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.
4
Constant-Diameter Curves ameters could determine the circular The famous American physicist ity of a section. Leaving aside the tricky mechanical Richard P. Feynman, who disappeared in 1988, was an inexhaustible source of problems, let us consider the mathe inspiration for anyone who lmew him matical aspects. What diameter is re personally, because of his driving en ferred to? Is it the distance between thusiasm when dealing with any kind two parallel tangents to the border, or of problem. His books are filled with is it the chord of the curve ofFeynman's acute observations and problems, of Fig. 17 [1]? He doesn't explain. If you ten mixed with jokes to test the smart use a gauge caliber to measure the di ness of the reader. In one of them, [1], ameter you get the distance between on pp. 167-168, Feynman tells us that two parallel tangent planes, but if you when investigating the causes of the want to measure the chord of the curve accident of the Challenger space shut as in his Fig. 17, you have to know the tle, it occurred to him to consider the position of the equichordal point. Actually there is an incongruity be properties of what can be called con stant-diameter curves. He even shows tween the description Feynman gives a sketch, drawn by his hands, of a and the sketch he draws. Furthermore, he cites the example of a Reuleaux tri curve of this kind. In the mathematical literature these angle, which is a constant-width curve curves are referred as equichordal [2] but not an equichordal curve! Dulcis in curves. A historical example is the li fundo, he tells us a story of when he ma9on studied by Etienne Pascal, the was a kid and saw in a museum a mech father of the famous Blaise. The name anism with constant-diameter curves "lima9on" was given by G. P. de Rober turning on shafts that wobbled but that val (1602-1675), Blaise Pascal's con made a gear rack move perfectly hori temporary and friend, who also pro zontally. To do this, the gears would posed the concept of generalized have to be constant-width curves not conchoids, to which category these equichordal curves. curves belong. Recently a long-stand ing problem related to constant-diam eter curves was solved using tech niques from dynamical systems [3]. These curves should not be con fused with the constant-width [4,5] curves, but it seems that in Feynman's description they are not clearly distin guished. 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 de grees. But three matching diameters won't guarantee that things will fit; six This figure has all its diameters the same diameters, or any other number of di length-yet it is obviously not round. (Figure 17 from R. P. Feynman, What Do You Care ameters, won't do, either." First of all, it is a bit odd that NASA What Other People Think?, W.W. Norton & technicians would believe that three di- Co., N.Y., 1988, page 168.)
THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
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,
1.
issue 1 (1 997), 1 4 1 -2 1 2 .
4. D. Hilbert, S. Cohn-Vossen, Geometria Jn tuitiva ,
Figure
Boringhieri, Turin, reprint 1 967.
5. M . Gardner, Giochi Matematici, vol. 4, San soni, Florence, 2nd reprint 1 979.
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
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,
6. B. F. Chao, "Feynman's Dining Hall Dy namics," Physics Today 42 (1 989), no. 2, p. 1 5. 7. A. Ruina, M. Kuzik, "Feynman: Wobbles, Bottles and Ripples," Physics Today 42 (1 989), no. 1 1 , 1 27-130. 8. B. Kawohl, "Symmetry or not?", Mathemat ical lntelligencer 20
(1 998), no. 2, 1 6-22.
Angelo Ricotta ISAC-CNR
Via del Fossa del Cavaliere 100
001 33 Rome
Italy e-mail:
[email protected]
Hardy's Duncan Prize Book G.H. Hardy attributed his initial inter est in mathematics to competitive in stincts. In his Apology he wrote I do not remember having felt, as a boy, any passion for mathematics... I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively. .
Figure 2.
VOLUME 25, NUMBER 4, 2003
5
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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
THE MATHEMATICAL INTELUGENCER
1
:w
:11.1
6
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.
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
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
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: The true significance of the Sagrada Familia magic square is thus that it is
a monumental blunder: Subirachs has immortalized his nescience in stone. Lee Sallows Johannaweg 1 2 6523 MA Nijmegen The Netherlands e-mail:
[email protected]
A Tragic Square
Dis ont nt
Wo
Hardship
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
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
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.
8
I
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, [Lefh 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.
IA fhile 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
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 Ex perience [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 Weyl-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. 3. 4.
Ancocbea. G .. University of Salamanaca. Spain Borsuk. K.• University of Warsaw, Poland Cramer. H.• University of Stockholm. Sweden
S. Hlavaty, V.. University of Praaue. Czechoslovakia
6. 7. 8.
9. 10.
Whitehead. J. H. C.. University of Oxford. En&)and Gardina, L. J., Princeton Riesz, M., University of Lund, Sweden Lefachetz, S .. Princeton
Veblen, 0.. Institute for Ad vanced S1udy
II. Hopf, H.. Federal Technical School, Switzerland
12.
13. 14.
Newman. M. H. A., University of Manchester, EnaJand Hodae. W. V. D.. Cambridge. En&land
Dirac. P. A. M.. Cambridge Uni versity, En&)and
IS. Hua. L. K., Tsing Hua University, China
16. 17. 18. 19.
20. 21. 21.
23.
Tukey, J. W.. Princeton Harrold.
0. G..
Princeton
Mayer, W., Institute for Ad vanced Study Mautner, F. 1., Institute for Ad vanced Study GOdel, K., Institute for Advanced Study Levinson, N., Massachusens In stitute of Technology Cohen, I. S., University of Penn sylvania Seidenberg. California
A.,
University of
24. Kline, J. R., University of Pennsylvania
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 Califor nia
31. Bargmann, V., 32. Jacobson, N.. 33. 34. 35.
Princeton The Johns Hop
kins University Kac, M., Cornrll University Stonr. M. H .. University of Chicago von Neumann, J., lnstitutr for Advanced Study
36. Hedlund, G. A., University of
37. 38. 39. -40. 41. 41. 43.
Virginia Zariski,
0.,
University of Illinois
Whyburn, G. T.. Univrrsity of
Virginia.
McShane, E. J.. University of Vir&inia Quine, W. V., Harvard Wilder, R. L., University of Michipn
Kaplansky, 1 .. lnstitutr for Ad vanced Study Bochner, S .. Princeton
44. Leibler. R. A .. Institute for Ad vanced Study
45.
Hildebrandt, T. H .. University of Michipn
46. Evans. G. C., University of Cal ifornia
47.
Widder, D. V .. Harvard
48. Hotelling, H., University of Nonh Carolina
49. Peck, L. G., Institute for Ad vanced Study
50. Synge. J. L., Carnegie Institute of Technology
Sl. Rosser, J. B .. Cornell
60. Hurewicz, W., Massachusetts In stitute 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 65.
dleton, Louisville, Ky.
Hille, E .• Yale University
66. Alben, A. A., University of Chicaao
67.
Rado, T., The Ohio State University
68. Whitney, H., Harvard 69. Ahlfors, L. V., Harvard 70. Thomas, T. Y., Indiana
Univer
71. Crosby, D. R.• Princrton 72. Weyl, H., lnstitutr for Advanced sity
Study
73. Walsh, J. L.. Harvard 74. Dunford, N., Yale 75. Spenser, D. C., Stanford Univer76. 77.
sity
Montgomery, D., Yale 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.
81. 83.
Steenrod, N. E.. University of Michipn Wilks, S. S., Princeton Boas, R. P., Mathematical Re views, Brown Univenity
52. Murnaghan,
84. Doob, J. L., University of Illi nois
53. 54.
86. Zygmund, A., University of Pennsylvania
F. D., The Johns Hopkins University Mac lanr, S., Harvard
Cairns, S. S., Syracuse Univrr sity
SS. Brauer. R.. University of Toronto, Canada
56. Schoenbei'J, I. J .. University of Pennsylvania
57.
58. 59.
85.
Feller. W., Cornell University
87.
Anin, E., Princrton
88. Bohnenblust. H. F.. California Institute of Trchnolosy
89. Allendoerfer, C. B.. Haverford College
Shiffman, M., Now York Univrr sity
90. Robinson, R. M.. Princeton
Milgram. A. N.. Institute for Ad vanced Study
92. Beglr, E. G.. Yale
Walker. R. J .. Cornell
91.
Jkllman, R.. Princeton
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 opening up of new fields and by increasing specialization. This has been bal anced by interludes of common activ ity among related fields and the de velopment in common of broad general ideas. Just as for science as a whole, so in mathematics. As many historical instances show, the bal anced development of mathematics re
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 Mathe matics in America duly noted:
The world war had just ended, math ematicians had returned to their uni versity positions, and large numbers of veterans were beginning or resum ing graduate work. It was a good time to take stock of open problems and to try to chart the future course of re search [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 intellec tual ravage caused by the war years, it seemed exceedingly desirable to have as many participants from abroad as possible. As the list of mem
quires both specialization and gener
bers shows, considerable success was
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 eas ily by doing both. The increasing tempo of modern research makes these interludes of common concern and as sessment come more and more fre quently, 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 unifi cation [Lefschetz 1947, p. 309}.
attained in this. Our conference be came, as it were, the first interna tional gathering of mathematicians in a long and terrible decade. The mani fold contacts and friendships renewed on this occasion will, we all hope, in the words of the Bicentennial an nouncement, "contribute to the ad vancement of the comity of all nations and to the building of a free and peace ful 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
11
Solomon Lefschetz was impulsive, frank and opinionated; enough so that many found him
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
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
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 Weyl 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 1 944} 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 un precedented pool of mathematical tal ent, 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 Lef schetz, in which William Hodge and Os car Zariski served as discussion lead ers. 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. Lef schetz, commenting on Zariski's pre sentation, 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?" Lef schetz 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 unify ing theory. He pointed to Hilbert's still unsolved sixteenth problem on the nesting configurations for the compo nents 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 prob lems. Oddly enough, Hilbert's tenth Paris problem, the decision problem for Diophantine equations (proved un solvable by Yuri Matijacevic in 1970) was not even mentioned, though it was only in the 1930s that the notion of a computable algorithm became tract able. Church called attention to the re cent 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 con jecture, noting that "our knowledge of these matters is practically nil." Further discussions on mathemati cal 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 prob ably denumerable. A philosophical de bate 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 informa tive. 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 or ganizers' 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 "alge bra would lose much power."
Taking Stock Having touched upon the overall at mosphere at this meeting shortly be fore 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 cente nary. The following year saw the pub lication of the second volume ofA Cen tury 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 pur pose of the Princeton meeting-which aimed to cast its eye on what the fu ture 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 dif ferent 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 practition ers forty years earlier. Several noted that some highly significant work al ready published before 1946 received no attention at the Princeton confer ence. Thus, Robert Osserman was as tonished 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 particu larly illuminating, given that he had par ticipated 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 mathe matical 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. Infact the relationship be tween the two subjects inspired heated fruitless discussions of "What is prob ability? " 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}.
by quoting himself in 1931 when he of fered these remarks at a conference in Bern:
[Duren 1 988] Peter Duren et at. , eds., A Cen
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
Before one can generalize, formalize, or axiomatize, there must be a math ematical 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.
[Duren 1 989] Peter Duren et a/. , eds. , A Cen
Despite the tumultuous political events that had intervened, Weyl's views in 1946 reflected much the same opinion:
sity Bicentennial Conferences, Series 2, Con
tury of Mathematics in America ,
vol. 1 , Prov
idence, R. I . : American Mathematical Soci ety, 1 988. 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 1 947] Solomon Lefschetz et at. , eds., Problems of Mathematics,
Princeton Univer
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,
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.
2 1 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 98 1 ] Nathan Reingold, "Refugee
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.
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 Trends and Old Images in the History of Mathematics,"
REFERENCES
[Birkhoff 1 938] G. D. Birkhoff, "Fifty Years of
in Vita Mathernatica. Historical Research and
American Mathematics, " in Semicentennial
Integration
Addresses of the American Mathematical
Calinger, MAA Notes Series, vol. 40, Wash
Society,
ington, D.C.: Mathematical Association of
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,
Teaching,
ed.
Ronald
America, 1 996, pp. 3-1 6. [Rowe 2003a]
--
, "Mathematical Schools,
Communities, and Networks," in [Nye 2003, pp, 1 1 3-1 32].
3-58]. [Browder 1 976] Felix Browder, ed. , Mathemat ical Developments Arising Problems,
from
Hilbert's
Symposia in Pure Mathematics,
vol. 28, Providence, R. 1 . : American Mathe matical Society, 1 976.
[Davis & Hersh 1 98 1 ] P. J. Davis and Reuben Hersh,
with
The
Mathematical
Experience,
Boston: Birkhauser, 1 981 . [Doob 1 989] J. L. Doob, "Commentary on Probability," in [Duren 1 989, 353-354].
[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
Co n seq uences of Revers i n g Prefe rences
ther than standard election disruptions involving shenanigans, strategic voting, and so forth, it is reasonable to expect that elections are free from difficulties. 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 1 first choice. 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
17
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
4
Prefer A>C>B
3
A>B>C
4
B>C>A
3
C>B>A Total
A
4w1
3w1 0 0
7w1
-
0
4w2
4w1
4w2
3w2 3w2
+ 6w2
0
(1)
3w,
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 4
Prefer
A
B>C>A
0
4
A>C>B
Number
3
C>B>A
3
A>B>C Total
0
4w,
3w1 7w1
B
c
4w1
4w2
0
4w2
3w2 3w2
4w1
+ 6w2
3w1
3w1
(2)
0
+ 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 = 4w i + 6w2 = 3w i + 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-
18
THE
MATHEMATICAL INTELLIGENCER
./(ui,j (p))
=
ffi, j Cf(p)).
(3)
In words, if 2 1 1 voters confused Sue with Mary when mark
ing the ballot (instead of the correct profile p, they used
c
B
4w1
metry embedded in the profile's two pairs of two rankings. The flrstpairis the rankingA>C>B with 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 and fs 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,
(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 = Wi i
(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 w 1 - 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, W t + Wn, Wz + Wn- b Ws + Wn - 2, · · · W t + 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) + 2w t ] > 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 trun cated 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 " 1 1" 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.
Ranking
No.
Ranking
5
A>B>C
3
C>B>A
4
A>C>B
11
B>C>A
C>A>B
0
B>A>C
VOLUME 25, NUMBER 4, 2003
(6)
19
C 5 + 15s
4 + 15s C
A 9+s
B 11 + 8s
"'---..._--�
10 14
a. Profile
A 14 + s
B 5 + 8s
---..._--�
14 10
b. Reversed 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, ,Q), and the (7,5,0) method becomes (l, ,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 = 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.
t
t
t•
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
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' second place 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 12. age of Borda points assigned to the candidates is 36/3 =
Profile Original, Fig. 1a Reversed, Fig. 1 b
8
A
c
9.5 - 1 2
=
-2.5 1 5 - 1 2
1 4.5 - 1 2
=
2.5
9 - 12
=
3
1 1 .5 - 1 2
=
-0.5 (7)
=
-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 pro file p, the average Borda score is C n)/3 = �· so point (�, �' � ) on the diagonal x = y = z indicates each candi
%
date'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) pro cedure 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-
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
(2��,
���' 2�0). Geometrically, the normalized tally is a point in l
the simplex { (x, y, z) x + 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
A
a. Profile line and transfer
-
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
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 representa tion. The length of the line connecting the (1, s, 0) out comes is 1 1- times the length of the line connecting plu+s rality outcomes, or 2 1 times the length of the line con( s) 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 candidates 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.
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 antiplurality 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)
{6) , 6 c
A &......
{ 14) , 4
a.
Figure 3. Creating a profile.
22
THE MATHEMATICAL INTELLIGENCER
_..;a B
____....._ ..__
_
{ 10), 5
b.
R (p) .
a.
b.
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 dot ted line and 3/2 as far as the distance between Borda tal lies. As these two points define a straight line, they deter mine (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 ori entation 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 posi tional 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 al lowed 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-can didate 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 top two 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 top winner reversal bias. That is, the same candidate can win with p and (!h(p) but otherwise the rankings dif fer; 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 pro file p and (!h(p) is that the Borda ranking be a com plete tie.
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 speci fied number of regions. To verify the second part, which asserts there is a profile p with an A>B>C plurality out come 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 con struct. 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 suf ficiently 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 pro cedure'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 re versal 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 pro file 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 sec ond-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 (s 1 = 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, 1 1 , 4), while the antiplurality tallies for \Yt(p) are (15, 13, 20). It is immediate that (9, 1 1 , 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 con structed 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 re lationships 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 de termine 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 re versal effects, and the number of positional method out comes (the number of regions the positional line crosses) are also specified. Number of outcomes
p Antlplurality
01(p} Plurality
A>B>C
C>B>A
3
A>C>B
B>C>A
no reversal effects
5
C>A>B
B>A>C
two-winner reversal
Reversal biases no reversal effects
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 out comes2, we obtain the likelihoods of different reversal be haviors. 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 com mon variance, and the mean has an equal number of vot ers of each type. 1. A necessary and sufficient conditionfor 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 behav ior is 0.31. 2. A necessary and sufficient condition for a reversal ef fect to occur for the plurality outcome is that a pro file'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 as sertion and smaller likelihoods for the other two. To ex tend 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
components dominate a profile, the more dramatic the re versal 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
Three-candidate Profi le Subapacea
Tr
at an n-voter profile for a t h r e-candidate election as a v tor in R' ,.,;th non-negative int ger c mpon nt. . Pr fil ha n d mp com n nt d int U1 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 app r priat c . ordinat ·y m which
•
.
·
M
· -
.
•
i,(3!)
=
{p
=
CPt,
. • .
P ) E
6
/(>
- Pi =
J
I
1t ,
PJ �
0],
thro ugh 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.
O + Os C
C 1 1 - 30s
The first h
files. i
o-dimen ional u bspace th Basic Pro panned by B,1 and Ba.
- 1 + Os C
2
c -1 + 0
A .a.::;...--.L.....-...;::,jO B 2 Os -1 +0 a. A-Basic, B A
Th
A ""'----:-L....:-...,. B
0 + Os
e. Condorcet, C
0 + Os
A �---'L---� B - 1 3 + 36s - 1 1 2-6 r. An example
.. _.L..._ .. .....;::,jO B A .a.::;._ 2 Os - 1 + Os b . B-Bs.sic, Bs
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 itional out om for Basic pro an oc alway. ar consi "t nt-no voting conflic ur h r . onfli t tart wi1h lhc two-dim nsional Re1•r.rsal ub. pact> pann d b R., and RH. •
-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
.
1+2
2
c
A 4
C I
B
1 + 2s c.
A-Rever81ll , R"
2s
Z.•
'}
-
A 1 +2
B 2 - 4s d. B-Revcrsal, Rs
VOLUME 25, NUMBER
4,
2003
25
procedures are affected by reversal effects. This assertion
profile. The average number of assigned points per candi
follows from a convenient decomposition of profiles which
date for (1,
allows
us
to analyze all possible positional and pairwise
elections [Saari 1999].
s, 0) is [(6 + 3s) + (3 + 6s)]/3 = 3 + 3s, so each (3, 3s, - 3 - 3s). Be cause these differences change with the s value, it means
candidate's tally minus this average is
This decomposition expresses any profile as a union of
that the profile has a Reversal component. (This is not ob
profiles of four types. To start, a "Neutral" configuration
vious; it illustrates a case with a fractional coefficient.) By
t) pivot point,
has the same number of voters assigned to each of the six
comparing the differences for the antiplurality (s = 1) with
rankings. Second, a "Condorcet" profile configuration af
the Borda
fects only pairwise rankings; it is given by Condorcet
-4.5) = (1, 1.5, - 1 .5) components show that the effect of
(s
=
the
(3, 3, - 6) - (3, 1.5,
triplets such as A >B>C, B>C>A, C>A>B, or its reversal.
this hidden Reversal term is to create a bias for the an
While such a triplet (a Z3 orbit of a ranking) has no effect
tiplurality outcome favoring B at the expense of A and C.
on positional rankings (because each candidate is ranked
Indeed, this distortion causes the antiplurality ranking of
first, second, and third once), it causes pairwise cycles; in
A - B>C to conflict with the A >B>C conclusion for all
part these cycles arise because z3 does not admit symme
other positional procedures.
tries (that is, a subgroup) of order two. More is said about
To make these statements more concrete, we modify the
this below. The "Reversal" configuration is created by us
Fig. 5a profile to create a p with a plurality top-winner re
ing pairs consisting of a ranking and its reversal; these pro
versal bias; p 's plurality ranking will be B>A > C and 0Jt(p )'s
file components (the Z2 orbit of a ranking) do not affect
plurality ranking will be B>C>A. To achieve this goal, we
pairwise rankings but, as demonstrated above, they affect
need to add reversal terms. That is, select y and z values
positional outcomes. The remaining "Basic" configuration
from Fig. 5b so that adjoining it to the profile of Fig.
requires all positional and pairwise rankings and tallies to
5a gives the plurality B>A > C and antiplurality A >C>B
agree.
rankings.
Using a vector approach, all three-candidate profiles can
Adding each candidate's tallies from the Figs. 5a, b tri
be decomposed into components of these four types where
angles, the desired B>A > C plurality outcome and A>C>B
the coefficients may be fractions. As demonstrated by Eq.
antiplurality outcomes occur, respectively, if and only if
5, only the Reversal and Basic directions affect positional
z + y +
outcomes; only the Basic directions affect the Borda Count. Consequently all differences between the Borda and any
6 + y >z >
[Saari 1999].
y
x=
= z diagonal. So, for all positional methods the difference
(10)
9 + y + 2z > 2y > 9 + y + z,
cedure line. (Details can be verified by using [Saari 1999].) ated by the Basic portion of a profile is parallel to the
3,
and
components geometrically affect the positioning of the pro The procedure line (in a three-dimensional space) gener
> y + 6 > z,
or
other positional ranking are strictly due to Reversal terms The new twist added here is a description how these
3
or 9 + z > y > 9.
( 1 1)
of each candidate's Basic tally from the average number of
The simplest choice of y = 10, z = 4 defines the p in Fig.
assigned points is the same. All differences in election out
6a; 0Jt(p) is in Fig. 6b. While the Fig. 6a plurality ranking
comes, then, are introduced by Reversal terms; they pivot
changes from that of Fig. 5a, the Borda ranking remains the
the procedure line about the Borda outcome. If a strong
same, reflecting Borda's immunity to reversal terms.
Reversal component creates a B>C>A plurality outcome,
This construction can also be used to demonstrate how
for instance, then the pivoting of the procedure line cre
the "size" of the Reversal term affects the 0Jt(p) outcomes.
ates a tendency for the other endpoint-the antiplurality
While all Eq. 10 choices define a p with a plurality B>A > C
outcome-to define the opposite A >C>B outcome.
ranking, w e know from the properties o f the procedure line
To illustrate by creating examples, start with the Fig. 5a
O + Os C
A &....-----""---.....;::a B 6 3 3 + 6s 6 + 3s a. Starting profile
Figure 5. Creating examples.
26
THE MATHEMATICAL INTELLIGENCER
that different Reversal components generate different 0Jt(p)
C z + 2ys
A &....-----""---.....;::a B y + 2zs y+z b. Reversal terms
C13 + 20s
4 + 20s C
A c...-. ....--� ... B
20 17 16 + 21s a. Profile p
17 + 6s
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 � 1 4 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
!'lt(p) plurality ranking
0 :S y :S 8
C>B>A
y=9
C - B>A
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 of A > 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 P I 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 ·
·
·
-
Reversal behavior
1 0 :S y :S 1 2
B>C>A
Top-winner
y= 13
B>C - A
Top-winner
y :0: 1 4
B>A>C
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 -
VOLUME 25, NUMBER 4, 2003
27
3s C
C 3 + 6s
A �----'---� B 2 + 2s 4+s a. Profile P l is where x = 0
A �--�--� B 7 8 7 + 4s 5 + 5s b. Profile p = Pl + P2
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 partic ular, 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 re versal problem. So while a Borda runoff can suffer a top reversal 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 R m . 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 out comes 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 R m centered at the origin 0 called the representa tion 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 coor dinates 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 corre sponds to A>B, B>C, A>C or A>B>C. The two remain ing vertices, (1, 1, 1) and ( - 1, - 1, - 1 ), correspond to cyclic rankings.
The representation cube is the convex hull of the six la beled 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 neg ative orthants; the points in these two orthants are the pair wise cyclic outcomes that can cause problems. Indeed, the
� � 7b choice of p defines the point C 8 7 8 � 1 1 ) ; as all components are negative the election, rank, 5 ings 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 com15-voter Fig. 4
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 candi date. 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 in stance, 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 R C �) = R3 de termines a specific agenda winner. But 3! = 6 of these eight orthants represent transitive rankings where the top ranked 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
28
THE MATHEMATICAL INTELLIGENCER
=
3, and a similar approach holds for all
n.
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 dis covered a surprising array of unexpected behaviors al lowed 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 candi dates). 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 dif fers 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 met ric. 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 substan tial differences. Such a situation requires cyclic symmetries [Saari 2000a] . Combining these two notions, examples are immediate; e.g., the next profile repeatedly uses the Con dorcet {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 pref erences, and she is bottom-ranked in the others. Number 10
Ranking
Number
Ranking
A>B>C>D
9
B>C>D>A
10
A>C>D>B
9
C>D>B>A
10
A>D>B>C
9
D>B>C>A
(1 3)
table 14, the revised rankings allow A to beat each of the other candidates by 29:28 to become the Condorcet win ner. Thus, Dodgson's method admits a top-winner rever sal bias. Other methods, such as the ones developed by the math ematicians Borda [ 1781], Copeland [ 1951], and Kemeny [ 1957], and Dodgson's method for n = 3 do not have a re versal bias, because these methods replace the actual pair wise rankings with the "nearest" transitive ranking. For in stance, 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 be tween 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 transi tivity 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 rep resentation 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 be cause, seemingly, only counting is involved. From a math ematical perspective, however, everything becomes delight fully 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-
A is the Condorcet winner by beating the other candidates with a 30:27 tally. Here, ffi(p) is Number
Ranking
Number
Ranking
10
D>C>B>A
9
A>D>C>B
10
B>D>C>A
9
A>B>D>C
10
C>B>D>A
9
A>C>B>D
(1 4)
where A is the Condorcet loser since she loses to each op ponent 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
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, Gib bard [ 1973] and Satterthwaite [ 1975] proved the amazing result that all reasonable election procedures for three or more candidates admit situations where some voter, by vot ing 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 sus ceptibility decreases as s (According to this theorem, the plurality vote is highly susceptible to strategic behav ior. We know this; just recall those "Don't waste your vote" calls for strategic action voiced during close elections in volving more than two candidates.) But as s - a proce dure becomes less susceptible to the Reversal components. Is there a connection? Probably, but it has not been estab lished.
i·
i
,
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
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
A U T H O R S
DONALD G. SAARI
USA
30
THE MATHEMATICAL INTELLIGENCER
voix, Paris.
[3) Copeland, A. H. 1 951 , A reasonable social welfare function.
.,..,_.. BARNEY
n..�I11V'..,I of t.A.�IIv>rn. ..ll...,.,_
USA
[ 1 3] Riker, W. H . , 1 982, Liberalism Against Populis m, W. H. Freeman,
[21 ] Saari, D. G . , and V. Merlin 2000, A geometric examination of Ke meny's rule, Social Choice & Welfare 17, 403-438.
