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: a.ricotta@isac.cnr.it

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: L.Sallows@nici.kun.nl

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.

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Tool for ScienJific CreatiVJI)• J/1/Ce /98/ Free: 877-724-9673 • Email: info@mackidlan.com

[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: Colin.C.Adams@williams.edu

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.

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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

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34 THE MATHEMATICAL INTELLIGENCER

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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: hauser@math.tu-berfin.de

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: senechal@smith.edu

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.

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46 THE MATHEMATICAL INTELUGENCER

�.u�

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_ J/kr�hmt. �/-. ·�.Y "I' �I.Y/,.'.1/.)

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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'

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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

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B E R L I W . lO D O R ' B E R G - S T � A s e 2

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ft'-'W'tk .. � �� rp_ay.. 4-td � J- ..u tnu� .uJ� �'l 1.

48 THE MATHEMATICAL INTELUGENCER

••nt L L

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(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

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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

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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: lkloSinski@math.scu.edu

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: vakil@math.stanford.edu

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: pdehaye@stanford.edu

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. dford@math.stanford.edu

(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: imj@maths.ox.ac.uk

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:

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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 : dirk.huylebrouck@ping.be

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: pekonen@mit.jyu.fi

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: jean-pierre.kahane@math.u-psud.fr

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: cnash@thphys.may.ie

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: aslaksen@math.nus.edu.sg

84 THE MATHEMATICAL INTELLIGENCER

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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: r.j.wilson@open.ac.uk Paul Dirac Cloud chamber track

88 THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK