi(p) has no Condorcet winner, so we need to invoke Dodgson's metric. The problem arises if A barely is a Condorcet winner with p-so she barely is a Condorcet loser with ffi(p )-and the tallies for all other pairwise rankings involve substantial differences. Such a situation requires cyclic symmetries [Saari 2000a] . Combining these two notions, examples are immediate; e.g., the next profile repeatedly uses the Condorcet {B, C, D) triplet to create sizeable differences in their pairwise tallies. To ensure that A barely is the Condorcet winner, she is top-ranked in slightly over half of the preferences, and she is bottom-ranked in the others.
Number Ranking Number Ranking
1 0 A>B>C>D 9 B>C>D>A
1 0 A>C>D>B 9 C>D>B>A (1 3)
1 0 A>D>B>C 9 D>B>C>A
A is the Condorcet winner by beating the other candidates with a 30:27 tally. Here, ffi(p) is
Number
1 0
1 0
1 0
Ranking
D>C>B>A
B>D>C>A
C>B>D>A
Number
9
9
9
Ranking
A>D>C>B
A>B>D>C
A>C>B>D
(1 4)
where A is the Condorcet loser since she loses to each opponent with a 30:27 tally. The remaining rankings define the B>D, D>C, C>B cycle with 38: 19 tallies. Without a ffi(p) Condorcet winner, we need to invoke Dodgson's metric; the Dodgson winner is A. Indeed, interchange the last pair for two individuals in each ranking on the left of
table 14, the revised rankings allow A to beat each of the other candidates by 29:28 to become the Condorcet winner. Thus, Dodgson's method admits a top-winner reversal bias.
Other methods, such as the ones developed by the mathematicians Borda [ 1781] , Copeland [ 1951] , and Kemeny [1957], and Dodgson's method for n = 3 do not have a reversal bias, because these methods replace the actual pairwise rankings with the "nearest" transitive ranking. For instance, Saari and Merlin [2000) showed that the Kemeny method can be viewed as fmding the nearest transitive ranking with an h-metric-the sum of the difference between coordinates. With n = 3 and Dodgson's metric, the nearest region with a Condorcet winner is either transitive, or on the boundary of a transitive orthant. The other two methods use the transitivity plane introduced in [Saari 1999, 2000b] ; it is a lower-dimensional plane symmetrically positioned in the representation cube passing through the origin and transitive orthants. Borda's method can be viewed as replacing a point in the representation cube with the nearest (l2 or Euclidean distance) point on the transitivity plane. Copeland's method converts each pairwise tally into a 1 or - 1, indicating who won or lost, and sums the tallies; i.e., it replaces a point in an orthant of the representation cube with the outside vertex of that orthant. Then, Copeland's method replaces the vertex with the l2 nearest point on the transitivity plane.
It is easy to show that the distance from a point in the representation cube defined by p to one of these regions is the same as the distance from the point defined by ffi(p) to the reversal of these regions. But these reversed regions define reversed transitive rankings, so the ranking of ffi(p) reverses that given by p.
Final Comments On first glance the study of elections seems to be trivial because, seemingly, only counting is involved. From a mathematical perspective, however, everything becomes delightfully complex. As we have recently learned, an important source of the mathematical complexity is that profiles can be full of hidden symmetries from higher-dimensional spaces; symmetries which cause all sorts of unanticipated problems and difficulties for election procedures. The re-
Figure 8. Representation cube.
VOLUME 25, NUMBER 4, 2003 29
versal problems identify only a small portion of the tip of a very big iceberg of complexity. Of interest, this structure extends to problems from statistics, probability, and other aggregation methods; different symmetry groups are needed, but the ideas are similar.
As an illustration of related issues, consider strategic voting-something all of us have done. For instance, if you have A>B>C preferences in a close election between A and B, you might be tempted to mark your ballot as A>C> B to increase A 's point spread over B. More generally, Gibbard [ 1973] and Satterthwaite [ 1975] proved the amazing result that all reasonable election procedures for three or more candidates admit situations where some voter, by voting strategically, gets a better election outcome. But if all methods admit strategic options, the next natural question is to determine which (1, s, 0) method is least susceptible to a small number of strategic voters being successful. The answer [Saari 1995] is Borda's method; the level of susceptibility decreases as s - i· (According to this theorem, the plurality vote is highly susceptible to strategic behavior. We know this; just recall those "Don't waste your vote" calls for strategic action voiced during close elections involving more than two candidates.) But as s - i a procedure becomes less susceptible to the Reversal components. Is there a connection? Probably, but it has not been established.
, A U T H O R S
DONALD G. SAARI
USA
30 THE MATHEMATICAL INTELLIGENCER
REFERENCES
[ 1 ) Borda, J. C. 1 781 , Memoire sur les elections au scrutin , Histoire
de I'Academie Royale des Sciences, Paris.
[2) Condorcet, M. 1 785. Essai sur !'application de !'analyse a Ia prob
abilite des decisions rendues a Ia pluralite des voix, Paris.
[3) Copeland, A. H. 1 951 , A reasonable social welfare function.
Mimeo, University of Michigan.
[4) Gibbard, A, 1 973, Manipulation of voting schemes: a general re
sult. Econometrica 41 , 587-601 .
[5] Kemeny J . , 1 959, Mathematics without numbers. Daedalus 88, 571-591 .
[6) Merlin, V. , M. Tataru , and F. Valognes 2000, On the probability
that all decision rules select the same winner, Journal of Mathe
matical Economics 33, 1 83-208.
[7] Nanson, E. J . , 1 882, Methods of elections, Trans. Proc. R. Soc.
Victoria 18, 1 97-240.
[8) Nurmi, H . , 1 999, Voting Paradoxes and How to Deal with Them,
Springer-Verlag, NY.
[9] Nurmi, H . , 2002, Voting Procedures under Uncertainty Springer
Verlag, Heidelberg.
[1 OJ Ratliff, T. 2001 , A comparison of Dodgson's method and Kemeny's
rule, Social Choice & Welfare 1 8, 79-89.
[1 1 ) Ratliff, T. 2002, A comparison of Dodgson's Method and the Borda
Count. Economic Theory 20, 357-372.
[1 2) Ratliff, T. 2003, Some starting paradoxes when electing commit
tees, to appear Social Choice & Welfare
.,..,_.. BARNEY n..�I11V'..,I of t.A.�IIv>rn. .. ll...,.,_
USA
[ 13] Riker, W. H . , 1 982, Liberalism Against Populism, W. H. Freeman,
San Francisco.
[1 4] Saari, D. G. , 1 992, Millions of election rankings from a single pro
file, Social Choice & Welfare (1 992) 9, 277-306.
[1 5] Saari, D. G. 1 994, Geometry of Voting, Springer-Verlag, New
York.
[1 6] Saari, D. G. 1 995, Basic Geometry of Voting, Springer-Verlag, New
York.
[1 7] Saari, D. G. 1 999, Explaining all three-alternative voting outcomes,
Journal of Economic Theory 87, 31 3-355.
[1 8] Saari, D. G. 2000a, Mathematical structure of voting paradoxes 1 ;
pairwise vote, Economic Theory 15, 1 -53.
[1 9] Saari , D. G. 2000b, Mathematical structure of voting paradoxes 2:
positional voting. Economic Theory 15, 55-1 01 .
[20] Saari, D. G. 2001 , Chaotic Elections! A Mathematician Looks at
Voting, American Mathematical Society, Providence, Rl .
cKichan JOtrWAII. ltiC
The Gold Standard for Mathematical Publishing Sciemific WorlcP/a c and ientific H'cmJ make writing,
publi hing, and d ing malhemati c ier than you ever imagined possible. You compose and edit your document directly on the
creen without being forced to think in a programming language.
A click of a bunon allows you to typeset your documents in It\'fEX
And, wilh Scientific WorlcP/ace, you can compute and plot
lutions wilh lhe integrated computer algebra sy tern.
I"ISil our website for Jn!e trial •oenwns af all ow oft>oYJre.
Tool for ScienJific CreatiVJI)• J/1/Ce /98/ Free: 877-724-9673 • Email: [email protected]
[21 ] Saari, D. G . , and V. Merlin 2000, A geometric examination of Ke
meny's rule, Social Choice & Welfare 17, 403-438.
[22] Saari, D. G . , and J. Van Newenhizen, 1 988, Is Approval Voting an
"unmitigated evil?" Public Choice 59, 1 33-1 4 7 .
[23] Saari, D. G. and M. Tataru 1 999, The likelihood of dubious elec
tion outcomes, Economic Theory 13, 345-363.
[24] Satterthwaite, M. , 1 975, Strategyproofness and Arrow's condi
tions, Jour. Econ. Theory 10, 1 87-21 7.
[25] Tabarrok, A. , 2001 , Fundamentals of voting theory illustrated with
the 1 992 election, or could Perot have won in 1 992? Public Choice
106, 275-297.
[26] Tabarrok, A. and L. Spector 1 999, Would the Borda Count have
avoided the Civil War? Journal of Theoretical Politics 1 1 , 261 -288.
[27] Tataru, M . , and V. Merlin 1 997, On the relationships of the Con
dorcet winner and positional voting rules, Mathematical Social Sci
ences 34, 81 -90.
VOLUME 25, NUMBER 4. 2003 31
M athematic a l l y Bent
The proof is in the pudding.
Opening a copy of The Mathematical
Intelligencer you may ask yourself
uneasily, "What is this anyway-a
mathematical journal, or what?" Or
you may ask, "Where am I?" Or even
"Who am I?" This sense of disorienta
tion is at its most acute when you
open to Colin Adams's column.
Relax. Breathe regularly. It's
mathematical, it's a humor column,
and it may even be harmless.
Column editor's address: Colin Adams,
Department of Mathematics, Bronfman
Science Center, Williams College,
Williamstown, MA 01 267 USA
e-mail: [email protected]
C o l i n Adam s , Ed itor
Don't Touch the Button Colin C. Adams
'' I just want to say how pleased we
are to have the three of you join
ing the department. Let me quickly go
over a few things at this first meeting.
You have all received your teaching as
signments. I apologize if the times that
your classes meet are not your first
choice, but the administration requires
us to spread our classes over the avail
able time slots.
Phones are for business purposes
only, not for personal calls. And all Fed
Ex packages must be paid for yourself.
The department cannot afford to pay
for overnight delivery. You can help
yourself to supplies such as pads, pens,
stapler, scissors, etc. from the supply
closet, but they are for office use only.
And whatever you do, do not ever
touch my belly button. The depart
mental secretary, Karen, can show you
how the copier works. If you give more
than 10 pages of copied material to
each student, you must charge the
costs to their college accounts. Karen
can show you how to do that. Well,
we're thrilled to have you on board.
And remember, there is a departmen
tal party at my house on Friday night
at 8:00. That will be a great chance to
meet everyone. Well, then, good luck.
Any problems, just let me know."
Karl Fustrum, chair of the Mathe
matics Department, rose, and the three
junior faculty members followed suit.
He ushered them out the door of his of
fice, and then shut it behind them. Lisa
Karman, the logician, turned to the
other two and said, "Was I dreaming,
or did he say something about his belly
button?"
Arthur Delafield, young ergodic
theorist, said, "Yes, I thought he said
32 THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
something about not touching it. But it
went by really fast."
Misha Dimianowski, foliations ex
pert said, "Belly button. Belly button.
What is this belly button?"
When Lisa stuck her key into the
lock on the door of her office, Jamal
Kierman stuck his head out of the
neighboring office. "Hey, did you get
the belly button speech?" he asked.
"Yeah, what's that all about?"
"Who knows for sure. No one has
ever touched it. But he warns every
body once. And then that's it. Other
wise, he's normal enough. Liked and
hated, much as any chairman is. You
going to the department party tomor
row night?"
"Do I have any choice?"
"Not if you want tenure," Jamal said,
smiling. He ducked back into his office.
On Friday evening, Lisa drove over
to the chairman's suburban address.
She parked on the street in front of the
Fustrum's house in a line of cars. At the
door, she was met by the chair's wife,
Dahlia, call me Dahl, Fustrum. Lisa was
ushered into the living room, handed a
glass of wine, and then left to her own
devices. She knew a few faces from the
interview process, but most were new.
Over by the kitchen, she noticed Fus
trum talking heatedly to the vice-chair
Bob Lindstrom. She found herself
staring at Fustrum's stomach, which
bulged out above his pants, pulling his
black turtleneck taut. She almost imag
ined she could see a protrusion right
where his belly button should be, but she
wasn't sure. She was working her way
forward, from nut bowl to cheese plate,
when Jamal stepped in front of her.
"Don't do it."
"Do what?" she asked.
"You were headed straight for his
belly button. You haven't taken your
eyes off it."
"Oh, come on," she said. "I was, urn,
looking at that bowl on the coffee
table. Very pretty."
"Yeah, right. You're hooked. I can
see it already. I've seen it before."
"Seen what before?"
"The belly button obsession. Arman
Lobindi, used to be in your office. Did
PDE's, or at least used to until he got
hooked."
"What do you mean, he got
hooked?"
"Well, after he got here and heard
the belly button speech, he became ob
sessed. Couldn't concentrate on his
work anymore, just thought about the
button more and more. That's what he
called it, the button.
"I remember stopping by his office
when he wasn't there, and seeing his
blackboard covered with pictures.
Took a while for me to figure out what
they were."
"Yes, what were they?"
"Hypothetical belly buttons, of all
types. Innies, outies, normal, de
formed. He had them listed in cate
gories with probabilities associated to
each one. It was sad."
"So what happened to him?" asked
Lisa.
"He didn't get renewed. Ostensibly
because his research came to a stand
still. But I think it had to do with an in
cident in the men's room."
"What incident?"
"He used to follow the chair in there,
hoping to get a glimpse. I can only
guess he peeked over the stall and Fus
trum decided that was enough. But no
body knows for sure."
Misha, who had been talking to a
topologist across the room, waved and
joined them.
"I know now what is this belly but
ton. My wife, she explain this to me."
Misha pointed to a bored-looking
woman seated in the comer reading a
magazine and ignoring the people
around her.
Fustrum turned away from the vice
chair and strode toward them, a wel
coming smile on his face. Lisa forced
herself to look him in the eye, and not
to look down toward his advancing
belly.
"Hello, Lisa, Misha. Found your way
here all right? I hope you are getting to
meet some of the other faculty. Jamal,
are you introducing them around?"
"Oh, don't worry. I'll keep an eye on
them."
Misha smiled at the chair. "Yes, I un-
derstand now what you said to us to
day. I talk to my wife. I know now what
you mean." And Misha winked at him.
Fustrum darkened.
"Help yourself to more drinks," he
said, and he was gone.
Misha looked confused. "Subject is
verboten, Misha," said Jamal. "Don't
bring it up."
"Oh", said Misha, looking worried.
"We do not speak of this anymore?"
"Right," said Jamal. ''I'm going to get
some hors d'oeuvres."
The first semester went well. Lisa
was teaching one large lecture of mul
tivariable calculus and a small gradu
ate class on set-theory. She only saw
the chair at department meetings and
at colloquia. At those events, she would
always sit in a position that allowed her
furtive glances of Fustrum's midriff. At
one colloquium he was wearing a
button-down shirt. She sat in the same
row as he did, five seats to his right. At
one point, he shifted in his seat, and
the front of his shirt separated just a
bit between the two bottom buttons.
Lisa found herself staring into a dark
ened opening that contained the infa
mous belly button. She leaned forward
with her mouth open in anticipation.
Then Fustrum shifted again. As she
looked up, she found him staring back
at her, with a grim face.
In the spring, she taught two sec
tions of linear algebra. With only one
course prep, she found more time to
get her own research done. One day,
when trying to find a preprint in her fil
ing cabinet, she noticed a manila folder
wedged in the back of the drawer. It
was labeled "Button" in large block let
ters. She pulled it out and opened it.
Inside were a variety of photos. Most
seemed to have been taken through a
small hole. Several were pictures of
Fustrum at various events. Others were
not obviously of Fustrum, but there
could be no question. Two or three
were shots of a slight crack between
the buttons in the front of a button
down shirt. At least one photo ap
peared to have been taken through the
crack of the door of a bathroom stall.
In all the shots, the belly button was
impossible to make out in any detail
whatsoever. But Lisa found herself fas
cinated nonetheless.
Spreading them on her desk, she
stared at them, trying to see something
in the belly that would make it extra
ordinary. As she rearranged the pic
tures, she suddenly had the feeling
there was someone behind her. Turn
ing quickly, she found the vice-chair of
the department in her doorway. She
stood, hoping to block the photos from
his view.
"Hi, Bob," she said, a little too
brightly.
"The chair wants to see you," said
Lindstrom evenly, with the slightest
smile at the edges of his mouth.
"Oh, okay", said Lisa, "I'll be there
in just a minute."
As soon as he was gone, she
scooped the pictures into the file and
slipped it back into the rear of the file
drawer where she had found it.
Then she shut her office door, and
walked down to the chair's office. The
secretary told her to have a seat on a
bench in the hall. She had been there
about fifteen minutes when Lindstrom
came by.
"Oh, I'm sorry", he said, still smiling
just slightly, "but Karl was going to
meet with you at your office." Lisa
jumped up. "Oh", she said nervously, as
she headed as quickly as she could
back to her office.
She found the door open. Fustrum
was leaning back in her office chair
with his feet up on the desk He had his
hands behind his head and seemed to
be sleeping. The file drawer that con
tained the pictures was open.
She tried to slip carefully behind
him to see if the file was still in the
drawer. But as she did, he was startled
awake. He lost his balance and the
chair he sat in slid out from beneath
him. Lisa watched in horror as his head
caught the comer of the drawer. The
office chair careened across the room,
knocking her to the floor, and Fustrum
fell solidly next to her.
She lifted her head slowly off the
floor. The chair's turtleneck had be
come untucked in the fall, and his
belly, the one that always kept the shirt
tight as a drum, was exposed. There be
fore her, just at the crest of his stom
ach, was the famous belly button. She
gulped as she stared at it. It appeared
normal enough, standard size, just a bit
VOLUME 25, NUMBER 4, 2003 33
protruding out through the hole. She
shook her head once to clear it. The
belly button stared at her balefully.
"So you're the belly button," she
said. Fustrum remained motionless on
the floor. "I don't think I have ever
talked to a belly button before." It
stared at her unblinking.
She felt its pull. Her right hand rose
up from the floor. At that instant, Ja
mal ran in the office. "Are you all
right?" he asked.
She pointed at it. "Look," she said.
He froze.
"Get away from it," he said.
"It's just a belly button," she replied.
But she was transfixed, unable to take
her eyes off it.
"It's not just a belly button. It's the belly button. Get away from it."
"I can't," she said. Her fmger started
to move forward, trembling at the tip.
"Stop it, what are you doing?" said
Jamal in horror. His feet were fastened
to the floor.
"I want to touch it. I need to touch
it. He's out cold. He'll never know I
touched it."
She expected there to be some kind
of electrical jolt when she actually
came into contact, but there was none.
It felt warm like any other part of the
body. As she felt her fmger make con
tact, she looked up and saw that Fus
trum had lifted his head off the floor
and was looking at her with a crazed
smile.
"Was it worth it?" Jamal asked. Lisa
was putting files in a box.
"Yes, it was," she replied. "I couldn't
have gone on living knowing that I had
been that close to it but hadn't touched
it. It was something I just had to do."
"And you don't care that you were
denied renewal? You don't care that
you're out of a job?"
"Well, I care. It's inconvenient. But
I'll get another job. Wrankle has
promised to write me a letter of rec
ommendation that explains the whole
belly button thing and the reason I
didn't get renewed. He knows some
people at Purdue. I should be fine.
You're the one that has the problem."
"What do you mean?''
"Well, you have tenure now. But you
have to live with the belly button. You
have to confront it every day. And you
still have to worry about promotion. You
touch it once over the next 7 years, and
you're an associate professor for life."
Jamal looked sick. Lisa smiled as
she lifted the box off the desk. "Good
luck," she said, as she walked out the
door.
NEW f6 FO RT H CO M I N G from Birkhiiuser Stochastic Calculus Applications in Science and Engineering M. GRIGORIU, Cornell University, Ithaca, NY This work analyzes and presents solutions for a wide range of stochastic problems in applied mathematics, physics, engineering, finance, and economics. A userfriendly, systematic exposition covers the essentials of probability theory, random processes, stochastic integration, and Monte Carlo simulation. The Monte Carlo method is used to illustrate theoretical concepts and numerically solve problems; this text approaches stochastic problems both analytically and numerically. It may be used in the classroom or for self-study by researchers and graduate students.
TABLE OF CONTENTS: Introduction • Probability Theory • Stochastic Processes • Ito's Formula and Stochastic Differential Equations • Monte Carlo Simulation • Deterministic Differential, Algebraic, and Integral Equations • Deterministic Systems with Stochastic Input • Stochastic Operator and Deterministic Input • Bibliography • Index
2002/792 PP., 1 77 1LLUS./HARDCOVER/S79.95 ISBN O·B1 76-4242·0
Resampling Methods A Practical Guide to Data Analysis Second Edition P. GOOD, Information Research, Huntington Beach, CA
This work is a practical, table-free introduction to data analysis using the bootstrap, cross-validation, and permutation tests. The book's many exercises, practical data sets, and use of free shareware make it an essential resource for students and teachers, as well as industrial statisticians, consultants, and research professionals. Topics and features: • Uses resampling approach to introduction statistics • Includes systematic guide to help one select correct
procedure for a particular application
• Detailed coverage of all three statistical methodologies--classification, estimation, and hypothesis testing
• Numerous practical examples using popular computer programs such as SAS, Stata, and StatXact
• Downloadable freeware from author's web site: http://users.oco.net/drphilgood/resamp.htm
2001/256 PP., 33 1LLUS./HARDCOVER/S69.95 ISBN 0-81 76·4243-9
CALL: 1 -800-777-4643 • FAX: {201 ) 348-4505 • E-MAIL: [email protected] Please mention promotion *Y7261 when ordering. Prices are valid in the Americas only and are subject to change without notice.
For price and ordering information outside the Americas, please contact Birkhauser Verlag AG, P.O. Box 133, CH-4010 Basel, Switzerland. Phone +41-61-205 0707; Fax +41-61 -205 0792 or E-mail: [email protected]
6/03
34 THE MATHEMATICAL INTELLIGENCER
Analysis of Variance for Random Models Theory, Methods, Applications, and Data Analysis M. OJEDA, University of Veracruz, Xalapa, Mexico; and H. SAHAI, University of Puerto Rico, Mayaguez, PR
This two-volume work is a comprehensive presentation of methods and techniques for point estimation, interval estimation, and hypotheses tests for linear models involving random effects. Volume l examines models with balanced data (orthogonal models); Volume l l studies models with unbalanced data (non-orthogonal models). Accessible to readers with a modest mathe· matical and statistical background, the work will appeal to a broad audience of graduate students, researchers, and practitioners.
VOLUME 1: Balanced Data 2003/APPROX. 400 PP., 9 ILLUS., 50 TABLES HARDCOVER/$79.95 (TENT.l/ISBN 0-81 76-3230-1
VOLUME II: Unbalanced Data 2003/APPROX. 400 PP., 1 4 I LLUS., 60 TABLES HARDCOVER/$79.95 (TENT.)/ISBN 0-81 76-3229-8
Birkhiiuser @ Boston · Basel · Berlin
Promotion *Y7261
KAI HAUSER AND REINHARD LANG
On the Geometrica and Phys ical Mean i ng of Newton 's So ut ion to Kep er' s Prob em
• n the short treatise De Motu (1 684), which serves as a precursor to the Principia Mathe�matica (1 687), Newton essentially deals with the following two problems.
Problem A. Given that the orbit of a planet or a comet P is a conic section with one focus at the sun S, what is the centripetal force by which P is attracted to S?
Problem B. Conversely, given that the centripetal force by which the sun S attracts a mass point P is inversely proportional to the square of the distance SP, what is the orbit of P?
Solutions to these problems appeared for the first time in De Motu (6, p. 38-39 and p. 46-48]. There Problem B reads as follows.
Problem 4. Supposing that the force is inversely proportional to the square of the distance from its centre, and with the quantity of that force known, there is required the ellipse that a body will describe if it is launched from a given place with a given velocity along a given straight line.
In this article we concentrate on Newton's solution to Problem 4 resp. Problem B. Even though it seems as if Newton is concerned only with ellipses, his reasoning is nevertheless basically correct. The proof reappears, in slightly modified form, as Proposition XVII in Book I of the Principia [7, p. 65-66]. In what follows we refer to this version of the
solution of Problem B. As a service to the reader a reprint of Proposition XVII is appended at the end of this article. We are using the translation of Motte and Cajori [7] , which allows us to refer directly to the passages quoted in [4].
This article is not meant as a historical study of possibly earlier, alternative solutions by Newton of Problem B. (A reconstruction of such ideas can be found, for example, in [9].) Instead, we focus on the solution which appeared in print for the first time. It covers all essential points, thus there is no need for speculation. Our intention is to present Proposition XVII to a general audience as an introduction to a central piece of the Principia. Newton's proof is as austere as it is beautiful. Once the ideas are explained, it becomes transparent. Greek geometry is all that is required, without any use of vector analysis or calculus. And yet Newton's reasoning is the beginning of what is now called analytic mechanics; cf. , for example, [ 1 ] , where Lecture Four takes Newton's proof as its point of departure.
Before entering into the details we give an introductory survey of various aspects of Proposition XVII.
Logical Structure of the Proof of Proposition XVII Assume a centripetal force as in Problem B and the position and velocity vector of a mass point P are given. From
© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 4, 2003 35
these initial data Newton derives a criterion according to
which the orbit of P is an ellipse resp. hyperbola resp.
parabola. The two main ingredients of the proof are (a) his
solution of Problem A in Proposition XI resp. Proposition
XII resp. Proposition XIII [ 7, p. 56-61] , and (b) the appli
cation of the uniqueness theorem for the initial-value prob
lem. The latter states that, given the initial data, there can
not be more than one orbit of P that goes through the initial
position of P with the given velocity vector. (This fact must
have been known to Newton, as pointed out below.) Re
lying on (a) and (b), Newton proceeds as follows. He con
structs a conic section with focus S from the initial data
such that (i) its tangent in the point P is the given veloc
ity vector, and (ii) that the centripetal force associated to
this conic section according to (a) coincides with the given
centripetal force. Hence the conic section so constructed
is an orbit of P, and, by virtue of (b), it is indeed the orbit
sought.
Corollary after Proposition XIII Although not relevant for the purpose of this article, we
mention this Corollary [7, p. 61] because it has been inten
sively debated in the literature. Having proved Propositions
XI, XII, and XIII, Newton remarks,
Cor. I. From the three last Propositions it follows, that
if any body P goes from the place P with any velocity in
the direction of any right line PR, and at the same time
is urged by the action of a centripetal force that is in
versely proportional to the square of the distance of the
places from the centre, the body will move in one of the
conic sections, having its focus in the centre of force;
and conversely. For the focus, the point of contact, and
the position of the tangent, being given, a conic section
may be described, which at that point shall have a given
curvature. But the curvature is given from the centripetal
force and velocity of the body being given; and two or
bits, touching one the other, cannot be described by the
same centripetal force and the same velocity.
At this place in the Principia no further details are given.
However, Newton himself later makes the following com
ment on Corollary I [5, p. 136].
The Demonstration of the first Corollary of the XIth,
XIIth & XIIIth Propositions being very obvious, I omit
ted it in the first edition & contented myself with adding
the XVIIth Proposition whereby it is proved that a body
going from any place with any velocity will in all cases
describe a conic Section: which is that very Corollary.
Following Newton, Chandrasekhar [4, p. 102f] also con
ceives Corollary I in the light of Proposition XVII. On this
view mention of the notion of curvature in Corollary I is
not at all astonishing. Without a doubt, Newton (cf. [4, p.
1 10]) was aware of the simple relation between the curva
ture and the semilatus rectum of a conic section (for the
latter term see below); and in the proof of Proposition XVII
36 THE MATHEMATICAL INTELLIGENCER
the conic section is identified via its semilatus rectum from
the given central force and the initial data.
In [9] a different interpretation of Corollary I is proposed
without reference to Proposition XVII. Relying on the no
tion of curvature, an alternative construction of the conic
section is carried out.
Is Proposition XVII a Solution to Problem B? Two objections have repeatedly been brought forward. (i) It is alleged that Newton does not prove that the orbit is a conic
section, but rather assumes this from the outset (see, e.g., [5,
pp. 133, 134, 136]). However, this is incorrect. For the proof
strategy sketched above clearly shows that Newton con
structs a conic section which-by virtue of the uniqueness
theorem (b)-must be the sought-after orbit. This argument
is emblematic of a method often applied in physics when an
initial-value problem is solved by adjusting the parameters of
a putative solution to the initial data. (ii) The second objec
tion holds that Newton did not know the uniqueness theo
rem (b). This goes under the title: "Did Newton prove that
orbits are elliptic?" [3, p. 30ff]. As vigorously argued by
Arnold, this is a chimerical problem, because in the times of
Newton the function defining a differential equation was au
tomatically supposed to be analytic in its domain of defini
tion, and therefore uniqueness was no problem ( cf. [3, p. 31f]
and [4, p. 1 12]). Besides, what Newton proved in Propositions
XLI and XLII goes much further than uniqueness.
Relation of the Proof of Proposition XVII to the Notion of Energy Even though Newton's geometrical construction of the or
bit of P is fully correct, its physical meaning is not obvious
at first sight. As pointed out by De Gandt [6, p. 49],
This quotient depends on the initial conditions, as may
be seen from what happens to the point K in different
possible configurations. But it is not easy in this case
in contrast to an analytico-algebraic mode of presenta
tion employing differential equations and constants of
integration-to see how the initial position and velocity
enter into the expression (2SP + 2KP)IL.
It is convenient to abbreviate the inverse of the quotient
mentioned in this passage by PN· This coefficient, which is
decisive for the construction of the conic section to be
found, suggests a connection with kinetic energy. Arguing
in the spirit of Newton's proof, we will show that PN is in
fact the absolute value of the ratio of kinetic to potential
energy. In particular, this expresses Newton's criterion
whether PN < 1 resp. PN > 1 resp. PN = l-in rather simple
form, and makes it physically transparent. The coefficient
PN is of course not a function of the additive total energy,
and conservation of energy plays no role in Newton's proof
of Proposition XVII. This idea enters only later in the much
more general context of the initial-value problem for cen
tral forces of any kind, see [7, p. 128-134].
In the next section we recall the area law. After that, we
note elementary properties of conic sections and state
Newton's answer to Problem A without proof. (An exposition of the proof can be found, for example, in [ 10].) In the following section we concentrate on Newton's Proposition XVII and explain his proof step by step, emphasizing motivation. The connection with energy is described in the last section. Finally, Newton's original proof of Proposition XVII is reprinted in the appendix.
This article is mainly based on a study of Newton's work itself. In addition, the authors have relied on Chandrasekhar's book [4] for the mathematics and physics in the Principia, and on De Gandt's book [6] for the conceptual meaning of Newton's dynamics.
The Area Law Suppose we are given a center S and a centripetal (not necessarily gravitational) force acting on a mass point P. Then the area swept out by the line segment SP during the time interval t is proportional to t. Let us denote the proportionality factor by i c ( c ::::: 0). For infinitesimal intervals !:l.t, the area law can be formulated as follows: Let P1 denote the position of the mass point P at time t1 , and P2 its position at time t1 + !:l.t. The area swept out by the line segment SP during the time interval !:l.t is approximately equal to the area of the triangle SP1P2. Denote this area by ISP1P2 I - Then
(1) 1
ISP1P2 1 = 2 c . !:l.t.
Newton justifies this as follows: During the time interval !:l.t the trajectory of the point P is approximated by the line segment connecting P1 with P2. Fix R on the extension of the line segment P1P2 such that IP�I = IP1Pd . Without the central force, P2 would move to R in the interval !:l.t. But under the influence of the central force P2 moves to P3 where RP3 is parallel to SP2 (see Fig. 1). It follows
ISP�31 = ISP�I = ISP1P2i
and hence (1). In the sequel we assume that the constant c is strictly positive, i.e., the mass point P is not moving along the line connecting it to S.
Newton used ( 1) to express dynamical parameters in
+pi31i;IIM R
s
ijlijil;ifM
Q
M
s
purely geometric terms. Consider for example the velocity of the mass point P and its acceleration. Denote the absolute value of the velocity by v and the absolute value of the acceleration by a. When P moves to Q during the time interval !:l.t ( cf. Fig. 2), v = PQI !:l.t. To describe this geometrically, drop the perpendicular through S onto the line PQ. Let M be the pedal point of this perpendicular. With (1) we obtain
_!_ c . !:l.t = ISPQI = _!_ PQ . SM = _!_ ( v !:l.t) . SM 2 2 2
and from that
(2) c v =
SM.
In similar fashion Newton uses the area law for expressing the acceleration a geometrically, in this case via a comparison with Galileo's law of falling bodies and subsequent elimination of (!:l.t)2 by virtue of (1). This yields an expression of a in terms of the geometry of the orbit of P. Details can be found, for example, in [7, p. 48], [4, p. 77] , or [9, p. 15].
Theorem A Before formulating Newton's answer to Problem A, we note a few elementary properties of conic sections.
In the case of an ellipse (Fig. 3), we denote the foci by S and S' and the semimajor axis by a = OA. A point P lies on the ellipse if and only if SP + PS' = 2a. The eccentricity E (0 ::5 E < 1) is defmed by OS = E · a. Given the semimajor axis a and the eccentricity E, the semiminor axis b is obtained via the equation b2 = (1 - e)a2.
In the case of a hyperbola (Fig. 4), the foci are denoted by S and S' and its semiaxis by a = OA. A point P belongs to the branch of the hyperbola with focus S if and only if PS' - PS = 2a. The eccentricity E > 1 is defined by OS = E · a. The semiaxis b is now given via b2 = (e - 1)a2.
In the case of a parabola (Fig. 5), we denote the focus by S and the apex by A. The directrix of the parabola is perpendicular to the axis of the parabola, and its distance
VOLUME 25, NUMBER 4, 2003 37
A
from S is SB = 2 · SA. A point P lies on the parabola if and only if its distance SP from the focus equals its distance PV from the directrix.
A measure for the "width" of a conic section is the parameter p defmed simultaneously for ellipses, hyperbolas, and parabolas as p = SC (see Figs. 3-5). The parameter p is called the semilatus rectum. For parabolas we have p = 2 · SA. In the case of an ellipse or hyperbola, p is related to the semiaxes a and b via
(3) b2 p = -. a To prove this, apply Pythagoras's Theorem to the triangle SCS'. With the upper (lower resp.) sign representing the case of an ellipse (hyperbola resp.), one obtains
(2a ::;: p)2 = 4€2a2 + p2
or
4€2a2 = 4a2 ::;: 4ap,
which yields (3) if one solves for p using b2 = :±:(1 - €2)a2. For future use we present the last equation in somewhat modified form. Substituting SS' = 2m and SP :±: PS' = :±: 2a, gives
( 4) SS'2 = (SP :±: PS')2 - 2p (SP :±: PS') THEOREM A. Suppose S is the center and P is a mass point moving under the influence of a centripetal force. Assume that the orbit of P is a conic section with focus S. (In the case of a hyperbola the orbit of P is assumed to be the branch with focus S.) Denote the semilatus rectum of this conic section by p and let c > 0 be the area constant associated to this motion as defined in (1). Then, the acceleration of P towards S is in-
+iiriii;liM
38 THE MATHEMATICAL INTELLIGENCER
+ijiijiiijiW c
A B
versely proportional to the square of the distance SP. More precisely, the absolute value of the acceleration is k/SP2 with proportionality factor k given by
(5) k = c2Jp. Proof Newton's point of departure for his proof is the description of the acceleration in geometric terms mentioned in the previous section. In the case of a conic section, the acceleration is explicitly computable, leading eventually to the formula (5). The details of this computation can be found, for example, in [7, p. 56-61], [4, p. 93-103], or [10, p. 11] .
Remarks. (a) Let a conic section C with focus S be given. We would like to apply Theorem A To this end we introduce a mass point P on C and define its movement along C as follows. Assume c > 0 is a given constant. Denoting the position of P at time t 2: 0 by Pt, we stipulate that the area of the segment of the conic section bounded by the rays SP and SPt equals i ct. The reason for this stipulation is that the validity of an area law for the movement of the mass point P on its orbit guarantees that the underlying force is centripetal. This is just the converse of the area law proved above. To prove it, one only needs to consider Figure 1 and to reverse the corresponding argument ( cf. [7, p. 42]).
The foregoing existence proof shows that a given conic section is the orbit of a mass point P moving under the influence of a centripetal force. This suffices for the purposes of the subsequent section. Newton actually studied the much deeper problem how to calculate at any assigned time the location of the body moving along the given conic section [4, p. 127-142] . His profound mathematical investigations are once more motivated by the area law. Naturally, with respect to physics, the latter is of the utmost importance in the Principia as well, for example, for Newton's understanding of the concept offorce (see, e.g., [6, p. 272]). Thus referring to the area law P6lya rightly states [8, p. 1 1 1] , "Seldom has such a simple argument had such important repercussions."
(b) The proof of Theorem A makes use of the reflection property for conic sections in an essential way. In the case of an ellipse this property states that the continuation of the ray SP after its reflection at the tangent in P must go through the conjugate focus S'. This means that the angle between the tangent and the line SP equals the angle between the tangent and the line PS'. The situation is analogous for parabolas if we imagine that the conjugate focus S' has wandered towards infinity and interpret PS' as the
parallel to the parabola's axis. In the case of a hyperbola
the extension of the reflected ray to the opposite side of
the tangent must go through S' .
Theorem B
Proof strategy
Suppose P is a mass point attracted by the sun S resulting
in an acceleration kfSp2 with a positive constant k. Denote
the vector of the initial velocity of P by v and its absolute
value by v = lvl. We seek to determine the orbit of P from
the initial data {P,v}. Newton shows that the orbit is a conic
section. The point of departure of Newton's argument is the
solution of Problem A. Newton makes two observations:
first, that the conic section parameter entering into the
statement of Theorem A is the semilatus rectum; second,
that the reflection property of conic sections is used in the
derivation of the law of attraction in an essential way. New
ton realized that conversely the solution of Problem B may
be based on these two observations: 1. The law of attraction
with proportionality factor k together with the initial data
{P,v} allows one to determine the semilatus rectum of the
desired conic section. 2. From the initial data {P,v} one can
construct, using the reflection property, the line containing
the cof\iugate focus of the conic section to be determined.
Armed with these facts, the solution of problem B is al
most automatic. The course of the argument may be sum
marized as follows:
semilatus rectum �ii) conic section M.. orbit o/ k, {P, v}
(i� reflection property �ii) We commence by explaining steps (i) through (iv). The de
tails of step (iii) will be deferred to a separate subsection.
(i) The initial data {P,v} determine the tangent at the
point P. Let M be the orthogonal projection of S onto the
tangent (Fig. 2). By virtue of (2), the area constant of the
conic section to be determined is given as c = v · SM. In
view of (5) we set
(6)
We shall use p in the construction of the desired conic sec
tion in such a way that its semilatus rectum, as expected,
will be p.
(ii) The focus S of the desired conic section and its tan
gent in P are known. The conjugate focus S' lies on the re
flection of the line SP at the tangent. In the case of an el
lipse, S' is on the same side of the tangent as S. In the case
of a hyperbola, S' is on the opposite side. Our task is to de
cide from the initial data which of the two cases applies,
including the limit case of a parabola (when S' ----') oo) . To
that end we introduce an orientation on the line obtained
from reflection at the tangent. For a point Q on that line
count the distance PQ as positive (negative resp.) if Q sits
Mjlriil;ljM 'i@iliji+ H
s PH > 0 (--> ellipse)
PH < 0 (--> hyperbola)
on the same (opposite resp.) side of the tangent as S. The
still-unknown conjugate focus S' we provisionally denote
by a different letter, say H, regarding the distance PH as a
signed value (Fig. 6 and Fig. 7). Otherwise we retain the
notation from the previous section, still treating PS' as the
unsigned distance of P and S'. Thus in the case of an el
lipse versus that of a hyperbola we have PH = ±PS'. (iii) How is the signed value PH to be found? For the
sake of simplicity the limit case PH----') oo is disregarded for
the moment. We consider the triangle SPH with the un
known values PH and SH. The cosine rule in the triangle
SPH yields one equation for these unknowns. Is there an
other relationship between PH and SH? Here step (i) is of
help: substituting the tentative semilatus rectum defined in
(6) for the still unverified semilatus rectum in ( 4) yields an
other equation. Thus we have two equations determining
PH and SH. This fixes the position of S' on the reflection
of the line SP at the tangent in P. In the next subsection
we shall see how Newton carried out this argument in geo
metric terms to derive a criterion for deciding which kind
of conic section results as the orbit of P. From this proof
it will be immediately obvious that the constructed conic
section will indeed have the parameter p defmed in (i) as
its semilatus rectum.
(iv) The conic section constructed in step (iii) is the or
bit of a mass point P moving under a centripetal force (see
Remark (a) following Theorem A). Let c be the associated
area constant as in step (i). By virtue of Theorem A, the ac
celeration of P is £ · .� . Because of (6), this equals the ac-P Sr-
celeration 8� given by assumption. But the law of accel-
eration, together with the initial data, uniquely determines
the motion of P. Therefore the conic section constructed
in step (iii) is indeed the orbit of P.
Remark. Each of these four steps rests on profound intu
itions. By contrast the execution of the argument requires
only a minimum of technical effort.
Newton's Criterion
We simultaneously consider the cases with H lying on ei
ther side of the tangent. Recall Euclid's proof of the cosine
rule in the triangle SPH. To be able to apply Pythagoras's
theorem, drop the perpendicular from S onto the line ob
tained by reflection at the tangent. Denote the pedal point
of this perpendicular by K, and let the line segment PK be
signed in accordance with the convention adopted above.
VOLUME 25, NUMBER 4, 2003 39
H s PK < O
PK > O
Simultaneously for PH > 0 and PH < 0 (for the case PH > 0 see Figures 8 and 9; the proof in the case PH < 0 is com
pletely analogous), we obtain
(7) SH2 = (PH - PK)2 + (SP2 - PK2) = SP2 + PH2 - 2PH · PK.
In addition we have equation ( 4), which takes the follow
ing form after substituting :± PS' = PH:
SH2 = (SP + PH)2 - 2p(SP + PH) (8) = SP2 + PH2 + 2 SP · PH - 2p(SP + PH).
Comparing (7) with (8) yields
PH(SP + PK) = p(SP + PH),
in other words,
p PH 1 (9) SP + PK SP + PH 1 + SP/PH' Since all values on the left-hand side are given, we can
compute PH. Moreover, we read off the following criterion
from (9):
SP f PK < 1 � PH > 0 � conic section = ellipse
SP f PK = 1 � PH = 00 � conic section = parabola
SP f PK > 1 � PH < 0 � conic section = hyperbola.
Finally, let us verify that the resulting conic section is
uniquely determined by (9) and that it has the desired prop
erties. To abbreviate, set
(10) - p PN - SP + PK' For PN < 1 (resp. PN > 1) the conjugate focus S' of the el
lipse (resp. hyperbola) is fiXed by (9). The ellipse (resp. hy
perbola) is uniquely determined by the parameters a and
E, which, by construction, are given via SP :± PS' = ±2a and SS' = 2e · a. Furthermore the parameter p defmed in
(6) is indeed the semilatus rectum of the conic section. This
is seen by going backward from (9) and (7) to (8) and com
paring with (4). For PN = 1 the parabola is fiXed by the focus S, the axis
parallel to PK, and the semilatus rectum p = SP + PK (Figs. 10 and 1 1).
40 THE MATHEMATICAL INTELLIGENCER
i#'dii;!IUI
PK > O
This defmition of the semilatus rectum is consistent be
cause the assumption p · (SP + PK)- 1 = 1 implies
p = SP + PK = PV + PK = SB = semilatus rectum.
This concludes the proof of step (iii) and the solution of
Problem B. Let us summarize the result.
THEOREM B. Suppose we are given the center S, the initial data {P, v} (where v does not point in the direction of S), and the factor k by which the acceleration is inversely proportional to the square of the distance SP. Then the following holds: If PN < 1 the orbit of P is an ellipse with focus S. The conjugate focus of this ellipse is given via (9), and its semilatus rectum p is as
in (6). For PN = 1 the orbit of p is a parabola with focus S and semilatus rectum p = SP + PK For PN > 1 the orbit of P is a hyperbola with focus S. The conjugate focus of this hyperbola is given via (9), and its semilatus rectum p is as in (6).
Remarks. (a) In step (i) we derived the parameter p geo
metrico-algebraically from the data k and {P, v}. Newton
executed this step somewhat differently in geometric
fashion, the reason being that he had formulated the law
of acceleration as a proportion without explicitly speci
fying the proportionality factor k. To determine p he pro
ceeds as follows. Choose some arbitrary conic section,
say an ellipse, with a known semilatus rectum p* and with
one focus at the center S. Regard this ellipse as the orbit
of some mass point P*. Arguing geometrically, one can
specify the velocity v* of P* in such a way that the pro
portionality factor in the acceleration law for this ellipse
according to Theorem A equals that in the given acceler-
lpiiil;i+i+
PK < O
ation law. Let M* denote the orthogonal projection of S onto the tangent in P*. To get rid of the proportionality factor k, apply the relationship in (6) to both p and p* and form the quotient
pip* = (v · SM)ZI(v* · SM*)Z.
This allows one to derive p from the given data in a geometric way. (b) Newton's coefficient PN looks somewhat puzzling at first. Neither from definition (10) nor from the right-hand side of (9) is it transparent how PN depends on the scalar initial data SP and v. (c) At first glance it is not clear whether PN carries any physical meaning. Nevertheless Newton argues in physical terms when he writes that the type of conic section obtained depends on the initial velocity v. With given tangent direction at P, PN is indeed an increasing function of v:
(11) 1 1 c2 PN =
SP + PK . p =
SP + PK . k
1 28M2 1 2 . - v . k SP + PK 2
The quadratic appearance of v suggests a connection with kinetic energy. We shall examine this question and the related remark (b) in the next section.
Energy Our aim in this section is to simplify the purely geometric factor on the right hand side of (11). It will turn out that Newton's coefficient PN depends in a simple way on the kinetic and the potential energy. We use the following notation. Let S be a center and P a mass point with mass m. Assume that the law of acceleration is kfSp2. The kinetic energy of P is given as Ekin = t mv2 and the potential energy as Epot = -km/SP.
THEoREM C. With the same assumptions as in Theorem B,
Proof For abbreviation we write ( 1 1) as PN = 2� u · v2 with
2 8M2 (13) u = SP + PK
.
Here the term PK does not look very natural. There are two ways to eliminate it. Method 1 is formulated in trigonometric language and yields a one-line proof of (12). Method 2, on the other hand, is based on similarity arguments and is inspired by Newton's mode of reasoning in proportions (see Remark (a) below).
Method 1. Let y denote the angle of incidence, which equals the angle of reflection. Hence LSPH = 2y and LPSM = y (Fig. 12). Substituting PK = SP · cos 2y and SM = SP · cos y in (13) and applying the addition theorem cos 2y = 2 cos2y - 1, we obtain
u = 2 SP2 cos2y = SP. SP + SP cos 2y
H s
Method 2. We distinguish the cases that the conic section is a parabola or an ellipse resp. hyperbola. Case 1: parabola. Substituting SP + PK = p into (13) and using the similarity of the triangles SPM and SMA (Fig. 13), we obtain
(14) u = 2 SM2 = 2 SM2 = SM . SM . SP = SP. p 2 SA SA SP
Case 2: eUipse resp. hyperbola. We use the very relation (9) on which Newton based his proof of Theorem B:
1 SP + PK
1 PH p SP + PH
.
Plugging this into (13) and recalling (3), we get
2 8M2 PH (15) u = -p- .
SP + PH
2a · SM2 PS' SM2 · PS' · -- = b2 2a b2
Denote the orthogonal projection of S' onto the tangent at P by M' (Fig. 14 resp. Fig. 15). For the purpose of relating SM to S' M' notice that M and M' both have distance a from 0. The reason is that OM II S'V (Fig. 14 resp. Fig. 15), whence OM = l (PS' ::!:: PV) = l (PS' ::!:: PS) = a; similarly
2 2 OM' II SV' whence OM' = a. In the case of an ellipse resp. hyperbola the foci S and S' lie within resp. outside the circle with center 0 and radius a. Now apply the chord theo-
ii'riil;ii+
B
VOLUME 25, NUMBER 4, 2003 41
+jiijil;li§l V'
rem to the chords MM' and AA' resp. the secant theorem
to the secants SM' and SA' (Fig. 16 resp. Fig. 17).
With OA = a and OS = e · a we have SA = (1 - e)a, resp. SA = ( E - 1) a, whence
SM · S'M' = SM · SM' = SA · SA ' = ::t::: (1 - e)a · (1 + e) a = ::t::: (l - e2)a2 = b2.
Substituting b2 = SM · S'M' into (15) and using the simi
larity of the triangles SPM and S' PM', we obtain
SM2 · PS' SM PS' u
= SM . S'M'
= SP . SP . S' M' = SP.
This finishes the proof of Theorem C.
Remarks. (a) Although the statement and proof of Theo
rem C are not due to Newton himself, Method 2 is remi
niscent of an argument following Prop. XVI in Corollary VI [7, p. 64]. Newton's point of departure in this Corollary is
the case of a parabola (Fig. 13), and he shows in quite sim
ilar fashion as (14) that SM2 is proportional to SP, i.e., that
v2 is proportional to __!____ In other words, in the case of a
I � 1 " parabola jEkin/Epot is a constant (whose exact value = IS irrelevant for Newton's argument). Moreover, Newton
proves that in the case of an ellipse resp. hyperbola the en
ergy quotient PE : = jEkin/Epotl is less than resp. greater than
this constant. Theorem C may be regarded as an extension
of this observation, because (12) together with (9) implies
PE = ___!2!__. Therefore Newton's estimate of the energy SP + PH . · al
"ty quotient is made precise by this geometnc equ 1 .
In the proof of Theorem C we started from Newton's co
efficient PN and found a relation with PE· Inspired by Corol-
s·
42 THE MATHEMATICAL INTELLIGENCER
M'
A' 0 A
M'
lary VI, one might as well begin with the energy quotient
PE and try to establish its geometric meaning. Of course, in
this order one would arrive at the same result. For the
proof, suppose the orbit of P is a conic section, and replace
in the kinetic energy v by c!SM using (2), and in the po
tential energy k by c2/p using (5). It follows that PE = t p .
SP2 • Proceeding in analogous fashion as in Method 2
abcf:e, the simple geometric meaning for the energy quo
tient is revealed, namely
PH (16) PE = SP + PH { 1 for a parabola
- PS' --- for an ellipse resp. a hyperbola. 2a
(b) In Corollary VI [7, p. 64] as well as in Prop. XVII [7, p.
65/66], Newton argues geometrically in terms of propor
tions. It is therefore not astonishing that PN is in fact the
ratio of Ekin and jEpotl rather than a function of the addi
tive total energy E = Ekin + Epot· Conservation of energy
does not play any role in Newton's proof of Prop. XVII. This
idea enters only later in Propositions XL, XLI, and XLII,
where the initial-value problem is studied for central forces
of an arbitrary kind. There the energy theorem is formu
lated in terms of proportions as well, but as a proportion
ality between the increment of the square of the velocity
and the increment of Epot· The criterion PN < 1 resp. PN = 1 resp. PN > 1 in Theorem
B is of course formally equivalent to the condition E < 0 resp. E = 0 resp. E > 0. However, it is the energy quotient
PE rather than E which enters into the construction of the
conic section. Let S, the tangent line at the point P, and the
scalar value PE be given. If PE < 1 resp. PE > 1 one obtains
the cor\iugate focus S' as follows. Fix the point V' on the ex-
ljiijii;liil
s S'
tension of SP beyond P resp. beyond S so that PV' ISV' = PE, and reflect V' in the tangent (see Fig. 14 resp. Fig. 15).
(c) In the modern proof of Theorem B conservation of en
ergy is employed, and the orbit of the mass point is pa
rametrized by E, see for example [2, p. 38-40]. It follows
that for E =f. 0 the principal axis 2a of the ellipse resp. hy
perbola depends merely on the scalar value E. This is an
immediate consequence of (16). To see this, substitute
Ekin/Epot = -PH · (SP + PII)-1 into E = Ekin + Epot , yielding
E = E ( Ekin + 1) pot
E pot
_ _ km ( PH + 1) E - SP SP + PH km
SP + PH 1 - km for an ellipse 2a
= 0 for a parabola
km +- for a hyperbola. 2a
Appendix: Reprint of Newton's Proposition XVII (cf. [7, p. 65-66D.
PROPOSITION XVII. PROBLEM IX Supposing the centripetal force to be inversely proportional to the squares of the distances of places from the centre, and that the absolute value of that force is known; it is required to determine the line which a body will describe that is let go from a given place with a given velocity in the direction of a given right line.
Let the centripetal force tending to the point S be such
as will make the body p revolve in any given orbit pq; and
suppose the velocity of this body in the place p is known.
Then from the place P suppose the body P to be let go with
a given velocity in the direction of the line PR; but by virtue
of a centripetal force to be immediately turned aside from
that right line into the conic section PQ. This, the right line
PR will therefore touch in P. Suppose likewise that the right
line pr touches the orbit pq in p; and if from S you suppose
perpendiculars let fall on those tangents, the principal la
tus rectum of the conic section (by Cor. I, Prop. XVI) will
be to the principal latus rectum of that orbit in a ratio com
pounded of the squared ratio of the perpendiculars, and the
squared ratio of the velocities; and is therefore given. Let
this latus rectum be L; the focus S of the conic section is
also given. Let the angle RPH be the supplement of the
angle RPS, and the line PH, in which the other focus H is
placed, is given by position. Let fall SK perpendicular on
PH and erect the conjugate semiaxis BC; this done, we '
shall have
SP2 - 2PH · PK + PH2 = SH2 = 4CH2 = 4(BH2 - BC2) = (SP + PH)2 - L(SP + PH)
= SP2 + 2PS · PH + PH2 - L(SP + PH).
Add on both sides
2PK · PH - SP2 - PH2 + L(SP + PH),
and we shall have
L(SP + PH) = 2PS · PH + 2PK · PH, or
(SP + PH) : PH = 2(SP + KP) : L. Hence PH is given both in length and position. That is, if the
velocity of the body in P is such that the latus rectum L is
less than 2SP + 2KP, PH will lie on the same side of the tan
gent PR with the line SP; and therefore the figure will be an
ellipse, which from the given foci S, H, and the principal axis
SP + PH, is given also. But if the velocity of the body is so
great, that the latus rectum L becomes equal to 2SP + 2KP, the length PH will be infinite; and therefore, the figure will
be a parabola, which has its axis SH parallel to the line PK, and is thence given. But if the body goes from its place P with a yet greater velocity, the length PH is to be taken on
the other side the tangent; and so the tangent passing be
tween the foci, the figure will be an hyperbola having its prin
cipal axis equal to the difference of the lines SP and PH, and
thence is given. For if the body, in these cases, revolves in
a conic section so found, it is demonstrated in Prop. XI, XII,
and XIII, that the centripetal force will be inversely as the
square of the distance of the body from the centre of force
S· and therefore we have rightly determined the line PQ, �hich a body let go from a given place P with a given ve
locity, and in the direction of the right line PR given by po
sition, would describe with such a force. Q.E.F.
Acknowledgment
The authors are grateful to an anonymous referee for his com
ments. Many thanks go to Professor Gerhard Winkler (GSF
Forschungszentrum ftir Umwelt und Gesundheit GmbH,
Neuherberg) for providing generous secretarial support.
REFERENCES
[1 ] Albouy, A.: Lectures on the Two-Body Problem , in H. Cabral and
F. Diacu (eds.), Celestial Mechanics -The Recife Lectures, Prince
ton University Press: Princeton (2002).
[2] Arnol'd, V. 1 . : Mathematical Methods of Classical Mechanics,
Springer-Verlag: New York, Heidelberg, Berlin (1 978).
VOLUME 25, NUMBER 4, 2003 43
A U T H O R S
KA1 HAUSER
Tec:hnische UnlllerSJtl!t MA 8· t StraBe des 17. Junt 136
0·1 0623 Ber11n Germany
e-mail: [email protected]
Kai Hauser stud1ed mathematiCS and philosOphy at the University
of Heidelberg, where he did his Hablitation 1n 1 993. after receiv· tng a PhD from the Galiforma lnstrtute of TechnolOgy tn 1 989. He
wor\<S tn log1c and foundations of mathematics and in philosophy.
[3] -- : Huygens and Barrow, Newton and Hooke, Birkhauser:
Basel, Boston, Berlin (1 990).
[4] Chandrasekhar, S. : Newton's Principia for the Common Reader,
Clarendon Press, Oxford (1 995).
[5] Cohen, I. B. : A Guide to Newton's Principia, in Newton 1 . : The Prin·
cipia, new translation by I. B. Cohen and A Whitman. University
of California Press, Berkeley (1 999).
[6] De Gandt, F.: Force and Geometry in Newton's Principia, Prince·
ton University Press, Princeton (1 995).
REINHARD LANG
lnstitut tOr wanote Mathemall
lm euenheimer Feld 294 0·69120 Heidelberg
Germany a-mall: amOmath.unt·heidelberg.de
R91nhard Lang studied mathematiCs and physics at the UniverSity
of Heidelberg, where he received a doctorate tn mathematics tn
1 976 and became Privatdozent In 1 983. His Interests he lfl math
ematical physics �n partiCUlar potential theo!y and statistical mechanics) and n Greek mathematiCS and philosophy.
[7] Newton 1 . : Principia, Motte's translation, revised by F. Cajori, Uni
versity of California Press, Berkeley (1 962). (For a new transla
tion see [5].)
[8] P61ya, G.: Mathematical Methods in Science, Mathematical Asso
ciation of America: Washington, D.C. (1 977).
[9] Pourciau, B . : Reading the Master: Newton and the Birth of Ce
lestial Mechanics, Amer. Math. Monthly 104 (1 997), 1 -1 9.
[ 10] Stein, S. K. : Exactly How Did Newton Deal with His Planets?, Math.
lntelligencer 18 (1 996), no. 2, 6-1 1 .
TEX Devil In the course of editing a book, I came upon the sloppy usage "thm" where "theorem" was intended. I asked an efficient assistant to correct it, which he did by writing a simple 'fEX command. Several weeks later, I found in the new version of the book the phrase "division algoritheorem." Whom do I congratulate for the nice neologism-the author?
-Rajendra Bhatia
44 THE MATHEMATICAL INTELLIGENCER
MMMj.I§,Fiilflld·l,ii.;:;,;;nfJ Marjorie Senechal , Ed itor [
Mathematicians' Visiting Cards G. L. Alexanderson and
Leonard F. Klosinski
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. Jfhat 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]
We recently came across a tattered
and badly worn manila envelope
that contained 121 visiting cards that
had been given to George P6lya and his
wife Stella during their years in ZUrich,
Oxford, Cambridge, Harvard, Prince
ton, and Stanford. Included are cards
from some of the most important math
ematicians of the early twentieth cen
tury. These turned up in a collection of
manuscripts, letters, and reprints long
stored in the P6lyas' house in Palo Alto.
P6lya died in 1985, his wife in 1989. The
person who bought the house from the
P6lya estate, Dan Comew, found the
material in two suitcases and a box in
the attic and delivered them recently to
the Stanford Mathematics Department.
P6lya was a Hungarian-Swiss-Amer
ican mathematician known for his
deep research in a variety of fields
real and complex analysis, number
theory, geometry, combinatorics, and
applied mathematics-and for his con
tributions to heuristics and problem
solving, most notably for his best
selling book, How To Solve It. A collection of visiting cards is of no
mathematical interest and, probably,
of little interest in the history of math
ematics. But the cards do provide a
glimpse into the culture of the mathe
matical community not so many years
ago. Visiting cards (as distinct from
business cards, which are a quite dif
ferent thing) probably first appeared in
the eighteenth century, but the full rit
ual of the visiting card (the social carte de visite) did not flourish until the Vic
torian era. The practices varied some
what from country to country, but very
significant subtle messages were con
veyed by how the card was presented,
which comer of the card was turned
up, whether the card was folded once
vertically, whether it was presented
face up or face down, and so on. For
example, a folded top left comer indi
cated that the caller had delivered it in
person, whereas an unfolded card had
been delivered by a servant. A folded
lower right comer was an expression
of sympathy, but a folded right top cor
ner extended congratulations. Cards
could also be used as thank-you notes
after balls and dinner parties. They
could be delivered at any time, but, if
an actual visit was anticipated, only at
hours when the recipient was "at
home," and this was during rigidly reg
ulated times of the day. One was never
"at home" on Sunday, a day reserved
for immediate family or closest friends.
Traditions in the military (where the
practice of junior officers leaving visit
ing cards with senior officers contin
ued longer than in most professions)
were quite rigid, at least in the early
years: "U.S.A." in the lower right cor
ner meant the person was in the United
States Army, for example. One's rank
was given only by those with the rank
of lieutenant or higher. Japanese cards
traditionally had the name in Japanese
on one side, English or some other
Western language on the other.
Visiting cards were a part of the ritual
of social and professional life for much
of the nineteenth and early twentieth
centuries. In some countries there are
museums devoted to displaying visiting
cards (in St. Petersburg and Budapest,
for example: see http://origo.hnm.hu/
english/ and http://www.cityvision2000.
com/sightseeing/muse_abc.htn). P6lya
used to tell that in Gottingen when he
arrived there in 1912, there was a long
tradition of new junior faculty donning
black frock coat and top hat to pay a
call to each senior faculty member. He
(and it was certainly a man in those
days) would present a visiting card at
the door and, if the senior faculty mem
ber and his wife were "at home," he
would be received and there would fol
low a visit, by tradition quite brief
certainly less than half an hour. There
was always a small table near the front
door with a vase of flowers and a sil
ver, crystal, or pewter tray for visiting
cards.
P6lya was educated at Budapest,
Gottingen, and Paris before taking a
position at the Eidgenossische Tech-
© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25, NUMBER 4, 2003 45
� H
cJ �� llt,...a.
� .
1' . � .
. .H �•f'1u/ !fl, ,.;, (C,.,r.,f�,.
. r; . /t/;,7 �fl./ "--
/1 " A /'. J '}" < 1 1 Loll' t /. t ,,:,, ,k ll ),i� 'IU • I I� , ,/.. ,...; 'l'd-h. t,
46 THE MATHEMATICAL INTELUGENCER
�.u�
�\ . � rl'
��
_ J/kr�hmt. �/-. ·�.Y "I' �I.Y/,.'.1/.)
lt
Dr. OOROTHY WRINCH
Ledy l'\ArQ ret ffeJI, Ox.!om
· �,/ . .YC.raa _../ Professor ved Umversrtetet
Elmholtsvn. 29 Bestun, Oslo
. 4 � /. fi?. '!f ·'?·7"- at 2m& tin.� .... &:.? �/;/,��� - � �r;/7 ,<. /. � /, NN' /ll 7 /u /IN'
fROPES R pR . . TfuRWJTZ
ZuRicH
EL'�I ,· J NO
VOLUME 25, NUMBER 4, 2003 4 7
P R O F. DE; FR ITZ O ETH R
K A R L R U H E
TT
B E R L I W . lO D O R ' B E R G - S T � A s e 2
R. 'SPRP.CHER: LUnow 17:\
I. l ' '1 �. ' T H 1. I ' 'D F I:{ A I
<...l E A N D I E U D O N N E 4u.: ,-, � ��U..Vt".,� 2...-t. , ,. '<l • �.,.,.,.....,., //l:x;,;{..,, ai''J'""'
ft'-'W'tk .. � �� rp_ay.. 4-td � J- ..u tnu� .uJ� �'l 1.
48 THE MATHEMATICAL INTELUGENCER
••nt L L
•
(EJ f'"fr) 7 0/ . ra
r //'/N _;/ V/N//l �'r ..h.., ... :., ,. # . --";/,� �J . ��.., ' �r--H 41 / " � 9..../. ,...{,�-�
Doct r
PROF. NICOLAS KRYLOFF In enieur d an
Sdenco ath#m tiquea honord aus d I'Unl ers t ' Je Ka !I S<aenc d'Ukraan
encn de I'CR. S
MO ' lEUR LE PROI ES EUR ET M DAME FERDI A. D GO 'SET! I
, I .-/_ � � bv.f � j-.4��..:. � �£�� ��- � ;� 4 Zurich chtuch: r r. 7
G . no o lbt •
rue.-r.J .. � ��-� k.; ZUjtc-G,t::UA: ; ��·� d-
d 4t:td� j�CL- d- � a.&tu� :�u �fo �Mu�eu.-1;-� t! J?� �- e./.._ � � 13 1�--- � &.� .;; o �- i_.· � PI � ;/-� .
N icolas lusin pro•o S r adj nt I' Un v r$ I d OICOU
l ) � .
VOLUME 25, NUMBER 4, 2003 49
R O .l·' . V I T O O L E R
D R . JO 'E H I.E ' RD W A L. H .6 · I T .A ' T l' K O l'B K 0 M 'l' H J! l'1 ' l ll fl Y R D U .N I VJI:B ITT
50 THE MATHEMATICAL II'ffELUGENCER
CAND PH!L. O & O n T n U S KY
MR.JA ES"\V. LI':XA ' D EH . MD
29 CLE �=" "- ""' N O L A N £ Rl CCTON, It -.J R81=.v'
� .. �. (�� tZ
f Rs . • JAM F:s \ . r.EXA ' DE 0 , 2 � '.'
. I N , I . ...... I'l l 1 1.1 1' no., . . Jw.
f· · ( .
nische Hochschule (ETH) in Ziirich in 1914. There he remained until he emigrated to the United States in 1940 where he taught at several institutions before arriving at Stanford University in 1942. He taught there until his retirement. These cards were left by mathematicians from France, Germany, Switzerland, Italy, Sweden, Denmark, Norway, Finland, Latvia, Belgium, The Netherlands, Russia, Hungary, Bulgaria, Serbia, Austria, Poland, Spain, Great Britain, Japan, and the United Statesthe mathematical world of that time. Some contain handwritten notes. Alas, to varying degrees they now appear foxed and sometimes brittle from damp and the dust of decades. But with some computer wizardry the second author has cleaned some of them up, so a sample of them can be shown here.
The cards fall into several welldefined periods. There are the Ziirich years where cards were delivered to the two apartments the P6lyas lived in
in Zurich and to their chalet in Engelberg (a few have short handwritten thank-you notes for pleasant weekend stays at the chalet). Some of the English cards date from their time at Oxford and Cambridge in 1924, and others are from their visits to Harvard, Princeton, and Stanford in 1933.
There are two cards from Jean
Dieudonne, and one from Nicolas Bourbaki. So much for the notion that Bourbaki did not really exist! There is one from "G. H. Hardy and J. E. Littlewood", written in Hardy's distinctive hand, on a Cambridge University card. Some names are instantly recognizable to anyone interested in twentieth century mathematics: Henri Cartan (the son of Elie Cartan), Alfred Haar (written as Haar Alfred in the Hungarian manner of giving the family name first), Adolf Hurwitz (who brought P6lya to Ziirich in 1914), Gaston Julia (widely known today because of Julia sets), Solomon Lefschetz, Nicolas Lusin, Gosta Mittag-Leffler (with an accom-
:!9 t.KVRio U t.A 1': PnlhC'KTtH"-'· t,;.WJ I;Rttt::
panying notation, "Djursholm," his home and now the site of the MittagLeffler Institute outside Stockholm), Louis J. Mordell, John von Neumann ("Neumann Janos"-an early card, for the name is written in the Hungarian manner and before von Neumann introduced to his name the aristocratic German "von"), Issai Schur, Oswald Veblen, and many others. (For additional identifying information about these, see [ 1 ] or [2].)
Of the two cards of the Riesz brothers, Marcel and Frigyes, the first dates from Marcel Riesz's Stockholm years so the card reads "Marcel Riesz," but the other reads "Riesz Frigyes" because Frigyes had remained in Hungary, at Szeged. The card of Olga Taussky reads "Can d. Phil. Olga Taussky," obviously before she had earned her doctorate and before she married John Todd. There is a card of M�or Percy MacMahon, author of Combinatory Analysis and well-known combinatorialist who did many calculations-finding, for exam-
VOLUME 25, NUMBER 4, 2003 51
ple, the number of partitions of 200 in order to check Ramanujan's famous formula for the partition function, long before computers made this kind of calculation routine. MacMahon's card includes the fact that he is "late Royal Artillery," and lists not only his Cambridge address but also his London club, the Athenreum.
Another card is that of Alfred Errera, not perhaps as well-known as many of the others, but nevertheless interesting. Errera was Belgian and contributed, along with G. D. Birkhoff, some valuable work on the Four-Color Problem. He came from a wealthy fam� ily, and invitations to the Errera home in Brussels were much sought after because the food served was known to be extraordinary. Anne Davenport, widow of the Cambridge number-theorist, Harold Davenport, in correspondence with the first author, wrote that when the Erreras came to Cambridge and the Davenports invited them to dinner, she was so nervous she "forgot to put any salt in the vegetables!"
The collection contains not only cards from James W. Alexander, 2nd, and Solomon Lefschetz, it also contains cards from their wives. Alexander came from an old, distinguished, and wealthy family (of the Equitable Insurance firm), and is remembered in mathematics for the Alexander polynomial in knot theory. Solomon Lefschetz was the legendary topologist, long influential in the Princeton department. Wives did not always have their own visiting cards. The card of the well-known analyst, Otto Blumenthal, reads "Professor Otto Blumenthal und Frau." A lady's card was traditionally somewhat larger than that of a gentlemen, a distinction apparent from the examples in this collection; the gentleman had to have cards that would fit into a convenient pocket, say in a waistcoat, but a lady could carry a card case in her purse.
Harvard is represented by "Mr. Ralph Philip Boas" and "Assistant Professor" Joseph Leonard Walsh, obviously early in Walsh's distinguished career at Harvard. Another telling observation is the reverse snobbery connoted by the fairly consistent use on the Harvard and Princeton cards of the
52 THE MATHEMATICAL INTELLIGENCER
modest title "Mr." or "Mrs.", whereas the German cards, in particular, might use "Professor Dr." The latter leave no doubt about the person's rank and station in life. Ferdinand Gonseth at the ETH used "Monsieur le Professeur et Madame Ferdinand Gonseth."
French cards are usually quite simple, sometimes with the university affiliation. Szolem Mandelbrojt (uncle of the younger Benoit Mandelbrot of fractal fame), later a member of the College de France, has a card reading "S. Mandelbrojt/Maitre de Conferences de l'Universite de Lille." Russian cards seem to be printed in Roman letters (at least those for use outside Russia), with additional information-titles and affiliations-in French, for example, "Nicolas Lusin, professeur-adjoint a l'Universite de Moscou," with a handwritten addition "Rue Stanislas, 14." German cards, however, can be quite elaborate: "Dr. phil. I. Schur/ord. Professor an der Universitat/Mitglied der
A U T H O R S
GERALD L. ALEXANDERSON
Department of MathematiCS and Computer Science
Santa Clara U111Verstty
Santa Clara. CA 95053·0290 USA
e-mail: gaJexand math.scu.edu
Gerald L. Alexanderson was a long-time
fnend of George P61ya and his w1fe
Stella, and he Is the author of the biog
raphy Random Walks of George PO/ya
(Mathematical Associat1on of America, 1992). He has been at Santa Clara Uni
versity for 45 years-for most of that
time as Department Chair. He is a past
President of the MAA He now edits the
MAA's Spectrum series, and collects
1 7th-1 9th-century mathematics books.
Preussischen Akademie der Wissenschaften!Berlin Schargendorf/Ruhlaerstr. 14."
These days mathematicians, among academics, have a reputation for rather casual attire and lifestyle. It would be difficult to imagine a revival of the ritual of the mandatory visiting card and the prescribed social calls they imply. But this little collection, found in an attic, recalls a mathematical world not that distant in the past. Many living mathematicians will recall personal contacts with some of the people we've mentioned here. The social customs the cards recall are, however, long gone.
REFERENCES
[ 1 ] Alexanderson, G. L. , Random Walks of
George P6/ya, Washington, Mathematical
Association of America, 2000.
[2] P61ya, George (G. L. Alexanderson, Editor),
The P6/ya Picture A/bum/Encounters of a
Mathematician, Basel, Birkhauser, 1 985.
LEONARD F. KLOSINSKI
Department of MathematiCS and Computer Science
Santa Clara UnM!fSity
Santa Clara. CA 95053-0290 USA
a-mall: [email protected]
Leonard F. Klosinski has been at Santa
Clara University for 42 years; to be sure,
for a few of those he was an under
graduate student. He has been d�rector
of the William Lowell Putnam Ma he
matical Competihon for 28 years
longer than any other director since the Competition's found1ng in 1 938. He re
ceived the MAA's Haimo Award for
mathematics teaching in 2000.
l$@il:i§j:@hl$il§:h§4fii,i,i§,id M ichael Kleber and Ravi Vaki l , Ed itors
One Hundred Prisoners and a Lightbulb Paui-Oiivier Dehaye, Daniel Ford ,
and Henry Segerman
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.
Contributions are most welcome.
Please send all submissions to the
Mathematical Entertainments Editor,
Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380,
Stanford, CA 94305-21 25, USA
e-mail: [email protected]
Asingle lightbulb flickers into life in the center of the room. 100 pris
oners shade their eyes from the glare, then focus on the prison warden standing by the lightswitch, with a standard evil-puzzler's glint in his eye. He begins to speak:
In one hour, you will all be taken to your cells to be kept in solitary confinement, with no possibility of communicating with any of your fellow inmates.
Well, almost no possibility . . . . every night from now on, I will choose one of you at random, retrieve you from your cell, and take you to this room, where you may see if the lightbulb is on or off, and you may turn it on or off as you wish.
A murmur ripples around the room as the prisoners consider the prospect of having such an effect on their hitherto impotent and externally controlled existences.
If at some point, I take you to this room and you believe that all 100
prisoners have been chosen and taken here at some time, then you may tell me this.
ff you are correct, I will free you all. If of course you are incorrect
. . . well let's say none of you will live to flip any more lightbulb switches in this world.
He exits with a flourish of his cloak, thoughtfully leaving the lightbulb on.
The prisoners are in the dark as to how to get free, but they are perfectly clear about wanting to be able to at least flip light switches into old age (and it looks like they might need to!). So they must come up with a strategy that will announce that all 100 prisoners have been chosen only if they actually have, with 100% certainty, preferably before they all die of old age.
At first it seems impossible that any
one prisoner could know about what the other 99 have been up to. Coming into the room and seeing the lightbulb is on doesn't seem to give you much information. You don't know who set it, and if you flip the switch you have no idea who will see that you flipped it. There seems to be no way to send a message to anyone in particular. It seems hopeless that they will get out at all. But in fact:
Amazing fact 1. They can get out.
Here is how. (You may wish to ponder on your own before reading on.)
If at First You Don't Succeed . . .
Strategy 1. Cut the sequence of days into blocks of length 100. The first prisoner to enter the room in a given block turns the lightbulb off. If a prisoner enters the room a second time in the same 100-day block, then he turns the lightbulb on. If a prisoner enters the room on the last day of a 100-day block, and it is his first time, and the lightbulb is still off, then that prisoner knows that every prisoner has been chosen exactly once in this 100-day block He then correctly declares that all prisoners have been in the central room at least once. If the lightbulb is on on the last day in a block then we have failed this time so we try again in the next block of 100 days, and keep trying until someone announces.
Expected results for strategy 1. The probability of succeeding in any given block is the number of orderings of the 100 prisoners divided by the number of possible ways the prisoners could have entered the room. With n prisoners, that is (n!/nn).
The expected number of blocks which must be used before succeeding is equal to lip where p is the probability of succeeding with one block To see this, suppose p is the chance of succeeding in any given block Then the expected number of blocks until we
© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25. NUMBER 4, 2003 53
succeed is equal to Ik kpqk- l where q = 1 - p. This is equal to
PJ!_ (ll(l - q)) = p!p2 = lip dq
Thus the expected number of blocks is nnln!. Each block has length n, so the expected number of days until freedom is nn+ 1/n!, which is O(n112en) (using Stirling's formula). For 100 prisoners, the expected value is 100101/100! or approximately 1044.
They can get out, although this is a disappointingly large number for the prisoners: about 1041 years. Sadly the universe may have ended long before they are free [4].
Amazing fact 2. They can get out before the universe ends.
Soul-Collecting Strategy 2. One prisoner, who will be known as The Countess, will be responsible for announcing that every other prisoner has entered the room at some time. The other (n - 1) prisoners will be ordinary.
Each ordinary prisoner starts with a
to have a sequence of events happen. We first need a soul dropped in the room, then for the Countess to pick it up, then another soul dropped, etc. As the number of uncounted souls goes down, the probability of a new one turning up on the next day goes down from ( n - 1 )In for the first soul to lin for the last soul. Meanwhile, the probability for the Countess to show up on the next day is constantly lin.
Since the expected time needed for an event occurring with probability p on the next day is lip, we immediately get that the time is
n I - + (n - l)(n), (n- 1 1 )
k = l k which is between n2 and 2n2. Therefore the expected number of days until the prisoners escape is 0( n2).
This is much better than our previous exponential solution. The 100 prisoners should get out in around 10,400 days, or about 29 years. They will be past their best, but they will live to see the outside world. However, they can do much better than that:
single token, called a soul, and each Amazing fact 3. They can get out bewill try to leave that token in the cen- fore they are ineligible for the Fields tral room. The Countess will collect medal. souls from the central room until she has all of them. She may then declare success.
We may assume that the lightbulb starts in the off position (as the prisoner who enters on the frrst day may turn it off before doing anything else).
When an ordinary person enters the room and finds the lightbulb off, he may drop his soul in the room, if he has not already done so, by turning the lightbulb on. If the lightbulb is already on then he leaves it alone.
When the Countess enters the room, if she finds the lightbulb on then she turns it off and adds one
to her soul count. If her count is now n - 1, then she knows that everyone must have entered the room, so she can declare. If the lightbulb is off when she enters, she leaves it off
Expected results for strategy 2. For the strategy to complete we need
54 THE MATHEMATICAL INTELLIGENCER
Pyramid Scheme This is again a method for collecting souls. This time there is no single counter. Rather everyone is involved in a process of collecting souls together. The lightbulb will be worth different numbers of souls on different nights.
Strategy 3. A sequence is given which describes how many souls the lightbulb is worth on each night, which is always a power of two. Let V( n) denote the number of souls that the lightbulb is worth if it is left in the on position on night n, or discovered on on night n + 1.
Assume that the number of prisoners is a power of two. This will turn out not to matter in the end.
A prisoner enters the room on a night and collects however many souls have been left there the night before (so if it is night n and the lightbulb is on he picks up V( n - 1) souls) and turns the lightbulb off. He now looks at the number of souls M that he has collected, but
represented in base 2. If the coming night is worth V( n) = 2k souls then he looks at the binary bit of M worth 2k souls. If this bit is 1 then he drops 2k souls by turning the lightbulb on and subtracting 2k from M, his own total of souls collected. If the 2k-bit is zero then he leaves the lightbulb off.
Notice that this has the effect that souls are "glued together" into lumps of size 2k which can be transferred on nights which are worth 2k. Whenever a prisoner has two lumps of size 2k he glues them together into a lump of size 2k+ 1 . This may occur if he has just picked up a 2k lump or has just glued two smaller 2k- 1 lumps together. When all the souls have been glued together into one lump of size n = 210g2 n then the prisoner who holds this lump declares success.
We have yet to say what would be an efficient choice of values for the V(n). Starting with a block of nights worth 1 is a good idea, to hopefully glue all the single souls into pairs. Then follow with a block of nights worth 2 to glue into blocks of 4 souls, and so on. We want the lengths of the blocks to be long enough to give a good chance of gluing all the lumps of size 2k- l into lumps of size 2\ but not too long as we don't want to waste time once they have all been glued.
Expected results for strategy 3. In order to achieve a good asymptotics, we start with a block of (n log n + n log log n) 1-nights, then ( n log n + n log log n) 2-nights, then ( n log n + n log log n) 4-nights, all the way up to (n log n + n log log n) (log2 n)-nights. If we have failed after this number of days, then we can simply throw up our hands and start over again. In other words, the sequence V( n) repeats.
The probability of gluing all lumps of 2k- l into lumps of size 2k within n log( n) + en nights (where c is some constant) is bounded by e-e-c asymptotically. This is known as the coupon collector's problem [2]. With some careful estimation this result can be extended by changing c to a function c(n) :::; log(n). This gives us a probability of successfully completing each stage of at least e- lllog(n)E(n), where E( n) is an error factor such that E(n)10g2(n) tends to 1.
The chance of successfully completing all log2( n) stages is at least eiog2(e)fE(n)'0g2(n). Thus the expected number oftimes we need to go through the whole cycle is less than e'0g2(e) 11E(n)10g2(n). This gives that the expected number of days is of order eo�(e) [n log(n)(log(n) + log(log(n)))] , which is O(n log(n)2).
It can be proved that no changes to the lengths of blocks of nights V( n) can improve the asymptotics; but what if we want the best sequence for precisely 100 prisoners?
Our assumption that the number of prisoners is a power of two can be relaxed. To apply our strategy, we just need that everyone starts with at least one soul. So, with 100 prisoners for example, one prisoner could be given 29 souls and the other 99 prisoners given 1 soul to start with. The prisoner who first collects 128 souls declares.
To try to get a good upper bound on the expected number of days to freedom, we used a computer simulation to search through the choices for V(n). Our best give around 4400 days, or 12 years. One sequence of block lengths which has about this average is [730, 630, 610, 560, 520, 470, 560, 720, 490, 560, 570, 560, 590, 590], that is to say: 730 1-days, then 630 2-days, . . . then 560 64- days, then 720 1-days, . . . then 590 64-days, then (back to the start) 730 1-days, . . . The optimisation algorithm works by trying to optimise the V( n) for two passes through the types of days (1-days, then 2-days, then 4-days etc.) then just repeats that sequence for the unlikely cases in which
the prisoners have still not finished after 2 passes. It is not entirely clear why the above sequence is good, though it makes sense that the first six terms are decreasing because fewer people have to "meet" in later stages. It also makes sense that the seventh term is larger, since one would want to give a lot of time for the last two blocks of 64 souls to meet. Giving up at this stage means having to continue through the next seven stages to finish up.
Can they do better with some
other strategy? For 100 prisoners we suspect that some sort of hybrid algorithm is probably the best, to use good points of more than one strategy. Col-
lecting together souls as in the pyramid scheme is certainly a good idea to start with, but something else may be better in the endgame. A hybrid given by B. Felgenhauer [ 1 ] uses the pyramid scheme to start with, but has a Countess start collecting midway through. His sequence of block lengths (chosen by hand) has expected days of around 3949, and running our optimisation program on the variables for his strategy gives around 3890.
Asymptotically, they cannot do better than O(n log n) expected number of days to freedom, because that is the expected number of days for everyone to have visited the room, ignoring that all prisoners actually have to communicate!
Variations Here are some variations you might like to think about. Each variation assumes all the conditions in the original problem, but with some aspects altered. In each case, you might like to ask yourself whether the prisoners can escape, and if so what is an efficient way to do this. We assume that the lightbulb always starts off.
1. Multiple bulbs-The central room contains two (or more) lightbulbs (the communication channel is wider).
2. Multiple rooms-There are two (or more) identical rooms. The prisoners are taken to one at random but don't know which they are in.
3. Separate transmitter/receiver
The warden turns the lightbulb off at 12 AM, chooses one prisoner to visit at 1 AM, and chooses again for someone to visit at 2 AM. The visitors only transmit or receive, not both.
4. Malicious Warden-The warden is malicious and knows the strategy that the prisoners will use (he listens to them agreeing on what to do). Each day he will choose which prisoner to allow into the room. His conscience demands that he allow every prisoner to visit the room infinitely often.
5. All prisoners have to an
nounce-The condition for everyone to be freed is that every pris-
oner must correctly announce (at some time). In other words: every prisoner must be sure that all prisoners have been to the room.
6. Simultaneous announcing-Any
one can announce on any day, not just the prisoner who was selected that morning. The condition for everyone to be freed is that everyone must correctly announce on the same day. If they are incorrect or if some announce while some do not, then they all die.
7. Prisoners are freed when they
announce-Everyone must correctly announce at some time. When someone announces, he is freed (never to visit the room again), but the others stay until they too announce. Visitors are chosen uniformly among the remaining prisoners. Note that the prisoners are still most interested in everyone escaping, rather than in minimizing their own time to escape.
8. Red/Blue cells (one announcer)
The prisoners are allocated red or blue prison cells. The announcing prisoner must correctly say how many red cells there are in order for them all to be released.
9. Numbers in cells (one an
nouncer )-Each cell has a natural number written on the wall. The announcer must give all the numbers.
10. All prisoners send messages to
all prisoners-( this is a combination of 5 and 9).
11 . Random visiting times-The prison is subterranean, with no clocks, calendars, or any other information as to what the time is. The prisoners lose all track of time, and the warden chooses prisoners at random times. In other words, the prisoners have no idea how many people have visited the room since they were last there, and they cannot use strategies which count days. They only know the order in which events occur.
12. Random times, all prisoners
must announce-Combine variations 5 and 11 . Every prisoner must announce that everyone has visited at some time, and they cannot use day-counting strategies.
VOLUME 25, NUMBER 4, 2003 55
13. Random times, message from
one to one-There are only two prisoners and the transmitter has to send a message (a natural number) to the receiver, but as in 1 1 they cannot count days.
14. Random times, messages from
many to one-Combine variations 9 and 1 1. One announcer must give all numbers written on the walls, and the prisoners lose track of time.
15. Random times, message from
many to many, 2 lightbulbs
16. Random times, messages from
many to many, l lightbulb
We now give a spoiler for most of these problems. It turns out that the strategies listed above (or slight modifications of them) are suitable for most of these variations.
1. Multiple bulbs-Counting souls (strategy 2) will still work, and can be made even faster as 2k souls can be left in a room which has k distinct lightbulbs, log2n lightbulbs allow for the best possible time to escape-as soon as everyone has actually been in the room, then the prisoner in the room can declare. Strategy 3 can also be improved by allowing gluings of souls into larger lumps, such as lumps of size (2k)Z if there are k distinct lightbulbs.
2. Multiple rooms-Counting
souls (strategy 2) will still work It will be slower, although the expected time until escape (for number of rooms independent of n) is still O(n2) days.
3. Separate transmitter/re-
ceiver-A strategy similar to soul-collecting (strategy 2) works. The Countess always picks up and never drops souls. Everyone else drops souls at every opportunity (though they are forced to pick them up if they find them). This strategy has expected time between n2log2n and n3. (If there are k souls outstanding, then the chance of the countess picking one up the coming night is between !_ X _!_ and l/n2, depend-
n n
56 THE MATHEMATICAL INTELLIGENCER
ing on how those k souls are distributed. This gives a total expected time of between n2 log2 n and n3.)
4. Malicious Warden-Strategy 2 will work, although there is clearly no bound on the time until escape; it depends on how mean the Warden wants to be.
5. All prisoners have to an
nounce-"Try-try-again" (strategy 1) works. Interleaving cycles of strategy 3 will also work: Each prisoner has one type of soul for each prisoner who will have to announce. One cycle is given to each prisoner's attempts to collect the souls destined for her, then after n of these a second cycle is devoted to each prisoner, and so on. This gives an expected time of n2log2( n ).
6. Simultaneous announcing
The prisoners cannot be sure of escaping. Suppose they will announce on day A. There is a first day, D, on which they all know this. The prisoner who enters on day D knows that she has entered and the state of the lightbulb. Every other prisoner only knows that he did not enter the room on that day. If a different prisoner entered on day D then all of the other prisoners who did not enter would have the same information, and so would have to come to the same conclusion: that they should announce on day A (provided there are at least 3 prisoners). Therefore it cannot matter who enters on day D, so they must all know on day D - 1. This contradicts the assumption that D was the first day they all knew they would announce on day A.
7. Prisoners are freed when
they announce-"Sloshy" soulcollecting (as in the answer to variation 12 below) will work When a prisoner has collected 100 souls and then given them all away again she may declare and be set free.
8-10. Red/Blue cells, or Numbers
in cells (one or all announ
cer(s))-The prisoners can es-
cape. See the Uber-theorem (below) for a strategy and proof.
11 . Random visiting times
Counting souls (strategy 2) will still work
12. Random times, all prisoners
must announce-"Sloshy" soul counting will work The lightbulb is always worth one soul. Any prisoner who has not announced does the following: If the lightbulb is on when he enters, then he collects the soul and turns the lightbulb off. lfthe lightbulb is off when he enters and he has one or more souls, then he drops one soul and turns
the lightbulb on. Any prisoner who has already announced always drops any souls that he has, and leaves any that are in the room. This strategy has expected time order less than or equal to en. This can be shown by constructing an appropriate Markov chain and giving lower bounds for the chance that a given prisoner will announce in the next 200 days. Notice that when there is only one prisoner left to announce, this strategy reduces to Strategy 2, soul-collecting with one Countess.
Note that this strategy would also work (less efficiently) if prisoners who have already announced just continue to slosh souls around (give and take souls rather than just give). This is because a random walk in a fmite space will eventually get everyplace. We will use this fact extensively later on.
Another strategy is that each prisoner who is not a soul-collector has a (very small) chance each day he enters of becoming one. After a number of visits to the room as a soul-collector he gives up and goes back to being an ordinary soulgiver. Any prisoner who has already announced always gives souls and never collects.
Can you think of a variation where the best strategy is worse than exponential in the number of prisoners?
13. Random times, message
from one to one-We have
two prisoners, one of whom is
trying to send a message to the other. The transmitting prisoner encodes the message as a natural number, M. He tries to give the receiving prisoner M souls. The problem now is how the receiver knows when the message has been sent-how does she know when she has received all of the souls? To do this, she occasionally puts a soul back into the room when she finds it empty. Hopefully the transmitter, having dropped all of his souls, will take the last soul back-thus indicating that he is finished. The receiver will then see that the soul has been taken and know that all of the souls have been sent, because the transmitter will only pick up a soul when he is done.
To do this reliably, the two prisoners behave as follows:
The transmitter drops all of his souls until he has none left. When he has no souls left he will take one soul from the room if he can. When he has one soul he will drop it in the room if he can.
The receiver takes every soul that she can, although she occasionally drops one back in the room ("pings''). If when she next enters the room she finds that the soul she has dropped has been taken, then she knows that the transmitter is finished and so knows the total number of souls sent.
14. Random times, messages
from many to one-For n prisoners transmitting and one receiving, the transmitters all behave as in variation 13. First suppose that the receiver wants to know the sum of the numbers of the transmitters, M.
This time the receiver occasionally tries to drop n souls back at the same time. The only way that all n pings will be taken is if all n of the transmitting pris-
oners are finished. When she succeeds then her maximum
value was the sum of the numbers of the transmitters, M.
What happens is that the receiver's total collected souls usually increases, but never falls back as much as n from the current all-time maximum unless that maximum is the total number of souls being transmitted. When a transmitter finishes, the receiver's total is allowed to slosh back by one more than before. When all transmitters are finished, then the receiver's to
tal will slosh between M and M - n, and when she sees both extremes in that order then she knows it is done.
Knowing the total, M, is enough to allow all the n transmitters to send arbitrary messages. Choosing base 2, give the i-th transmitter digits i,i + n, i + 2n, . . . in which to encode his message.
15. Random times, messages
from many to many, 2 light
bulbs-We can use the solution to variation 13 together with a way to pass around who is transmitter and who is receiver. To be precise, they use lightbulb one just as in 13. Some prisoner is chosen to be the first transmitter. We assume lightbulb two is on to start with. Whoever turns it off (picks up the "listening stick") is the first receiver. The transmitter sees that the listening stick has been picked up, and starts transmitting on lightbulb one. When the receiver knows the message is done, he puts down the listening stick and becomes the new transmitter. The new receiver is whoever next picks up the stick The prisoners keep sending messages around (without knowing whom they are transmitting to), and eventually each prisoner collects all the messages.
16. Random times, messages
from many to many, ! light
bulb-See the Dber-Dber-theorem below.
Ober-Theorem We will now give our method for variations 8, 9 and 10.
It turns out that each prisoner can
transmit an arbitrary message to all of the other prisoners, using only the one light.
We will start with one prisoner transmitting one bit to every other prisoner. If the transmitter wants to send a O-bit, then on any even-numbered day he leaves the lightbulb on
and on any odd-numbered day leaves the lightbulb off. If he wants to send a 1-bit, then on any even day he leaves the lightbulb off and on any odd day he leaves the lightbulb on. Every other prisoner leaves the lightbulb off. Now any prisoner who finds the lightbulb on
when he enters the room will know for sure that the transmitted bit is a 0 or a 1, depending on whether the previous day was even or odd. Every prisoner will fmd the lightbulb on at some time (with probability 1 ), and so will receive the message. Of course, there is nothing special about even and odd days. Any bijection between N and { 0, 1 } X N would do just as well. For example, j � (O,k) would mean that day j is the kth O-bit day. Those days whose number correspond to (O,n) are "even days" and those which correspond to (l,m) are "odd days."
To send two bits, divide the days into four sets. In other words, provide a bijection between N and {0, 1 } X {0, 1 } X N. The first bit is represented by the first two types of day, 0 and 1
mod 4 say, and the second bit by the other two types of days, 2 and 3 mod 4 say. Any prisoner who finds the lightbulb on will know for sure one of the bits being transmitted.
To transmit a message of arbitrary length, provide a bijection between N and {0, 1 } X N X N.
To allow every prisoner to transmit a message to every other prisoner, first divide the days among the prisoners
(so that each is allocated an infinite number) and then run the above algorithm with prisoner k transmitting on days which are allocated to her. For M prisoners, this can be thought of as given by a bijection between N and { 1, · · · , MJ X {0, 1 } X N X N.
To speed up transmission, if another
VOLUME 25, NUMBER 4, 2003 57
prisoner knows a given bit in one of the
messages being transmitted then he can
retransmit this bit by acting as the trans
mitter would-"echoing" the message.
Ober-Ober-Theorem We will now discuss a method that
allows each of the prisoners to send a
set of arbitrarily long messages, one to
each other prisoner. We assume fur
ther that we are in the setting of vari
ation 11 (Random visiting times), and
hence that the prisoners have no time
reference other than the order in which
events occur. Unlike all the variations
discussed up until now, this one could
not be solved using direct modifica
tions of strategies 1 or 2. One of the au
thors (D.F.) came up with what we
think is an original strategy.
The idea of the method
• The n prisoners will have agreed
upon an ordering among them ahead
of time.
• Prisoner 1 will be the observer, look
ing at the system formed by all the
other prisoners (and the lightbulb ).
Those non-observers will be called
robots because they will follow a
simple rule.
• Before starting his rule, the first
transmitter, say prisoner n, intro
duces 0 or 1 souls into the system.
• The observer will try to deduce how
many souls were originally intro
duced from the behavior of the ro
bots. For this, prisoner 1 has differ
ent procedures at his disposal:
-Two testing procedures Po, P1 that allow prisoner 1 to conduct experiments. He is trying to answer positively to one of the two questions Q0,Q1: "Did prisoner n introduce I souls (i = 0 or 1) in the system?" However, both Po and P1 can only produce positive results, or be inconclusive. Hence prisoner 1 will only answer negatively to Q1 when Po is conclusive.
-A resetting procedure that allows prisoner 1 to set the system back to its original position (the number of souls in the system is as
the transmitter left it). This al-
58 THE MATHEMATICAL INTELLIGENCER
lows him to proceed with additional experiments.
The two testing procedures will even
tually give an answer to the observer.
• Now prisoner 1 triggers prisoner 2
into an observing phase. That is, they
(more or less) exchange roles, and
prisoner 2 becomes an observer,
while prisoner 1 starts following a
simple rule and so becomes a robot.
Eventually, from the experiments he
will conduct on the system formed by
the other prisoners, prisoner 2 will
find out which bit prisoner 1 left in the
system and then become a robot.
• This continues, cycling through all
the prisoners. We have each prisoner
i sending a first bit to prisoner i + 1
mod n, then all of them sending a
second bit, etc . . . .
• Using intermediates, any prisoner
can send a message to any other, and
not only to his follower in the or
dering.
The simpler case n = 3 We now describe each step in full for
the case n = 3. Simple rules. The behavior of the
prisoners who are not currently observ
ing will be given by the directed graphs
cpk> with k a positive integer (see diagram
1 ). These graphs describe the number of
souls each prisoner is eager to have at
any time, and hence determine whether
he wants to drop or grab a soul each
time he enters the room. The graphs are
to be read left to right, and considered
to repeat (the dashed line). At any time
where more than one option is offered,
the prisoner chooses which option to try
with equal probability.
k·l
-----l�f-Tr-lgg_•• __
k·2 ------.3-�---
Diagram 1
slosh
To start with, one of the robots will
follow cpko and one will follow cpk1• As
sume that ko is big, and k1 is bigger.
This will be made precise later. We play
the role of prisoner 1, and (for now)
only observe prisoner 2 (and 3) running
the instructions cpko (resp. I{Jk1). More
precisely, when we get a chance to go
into the room, we note whether the
state of the lightbulb has changed from
the last time we were there (what we
call ajlickering). If the total number of souls in the
system is 0 (remember we include the
lightbulb in the system!), nothing can
happen because both prisoners are ea
ger to get more souls, but none are
available. If the total is 1, the lightbulb
might be switched on and off some
small number of times (if the prisoner
who starts with the soul is initially ea
ger to get rid of it), but eventually one
of the two prisoners will have 1 soul
and be eager to have 2, and the other
will have 0 and be eager to have 1. So
the situation will stall after a finite
number of flickerings. Similar stops
will occur if there are 4 or 5 souls in
the system.
On the other hand, if 2 souls are
available in the system, the system
might stop in a situation where each
has 1 soul and is eager to have 2, but
more importantly, the lightbulb might be turned on and off an arbitrarily
large number of times, if they both
keep going through a sequence 2, 1, 0,
1, 2, 1, 0, . . . (with a delay in their
phases). The lightbulb is then said to
flicker indefinitely. The same thing
could happen if there are 6, 7, . . . souls
in the system. Finally, in the case of 3 souls, the system might produce indef
inite flickering in the lamp in a more
complicated fashion.
This behavior is summarized in the
accompanying chart.
Number of souls
in the system
6, 7, . . . , k0 + k1 4, 5
2, 3
0, 1
Indefinite
flickering
possible
impossible
possible
impossible
It is also worth noting that there ex
ists an integer M such that if there are
0, 1, 4, or 5 souls in the system, the sys-
tern will stall after fewer than M flick
erings. Hence, obse:rvi.ng M + 1 flick
erings will guarantee that we are not in
any of the cases 0, 1 , 4, or 5, what we
call a positive result.
Experimentation. Assume the sys
tem contains either 0 or 1 soul, and con
duct one of the following procedures:
P1 Add 1 soul to the system. Wait for a positive result for some time. If this positive result arrives, return Yes, otherwise return unknown.
Po Add 3 souls to the system. Wait for a positive result. If this positive result arrives, return Yes, otherwise return unknown.
The waiting times should be taken so
that we can potentially observe at least
M + 1 flickerings and hope to get a
positive result.
We have the following chart of out
comes:
# of souls 0 originally
# of souls 3 4 after adding· step in Po
Possible unknown, Yes unknown outputs
for Po # of souls 2
after adding-step in P1
Possible unknown unknown, Yes
outputs for P1
Hence, a positive result to Pi guaran
tees a positive answer to Qi.
Assuming that we did not get a con
clusive result, we would certainly like
to run further experiments, but the sys
tem has probably stalled. What should
we do now?
Resetting. If we could return the
system to its original state with 0 or 1
soul (as set up by prisoner 3), we could
experiment further. To do this, we
would like to take souls out of the sys
tem. It seems hopeless, if for instance
one robot has no souls, the other has
5, and they are both eager to have
more. But if we are ready to give them
some, they will eventually have 6 and
be willing to drop the souls again. We
can then grab those leftovers, until we
are back to the initial number (0 or 1).
This allows us to conduct other exper
iments, and hence to determine even
tually whether prisoner 3 left behind a
soul or not. Note that we never have to
raise the number of souls added to the
system to more than 12 to get it mov
ing again, because with 12, at least one
robot prisoner is at the start or into his
"slosh" region, and is willing either to
take or give souls.
Triggering. Now that we know what
the bit sent by prisoner 3 was, we pre
pare our message for prisoner 2 by set
ting the total number of souls to 0 or 1 .
After that, we would like to signal pris
oner 2 to start his role of observer. This
is where the numbers ki come into play.
Prisoner 2 has agreed beforehand
that he will be "triggered" when he has
exactly 18 souls (ko = 18). Note first
that we never needed to go that high
during our experimentation phase (we
needed to go at most up to 12). So we
can be sure that we have not triggered
prisoner 2 before now. We drop those
18 souls in the system, and then start
to apply the rule 'Pk2 for some kz agreed
on ahead of time, bigger than k1.
We now have 18 or 19 souls in the
system, and each prisoner is running a
rule 'P*· We only have robots running
the place! So the whole system evolves
according to a random walk. Since there
are only 18 or 19 souls, there are finitely
many possible states. Moreover, we
know that one of the prisoners has at
least 6 souls, and hence the option of in
creasing or decreasing his number of
souls. This guarantees that our random
walk never stops, and there is a non
zero probability of getting from any
state to any other state. Hence prisoner
2 will eventually end up with 18 souls.
Now that he has his 18 triggering
souls, prisoner 2 just needs to erase
them in his mental count of souls. He
is back to 0 souls, and there might be
1 soul left somewhere else in the sys
tem. He becomes an observer and his
situation is similar to the one enjoyed
by prisoner 1 at the start.
In the case of 3 prisoners, we can
actually take ko = 18, k1 = 20, kz = 22,
k3 = 24, · · · , and in general the k's will
grow incrementally by 2 each time. The
only requirements are that they be big
enough that with that many souls in the
system (or one more if the message is
a 1) the system never gets stuck (when
all prisoners are robots), and that pris
oners are not triggered too early when
one is trying to trigger someone else.
Increments of 2 give just enough lee
way so that the 1-bit message doesn't
set someone else off too early.
Cycling. Now the prisoners just
have to cycle through that algorithm,
and give further bits to the prisoner fol
lowing them in the ordering. This will
eventually allow them to exchange ar
bitrarily long messages with the other
prisoners too.
The case of more prisoners
We would like first to identify the im
portant properties that the rules 'Pk have that allow the algorithm to work.
Really all we care about is the behav
iour of the system as a whole. Specifi
cally, we want it to behave in different
ways depending on the number of
souls in the system, as shown in dia
gram 2. In the case of 3 prisoners, the
test for P1 is done at the boundary be
tween 1 and 2 souls and the test for Po
at the boundary between 3 and 4 souls.
The maximum number of souls that the
observer needs to add to the system to
reset it is 12. Also, in sending the first
bit, the trigger value k is 18.
trigger -
upper bound for resetting -procedure
test for P.
test for P, -
will run indefinitely
can run indefinitely
will stop after at most M flickers
can run indefinitely
will stop after at most M flickers
Diagram 2
0
Note that in the case of n = 3, the
fact that all the rules used are of the
same type is not really important. In the
general case, we will have n - 1 types
of rule, all with different trigger values
k, and we require the triggering prisoner
to adopt the same type of rule as the
one the triggered prisoner is running.
VOLUME 25, NUMBER 4, 2003 59
Trigger
hn-3
h, ---,����--�----�����---h, --�������----��·�·r-�---- � r 0 --�-�-�-�-�-�-�-�---L-�-���-----------L---�---�-�-�-----
fl'k,O �T " :T _� ;T
� 9'k,2
He knows which rule is running just by
counting the number of cycles all the
prisoners have gone through.
The rules. A set of rules that will
work for general n is shown in dia
gram 3.
We set h1 to 2, and the other hi are de
fined recursively, so that � ;:::: 2 Ij:} hi" The value T refers to a number of soul
exchanges required to cover that sec
tion of the graph, rather than the num
ber of peaks. Take the number of peaks
si to be such that the total "length"
2sihi ;:::: T. The value of T will be spec
ified later.
All experimentation happens for
values of souls less than H, so once a
prisoner starts up on the long journey
towards H, she will never be able to
come back down until the observer
wants to reset the system. H has to be
set larger than hn-2 + I�:;}hi (the sum
of the highest peak in each 'P*, i) so that
the normal running of the system, with
other prisoners on their zig-zags down
below will not bring an escapee to H
and allow him to go back down. Again,
we defme the exact value of H a little
later, but assume for now that it is big.
The trigger values k are different for
each prisoner. The algorithm will work
with k ;:::: (n - 1)H so that with that
many souls in the system, at least one
prisoner is into his slosh range and
therefore the system cannot get stuck
when that many are added. They need
only increment by 2 each time, as in the
case for 3 prisoners. To trigger the pris
oner running rule 'Pk,*' simply add k souls, and your message (0 or 1), and
become a robot. As in the case of 3
60 THE MATHEMATICAL INTELLIGENCER
Diagram 3
prisoners, the random walk will even
tually end up with the prisoner being
triggered on k souls, and all other pris
oners have triggers of at least 2 more
than k and so will not be triggered pre
maturely.
Clearly this system will not run in
definitely with 0 souls. It is also clear that
it might run indefinitely with I.'f:12 hi souls. Here is one sequence that, if fol
lowed, will run forever: Call the pris
oner applying the rule 'P*,i robot i. To
start, set robot 0 to be at the bottom of
any valley on his cycle, just before a
peak of altitude hj, and give to each ro
bot i exactly hi souls (necessarily start
ing at the peak of his cycle). If robot j
gives his hj souls to robot 0 and then
takes them back, we are in a similar po
sition to the one we started with. We
can continue doing this indefinitely.
We need to show that the system
gets stuck for some higher value of
souls. This will require us to prove that
no proper subset of the robots can run
indefmitely, if there are fewer than H
souls in the system, which is proved
later on. Given this, it only takes one
robot on his way up to H to stall the
system. We can ensure this will happen
by putting in hn-2 + I�:[ hi + 1 souls.
We can now take H to be any number
larger than this number, say hn-2 +
I�:[hi + 2. The power of the Collective. We
now show that with fewer than H
souls, the system cannot run indefi
nitely if not all robots are involved.
Assume the system is running indefi
nitely with a minimal number of souls
changing hands. Once a robot starts
k Trigger
s }·� H
hn-2
hn-3 hn-4
h, h, 0 .....
lPk,n-2
up towards H, she can only take souls
and never return them. By minimal
ity, she never takes any new souls
and might as well not be there. So
we can assume that our subset of ro
bots must be able to run indefinitely
without anyone leaving towards H or
giving souls to any robot going to
wards H. First, we will assume that 'P*,o is
missing. Assume a subset not including
'P*,o runs; then there is a minimal sub
set not including 'P*,o which runs.
Now, let m be the largest number
such that 'P*,m is included in this sub
set. As the subset is minimal, 'P*,m must
complete a full cycle, for if it did not,
then we could simply leave robot m out.
Thus, at some time robot m must have
hm souls. However, by the choice of the
sizes of the peaks hm > Ir;,:(/ hi, it is
clear that he can never get rid of them
all without pushing one of the other ro
bots onto its path towards H. As there is a finite number of ini
tial states (looking only at the robots
below their peaks hi), there is a
global bound on the number of ex
changes, L, which can occur before
the system halts. T is chosen to be
larger than L. So we are left with the case when
'P*,o is included in the subset, but some
'P*,m is missing. As 'P*,o is included, it
must complete a full cycle (otherwise
it could be left out and we would have
the previous case). Once it has reached
the beginning of the series of peaks of
height hm it will, for at least the next T soul-exchanges, behave exactly as
'P*,m did in the previous case. But this
subsystem is guaranteed to stop before
cp*,o finishes its height-hm peaks, as for
at least the next T transitions this
subsystem behaves exactly as in the
previous case. Thus cp*,o will never
complete its hm peaks, and so never
complete a full cycle.
Epilogue. We have now proved ex
istence of a strategy. To apply this
strategy, we would need to calculate
precisely the value of several constants
used in our algorithm. For instance, the
constants T and M and the values at
which we test for Po, P1 are hard to
find, particularly within the hour that
the warden has given us.
Acknowledgements and Sources The origins of this problem are not
clear. According to legend [6,7), similar
problems have been the delight(bulb)
of Hungarian mathematicians. The first
written occurrence we could find of
the problem involving 100 prisoners
and a unique lightbulb is on an online
A U T H O R S
PAUL-OLMER DEHAYE
Department of Mathemahcs
Stanford Universtty
Stanford, CA 94305·2125
USA
e-mail: [email protected]
forum hosted at Berkeley [8). Another
variant involves 23 or 24 prisoners, two
lightbulbs, and the obligation that the
prisoners always change exactly one of
the lightbulbs. This one appeared on the
Ponder This website of IBM Research
[7). Those two online occurrences, on
the Berkeley forum in February 2002
and the IBM website in July 2002, were
followed by several others, in either of
the two versions. A thread was imme
diately started on rec. puzzles [5]; it was
published in the Fall issue of the Math
ematical Sciences Research Institute
newsletter Emissary [3], and the prob
lem was finally posed on the popular ra
dio show "Car Talk" [6) !
We thank Andrew Bennetts for in
troducing us to the original problem.
REFERENCES
[1 ] Bertram Felgenhauer. 1 00 prisoners and a
lightbulb. Newsgroup rec. puzzles, available
through http://groups.google.com, July 28
2002.
DANIEL FORD
Department of MathematiCS
Stanford Universtty
Stanford, CA 94305-2 1 25
USA
e-mail. [email protected]
(2] William Feller. An introduction to probability
theory and its applications. Vol. I, pages
46,59. Second edition. John Wiley & Sons
Inc . , 1 968.
[3] Mathematical Sciences Research Institute.
Emissary newsletter, November 2002. Also
available at http://www.msri.org/publications/
emissary/.
[4] Renata Kallosh and Andrei Linde. Dark en
ergy and the fate of the universe. 2003.
http:/ /arxiv.org/abs/astro-ph/030 1 087.
[5] "Oieg". 1 00 prisoners and a lightbulb.
Newsgroup rec.puzzles, available through
http://groups.google.com, July 24 2002.
[6] National Public Radio. Car Talk Radio
Show. Transcription available at http://
cartalk.cars.com/Radio/Puzzler/Transcripts/
200306/index.html
[7] IBM Research. Ponder This Challenge.
http :/ /domino. watson . ibm .com/Comm/
wwwr _ponder.nsf/challenges!July2002 .html,
July 2002.
[8] William Wu. Hard riddles. http ://www.ocf.
berkeley. ed u/ -wwu/ridd les/hard . shtml
#1 OOprisonerslightBulb, February 2002.
HENRY SEQERMAN
Department of MalhematJCS
Stanford Untverstty
Stanford. CA 94305-21 25
USA
e-mail: segerman@stanf()(d.edu
Paui-Oiivter Dehaye, Daniel Ford, and Henry Segerman are graduate students - indeed, all three started work on their doctorate at he same time, in September 2001 .
Oehaye. a Belgtan, did his undergraduate work at the Unrversite Ubre de Bruxelles. His specialty Is number theory. His tastes run to htking, traveling, beer. and chocolate.
Ford. an Australian, was an undergraduate at Sydney but makes special mentiOn of the Australian National Mathemattcs Summer School. He hkes all sorts of mathemattcs. ncluding algonthms and fast anthmetic. Hobbtes include juggling and htking.
Segerman was an undergraduate at Oxford. He specializes in low·dlmenstonal topology. but not to the exclusiOn of juggltng, the game of GO. and mathematical (or mathematics-Inspired) art.
VOLUME 25, NUMBER 4, 2003 61
lOAN JAMES
Autism Mathematic ians
he cause of autism is mysterious, but genetic factors are important. It takes a variety
of forms; the expression autism spectrum, which is often used, gives a false impres-
sian that it is just the severity of the disorder that varies. Different people are affected
in different ways, but the core problems are impairments of communication, social
interaction, and imagination. Mild autistic traits can pro
vide the single-mindedness and determination which en
able people to excel, especially when combined with a high
level of intelligence. This is particularly true of those with
the autistic personality disorder known as the Asperger
syndrome.
The Asperger syndrome is recognisable from the second
year, although not obvious until later, and endures through
out life. About half of those who have it succeed in mak
ing a success of their lives; the others find it too much of
a handicap. Very briefly, the criteria for Asperger's include
severe impairment in reciprocal social interaction; all-ab
sorbing narrow interests; imposition of routines and inter
ests on self and others; problems of speech, language, and
nonverbal communication; and sometimes motor clumsi
ness. The casual observer may notice an aversion to direct
eye-contact, peculiarities of expression, difficulty in coping
in social situations, and an obsession with a particular sub
ject, such as computer science. The syndrome is not un
common: more than one person in a thousand may have it.
The recent guide [ 10] by the psychiatrist Christopher Gill
berg is a good introduction to the subject.
Hans Asperger was a Viennese paediatrician who, in his
doctoral thesis [2] of 1944 (see [8] for a translation), de
scribed how among the people he had examined there were
a large proportion whom he regarded as mildly autistic but
who were otherwise remarkably able. He was struck by the
fact that they usually had some mathematical ability and
62 THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
tended to be successful in scientific and other professions
where this was relevant:
To our own amazement, we have seen that autistic indi
viduals, as long as they are intellectually intact, can al
most always achieve professional success, usually in
highly specialized academic professions, often in very high
positions, with a preference for abstract content. We found a large number of people whose mathematical ability de
termines their professions; mathematicians, technologists,
industrial chemists and high-ranking civil servants.
Asperger went on to write:
A good professional attitude involves single-mindedness as well as a decision to give up a large number of other
interests. Many people find this a very unpleasant deci
sion. Quite a number of young people choose the wrong
job because, being equally talented in different areas, they
cannot muster the dedication to focus on a single career.
With the autistic individual the matter is entirely dif
ferent. With collected energy and obvious corifidence and,
yes, with a blinkered attitude towards life's rich rewards, they go their own way, the way in which their talents have directed them since childhood.
Only a few years ago it emerged that essentially the same
phenomenon had previously been described by the Rus-
sian neurologist G. E. Saucharewa under the name schizoid
personality disorder. It was a considerable time before As
perger's research attracted much attention, but when it did
the term Asperger syndrome was introduced to describe
the kind of people he was referring to. Although there have
been changes in the definition, the description is still used
for a high-functioning variant of autism with predominantly
good language and intelligence and better social insight
than other forms of autism. A recent survey [4] of Cam
bridge University undergraduates confirmed the impres
sion that a much higher proportion of Asperger people is
to be found among the students of mathematics, physics,
and engineering than students of the humanities. It seems
likely that whereas in the past many people with Asperger
syndrome were particularly attracted to professions where
mathematical ability was an advantage, nowadays ability in
computer science has become equally important if not
more so.
Possible cases of the syndrome can be found through
out the arts and sciences. For instance, the painters Kandin
ski, Turner, and Utrillo, the
to have had Asperger syndrome? Michael Fitzgerald [6] has
argued the case of Ramanujan, and, with M. Arshad [ 1] , that
of the Nobel Laureate John Nash. Banach and Riemann
might also be considered. Among mathematicians with the
syndrome alive today, one (see [3]) has been awarded the
prestigious Fields medal. It seems easier to find manic-de
pressive mathematicians [ 12]-for example, Abel, Sylvester,
and Cantor-although these are more common in the arts
than the sciences. However, it is the association of the syn
drome with mathematical ability, observed by Asperger him
self, which makes it of special interest.
There is some doubt as to whether there is a sex dif
ference. Women appear to be less seriously affected by the
syndrome than men, and perhaps are less likely to present
themselves for assessment. Simon Baron-Cohen, the psy
chiatrist who heads the autism research centre at Cam
bridge University, believes that autistic adults show an un
usually strong drive to "systematise" the world around
them. Even in normal populations, men are more prone to
systematise than women; conversely, women are more able
to empathise than men. Social
composers Bartok and Bruck
ner, the philosopher Wittgen
stein, the chemist Marie Curie
and her elder daughter the
atomic physicist Irene Joliot
Curie have all been suggested.
In fact Asperger himself went
" It seems that for success interaction usually depends on
empathy, although the autistic
often learn to compensate for
the lack of it and succeed in
presenting the appearance of
normal interaction.
i n science or art a dash of autism is essential . "
so far as to conjecture: "It seems that for success in sci
ence or art a dash of autism is essential. For success the
necessary ingredient may be an ability to turn away from
the everyday world, from the simple practical, an ability to
rethink a subject with originality so as to create in new un
trodden ways, with all abilities canalised into the one spe
ciality."
Retrospective attempts at diagnosis are inevitably some
what speculative; the information on record does not an
swer all the questions that would be asked in a clinical in
vestigation today. Bearing this in mind, there does not seem
much doubt (see [ 14], [ 16]) that among physicists Newton,
Cavendish, Einstein, and Dirac had the Asperger syndrome;
in fact Newton appears to be the earliest known example
of a person with any form of autism (a convenient outline
of Newton's life has been given by Milo Keynes in [17], but
the articles of Michael Fitzgerald [6] and Anthony Storr [23]
are most relevant). It seems to be widely accepted that Ein
stein had Asperger syndrome, although none of the many
detailed biographies mentions this. Since autism became
generally recognised by psychiatrists only within the last
sixty years, there must be numerous past cases which have
gone unrecognised, although it may seem surprising that
even recent biographers should pass over what must be one
of the main features of the life-stories of their subjects.
Newton and Dirac can reasonably be counted as mathe
maticians, although they are generally classed as physicists;
Cavendish and Einstein also made extensive use of mathe
matics. What other well-known mathematicians are likely
It is probably impossible for
the non-autistic to understand what it must be like to be
autistic, but the personal studies of college students with
autism collected by the anthropologist Dawn Prince
Hughes [20] give some idea. Asperger people who write
about their experience, as several have done, describe the
great feeling of relief they experienced at discovering they
were not "from another planet" (one of the Web sites is
called Oops . . . . Wrong Planet!), but that there were many
others in the world just like themselves. The Internet and
its many chat groups dedicated to people diagnosed with
autistic spectrum disorders has encouraged the rapid
growth of a thriving community, where normal social con
tact is unnecessary.
Not all psychologists recognise the Asperger syndrome
as a distinct condition in the autistic spectrum; even those
that do may still prefer different terminology, such as
"autistic psychopathy" or "autism spectrum disorder." Oth
ers, such as Anthony Storr [22], prefer to use the term
"schizoid personality" for a condition which seems, to the
lay person, to be somewhat similar. Although certain of the
symptoms can be alleviated, there is no cure for the As
perger syndrome, and some of those who have it, such as
Luke Jackson [ 13], say that on the whole they are glad of
this (one of the e-mail groups is called AS-and-proud-of-it).
What would be appreciated is more understanding of their
difficulties from other people, such as fellow-students,
teachers, and colleagues, so that their lives are not made
unnecessarily difficult. The syndrome is not properly un
derstood by otherwise well-informed people, who find it
VOLUME 25, NUMBER 4, 2003 63
hard to realise what some of those who have it may be ca
pable of achieving.
Francis Galton, in his well-known book on Hereditary
Genius [9], discusses the tendency for intellectual dis
tinction to run in families. There is some evidence that
mathematical ability is inherited;
bly the old stereotype has lingered on in the case of math
ematicians.
Why do people who interview students sometimes
claim that they can spot a mathematician the moment he
or she enters the room? Why are mathematicians, along
with computer scientists, com
the case of the Bernoullis seems
exceptional, but one might also
instance the Artins, the Ascolis,
the Birkhoffs, the Cartans, the
Knesers, the Neumanns, the
Noethers, the Novikovs and
many more. Of course this may
be partly a matter of upbringing
(although a number of the great
The precocious usual ly excel , at an early age , either in mathematics , languages , or music.
monly regarded as loners and
placed in a group with geeks
and nerds? Could it be that the
type of personality which in
clines people towards mathe
matics has something to do
with this? And could it also be
that here is part of the explana-
mathematicians, including Banach, d'Alembert, Hamilton,
Kolmogorov, and Newton, were adopted or fostered).
Even so, there may be a genetic factor at work, possibly
causing a disposition towards abstract thought and visual
thinking (Temple Grandin explains what this means in
[11]).
According to Camilla Benbow [5], American high school
students with exceptionally high mathematical or verbal
reasoning ability are more likely to be myopic, left-handed,
or allergic than are students generally; the difference is
most striking in the case of myopia, which occurs four
times as often. Myopia affects the personality as well as the
eyesight (Patrick Trevor-Roper describes some famous my
opes in [24]). Among the great mathematicians of the past,
Sophus Lie, Henri Poincare, Tullio Levi-Civita, and Emmy
Noether were strongly myopic. Other ocular defects, such
as cataracts, do not appear to be particularly common
among mathematicians. There is certainly a genetic factor
in myopia; and it has been suggested that the condition may
be genetically related to autism.
The precocious usually excel, at an early age, either in
mathematics, languages, or music. Some famous mathe
maticians who had such a gift for mathematics include
Abel, Jacobi, Galois, Borel, Wiener, and von Neumann. Oth
ers were also calculating prodigies, for example Euler,
Gauss, Hamilton, Poincare, Ramanujan, and Banach. Such
savant skills [12] are often related to autism, but are more
striking when they occur in individuals of generally low in
telligence.
There is an extensive literature concerning the psychol
ogy of mathematical ability in schoolchildren. For example
Thomas Sowell [21] writes about exceptionally bright chil
dren who are also exceptionally slow to develop the abil
ity to speak, which he calls the Einstein syndrome. Ac
cording to V. A. Krutetskii [18], a hundred years ago it was
believed in the United States that gifted children were in
ferior to ordinary, normal children in every respect except
intelligence. Gifted children were alleged to be physically
weak, sickly, unattractive, emotionally unstable, and neu
rotically inclined. Subsequent study by psychologists not
only failed to confirm this but led to the establishment of
what was in almost every way the opposite picture. Possi-
64 THE MATHEMATICAL INTELLIGENCER
tion for the difference in the rei-
ative numbers of men and women to be found in mathe
matics? I hope to discuss such questions in another
article, but first would like to hear what readers of The
Intelligencer think about what I have said so far. I would
be particularly interested to hear from people with As
perger syndrome.
REFERENCES
1 . Arshad, M . , and Fitzgerald, Michael. Did Nobel Prize winner John
Nash have Asperger's syndrome and schizophrenia? Irish Psychi
atrist 3 (2002), 90-94.
A U T H O R
lOAN M. JAMES
Mathemahcal lnsbtute
24-29 St. G1les Oxford OX1 3LB
England
e·mall: [email protected]
loan James, F.R.S., was until 1 995 Savilian Professor of
Geometry at Oxford, and 1s now Emeritus: he has also long
association WI h New College Oxford and With the Mathe·
matical Institute. He has held viSiting positions at numerous
other universities, including HaNard, Yale, Princeton. Paris,
Kyoto, Madras, and British Columbia. He is known pnmanly
for his many research publications in topology and his edit·
1ng, including the JOUmal Topology; he also has a continuing
interest tn history or mathematiCS.
2. Asperger, H. Die 'autischen Psychopathen ' im Kindesalter. Archiv
fUr Psychiatrie und Nervenkrankheiten 1 1 7 (1 944), 76-1 36.
3. Baron-Cohen, S . , Wheelwright, S . , Stone, V,. and Rutherford, M .
A mathematician, a physicist and a computer scientist with As
perger syndrome. Neurocase 5 (1 999), 475-483.
4. Baron-Cohen, S . , Wheelwright, S . , Skinner, R . , Martin, J. and Club
ley, L. The Autism-Spectrum Quotient (AQ): Evidence from As
perger Syndrome/ High-Functioning Autism, Males and Females,
Scientists and Mathematicians. Journal of Autisrn and Develop
mental Disorders 31 (2001 ), 5-1 7 .
5 . Benbow, C.B. Possible biological correlates of precocious mathe
matical reasoning ability. Trends in the Neurosciences 1 0 (1 987),
1 7-20.
6. Fitzgerald, Michael. Did Isaac Newton have Asperger's syndrome?
European Journal of Child and Adolescent Psychiatry 9 (1 999), 204.
7. Fitzgerald, Michael. Did Ramanujan have Asperger's disorder or As
perger's syndrome? Journal of Medical Biography 1 0 (2000), 1 67-1 69.
8. Frith, Uta (ed.) . Autism and Asperger Syndrome. Cambridge Uni
versity Press, Cambridge, 1 991 .
9. Galton, Francis. Hereditary Genius. Macmillan, London, 1 869.
1 0. Gillberg, Christopher. A Guide to Asperger Syndrome. Cambridge
University Press, Cambridge, 2002.
1 1 . Grandin, Ternple. Thinking in Pictures. Vintage Books, New York,
1 996.
1 2 . Herrnelin , Beate. Bright Splinters of the Mind. Jessica Kingsley,
London, 2001 .
CAMBRIDGE
1 3. Jackson, Luke. Freaks, Geeks and Asperger Syndrome: a User
Guide to Adolescence. Jessica Kingsley, London, 2002.
1 4 . Jarnes, loan. Singular Scientists. Journal of the Royal Society of
Medicine 96 (2003), 36-39.
1 5. Jarnes, loan. Remarkable Mathematicians. Cambridge University
Press, Cambridge, and Mathematical Association of Arnerica,
Washington, DC, 2002.
1 6 . Jarnes, loan. Remarkable Physicists. Cambridge University Press,
Cambridge, 2003.
1 7 . Keynes, Milo. The personality of Isaac Newton. Notes and Records
of the Royal Society 49 (1 995), 1 -56.
1 8. Krutetskii, V.A. The psychology of mathematical abilities in school
children, (ed. by Kilpatrick, J. and Wirzup, 1 . , trans. by Teller, J . ) .
University of Chicago Press, Chicago, IL, 1 976.
1 9. Pickering, George. Creative Malady. George Allen & Unwin, Lon
don, 1 974.
20. Prince-Hughes, Dawn (ed.). Aquamarine Blue: Personal Studies of Col
lege Students with Autism, Ohio University Press, Athens OH, 2002.
21 . Sowell, Thornas. The Einstein Syndrome. Basic Books, New York,
NY, 2001 .
22. Storr, Anthony. The Dynamics of Creation. Martin Seeker and War
burg, London, 1 972.
23. Storr, Anthony. isaac Newton. British Medical Journal 291 (1 985),
1 779-1 784.
24. Trevor-Roper, Patrick. The World Through Blunted Sight. Allen
Lane, London, 1 988.
New from C a m bridge i n 2 004 With a 3 3 % reduction in subscription prices as of 2004
COM POSITIO MATH EMATICA Foundation Compositio Mathematica london Mathematical Society Managing editors:
Bas Edixhoven, Leiden University Gerard van de Veer, University of Amsterdam
COMPOSITIO MATHEMATICA
Compositio Mathematica provides first-class research papers in
the mainstream of pure mathematics, includ ing such areas as
algebra, number theory, topology, algebraic and analyt ic geometry, and geometric analysis .
In 2004, the pricing is as follows: Institutions print and electronic: £750/S 1 200 Institutions electronic only: £712/S1140 Further information about the journal can be found at: http://www.compositio.ni
To order, please contact Cambridge University Press at tel : +44 (0) 1 223 326070
www.cambridge.org � CAMBRIDGE � UNIVERSITY PRESS
VOLUME 25, NUMBER 4, 2003 65
ii,IM\!Jffij.J§.rblh£111.Jihhl Dirk H uylebro u c k , Ed itor I
A Mathematician in Lhasa Michele Emmer
Does your hometown have any
mathematical tourist attractions such
as statues, plaques, graves, the cafe
where the famous conjecture was made,
the desk where the famous initials
are scratched, birthplaces, houses, or
memorials? Have you encountered
a mathematical sight on your travels?
If 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 : [email protected]
Tibet, Lhasa
I fell in love with Tibet while reading
the account of Fosco Maraini's trav
els. Maraini set out for Tibet in 1937 to
gether with a noted expert on oriental
matters, Giuseppe Tucci. Those were
the years when that far-off and inac
cessible country began to open up to
visitors. Maraini was the photographer
for the expedition and took some won
derful photos. After going back there
in 1948, he wrote an account of his
travels, Segreto Tibet, first published in
1951 [1 ] . He returned again a few years
later and updated his book to the
present day [2]. That first expedition,
in 1937, arrived in Tibet overland from
India along the caravan route for
Lhasa, passing through Sikkim and the
capital, Gantok. They were guests of
the Maharaja Tashi N amgyal, and
Maraini was allowed to meet the Ma
harajah's second daughter, Perna
Choki (Lotus of Joyful Faith), who was
twenty-two years old at the time. "She
was as fascinating as her mystical
name-intelligent, high-strung and
quick-witted. Her hair was jet-black,
worn in a braid (like most Tibetan
women), framing a thin pale face with
eyes that were intense and penetrating,
but could also be soft and languid."
Maraini managed to take an extraordi
nary photo of her.
Looking at that photo of the
princess covering her face with her
hand as she looked up at the sky, I de
cided then and there that one of my am
bitions in life would be to visit Tibet,
even though Sikkim was not Tibet ( al
though the costumes and traditions
were similar), even though that photo
was taken almost seventy years ago,
even though that intact "medieval"
world no longer existed.
All this was in my mind, as was the
dream of seeing "Potala set amongst
its mountains," the great palace of the
sovereign-god, overlooking the city of
Lhasa, the home of the Lama before he
went into exile. The palace was
founded in the 7th century, and en
larged in the 17th century.
The name Lhasa is of Indian origin
and harks back to the legendary palace
of Bodhisattva Avaloki-tesvara (the
present Dalai Lama is held to be his
reincarnation). A wonderful book has
recently been published on the city of
Lhasa and the Potala palace. Titled The
Lhasa Atlas [3], it is entirely devoted
to the traditional architecture of Tibet,
and contains photos and plans of many
buildings in the Tibetan capital.
When I received an invitation to visit
the University of Tibet to take part in
a conference on mathematics educa
tion, I didn't think twice about accept
ing. This trip to Lhasa was not as a
tourist but for work reasons, and I had
the chance to encounter several Ti
betan members of the university teach
ing staff.
On the plane, the sight of those im
mense mountains (6,000 to 7,000 me
ters high) and their glaciers was almost
overpowering. The airport at Gongkar,
about 90 km from Lhasa, is high in the
mountains, and the plane has to zigzag
through the peaks as it comes in to
land.
The bus from the airport takes the
only road; it runs alongside a river, the
Kyichu, which eventually flows into the
Tsangpo, the Brahmaputra. The small
houses here and there along the road
seem to be built of sand and clay.
Nearer the capital, there are a few
more modem buildings, and often sol
diers of the People's Army of China,
standing to attention under their regu
lation sunshades, guard the entrance.
On the outskirts of Lhasa, the road
becomes wider and runs through the
city, passing close to the imposing
Potala palace. Fosco Maraini never got
as far as Lhasa, but mountaineer Hein
rich Harrer lived here for many years.
Famous for his participation in the first
Eiger North Face climb ever, the mem
ber of the Nazi party had to escape
from India to Tibet during the war. He
stayed on and became a close friend of
© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 4, 2003 67
Potala "set amongst its mountains." Photo taken by the author.
the Dalai Lama. Harrer's book Lost
Lhasa [4] shows what Lhasa looked like in the 1940s, how isolated the Potala palace was-a sort of apparition in the valley-and the entrance to the holy city through a sacred gateway. There is also an interesting city map of Lhasa at that time. Later, Harrer wrote the book Seven Years in Tibet on which the film by the same title was based (shot in Peru, though), as well as another book called Return to Tibet [5,6,7].
Today, the road passes in front of the Potala palace, and close by there are an amusement park, an open-air market with dozens of stalls, and a car park Fosco Maraini only saw a documentary about Lhasa, and the Jokhang temple, the spiritual focus of the Tibetan religion, did not impress him. The temple was begun in AD 647
during the reign of Songsten Gampo, and construction continued for many years. What we see today is a more or less 10th-11th-century version of the temple.
The huge bazaar, which is surrounded by a maze of narrow alleyways, is packed with stalls and shops selling mainly religious articles. Here
shoes, called Khata, with their characteristic long pointed shape. Tibetans use them as a sign of respect when welcoming visitors to their homes; they'll give you a pair when you arrive or when you leave.
Before going into the temple, there's a chance to rotate one of the hundreds of prayer wheels along the outside walls and in the courtyards.
Once you get into the building, past the garbage and rats all around it, you see groups of monks reciting the ancient texts and crowds of pilgrims everywhere. The shadowy interior has a mystical atmosphere about it, while at the same time being open to all comers, as befits Buddhism. It is full of visitors from all over the world, but still a very spiritual place. It is here in Jokhang that you understand why the Chinese government attaches such importance to the question of the personality of the Dalai Lama. In June 2002, talks began between Dalai Lama's representatives and the Chinese authorities, with a good chance of reaching an agreement.
Tibet is a combination of a medieval country in many respects with some aspects of modernity and globalization. The Chinese government has made changes since Maraini wrote, "There were no roads, railways, vehicles, or airfields; there was no energy supply of any sort and only traditional medicine was available. Aristocrats and monks shared the task of governing, and reli-
you will find the traditional white silk The Jokhang temple. Photo taken by the author.
68 THE MATHEMATICAL INTELLIGENCER
Prayer wheels outside the Jokhang temple. Photo taken by the author.
gion dominated every aspect oflife." In
any case Lhasa is not all of Tibet.
Mathematics Education in Tibet The vice-chancellor of the University of
Tibet, Da Luosang Langjie, took part
in the conference on mathematics in
Lhasa, and part of his speech was de
voted to mathematics in ancient Tibet
[8]. In particular, he described the math
ematical education of civil servants,
which took place in a special training
school called "Tsikhang Loptra,"
founded in 1751 by the seventh Dalai Lama, Luo Sang Jia Cuo. The aim was
to train these officials to govern the
country. The students came from noble
families, and they had to pass a simple
entrance examination in languages and
mathematics, the latter word being a
modem interpretation of the original
term Kalacakra (meaning, literally,
"wheel of time"). There was no set du
ration for the training courses, which
could last from one to five years.
Students learned to carry out calcu
lations using what were known as
"chips." The various systems of weights
imal system but other number systems
as well. For this reason, an elaborate
conversion table had been devised,
which students had to learn by heart.
Another important institution was
the school of medicine and astronomy,
and measures used in ancient Tibet <9 I() ( 1 ::: � c 2. a were rather elaborate, so conversion
from one to another was extremely
complicated, involving not only the dec- Conversion table, from [8].
called "Mentsi Khang," which was
founded in 1695. Education was based
on traditional medicine, which still
plays an important part in Tibetan cul
ture, and on the calculation of the as
tronomical calendar, an extremely im
portant aspect of social organization,
as it is in so many other countries. As
tronomical calculations were used to
work out the dates for important reli
gious festivals and for the seasons, as
in all cultures. The Tibetan temple with
a school for the future monks of Bei
jing was the Yonghe-Gong, which orig
inally was not a religious institution. It
was built in the 33rd year of the reign
of the emperor Qing Kangxi and was
the official residence of the third em
peror of the Qing dynasty, Yong Zheng,
before he acceded to the throne.
Only in 17 44 did this temple become
a holy place for the cult of the Lama.
Among the many buildings in the tem
ple complex, there are four pavilions
devoted to study: one for the teaching
of Buddhist writings, one for esoteric
Buddhism, one for medicine, and one
for mathematics, called Kalacakra Hall
(mathematics pavilion). It contains
many scrolls with the calculations for
the astronomical calendar [9] .
I had the good fortune to visit both
VOLUME 25, NUMBER 4, 2003 69
70 THE MATHEMATICAL INTELLIGENCER
temples, the one in Lhasa, considered
to be the spiritual focus of Tibetan
Buddhism, and the one in Beijing. I
couldn't help noticing that the Beijing
temple was more like a museum than
a temple, while in Lhasa, religion is still
the focal point of Tibetan life. Although
other cultures are also prominent,
the Muslims, for instance, seem to live
a rather independent life in their
quarters.
The conference gave much impor
tance to the question of mathematical
education, and there were participants
from many countries. There was a spe
cial focus on mathematics education in
Tibet in recent times, too, and the hope
is that this will be the beginning of
The Yonghe-Gong temple. Photo taken by the
author (left). The Yonghe-Gong temple, lay
out (edited by Niu Song, "Yonghe-Gong," Bei
jing, 2001) (below).
1 . .�tl 2 . • il 3. 2;li] 4. IIB•n 5 . •• 6 . •• 7. illi'** 8. *•* 9. i!li..rlllin 10. :a<..rlllifl I I . Jlif!lfl 12. ll!lf*"* 13. Uf:�ll
14. l!l�JII: 15. 1:"11 16. lliWi'JII: 17. Jlif!l'il'JII: 18. �df;ll 19. i!!i£111: 20. :a<£11 21. ��-22. Jl!t-Ettl\ 23. ffl!Jtl 24. JJtlftl 25. Jitlftl 26. 7l<.Jitftl 27. ll*il;llftl\ 28. !!@�it 29. �lilttt 30. i!lillllLIJI\ 31 . *!GiLlltl 32. 1JUUim
!. Arches
2. Imperial path 3. Toilet 4. Gate of Luminous Peace
5. Drum Tower 6. Bell Tower
7. West Stele Pavilion
8. East Stele Pavilion
9. West Outer Gate
10. East Outer Gate
II . Gate of Harmony and Peace
12. Imperial Stele Pavilion
13. Scripture-Lecturing Hall
14. Esoteric Hall
15. Kllacakra Hall
I 6. Bhaisajya Hall
17. Hall ofHannony and Peace
18. Hall of Eternal Blessings
19. West Side Hall
20. East Side Hall
21 . Hall of he Wheel of the Dhanna
22. Initiation Tower
23. Panchen's Tower
24. Hall of Infinite Happiness
25. Pavilion of Perpetual Tranquility
26. Pavilion of Eternal Health
27. Yamudage Tower
28. Hall of Buddha's Light
29. Tower of Peaceful Accomplishment
30. West Shunshan Tower
31 . East Shunshan Tower
1. M 2 . • il 3. H v (flf4) 4. 111':!.t"! 5. u 6 . •• 7. N•* 8. ••* 9. N•M <�mo to. ••P� <�m u. JlfDP" 12. ll!JP* 13 . ... 14. l!l*JI
15. lt!f:JI 16. l!afilt 17. Jlfii�Jit 18. 7!<.�· 19. l§'i.!JI 20. �
21 . l'ir'*JII:
22. lll!rtl 23. rl:.---7'-z :.---ji)! 24. Hila 25. �-llG 26. 7!<..1111 27. "rA�'f!ji)! 28. I!!H.&tt 29. .lit Ill 30. i!illlll!Jtt 31 . JI(MI!Jtl 32. H v (:fff4)
33. i!i;&it 34. �R'l,lfll ·�*:l!liiilll .
32. Paid toilet 33. Ajacang 33. 7�-'1' 7 ""'(1){1
34. �m PLANE FIGURE TO YONG·HE·GONG 34. Souvenir shop
JUIJ!: lJt iii !II The Yonghe-Gong temple (seen from above) with layout (edited by Niu Song, "Yonghe-Gong," Beijing, 2001).
awareness and collaboration between
Tibetan mathematicians and the rest of
the world's scientific community. The
effort to organise a conference, to
gether with the East China Normal Uni
versity, in faraway Shanghai, certainly
was a positive step in the mathemati
cal direction. The presentation of this
unusual mathematical meeting has to
highlight this joint effort. The fact is
that so little is known about the situa
tion and the problems, including edu
cation, in this fascinating country.
Note
The author wishes to thank the refer
ees for the numerous remarks to im
prove this present paper. Still, he is
aware that this report remains a per
sonal testimony of an extraordinary
scientific trip to Tibet, and that it thus
has a rather subjective character; but
is not this often the case when very dif
ferent cultures meet, such the Euro
pean, Chinese, and Tibetan, in the
short period of time of a scientific con
ference? As for any contribution in The
InteUigencer, the journal invites its
readers to add to the statements given
here.
REFERENCES
1 . F. Maraini, Segreto Tibet, Leonardo da Vinci
publishing house, Bari, 1 951
2. F. Maraini, Segerto Tibet, new edition , Cor
baccio publishers, Milan, 1 998
3. K. Larsen & A. Sinding-Larsen, The Lhasa
Atlas: Traditional Tibetan Architecture and
Townscape, Serindia Publications, London
2001
4. H. Harrer, Lost Lhasa, H. N. Abrams, Inc.
Publ . , New York, 1 992
5. H. Harrer, Seven Years in Tibet, The Put
nam Publ. Group, New York, 1 982.
6. H . Harrer, Return to Tibet, F.A. Thorpe
Publ . , London, 1 984.
7 . Seven Years in Tibet, film by Jean-Jacques
Annaud, cast: Brad Pitt, David Thewlis, B.D.
Wong, Danny Denzongpa, script Becky
Johnston, USA, 1 997.
8. From Luosang Langjie 'Mathematics Edu
cation in Tibet: History, Current Situation
and Future Development" in Abstracts,
Satellite Conference on Mathematics Edu
cation, Tibet University & East China Normal
University, Lhasa. 2002, pp. 26-33.
9. "Yonghe Gong," guide to the temple, pro
duced by the temple admin office (in Eng
lish), China Picture-book Publishing House,
Beijing, 1 995.
VOLUME 25, NUMBER 4, 2003 71
FRANCESCO CALOGERO
Cool l rrat iona Num bers and The i r Rather Coo Rat iona Approxi mations
Pretty cool?
1000/998001 = 23 . 53/(36 . 372) = 0. [001 002 003 . . . 009 010 011 . . . 099 100 101 . . . 996 997 999 000]
c10oo - 10-2997 • 999 ool)/9992 = o.o01 oo2 oo3 . . . oo9 010 011 . . . 099 100 101 . . . 997 998 999
Try the following parlor game: The fraction 10/81, when
written out as a decimal number (in the standard base 10)
to the ninth decimal place, reads as follows:
10/81 = 0. 123456790 . . . ' (0. 1)
as even nonmathematical party guests can readily verify
provided they know how to perform elementary divisions,
or have handy a pocket calculator (but in the latter case
they will miss the thrill of seeing the sequence of integers
emerge one by one via the division algorithm). They will
immediately spot the "missing" digit 8 in the right-hand side
of (0. 1). Clearly to correct this "defect" one should subtract
from 10/81 a number of order 10-9, so as to change the last
two digits shown from 90 to 89. Hence you suggest, as an
educated guess, to subtract the number 10-9 (3340/3267);
and you then write out, digit after digit, the resulting deci
mal number, say, up to its lOOth decimal digit, to wit (with
out rounding off of the last digit):
10/81 - 10-9 (3340/3267)
= 0. 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50 51 52 53 54 5 . . . (0.2)
72 THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
(here, and occasionally below, the display of a number on
more than one line is of course merely for typographical
reasons; note moreover that here, and always below, blank
spaces have no mathematical significance; they are merely
inserted as visual aids). Who will believe you? Who will
take the chance to predict the next, say, fifty digits? The
mathematically educated guests are the least likely to be
correct in their reaction and guesses.
The purpose of this paper is to provide an explanation
for this numerology, as well as additional material on num
bers displaying a remarkable pattern when written out in
decimal form.
Main Result I recall the definition of the Champemowne constant-with
thanks to the Referee, who pointed out that this number,
which I had naively invented, was already well known (at
least) since 1933 [1] . (The Referee contributed many other
items in the References, as well.) In the standard decimal base 10 it reads as follows:
c(10) = 0.1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15. . . . (1.1)
Here and throughout the rest of this paper (though not in (0. 1) and (0.2)), dots indicate digits to be filled in by a selfevident rule-in this case, continuation ad infinitum according to the infinite sequence of the integers. This number c(lO) is obviously irrational, for its decimal expansion neither terminates nor becomes periodic. I moreover call it "cool" (as in the title of this paper), in the modem sense of "remarkable, attractive."
The defmition (1.1) calls into play the base used to represent numbers, for instance, the Champemowne constant c(2) written in base 2 reads
c(2) = 0.1 10 1 1 100 101 110 111 1000 1001 1010 1011. 0 0 0 (1.2)
And of course the value of the Champemowne constant in base b, c(b ), depends on the base b (b = 2,3,4, . . . ); for instance obviously 1110 = 0. 1 < c(lO) < 2/10 = 0.2, while 1/2 = 0.5 < c(2) < 1 (here and hereafter all numbers explicitly written out in decimal form are in the standard base 10).
In the next section I obtain the following representation of the Champemowne constant c(b) as an infinite series of rational numbers:
X
c(b) = b(b - 1r2 - (1 - b- 1) I f3k(b)b-ak(b), (1.3a) k=l
ak(b) = kbk - (bk - 1)/(b - 1) k- 1
= Ck - 1)bk + Cb - 2) I b1 + Cb - 1), Ct3b) !=1
f3kCb) = Cbk _ 1)-2 Cbk+ 1 _ 1r2 bk+ 1 [b2k Cbk+ 1 _ 1) - (bk+ 1 - 2)(bk - 1)]. (1.3c)
The series (1.3a) converges very fast when b is large due to the rapid growth with k of the exponent ak(b ), for instance,
a1(10) = 9, az(10) = 189, a3(10) = 2 889, a4(10) = 38 889, a5(10) = 488 889. (1.4)
Note that the explicit representation of the positive integer ak(b) in base b can be read directly from the second expression in the right-hand side of (1 .3b ): it presents the digit(s) of the number k - 1 written in base b, followed by the digit b - 2 repeated k - 1 times, followed finally by the digit b - 1 ; for this sequence of integers, (1.3c) and (1.4), see the entry A033713 in [2]. As for the coefficient f3k(b), it has been defined so that it tends to unity both as k � oo
and as b � oo, and it is indeed close to unity for all (positive integer) values of k when b is large, for instance,
/31(10) = 33 400/29 403 = 23 0 52 0 167/(35 ° 1 12) = (10/9)(3 340/3 267), (1.5a)
f3z(10) = 1 099 022 000/1 086 823 089 = 24 0 53 0 549 51 1/(38 0 1 12 0 372) = (10/9)(109 902 200/120 758 121). (1.5b)
The fact that the series (1.3) converges very fast when b is large implies that its truncation yields rational numbers
that provide very good approximations to the Champernowne constant c(b ), and this entails that these rational numbers, when written out as decimals in base b, feature initially a lot of digits that reproduce the pattern characteristic of c(b ), namely the sequence of the positive integers (written in base b). For instance, calling rn(b) the rational approximation to c(b) obtained by truncating the series in the right-hand side of (1.3a) at its nth term, one gets
r0(10) = 10/81 = 2 · 5/34 = 0. [ 123456790], (1.6a)
as well as (see (1.4) and (1.5a))
r1(10) = 10/81 - 10-8 (334/3 267) = 60 499 999 499/490 050 000 000 = 71 ° 389 ° 2 190 5211(27 0 34 0 58 ° 1 12) = 0. 1 2 3 4 5 6 7 8 [9 10 1 1 12 13
0 0 0 94 95 96 97 99 00 01 02 03 04 05 06 07 08 0]. (1.6b)
In these formulas (and always below), in the decimal expression of a number written out in decimal form the digits enclosed within square brackets are meant to be periodically repeated thereafter (they correspond to the recurring part of the decimal expression), while the interspersed dots, as already intimated above, indicate digits that the reader shall fill in by obvious interpolation (hence in (1.6b) dots stand for the sequence of the positive integers from 14 to 93: we omitted them to save space). The lengths of the non-recurring parts are of course consistent with the simple rule (see, for instance, [3]) according to which the fraction NID, with N and D coprimes, when written out in decimal form in base 10 shows a non-recurring sequence of digits the length of which coincides with the largest one of the two exponents of the primes 2 and 5 in the decomposition as a product of primes of the denominator D. The diligent reader will also verify that the lengths 9 and 198 of the recurring part in the decimal expression ofro(lO) and r1(10), see (1.6a,b), are as well consistent with the general rule, as described for instance in [3] . Also note that these formulas, (1.6a,b ), provide the exact expressions to replace (0. 1,2)-hence the correct solution to the parlor game described at the beginning.
Finally let me call attention to the cool look of the last part of the recurring sequence of integers in the right-hand side of (1.6b)-the part which does not reproduce the pattern of digits of c(10), yet clearly is itself remarkably neat. This is in fact natural, for only a well organized sequence of integers can be expected to reproduce a much longer concatenation of the integers via the addition to r1(10) of the next term in the expansion (1.3a). Indeed this phenomenology is relevant to the overall cool look of all the examples displayed below.
I call numbers such as those just displayed "rather cool rational numbers (of order p, in base b)." Here p is the number of initial digits that coincide with those that correspond to a neat pattern when the number is written out in decimal form in base b (b = 10 in these examples, in which the pattern is the concatenation of the integers). For instance,
VOLUME 25, NUMBER 4, 2003 73
r0(10), see (1.6a), is a rather cool rational number of order 7, while r1(10), see (1.6b ), is a rather cool rational number of order 186 (but I also re-emphasize the overall cool appearance of the recurring part). And the following rather cool rational numbers,
r2(10) = r1 (10) - 10- 187 (1 099 022/120 758 121), (1.6c)
r3(10) = r2(10) - 10-2 886 (1 109 890 222/1 231 853 592 321), (1.6d)
r4(10) = r3(10) - 10-38 885
(123 443 2 1 1 358/1 371 440 348 559 369), (1.6e)
may be expected, on the basis of the expansion formula (1.3), to have at least orders 2 880, 38 880, 488 880, respectively (see (1.4)). Indeed r2(10), when written out in base 10, displays after the decimal point the sequence of the integers from 1 to (and including) the number 997, which is then followed by 999 (rather than 998); hence, according to the definition given above, r2(10) is a rather cool rational number of order 2885, and it reads as follows:
r2(10) = 0.1 2 3 . . . 97 98 [99 100 101 . . . 995 996 997 999 000 001 002 . . . 009 010 0 1 1 . . . 096 097 098 0]. (1.7)
Note again the cool overall appearance, also including the recurring part, which clearly has length 2997 = 3 · 999, while the non-recurring part has length 187 = 9 + 2 · 89.
I end this section by emphasizing that, as clearly implied by the above treatment, a rather cool rational number, in addition to having in its decimal representation an initial long string of integers (or some other neat pattern of digits), should also have a reasonably neat appearance (being cool has to do with appearances!) when written as a single
rational number, or as a sum ofjew rational numbers, which should be characterized by numerators and denominators that only involve few digits (in both instances the term "few" refers to a comparison with the order p of the rather cool rational number under consideration). For instance, r1(10) qualifies as a rather cool rational number because it is written as the sum of 2 rational numbers, that can themselves be written in fractional form using (in addition to appropriate powers of the base 10) numerators and denominators involving no more than 4 digits, or as a single fraction the numerator and denominator of which involve respectively 1 1 and 12 digits (and both 4 and 12 are small numbers in comparison to 186). Analogous considerations apply to all the other rather cool rational numbers rn(10), n = 2,3,4, . . .
A proof of (1.3) is provided in the next Section.
Proof The starting point is the observation that the definition of the Champemowne constant c(b ), as given above, entails the representation
00
c(b) = I n b-p(b,n), (2. 1a) n= 1
where p(b,n) is the positive integer that identifies the position at which the last digit of the expression (written out
74 THE MATHEMATICAL INTELLIGENCER
in base b) of the positive integer n appears in the decimal expression (written out in base b) of c(b )-a position assessed by counting all decimal digits in that expression of c(b ), starting from the first digit 1 (see (1. 1) and (1.2)).
To evaluate p(b,n ), partition the infinite sum in the righthand side of (2. 1a) into portions, in each of which the number n has the same number, say k, of digits:
with
oo nk- 1 c(b) = I I nb-p(b,n,k),
k=1 n=nk- l (2. 1b)
(2.2)
Here p(b,n,k) is merely a redundant (but more convenient) notation for p(b,n).
It is now plain that
p(b,n, 1) = n, (2.3)
and that
p(b,n,k) = p(b,nk- 1 - 1 ,k - 1) + k[n - (nk-1 - 1)]. (2.4)
This crucial formula, (2.4), may be justified verbally: when n gets above nk-1 - 1, it has k digits, so p(b,n,k) increases by k for each unit increase in n.
Clearly the recursion relation (2.4), together with the initial datum (2.3), determines uniquely the numbers p(b,n,k), and it is easy to verify that they are given by the formula
p(b,n,k) = kn - q(b,k), (2.5)
where the integers q(b,k) satisfy the recursion relation
q(b,k) - q(b,k - 1) = nk-1 - 1 = bk- 1 - 1 (2.6a)
with the initial datum
q(b, 1) = 0. (2.6b)
This recursion relation (2.6) is easily solved: q(b,k) = -k + (bk - 1)/(b - 1); hence
00
c(b) = I b-k+(bk - 1)/(b - 1) S(b,k), k=1
bk- 1 S(b,k) = I nb-kn.
n = bk- l
(2.7a)
(2.7b)
The evaluation of S(b,k) is now a matter of standard algebra; inserting it in (2. 7a) and simplifying leads to (1.3).
Some Afterthoughts Exercise 3. 1 . Define, in analogy to the Champemowne constant c(b ), another, more general, class of cool irrational numbers, c(b,s), the decimal representations of which are analogous to that of c(b ), except that now every integer is preceded by s zeros (here and below s is of course a nonnegative integer), so that of course c(b,O) = c(b), and for instance c(10,3) written out in the standard base 10 reads
c(10,3) = 0.000 1 000 2 000 3 000 4 . . . 000 9 000 10 000 1 1 . . . 000 99 000 100 000 101 . . . (3. 1)
Show that the corresponding generalization of the series
representation (1.3) reads
c(b,s) = b8+1f(b8+ 1 - 1)2 X
- (1 - b-1) I f3k(b,s)b-ak(b,s), (3.2a) k�1
ak(b,s) = (k + s)bk - (bk - 1)/(b - 1) k - 1
= Ck + s - 1)bk + Cb - 2) I b1 + Cb - 1), (3.2b) l�1
f3k(b,s) = (bk+s+ 1 _ 1) -2 (bk+s _ 1) -2 bk+s+ 1
[(b2k+s _ bk + 1)(bk+s+ 1 _ 1) + bk+s _ 1] , (3.2c)
so that, for instance for s = 1, the analogs of (1.6) read
r0(10,1) = 100/9801 = 22 · 52/(34 · 112)
= 0.[0 1 0 2 0 3 . . . 0 9 10 11 12 . . . 96 97 99 00], (3.3a)
r1(10, 1) = 10-14 (25 456 611 570 247 933 657/24 950 025) = 25 456 611 570 247 933 657/(214 . 36 . 516 . 112 . 372) = 0.0 1 0 2 0 3 . . . 0 8[0 9 0 10 0 11 0 12 . . .
0 98 0 99 100 101 102 . . . 996 997 999 000 001 002 . . .
007 008 0]. (3.3b)
Exercise 3.2. Note that the rational numbers
(3.4)
occurring in the previous exercise have a remarkably neat
periodic expansion, see (3.3a) and, for instance,
ro(10,2) = 1000/998001 = 23 · 53/(36 · 372)
= 0. [001 002 003 . . . 009 010 011 . . . 099 100 101 . . . 996 997 999 000]. (3.5)
Exercise 3.3. Show that the rational number
r(b,k - 1)
= bk (bk - 1)-2[ 1 - b-kbk (b2k - bk + 1)], (3.6)
with b and k positive integers, when written out as a deci
mal number in base b, terminates, and has quite a neat look,
exemplified by
r-c10,2) = c10oo - 10-2997 . 999 OOI)/9992
= 0.001 002 003 . . . 009 010 011 . . . 099 100 101 . . . 997 998 999. (3.7)
Remark 3.4. The cool irrational number ()(b,n), the deci
mal representation of which (in base b) has after the dec
imal point the digit n (of course with n < b) followed by a
zero and then again n followed by 2 zeros and then again
n followed by 3 zeros and so on endlessly,
()(b,n) = O.nOnOOnOOOnOOOOn . . . , (3.8)
is expressed by the following formula in terms of the (Ja
cobian) theta function ()2(z,q):
()(b,n) = n[ - 1 + (1/2) b118 e2(0,b- 112)] . (3.9)
See p. 464 of [4], or equivalently eq. (16.27.2) of [5], or equiv
alently eq. 13. 19(6) of [6] , or (not equivalently, due to a mis
print) eq. (8.18.3) of [7].
Problem 3.5. Investigate the cool irrational number c(z), as
defined by (1.3) with b replaced by z (with Re(z) > 0), con-
sidered as a function of the complex variable z; and, more
generally, the cool irrational number c(x,y), as defined by
(3.2) with b,s replaced by x,y (with Re(x) > 0, Re(y) > 0),
considered as a function of the two complex variables x,y.
(Hints: c(O,s) = 0, lim [(z - 1)c(z)] = 7T2/3.) Z--->1
Problem 3. 6. Are the cool irrational numbers c(b,s) tran
scendental? For the Champernowne constant c(10) =
c(10,0), see (1 .1), this result was proved in 1961 by K.
Mahler, see for instance the "Champernowne constant" en
try in [8] .
Remark 3. 7. The "prime cool irrational number" p(b ), writ
ten in base b, has, after the decimal point the (endless) con
catenation of prime numbers written out in base b, so that
for instance
p(10) = 0.2 3 5 7 11 13 17 . . . . (3 . 10)
This number, already mentioned in [1 ] , is generally known
as the "Copeland-Erdos constant" [9]-see below and, for
instance, this entry in [8] .
Conjecture 3.8. Both the Champernowne constant and the
Copeland-Erdos constant were introduced [ 1 ,9] in the con
text of the investigation of "normal numbers," namely those
(irrational) numbers the decimal expansions of which fea
ture all (fmite) sequences of digits with the frequency ap
propriate to their length (see [ 1 ] and [9] and the entry "Nor
mal Number" (in [8]). It is obvious that ()(b,n), see (3.8) and
(3.9), is not normal; it is presumed, but not yet proven, that
7T is normal; it is known that the Champernowne constant
and the Copeland-Erdos constant are both normal [ 1,9]. It seems reasonable to conjecture that the cool numbers
c(b,s) introduced above are normal even if s > 0, in spite
of the fact that the frequency of the digit 0 in their decimal
representations exceeds that of the other digits for any fi
nite truncation.
Remark 3.9. A leitmotif of this paper has been the inves
tigation of (irrational) numbers featuring remarkable pat
terns when represented in decimal form. An analogous,
much studied, problem concerns (irrational) numbers fea
turing remarkable patterns when represented as continued
fractions: see the entry "Continued Fraction Constants" in
[10].
Exercise 3.1 0. Define, in analogy to the Champernowne
constant c(b ), the cool irrational number C(b ), the decimal
representation of which is analogous to that of c(b ), except
that now every integer is preceded by as many zeros as its
own length (when written out in base b), so that for in
stance C(10) written out in the standard base 10 reads
C(10) = 0.0 1 0 2 0 3 . . . 0 9 00 10 00 11 . . .