San Francisco. [1 4] Saari, D. G., 1 992, Millions of election rankings from a single pro
[22] Saari, D. G . , and J. Van Newenhizen, 1 988, Is Approval Voting an
[1 5] Saari, D. G. 1 994, Geometry of Voting, Springer-Verlag, New
[23] Saari, D. G. and M. Tataru 1 999, The likelihood of dubious elec
file, Social Choice & Welfare (1 992) 9, 277-306.
"unmitigated evil?"
Public Choice 59,
1 33-1 4 7 .
tion outcomes, Economic Theory 13, 345-363.
York. [1 6] Saari, D. G. 1 995, Basic Geometry of Voting, Springer-Verlag, New
[24] Satterthwaite, M . , 1 975, Strategyproofness and Arrow's condi tions, Jour. Econ. Theory 10, 1 87-21 7.
York. [1 7] Saari, D. G. 1 999, Explaining all three-alternative voting outcomes, Journal of Economic Theory 87,
[25] Tabarrok, A., 2001 , Fundamentals of voting theory illustrated with the 1 992 election, or could Perot have won in 1 992? Public Choice
31 3-355.
106,
[1 8] Saari, D. G. 2000a, Mathematical structure of voting paradoxes 1 ; pairwise vote, Economic Theory 15, 1 -53.
avoided the Civil War? Journal of Theoretical Politics 1 1 , 261 -288.
[1 9] Saari, D. G. 2000b, Mathematical structure of voting paradoxes 2:
positional voting.
Economic Theory 15,
[27] Tataru, M . , and V. Merlin 1 997, On the relationships of the Con
55-1 01 .
[20] Saari, D. G. 2001 , Chaotic Elections! A Mathematician Looks at Voting,
275-297.
[26] Tabarrok, A. and L. Spector 1 999, Would the Borda Count have
American Mathematical Society, Providence, Rl.
dorcet winner and positional voting rules, Mathematical Social Sci ences 34,
81 -90.
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VOLUME 25, NUMBER 4. 2003
31
M a the m a tic a l l y B e n t
C o l i n Ad am s , Ed itor
Don't Touch the Button Colin C. Adams
The proof is in the pudding.
Opening a copy of The Mathematical Intelligencer you may ask yourself
uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am I?" Or even "Who am I?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01 267 USA e-mail:
[email protected]
32
''
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
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?"
derstand now what you said to us to
Spreading them on her desk, she
"The belly button obsession. Arman
day. I talk to my wife. I know now what
stared at them, trying to see something
Lobindi, used to be in your office. Did
you mean." And Misha winked at him.
in the belly that would make it extra
PDE's, or at least used to until he got
Fustrum darkened.
ordinary.
"Help yourself to more drinks," he
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
there was someone behind her. Turn
said, and he was gone. Misha looked confused. "Subject is
ing quickly, she found the vice-chair of
verboten, Misha," said Jamal. "Don't
the department in her doorway. She stood, hoping to block the photos from
bring it up. " "Oh", said Misha, looking worried. "We do not speak of this anymore?" "Right," said Jamal. ''I'm going to get
"I remember stopping by his office
The first semester went well. Lisa
when he wasn't there, and seeing his
was teaching one large lecture of mul
blackboard
pictures.
tivariable calculus and a small gradu
Took a while for me to figure out what
ate class on set-theory. She only saw
covered
with
they were." "Yes, what were they?" "Hypothetical belly buttons, of all types.
Innies,
outies,
normal,
de
his view. "Hi,
Bob,"
she said, a little too
brightly. "The chair wants to see you," said
some hors d'oeuvres."
called it, the button.
As she rearranged the pic
tures, she suddenly had the feeling
Lindstrom evenly, with the slightest smile at the edges of his mouth. "Oh, okay", said Lisa, "I'll be there in just a minute."
the chair at department meetings and
As soon as he was gone, she
at colloquia. At those events, she would
scooped the pictures into the file and
always sit in a position that allowed her
slipped it back into the rear of the file
furtive glances of Fustrum's midriff. At
drawer where she had found it.
formed. He had them listed in cate
one colloquium he was wearing a
Then she shut her office door, and
gories with probabilities associated to
button-down shirt. She sat in the same
walked down to the chair's office. The
each one. It was sad."
row as he did, five seats to his right. At
secretary told her to have a seat on a
one point, he shifted in his seat, and
bench in the hall. She had been there
"So what happened to him?" asked Lisa.
the front of his shirt separated just a
about fifteen minutes when Lindstrom
"He didn't get renewed. Ostensibly
bit between the two bottom buttons.
came by.
because his research came to a stand
Lisa found herself staring into a dark
"Oh, I'm sorry", he said, still smiling
still. But I think it had to do with an in
ened opening that contained the infa
just slightly, "but Karl was going to
cident in the men's room."
mous belly button. She leaned forward
meet with you at your office." Lisa
"What incident?"
with her mouth open in anticipation.
jumped up. "Oh", she said nervously, as
"He used to follow the chair in there,
Then Fustrum shifted again.
As she
hoping to get a glimpse. I can only
looked up, she found him staring back
guess he peeked over the stall and Fus
at her, with a grim face.
trum decided that was enough. But no body knows for sure." Misha, who had been talking to a
In the spring, she taught two sec
she headed as quickly as she could back to her office. She found the door open. Fustrum was leaning back in her office chair
tions of linear algebra. With only one
with his feet up on the desk He had his
course prep, she found more time to
hands behind his head and seemed to
topologist across the room, waved and
get her own research done. One day,
be sleeping. The file drawer that con
j oined them.
when trying to find a preprint in her fil
tained the pictures was open.
"I know now what is this belly but
ing cabinet, she noticed a manila folder
She tried to slip carefully behind
ton. My wife, she explain this to me."
wedged in the back of the drawer. It
him to see if the file was still in the
Misha
bored-looking
was labeled "Button" in large block let
drawer. But as she did, he was startled
woman seated in the comer reading a
pointed
ters. She pulled it out and opened it.
awake. He lost his balance and the
magazine
Inside were a variety of photos. Most
chair he sat in slid out from beneath
seemed to have been taken through a
him. Lisa watched in horror as his head
and
to
a
ignoring the
people
around her. Fustrum turned away from the vice
small hole. Several were pictures of
caught the comer of the drawer. The
chair and strode toward them, a wel
Fustrum at various events. Others were
office chair careened across the room,
coming smile on his face. Lisa forced
not obviously of Fustrum, but there
knocking her to the floor, and Fustrum
herself to look him in the eye, and not
could be no question. Two or three
fell solidly next to her.
to look down toward his advancing
were shots of a slight crack between
She lifted her head slowly off the
belly.
the buttons in the front of a button
floor. The chair's turtleneck had be
"Hello, Lisa, Misha. Found your way
down shirt. At least one photo ap
come untucked in the fall, and his
peared to have been taken through the
belly, the one that always kept the shirt
meet some of the other faculty. Jamal,
crack of the door of a bathroom stall.
tight as a drum, was exposed. There be
are you introducing them around?"
In all the shots, the belly button was
fore her, just at the crest of his stom
here all right? I hope you are getting to
"Oh, don't worry. I'll keep an eye on them." Misha smiled at the chair. "Yes, I un-
impossible to make out in any detail
ach, was the famous belly button. She
whatsoever. But Lisa found herself fas
gulped as she stared at it. It appeared
cinated nonetheless.
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
"And you don't care that you were
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.
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.
"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."
N EW f6 FO RT H C O M I N G from Birkhiiuser Stochastic Calculus
Applications in Science and Engineering M. GRIGORIU, Cornell University, Ithaca, NY
This work analyzes and presents solutions for a wide range of stochastic problems in applied mathematics, physics, engineering, finance, and economics. A user friendly, systematic exposition covers the essentials of probability theory, random processes, stochastic integra tion, and Monte Carlo simulation. The Monte Carlo method is used to illustrate theoretical concepts and numerically solve problems; this text approaches stochastic problems both analytically and numerically. It may be used in the classroom or for self-study by researchers and graduate students. TABLE OF CONTENTS: Introduction • Probability Theory • Stochastic Processes • Ito's Formula and Stochastic Differential Equations • Monte Carlo Simulation • Deterministic Differential, Algebraic, and Integral Equations Input •
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Deterministic Systems with Stochastic
Stochastic Operator and Deterministic Input
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On the Geometrica and Physical Mean i ng of Newton ' s So ution to Kep er' s Prob em
•
n the short treatise
�matica
De Motu
(1 684) , which serves as a precursor to the
Principia Mathe
(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 (6, p. 38-39 and p. 46-48]. There Problem B reads as follows.
De Motu
Problem
4. Supposing that the force is inversely pro portional to the square of the distance from its centre, and with the quantity of that force known, there is re quired 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 New ton is concerned only with ellipses, his reasoning is never theless 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 al lows us to refer directly to the passages quoted in [4]. This article is not meant as a historical study of possi bly 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 posi tion 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
the conic section is identified via its semilatus rectum from
which the orbit of P is an ellipse resp. hyperbola resp.
the given central force and the initial data.
parabola. The two main ingredients of the proof are (a) his
In [9] a different interpretation of Corollary I is proposed
solution of Problem A in Proposition XI resp. Proposition
without reference to Proposition XVII. Relying on the no
XII resp. Proposition XIII [ 7, p. 56-6 1 ] , and (b) the appli
tion of curvature, an alternative construction of the conic
cation of the uniqueness theorem for the initial-value prob
section is carried out.
lem. The latter states that, given the initial data, there can not be more than one orbit of P that goes through the initial
Is Proposition XVII a Solution to Problem B?
position of P with the given velocity vector. (This fact must
Two objections have repeatedly been brought forward. (i) It
have been known to Newton, as pointed out below.) Re
is alleged that Newton does not prove that the orbit is a conic
lying on (a) and (b), Newton proceeds as follows. He con
section, but rather assumes this from the outset (see, e.g., [5,
structs a conic section with focus S from the initial data
pp. 133, 134, 136]). However, this is incorrect. For the proof
P is the given veloc
strategy sketched above clearly shows that Newton con
ity vector, and (ii) that the centripetal force associated to
structs a conic section which-by virtue of the uniqueness
this conic section according to (a) coincides with the given
theorem (b)-must be the sought-after orbit. This argument
centripetal force. Hence the conic section so constructed
is emblematic of a method often applied in physics when an
is an orbit of P, and, by virtue of (b), it is indeed
initial-value problem is solved by adjusting the parameters of
such that (i) its tangent in the point
sought.
the orbit
a putative solution to the initial data. (ii) The second objec tion holds that Newton did not know the uniqueness theo
Corollary after Proposition XIII
rem (b). This goes under the title: "Did Newton prove that
Although not relevant for the purpose of this article, we
orbits are elliptic?" [3, p. 30ff]. As vigorously argued by
mention this Corollary [7, p. 6 1 ] because it has been inten
Arnold, this is a chimerical problem, because in the times of
sively debated in the literature. Having proved Propositions
Newton the function defining a differential equation was au
XI, XII, and XIII, Newton remarks,
tomatically supposed to be analytic in its domain of defini tion, and therefore uniqueness was no problem (cf. [3, p. 31f]
Cor. I. From the three last Propositions it follows, that
and [4, p. 1 12]). Besides, what Newton proved in Propositions
if any body P goes from the place P with any velocity in
XLI and XLII goes much further than uniqueness.
is urged by the action of a centripetal force that is in versely proportional to the square of the distance of the
Relation of the Proof of Proposition XVII to the Notion of Energy
places from the centre, the body will move in one of the
Even though Newton's geometrical construction of the or
the direction of any right line
PR, and at the same time
conic sections, having its focus in the centre of force;
bit of P is fully correct, its physical meaning is not obvious
and conversely. For the focus, the point of contact, and
at first sight. As pointed out by De Gandt [6, p. 49],
the position of the tangent, being given, a conic section may be described, which at that point shall have a given
This quotient depends on the initial conditions, as may
K in different
curvature. But the curvature is given from the centripetal
be seen from what happens to the point
force and velocity of the body being given; and two or
possible configurations. But it is not easy in this case
bits, touching one the other, cannot be described by the
in contrast to an analytico-algebraic mode of presenta
same centripetal force and the same velocity.
tion employing differential equations and constants of integration-to see how the initial position and velocity
At this place in the
Principia no further details are given.
enter into the expression (2SP +
2KP)IL.
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
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
ted it in the first edition & contented myself with adding
in the spirit of Newton's proof, we will show that PN is in
the XVIIth Proposition whereby it is proved that a body
fact the absolute value of the ratio of kinetic to potential
going from any place with any velocity will in all cases describe a conic Section: which is that very Corollary.
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
Following Newton, Chandrasekhar [4, p. 102f] also con
PN is of course not a function of the additive total energy,
ceives Corollary I in the light of Proposition XVII. On this
and conservation of energy plays no role in Newton's proof
view mention of the notion of curvature in Corollary I is
of Proposition XVII. This idea enters only later in the much
not at all astonishing. Without a doubt, Newton (cf. [4, p.
more general context of the initial-value problem for cen
1 10]) was aware of the simple relation between the curva
tral forces of any kind, see [7, p. 128-134].
ture and the semilatus rectum of a conic section (for the
In the next section we recall the area law. After that, we
latter term see below); and in the proof of Proposition XVII
note elementary properties of conic sections and state
36
THE MATHEMATICAL INTELLIGENCER
Newton's answer to Problem A without proof. (An exposi tion 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 moti vation. 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 Chandrase khar'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 nec essarily 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 propor tionality factor by 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 t 1 , and P2 its posi tion 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
i
1
ISP1P2 1 = 2
(1)
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 ab solute 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 geo metrically, 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)
v=
c . SM
In similar fashion Newton uses the area law for express ing the acceleration a geometrically, in this case via a com parison with Galileo's law of falling bodies and subsequent elimination of (!:l.t)2 by virtue of (1). This yields an ex pression 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 eccentric ity E (0 ::5 E < 1) is defmed by OS = E · a. Given the semi major 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 = 2 = (e - 1)a2. E · a. The semiaxis b is now given via b 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
+ijiijiiijiW
c
A 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 pa rameter 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
b2 p = -. a
(3)
To prove this, apply Pythagoras's Theorem to the triangle With the upper (lower resp.) sign representing the case of an ellipse (hyperbola resp.), one obtains
SCS'.
(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 mod ified form. Substituting SS' = 2m and SP :±: PS' = :±: 2a, gives
SS' 2
(4)
=
(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 centripetalforce. Assume that the orbit of P is a conic section with fo cus S. (In the case of a hyperbola the orbit of P is as sumed to be the branch with focus S.) Denote the semi latus 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
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 k = c2Jp.
(5)
Newton's point of departure for his proof is the de scription 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].
Proof
Remarks.
(a) Let a conic section C with focus S be given. We would like to apply Theorem A To this end we intro duce 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 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 cen tripetal. 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 in fluence 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 sec tion [4, p. 1 27-142]. His profound mathematical investiga tions are once more motivated by the area law. Naturally, with respect to physics, the latter is of the utmost impor tance in the Principia as well, for example, for Newton's understanding of the concept of force (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 im portant 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 be tween the tangent and the line PS'. The situation is analo gous for parabolas if we imagine that the conjugate focus S' has wandered towards infinity and interpret PS' as the
i
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
Mjlriil;ljM
'i@iliji+
H
S'.
Theorem B Proof strategy
P is a mass point attracted by the sun S resulting 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 Suppose
in an acceleration
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
1. The law of attraction
be based on these two observations: with proportionality factor
{P,v}
k together with the
initial data
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:
o/ k, {P, v} (i�
semilatus rectum
�ii)
PH < 0 ( --> hyperbola)
on the same (opposite resp.) side of the tangent as S. The
S' we provisionally denote 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 i s the signed value PH t o 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 still-unknown conjugate focus by a different letter, say
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
we shall see how Newton carried out this argument in geo
�ii)
M.. orbit
of conic section results as the orbit of
P.
From this proof
section will indeed have the parameter p defmed in (i) as
We commence by explaining steps (i) through (iv). The de
(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
metric terms to derive a criterion for deciding which kind it will be immediately obvious that the constructed conic
tails of step (iii) will be deferred to a separate subsection.
view of
PH > 0 (--> ellipse)
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 conic section
reflection property
s
=v
· SM. In
its semilatus rectum.
(iv) The conic section constructed in step (iii) is the or bit of a mass point
A). Let c be the associated area constant as in step (i). By virtue of Theorem A, the ac
� . Because of (6), this equals the ac-
8� given by assumption. But the law of accel-
celeration of P is £ · . celeration
(5) we set
P moving under a centripetal force (see
Remark (a) following Theorem
P
Sr-
eration, together with the initial data, uniquely determines (6)
the motion of
P. Therefore the
conic section constructed
in step (iii) is indeed the orbit of P.
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.
Remark.
Each of these four steps rests on profound intu
itions. By contrast the execution of the argument requires
(ii) The focus S of the desired conic section and its tan
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
only a minimum of technical effort.
gent in
cide from the initial data which of the two cases applies, including the limit case of a parabola (when
S' ----') oo) .
To
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
that end we introduce an orientation on the line obtained
tained by reflection at the tangent. Denote the pedal point
from reflection at the tangent. For a point
Q on that line PQ as positive (negative resp.) if Q sits
of this perpendicular by
count the distance
signed in accordance with the convention adopted above.
K, and let the line segment PK be
VOLUME 25, NUMBER 4, 2003
39
i#'dii;!IUI
s
H
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
SH 2
(7)
=
(PH - PK) 2 + (SP2 - PK 2) SP2 + PH 2 - 2PH PK. =
·
In addition we have equation (4), which takes the follow ing form after substituting :± PS' = PH:
SH2
(8)
= =
(SP + PH)2 - 2p(SP + PH) SP2 + PH2 + 2 SP PH - 2p(SP + PH). ·
Comparing (7) with (8) yields
PH(SP + PK)
=
p(SP + PH),
in other words,
p SP + PK
(9)
PH SP + PH
1 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
SP f PK SP f PK
<1 1
=
� PH >
0 � conic section
� PH = 00 � conic section
> 1 � PH < 0 � conic section
=
ellipse
=
parabola
=
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 PN -
-
(10)
p 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).
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
Suppose we are given the center S, the ini tial data {P, v} (where v does not point in the direc tion 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 el lipse is given via (9), and its semilatus rectum p is as in (6). For PN = 1 the orbit of p is a parabola with fo cus 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 semila tus rectum p is as in (6). THEOREM B.
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+
·
=
40
THE MATHEMATICAL INTELLIGENCER
semilatus rectum.
This concludes the proof of step (iii) and the solution of Problem B. Let us summarize the result.
=
=
=
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 geo metric 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 ob tained depends on the initial velocity v. With given tangent direction at P, PN is indeed an increasing function of v:
(11)
PN =
1
SP + PK
.p =
1
SP + PK
.
c2
k
1
28M2
k SP + PK
1 2 . v . 2
The quadratic appearance of v suggests a connection with kinetic energy. We shall examine this question and the re lated remark (b) in the next section.
s
H
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
Method 2.
(14) u =
2 SM2 2 SM2 = = 2 SA p
We use the very relation (9) on which Newton based his proof of Theorem B:
Case 2: eUipse resp. hyperbola. 1
SP + PK Plugging this into
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 ki netic and the potential energy. We use the following nota tion. Let S be a center and P a mass point with mass m. As sume that the law of acceleration is kfSp2. The kinetic energy of P is given as Ekin = mv2 and the potential en ergy as Epot = - km/SP.
t
THEoREM C.
With the same assumptions as in Theorem B,
Proof For abbreviation we write ( 1 1) as PN = (13)
u=
2 8M 2
SP + PK
.
�u
2
·
v2
with
SM . SM . SP = SP. SA SP
2 8M2 (15) u = -p
PH . SP + PH
1 p
(13) and recalling (3), we get .
PH SP + PH
2a
·
SM2
b2
PS' · --
2a
=
SM 2 PS' 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 case of an ellipse resp. 2 OM' = a. In the OM' I SV' whence hyperbola the foci S and S' lie within resp. outside the cir cle with center 0 and radius a. Now apply the chord theo-
ii'riil;ii+
Here the term PK does not look very natural. There are two ways to eliminate it. Method 1 is formulated in trigono metric 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 = 2 y and LPSM = y (Fig. 12). Substituting PK = SP · cos 2y and SM = SP · cos y in (13) and applying the addition theorem cos 2 y = 2 cos2 y - 1, we obtain u=
2 SP 2 cos2 y
SP + SP cos 2y
B
= SP. VOLUME 25, NUMBER 4, 2003
41
+jiijil;li§l
M'
V'
A'
A
0
M'
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 u
=
SM2 PS' SM . S'M' ·
=
SP .
SM PS' = SP. SP . S'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 = S p . P2 • Proceeding in analogous fashion as in Method 2 ab , the simple geometric meaning for the energy quo tient is revealed, namely
t
cf:e
(16)
PE
=
This finishes the proof of Theorem C. (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 v 2 is proportional to __!____ In other words, in the case of a " parabola jEkin/EpotI is a�constant (whose exact value = 1 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!_ estimate of the energy SP + PH _. Therefore Newton's " . quotient is made precise by this geometnc equal1ty. In the proof of Theorem C we started from Newton's co efficient PN and found a relation with PE· Inspired by Corol-
-
Remarks.
·
PH 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
42
THE MATHEMATICAL INTELLIGENCER
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-
=
Epot _
(
Ekin Epot
+ 1)
km ( PH +1 SP SP + PH
)
km SP + PH
Add on both sides
km 2a
and we shall have
1-
0
km +2a
for an ellipse for a parabola for a hyperbola.
Appendix: Reprint of Newton's Proposition XVII (cf. [7, p. 65-66D. PROPOSITION XVII. PROBLEM IX
Supposing the centripetal force to be inversely propor tional 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 de scribe that is let go from a given place with a given ve locity 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 + PH 2 = SH2 = 4CH 2 = 4(BH 2 - BC 2) = (SP + PH)2 - L(SP + PH) = SP2 + 2PS · PH + PH 2 - L(SP + PH).
2PK · PH - SP2 - PH 2 + L(SP + PH), 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 ] Albo uy, 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
REINHARD LANG
Tec:hnische UnlllerSJtl!t MA 8· t
lnstitut tOr
0 · 1 0623 Ber11n
Heidelberg Germany
0·69120
Germany
a-mall: amOmath.unt·heidelberg.de
e-mail:
[email protected] Kai Hauser stud1ed mathematiCS and philosOphy at the University
of Heidelberg, where he did his Hablitation
1n
1 993. after receiv·
tng a PhD from the Galiforma lnstrtute of TechnolOgy tn 1 989. He
wor\<S tn log1c and foundations of mathematics and in
[3]
--
philosophy.
: Huygens and Barrow, Newton and Hooke, Birkhauser:
Basel, Boston, Berlin (1 990). Clarendon Press, Oxford (1 995).
[5] Cohen, I. B.: A Guide to Newton's Principia, in Newton 1 . : The Prin·
new translation by I. B. Cohen and A Whitman. University
1 976 and became Privatdozent In 1 983. His Interests he
chanics) and
n
Greek mathematiCS and
[7] Newton 1 . : Principia, Motte's translation, revised by F. Cajori, Uni 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 Amer. Math. Monthly 1 04 (1 997), 1 -1 9.
[ 1 0] Stein, S. K.: Exactly How Did Newton Deal with His Planets?, Math. lntelligencer 18 (1 996), no. 2, 6-1 1 .
In the course of editing a book, I came upon the sloppy usage
"thm" where "theorem" was intended. I asked an efficient as sistant to correct it, which he did by writing a simple 'fEX com mand. Several weeks later, I found in the new version of the book the phrase "division algoritheorem." Whom do I con gratulate for the nice neologism-the author?
-Rajendra Bhatia
THE MATHEMATICAL INTELLIGENCER
math
philosophy.
TEX Devil
44
lfl
ematical physics �n partiCUlar potential theo!y and statistical me
lestial Mechanics,
of California Press, Berkeley (1 999). [6] De Gandt, F.: Force and Geometry in Newton's Principia, Prince· ton University Press, Princeton (1 995).
R91nhard Lang studied mathematiCs and physics at the UniverSity
of Heidelberg, where he received a doctorate tn mathematics tn
versity of California Press, Berkeley (1 962). (For a new transla
[4] Chandrasekhar, S.: Newton's Principia for the Common Reader,
cipia ,
wanote Mathemall
lm euenheimer Feld 294
StraBe des 17. Junt 1 36
MMMj.I§,Fiilflld·l,ii.;:;,;;nfJ
Marj o ri e Senechal , Ed itor
Mathematicians' W Visiting Cards G. L. Alexanderson and Leonard F. Klosinski
This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. Jfhat we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal,
Department
of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail:
[email protected]
e 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
[
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
ferent thing) probably first appeared in
tradition of new junior faculty donning
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
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
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THE MATHEMATICAL II'ffELUGENCER
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nische Hochschule (ETH) in Ziirich in 1914. There he remained until he emi grated to the United States in 1940 where he taught at several institutions before arriving at Stanford University in 1942. He taught there until his re tirement. These cards were left by math ematicians 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 States the 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 well defined 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 Engel berg (a few have short handwritten thank-you notes for pleasant weekend stays at the chalet). Some of the Eng lish cards date from their time at Ox ford and Cambridge in 1924, and oth ers 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. Little wood", written in Hardy's distinctive hand, on a Cambridge University card. Some names are instantly recognizable to anyone interested in twentieth cen tury mathematics: Henri Cartan (the son of Elie Cartan), Alfred Haar (writ ten as Haar Alfred in the Hungarian manner of giving the family name first), Adolf Hurwitz (who brought P6lya to Ziirich in 19 14), Gaston Julia (widely known today because of Julia sets), Solomon Lefschetz, Nicolas Lusin, Gosta Mittag-Leffler (with an accom-
panying notation, "Djursholm," his home and now the site of the Mittag Leffler 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 in troduced to his name the aristocratic German "von"), Issai Schur, Oswald Veblen, and many others. (For addi tional identifying information about these, see [ 1 ] or [2].) Of the two cards of the Riesz broth ers, 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 "Cand. 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, au thor 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 Cam bridge address but also his London club, the Athenreum. Another card is that of Alfred Er rera, not perhaps as well-known as many of the others, but nevertheless in teresting. Errera was Belgian and con tributed, 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 be cause 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 con tains cards from their wives. Alexander came from an old, distinguished, and wealthy family (of the Equitable Insur ance firm), and is remembered in math ematics for the Alexander polynomial in knot theory. Solomon Lefschetz was the legendary topologist, long influen tial in the Princeton department. Wives did not always have their own visiting cards. The card of the well-known an alyst, Otto Blumenthal, reads "Profes sor Otto Blumenthal und Frau." A lady's card was traditionally somewhat larger than that of a gentlemen, a dis tinction apparent from the examples in this collection; the gentleman had to have cards that would fit into a conve nient 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 Pro fessor" Joseph Leonard Walsh, obvi ously early in Walsh's distinguished career at Harvard. Another telling ob servation is the reverse snobbery con noted 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 sta tion in life. Ferdinand Gonseth at the ETH used "Monsieur le Professeur et Madame Ferdinand Gonseth." French cards are usually quite sim ple, sometimes with the university af filiation. Szolem Mandelbrojt (uncle of the younger Benoit Mandelbrot of frac tal fame), later a member of the Col lege 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 hand written addition "Rue Stanislas, 14." German cards, however, can be quite elaborate: "Dr. phil. I. Schur/ord. Pro fessor an der Universitat/Mitglied der
Preussischen Akademie der Wissen schaften!Berlin Schargendorf/Ruhlaer str. 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 rit ual of the mandatory visiting card and the prescribed social calls they imply. But this little collection, found in an at tic, 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.