00 99 000 100 000 101 . . . . (3. 1 1)
Show that the corresponding generalization of the series
representation (1.3) reads
C(b) = b2/(b2 - 1)2 �
- (1 - b-2) I Bk(b)b-2ak(b)-k, (3. 12a) k � l
VOLUME 25, NUMBER 4 , 2003 75
with ak(b) defined by (1.3b) and
Bk(b) = (1 - b-2)- 1 bk[(b2k - 1)-2 (b3k - bk + 1) - (b2(k+ 1) - 1)-2 (b3k+2 - bk + 1)], (3.12b)
so that the analogs of (1.6) read
Ro(10) = 100/9801 = 22 · 52/(34 · 112)
= 0.[01 02 03 . . . 09 10 11 12 13 . . . 95 96 97 99 00] , (3. 13a)
R1(10)
= 10- 14 (2 550 249 999 999 999 974 977/249 950 oo2 5) = 2 550 249 999 999 999 974 977/(214 . 34 . 516 . 1 12 . 1012) = 0.0 1 0 2 . . . 0 8[0 9 0010 001 1 0012 . . . 0098 0099 0100
0101 0102 . . . 9996 9997 9999 0000 0001 0002 0003
. . . 0008 00]. (3.13b)
Remark 3. 1 1 . There is a major dichotomy among the dy
namical systems that display a chaotic behavior ("deter
ministic chaos") and those that do not ("integrable sys
tems"). Likewise an irrational number, when written out in
decimal form, may feature an (endless) sequence of digits
that is chaotic, or instead one that displays an easily de
scribable pattern. An example of the former is w, an ex
ample of the latter is the Champernowne constant, see
(1. 1). These two (irrational) numbers are both transcen
dental (see above, under Problem 3. 6), and they are pre
sumably both normal (see Conjecture 3.8). Shall we say
that w, in contrast to the Champernowne constant, is not
cool?
Acknowledgments The research reported herein was mainly done while I was
visiting the Isaac Newton Institute for Mathematical Sci
ences in Cambridge in the framework of the Program on
Integrable Systems (second semester of the year 2001 ), and
was motivated by a chance encounter with (a biography
of) Paul Erdos in Shelliro. Useful discussions with Mario
Bruschi and the assistance of Matteo Sommacal in per
forming certain early numerical checks are gratefully ac
knowledged. I wish moreover to acknowledge with thanks
many suggestions, and some corrections, provided by a Ref
eree who was clearly much more knowledgeable than I on
some of the topics treated herein.
REFERENCES
( 1 ] D. G. Champemowne, "The construction of decimals normal in the
scale of ten," J. London Math. Soc. 8 (1 933), 254-260.
[2] N. J . A Sloane, The On-Line Encyclopedia of Integer Sequences,
http://www .research.att.com/ �enjas/sequences.
[3] J. R. Silvester, "Decimal deja vu," Math. Gaz. 83 (1 999), 453-463.
(4] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis ,
Cambridge University Press, 1 962.
76 THE MATHEMATICAL INTELLIGENCER
A U T H O R
FRANCE CO CALOGERO
n.v-.,rtrnNI'I · Ptl Uf1Mi,.,..<V ot Rome 'La Sapienza"
p Aldo Mofo 00185 Rome
laly
H hObbies are chess and
[5] M. Abramowitz and I. A Stegun, Handbook of Mathematical Func
tions, National Bureau of Standards, Applied Mathematics Series
55, U. S. Government Printing Office, Washington, D. C . , 1 965.
(6] Higher Transcendental Functions, edited by A Erdelyi, vol. I I , Mc
Graw-Hill, 1 953.
[7] I . S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
Products (edited by A Jeffrey) , Academic Press, 1 994.
[8] Resource Library, http://www.mathworld.wolfram.com.
[9] A H. Copeland and P. Erdos, "Note on normal numbers," Bull.
Amer. Math. Soc. 52 (1 946), 857-860.
(1 0] http://pauillac.inria.fr/algo/bsolve/constanVcntfrc/cntfrc .html.
I a §II l§i,'tJ Osmo Pekonen , Ed itor I
Feel like writing a review for The
Mathematical Intelligencer? You are
welcome to submit an unsolicited
review of a book of your choice; or, if
you would welcome being assigned
a book to review, please write us,
telling us your expertise and your
predilections.
Column Editor: Osmo Pekonen, Agora
Center, University of Jyvaskyla, Jyvaskyla,
40351 Finland
e-mail: [email protected]
Joseph Fourier, 1 768- 1 830: createur de Ia physique· mathematique by Jean Dhombres and
Jean-Bernard Robert
PARIS, BELIN, 1 998 768 pp. €36.50 ISBN 2· 701 1 ·1 21 3·3
REVIEWED BY JEAN-PIERRE KAHANE
The name Fourier is most familiar
to mathematicians, physicists, engi
neers, and other scientists. Fourier se
ries, Fourier coefficients, Fourier inte
grals, Fourier transforms, the Fourier
equation, and Fourier analysis are
everyday terms. Fourier series were
the source, and the test case, for all
fundamental notions of mathematical
analysis, including the general notion
of function, diverse notions of the in
tegral, and Cantor's set-theory. They
provide the first, and still the most im
portant, example of an orthonormal ex
pansion, leading to the main develop
ments in functional analysis. Fourier
transforms are essential in probability
theory. In mathematical physics, the
Fourier equation, or heat equation, is
the paradigm of a mathematical model
for a natural phenomenon.
The fast Fourier transform (FFT) is
now used in all fields of science, from
astrophysics to biology. Our knowl
edge of the Universe changed after the
FFT. Wavelets as a new avatar of
Fourier series has a spectacular impact
in image processing. The Fourier point
of view on the relation between nature,
science, and applications, is now again
fashionable among mathematicians. As
Fourier wrote in the Discours prelimi
naire, "The thorough study of nature is
the most fertile ground for mathe
matical discoveries"; and also, "Our
method leaves nothing vague in the so
lutions, it leads to the ultimate numer
ical applications that are the aim of any
research." From mathematical physics
to numerical analysis, the heritage of
Fourier is invaluable.
Fourier's work has not been widely
known, and his life did not attract
much attention until recently. He is not
a gloire nationale in France. After he
died in 1830, obituaries were read by
Fran<;ois Arago at the Academie des
Sciences and by Victor Cousin at the
Academie Fran<;aise; Joseph Fourier
was a member of both Academies.
Both obituaries contain interesting
pieces of information, but Fourier's
contribution to mathematics is essen
tially ignored.
Victor Hugo says a few words about
Joseph Fourier and the utopian
Charles Fourier in his novel Les Mis
erables when he evokes the year 1817:
"There was at the Academy of Sciences
a celebrated Fourier whose name is
forgotten now, and in some unknown
attic an obscure Fourier who will be re
membered in times to come."
The collected works of Joseph
Fourier were never published. When
Gaston Darboux published The Ana
lytical Theory of Heat and gathered the
material for a partial edition of his
other works, he left out the whole of
what Fourier called "Analyse indeter
minee," including what we now call lin
ear programming. Darboux says that
Fourier gave these things "an exagger
ated importance." Most symptomatic is
the fact that at the beginning of the
1970s the first editions of the Encyclo
paedia Universalis, a kind of French
Encyclopaedia Britannica, did not
contain an article about Fourier. The
dominant attitude toward Fourier
among French mathematicians of the
time was condescending: Fourier was
not a real mathematician, he did not
prove what he said, he wrote mean
ingless formulas, and he did not per
ceive any of the difficulties of the the
ory. For mathematicians he was too
much of a physicist, and for physicists
too much of a mathematician.
The general attitude has changed
since then. Reading The Analytical
© 2003 SPRINGER· VERLAG NEW YORK, VOLUME 25, NUMBER 4, 2003 77
Theory of Heat now, we see that
Fourier established a program and not
a textbook; and we know how impor
tant good programs are in mathemat
ics. The Fourier formulas are not valid
in every case, and it is precisely the
work of mathematicians to define the
right context for their validity. And that
is how Fourier analysis interacts with
the whole of mathematics.
Fourier had a very interesting life
and received diverse honors during his
lifetime. But his treatment of trigono
metric series encountered incompre
hension and hostility from Lagrange,
who was the most respected mathe
matician in France, and from Poisson
among the younger generation.
This explains why Arago was reluc
tant to discuss his contribution to
mathematics. While the French did not
recognize Fourier properly, his role as
a mathematician was appreciated in
other countries, mainly Germany.
Dirichlet and Sturm visited Fourier in
Paris, and they established Fourier se
ries, or their analogues, as an impor
tant mathematical topic. The main
recognition of the role of Fourier as a
pioneer is due to Riemann. Riemann in
his dissertation on trigonometric series
discusses the history of the subject and
says that Fourier was the first to un
derstand the nature of trigonometric
series in a completely correct way. He
also explains how mathematical physics
was linked with this new theory of rep
resentation of arbitrary functions. With
Fourier, Riemann says, a new epoch
began in this part of mathematics,
which proved essential in the spectac
ular development of mathematical
physics.
Before the book of Dhombres and
Robert, the main references on Fourier
were I. Grattan-Guinness and J. Ravetz,
Joseph Fourier 1 768-1830, a Survey
on his Life and Work (MIT Press,
1972), and J. Herivel, Joseph Fourier,
the Man and the Physicist (Oxford,
Clarendon Press, 1975). L. Charbon
neau wrote a dissertation on Fourier
and published a "Catalogue des manu
scrits de Joseph Fourier" (Cahiers
d'histoire et de philosophie des sciences 42, 1994).
The Dhombres-Robert book is a
most welcome addition. It was essen-
78 THE MATHEMATICAL INTELLIGENCER
tially completed in 1995 and should
have been published at that time. It is
referred to in the book by Kahane and
Lemarie-Rieusset entitled Fourier Se
ries and Wavelets (Gordon and
Breach, 1995) as "La chaleur mathe
matisee, Joseph Fourier, Et ignem re
gunt numeri (Paris, Belin, 1995)." Why
did it take so long for it to be printed?
The reason is simply that it is a very
ambitious and long book, and its extent
and content exceeded by far what the
publisher had expected. It belongs to
an excellent collection called "Un sa
vant, une epoque," whose editor is Jean
Dhombres. The preceding books were
about Abel, Arago, Bacon, Berthelot,
Boole, Boucher de Perthes, Branly,
Cardan, Cauchy, Darwin, Djerassi,
Duhem, Edison, Fleming, Gay-Lussac,
Geoffroy Saint-Hilaire, Hardy, Heisen
berg, Humboldt, Kovalevskaya, Langevin,
Linnaeus, Mendel, Peiresc, Planck,
Perrin, Tesla, Van Leeuwenhoek, Vidal
de la Blache, Von Frisch, Wegener,
Yersin, and Yukawa, each of them con
sidering only one aspect of the scien
tist and limited to about 200 pages. The
book on Fourier contains 767 pages
and considers all aspects of Fourier's
life and work When it fmally appeared
in 1998, the title was changed. It be
came simply Fourier, createur de la
physique mathematique.
The common theme of the books in
this collection is to connect the life,
work, and period of the subject. In the
case of Fourier, the life is worth a
novel, the work has both historical and
present value, and the period includes
the most fascinating events of French
history. For Fourier, his life, his work,
and his time are strongly connected.
The book was written by a mathemati
cian, Jean Dhombres, and a physicist,
Jean-Bernard Robert. Both of them
have devoted part of their scientific
lives working about and around
Fourier. Jean-Bernard Robert is a pro
fessor at Universite Joseph Fourier in
Grenoble; he was also the director of
studies at Ecole normale superieure de
Lyon when he was writing the book He
read letters, documents, and archives,
and also reproduced some of the ex
periments Fourier did when he was in
Grenoble. Jean Dhombres is a renowned
historian of mathematics, and has
worked, in particular, on the period of
the French Revolution. He published
the lectures and debates of the first
Ecole Normale, where Fourier was a
student, and he read carefully some lit
tle-known mathematical papers by
Fourier. Together, Dhombres and
Robert have produced a beautiful
work
The book contains ten chapters, a
series of letters, documents and quo
tations, a chronology, a list of refer
ences, and an index. The tenth chap
ter, titled "Epilogue," expresses in a
few pages the sympathy of the authors
for the man and the scientist, and in
particular for what they call "la severite
de Fourier," Fourier's gravity. Fourier's
letters and quotations are well chosen
and give an excellent idea of his char
acter. There are letters from his youth
and from the revolutionary period, and
notes by Fourier about his teachers at
the Ecole Normale. There is a steno
graphic report of his discussion with
Gaspard Monge concerning the defini
tion of the plane, written after Monge's
first lecture at the Ecole Normale, and
there is the beginning ("Discours pre
liminaire") and the end of his main
work, Theorie analytique de la chaleur,
the Analytical Theory of Heat. If you
want an introduction to Fourier, start
ing with this Epilogue is not a bad idea.
In Fourier's day, French mathe
maticians included Laplace, Monge,
Legendre, Lagrange, and Cauchy, while
the physicists included Haiiy, Coulomb,
Ampere, Biot, Malus, Fresnel, and
Arago. The first chapter describes this
heady scientific environment and the
significance of Fourier in this brilliant
cohort. "Et ignem regunt numeri"
means "also heat is governed by num
bers." Numbers and mathematical
analysis are a general tool for under
standing the real world. This chapter is
called "les regimes d'un monde sa
vant," the schemes of a scientific
world. It is a self-contained study, and
at the same time it is an introduction
to the rest of the book Like every chap
ter, it is followed by notes and refer
ences that are of very general interest.
In the central part of the book
(chapters 2 to 7), the life and works of
Fourier appear in chronological order:
the first years in Auxerre, the first
teaching experience and its involve
ment in the Revolution, the Ecole nor
male and Ecole polytechnique, Egypt
with a mixture of science and politics,
Grenoble where he is prefect and writes
his main work, Paris 1815-1830 where
he completes his scientific life as an
academician. Chapters 8 and 9 return
to the Analytical Theory of Heat: its
role as a piece of mathematical physics
and its influence on mathematics.
Fourier's life is fascinating, and the
central part of the book can be read
simply as a story, enriched by docu
ments and illustrations. His work is de
scribed carefully, with emphasis on the
theory of heat. The first and last chap
ters are in the nature of essays in sci
entific philosophy. But both aspects,
life and ideas, are linked with the ex
ceptional time of and around the
French Revolution.
Fourier, as Arago observed, is a
pure product ofthe French Revolution.
He was of a poor family and became
an orphan at the age of 10, but was no
ticed as a bright boy, taught by the or
ganist of the cathedral, and sent to the
Military College of Auxerre. The mili
tary colleges played an important role
in France at the time. They were run
by monks and led both to military and
to ecclesiastic careers. Fourier learned
Latin and discovered mathematics. He
graduated from college at 14, and at 16
was appointed as a teacher, a tempo
rary position.
There were two paths before him:
Army or Church. He decided that he
wished to serve in the artillery, the
most scientific ann of the military. He
had already written a paper on the
roots of algebraic equations and had at
tracted the attention of Legendre, who
supported his application. In spite of
this, his application was refused. The
Minister answered Legendre that, were
he even a second Newton, Fourier
could not enter artillery because he was
not a member of the nobility. Fourier
had to switch to the Church. He was
supposed to take his vows on Novem
ber 5, 1789. But the National Assembly
ordered a suspension of religious vows
on November 2. Fourier gave up the
Church, settled back as a teacher in the
military school in Auxerre, and contin
ued with algebraic equations.
Fourier became involved in public
affairs in 1793, the year Louis XVI was
executed and the war began between
the European Coalition and the French
Republic. Fourier took part in revolu
tionary committees and proved effi
cient in many ways. When the Ecole
normale was created he was chosen to
be one of its 1500 students. The pupils
were selected on the basis of involve
ment in teaching and devotion to rev
olutionary ideals. The teachers were
the most famous scholars of the time:
Hauy in physics, Lagrange, Laplace,
and Monge in mathematics. Fourier
was more prepared than any other stu
dent, and the debate between citizen
Fourier and citizen Monge on the foun
dation of geometry, which is described
in the book, is an example of elevated
discourse in a friendly atmosphere.
Then for a few years Fourier was as
sociated with Monge. When the Ecole
Polytechnique was created, Monge was
one of the main professors and Fourier
was elected as a lecturer. He lectured
on a wide variety of mathematical top
ics, including calculus, statics, dynam
ics, hydrostatics, and probability. He
published a memoir on statics and de
veloped his discoveries on algebraic
equations in some of his lectures.
When Bonaparte led the expedition in
Egypt, Monge founded the Institut
d'Egypte, modeled on the Institut de
France. Monge was president, and
Fourier was "secretaire perpetuel."
This was not exactly a rest cure.
Fourier then worked on a wide variety
of subjects, ranging from egyptology to
what he called "analyse indeterminee,"
which is still unpublished today. He
had contacts with the French officers
and with the Egyptian leaders. When
Bonaparte and Monge left Egypt, he
was left in charge of the French and
had to negotiate with the English and
the Egyptians for their return. All this
is worth a movie, and the book pro
vides exhaustive information on that
Egyptian episode.
Fourier returned to France in 1801,
and Bonaparte appointed him Prefet de
l'Isere in 1802. In 1802, "deja Napoleon
peryait sous Bonaparte," wrote Victor
Hugo. A long chapter of the book, a
hundred pages, is devoted to the
Grenoble period of Fourier's life. He
had to deal with the duty of a prefect,
the representative ofthe central power
in the department. This department,
l'Isere, was not the easiest one. It had
been the starting point of the Revolu
tion; there were problems to solve
about swamps, mines, roads, health,
and education. Fourier proved active
and efficient as a prefect. Moreover, he
had to write the introduction to De
scription de l'Egypte, an enormous re
port on what was seen in Egypt. This
introduction, entitled "Preface his
torique," is an important book in itself.
Last but not least, he worked on the
propagation of heat. How his work was
received is a long story, and worth the
detailed account given in this chapter.
The memoir was crowned by the Acad
emy in 1807, but it was not published
until 1822.
In the meantime, Napoleon was de
feated, came back, was defeated again;
the Bourbon monarchy was restored;
Fourier was dismissed, restored, dis
missed again as a prefect. Finally, he
settled in Paris, was elected to the
Academie des sciences, and became its
secretaire perpetuel. He received
recognition as a scientist and suc
ceeded in publishing his main work.
What he wrote now was mainly acade
mic obituaries. He met a few younger
people, like Dirichlet and Sophie Ger
main. He lived a lonely, unremarkable
life until his death in 1830.
This is the end of the novel but not
the end of the book. Chapters 8 and 9
are learned comments on the scientific
work of Fourier and his heritage.
Chapter 8, "le physicien-mathemati
cien," is 200 pages long, and it contains
a detailed exposition of the Analytical
Theory of Heat. What is most interest
ing for a mathematician is that it ex
presses the point of view of the physi
cist and emphasizes the importance of
the Fourier approach in physics. For
example, we can forget about dimen
sional analysis when we deal with
Fourier series as a mathematical ob
ject, but in writing the equations it
plays a crucial role. The mathematical
treatment of the equations and the in
troduction of Fourier series and inte
grals are excellent. This can be the out
line of a course in physics as well as in
mathematics.
VOLUME 25, NUMBER 4, 2003 79
Chapter 9, "un homme et Ia construction d'une posterite," is shorter. It contains a description of the main steps of Fourier analysis and related matters, and also examples of underestimation of Fourier among mathematicians (some were mentioned at the beginning of this review). It could be very useful in any course on harmonic analysis, just to provide a historical and critical flavor.
As I said before, there is thorough documentation within and at the end of every chapter. The authors give guides for further studies. It is impossible now to work on the time of Fourier without consulting this book
A book to read, to consult, to refer to; a real model of cooperation of a physicist and a mathematician in writing history. At last, the French have made a decisive contribution to our knowledge and appreciation of Joseph Fourier. Thank you, Jean Dhombres and Jean-Bernard Robert.
1 1 rue du Val-de-Grace
75005 Paris, France
e-mail: [email protected]
Four Colours Suffice: How the map problem was solved by Robin Wilson
ALLEN LANE, THE PENGUIN PRESS, 2002 262 pp. £1 2.99 ISBN 0 713 99670 6
REVIEWED BY CHARLES NASH
This is an excellent book It is a book for the layman on the history of the
famous four colour problem from its inception to the present day (i.e., 2003). It deals with its first proposal by Francis Guthrie in 1852, its solution by Wolfgang Haken and Kenneth Appel in 1976 using a lengthy computer programme, and also subsequent developments. As the author points out in his preface, 2002-the year of publication of this book-is the 150th anniversary of the posing of the problem and the
1What Kempe calls a linkage is now called a graph.
80 THE MATHEMATICAL INTELLIGENCER
25th anniversary of the publication of its solution.
The four colour problem-the prob
lem being to prove that four colours
are always sufficient to colour any map-was known empirically to map makers for a long time. Its origins, in cartography, mean it is easy for the non-mathematician, or indeed any lay person, to understand.
Though cartographical in origin, the problem and its proof are not particularly interesting to map makers since they already believed it to be true; and its validity and ultimate proof open up no new vistas in cartography. Hence it is essentially a problem of mathematical interest.
The Four Colour Problem and Graph Theory Arthur Cayley spoke about the four colour problem, and his own work on it, to the London Mathematical Society in 1878 and aroused the interest of another Cambridge trained mathematician A. B. Kempe.
The four colour problem is essentially a problem in graph theory, and this was realised by Kempe who published a "proof" of the four colour problem in 1879 in the American Journal
of Mathematics; the flaw in Kempe's "proof" was subtle and was only found eleven years later in 1890 by the Oxford mathematician P. J. Heawood then working in Durham.
Despite the flaw Kempe's work was very good; among the steps forward he made was to formulate the colouring problem using what we would now call graph theory: On p. 90 Wilson provides us with the following quote from Kempe:1
If we lay a sheet of tracing paper
over a map and mark a point on it
over each district and connect the
points corresponding to districts
which have a common boundary, we
have on the tracing paper a diagram
of a 'linkage', and we have the exact
analogue of the question we have been considering, that of lettering the points in the linkage with as few
letters as possible, so that no two di
rectly connected points shall be let
tered with the same letter.
Heawood, as well as fmding the mistake in Kempe's proof, made important contributions of his own. He proved the five colour theorem-the five colour theorem simply says that five colours suffice, and is not trivial to prove. He also proved seven colours suffice for maps on the torus; he went on to work out the correct formula for the number of necessary and sufficient colours for surfaces of higher genus. This number, if the genus of the surface is h, is the integral part of (1/2)(7 + Y1 + 48h), i.e., the integer
r-! (7 + v 1 + 48h) J valid for h 2: 1 where we are using the brackets [ ] to denote integral part. However he, in turn, had an error in his proof for h 2: 2 which was found by Heffter in 1891 . Heawood's error was in the necessary part of his proof: He did prove that his formula provided a sufficient number of colours for surfaces of genus h 2: 2, but he did not show that maps existed which required this number of colours-for the torus case he had indeed produced a map which needed 7 colours.
It is not difficult to see that the genus 0 case is equivalent to the standard planar case by stereographic projection, so one can also pose the problem on a sphere instead of a plane if one wishes.
The assertion that there exist maps on a surface of genus h, with h 2: 1 , which require [(112)(7 + Y1 + 48h)]
colours was christened the Heawood conjecture; it resisted proof until 1968 when a proof was supplied by Ringel and Youngs.
The Chromatic Polynomial P(,\) Further mathematical progress was made in America by G. D. Birkhoff, a founder of the subject of dynamical
systems, who, in 1912-13, introduced what are called reducible configurations and a polynomial, associated to
each map, known as the chromatic
polynomial P(A). Birkhoff proved that the number of
ways of colouring any map with A colours is a polynomial in A, and this
is called the chromatic polynomial
P(A) of the map; he hoped that P(A) would play an important part in a proof
of the four colour theorem. In 1930-32
he and another American mathemati
cian, Hassler Whitney, obtained more
results on P(A). An intriguing fact about P(A) men
tioned by Wilson is that Tutte has
proved that, if ¢ is the golden ratio,
then, for a given map,
P(cf) = 0,
in the sense that if a map has n coun
tries then
P(q?) ::::; ¢5-n.
Recall that ¢ = (1 + v5)!2 = 1.618 . . .
and so if n takes the values 10, 20, 30,
say, then we find respectively that
P(cf) :::::: 0.0901, 0.000733, 0.00000596.
The significance of this property for
the four colour problem is apparently
not well understood.
Heesch's Successful Strategy The final and successful strategy for a
proof was to use the idea of what is
called an unavoidable set of reducible
configurations: an unavoidable set of
configurations (not necessarily reduc
ible) is a collection of configurations at
least one of which must appear in
every map. It turns out that, if these
configurations are reducible and one
proves the result for this set then the
theorem is solved. We outline the logic
involved in the next paragraph.
First of all a map with n countries
is called a minimal criminal if it can
not be coloured with four colours, but
all maps with n - 1 countries can be
coloured with 4 colours. Clearly mini
mal criminals should not exist. A re
ducible configuration is one that can
not occur inside a minimal criminal;
when maps contain reducible configu
rations inside them somewhere, if
these are coloured successfully then
the colouring can always be extended,
with recolouring if necessary, to the
entire remainder of the map. This lat-
ter property means that if we have an
unavoidable set of reducible configu
rations then proving that these can all be coloured with four colours solves
the four colour problem.
The first obvious snag, when this
strategy was suggested by the German
mathematician Heinrich Heesch in
1948, was a quantitative one: this is that
the number of configurations in such
a set might be far too numerous to
check. Now such unavoidable sets
of reducible configurations are not
unique, but Heesch seemed to think in
terms of a set containing about 10,000
configurations, which he believed fea
sible to check.
However this method did work and
was the one used by Appel and Haken
in their proof in 1976. Though their un
avoidable set contained a mere 1936
reducible configurations-reduced in
their published proof to 1482-their
proof was accompanied by 450 micro
fiche pages of diagrams and explana
tions and, as is well known, used a com
puter to do almost all the checking.
The published proof, which ap
peared in 1977, in the Illinois Journal
of Mathematics, consisted of two pa
pers, the first by Appel and Haken, the
second by Appel, Haken, and J. Koch.
The first paper discussed their proof
and explained their methods, the sec
ond paper described the computer
work and listed all the elements of the
unavoidable set of reducible configu
rations.
Of course Heesch, who had worked
on the problem for four decades or
so, was a bit disappointed to be beaten
to it.
Some Other Dramatis Personae Since accessibility to a lay public is
rare for mathematical problems, when
the problems are famous for lying un
solved for years, decades or more, then
many and varied are the people who
are attracted to them. This problem is
no exception, and I shall mention some
of them now; as always the book itself
is the place to find the whole story.
In fact we learn in this book that, in
1840, i.e., before the four colour prob
lem was posed in 1852 by Francis
Guthrie, A. F. Mobius discussed and
solved what is called the five princes
problem. This bears, what transpires to
be superficial, similarity to the four
colour problem; this similarity was
to be the source of some confusion
among later workers on the four colour
problem.
W. R. Hamilton was told in a de
tailed letter of 1852 about the problem
by A. J. de Morgan, with whom he was
in regular correspondence for years.
However Hamilton's reply contained
the words
I am not likely to attempt your
"quaternion" of colours very soon.
and that was that.