A U T H O R S
GERALD L. ALEXANDERSON
LEONARD F. KLOSINSKI
Department of MathematiCS and
Department of MathematiCS and
Santa Clara U111Verstty
Santa Clara UnM!fSity
Computer Science
Santa Clara. CA 95053·0290
e-mail:
Computer Science
Santa Clara. CA 95053-0290 USA
USA
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
a-mall: [email protected] Leonard F. Klosinski has been at Santa
Clara University for 42 years; to be sure,
for a few of those he was an under
raphy Random Walks of George PO/ya
graduate student. He has been d�rector
1 992). He has been at Santa Clara Uni
matical
(Mathematical Associat1on of America,
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.
of the William Lowell Putnam Ma he 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
One Hundred Prisoners and a Lightbulb Paui-Oiivier Dehaye, Daniel Ford , and Henry Segerman
This column is a placefor 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.
M i c h ae l Kleber and Ravi Vaki l , Ed itors
Athe center of the room. 100 pris single lightbulb flickers into life in
oners shade their eyes from the glare, then focus on the prison warden stand ing 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. Amazing fact 1. They can get out. Well, almost no possibility. . . . Here is how. (You may wish to pon every night from now on, I will choose one of you at random, re der on your own before reading on.) trieve you from your cell, and take you to this room, where you may see If at First You Don't Succeed if the lightbulb is on or off, and you Strategy 1 . Cut the sequence of days may turn it on or off as you wish. into blocks of length 100. The first pris .
A murmur ripples around the room as the prisoners consider the prospect of having such an effect on their hitherto impotent and externally controlled ex istences.
If at some point, I take you to this room and you believe that all 1 00 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. .
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil,
Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 1 25, USA e-mail: [email protected]
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:
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 prison ers have been chosen only if they ac tually have, with 100% certainty, prefer ably before they all die of old age. At first it seems impossible that any
.
.
oner to enter the room in a given block turns the lightbulb off. If a prisoner en ters 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 prison ers 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 un til 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 probabil ity of succeeding with one block To see this, suppose p is the chance of suc ceeding 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 I kpqk - l where k q = 1 - p. This is equal to
P J!_ (ll(l - q)) dq
=
p!p2
=
lip
Thus the expected number of blocks is nnln!. Each block has length n, so the expected number of days un til 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 be fore the universe ends. Soul-Collecting Strategy 2. One prisoner, who will be
known as The Countess, will be re sponsible 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 single token, called a soul, and each will try to leave that token in the central room. The Countess will collect 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 pris oner 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, ifhe 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. Ifher 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
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 de clares 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.
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) de note the number of souls that the light bulb is worth if it is left in the on po sition on night n, or discovered on on night n + 1. Assume that the number of prison ers 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 num ber of souls M that he has collected, but
In order to achieve a good asymptot ics, 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 asymp totically. This is known as the coupon collector's problem [2]. With some care ful estimation this result can be ex tended by changing c to a function c(n) :::; log(n). This gives us a proba bility of successfully completing each stage of at least e lll g(n) E(n) , where E(n) is an error factor such that E(n)10g2(n) tends to 1.
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 prob ability 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
( nI-1 -1 ) k=l
k
+
(n - l)(n),
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 previ ous exponential solution. The 100 pris oners 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: Amazing fact 3. They can get out be fore they are ineligible for the Fields medal.
=
Expected results for strategy
-o
3.
The chance of successfully com pleting all log2 (n) stages is at least eiog2(e)fE(n)'0g2(n). Thus the expected number of times we need to go through the whole cycle is less than e'0g2(e) 11E(n)10g2(n). This gives that the ex pected 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 pre cisely 100 prisoners? Our assumption that the number of prisoners is a power of two can be re laxed. To apply our strategy, we just need that everyone starts with at least one soul. So, with 100 prisoners for ex ample, 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 free dom, 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 algo rithm works by trying to optimise the V(n) for two passes through the types of days (1-days, then 2-days, then 4days etc.) then just repeats that se quence for the unlikely cases in which the prisoners have still not finished af ter 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 algo rithm 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 Count ess 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 strat egy gives around 3890. Asymptotically, they cannot do bet ter 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 commu nicate! Variations Here are some variations you might like to think about. Each variation as sumes all the conditions in the original problem, but with some aspects al tered. 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 pris oners 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 visi tors only transmit or receive, not both. 4. Malicious Warden-The warden is malicious and knows the strat egy 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 al low every prisoner to visit the room infinitely often. 5. All prisoners have to an nounce-The condition for every one to be freed is that every pris-
6.
7.
8.
9.
10.
11.
12.
oner must correctly announce (at some time). In other words: every prisoner must be sure that all pris oners have been to the room. 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 every one must correctly announce on the same day. If they are incorrect or if some announce while some do not, then they all die. Prisoners are freed when they announce-Everyone must cor rectly 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. 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. Numbers in cells (one an nouncer )-Each cell has a natural number written on the wall. The announcer must give all the num bers. All prisoners send messages to all prisoners-(this is a combina tion of 5 and 9). Random visiting times-The prison is subterranean, with no clocks, calendars, or any other in formation 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. Random times, all prisoners must announce-Combine varia tions 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
ing on how those
13. Random times, message from prisoners and the transmitter has
expected
to send a message (a natural num
n2 log n and n3.) 2
ber) to the receiver, but as in
11
k
souls are
of
between
distributed. This gives a total
one to one-There are only two
time
cape. See the Uber-theorem (be low) for a strategy and proof.
1 1 . Random
visiting
times
Counting souls (strategy 2) will
4. Malicious Warden-Strategy 2 will work, although there is
still work
14. Random times, messages from
clearly no bound on the time
all prisoners must announce-"Sloshy" soul
many to one-Combine varia
until escape; it depends on how
counting will work The light
tions
One announcer
mean the Warden wants to be.
must give all numbers written on
5. All prisoners have to an
they cannot count days.
9
and
1 1.
the walls, and the prisoners lose
nounce
track of time.
egy
-
(strat
"
of strategy 3 will also work: Each
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 strategies listed above (or slight modi of these variations.
is always worth one soul.
nounced does the following: If
the lightbulb
is on when he en
ters, then he collects the soul and turns the lightbulb off. lfthe
each prisoner who will have to
lightbulb
is off when he enters
is given to
and he has one or more souls,
each prisoner's attempts to col
then he drops one soul and turns
lect the souls destined for her,
the lightbulb
then after n of these a second cy
who has already announced al
on. Any prisoner
is devoted to each prisoner,
ways drops any souls that he
and so on. This gives an ex
has, and leaves any that are in
cle
fications of them) are suitable for most
bulb
Any prisoner who has not an
prisoner has one type of soul for announce. One cycle
these problems. It turns out that the
the room. This strategy has ex
pected time of n2log (n). 2
bulbs-Counting
6. Simultaneous announcing
souls (strategy 2) will still work,
The prisoners cannot be sure of
equal to
and can be made even faster as
escaping. Suppose they will an
by constructing an appropriate
1. Multiple
pected time order less than or
en. This can be shown
A. There is a first
Markov chain and giving lower
on which they all know
bounds for the chance that a
log n lightbulbs allow for the 2 best possible time to escape-as
D,
this. The prisoner who enters on
given prisoner will announce in
day
knows that she has en
the next
soon as everyone has actually
D
tered and the state of the light
when there
been in the room, then the pris
bulb. Every other prisoner only
left to announce, this strategy re
oner in the room can declare.
knows that he did not enter the
2k
nounce on day
souls can be left in a room
which has
Strategy
day,
k distinct lightbulbs,
room on that day. If a different
3 can also be improved
days. Notice that
is only one prisoner
duces to Strategy 2, soul-collect ing with one Countess.
prisoner entered on day
larger lumps, such as lumps of
all of the other prisoners who
also work (less efficiently) if
size
did not enter would have the
prisoners who have already an
same information, and so would
nounced just continue to slosh
(2k)Z if there
are
k
distinct
D then
200
by allowing gluings of souls into
lightbulbs.
Note that this strategy would
rooms-Counting
have to come to the same con
souls (strategy 2) will still work
clusion: that they should an
souls rather than just give).
It will be slower, although the
nounce on day
A (provided there
This is because a random walk
2. Multiple
are at least
There
in a fmite space will eventually
number of rooms independent
fore it cannot matter who enters
get everyplace. We will use this
of n) is still O(n2) days.
on day
fact extensively later on.
D, so they must all know D 1. This contradicts assumption that D was the
on day
transmitter/re-
3 prisoners).
souls around (give and take
expected time until escape (for
3. Separate
-
ceiver-A strategy similar to
the
soul-collecting
first day they all knew they
(strategy
2)
would announce on day
works. The Countess always picks up and never drops souls.
7.
A.
Another strategy is that each prisoner who is not a soul-col lector has a (very small) chance each day he enters of becom ing one. After a number of vis
Everyone else drops souls at
Prisoners are freed when they announce-"Sloshy soul
its to the room as a soul-col
every opportunity (though they
collecting (as in the answer to
lector he gives up and goes
are forced to pick them up if
variation 12 below) will work
back to being an ordinary soul
they find them). This strategy
When a prisoner has collected
giver. Any prisoner who has al
has
100
expected
time
between
"
souls and then given them
n2log n and n3. (If there are 2 souls outstanding, then the
k
all away again she may declare
chance of the countess picking
8-10. Red/Blue cells, or Numbers
one up the coming night is be tween !_ X _!_ and l/n2, dependn
56
"Try try-again
-
1) works. Interleaving cycles
12. Random times,
n
THE MATHEMATICAL INTELLIGENCER
and be set free. in
cells (one or all announ cer(s))-The prisoners can es-
ready 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 pris oner 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 oc casionally 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 num bers of the transmitters, M. This time the receiver occa sionally 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 num bers of the transmitters, M. What happens is that the re ceiver's total collected souls usually increases, but never falls back as much as n from the cur rent all-time maximum unless that maximum is the total num ber of souls being transmitted. When a transmitter finishes, the receiver's total is allowed to slosh back by one more than be fore. 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 trans mitter and who is receiver. To be precise, they use lightbulb one just as in 13. Some prisoner is chosen to be the first transmit ter. 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 listen ing 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 be comes 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 eventu ally each prisoner collects all the messages. 16. Random times, messages from many to many, ! light bulb-See the Dber-Dber-theo rem below. -
Ober-Theorem We will now give our method for vari ations 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-num bered 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 noth ing 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 num ber 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 light bulb 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 algo rithm 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 1 1 (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 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 an swer 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 neg atively to Q 1 when Po is conclu sive. -A resetting procedure that allows prisoner 1 to set the system back to its original position (the num ber of souls in the system is as the transmitter left it). This al=
58
The two testing procedures will even tually give an answer to the observer. •
•
•
The idea of the method •
lows him to proceed with addi tional experiments.
THE MATHEMATICAL INTELLIGENCER
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
�
-Trlgg_ ••__ ---lf-
k·2 ------.3 �--slosh
To start with, one of the robots will follow cpko and one will follow cpk1 • As sume that ko is big, and k 1 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
Indefinite
in the system
flickering
6, 7,
Diagram
1
. . .
, k0
+
k1
possible
4, 5
impossible
2, 3
possible
0, 1
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
Po
Add 1 soul to the system. Wait for a positive result for some time. If this positive result ar rives, return Yes, otherwise return unknown. Add 3 souls to the system. Wait for a positive result. If this positive result arrives, return Yes, otherwise return un known.
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 originally
0
# of souls after adding· step in Po
3
Possible outputs for Po # of souls after addingstep in P1 Possible outputs for P1
unknown,
4
Yes
unknown
2
unknown
unknown,
Yes
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
will run indefinitely
can run indefinitely
test for P. test for P, -
will stop after at most M flickers can run indefinitely will stop after at most M flickers
0
Diagram 2
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
Trigger
k
s
}·�
H
hn -2 hn -3
h, -h, --�-,����--�----���---����----��·�·r-�--� 0 --� --L-� -�-�-�---L-�-��------��-----� -� �--�--� fl'k,O � " :T _�
hn-3 hn -
�
h, h, 0
r
�
9'k,2
;T
T
4
..... lPk,n-2
Diagram 3
He knows which rule is running just by
prisoners, the random walk will even
counting the number of cycles all the
tually end up with the prisoner being
and never return them. By minimal
prisoners have gone through.
k souls, and all other pris oners have triggers of at least 2 more than k and so will not be triggered pre
ity, she never takes any new souls
triggered on
The rules. A set of rules that will n is shown in dia
work for general gram 3.
maturely.
h1 to 2, and the other hi are de } 2 Ij: hi" The value T refers to a number of soul
definitely with 0 souls. It is also clear that
exchanges required to cover that sec
souls. Here is one sequence that, if fol
tion of the graph, rather than the num
lowed, will run forever: Call the pris
Clearly this system will not run in
We set
fined recursively, so that � ;::::
it
might
run indefinitely with
ber of peaks. Take the number of peaks
oner applying the rule
si to be such that the total "length" 2sihi ;:::: T. The value of T will be spec
start, set robot
ified later.
All experimentation happens for values of souls less than
H,
so once a
prisoner starts up on the long j ourney
I.'f:12 hi
'P* ,i robot
i.
To
0 to be at the bottom of
any valley on his cycle, just before a peak of altitude bot
hj, and give to each ro
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
up towards H, she can only take 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 'P*,o which runs.
set not including Now, let
m
be the largest number
'P*,m is included in this sub set. As the subset is minimal, 'P* ,m must
such that
complete a full cycle, for if it did not,
m out. m must have
then we could simply leave robot
she will never be able to
takes them back, we are in a similar po
Thus, at some time robot
come back down until the observer
sition to the one we started with. We
hm souls. However, by the choice of the
can continue doing this indefinitely.
sizes of the peaks
towards
H,
wants to reset the system.
�:;}
hm > Ir;,:(/ hi,
it is
hi (the sum of the highest peak in each 'P* , i) so that
We need to show that the system
clear that he can never get rid of them
gets stuck for some higher value of
all without pushing one of the other ro
the normal running of the system, with
souls. This will require us to prove that
other prisoners on their zig-zags down
no proper subset of the robots can run
As there is a finite number of ini
H
tial states (looking only at the robots
set larger than
hn - 2 + I
H has to be
below will not bring an escapee to
H
indefmitely, if there are fewer than
bots onto its path towards H.
hi),
and allow him to go back down. Again,
souls in the system, which is proved
below their peaks
we defme the exact value of H a little
later on. Given this, it only takes one
global bound on the number of ex
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
robot on his way up to
H
to stall the
system. We can ensure this will happen by putting in
[ hn - 2 + I�: hi +
We can now take
H to
be any number
larger than this number, say
[ I�: hi + 2.
1 souls.
hn - 2
+
changes,
there is a
L, which can occur before
the system halts. larger than
L.
T is
chosen to be
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
The power of the Collective. We now show that with fewer than H
must complete a full cycle (otherwise
souls, the system cannot run indefi
the previous case). Once it has reached
case for 3 prisoners. To trigger the pris
nitely if not all robots are involved.
the beginning of the series of peaks of
k
Assume the system is running indefi
height hm it will, for at least the next
and
nitely with a minimal number of souls
soul-exchanges,
become a robot. As in the case of 3
changing hands. Once a robot starts
'P*,m did in the previous case. But this
therefore the system cannot get stuck when that many are added. They need only increment by
2 each time, as in the
oner running rule
'Pk, *' simply add
souls, and your message
60
(0
THE MATHEMATICAL INTELLIGENCER
or
1),
it could be left out and we would have
behave
exactly
T
as
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
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.
(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,
REFERENCES
[1 ] Bertram Felgenhauer. 1 00 prisoners and a lightbulb. Newsgroup rec. puzzles, available
July 2002. [8] William Wu. Hard riddles. http://www.ocf.
through http://groups.google.com, July 28
berkeley. ed u/ -wwu/ridd les/hard . shtml
2002.
#1 OOprisonerslightBulb, February 2002.
A U T H O R S
PAUL-OLMER DEHAYE Department of Mathemahcs
DANIEL FORD
Department of MathematiCS
Stanford Universtty
Stanford Universtty
Stanford, CA 94305·2 125
Stanford, CA 94305-2 1 25
HENRY SEQERMAN Department of MalhematJCS Stanford Untverstty
Stanford. CA 94305-21 25
USA
USA
USA
e-mail: [email protected]
e-mail. [email protected]
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
Auti s m M ath emati c i an s
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 impressian 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 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 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
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 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. 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
" It seems that for success i n science or art a dash of autism is essential . "
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; 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 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-
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 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 explanation for the difference in the reiative 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.
The precocious usually excel , at an early age , either i n mathematics , languages , or music.
64
THE MATHEMATICAL INTELLIGENCER
REFERENCES
1 . Arshad, M . , and Fitzgerald, Michael. Did Nobel Prize winner John Nash have Asperger's syndrome and schizophrenia ?
Irish Psychi
atrist 3 (2002), 90-94.
A U T H O R
lOAN M. JAMES Mathemahcal lnsbtute
24-29 St. G1les
Oxford OX1 3LB England e·mall: [email protected] loan James, F.R.S., was until 1 995 Savilian Professor of
Geometry at Oxford, and
1s
now Emeritus: he has also long
associat ion 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
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
mental Disorders 31 (2001 ), 5-1 7. 5. Benbow, C.B. Possible biological correlates o f precocious mathe matical reasoning ability.
1 3. Jackson, Luke. Freaks, Geeks and Asperger Syndrome: a User
Trends in the Neurosciences 1 0 (1 987),
of the Royal Society 49 (1 995), 1 -56. 1 8. Krutetskii, V.A. The psychology of mathematical abilities in school children,
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
(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.
2 1 . Sowell, Thornas. The Einstein Syndrome. Basic Books, New York,
versity Press, Cambridge, 1 991 . 9. Galton, Francis. Hereditary Genius. Macmillan, London, 1 869. 1 0. Gillberg, Christopher. A Guide to Asperger Syndrome. Cambridge
NY, 2001 . 22. Storr, Anthony. The Dynamics of Creation . Martin Seeker and War burg, London, 1 972.
University Press, Cambridge, 2002. 1 1 . Grandin, Ternple. Thinking in Pictures . Vintage Books, New York,
23. Storr, Anthony. isaac Newton. British Medical Journal 291 (1 985), 1 779-1 784.
1 996. 1 2. Herrnelin, Beate. Bright Splinters of the Mind. Jessica Kingsley, London, 2001 .
24. Trevor-Roper, Patrick. The World Through Blunted Sight. Allen Lane, London, 1 988.
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65
ii,IM\!Jffi j.J§.rblh£111.Jihhl
D i r k H uylebro u c k , Ed itor
Tibet, Lhasa
A I Mathematician in Lhasa Michele Emmer
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where thefamous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck,
Aartshertogstraat 42,
8400 Oostende, Belgium e-mail: [email protected]
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 195 1 [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 Namgyal, 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
I
founded in the 7th century, and en larged in the 1 7th 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 docu mentary about Lhasa, and the Jokhang temple, the spiritual focus of the Ti betan 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 sur rounded by a maze of narrow alley ways, is packed with stalls and shops selling mainly religious articles. Here you will find the traditional white silk 68
THE MATHEMATICAL INTELLIGENCER
shoes, called Khata, with their charac teristic long pointed shape. Tibetans use them as a sign of respect when wel coming 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 an cient texts and crowds of pilgrims everywhere. The shadowy interior has a mystical atmosphere about it, while at the same time being open to all com ers, as befits Buddhism. It is full of vis itors 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 im portance to the question of the per sonality of the Dalai Lama. In June 2002, talks began between Dalai Lama's representatives and the Chi nese authorities, with a good chance of reaching an agreement. Tibet is a combination of a medieval country in many respects with some as pects 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-
The Jokhang temple. Photo taken by the author.
Prayer wheels outside the Jokhang temple. Photo taken by the author.
gion dominated every aspect of life." 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 and measures used in ancient Tibet were rather elaborate, so conversion from one to another was extremely complicated, involving not only the dec-
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,
<9 I() (
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 1744 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
1 ::: � c 2. a
Conversion table, from [8].
VOLUME 25, NUMBER 4, 2003
69
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).
70
THE MATHEMATICAL INTELLIGENCER
1 . .�tl
2. Imperial path 3. Toilet 4. Gate of Luminous Peace
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20. :a<£11
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22. Initiation Tower
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18. Hall of Eternal Blessings
19. i!!i£111:
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2 1 . Hall of he Wheel ofthe Dhanna
21. ��-
22. lll!rtl 23. rl :.--- 7'- z :.--- ji)! 24. Hila
23. Panchen's Tower 24. Hall of Infinite Happiness 25. Pavilion of Perpetual Tranquility 26. Pavilion of Eternal Health
23. ffl!Jtl 24. JJtlftl
25. Jitlftl 26. 7l<.Jitftl
27. ll*il;llftl\
25. �-llG 26. 7!<..1111
27. 28. 29. 30. 31. 32. 33.
27. Yamudage Tower
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28. Hall of Buddha's Light
29. �lilttt 30. i!lillll LIJ I\ 3 1 . *!GiLll tl
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2. 3. 4. 5.
29. Tower of Peaceful Accomplishment 30. West Shunshan Tower 3 1 . East Shunshan Tower 32. Paid toilet
7 � -'1' 7 ""' (1){1 34. �m
33. Ajacang
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PLANE FIGURE TO YONG·HE·GONG
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I!!H.&tt .litIll i!illlll!J tt JI(MI!Jtl H v (:fff4)
34. Souvenir shop
JUIJ!: lJtiii !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.
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.
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
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
REFERENCES
cation in Tibet: History, Current Situation
1 . F. Maraini, Segreto Tibet, Leonardo da Vinci
and Future Development" in Abstracts,
publishing house, Bari, 1 951 Note
4. H. Harrer, Lost Lhasa, H. N. Abrams, Inc.
2. F. Maraini, Segerto Tibet, new edition, Cor baccio publishers, Milan, 1 998
Satellite Conference on Mathematics Edu cation, Tibet University & East China Normal University, Lhasa. 2002, pp. 26-33.
3. K. Larsen & A. Sinding-Larsen, The Lhasa
9. "Yonghe Gong," guide to the temple, pro
Atlas: Traditional Tibetan Architecture and
duced by the temple admin office (in Eng
Townscape,
2001
Serindia Publications, London
lish), China Picture-book Publishing House, Beijing, 1 995.
VOLUME 25, NUMBER 4, 2003
71
FRANCESCO CALOGERO
C ool l rrationa N u m bers and Thei r Rather C oo Rationa Approxi mations Pretty cool? 1000/998001
=
23 . 53/(36 . 372) = 0. [001 002 003 . . . 009 010 0 1 1 . . . 099 100 101 . . . 996 997 999 000]
c10oo - 10- 2997 • 999 ool)/9992
=
o.o01 oo2 oo3 . . . oo9 010 0 1 1 . . . 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 2 1 2 2 2 3 2 4 2 5 2 6 27 2 8 2 9 3 0 3 1 3 2 3 3 3 4 3 5 3 6 3 7 38 3 9 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 self evident rule-in this case, continuation ad infinitum ac cording to the infinite sequence of the integers. This num ber 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 rep resent numbers, for instance, the Champemowne constant c(2) written in base 2 reads c(2) 0. 1 10 1 1 100 101 1 10 1 1 1 1000 1001 1010 1011. =
0
0
0
(1.2)
X
ak(b)
=
b(b - 1r 2 - (1 - b - 1 )
=
kbk - (bk - 1)/(b - 1 )
= f3kCb)
=
Cbk
_
I
k= l
Ck - 1)bk + Cb - 2) 1) -2 Cbk + 1
_
f3k(b)b - ak(b),
k- 1
I
!=1
b1
+
(1.3a)
Cb - 1),
Ct3b)
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 in stance, a 1 (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 inte ger 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 (posi tive integer) values of k when b is large, for instance, /31 (10) f3z(10)
=
=
33 400/29 403
=
23 52 167/(35 1 12) = (10/9)(3 340/3 267), 0
0
°
0
=
(1.5a)
1 099 022 000/1 086 823 089
= 24 53 549 51 1/(38 1 12 372) 0
0
0
(10/9)(109 902 200/120 758 121).
r0(10)
= 10/81
=
2 5/34 ·
= 0. [ 123456790],
(1.6a)
as well as (see (1.4) and (1.5a))
And of course the value of the Champemowne constant in base b, c(b ), depends on the base b (b = 2,3,4, . . . ); for in stance obviously 1110 = 0. 1 < c(lO) < 2/10 = 0.2, while 1/2 = 0.5 < c(2) < 1 (here and hereafter all numbers explic itly 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: c(b)
that provide very good approximations to the Champer nowne 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 charac teristic of c(b ), namely the sequence of the positive inte gers (written in base b). For instance, calling rn(b) the ra tional approximation to c(b) obtained by truncating the series in the right-hand side of (1.3a) at its nth term, one gets
(1.5b)
The fact that the series (1.3) converges very fast when b is large implies that its truncation yields rational numbers
r1 (10)
= 10/81 - 10-8 (334/3 267)
60 499 999 499/490 050 000 000 71 389 2 190 52 11(27 34 58 1 12) 0. 1 2 3 4 5 6 7 8 [9 10 1 1 12 13 94 95 96 97 99 00 01 02 03 04 05 06 07 08 0]. =
=
°
0
°
0
°
=
0
0
0
(1.6b)
In these formulas (and always below), in the decimal ex pression of a number written out in decimal form the dig its enclosed within square brackets are meant to be peri odically repeated thereafter (they correspond to the recurring part of the decimal expression), while the inter spersed 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 inte gers 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 writ ten 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 denomi nator 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 pat tern 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 phe nomenology 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 num ber of initial digits that coincide with those that correspond to a neat pattern when the number is written out in deci mal 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
is a rather cool rational number
7, while r1(10), see (1.6b ),
in base
b) of the positive integer n appears in the decimal b) of c(b)-a position as
expression (written out in base
also re-emphasize the overall cool ap
sessed by counting all decimal digits in that expression of
pearance of the recurring part). And the following rather
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 right
of order 186 (but
I
cool rational numbers, r2(10)
=
r3(10)
=
r1 (10) - 10 - 1 87 (1 099 022/120 758 121),
( 1.6c)
2 r2(10) - 10 - 886
hand side of (2. 1a) into portions, in each of which the num
ber
n has the same number, say k,
(1 109 890 222/1 231 853 592 321),
c(b)
(1.6d)
r4(10) = r3(10) - 1 0 - 38 885 (123 443 2 1 1 358/1 371 440 348 559 369),
(1 .6e)
=
I
k=1
of digits:
nk - 1
oo
I nb -p(b,n,k),
(2. 1b)
n=nk - l
with (2.2)
may be expected, on the basis of the expansion formula
p(b,n,k) is merely a redundant (but more convenient) p(b,n).
(1.3), to have at least orders 2 880, 38 880, 488 880, re
Here
spectively (see (1 .4)). Indeed r2(10), when written out in
notation for
base 10, displays after the decimal point the sequence of
It is now plain that
the integers from 1 to (and including) the number 997,
p(b,n, 1)
which is then followed by 999 (rather than 998); hence, ac cording to the definition given above, r2(10) is a rather cool
and that
rational number of order 2885, and it reads as follows:
p(b,n,k) = p(b,nk - 1 - 1,k -
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
=3 =9+2
recurring part, which clearly has length 2997 while the non-recurring part has length 187
· ·
999, 89.
I end this section by emphasizing that, as clearly implied by the above treatment, a rather cool rational number, in
p(b,n,k), and it is easy to verify that they are given by the formula
p(b,n,k)
=
with the initial datum
q(b, 1) (bk - 1)/(b
-
n
=
2,3,4, . . .
S(b,k) =
is the observation that the definition of
the Champemowne constant the representation
c(b),
as given above, entails
n= 1
where
(2. 1a)
p(b,n) is the positive integer that identifies the po
sition at which the last digit of the expression (written out
74
THE MATHEMATICAL INTELLIGENCER
S(b,k),
(2. 7a)
bk - 1
I nb -kn.
(2.7b)
n = bk - l
S(b,k) is now a matter of standard alge
bra; inserting it in (2. 7a) and simplifying leads to (1.3).
stant
c(b ),
another, more general, class of cool irrational
numbers, c(b,s), the decimal representations of which are analogous to that of preceded by
00
c(b) = I n b -p(b,n),
The evaluation of
1)
Some Afterthoughts Exercise 3. 1 . Define, in analogy to the Champemowne con
Proof The starting point
q(b,k) = -k +
00
numerator and denominator of which involve respectively
is provided in the next Section.
1); hence
c(b) = I b -k + (bk - 1)/(b -
involving no more than 4 digits, or as a single fraction the
A proof of (1.3)
(2.6b)
0.
k=1
priate powers of the base 10) numerators and denominators
the other rather cool rational numbers rn(10),
=
This recursion relation (2.6) is easily solved:
be written in fractional form using (in addition to appro
1 1 and 12 digits (and both 4 and 12 are small numbers in
(2.5)
q(b,k) satisfy the recursion relation q(b,k) - q(b,k - 1) nk- 1 - 1 = bk - 1 - 1 (2.6a)
refers to a comparison with the order p of the rather cool
comparison to 186). Analogous considerations apply to all
= kn - q(b,k),
where the integers
that only involvefew digits (in both instances the term "few"
ten as the sum of 2 rational numbers, that can themselves
(2.4)
Clearly the recursion relation (2.4), together with the
cool has to do with appearances!) when written as a single
qualifies as a rather cool rational number because it is writ
1)].
initial datum (2.3), determines uniquely the numbers
rational number, or as a sum ofjew rational numbers, which
rational number under consideration). For instance, r1 (10)
k [n - (nk- 1 -
n gets above nk- 1 - 1, it has k digits, so p(b,n,k) increases by k for each unit increase in n.
long string of integers (or some other neat pattern of dig
should be characterized by numerators and denominators
+
(2.3)
This crucial formula, (2.4), may be justified verbally: when
addition to having in its decimal representation an initial its), should also have a reasonably neat appearance (being
1)
= n,
s zeros
c(b ),
except that now every integer is
(here and below
negative integer), so that of course
s is
of course a non
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
k�1
ak(b,s)
=
f3k(b,s)b - ak(b,s),
=
(3.2a)
(k + s)bk - (bk - 1)/(b - 1)
=
Ck + s - 1)bk + Cb - 2 )
k-1
I
=
b1 + Cb - 1),
(3.2b)
l�1
f3k(b,s)
1) -2 (bk +s 1) - 2 bk +s+ 1 (bk +s+ 1 +s k k 2 b + 1)(bk + s + 1 [(b 1) + bk + s _
=
_
_
_
so that, for instance for s
=
_
1],
(3.2c)
1, the analogs of (1.6) read
r0(10, 1) 100/9801 22 · 52/(34 1 1 2) 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/(2 14 . 36 . 5 1 6 . 1 1 2 . 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) =
=
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)] 7T 2/3.) 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
=
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) b 118 e (0,b - 112)]. 2
(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-
p(10) = 0.2 3 5 7 1 1 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 1 1 . . . 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
k�l
Bk(b)b -2ak(b) -k,
(3. 12a)
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)],
A U T H O R
(3. 12b)
so that the analogs of (1.6) read Ro(10) 100/9801 22 52/(34 1 1 2) = 0.[01 02 03 . . . 09 10 1 1 12 13 . . . 95 96 97 99 00], =
=
·
·
(3. 13a)
R 1 (10) 10 - 14 (2 550 249 999 999 999 974 977/249 950 oo2 5) = 2 550 249 999 999 999 974 977/(2 14 34 . 5 1 6 . 1 1 2 . 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) =
.
=
FRANCE CO CALOGERO
n.v-.,rtrnNI'I
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.
· Ptl
" Uf1Mi,.,..
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?
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.
REFERENCES
(1] D. G. Champemowne, "The construction of decimals normal in the
(6] Higher Transcendental Functions, edited by A Erdelyi, vol. I I , Mc Graw-Hill, 1 953.
scale of ten," J. London Math. Soc. 8 (1 933), 254-260.
[7] I . S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
http://www .research.att.com/ �enjas/sequences.
[8] Resource Library, http://www.mathworld.wolfram.com.
[2] N. J . A Sloane, The On-Line Encyclopedia of Integer 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
Products
(edited by A Jeffrey), Academic Press, 1 994.
[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
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
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling
us
your expertise and your
predilections.
Column Editor: Osmo Pekonen, Agora Center, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail: [email protected]
T
he 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 ofHeat 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 sci ences 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 of the 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, 1 789. 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 1 793, 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 of the 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 con struction d'une posterite," is shorter. It contains a description of the main steps of Fourier analysis and related matters, and also examples of under estimation of Fourier among mathe maticians (some were mentioned at the beginning of this review). It could be very useful in any course on har monic analysis, just to provide a his torical 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 impos sible 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 writ ing history. At last, the French have made a decisive contribution to our knowledge and appreciation of Joseph Fourier. Thank you, Jean Dhombres and Jean-Bernard Robert. 1 1 rue du Val-de-Grace 75005 Paris, France
e-mail: [email protected]
Four Colours Suffice: How the map problem was solved by Robin Wilson ALLEN LANE, THE PENGUIN PRESS, 262 pp. £1 2.99
ISBN
2002
0 7 1 3 99670 6
REVIEWED BY CHARLES NASH
T
his 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 Fran cis Guthrie in 1852, its solution by Wolfgang Haken and Kenneth Appel in 1976 using a lengthy computer pro gramme, and also subsequent develop ments. 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 particu larly 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 mathemati cal 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 an other Cambridge trained mathemati cian A. B. Kempe. The four colour problem is essen tially a problem in graph theory, and this was realised by Kempe who pub lished a "proof" of the four colour prob lem 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 Ox ford 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 asfew
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 mis take 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 sur face is h, is the integral part of (1/2)(7 + Y1 + 48h), i.e., the integer
r-!
(7
J
+ v1 + 48h)
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 stan dard planar case by stereographic pro jection, so one can also pose the prob lem 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 configura tions 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 ex pected to fmish a proof in a year-and there are other relevant details which
could be an error in the proof that
away at all with ordinary computerless
would be almost impossible to find.
proofs which are the life blood on
Mathematicians like to get their
which all the rest feeds.
the avid reader can find for him or her
hands on all the details, ideas, and
self.
mechanisms of a proof, and in doing so
Conclusion
All this meant that Appel and Haken
often learn an enormous quantity of
The book under review is certainly pro
had to work hard and quickly to get
useful things. Sometimes this activity
fessionally done and repays careful
there first, which they indeed did.
leads to more progress than was cre
reading. There is a wealth of mathe
Finally in 1994, Robertson, Sanders, Seymour, and Thomas constructed a
ated by the proof they are examining.
matical, human, and circumstantial de
Alternative proofs are often pub
tail provided for all parts of the story.
revised and slightly simpler proof of
lished, though the same activity is oc
At the back there is a very useful
the four colour theorem using the Ap
casionally frowned on by the closely
pel and Haken methods and a com
related
theoretical
Chronology of Events, a no less useful Glossary of mathematical terms to
physics-unjustly, I think. In fact a
help the layman, and finally a good
common paper title in mathematics is
index.
puter.
discipline
of
Computer Proofs
something like
A new proof of so and
The book gives us a feel for how
The earliest reference I could find in
so's theorem. This is as it should be: the
much nearer the mathematical centre
the book to the use of a computer on
new proofs often illuminate what really
stage the four colour problem was in
the four colour problem refers to the
makes the theorem true and can be a
the past. Just to select one example
1960s. On p. 180 Wilson says
great benefit in understanding how the
from many: Birkhoff gave it consider
whole of mathematics fits together.
able attention having heard Veblen lec
Hence we prefer a proof we can Haken invited Heesch to the Uni versity of Illinois to give a lecture, check all of by hand, but we should not and raised the question of whether actually dismiss out of hand one we computers could be helpful in the can't. We must not label a computer examination of large numbers of proof with the pejorative epithet brute configurations. Infact this thought force unless we have very good reason had already occurred to Heesch, and an elegant illuminating non com and in the mid 1960's he had en puter proof to offer in its stead. Nevertheless there can be the gen listed the help of Karl Diirre, a mathematics graduate from uine concern that a large scale com Hanover who had become a sec puter proof has a practically unde ondary school teacher. tectable error. This may have been the
ture 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,
reason why, in 1986, Appel and Haken
that Birkhoff is said to have once re
Further down the page he goes on to
had to publish a paper refuting persis
marked that almost every great math
say that
tent rumours about errors in their
ematician had worked on the four
proof.
colour problem at one time or an
By November 1965, using the pro gramming 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.
Finally that kind of genuine concern
other. It was not so universal in its ap
is now considerably allayed, I think, by
peal later in the twentieth century
the second proof, mentioned above, of
due, in part, presumably, to the big
Robertson,
bang-like growth of so many new ar
Sanders,
Seymour,
and
Thomas. They used an unavoidable set
eas of mathematics as the century pro
of only 633 reducible configurations as
gressed.
compared with
and Haken's
However, graph theory, to which
1482. Their computer was a 1994 model
Appel
this theorem belongs, is a growing and
(instead of a 1976 one), used much less
vibrant theory with numerous links
computer time, and the result was eas
to abstract mathematics as well as
So we see that the computer entered
ier to check. In addition they have
physics and network theory. General
the fray in 1965 in a small way but was
made the programmes publicly avail
isations of the chromatic polynomial
able and many computers can dupli
P( A) such as the Tutte polynomial are
to go on to dominate. Some mathematicians were very
cate their runs. So the success of this
of great importance. Knot theory and
disappointed that the proof was by
second computer based proof is some
the Jones polynomial have now become
computer and could not, in practice, be
thing decidedly positive.
quite well known to many theoretical
checked by hand. Some refused to ac
There will surely be even larger
physicists; a whole new approach to the
cept it as a proof, some were unpleas
computer based mathematical proofs
Jones polynomial for a general three
ant and unfair about the matter. Some
in the future. However, I don't think
manifold was created by the quantum
were perfectly pleasant and fair but
that they should be banned, or that
field theoretic formulation of Edward
were just worried about whether there
their proliferation, if it happens, will do
Witten; perhaps the Tutte polynomial
82
THE MATHEMATICAL INTELLIGENCER
may also be amenable to a quantum
Mathematics and Art held in Mau
field theoretic construction.
beuge, France, in September
2000,
subjective and reflects my own views
and
and experiences. At the same time, I
I have tried to give an idea of the
as soon as I opened the book, I started
hope it may serve as a possible frame
sorts of mathematical, human, and his
wishing I had been there. It must have
of reference
torical detail that the author Robin Wil
been a killer conference! But does that
when picking up this book.
son has put into the book-I have not
make for a killer conference proceed
done it justice. Do go and read this
ings?
for your
expectations
How many proceedings from con ferences that you did not attend do you
Mathematics and Art is a very wide
have on your bookshelf? I think it's
venue. My background involves teach
Publisher-Allen Lane, The Penguin
ing a course on Mathematics in Art and
only fair to say that many of the arti
Press-with the exception of the dust
Architecture at the National University
If you are planning to teach a course
no colour illustra tions at all; this is a disgrace for a book
tion called "Art Figures: Mathematics
of Singapore, consulting for an exhibi
on mathematics in art for first-year
book, it is well worth it. I have one complaint to direct to the
jacket, there are
on the four colour problem, which
in Art" at the Singapore Art museum,
cles in the book are not easy reading.
general students, then I'm afraid you
will not fmd many articles that you can
Typograph
and numerous TV interviews and pub
ical justice has not been done to the
lic lectures at museums, libraries, and
ematics
subject. The book is copiously illus
schools.
ments-a Simple Toolkit for Experi
costs, incidentally,
£12.99.
trated, but the colour differences are
I personally like to subdivide dis
all indicated by varying shades of black
cussion of mathematics and art into the
and grey. I very much doubt that the
following four rough categories:
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.
•
Mathematics in art
•
Mathematical art
•
Mathematics as art
•
Mathematics is art
use directly. The article on "The Math of Tuning
Musical
Instru
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
"Mathematics in art" refers to topics like perspective in paintings, symmetry
in teaching undergraduate classes. In the interesting article "Mathe
in ornamental art, and musical scales.
matics and Art:
This is material that even the most anti
Michele Emmer says: "If it i s almost
National University of Ireland
scientific art-theorist would recognize
impossible to describe a film using
Maynooth
as relevant. You can approach virtually
words, it is good, because it means
Ireland
any art museum with an offer of a pub
that the film has been made really us
e-mail: [email protected]
lic lecture on such topics and be con
ing a visual technique, mixing images,
fident of a good turnout.
sounds, music in an essential and pos
Department
of
Mathematical Physics
"Mathematical
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
art"
includes
the
sibly unique way. " By the same token,
works of Escher and other mathemat
a good lecture on mathematics and
ically inclined artists, who while wor
art may not translate into a good ar
shiped by mathematicians are some
ticle. Many of the articles are written
times frowned upon or ignored by the
by people I admire deeply, who are
art community. When I was working on
I
excellent speakers and have wonder
the exhibition at the Singapore Art Mu
ful Web pages. Yet I sometimes do not
seum, I had to conform to a strict "no
get much out of their articles in this
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
REVIEWED BY HELMER ASLAKSEN
The Film Series, "
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
mathematics. With the advent of com
the reasons, the main one to my eyes, which solders the arts to mathematics
'm convinced the title of this book
puter graphics, mathematicians have
will intrigue most readers of the
been able to create stunning graphics.
is probably the following: the tangible
Mathematical Intelligencer. When you
Yet how many art museums would be
object, the living being, are not only
look at the list of contributors and see
interested in a public lecture about the
present in space, and are evolving in
names like Michele Emmer, Michael
Mandelbrot set?
space, but are moreover a highly elab
Field, George W. Hart, John Hubbard,
"Mathematics is art" refers to the
orated construction, obtained from the
Richard S. Palais, Konrad Polthier, and
view held by many mathematicians
unfolding of the properties of the pri
John Sullivan (to name but a few in al
that mathematics is an art, not a sci
mordial space. " I don't like it when I'm
phabetical order) , I'm sure you will be
ence. However, few art theorists share
"dead on arrival" on page
even more interested. The book is the
this view.
and when on page
proceedings
of the
Colloquium
on
This classification is of course very
1 of a book, 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 ARPAM pro ject. What is the ARPAM project? The 15-page article does not explain the acronym "Association pour Ia Realisa tion et Ia Gestion du Pare de Prome nade et d'Activites Mathematiques." Af ter 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 mu sic 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 bet ter in two colors than in black and white? And unfortunately, the color pictures from Maria Dedo's excellent article on "Machines for Building Sym metry" 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 ver sions in the back. The conference must have been spectacular, and the proceedings con tain a number of excellent articles that deserved better editing, both of the writing of the individual articles and the overall organization of the book.
NEW FROM BIRKHAUSER Sampling, Wavelets, and Tomography
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www.birkha u er.com
Department of Mathematics National University of Singapore
Singapore 1 1 7543 Singapore
e-mail: [email protected]
84
Birkhiiuser Bo ton
7/03 THE MATHEMATICAL INTELLIGENCER
·
Ba
I
·
Berlin
V O L U M E
2 5
The Mathematical lntelligencer I ndex
Authors
Calogero, Francesco. Cool Irrational Numbers and Their Rather Cool
Adams, Colin. A Difficult Delivery.
(1) 8-9. Adams,
Rational
Approximations.
Wiling
Away
the
Gotz, Ottomar. Regiomontanus. (3)
Chernoff, Paul R. "Some of the Peo
Groetsch, Charles. Hardy's Duncan
Crato,
Nuno.
Pedro
Nunes,
Por
tuguese Mathematician and Cos
(3) 27-28. Adams, Colin. Don't Touch the But
mographer. (1) 80.
Leonard F. Mathematicians' Vis
(4) 45-52.
Diane. The Card Game SET. (3)
33-40. Complaint. (2) 23. Davis,
Oxen Revisited. (3) 41-43. Aslaksen, Helmut. Review of Mathe
matics and Art: Mathematical Visualization in Art and Edu cation, edited by Claude P. Bruter. (4) 83-84.
Atzema, Eisso J. Into the Woods:
5-6.
en sont les matMmatiques?, edited 57.
Hauser, Kai, and Lang, Reinhard. On the Geometrical and Physical
Davis, Chandler. The Cosmological
Albinus, Hans-Joachim. Pythagoras's
Prize Book (4)
Guillemin, Victor. Review of Ou
by Jean-Michel Kantor. (3) 56-
Davis, Benjamin Lent, and Maclagan,
ton. (4) 32-34. Alexanderson, G.L., and Klosinski, iting Cards.
44-46.
ple, All of the Time." (1) 71-73.
Hours. (2) 18-19. Adams, Colin. The Three Little Pigs.
Chess and Logic? (3) 53-55.
(4)
72-76.
Colin.
George, Alexander. A Free Lunch in
Martin.
of Kepler's Problem. (4) 35-44.
Exponential
and
Trigonometric Functions-From the Book (1)
Meaning of Newton's Solution
5-7.
Hersh, Reuben. The Birth of Random Evolutions. (1) 53-60. Hickerson, Dean. Prime Maze. (1) 48.
Dawson, John W. Jr. Review of From
Trotsky to Godel: The Life of Jean van Heijenoort, by Anita Burdman Federman. (2)
78-79.
Hickerson, Dean. Prime Maze-The Solution. (2) 75-76. Hitotumatu, Sin. More Visible Sums.
(3) 4-5.
Dehaye, Paul-Olivier, Ford, Daniel,
Holbrook, John, and Kim, Sung Sao.
7-17.
and Segerman, Henry. One Hun
A Very Mean Value Theorem. (1)
Vi sualiser la quatrieme dimen sion, by Franc;ois Lo Jacomo.
dred Prisoners and a Lightbulb.
Norbert Wiener in Maine. (2) Banchoff, Thomas F. Review of
(3) 57-59.
Barney, Steven. See Saari, Donald G., and Barney, Steven. (4) Batchelor,
Marjorie.
17-31.
Undergradu
ate Training Revisited: Thoughts on an Unusual Reunion. (1)
1 7-
Elkies,
Noam
Bauer, Friedrich L. Why Legendre a Wrong Guess
about
1T(x), and How Laguerre's Con
D.,
and
Stanley,
Richard P. The Mathematical Knight. (1) 22-34.
Last Tango, a Musical,
Music
by Joshua Rosenblum, Book by
Joanne Sydney Lessner, Lyrics
77-78. Emmer, Michele. A Mathematician in
67-71.
tinued Fraction for the Loga
Ewing, John. Predicting the Future
rithmic Integral Improved It. (3)
of Scholarly Publishing. (2) 3-6. Fenske, Christian C. Extrema in Case
7-1 1. Booss-Bavnbeck,
Bernheim,
and
H0yrup, Jens. Mathematics and
of Several Variables. (1) 49-51. Ford,
Daniel.
See
Dehaye,
Bernheim, and H!Ziyrup, Jens. (3)
12-25. James, loan. Autism in Mathemati
Emmer, Michele. Review of Fermat's
Lhasa. (4)
42-47. H0yrup, Jens. See Booss-Bavnbeck,
by Lessner and Rosenblum. (1)
21. Made
(4) 53-61.
Paul
cians. (4) 62-65. Kahane,
Jean-Pierre.
Review
of
Joseph Fourier, 1 768-1830: Createur de la Physique Mathe matique, 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)
74.
Klosinski, Leonard F. See Alexan
War: An Invitation to Revisit. (3)
Olivier, Ford, Daniel, and Seger
derson,
12-25.
man, Henry. (4) 53-61 .
Leonard F. (4) 45-52.
G.L.,
and
Klosinski,
VOLUME 25, NUMBER 4, 2003
85
Lang, Reinhard. See Hauser, Kai, and Lang, Reinhard.
(4) 35-44.
Levy-Leblond, Jean-Marc. Columella's Formula.
Sallows, Lee. A Tragic Square.
(4) 7.
(2) 51-54.
(4) 6-7.
Segerman, Henry. See Dehaye, Paul
Longuet-Higgins, Michael S. Nested Triacontahedral Shells Or How
(4) 53-61.
Machover, Maurice. Cauchy Product
Shell-Gellasch, Amy E. Reflections of
(3) 43.
jamin Lent, and Maclagan, Di
(3) 33--40.
Petrovskil and Pontryagin.
55-73.
(2)
matics and Mathematicians.
35-41.
(1)
D. and Stanley, Richard P.
22-34.
Mathematics, Yaounde, Came
10-15, 2001. (3)
and
van
der
Waall,
Willem. The Christoffel Plaque in Monschau.
(3) 47-51 .
(1) 10-16.
dra, and van der Waall, Robert
Genese d 'une Theorie Mathe matique, by Alexei Sossinsky.
Wilson, Robin. The Philamath's Al
Pekonen, Osmo. Review of Noeuds:
(1) 75-77.
Ricotta, Angelo. Constant-diameter Curves.
(4) 4-5.
luctant Revolutionary.
(1) 61-70.
Rowe, David E. From Konigsberg to Gottingen: A Sketch of Hilbert's Early Career.
Future and Assessing the Past
1946 Princeton Bicentennial (4) 8-15.
Conference.
Saari, Donald G, and Barney, Steven. Consequences Preferences.
(3) 47-51.
phabet A.
of
Reversing
(4) 1 7-31.
viewed by Victor Guillemin.
56-57.
phabet B.
(2) 80.
blum, Joshua, and Joanne Syd ney Lessner. Lo
Fran<;ois. Visualiser la quatrieme dimension. Re
viewed by Thomas F. Banchoff.
(3) 57-59.
Robert, Jean-Bernard. Bernard.
(3) 64.
(4) 88.
Weighing the Odds: A Course in Probability and Statistics, by David (2) 77-78.
(4) 77-80.
Joshua,
and
Joanne
Fermat's Last Tango, a Musical. Reviewed by
Michele Emmer.
(1) 77-78.
Noeuds: Genese d'une Theorie Mathematique.
Sossinsky, Alexei.
Reviewed by Osmo Pekonen.
(1) 75-77.
Weighing the Odds: A Course in Probability and Statistics. Reviewed by Marc
Williams, David.
Books Reviewed Aigner, Martin, and Ehrhard Behrends
Alles Mathematik: Von Pythagoras zum CD Player. Re (eds).
viewed by Manfred Stem.
60-62.
(3)
Yor.
(2) 77-78.
Wilson, Robin. Four Colours Suffice:
How the Map Problem Was Solved. Reviewed by Charles Nash.
(4) 80-83.
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 car rying weapons of math instruction.
THE MATHEMATICAL INTELLIGENCER
See Dhom
Sydney Lessner.
At Heathrow Airport today, an individual, later discovered to
86
(1) 77-78.
Jacomo,
Rosenblum,
Yor, Marc. Review of
Williams.
(3)
bres, Jean, and Robert, Jean
Wilson, Robin. The Philamath's Al
(2) 44-50.
Rowe, David E. On Projecting the the
Willem.
Wilson, Robin. Anniversaries.
Rowe, David E. Hermann Weyl, the Re
Jean-Michel (ed). Oil en sont les mathematiques? Re
Robert
van der Waall, Helena Alexan
Missed Opportunities.
(2)
Lessner, Joanne Sydney. See Rosen
van der Waall, Robert Willem. See
Andrews,
Dawson, Jr.
78-79.
(3) 60-62.
and
Dyson,
by John W.
tin Aigner and Ehrhard Behrends.
Pak, Igor. On Fine's Partition Theo rems,
From Trotsky to Godel: The Life of Jean van Heijenoort. Reviewed
Kantor,
van der Waall, Helena Alexandra,
(4) 80-83.
(4) 77-80.
Anita Burdman.
Manfred. Review of Alles Mathematik: Von Pythagoras zum CD Player, edited by Mar
Charles. Review of Four Colours Suffice: How the Map Problem Was Solved, by Robin
Nash,
Wilson.
Jean-Pierre Kahane.
Feferman,
(1)
Stem,
Nana, Cyrille. Seminar-Workshop in
29-32.
1830:
Stanley, Richard P . See Elkies, Noam
Maritz, Pieter. Around the Graves of
roon, December
Joseph Fourier, 1 768Createur de la Physique Mathematique. Reviewed by Bernard.
(2) 20-23.
My Adviser: Stories of Mathe
Maclagan, Diane. See Davis, Ben ane.
Needs is an 18¢ Piece.
(4) 83-84. Dhombres, Jean, and Robert, Jean
Shallit, Jeffrey. What This Country
43.
reviewed by Helmut Aslaksen.
Olivier, Ford, Daniel, and Seger man, Henry.
(2) 25-
to Grow a Quasi-Crystal.
of Series.
( ed). Mathematics and Art: Mathematical Visual ization in Art and Education,
Bruter, Claude P.
Sallows, Lee. Not-so-magical square.
k1£1 1.19·h•t§i ..
Robin Wilson
Anniversaries T
he year 2002 saw three varied an niversaries: the bicentenaries of the birth of Niels Henrik Abel (180229) 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.
I
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
Niels Henrik Abel
Abel's collected works and rosette
(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.
Janos Bolyal
Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]
88
Paul Dirac
THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
Cloud chamber track