Tait also worked on the problem,
and in 1880, produced several "proofs"
which he thought improved on
Kempe's "proof"; this latter was still
believed to be correct until 1890. Tait,
though he didn't solve the problem,
was thought to have made a useful con
tribution to the matter.
H. Minkowski is mentioned as once
having tried to begin a proof of the four
colour problem in a lecture with some
dismissive remarks about the quality
of previous combatants. Some weeks
later he changed to a more muted tune.
Lebesgue took an interest. In 1940,
a year before he died, he published a
paper giving some new unavoidable
sets.
In April 1975 Martin Gardner, then
the author of the mathematical column
of the Scientific American, claimed in
his column, as an April Fool, that the
four colour theorem had been dis
proved. He gave a counterexample map
which he said required five colours
he then received lots of correspon
dence showing how to colour the map
with four colours.
In 1976, the year of the proof of Ap
pel and Haken, several other research
groups were also very near to success.
There was F. Allaire in Canada, of
whom Haken generously said, cf. p.
205, that his reducibility methods were
even better than Heesch's and much
better than ours.
Swart in Zimbabwe was also doing
great work and then joined with Al
laire.
VOLUME 25, NUMBER 4, 2003 81
W. Stromquist in Harvard was expected to fmish a proof in a year-and there are other relevant details which the avid reader can find for him or her self.
All this meant that Appel and Haken had to work hard and quickly to get there first, which they indeed did.
Finally in 1994, Robertson, Sanders, Seymour, and Thomas constructed a revised and slightly simpler proof of the four colour theorem using the Appel and Haken methods and a computer.
Computer Proofs The earliest reference I could find in the book to the use of a computer on the four colour problem refers to the 1960s. On p. 180 Wilson says
Haken invited Heesch to the Uni
versity of Illinois to give a lecture,
and raised the question of whether
computers could be helpful in the
examination of large numbers of
configurations. In fact this thought
had already occurred to Heesch,
and in the mid 1960's he had en
listed the help of Karl Diirre, a
mathematics graduate from
Hanover who had become a sec
ondary school teacher.
Further down the page he goes on to say that
By November 1965, using the programming language Algol 60 on the University of Hanover's CDC
1 604A computer, Diirre was able to
confirm that the Birkhoff diamond is D-reducible and soon established the D-reducibility of many more configurations of increasing com
plexity.
So we see that the computer entered the fray in 1965 in a small way but was to go on to dominate.
Some mathematicians were very disappointed that the proof was by computer and could not, in practice, be checked by hand. Some refused to accept it as a proof, some were unpleasant and unfair about the matter. Some were perfectly pleasant and fair but were just worried about whether there
82 THE MATHEMATICAL INTELLIGENCER
could be an error in the proof that would be almost impossible to find.
Mathematicians like to get their hands on all the details, ideas, and mechanisms of a proof, and in doing so often learn an enormous quantity of useful things. Sometimes this activity leads to more progress than was created by the proof they are examining.
Alternative proofs are often published, though the same activity is occasionally frowned on by the closely related discipline of theoretical physics-unjustly, I think. In fact a common paper title in mathematics is something like A new proof of so and
so's theorem. This is as it should be: the new proofs often illuminate what really makes the theorem true and can be a great benefit in understanding how the whole of mathematics fits together.
Hence we prefer a proof we can check all of by hand, but we should not actually dismiss out of hand one we can't. We must not label a computer proof with the pejorative epithet brute
force unless we have very good reason and an elegant illuminating non computer proof to offer in its stead.
Nevertheless there can be the genuine concern that a large scale computer proof has a practically undetectable error. This may have been the reason why, in 1986, Appel and Haken had to publish a paper refuting persistent rumours about errors in their proof.
Finally that kind of genuine concern is now considerably allayed, I think, by the second proof, mentioned above, of Robertson, Sanders, Seymour, and Thomas. They used an unavoidable set of only 633 reducible configurations as compared with Appel and Haken's 1482. Their computer was a 1994 model (instead of a 1976 one), used much less computer time, and the result was easier to check. In addition they have made the programmes publicly available and many computers can duplicate their runs. So the success of this second computer based proof is something decidedly positive.
There will surely be even larger computer based mathematical proofs in the future. However, I don't think that they should be banned, or that their proliferation, if it happens, will do
away at all with ordinary computerless proofs which are the life blood on which all the rest feeds.
Conclusion The book under review is certainly professionally done and repays careful reading. There is a wealth of mathematical, human, and circumstantial detail provided for all parts of the story. At the back there is a very useful Chronology of Events, a no less useful Glossary of mathematical terms to help the layman, and finally a good index.
The book gives us a feel for how much nearer the mathematical centre stage the four colour problem was in the past. Just to select one example from many: Birkhoff gave it considerable attention having heard Veblen lecture on the four colour problem to the American Mathematical Society in 1912. Wilson, on p. 153, says
From this time onwards Birkhoff
regarded solving the jour colour
problem as one of his greatest as
pirations, even though he was later
to regret the amount of time he
spent on it.
Wilson also points out, on p. 164, that Birkhoff is said to have once remarked that almost every great mathematician had worked on the four colour problem at one time or another. It was not so universal in its appeal later in the twentieth century due, in part, presumably, to the bigbang-like growth of so many new areas of mathematics as the century progressed.
However, graph theory, to which this theorem belongs, is a growing and vibrant theory with numerous links to abstract mathematics as well as physics and network theory. Generalisations of the chromatic polynomial P( A) such as the Tutte polynomial are of great importance. Knot theory and the Jones polynomial have now become quite well known to many theoretical physicists; a whole new approach to the Jones polynomial for a general threemanifold was created by the quantum field theoretic formulation of Edward Witten; perhaps the Tutte polynomial
may also be amenable to a quantum
field theoretic construction.
I have tried to give an idea of the
sorts of mathematical, human, and his
torical detail that the author Robin Wil
son has put into the book-I have not
done it justice. Do go and read this
book, it is well worth it.
I have one complaint to direct to the
Publisher-Allen Lane, The Penguin
Press-with the exception of the dust
jacket, there are no colour illustra
tions at all; this is a disgrace for a book
on the four colour problem, which
costs, incidentally, £12.99. Typograph
ical justice has not been done to the
subject. The book is copiously illus
trated, but the colour differences are
all indicated by varying shades of black
and grey. I very much doubt that the
author's original illustrations were not
in colour, but whether they were or
not, I'm sure that the publisher could
have supplied all, or a significant per
centage, of them in colour.
Finally, congratulations to Robin
Wilson.
Department of Mathematical Physics
National University of Ireland
Maynooth
Ireland
e-mail: [email protected]
Mathematics and Art: Mathematical Visualization in Art and Education edited by Claude P. Bruter
NEW YORK. SPRINGER-VERLAG. 2002. 497 pp. US$ 84.95
ISBN 35-4043-4224
REVIEWED BY HELMER ASLAKSEN
I 'm convinced the title of this book
will intrigue most readers of the
Mathematical Intelligencer. When you
look at the list of contributors and see
names like Michele Emmer, Michael
Field, George W. Hart, John Hubbard,
Richard S. Palais, Konrad Polthier, and
John Sullivan (to name but a few in al
phabetical order), I'm sure you will be
even more interested. The book is the
proceedings of the Colloquium on
Mathematics and Art held in Mau
beuge, France, in September 2000, and
as soon as I opened the book, I started
wishing I had been there. It must have
been a killer conference! But does that
make for a killer conference proceed
ings?
Mathematics and Art is a very wide
venue. My background involves teach
ing a course on Mathematics in Art and
Architecture at the National University
of Singapore, consulting for an exhibi
tion called "Art Figures: Mathematics
in Art" at the Singapore Art museum,
and numerous TV interviews and pub
lic lectures at museums, libraries, and
schools.
I personally like to subdivide dis
cussion of mathematics and art into the
following four rough categories:
• Mathematics in art
• Mathematical art
• Mathematics as art
• Mathematics is art
"Mathematics in art" refers to topics
like perspective in paintings, symmetry
in ornamental art, and musical scales.
This is material that even the most anti
scientific art-theorist would recognize
as relevant. You can approach virtually
any art museum with an offer of a pub
lic lecture on such topics and be con
fident of a good turnout.
"Mathematical art" includes the
works of Escher and other mathemat
ically inclined artists, who while wor
shiped by mathematicians are some
times frowned upon or ignored by the
art community. When I was working on
the exhibition at the Singapore Art Mu
seum, I had to conform to a strict "no
Escher" policy. An offer to an art mu
seum of a public lecture about Escher
may not necessarily be accepted.
"Mathematics as art" refers to visual
mathematics. With the advent of com
puter graphics, mathematicians have
been able to create stunning graphics.
Yet how many art museums would be
interested in a public lecture about the
Mandelbrot set?
"Mathematics is art" refers to the
view held by many mathematicians
that mathematics is an art, not a sci
ence. However, few art theorists share
this view.
This classification is of course very
subjective and reflects my own views
and experiences. At the same time, I
hope it may serve as a possible frame
of reference for your expectations
when picking up this book.
How many proceedings from con
ferences that you did not attend do you
have on your bookshelf? I think it's
only fair to say that many of the arti
cles in the book are not easy reading.
If you are planning to teach a course
on mathematics in art for first-year
general students, then I'm afraid you
will not fmd many articles that you can
use directly. The article on "The Math
ematics of Tuning Musical Instru
ments-a Simple Toolkit for Experi
ments" by Erich Neuwirth is one of the
exceptions.
The word "education" appears in
the subtitle, but it seems some of the
authors feel that as soon as you have a
couple of pictures, it is "educational."
Fortunately, Michael Field and Ronnie
Brown wrote about their experiences
in teaching undergraduate classes.
In the interesting article "Mathe
matics and Art: The Film Series, "
Michele Emmer says: "If it i s almost
impossible to describe a film using
words, it is good, because it means
that the film has been made really us
ing a visual technique, mixing images,
sounds, music in an essential and pos
sibly unique way. " By the same token,
a good lecture on mathematics and
art may not translate into a good ar
ticle. Many of the articles are written
by people I admire deeply, who are
excellent speakers and have wonder
ful Web pages. Yet I sometimes do not
get much out of their articles in this
book.
I must also confess that at times I
have problems with the writing style.
On page 1 of the book, it says, "One of
the reasons, the main one to my eyes,
which solders the arts to mathematics
is probably the following: the tangible
object, the living being, are not only
present in space, and are evolving in
space, but are moreover a highly elab
orated construction, obtained from the
unfolding of the properties of the pri
mordial space." I don't like it when I'm
"dead on arrival" on page 1 of a book,
and when on page 9 of the opening ar
ticle I read, "From there results that the
VOLUME 25. NUMBER 4. 2003 83
acquisition of the lrnowledge and the formation of the spirit, which have a phylogenesis, deserve to be conceived according to a process of ontogenesis which respects this phylogenesis," I
went into a shell-shock from which I
never fully recovered. One article is about the ARP AM pro
ject. What is the ARP AM project? The 15-page article does not explain the acronym "Association pour Ia Realisation et Ia Gestion du Pare de Promenade et d'Activites Mathematiques." After reading the article, it was unclear to me whether this was just a plan or whether the parks actually existed.
The articles follow the order of the talks at the conference. I think the book would have been more useful if for instance the three articles on music had been grouped together. There are also no biographies of the authors.
There are 57 pages of color plates at the end. Almost all of them appear in the main text in black and white. I must confess that I am color-blind, so my view may be biased, but for many of them I did not see a compelling reason to duplicate them in color. With all due respect to the late Fred Almgren, do we need to see a color picture of him in addition to the black-and-white picture in the text? Do the pictures from Bruce Hunt's excellent article "A Gallery of Algebraic Surfaces" look so much better in two colors than in black and white? And unfortunately, the color pictures from Maria Dedo's excellent article on "Machines for Building Symmetry" did not appear in the main text at all. I think the color pictures would have been more effective if they had been selected more carefully and if it were indicated clearly which of the black-and-white pictures had color versions in the back.
The conference must have been spectacular, and the proceedings contain a number of excellent articles that deserved better editing, both of the writing of the individual articles and the overall organization of the book.
Department of Mathematics
National University of Singapore
Singapore 1 1 7543 Singapore
e-mail: [email protected]
84 THE MATHEMATICAL INTELLIGENCER
NEW FROM BIRKHAUSER
Sampling, Wavelets, and Tomography IJ. BENEDETTO, UNIV£RSITY OF IMRVIAND. COLLEGE �K. MD. AND A.J. IAYEO, DEPAUL UNIVERSITY. OIICAGO. IL (EDS) The actJve research areas of sampling. wavelets, and tomography have numerous applicatiOns to signal/1mage processtng and med1cal •mag1ng technology. This state-of-the-art work, the first publtcahOn to treat the 1ntersecbon of these expand· 1ng fields, reflects he contributions of mathematiciartS and engineers and stress· es the Interdependence of all three areas and the1r common roots at the heart of harmoniC. and Founer analysis. 2001/APPROX. 152 PP., 50 ILLUSJHARDCOYIR
$69.95 (Ttnt.)IISIN H17Hl04-4 APPliED AND NU IRICAL IIARMONIC ANALYSIS
The Evolution of Applied Harmonic Analysis Models of the Real World E. PREST1Nl, UNIV£RSII'Y OF ROM£. "lOR 1/f/\'GATA" ROM ITAlY This dearly wrmen, extenSIVely illustrated boo provides a development of the basiC concepts of harmonic analysis and presents actwe diSOpltnes of modem science through the umfymg framework of Founer analysis. The work grves a hJstoncal lntr� duoion to a number of soentific fields and includes canoe e applications of harmonic analysis to signal process1ng. computenzed music. Fourier opucs. rad1o astronomy, crystallography, computenzed tomography, nuclear magnetic resonance, and a scanning, 20011APPROX. 120 PP., 169 1UUSJSOFTCOVER
$.49.95 CT•nt.)IISIN H17H1U-4 APPLIED AND NUMERICAl HARMONIC ANALYSIS
Wavelets Through a Looking Glass The World of the Spectrum 0. 8RATTELt UN/VfRSITY OF OSiQ I\IORIIIIIY. AND P. JORGENSEN. UN/VfRSITY OF lOW\ IO'r� CITY. /A ThiS self-<Onta1ned book grves a d tailed, rich, and entertaining tour of wavelets and theu applica ons for mathematicians, computer soennsts. and eng1neers. Connections are made to areas such as computer graphics algonthms, commu· mcations engmeering. quanrum computIng. and advanced computer soence wtual·reahty appl1cat1ons such as audio-systerrtS processing. 10011424 PP., 122 IUUSJHUDCDVIIV$S9.9S
ISBN H17H210-) APPUID AND NUMIRIW IIAIUIONIC ANALYSIS
www.birkhau er.com
Birkhiiuser Bo ton · Ba I · Berlin
7/03
V O L U M E 2 5
The Mathematical lntelligencer Index
Authors
Adams, Colin. A Difficult Delivery.
(1) 8-9.
Adams, Colin. Wiling Away the
Hours. (2) 18-19.
Adams, Colin. The Three Little Pigs.
(3) 27-28.
Adams, Colin. Don't Touch the But
ton. (4) 32-34.
Alexanderson, G.L., and Klosinski,
Leonard F. Mathematicians' Vis
iting Cards. ( 4) 45-52.
Albinus, Hans-Joachim. Pythagoras's
Oxen Revisited. (3) 41-43.
Aslaksen, Helmut. Review of Mathematics and Art: Mathematical Visualization in Art and Education, edited by Claude P.
Bruter. ( 4) 83-84.
Atzema, Eisso J. Into the Woods:
Norbert Wiener in Maine. (2)
7-17.
Banchoff, Thomas F. Review of Visualiser la quatrieme dimension, by Franc;ois Lo Jacomo.
(3) 57-59.
Barney, Steven. See Saari, Donald G.,
and Barney, Steven. ( 4) 17-31.
Batchelor, Marjorie. Undergradu
ate Training Revisited: Thoughts
on an Unusual Reunion. (1) 17-
21 .
Bauer, Friedrich L. Why Legendre
Made a Wrong Guess about
1T(x), and How Laguerre's Con
tinued Fraction for the Loga
rithmic Integral Improved It. (3)
7-1 1.
Booss-Bavnbeck, Bernheim, and
H0yrup, Jens. Mathematics and
War: An Invitation to Revisit. (3)
12-25.
Calogero, Francesco. Cool Irrational
Numbers and Their Rather Cool
Rational Approximations. (4)
72-76.
Chernoff, Paul R. "Some of the Peo
ple, All of the Time." (1) 71-73.
Crato, Nuno. Pedro Nunes, Por
tuguese Mathematician and Cos
mographer. (1) 80.
Davis, Benjamin Lent, and Maclagan,
Diane. The Card Game SET. (3)
33-40.
Davis, Chandler. The Cosmological
Complaint. (2) 23.
Davis, Martin. Exponential and
Trigonometric Functions-From
the Book (1) 5-7.
Dawson, John W. Jr. Review of From Trotsky to Godel: The Life of Jean van Heijenoort, by Anita
Burdman Federman. (2) 78-79.
Dehaye, Paul-Olivier, Ford, Daniel,
and Segerman, Henry. One Hun
dred Prisoners and a Lightbulb.
(4) 53-61.
Elkies, Noam D., and Stanley,
Richard P. The Mathematical
Knight. (1) 22-34.
Emmer, Michele. Review of Fermat's Last Tango, a Musical, Music
by Joshua Rosenblum, Book by
Joanne Sydney Lessner, Lyrics
by Lessner and Rosenblum. (1)
77-78.
Emmer, Michele. A Mathematician in
Lhasa. (4) 67-71.
Ewing, John. Predicting the Future
of Scholarly Publishing. (2) 3-6.
Fenske, Christian C. Extrema in Case
of Several Variables. (1) 49-51.
Ford, Daniel. See Dehaye, Paul
Olivier, Ford, Daniel, and Seger
man, Henry. (4) 53-61 .
George, Alexander. A Free Lunch in
Chess and Logic? (3) 53-55.
Gotz, Ottomar. Regiomontanus. (3)
44-46.
Groetsch, Charles. Hardy's Duncan
Prize Book (4) 5-6.
Guillemin, Victor. Review of Ou en sont les matMmatiques?, edited
by Jean-Michel Kantor. (3) 56-
57.
Hauser, Kai, and Lang, Reinhard. On
the Geometrical and Physical
Meaning of Newton's Solution
of Kepler's Problem. ( 4) 35-44.
Hersh, Reuben. The Birth of Random
Evolutions. (1) 53-60.
Hickerson, Dean. Prime Maze. (1) 48.
Hickerson, Dean. Prime Maze-The
Solution. (2) 75-76.
Hitotumatu, Sin. More Visible Sums.
(3) 4-5.
Holbrook, John, and Kim, Sung Sao.
A Very Mean Value Theorem. (1)
42-47.
H0yrup, Jens. See Booss-Bavnbeck,
Bernheim, and H!Ziyrup, Jens. (3)
12-25.
James, loan. Autism in Mathemati
cians. ( 4) 62-65.
Kahane, Jean-Pierre. Review of
Joseph Fourier, 1 768-1830: Createur de la Physique Mathematique, by Jean Dhombres and
Jean-Bernard Robert. (4) 77-80.
Kim, Sung Sao. See Holbrook, John,
and Kim, Sun Sao. (1) 42-47.
Kleber, Michael. Capitalism Over
turned. (1) 52.
Kleber, Michael. Capitalism Over
turned-The Solution. (2) 7 4.
Klosinski, Leonard F. See Alexan
derson, G.L., and Klosinski,
Leonard F. ( 4) 45-52.
VOLUME 25, NUMBER 4, 2003 85
Lang, Reinhard. See Hauser, Kai, and
Lang, Reinhard. ( 4) 35-44.
Levy-Leblond, Jean-Marc. Columella's
Formula. (2) 51-54. Longuet-Higgins, Michael S. Nested
Triacontahedral Shells Or How
to Grow a Quasi-Crystal. (2) 25-43.
Machover, Maurice. Cauchy Product
of Series. (3) 43. Maclagan, Diane. See Davis, Ben
jamin Lent, and Maclagan, Di
ane. (3) 33--40. Maritz, Pieter. Around the Graves of
Petrovskil and Pontryagin. (2) 55-73.
Nana, Cyrille. Seminar-Workshop in
Mathematics, Yaounde, Came
roon, December 10-15, 2001 . (3) 29-32.
Nash, Charles. Review of Four Colours Suffice: How the Map Problem Was Solved, by Robin
Wilson. (4) 80-83.
Pak, Igor. On Fine's Partition Theo
rems, Dyson, Andrews, and
Missed Opportunities. (1) 10-16. Pekonen, Osmo. Review of Noeuds:
Genese d 'une Theorie Mathematique, by Alexei Sossinsky.
(1) 75-77. Ricotta, Angelo. Constant-diameter
Curves. (4) 4-5. Rowe, David E. Hermann Weyl, the Re
luctant Revolutionary. (1) 61-70. Rowe, David E. From Konigsberg to
Gottingen: A Sketch of Hilbert's
Early Career. (2) 44-50. Rowe, David E. On Projecting the
Future and Assessing the Past
the 1946 Princeton Bicentennial
Conference. (4) 8-15. Saari, Donald G, and Barney, Steven.
Consequences of Reversing
Preferences. (4) 17-31 .
Sallows, Lee. A Tragic Square. (4) 7. Sallows, Lee. Not-so-magical square.
(4) 6-7. Segerman, Henry. See Dehaye, Paul
Olivier, Ford, Daniel, and Seger
man, Henry. (4) 53-61 . Shallit, Jeffrey. What This Country
Needs is an 18¢ Piece. (2) 20-23. Shell-Gellasch, Amy E. Reflections of
My Adviser: Stories of Mathe
matics and Mathematicians. (1) 35-41.
Stanley, Richard P. See Elkies, Noam
D. and Stanley, Richard P. (1) 22-34.
Stem, Manfred. Review of Alles Mathematik: Von Pythagoras zum CD Player, edited by Mar
tin Aigner and Ehrhard Behrends.
(3) 60-62. van der Waall, Helena Alexandra,
and van der Waall, Robert
Willem. The Christoffel Plaque
in Monschau. (3) 47-51 . van der Waall, Robert Willem. See
van der Waall, Helena Alexan
dra, and van der Waall, Robert
Willem. (3) 47-51. Wilson, Robin. The Philamath's Al
phabet A. (2) 80. Wilson, Robin. The Philamath's Al
phabet B. (3) 64. Wilson, Robin. Anniversaries. (4) 88.
Yor, Marc. Review of Weighing the Odds: A Course in Probability and Statistics, by David
Williams. (2) 77-78.
Books Reviewed
Aigner, Martin, and Ehrhard Behrends
(eds). Alles Mathematik: Von Pythagoras zum CD Player. Re
viewed by Manfred Stem. (3) 60-62.
Bruter, Claude P. ( ed). Mathematics and Art: Mathematical Visualization in Art and Education, reviewed by Helmut Aslaksen.
(4) 83-84.
Dhombres, Jean, and Robert, Jean
Bernard. Joseph Fourier, 1 768-1830: Createur de la Physique Mathematique. Reviewed by
Jean-Pierre Kahane. (4) 77-80. Feferman, Anita Burdman. From
Trotsky to Godel: The Life of Jean van Heijenoort. Reviewed
by John W. Dawson, Jr. (2) 78-79.
Kantor, Jean-Michel ( ed). Oil en sont les mathematiques? Re
viewed by Victor Guillemin. (3) 56-57.
Lessner, Joanne Sydney. See Rosen
blum, Joshua, and Joanne Syd
ney Lessner. (1) 77-78. Lo Jacomo, Fran<;ois. Visualiser
la quatrieme dimension. Re
viewed by Thomas F. Banchoff.
(3) 57-59. Robert, Jean-Bernard. See Dhom
bres, Jean, and Robert, Jean
Bernard. (4) 77-80. Rosenblum, Joshua, and Joanne
Sydney Lessner. Fermat's Last Tango, a Musical. Reviewed by
Michele Emmer. (1) 77-78. Sossinsky, Alexei. Noeuds: Genese
d'une Theorie Mathematique. Reviewed by Osmo Pekonen.
(1) 75-77. Williams, David. Weighing the Odds:
A Course in Probability and Statistics. Reviewed by Marc
Yor. (2) 77-78. Wilson, Robin. Four Colours Suffice:
How the Map Problem Was Solved. Reviewed by Charles
Nash. (4) 80-83.
At Heathrow Airport today, an individual, later discovered to be a public school teacher, was arrested trying to board a flight while in possession of a compass, a protractor, and a graphing calculator. Authorities believe he is a member of the notorious al-Gebra movement. He is being charged with carrying weapons of math instruction.
86 THE MATHEMATICAL INTELLIGENCER
k1£1 .. 1.19·h•t§i Robin Wilson I
Anniversaries The year 2002 saw three varied an
niversaries: the bicentenaries of
the birth of Niels Henrik Abel (1802-
29) and Janos Bolyai (1802-60), and
the centenary of the birth of Paul
Dirac (1902-84).
Abel's greatest achievement was to
prove that the general quintic equation
has no solutions by means of radicals.
Niels Henrik Abel
Abel's collected works and rosette
Please send all submissions to
the Stamp Corner Editor,
Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes,
MK7 6AA, England
During travels to Germany and France
he obtained fundamental results on el
liptic functions, the convergence of se
ries and "Abelian integrals," many of
which appeared in his 1826 "Paris
memoir." The story of Abel's attempts
to be recognised by the mathematical
community and his lack of success in
securing an academic post is a sorry
one. Tragically, his memoir was lost for
a time, and letters informing him that
it had been found and offering him a
job in Berlin arrived just two days af
ter his early death from tuberculosis at
the age of 26. The Abel stamps below
were issued for the Abel bicentennial
conference in Oslo in June 2002.
Bolyai was also slow in gaining
recognition. Along with Lobachevsky
Janos Bolyal
(but independently), he constructed a
"non-Euclidean geometry" -a geome
try that satisfies four of Euclid's five
basic postulates, but not the so-called
parallel postulate that there is exactly
one line through a given point and par
allel to a given line; in Bolyai's geome
try there are infinitely many such lines.
For almost two thousand years it had
generally been believed that no such
geometry can exist, yet the importance
of Bolyai's achievement was not fully
recognised until after his death. The
Bolyai stamps below were issued in
1960 and 2002.
In 1928 Dirac effectively completed
classical quantum theory by deriving
an equation for the electron that ( un
like those of Schrodinger and Heisen
berg) was consistent with Einstein's
theory of relativity. This equation ex
plained electron spin and led Dirac to
predict the existence of "anti-parti
cles," such as the positron, which was
detected four years later. The Dirac
stamps below were issued in 1982 and
1995.
e-mail: [email protected] Paul Dirac Cloud chamber track
88 THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK