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.
- W h i c h Mathematical Societies are the O l d e s t ? The Finnish Mathematical Society (Suomen Matemaattinen Yhdistys) celebrated in 1993 the 125th anniversary of its foundation. On this occasion, Mika Sepp/il~, the President of the society, compiled the following list of the eleven oldest mathematical societies of the world according to the information that was available to him (the place and the date of foundation are in parentheses).
1) Mathematische Gesellschaft in Hamburg (Hamburg 1690) 2) Wiskundig Genotschap (Amsterdam 1778) 3) Jednota Ceskoslovensk~ch Matematik6 a Fysik6 (Prague 1862) 4) Moskovskoe Matematicheskoe Obshchestvo (Moscow 1864) 5) London Mathematical Society (London 1865) 6) Suomen Matemaattinen Yhdistys (Helsinki 1868) 7) Soci6t6 Math6matique de France (Paris 1872) 8) Dansk Matematisk Forening (Copenhagen 1873) 9) Edinburgh Mathematical Society (Edinburgh 1883) 10) American Mathematical Society (New York 1888) 11) Deutsche Mathematiker-Vereinigung (Berlin 1890)
Such a list must of course be taken with caution because information is incomplete. We have not investigated how the list continues. Can someone generate the complete list? The Finnish Mathematical Society ranks rather high on the list. The early history of our society has been described in the delightful monograph:
Gustav Elfving, The history of mathematics in Finland 1828-1918, Societas Scientiarum Fennicae, Helsinki, 1981. Osmo Pekonen Department of Mathematics University of Jyv~skylfi P.O. Box 35 FIN-40351 JyviCskyli~" Finland e-maih
[email protected]
A Euphemism? Saunders Mac Lane 1 recalls the expression "Princeton ist ein kleines Negerdorf" as evidence of anti-Semitism among some mathematical circles in the Berlin of the 1930s and 1940s. What I heard at the Institute for Advanced Study in Princeton during the late 1940s, with Berlin understood as the source, was "Ach unser liebes verjudetes Negerdorf" ("Ach, our dear Jew-infested nigger village"), which brings the double insult into sharp focus. I wonder whether Professor Mac Lane felt impelled to soften the shock of this heinous expression. Verena Huber-Dyson R.R. 1 Pender Island, BC VON 2MO Canada
Editor's Note: A reader has warned that a phrase in George E. Andrews's article (Mathematical Intelligencer, vol. 16, no. 4 (1994)) is subject to misinterpretation. Andrews, citing an example by R. D. North in an article by Borwein, Borwein, and Dilcher, refers to "the astounding accuracy of 30 of the next 33 terms" in a decimal approximation to 7r. The terms referred to are of course the decimal digits, rather than the terms of Gregory's series. 1 MathematicalIntelligencer,vol. 16, no. 1 (1994),10.
THE MATHEMATICALINTELLIGENCER VOL. 17, NO. 2 (~)1995 Springer-Verlag New York 3
Complex Analysis in "Sturm und Drang Reinhold Remmert
In 1960 when I came to Erlangen and got to know Otto and Edith Haupt, the French Revolution in complex analysis was already history. Grauert's coherence theorem, which the high-fliers in Paris had not been able to master, was already a classical theorem. I have vivid memories of my first visit with the Haupts: we also discussed the latest progress in mathematics. Otto Haupt was very well informed about the developments of the fifties. We talked about the theorem of Riemann- R o c h Hirzebruch--Hirzebruch well known to him from his days as an assistant here in Erlangen-- about Stein manifolds, about Grauert's coherence theorem, and about the adoption of the language of agronomy in mathematics with expressive terms like sheaf, bundle, stalk, fibre, germ, and section. Haupt's opinion of the cult word "coherent sheaf" was: Whoever has to deal with such natural products, should be able to ponder d e e p l y - - a n d coherently. I have been entrusted with the honorable duty of delivering the Laudatio for Hans Grauert, who is the winner of the Karl Georg Christian von Staudt Prize. This prize - - today awarded for the first t i m e - - was endowed by Edith and Otto Haupt. In little less than 40 years of work Hans Grauert wrote nearly 90 papers. Whoever studies this oeuvre experiences the evolution of mathematics and a revolution in mathematical thinking. I have arranged a florilegium and I want to show how the laureate thinks and how during the "Sturm und Drang" of the fifties he became the mathematician who is being honored today. It is hard for mathematicians to make a speech on their research for an audience of not very close colleagues. 2 Mathematicians are deeply affected by the value and the beauty of their knowledge, but it is hardly possible for them to give others insight into their work. Our society regards mathematics with a sense of high esteem, but gets goose pimples at the same time: it does not deny
respect to mathematics, but does not want any closer contact with it; it praises mathematics as the safest form of knowledge, but without any envy toward its practitioners. The embarrassment that seizes speakers in situations like the one today has two sources. First of all, since the days of Felix Klein in Erlangen, mathematicians have lost the ability to give precollege mathematical instruction a 1 An edited version of the Laudatio for Professor Hans Grauert for the awarding of the von Staudt Prize, Erlangen, 12 November 1991. Translated from the German by P. Baptist and P. Hilton. 2 My remarks on this problem follow the Inaugural Lecture of K. Knopp, "Mathematics and culture," delivered on 27 January 1927 at the University of Tiibingen (published in Preuss. Jahrb. 211 (1928), 2833O0).
4 THEMATHEMATICALINTELLIGENCERVOL.17,NO. 2 (~)1995Springer-VerlagNewYork
usable form that guarantees a genuine effect on education in general (we all remember with a shudder the wild extravagances of set theory, much vaunted by educationists). Mathematicians have been still less able to maintain for mathematical science the position in public life which its form and significance merit. The mathematical theories themselves are denied to the citizen unless he boldly conquers them for himself. Mathematics has a tendency quickly to look ridiculous when it is made popular, as examples in recent times show. Therefore the true mathematicians prefer to remain silent, whereas on ceremonial occasions other scientists tell the public which problems they have solved and which problems still have to be solved. Only in very general terms--sometimes in parables--can we tell others what is happening in mathematics. Mathematical science has undergone many periods of change. The cradle of a great perestroika that also reached complex analysis, belatedly, and nurtured Grauert's work, was in Erlangen. Here Amalie Emmy Noether was born on 23 March 1882; here she was a welcome guest with the Haupt family. Today her name stands for conceptual abstract structural thinking; she prepared the way for functorial thinking. Her mathematical development is all the more remarkable, since she took her doctor's degree in 1908 with P. Gordan, a man who could only think formally. (In his papers he himself only wrote the formulas, linking words were inserted by friends.) In 1916 Emmy went to G6ttingen where she was soon called "der Noether." Heinz Hopf once related how in 1925 E. Noether-- truly no topologist-- gave him and P. Alexandroff a piece of advice which today is self-evident, but in those days was visionary: Create the homology theory of simplicial complexes basisfree (without incidence matrices) by means of the boundary operator. Instead of Betti numbers and torsion coefficients, focus on the homology groups themselves and the homomorphisms between them. It is pointless--though t e m p t i n g - - t o speculate whether under normal circumstances in Germany in the 1930s a Bourbaki circle around E. Noether, E. Artin, and H. Hasse could have developed with "membres fondateurs" like W. Krull and E K. Schmidt (both of whom spent some years in Erlangen), B. L. van der Waerden, G. K6the, and the "Noether-boys" F. A. H. Grell, M. Deuring, H. Fitting, and E. Witt. Emmy Noether went into exile to Bryn Mawr (Pennsylvania); but her ideas flew over the river Rhine into Alsace, duty-free. Today the name Bourbaki stands for the mathematics of France with representatives like H. Cartan, C. Chevalley, J. Dieudonn6, A. Weil, and many younger colleagues. Emmy Noether and Nicolas Bourbaki laid the foundations on which in the "Sturm und Drang" of the 1950s the young Grauert and others built skyscrapers. Is it due to coincidence or to the genius loci
that Bourbaki, like the German "Stiirmer and Dr/inger," had key experiences in Strasbourg? In 1770 Goethe and Herder met there; at Strasbourg in the thirties of this century Cartan and Weil held discussions; it is probable that, on a winter's day in late 1934, Bourbaki was born there (cf. A. Weil: Souvenirs d'apprentissage, Birkh/iuser, 1991, p. 104). And finally in Strasbourg in May 1957, H. Grauert and A. Grothendieck met for the first time and exchanged their independently developed ideas about spaces with nilpotent elements. Until 1949-1950 complex analysis, o r - - a s one said in those d a y s - - t h e theory of functions of several complex variables, was a tranquil mathematical theory. It could be learnt if one understood some German and some French. There were only two books, both already somewhat out of date: a so-called textbook by W. F. Osgood from Harvard (Teubner, 2nd edition 1929) and an Ergebnisbericht by H. Behnke and P. Thullen from Mi.inster (Springer-Verlag, 1934). In addition, there existed some original papers in German and French [Behnke, Carath6odory, Cartan, Hartogs, H. Kneser, Oka (in Japanese French), Stein]. Osgood, however, even then thought the theory so complicated that one could write about it only in German. And Cartan is said to have always asked students who wanted to learn several complex variables: Can you read German? If answered in the negative, his advice was to look for a different topic. The situation in those days is accurately represented by three quotations of the masters: Malgr6 le progr6s de la th6orie des fonctions analytiques de plusieurs variables complexes, diverses choses importantes restent plus ou moins obscures. [K. Oka, J. Sci. Hiroshima Univ. 6 (1936)] Trotz der Bemiihungen ausgezeichneter Mathematiker befindet sich die Theorie der analytischen Funktionen mehrerer Variablen noch in einem recht unbefriedigenden Zustand. [C. L. Siegel, Math. Ann. 116 (1939)] The theory of analytic functions of several complex variables, in spite of a number of deep results, is still in its infancy. [H. Weyl, Am. Math. Monthly 58 (1951)l In summer 1949 Hans Grauert began his studies in Mainz. For the winter semester 1949-1950 he arrived in the sleepy provincial town of M/.inster. He came into a Biedermeier idyll of the theory of functions with Reinhardt domains, midpoint-invariant automorphisms, and notched dicylinders. In 1957 he left Miinster and travelled to the new world, to Princeton. The tranquil life in the narrow number space C '~ ceased to exist. Complex analysis now had a different look a n d - - with Grauert-set out for new shores. In December 1949 H. Cartan lectured in M~inster for the first time since the war. He gave a moving account of this visit in Quelques souvenirs [address for the 80th anniversary of Heinrich Behnke, Unikat (Springer-Verlag, 1978)]. He was proselytizing in those days for the great, THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995 5
new, still unformed ideas of fibre bundles and coherent sheaves to Westphalia, where they were hesitantly taken up by us, the young ones. In 1953 this French Revolution with the motto "il faut faisceautiser" was already completed; at a colloquium in Brussels, H. Cartan and his student J.-P. Serre presented to a dumbfounded audience their theory of Stein manifolds, culminating in two theorems on cohomology groups with coefficients in coherent analytic sheaves. A German participant--today here among u s - - c o m m e n t e d tersely at that time, "We have bows and arrows, the French have tanks." Whoever wants to recapture this struggle for mastery of the new ideas should read Serre's letters of 1952 to Cartan, recently published under the title "Les petits cousins" in the volume Miscellanea mathematica (edited by P. Hilton, E Hirzebruch, and R. Remmert; Springer-Verlag, 1991). In 1952 K/ihler manifolds began their triumphant advance. In 1953 Hans Grauert received a small grant to study these manifolds in Zfirich with B. Eckmann. After five months he returned to Mfinster and delivered his visiting card: "Charakterisierung der Holomorphiegebiete durch die vollst/indige Kahlersche Metrik," Math. Ann. 131 (1956), 38-75, announced in C. R. Acad. Paris 238 (1954), 2048 -2050. Quite a few of you know this treatise. Using among others properties of minimal surfaces, it is shown: THEOREM. Let G be a pseudoconvex domain in C n with smooth real analytic boundary; suppose that there exists in G a complete I@'hler metric. Then G is a domain ofholomorphy. The thesis shows a young differential geometer and a mature complex analyst. Whoever has doubts has only to read the proof of Hilfssatz 8, necessary for establishing Satz 18. In his official report, Behnke leniently wrote, "In some parts the presentation is not easy to read." In the oral examination for the doctorate, complex manifolds, Riemann's period relations, modifications, and the Hopf-c~-process were examined; Grauert got his Ph.D. on 30 July 1954. This Ph.D. thesis already was a trumpet blast, but the next paper "Charakterisierung der holomorph vollst/indigen komplexen R/iume," Math. Ann. 129 (1955), 233-256, was a drumroll. The main result was completely unexpected: THEOREM. Stein spaces are exactly the holomorphic convex domains over number spaces. X is already Stein if it is holomorphic convex and holomorphic spreadable. In May 1955 H. Cartan promptly reported on the mdmoire inddit in the S6minaire Bourbaki (Exp. 115); this amounted to an act of ennoblement by the mathematicians of the Grande Nation. Even older German profess o r s - averse to the cheeky youth looking for deeper understanding by abstraction--were taken with these 6
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
results. This was no transcendental nonsense or no picket fence (Lattenzaun) mathematics: Es war einmal ein Lattenzaun mit Zwischenraum, hindurchzuschaun. Ein Architekt, der dieses sah, stand eines Abends pl6tzlich da-und nahm den Zwischenraum heraus und baute draus ein grot~es Haus. However, it was no longer function theory, as understood by Behnke, Carath4odory, and Hartogs. Where have the functions gone? What actually is complex analysis? The answer is given by the laureate in 1991: It is more difficult to construct holomorphic and meromorphic functions of several complex variables than those of one variable. Complex analysis (of several complex variables) is rather a special kind of geometry than an analysis of properties of functions (H. Grauert: "The methods of the theory of functions of several complex variables," MiscellaneaMathematica, Springer-Verlag, 1991, pp. 129-134). From 1956-1957 on, Grauert studied the Oka principle which had attracted attention through earlier contributions by Oka and Stein. This principle can be stated (imprecisely) as follows: Complex analytic problems which have continuous solutions also have holomorphic solutions. Obstructions to solutions are of a topological and not an analytic nature. The problem of giving this principle as precise and broad a form as possible had been of interest since the Brussels colloquium of 1953; in 1955 J. Frenkel showed its validity for holomorphic fibre bundles over Stein spaces with a solvable structure group. With an additional drumroll, Grauert imparted a startling generality to the Oka principle. The Habilitationsschrift with the innocent title "Rungesche Approximationss/itze und analytische Faserr/iume" appeared as a trilogy in a rapid succession in the Mathematische Annalen (1957-58). What had been proved? On Stein spaces, topological fibre bundles are always holomorphic fibre bundles. To put this more precisely, one first considers a holomorphic fibre bundle E over a (reduced) complex space X, whose fibre is a complex Lie group G. One forms the sheaf ga, resp., gc of germs of holomorphic, resp., continuous sections in E and the associated cohomology sets H 1(X, ga), resp., H i (X, gc). The natural inclusion ga ~-* g~ induces a mapping H 1(X, ga) _. H i (X, g~). Oka-Grauert Principle: If X is a Stein space, then H 1(H, g a) Z~ H 1(X, gO)is a bijection. The cohomology classes from g~, resp., gc are represented by analytic, resp., continuous E-principal bundles. An important step in the proof of the fundamental theorem is the following: LEMMA. In a holomorphic E-principal bundle over a Stein space every continuous section is homotopic to a holomorphic section.
Here are some applications of the principle. COROLLARIES. (1) IrE has continuous sections, then E also has holomorphic
sections. (2) Every topologically parallelizable Stein manifold is
propositions are reduced to homotopy relations which thus concern the continuous, resp., analytic deformation of mappings. It is a matter of proving that a particular effectwhich is produced by continuous functions can also be produced by analytic functions .... The existence proofs are carried out using analytic constructions. Thereby the arguments condense to patient and pleasingly concrete, though complicated, epsilontics . . . .
complex-analytically parallelizable. (3) All holomorphic vector bundles over C ~ are analytically
trivial. Grauert's proof is based on approximation techniques (first Annalen paper). The understanding of his results and the techniques of the calculation make the highest demands even on the expert. However, there is the force of habit. One remembers Jacobi-- the great man of K6nigsberg and Berlin, the republican of 1848--who once remarked, As in mathematics it is certainly important to accumulate conclusion after conclusion, so it will be good to gather together as many conclusions as possible in one symbol. For if the meaning of the operation has been established once and for all, then the sensory perception of the symbol will replace the whole line of reasoning that previously had to be started from scratch each time. For the Oka-Grauert principle this symbol may well be the arrow -%. Incidentall~ this principle was recently generalized to elliptic bundles by Gromov. On 16 October 1956 Hans Grauert presented his Habilitationsschrift to the faculty and made an application for his Habilitation. Already on 19 July 1956 a member of the faculty had written approvingly on the occasion of the first presentation in the faculty: "Dr. Grauert has achieved the scientific level on which the Habilitation represents the natural goal." On 8 February 1957--an important date for the Habilitand for several reasons-the venia legendi was granted, after a colloquium on "vector fields on spheres." He gave his inaugural lecture 1957. I want to read you from the reports of two referees. The first one writes (12/18/1956): Dr. Grauert has succeeded in establishing quite generally an isomorphism between topological and analytic fibre spaces with a holomorphically complete complex space as basis... thereby carrying out the program of our Paris colleagues. ... How highly this work is esteemed by the best expert in this field, our honorary doctor Henri Cartan, one may gather from the following. In July 1956, when Cartan had the opportunity to study the present manuscript he changed the title of his own course of lectures announced for the international symposium in Mexico in August 1956. He would lecture on Grauert's Habilitationsschrift. The other referee writes rather more about the contents (12/29/56): The proofs of these far-reaching theorems are based on propositions about the approximation of functions holomorphic in a holomorphically complete space X by functions which are holomorphic in a suitably extended space 9~. The
My anthology contains no contributions that Grauert has written together with others. So I am going to mention next his solution of the Levi problem: In 1911 E. E. Levi showed that boundaries of domains of holomorphy have properties which are associated with convexity. We speak of pseudoconvex domains. For a long time it was an open problem whether Levi's local pseudoconvexity of the boundary characterizes domains of holomorphy. In 1942 this so-called Levi problem was solved for domains in C 2 by K. Oka, and in 1954 for domains in all spaces C n, 2 < n < ~ , by H. Bremermann and F. Norguet. In 1958 Grauert moved away from number spaces completely. In a short paper "On Levi's problem and the embedding of real-analytic manifolds" [Ann. Math. 68 (1958), 460-472] he proved the following: THEOREM. If M is any complex manifold, then every rela-
tively compact strongly pseudoconvex domain G lying in M is holomorphically convex. He approximated real-analytically, and routinely used sheaf theory; for example, it is essential that all cohomology groups Hq(G, 0), q > 1, are finite-dimensional C-vector spaces. Since it is possible to surround every real-analytic manifold having a countable topology by strongly pseudoconvex complex tubes, the solution of the Levi problem has the following consequence: Every real-analytic manifold with countable topology is real-analytically embeddable in some I~N. Thus was a problem solved that H. Whitney had described as early as 1936 in his Annals paper "Differential manifolds" as the fundamental problem: Can any analytic
manifold be mapped in an analytic manner into Euclidean space? Grauert's solution of the Levi problem is a paradigm of Felix Klein's conviction that mathematics grows "as old problems are thought through with new methods." In Princeton in 1957 Grauert got to know the deformation theory of complex manifolds in the "nothing seminar" of Kodaira and Spencer. These two had dev e l o p e d - p a r t l y in collaboration with Nirenberg--a "Teichmfiller theory" for higher dimensions and, using the theory of harmonic integrals, had obtained deep results, e.g., semicontinuity theorems for dimensions of coTHE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
7
h o m o l o g y groups. Grauert quickly realized that m a n y theorems of this theory are a consequence of simple algebraic arguments, provided the coherence of image sheaves is ensured. From this follow further interesting properties of flat analytic families of complex structures over arbitrary complex spaces. So he started to prove the coherence theorem. After some m o n t h s of constant effort, he was rewarded with success; in his p a p e r "Ein Theorem der analytischen Garbentheorie u n d die M o d u l r ~ u m e komplexer Strukturen" (Publ. Math. IHES, no. 5 (1960), 233-292) he showed by a n e w p o w e r series technique, and by using "Met~fiberdeckungen':
M A I N T H E O R E M I (p. 287): Let 7r : X ~ Y be a proper holomorphic mapping of a complex space X into a complex space Y, and let S be a coherent analytic sheaf over Y [sic!]. Then the image sheaves 7re(S), g = 0, 1 , 2 , . . . , are coherent analytic sheaves over Y . This theorem and the m e t h o d s developed for conquering it attracted w o r l d w i d e attention. The finiteness theorems of Cartan and Serre are trivial corollaries. Even G6ttingen, where C. L. Siegel was resentful of anything as m o d e r n as the past 100 years, s h o w e d its respect: In 1959 Grauert went as Siegel's successor to the G e o r g i a Augusta University.
4~
HANS
GRAUERT
ein Doppelkomplex, da alle Durchschnitte U~....~, V~...,x holomorph-vollst~indige R~iume find, folgt aus dem Theorem B yon H. C~gxm% dab ~ und a sogar exakt find. Wit definieren fdr ~ eine Pseudonorm. Jede Kokette ~={~...%~...,~}r 'x ist, wenn /0. . . . . i~ festgesetzt werdeu, eine Kokette ~...~,eC~(*UnV,(p),S"), I,o=.~1 Aus Satz 6 folgt sofort : *U----U~!..,,(p). Wir setzen 1141" " ~'"*k
(t) Es sei Xveo. Dana gibt es zu jed~r endlichen KoketU ~eC~ "x mit ,~F,=o einz .x--t gilt. ~uordnung izt in bezug auf die Normen [l~lk+~.o , , , ~ . g i g oo~ (~) d) linear beschriinkt. IIKII;;, " Sam 7 ergibt : (~) Di~ ~uordramg [ ~ r ' eC*(U(p), S~) mit ~aC~.,~ und ~ = o ~vo~
IIKII;o, I W l h . , , , ~ g i g
ist in beeug auf die
~,, (~, d) zi,,,~ b.~a,a,~t.
Wir definieren : z~.'={~"
: ~=~r
k_>o, x>o,
B,~=~C~ -~-~'-~, t>o, X>o, ~'-' : ~r
~,.'=qr
B;" = ~ { ~ c ~ - ~ '
: ~r
x>o, ~>o,
und setzen (i) p~x
~ei r
..... } ~ z ' c v ( p ) ,
s,)
7~x/RKx
~i- K,,~y~a,~ mit
Jlr
w i t definier~-
einige Folgen yon Koketten : L = { k . . . ~ , , , ....... }
~z~'-',
~ = o , . . . , t.
~'={~, ~ , ....... } ~z~,,--(v,,(p), v,,(p)), ~-- ~. . . . . t. ~. = {n~....~._,. ,. ....... } E ~ - ~ " - " ~" = { ~ . . _ , .... ~ ....... } ~c--~.'-,CV,._,Cf,), v,,_tCp))) M= x. . . . .
t. .r ={.r~...,,_,, ~......... }~&-~.~---'(v,,(p), v,(0)), ~= t, ..., l--z.
r, = {r,...,3
~~(v,,(p)).
Wit setzen znniichst ~.~ ..... = ~ . ..... und erhalten ~. Es werde darm ~, so bestimmt, dab ~ , = ~,-1 ist. F,, sei ~,. Die Definition yon ~'~, ist besonders einfach : . . , ~ = ( __ ,) ~ {~....,,_,.,, ....... } ,,,it k...~,_,.,. ....... = ~ . . . . ~ - , ' , , , ....... " Setzen wir ~'~-----8"~',, so gilt offenbar : a'~,='~,_,, ,~'~z=~, (Man beachte, dab alle Koketten antikommutativ in ihren Indizees find). T, wird so gew~ihlt, dab
(z) Die Bewehidee itt, dab wit zeigen : Ht(V(,), Sa) = H~'~ = H~ - z , l ~ . . . ~ H ~
8
THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 2, 1995
Hk(U(p), Sa).
In the IHES paper there appear, for the first time, com- It is rich in new ideas and deep theorems; I cite four plex spaces whose holomorphic functions remain invis- highlights: ible to the geometric eye: they may be nilpotent. Grauert (a) LEMMA on the deflation of analytic sets. A nowhere and Grothendieck had the idea in 1957, independently of discrete compact analytic set A in a complex space X can each other, of introducing such spaces: One has to permit be "deflated complex analytically to a point" iff there is a nilpotent elements if one wants to utilize the full power strictly pseudoconvex neighbourhood U (A ) c X such that of the methods of algebraic and analytic geometry (inA is a maximal compact analytic set in U(A). finitesimal neighborhoods) which go back to E. Noether. At their meeting in Strasbourg in May 1957 Grauert and Grothendieck exchanged ideas which were soon afterwards to revolutionize analytic and algebraic geometry. The air got thinner once more, but at the same time clearer. The best mathematicians had difficulties in masl) tering the new thinking. Thus during his lectures in Harvard in 1958, Grothendieck always carried a small card in his breast pocket inscribed by J. Tate that he pulled out during the discussion periods: "There may be nilpotent elements in it." (Source: The Unreal Life of Oscar Zariski by Carol Parikh, Academic Press, 1990, p. 155.) By the early sixties everything was already history: The new spaces Thus a problem was solved that had been of great interare again called simply algebraic, resp., complex spaces. est since Hopf's studies on the rr-process (1948) and has In generalizing basic concepts, the utmost care is contributed to the founding of the theory of modificaneeded. There is such a thing as cheap and easy general- tions. ization that only performs jumps in the air and does not enhance the substance, that merely dilutes good wine. (b) PROJECTIVITY CRITERION. A reduced compact complex space X is projective-algebraic iff there is a weakly Fruitful generalizations go to the heart of the matter and negative vector bundle V over X. deepen one's understanding. That is what happens in the work of Grothendieck and Grauert. Here a holomorphic vector bundle V is called weakly negSince the Greeks nilpotency had lived hidden a w a y - - ative if the zero section in V possesses relatively compact e.g., on the parabola y = x 2 with the Artin ring II~[x]/(x2); open strictly pseudoconvex neighbourhoods. Nowadays, now, after more than 2000 years, it is brought to the sur- this criterion and its many variants are everyday tools of face of the rational world, and overnight it becomes a the complex-algebraic geometer. notion without which algebraic and analytic geometry are no longer imaginable. (c) F U N D A M E N T A L THEOREM. Every normal Hodge In H. Weyl's review of the second Grundlehren volume space is projectively algebraic. of Courant and Hilbert, we find the beautiful sentence: By a H o d g e space here we understand a compact comWhen one has lost oneself in the flower gardens of abstract plex space X with a K/ihler metric ds 2 whose class of algebra and topology, as so many of us do nowadays, one becomes aware here once more, perhaps with some surprise, of periods lies in the image of H2(X, Z). The fundamental theorem is part of a great tradition: how mighty and fruitbearing an orchard is classical analysis. it crowns investigations that Riemann began in 1857 and [Bull. AMS 44 (1938)] that Poincar6 and Wirtinger brought to completion (after In my opinion these lines could have been written in a fruitless efforts by Weierstrat~): review of Grauert's IHES paper. The excerpt (opposite) An n-dimensional compact complex toms cn/F2nis an gives an idea of how strenuously Grauert worked in his Abelian (= projectively algebraic) variety iff the lattice orchard. F2,~ satisfies the Riemann period relations. Perhaps here a variation of a maxim of Plutarch is comforting: In 1921 Lefschetz thought through this mysterious theorem anew. But in their core, the period relations were not Though a work may delight you by its depth and grace, understood until 1954: at that time, Kodaira, using ideas it does not of necessity follow that you will understand it of Hodge, proved his great theorem that Hodge manicompletely. folds are always projectively algebraic; the torus theorem was the catalyst and became an appendage. Grauert's The paper "Uber Modifikationen und exzeptionelle generalisation of Kodaira's theorem again required new analytische Mengen," Math. Ann. 146 (1962), 331-368, methods; they are so strong that they also proved Kopointed the way for transcendental algebraic geometry. daira's result. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
9
(d) COMPACT ALGEBRAIC SURFACES. There are connected compact 2-dimensional normal complex spaces X with a single singularity such that there exist two analytically independent meromorphic functions on X but nevertheless X is not projectively algebraic. For many, such examples were unexpected: according to a theorem of Chow and Kodaira, they do not exist for smooth surfaces. Later Hironaka found particularly simple examples; he blows up 10 points on an elliptic curve in P2 and then he deflates the elliptic curve. The category of Grauert-Grothendieck spaces has proved to be too narrow in the deformation theory of complex spaces. Grauert gave a theorem of Kuranishi its final form in "Der Satz yon Kuranishi ffir kompakte komplexe R/iume," Inv. Math. 25 (1974), 107-142, namely, THEOREM. Every compact complex space X has a holomorphic deformation (XT,lr , B) that is versal in 0 E B and complete in all points t E B. This had been proved for manifolds already in 1961 by Kuranishi (Ann. Math. 75 (1966), 536-577); his methods used almost-complex structures and harmonic analysis. If singularities occur, these techniques fail. Grauert works in the category of Banach-analytic spaces with nilpotent elements; he considers "astral spaces" which contain "complex spaces without points" [Ann. Math. 75 (1966), p. 119]; really a wonderful example of Dedekind's thesis that mathematical creations are God's free children, creations of the inventive mind, virtually without any relationship to external objects. About the same time, Douady, Hubbard, and Pourcin gave a proof. The transfer of the Kuranishi theorem from manifolds to spaces was considered to be very difficult. In the late sixties a prominent colleague gave his opinion in The New Encyclopaedia Britannica: Despite the efforts of many people, the corresponding problem for an arbitrary compact complex space remains unsolved. This is certainly among the most important questions in complex analysis. Grauert solved the pivotal problems of deformation theory with such convincing power that the whole theory nearly ran out of steam. Grauert's work is not restricted to complex analysis. Here I quote only two topics: first there is rigid analytic geometry whose development he decisively influenced after publication (1962) of the private notes Rigid Analytic Spaces by J. Tate. Further there is his paper "Mordells Vermutung fiber rationale Punkte auf algebraischen Kurven und Funktionenk6rpern," Publ. IHES, no. 25 (1965), 364-380. In the papers on rigid algebraic geometry the methods still have their complex-analytic origin in the Weierstrat~ preparation theorem, but in his second IHES paper, Grauert is a complete algebraist. There he proves for function fields the theorem which in 1984 was shown for number fields by G. Faltings. In 1966 in a lecture at the 10
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
Tata Institute in Bombay, P. Samuel wrote with obvious surprise: The algebraist Manin has given an analytic proof, in which k = C .... The analyst Grauert gives a purely ~gebrogeometric proof, a large part of which is valid in characteristic p # 0. In recent years Grauert has worked intensively on analytic equivalence relations. In this theory one starts with a reduced complex space (X, 0x) and an equivalence relation --~ on X. To ,-~ belongs in a canonical way a topological space Y : = X / ~ with a structure sheaf 0y and a morphism 7r : X --* X / ~ of ringed spaces. One asks: When is (Y, 0y) itself a complex space? Certainly the following properties are necessary: ,-~ is an analytic equivalence relation: the graph of ~ in X x X is an analytic set. X/~,, is locally holomorphically spreadable: Every point y E Y has an open neighbourhood V, such that the equivalence relation of the separation " y ~ y ' ~=~f(y) = f(y') for all f E Oy(V)" is discrete on V. Added in Proof: In 1994 the two-volume Selected Papers of Hans Grauert was published by Springer-Verlag.
Already in 1960 Cartan had shown that for proper equivalence relations these properties are also sufficient for (Y, 0y) to be a complex space. Grauert now showed in his paper "Set theoretic complex equivalence relations," Math. Ann. 265 (1983), 137-148: THEOREM. Let (X, 0x) be semi-normal with countable topology; let N be a semi-proper analytic equivalence relation, such that X / N is locally holomorphically spreadable. Then (Y, Oy ) is a weakly normal complex space. But this is only a prelude to the deeper theory of meromorphic equivalence relations, Here also Grauert succeeded in getting definitive results, bringing to fruition Stein's old ideas. An encomium should also say something about the man. The noble saying "Uhomme n'est rien, l'ceuvre tout" by G. Flaubert (letter to G. Sand, 31 Dec. 1875) may not be valid today. Grauert is an aristocrat among mathematicians, simple and modest in his habits. In 1969 when he became editor-in-chief of the Mathematische Annalen, he accepted the challenge. The situation of the Annalen in those days can be described in two sentences: In 1970 at the Bonn Arbeitstagung an outstanding American colleague asked me: "Do you believe that M A will improve now?" At the 1972 Arbeitstagung he gave the answer: "Congratulations, one has to read M A again." Already the young d o c t o r - - f o r m e d by outstanding t e a c h e r s - had something that one best paraphrases as "mathematical culture." As required, he is able to think
analytically or geometrically or algebraically. His mathematics is n o t - - as Jacobi once expressed i t - - a science of the self-evident. He has forced us to regard those problems as important that he considered important. From the wealth of his ideas, students and colleagues could always draw anew. Many a theorem that today carries another name, goes back to him. Henri Cartan sends regrets that he can not be here today. Let me quote this from his letter to the prize winner: Vous avez r6ussi ~ r6soudre des probl6mes de la th6orie des fonctions analytiques de plusieurs variables complexes qui 6taient consid6r6s comme pr6sentant des difficult6s insurmontables. You would also like to hear a critical note? Well, rather than going out of my way to find some reservations, I will just recall what Lessing wrote in La6coon about his much admired Herr Winkelmann: It is no minor praise to have made only such mistakes as everybody else had been able to avoid. Everywhere there are kings and carters. Hans Grauert i s - to use Kronecker's w o r d s - - k i n g and carter simultaneously. Undisturbed by the Zeitgeist he goes his way, always following the Chinese saying: "Whoever wants to reach the source has to swim against the tide."
Mathematisches Institut Einsteinstrasse 62 D-48149 Mfinster Germany
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
11
Already in 1960 Cartan had shown that for proper equivalence relations these properties are also sufficient for (Y, 0y) to be a complex space. Grauert now showed in his paper "Set theoretic complex equivalence relations," Math. Ann. 265 (1983), 137-148: THEOREM. Let (X, 0x) be semi-normal with countable topology; let N be a semi-proper analytic equivalence relation, such that X / N is locally holomorphically spreadable. Then (Y, Oy ) is a weakly normal complex space. But this is only a prelude to the deeper theory of meromorphic equivalence relations, Here also Grauert succeeded in getting definitive results, bringing to fruition Stein's old ideas. An encomium should also say something about the man. The noble saying "Uhomme n'est rien, l'ceuvre tout" by G. Flaubert (letter to G. Sand, 31 Dec. 1875) may not be valid today. Grauert is an aristocrat among mathematicians, simple and modest in his habits. In 1969 when he became editor-in-chief of the Mathematische Annalen, he accepted the challenge. The situation of the Annalen in those days can be described in two sentences: In 1970 at the Bonn Arbeitstagung an outstanding American colleague asked me: "Do you believe that M A will improve now?" At the 1972 Arbeitstagung he gave the answer: "Congratulations, one has to read M A again." Already the young d o c t o r - - f o r m e d by outstanding t e a c h e r s - had something that one best paraphrases as "mathematical culture." As required, he is able to think
analytically or geometrically or algebraically. His mathematics is n o t - - as Jacobi once expressed i t - - a science of the self-evident. He has forced us to regard those problems as important that he considered important. From the wealth of his ideas, students and colleagues could always draw anew. Many a theorem that today carries another name, goes back to him. Henri Cartan sends regrets that he can not be here today. Let me quote this from his letter to the prize winner: Vous avez r6ussi ~ r6soudre des probl6mes de la th6orie des fonctions analytiques de plusieurs variables complexes qui 6taient consid6r6s comme pr6sentant des difficult6s insurmontables. You would also like to hear a critical note? Well, rather than going out of my way to find some reservations, I will just recall what Lessing wrote in La6coon about his much admired Herr Winkelmann: It is no minor praise to have made only such mistakes as everybody else had been able to avoid. Everywhere there are kings and carters. Hans Grauert i s - to use Kronecker's w o r d s - - k i n g and carter simultaneously. Undisturbed by the Zeitgeist he goes his way, always following the Chinese saying: "Whoever wants to reach the source has to swim against the tide."
Mathematisches Institut Einsteinstrasse 62 D-48149 Mfinster Germany
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
11
Jeremy Gray*
Landau and Teichmiiller M. R. Chowdhury Readers of the Mathematical Intelligencer who, like me, h a v e a nostalgia for G6ttingen, should be thankful to the editors for reprinting N o r b e r t S c h a p p a c h e r ' s address [1] at the dedication of The E d m u n d L a n d a u Center for Research in Mathematical Analysis, in the D e p a r t m e n t of Mathematics of the H e b r e w University at Jerusalem. It is difficult for us e v e n to imagine the intense humiliation that w a s m e t e d out to L a n d a u (1877-1938) when, on 2 N o v e m b e r 1933, his a t t e m p t to r e s u m e teaching his calculus course in p e r s o n w a s nullified b y the b o y cott staged a n d led b y Teichm/iller. 1 For L a n d a u loved teaching, " m o r e p e r h a p s even than he realized himself," says H a r d y (1877-1947) in his obituary, 2 a n d continues, "His enforced retirement m u s t have b e e n a terrible b l o w to him; it w a s quite pathetic to see his delight w h e n he f o u n d himself again before a blackboard in C a m b r i d g e , 3 a n d his s o r r o w w h e n his o p p o r t u n i t y c a m e to an end" [2, p. 309]. Until very recently Teichm/iller's letter to L a n d a u "explaining the boycott" w a s t h o u g h t to be irretrievably lost. Rather unexpectedly, a t y p e d carbon c o p y of its t e x t - -
w i t h o u t the w r i t e r ' s n a m e - - w a s discovered in 1991 in the Nachlass of Eric K a m k e 4 (1890-1961). Its original 4 Kamke was forcibly retired in 1937 because of his opposition to national socialism, and was reinstated in 1945 after the war.
1 Landau was mortally wounded (t6dlich verletzt), said Gustav Herglotz (1881-1953) in a letter (dated 2 January 1934) to E. Kamke; see footnote 14 in [3] (pp. 5-6). 2 H. Heilbronn (1908-1975), a student of Landau's who had taken refuge in England in 1933, is listed as co-author, but the language is unmistakably Hardy's. 3 In 1935,when Hardy had arranged for Landau to be invited as visiting Rouse Ball lecturer. * C o l u m n Editor's a d d r e s s : Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England. 12
THE MATHEMATICALINTELLIGENCER VOL. 17, NO, 2 (~)1995 Springer-Verlag New York
m a y well have been the "copy of the text of Teichmiiller's letter" that Landau had enclosed with his letter to the ministry asking for his early retirement [1, p. 18]. This long-sought letter of Teichm/iller's has just appeared as an appendix to an extensive article [ 3 ] - in G e r m a n - - o n the life and work of Teichmfiller, edited by N. Schappacher and E. Scholz (with contribution by E. Scholz on Teichmiiller's life and by K. Hauser, F. Herrlich, M. Kneser, H. Opolka, and N. Schappacher on various aspects of his work). This important report, as the editors call it,5 comprises six sections, besides a foreword by the editors and an appendix containing two letters by Teichmiiller (including the one addressed to Landau). It is s t u d d e d with 67 footnotes which give additional valuable information. Scholz makes no attempt to characterize the tense and contradictory personality of Teichmiiller, 6 confining himself to a description of documented events of his short life7 (1913-1943). Teichmfiller's letter (dated 3.XI.1933) to L a n d a u is a lengthy affair; in [3] it takes two pages in small print (pp. 2 8 - 30). Neither its language nor its meaning is easy to follow. The opening sentence confirms "oral history" that the letter was written at Landau's request to sum up the standpoint represented by Teichmffiler in his conversation with L a n d a u the previous day. Teichm/.iller emphasizes at the outset that the letter reflects to some extent his personal view on "the difficult questions which surround yesterday's happenings." The second paragraph opens with the observation that "an action by students which t a r n i s h e s - - o r even just threatens to t a r n i s h - - the relation between a teacher and his students, can arise out of two causes," which he goes on to enumerate. "First, great non-academic successes of the spirit to which the larger part or a s o m e h o w decisive part of the students belong 8 can make situations which they hitherto implicitly accepted as unalterable, even though unsatisfactory, become out of date. Second, a provocative conduct, 9 or else a conduct attributable to a lack of interest f o r - - or perhaps of a thorough knowledge o f - - t h e mentality of the majority of the audience, m a y make the same impression upon the s t u d e n t s - though it be based on m i s u n d e r s t a n d i n g s - - a n d provoke them to resistance." "Which of the two causes here predominates, is not easy to decide," opines Teichmffiler. 5 It is more an obituary, albeit belated, of Teichmfiller, with detailed appreciation of his mathematical work. 6 Such an attempt was m a d e earlier in these pages by Abikoff [4], w h o also discussed Teichmffiler's mathematical work, except that on algebra. 7 Teichmiiller perished while on active military duty in an infantry division. Neither the date or place, nor the exact circumstances of his demise, is known. See footnote 44 in [1] (p. 14). s TeichmiiUer thus practically concedes that the followers of national socialism among the students were (at that time) very much a minority, albeit a disrupting (and, as he maintains, a decisive) one. 9 "This does not, naturally, come into question in your case," Teichm/iller gallantly assures Landau, in brackets.
Teichmtiller then (third paragraph) reminds L a n d a u that in the previous semester (Summer 1933) he had been a d v i s e d - - a d v i c e which he had f o l l o w e d - to give his lectures and exercises through an assistant. "As a result, we had become accustomed to treat the regulation as a natural consequence of political events and were astonished w h e n the position of the years before our revolution was sought to be reestablished," says Teichmiiller,
... Teichmiiller w a s instrumental in perpetrating a heinous crime ... w h i c h destroyed n o t only a truly great man and mathematician but also a great mathematical center.
and continues, "Before the discussion with you we h a d just assumed that this might be attributable to your thinking that y o u could n o w face us differently as we had not remained the old revolutionary fighters. Only thus are yesterday's occurrences explainable." In the discussion, however, Teichmiiller learned that other grounds were responsible for Landau's decision to try to resume teaching. We have perhaps no means of k n o w i n g for sure w h a t Landau said to Teichmfiller on that fateful day; in essence he must have insisted on his right and duty to teach, irrespective of the prevailing circumstances. The raison d'Stre of Teichmfiller's letter is to be found in the next two (fourth and fifth) paragraphs. Through yesterday's action a completely new situation has now been created. In order to restore peace in our institute it is necessary, above all, to clear up the fundamentals behind it. You spoke of your belief that what happened yesterday was an anti-Semitic demonstration. My standpoint was, and continues to be, that an anti-Jewish individual action might rather be directed against everyone else than against you. I am not concerned with making difficulties for you as a Jew, but only with protecting--above all--German students of the second semester from being taught differential and integral calculus by a teacher of a race quite foreign to them. I, like everyone else, do not doubt your ability to instruct suitable students of whatever origin in the purely abstract aspects of mathematics. But I know also that many academic courses, especially the differential and integral calculus, have at the same time educative value, inducting the pupil not only to a new conceptual world but also to a different frame of mind. But since the latter depends very substantially on the racial composition of the individual, it follows that an Aryan student should not be allowed to be trained by a Jewish teacher. The rest of the (fourth) paragraph is devoted to an elaboration of this atrocious doctrine. The fifth paragraph opens with the declaration that Teichmiiller would have little objection if Landau wished to hold, with the full consent of the students concerned, advanced lectures or seminars. Very few of Teichmiiller's comrades had agreed with this view though; the overwhelming majority had felt that a n y teaching activity THE MATHEMATICALINTELLIGENCERVOL.17, NO. 2,1995 13
b y Landau was intolerable. 1~ Teichmffiler concedes that in his opinion such an attitude could only be derived from anti-Semitism. Having said this, he loses no time in telling L a n d a u that "the difference between these two opinions is for the m o m e n t completely immaterial." He forcefully asserts that in no case should one think that there was a division a m o n g t h e m along "radical vs. m o d e r a t e " lines. "We all have a p r o g r a m and are good comrades; only on the purely theoretical question whether yesterday's action had anti-Semitic or proGermanic character do we have different views," says Teichm/.iller. Coming to the sixth and final paragraph, Teichmffiler tells Landau that he and his comrades were totally united on the p u r p o s e of the "action," the essential point of which was to restore the situation of the previous semester. As Dr. Weber was ready to represent L a n d a u in lectures and exercises, and as the uncertainty u of the previous semester no longer existed, it would not be necessary that L a n d a u should discuss the content of each individual lecture with Dr. Weber. H e (Weber) would give the lectures completely on his own; the students would prefer that. Because Dr. Weber w o u l d be the only person w h o w o u l d make a sacrifice in the entire affair, in that he w o u l d have to double his load in the interest of his y o u n g e r fellow-students, while Landau would merely have to stay a w a y from the lectures without any disadvantage w h a t e v e r - - pecuniary or othe r w i s e - Teichmffiler believes he has m a d e L a n d a u an "easy to accept" proposition. Thus ends Teichmffiler's infamous letter. There is no need to point out the m a n y absurdities, inconsistencies, and insults contained in this letter. The editors of [3] have called it a "bizarre" letter. I find nothing bizarre about it; I find it an extraordinary piece of writing, shamelessly u p h o l d i n g an indefensible attitude and an ignominious action, wherein the brilliant but thoroughly indoctrinated m i n d of the writer shines through. N o one should d e n y Teichmfiller his d u e as a mathematician because of his rabidly anti-Jewish activities (just as no one w o u l d wish to d e n y Andr6 Bloch his d u e as a mathematician because he was a multiple m u r d e r e r [5]); it is indeed a healthy sign that Teichmffiler's collected papers have been published of late [6] and that his life and work are being freely discussed. We m u s t not, however,
forget that Teichmtiller was instrumental in perpetrating a heinous crime, the L a n d a u boycott, which d e s t r o y e d not only a truly great m a n and mathematician b u t also a great mathematical center. N o one had more admiration for L a n d a u than Hardy. H e characterized [2, pp. 307-308] Landau's Handbuch der Lehre vonder Verteilung der Primzahlen (2 volumes, 1909) as his most important book, Vorlesungen fiber Zahlentheorie (3 volumes, 1927) as his greatest book, and
Darstellung und Begrfindung einiger neuerer Ergebnisse der Funktionentheorie 12 (1916 and 1929)~as his finest book. Of the Handbuch H a r d y said (writing in 1938), "Almost everything in it has been superseded, and that is the greatest tribute to the book"; of the Vorlesungen he said, "The richness of content of the book and the p o w e r of condensation it shows, are astonishing"; of the Ergebnisse he said, "It is one of the most attractive little volumes in recent mathematical literature, and the most effective answer to any one who suggests that Landau's mathematics was dull." We conclude with the following quote from [2, p. 309], as a fitting tribute to Landau. No one was ever more passionately devoted to mathematics than Landau, and there was something rather surprisingly impersonal, in a man of such strong personality in his devotion. Everybody prefers to do things by himself, and Landau was no exception; but most of us are at bottom a little jealous of progress by others, while Landau seemed singularly free from such unworthy emotions. He would insist on his own rights, even a little pedantically, but he would insist in the same spirit and with the same rigour on the rights of others.
References 1. N. Schappacher, Edmund Landau's G6ttingen: From the Life and Death of a Great Mathematical Center, Mathematical Intelligencer, vol. 13, no. 4 (1991), 12-18. 2. G.H. Hardy and H. Heilbronn, Edmund Landau, Journal of the London Mathematical Society (1), 13 (1938), 302-210. 3. N. Schappacher and E. Scholz (editors), Oswald Teichm/iller--Leben und Werk, Jahresbericht der Deutschen Mathematiker-Vereinigung, 94(1) (1992), 1 - 35. 4. W. Abikoff, Oswald Teichmiiller, Mathematical Intelligencer, vol. 8, no. 3 (1986), 8-16, 33. 5. H. Cartan and J. Ferrand, The Case of Andr6 Bloch, Mathematical Intelligencer, 10, no. 1 (1988), 23-26.
6. O. Teichmfiller, GesammelteAbhandlungen--Collected Papers, L. V. Ahlfors and E W. Gehring (eds.), Berlin: SpringerVerlag (1982).
10 Having read this, I doubt Shappacher's surmise that Landau "did not communicate Teichmiiller's name to the ministry, evidently in order not to create problems for this young and extremely talented student." Landau must have realized that communicating Teichmfiller's name would not only have been pointless but might have created more problems for Landau; at the least it might have hindered the smooth processing of Landau's request for early retirement. I have no doubt that, after TeichmfiUer's letter, Landau's own letter to the ministry was brief indeed. Needless to say, I fully share Schappacher's belief in Landau's "incredible sense of duty and correctness." 11 "What uncertainty?," one would like to ask Teichmfiller.
14 THEMATHEMATICALINTELLIGENCERVOL.17,NO. 2, 1995
Department of Mathematics University of Dhaka Dhaka-l O00 Bangladesh
12 A new edition of this still important work, revised, enlarged and brought up to date by D. Gaier was published by Springer-Verlag in 1986 (with Gaier cited as co-author).
The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author
and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editorin-chief, Chandler Davis.
Memories and Memorials 1 B. Booss-Bavnbek Remembering Teichmiiller The international mathematical community possibly has more urgent concerns than discussing the Nazi involvement of a German mathematician who died a long time ago. Should we not leave to our German colleagues the necessary work of preserving the history? Reading a series of memorials published by the official journal of the German Association of Mathematicians, the Jahresbericht der Deutschen Mathematiker-Vereinigung, I do not think so. They suggest to me that, on the contrary, renewed international interest should be taken in the German attitude toward the Nazi past. A recent perspective on the "life and work" of the German mathematician Oswald Teichmiiller (19131943) was provided by the Jahresbericht in [10]. 2 N o w Teichm/iller is remembered by some mathematicians and physicists for his original contributions especially to the theory of Riemann surfaces. Others, some of them refugees from Nazi Germany, remember Teichmiiller primarily as a member of the Nazi party since 1931 and in
1933 deputy leader of the science students' council at G6ttingen University, who with earnest resolution initiated and led the campaign to expel from academic life Richard Courant, Edmund Landau, Emmy Noether, Otto Neugebauer, and others. At age 20, heading a mob of brownshirts drummed together by him, he blocked Landau from entering the auditorium and told him the students refused to take instruction from a Jew. 3
1 To the memory of a dear friend, Ludovica Koch (1941-1993), who was vibrant with intelligence, knowledge, and compassion. 2 Some years ago, a summary of Teichmi.iller's political engagement and a survey of some aspects of his mathematical thinking for the nonspecialist were given in this magazine [1]. 3 In a letter to Landau, Teichmfiller added that he personally would have no objections to being taught by Landau in advanced subjects. Wolfgang Fuchs of Cornell recalls another "example of Teichmiiller's utter insensitivity and self-centeredness: In the summer of 1933, after Professor Emmy Noether had been dismissed due to Teichm~ller's henchmen, he approached her and suggested that she should give a private seminar to him and a few of his fellow students. And she, being a saint and utterly selfless, gave this seminar." THE MATHEMATICALINTELLIGENCER VOL. 17, NO. 2 (~)1995 Springer-Verlag New York 1 5
The Jahresbericht memorial contains 11 pages of biography by E. Scholz; comments on Teichm/.iller's mathematical publications b y various authors, particularly b y E Herrlich on the problem of moduli of Riemann surfaces ("Teichmiiller theory"); two letters b y Teichmiiller, one of Nov. 3, 1993 to Landau, the other of Dec. 12, 1938 to a former classmate from G6ttingen; a list of Teichm/.iller's publications; and a bibliography on the impact of Teichmiiller's work. The memorial comprises material from two seminars: one organized b y M. Kneser in G6ttingen, the other by H. Helling in Bielefeld. When I first read the memorial I was appalled and wrote a rather h a r s h - - m a y b e too h a r s h - - a d d e n d u m [3], which I circulated a m o n g mathematicians and historians. I'm grateful for the m a n y reactions I received, which have been taken into account in the present paper.
M y Objections The Teichmffiler memorial is at any rate more candid than m a n y other memorial articles published in the Jahresbericht, in that it does not pass silently over Teichmiiller's s u p p o r t for Nazism. What then were m y objections? These objections consisted partly of omissions a n d / o r questionable appraisals in the historical presentation of
w h a t is n o w called Teichmiiller theory. I compared the account of Lipman Bers: The theory of quasi-conformal mappings is about half a century old. Its originators were Ahlfors, Gr6tzsch and Lavrentyev.... The celebrated theorem by Teichmiiller, obtained about ten years after Gr6tzsch's results, should be considered as a far-reaching deepening and extension of Gr6tzsch's beautiful but simple papers. 4 N o t e too that Schiffer's interior variation of a Riem a n n surface (1938) is the same as a quasi-conformal variation, s In the memorial of 1992, however, the contributions of Teichmiiller's predecessors M.A. Lavrent'ev and M.M. Schiffer were not mentioned at all and H. Gr6tzsch only in passing. Lavrent'ev was a Soviet mathematician; Schiffer was a Jew forced to emigrate b y the Nazis; Gr6tzsch was dismissed as an anti-Nazi and w o r k e d at Halle after the War. Second, I felt that the Nazi state was represented as an a n o n y m o u s machine which could act as it did ind e p e n d e n t l y of support from individuals. Scholz wrote, for instance, on the boycott organized b y Teichmiiller
4 This is from his foreword to [6]. 5 See [5] and [9] (p. 276). Schiffer's paper [11]was carefully studied by Teichmiiller,according to some witnesses, and grudgingly quoted by him in another context.
Book-burning under the windows of the Mathematics Department of Berlin University, 1933. Reprinted by permission of Bundesarchiv. 16
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
to Jews or the struggle against them stands at the center of his thought and activity." Clearly, under this strict criterion Teichmfiller is (like Heidegger in Nolte's understanding) absolved from anti-Semitism, for he never linked any of his mathematical thoughts with it. For all the parallels between Nolte's Heidegger book and the Teichmfiller memorial, one must not overlook the differences between them: the philosopher Martin Heidegger surely was a pivotal figure, so any investigation of the relations between his thought and action as philosopher and as Nazi can be fruitful; Oswald Teichm/iller was, however brilliant a mathematician, a relatively minor figure in history, and there may not be 9 it does not necessarily follow that, if the work delights very much to be learned from looking more closely into you with its grace, the one who wrought it is worthy of your his Nazi engagement. esteem9[2] From the memorial, at least, nothing could be learned. Larry Zalcman (Bar-Ilan) elaborates: As Horst Tietz (Hannover) puts it in a letter to the German Association of Mathematicians [14], it "did not penIn ancient and medieval times it was widely recognized that a bad man could still do notable things and that high achieve- etrate the fog of incomprehension which lies over the ment was no guarantee of moral virtue. My conjecture would inferno." To do so it needed, at least, an addendum. be that Calvinist theology and German Romantic thought played a role in the development of the equation "genius = virtue". In any case, it would be of some interest to investi- A d d e n d u m a n d A f t e r m a t h gate this matter. I believe that this is behind the attempt to rehabilitate Teichmfller, if there is such an attempt, rather The addendum I submitted to the Jahresbericht was rethan a sinister neo-Nazi agenda. jected. In his letter, the editor W.D. Geyer criticized Jens Hoyrup (Roskilde) gives another possible read- me for providing only a few new "facts" about the life ing of the Jahresbericht memorial: "The emphasis on and work of Teichmiiller, for making "offensive evaluaTeichmfiller's distinction between politics (incl. the tions," for "questioning the scientific honour of the ofplanned takeover of professional power) and subject (i.e., fended persons," and for a "hardly tolerable attitude of theorems 6) functions - - no matter how it may have been moral superiority." He made it clear that any further disintended - - a s a legitimation of Teichmiiller's admission cussion of my article and its rejection was unwanted. on equal terms to the club." Herbert Mehrtens (Braunschweig) described the Jahresbericht's letter of rejection as "a mix of arrogance and political ignorance." R e v i s i o n of t h e Past Of course historical judgments may vary radically. One is reminded of the "historians' dispute" of the late Opinions may also differ on the style in which critieighties. E. Nolte, a prominent German historian, pro- cisms should be expressed. But here there was no offer voked the dispute by his controversial interpretations of help in adapting the style, only a blunt rejection, and of Nazism and the Final Solution, and his defense of I was made to feel that deviating perspectives were not the German philosopher M. Heidegger's deep and long- wanted. I therefore published my proposed addendum sepalasting commitment to Nazism and anti-Semitism. Nolte claims (I quote [12]) that he is not trying to justify the rately in a little booklet [3] which colleagues can order Holocaust and other Nazi atrocities but only to stop from me, without cost. Many of my colleagues have told me I should have the "demonization of the Third Reich," to free Germans from their "pathological condition" of still living in the written a different article, focusing more on what is hapshadow of Nazism, and to help Germany, with a clear pening now in Germany. Thus William Abikoff (Storrs, conscience, to "become a spiritually vital nation again." Connecticut) wrote, "I don't think I would write the corNolte insists that Heidegger was "a normal Nazi . . . . only rection as you did. I would write of historical evils that a middle-man" in establishing Nazi power and persecu- may make possible the expulsion (by the government) tions. He can also distinguish between a "normal anti- and killings (by the neo-Nazis) that are now [1992] ocSemite" and the more nefarious kind "for whom aversion curring." Similarly, Wolfgang Fuchs (Cornell): "At a time in which there are so many important things to criticize in modern Germany, we should concentrate our criti6 H o y r u p ' s own footnote: "Here, I think, one could dig deeper. cisms on these. To pour scorn on an effort that may be Teichmfiller must have found the ideas of special intuitive features of maladroit, but is well intentioned is a waste of ammuniGerman mathematics convenient to himself, so he could defend writing tion." his big treatise, bypassing all details in semi-Ramanujan style." in November 1933 against the number-theorist Edmund Landau: "His activities could only attain their dreadful effect to the extent that they were parallel to the institutional intervention of the Nazi state." I also missed any awareness of the broader impact of attitudes like Teichmfiller's for the Nazi ideology: racial supremacy, the Blitzkrieg concept, the Holocaust. Finally, I objected to the authors' deliberate renunciation of moral statements, as when they announced their intention "not to make an arch-villain of Teichmiiller." Teichmfiller was mathematically brilliant, but as Lipman Bers says, quoting Plutarch,
THE MATHEMATICALINTELLIGENCERVOL.17, NO. 2,1995 17
The "Courant-Landau Clique," the main target of the Germanification campaign. Foreground: Courant, Landau, Weyl. I accept this criticism. Today I would write my essay in obituaries," ever to publish an obituary for the statisdifferently. My message did not really get across, namely tician and pacifist Emil Julius Gumbel, while a compre(i) that the Jahresbericht memorial should not stand as hensive Bieberbach memorial celebrating Bieberbach's the last authoritative word on the life and work of mathematical work passed over his Nazi activities in 3 Teichmfiller, and (ii) that the editors of the Jahresbericht sentences out of 16 printed pages. 7 Mehrtens's comment: are strangely stubborn in defending the nationalist tone in some articles and in the editorial line. 7 Bieberbach became notorious for providing a theoretical justification,
Comments from Correspondents I was not the only one made uncomfortable by the Jahresbericht memorial. Irwin Kra (Stony Brook) in a letter to editor W. D. Geyer: "The original article on Teichmiiller has left a bitter taste in many mouths, mine included. The picture presented there is far from historical accuracy .... It would certainly be unfortunate if the Jahresbericht article were the last word on TeichmiiUer's life to appear in the Jahresbericht." I mentioned before that the Teichm/iller article was far more candid than some other memorials in the Jahresbericht. Herbert Mehrtens [8] was the first to comment in public that there is a bias in these memorials. He cited as characteristic the failure of the Jahresbericht, "not lacking 18
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
based on a pseudo-psychological typology of mathematical work, for the persecution of Landau: "Representatives of too different h u m a n races do not fit together as teacher and student. The instinct of the G6ttingen students felt in Landau an epitome of an un-German approach to the topics." And such calls from Bieberbach were published again and again: in the reports of the Prussian Academy of Sciences, in the widely read notices of the German Medical Doctors and Science Teachers, in a newly founded journal Deutsche Mathematik, and reprinted or summarized in numerous articles in German newspapers. After the end of the Third Reich, Bieberbach denied for 15 years the truth of the Holocaust and other Nazi crimes. Nothing of this is mentioned in the Centennial. Its author, the late H. Grunsky, mentions that Bieberbach was a Nazi, declares that he later recognized and deeply regretted these "errors," and asserts that "this is not the place" to go deeper into the matter. Bieberbach himself was not so tactful when, on January 11,1938, he demanded of the editor Grunsky of the reviewing journal Jahrbuch fiir die Fortschritte der Mathematik "to finally get rid of the Jews (and homosexuals) among the reviewers." [See ]7] and [13] (p. 217).]
"The official m e m o r y registers the person 'as a mathematician'; for anything else "this is not the place'; neither political nor moral judgment enters into it... The collective m e m o r y of the s c i e n c e s - - n o t only of mathematics - - produces the concept of the pure scientist." M a n y mathematicians were puzzled b y the pattern of belittling or concealing the Nazi past as in the centenary article for Bieberbach and the obituary for K. Strubecker. 8 Sanford L. Segal (Rochester), regarding the Strubecker obituary: "The characterization of the liberation of Strasbourg as Besetzung is particularly offensive. The city has, of course, been fought over m a n y times, but this sort of unreconstructed arch-nationalism should not find a place in an obituary. Once again, the editors did not edit appropriately and this m a y well reflect a bias on their p a r t - - i n c i d e n t a l l y , w h y did Strubecker merit this obituary in the first place, was he that p r o m i n e n t as a geometer?" 8 The obituary for Strubecker, to quote [4], "casts a warm glow over his appointment and services as Professor in the 'newly-founded Reichsuniversit~t' in Strasbourg during the Nazi occupation of that city. It describes those years as the happiest and most productive of his life. To top it off,the obituary characterizes the liberation of Strasbourg in 1944by the French as an 'occupation' (Besetzung)by allied troops! This evaluation of his life is completely silent on the content of his lengthy obituary on E. A. Weiss... In it, Strubecker glorifies the Hitler regime and praises Weiss fulsomely for his activism and leadership in the storm troops (S.A.)..."
Even more recently the Jahresbericht has published an obituary for AvakumoviG a Yugoslav mathematician w h o emigrated to West G e r m a n y in the 1960s. The authors of the obituary, J. Briining (Augsburg), entrusted with the liquidation of the A c a d e m y of Sciences of the former GDR by the German federal g o v e r n m e n t after reunification, and W. Eberhard (Duisburg) k n o w to report, "The end of the w a r brought also for the Avakumovi4 family deep-going slashes with the communist seizure of p o w e r u n d e r Tito." Lee Lorch (York University) notes concerning the Avakumovi4 obituary, "There is n o w h e r e any reference to the G e r m a n occupation, nor even any indication that Yugoslavia was in any w a y involved in the war. There is no statement as to what h a p p e n e d to Avakumovi4 d u r i n g the war, nor even w h e r e he was then. The bibliography excites m y curiosity on this; it lists five publications in 1940 (one in American Journal of Mathematics, four in Mathematische Zeitschrift), one in 1941 (apparently early in the year), in Yugoslavia. Then there is none until 1945, again in Yugoslavia. What was h a p p e n i n g in his life during those several intervening harsh years of the (unmentioned) G e r m a n occupation? The obituary does not tell us. The authors do not hesitate to slap at the left again and even to praise the monarchy, to say nothing of bypassing the fascists. The only mention of Tito is intended as a condemnation, nothing of his great war leadership which did so m u c h to h a m p e r the Nazi military."
The GOttingen mathematicians in happier times. Left: Emmy Noether on the steamer for KOnigsberg, 1930. Right: Edmund Landau's visit to Hollywood, 1931. (Left to right: G.T. Whyburn, Marianne Landau, Edmund Landau, Earl Hedrick, Susanne Landau, Karl Menger.) THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
19
Detlev Laugwitz (Darmstadt) writes, "I have been reading the obituaries in the Jahresbericht also with occasional astonishment, especially since I a m well acquainted with a few of the authors and respect them as otherwise quite reasonable people. It is not clear for me from which visual angle the editors select those a m o n g the m a n y deceased members of the G e r m a n Association of Mathematicians for a hagiography. (Probably old Kamke turns over in his g r a v e - - h e w h o brought the G e r m a n Association of Mathematicians after 1945 to new democratic life and watched carefully lest mathematicians 'implicated in the system' take over again.) Some years ago I wrote a short biographical note on the occasion of the 100th birthday of m y teacher in G6ttingen, Th. Kaluza (creator of the 5-dimensional field theory and highly regarded b y Einstein), but the Jahresbericht was not specially interested." It was published elsewhere. "Of course the years 1933-45 were not omitted. I think I treated them quite sensibly." Another c o m m e n t reads, "There seems to be a spate of obituaries appearing in the Jahresbericht, on mathematicians w h o s e passing was for one reason or another not previously acknowledged in that way. There must have been some conscious decision to solicit such articles, which a p p e a r to reflect a change of policy aimed to some extent at removing derogatory connotations to particular names." In a letter [4] to the Notices of the American Mathematical Society, David Brillinger, Chandler Davis, James A. Donaldson, Robert Finn, Wolfgang H. Fuchs, M a r y Gray, R a y m o n d L. Johnson, Jean-Pierre Kahane, Linda Keen, Irwin Kra, Klaus Krickeberg, Lee Lorch, B. Prum, Cora Sadosky, and Michael Shub presented objections similar to mine against the Jahresbericht's editorial policy. Many mathematicians, myself included, feel like H. Mehrtens: "Personally, I'm slowly getting tired of dealing with G e r m a n fascism in science history . . . . " But he continues, " . . . but I can't escape it because the topic is not at all closed but repressed as ever." That's the problem. It is not easy to write a biography. If one chooses to, here is the methodological approach that Thomas S6derqvist (Roskilde), a science historian, suggests: The life and work of a thinker cannot be treated as independent entities; nor can the life of the thinker be treated without a thorough understanding of the social, cultural, and political context in which his life and ideas were embedded. To pursue intellectual biography without these qualifications is not only scientifically unsatisfying--in the case of thinkers who devoted their lives to totalitarian causes it is also morally suspicious. History of science was founded on the belief that science is a progressive force towards democracy and human understanding. Even if this naive belief has been brutally refuted many times in this century, it is nevertheless an ideal which historians of science should keep in mind when they evaluate the lives and achievements of past generations. Without commitment to such moral standards of evaluation, history of science easily turns into cynicism. Although sympathetic to S6derqvist I w o u l d incline still more to the view of Jens Hoyrup: 20
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
9 [the perspective of the weekend seminars on Teichmffiler] dictates that the presentation of his work glorifies the originality of his achievements at the expense of pointing out other peoples' prior and independent discoveries. And so if the aim of the seminar was indeed historic-- and not hagiographic - - it should not have focused on Teichmffiler's mathematics but, e.g., on Teichmiiller theory and the associated history of ideas .... On the given conditions [of the seminars[, all that Scholz and Schappacher could d o . . . was probably to include the Teichmiiller letters, which, when carefully read, reveal him as one who would gladly make soap out of his victims and enjoy the profits.
References 1. W. Abikoff, Oswald Teichm611er,Mathematical Intelligencer 8(3) (1986), 8-16, 33. 2. L. Bers, Quasiconformal mappings and Teichmiiller's theorem, in L. Ahlfors, H. Behnke and H. Grauert, L. Bers, M. Heins, J. A. Jenkins, K. Kodaira, R. Nevanlinna, and . D. C. Spencer, Analytic Functions, Principal addresses deliv-
3. 4. 5. 6. 7.
8.
9. 10. 11. 12. 13. 14.
ered at the Conference on Analytic Functions (Princeton 1957), Princeton, NJ: Princeton University Press (1960), pp. 89120. B. Booss-Bavnbek (ed.), Perspectives on Teichmfiller and the Jahresbericht, IMFUFA tekst no. 249"/94, Roskilde, ISSN 0106-6242. Collective Letter, Notices Am. Math. Soc. 41(7) (1994), 571 572. F.P. Gardner, 1975: Schiffer's interior variation and quasiconformal mapping, Duke Math. J. 42 (1975), 371-380. S. L. Krushkal', Quasi-conformal Mappings and Riemann Surfaces, J. Wiley & Sons/V. H. Winston & Sons, New York/Washington, D. C. (1979) [Translated from Russian.[ H. Mehrtens, Ludwig Bieberbach and "Deutsche Mathematik," in Studies in the History of Mathematics, E. R. Phillips (ed.), Providence RI: Mathematical Association of America (1987), pp. 195-241. H. Mehrtens, Irresponsible purity: On the political and moral structure of the mathematical sciences in the National Socialist state, in Wissenschafl in der Verantwortung, Georges Fiillgraf and Annegret Falter (eds.), Frankfurt: Campus-Verlag, York (1990), pp. 37-55. [In German; English translation in Scientists, Engineers, and National Socialism, Monika Renneberg and Mark Walker (eds.), Cambridge: Cambridge University Press (1994), pp. 324-338.] S. Nag, The Complex Analytic theory of Teichmfiller Spaces, New York: Wiley-Interscience (1988). N. Schappacher and E. Scholz (eds.), Oswald Teichm611er--life and work, Jber. Deutscher Math.-Verein. 94 (1992), 1-39. [In German.] M. Schiffer, A method of variation within the family of simple functions, Proc. London Math. Soc. (2) 44 (1938), 432449. T. Sheehan, A normal Nazi, The New York Review of Books, 14 January 1993, 30- 36. R. Siegmund-Schulze, Mathematical Reviewing in Hitler Germany (Mathematische Berichterstattung in Hitlerdeutschland), G6ttingen: Vandenhoek and Ruprecht (1993). [In German.] H. Tietz, Who is a negative hero?, DMVMitteilungen (1994), no. 1, 41-42. [In German.[
Institut for matematik og fysik RUC/IMFUFA Postboks 260 DK-4000 Roskilde, Denmark
Shape and Size Through Hyperbolic Eyes Ruth Kellerhals
Introduction Hyperbolic geometry is a versatile and dynamic area of mathematics. It has deeply influenced and been influenced by fields such as topolog35 function theory, number theory, algebraic K-theory, and theoretical physics. It is, however, out of the scope of this article to survey hyperbolic geometry from its conception last century by Bolyai, Gauss, and Lobachevskii up to modern developments initiated by Thurston, Gromov, and others (see [M] and the references therein). The aim is rather to focus on one topic of hyperbolic g e o m e t r y - - t h e problem of determining volumes of hyperbolic n-dimensional manifolds M n - a n d to give a guide to recent results about the volume spectrum Vol~, the set of all volumes vol,~(M '~), which is actually of topological relevance. Finding Voln volumes is a true challenge. In contrast to the Euclidean situation, non-Eudidean volumes involve very special and transcendental functions in the form of iterated logarithm integrals called polylogarithms. For lower orders, these functions, although still mysterious, satisfy a certain cocycle equation explaining their importance for characteristic classes of singular spaces, for group cohomology and Hilbert's third problem on scissors congruence, for Zagier's conjectures about values of zeta functions, and for regulators for algebraic K-groups (for further discussion and references, see [Le]). Hyperbolic geometry has a certain priority, in that it incorporates both spherical and Euclidean geometry, as intrinsic geometries on embedded spheres and horospheres. In this article, working all the way up from the basics, I will describe the shape and size of hyperbolic bodies giving rise to hyperbolic n-manifolds, and insight into Vols.
I also try to plead for the importance of the two ancestors of this subfield of non-Euclidean geometry, N. I. Lobachevskii and L. Schl~fli.
A Hyperbolic Picture-Book There are various models for visualising hyperbolic space H '~ (as a general reference, see [B, p. 326ff]). The conformal models of Poincar6 in the upper half-space and in the unit ball are ideally suited to measure angles, since they appear as Euclidean angles. For example, three-dimensional hyperbolic space can be seen as H 3 = C X R + with coordinates z = x + iy, t > 0, and hyperbolic metric ds 2 = ( d x 2 + d y 2 + d t 2 ) / t 2. Its bounda r y O H 3 = PI(C) = C U {c~} is called the sphere at
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 @ 1995 Springer-Verlag New York 2 1
V
Figure 1.
infinity. The geodesics in H 3 are of two sorts; they are either straight lines or circular arcs both orthogonal to the bounding plane t = 0. (See Fig. 1.) The group of orientation-preserving isometries of H 3 corresponds to P S L ( 2 , C) = P G L ( 2 , C), which acts on the boundary sphere by z H (az + b)/(cz + d) taking Figure 3. Measuring distance in Klein's projective model. circles into circles. These M6bius actions can be extended to H 3 by passing from circles to hemispheres orthogonal or elliptic. Note that the orthogonal complement V • of a hyperbolic subspace V is elliptic, and vice-versa. Moreto the boundary. Better suited for drawings in higher-dimensional hy- over, k-dimensional planes in H n c E l'n are the interperbolic spaces, however, are the projective models, sections of H '~ with hyperbolic subspaces in E TM of diand especially the model of Klein. Let E 1,n denote mension k + 1. The isometry group in this picture is given the Lorentz-Minkowski space of real (n + 1)-tuples by the group O~,n c G L ( n + 1, R) consisting of matrices x = (x0, x l , . . . , x n ) equipped with the bilinear form A = (aij)o<_i,j<_n with (Ax, A y ) = (x, y} and mapping (x, y) = -xoYo + x l y l + " " + XnYn (x, y E R n+l) of sig- each of the two hyperboloid sheets onto itself, which nature (1, n). Then the submanifold {x c El'n I - X2o + translates to a00 > 0. In Klein's projective model (Fig. 3), H n is the interior x 2 + ... + xn2 = --1, X0 > 0} together with the metric - d x 2 + dx 2 + . . . + dx 2 is another interpretation for hy- of the quadric given by the equation (x, x} = 0 in real perbolic n-space, the hyperboloid or vector-space model. projective space Pn (R). Hyperbolic lines are Euclidean segments; hyperbolic angles, however, appear distorted. (See Fig. 2.) The restriction of the product in E 1,n to a (k + 1)- Finally, the hyperbolic distance dist(p, q) between two dimensional subspace V has signature (1, k), (0, k), or points p, q E H n is related to the cross-ratio [p, q; u, v] of (0, k + 1), in which cases V is called hyperbolic, parabolic, p, q and the intersection points u, v of the line through p, q with the above quadric: dist(p, q) = ~ Ilog([p, q; u, v]) I = ~ log \ ~ / 1 ' where x y denotes the Euclidean length between x and y. Polytopal Shapes
Figure 2. The hyperboloid model with a tesselated Poincar4 plane between the two sheets. (All computer graphics in this article were created by Dr. Konrad Polthier.) 22
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
One possible approach to constructing hyperbolic manifolds H n / F , where F is a discrete and fixed point-free group of hyperbolic isometries, is to find candidates for fundamental polytopes together with suitable face identifications according to the Poincar6 program. Indispensable for this geometrical procedure is a solid knowledge of hyperbolic polytopes. H o w does one describe polytopes and their combinatorial and geometrical properties? I will undertake to give a brief and certainly incomplete introduction to non-Euclidean polyhedral geometry.
Let X '~ be either the n-sphere S n e m b e d d e d in the Euclidean space E ~+1, or E '~ C E n+l, or the hyperbolic space H '~ e m b e d d e d in E 1'~, equipped with the bilinear forms (,) coming from their ambient space. A convex polytope P in X n is the intersection of finitely m a n y closed half-spaces b o u n d e d by hyperplanes Hi, i E I. For a compact Euclidean or hyperbolic polytope P, this is equivalent to saying that P is the convex hull of finitely m a n y points in X ~. We can write Hi = e~ in terms of the normal vector ei with (e~, e,) = 1 directed a w a y from the half-space (Fig. 4). The set of all convex polytopes P in X '~ is immense. Usuall}5 one restricts to manageable subfamilies by imposing conditions such as finite volume; or that all dihedral angles aij = / ( H i , Hj), i E I, be "acute," that is, less than or equal to 1r/2; or that the number ]I I of bounding hyperplanes be prescribed (for example, in the case of ]I[ = n + 1, that P be an n-simplex); or (even more conveniently) a certain minimal orthogonality among them. The first two conditions will be assumed henceforth as general hypotheses. These assumptions give rise to a partial classification in terms of the Gram matrix G(P) = ((ei, ej))i,jffl of P; namely, if G = (gij) is a symmetric ra x m matrix of rankn+lwithgii = l a n d g ~ j < O f o r i # j, a n d G is indecomposable (cannot be written as a direct sum), then G is the Gram matrix G(P) of a polytope P of finite volume and with acute dihedral angles such that
Therefore, a hyperbolic n-polytope P can have an arbitrarily large number m of facets as long as its Gram matrix G(P) has, among its m eigenvalues, precisely one negative and n positive ones. The vertices of P are described by their vertex polytopes of one dimension less. The vertex figure of P at the vertex q E P is the intersection of a sufficiently small sphere around q with P. If q E H n is a finite (infinite) point, it has a spherical (Euclidean) vertex polytope described by an elliptic (parabolic) principal submatrix of G(P) of rank n - 1. From the Gram matrix G(P) one can also read off whether P is compact or of finite volume. If P c X n has many dihedral angles equal to 7r/2, all combinatorial and metrical properties of P can be more conveniently visualized by means of its scheme or diagram ~ (P). G (P) is a weighted graph whose nodes ni correspond to the bounding hyperplanes Hi = e~ (i 6 I) of P. Two nodes ni and nj in Z (P) are disjoint if Hi and Hj are perpendicular in X n. Otherwise, they are joined by an edge with the positive weight cij := - ( e i , ej), which has the following geometrical meaning (Fig. 5):
(a) if G is positive definite (G elliptic), then m = n + 1, and P is a spherical n-simplex defined uniquely up to isometry; (b) if G is positive semidefinite (G parabolic), then m = n + 2, and P is a Euclidean (n + 1)-simplex defined uniquely u p to similarity; (c) if G is of signature (1, n) (G hyperbolic), then P is a convex hyperbolic polytope in H '~ = H n U cgHn with m facets defined uniquely u p to is0metry.
Moreover, by putting cii --= - 1 , and cij = 0 for Hi s Hi, ~(P) inherits all attributes such as order, determinant, and definiteness from the Gram matrix G(P) =
COS
cid =
O~ij 1,
cosh l~j,
if Hi and Hj intersect on P at the angle aid if Hi and Hj are parallel (for P c E n or H n, only) if Hi and Hi, admit a common perpendicular of length lij in P c H n.
(--Cij)i,j61. For the diagrams of Coxeter polytopes Pc c X '~ whose dihedral angles are all of the form rc/p with p > 2 an integer (and therefore acute), w e adopt the usual conventions coming from Lie theory: If two nodes are related by the weight cos (re~p), then they are connected
"~x,. x COS O~1
o
cosho/3~
o
E
o
~sh
lo
o
cosh 11 Figure 4. A simplex S in X 3 with facet Si = e#.
Figure 5. A Lambert cube in H 3 with its graph ~ (all dihedral angles equal to 7r/2 are omitted). THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
23
5
0
5
0
0
0--0--0--0--0
I
0
I
0 .......
0
Figure 6. by a (p - 2)-fold line for p = 3, 4, and by a single line marked p for p > 5. Parallel hyperplanes are characterized by ~ ~ o, and two nonintersecting or "ultraparallel" hyperbolic ones with common perpendicular by o . . . . o discarding the weight > I for short (Fig. 6). Coxeter polytopes are the fundamental domains of discrete reflection groups (Coxeter groups) acting on X n and giving rise to tesselations of X n (see Fig. 7). They are n a m e d after H. S. M. Coxeter who, in 1934, classified all spherical and Euclidean Coxeter groups. Hyperbolic Coxeter groups with quotients of finite volume, however, are far from being classified. One of the major obstacles is the combinatorial freedom w h e n constructing their fundamental polytopes [see (c) above]. Nevertheless, there are some partial results. For example, in 1950, E Lann@r found all compact hyperbolic Coxeter simplices. They exist only up to dimension n = 4 and are contained in the following list:
Figure 7. Tesselation of the hyperbolic plane H 2 by triangles with Coxeter graph
0
n=2
0
:
k
1
1
1
(#H-T+--
0
0
0
0
0
I
5 n=3
0--0
0
0--0
5
0
I I
0
0
6 0 ~ 0 - - 0
0
5 0
0
0
0
I I
0
I
0
0
59 0
0
5 0--0
0
5 o
o
o
o
8 n =4
0--0--0--0
0
5 0-~-- 0 - - 0 - - 0
0
5 0
0
0--0--0
'~
O--
0
/ 0
o
List of Lann~r simplices in H n. 24
/
.5
T H E M A T H E M A T I C A L INTELLIGENCER VOL. 17, NO. 2, 1995
o
o
\
I
E. B. Vinberg proved that compact hyperbolic Coxeter polytopes exist up to at most dimension 29; for noncompact ones of finite volume, his nonexistence bound equals 995. In contrast to this, concrete examples are known only for dimensions n _< 8 in the compact case (see Fig. 6, for example), and for n < 21 in the finitevolume case. The top-dimensional examples were found from arithmetical considerations. Actually, one has much more insight into the morphology of arithmetical Coxeter groups, but I don't want to go into this direction. For excellent surveys, see [V] and [VB].
The Size of Hyperbolic Figures Volume is one natural w a y of measuring the size of hyperbolic polytopes, though there are various others. I want to display some of the difficulties associated with the easily formulated problem "compute the volume of a hyperbolic simplex." Most of our knowledge about volumes of non-Euclidean polytopes is based on the works of N. I. Lobachevskii [L] and L. Schl/ifli [S] [boxes (i) and (ii)] and the methods they developed.
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
25
One common feature of their contributions is viewing non-Euclidean polytopes in terms of their dihedral angles which, in case they are acute, form a complete system of invariants. In comparison to the coordinate description, the angular characterisation has many advantages. One is Schl/ifli's beautiful formula for the volume differential. Consider an n-dimensional convex non-Euclidean polytope P. Deforming P a little such that its combinatorial structure is maintained, while its dihedral angles vary, the volume, with vol0(pt) :-- 0, behaves according to: dvoln(P) - n - - 1 E voln-2(F) daF; (1) F
here, n equals +1 or - 1 according to whether P is spherical or hyperbolic. The sum in Eq. (1) runs over all faces F C P which are intersections of two facets (faces of codimension one) of P forming the dihedral angle aF
(see Fig. 4).
Schl/ifli proved this formula only in the case ,~ -- +1 - surprisingly, he did not know about the existence of hyperbolic geometry! He formulated Eq. (1) in terms of his normalised volume function fn := w,~ voln, where wn -- 2n+l/vol,~(S n) is such that fn = 1 for a totally orthogonal simplex in S n. The three-dimensional hyperbolic version of Eq. (1) for orthogonal tetrahedra (orthoschemes) was already known to Lobachevskii. Another aspect common to Lobachevskii and Schl/ifli is the observation that every acute-angled polytope P can be genuinely dissected into simplices with a prescribed minimal number of right dihedral angles. Schl/ifli called these simplices orthoschemes and studied them on the sphere very thoroughly in his long and very rich article "Theorie der vielfachen Kontinuit/it" [S], the first treatise about (polyhedral) geometry in higher-dimensional (spherical) spaces. The dissection of P into orthoschemes can be obtained by cutting P first into arbitrary simplices, 9 and then, for each simplex separately, dropping perpendiculars successively to opposite and lower-dimensional faces starting from one vertex (Fig. 8). Orthoschemes are therefore the most basic objects in polyhedral geometry, and it suffices to determine size and volume of orthoschemes. An orthoscheme R c X '~ is bounded by n + 1 hyperplanes H 0 , . . . , Hn such that Hi 3_ Hj for [i - j[ > 1. Its n dihedral angles ai = /(Hi_I,H~) < 7r/2(1 < i < n) determine R up to congruence, and in hyperbolic space, they are all strictly smaller than 7r/2 (Fig. 9). The graph of R is given by the linear diagram ~(R):
Figure 8. Decomposition of a simplex into orthoschemes.
Figure 9. An orthoscheme in X 3 whose nonright dihedral angles sit at the dark edges. 26
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
o ~1 o . . . . .
o ~o,
where, by abuse of notation, we put as weights ai instead of cos ai. Orthoschemes generalize to higher dimensions the notion of the right triangle, which has easier trigonometry than a general triangle. Let us come back to Schl/ifli's volume differential representation. Formula (1) tells us that volume appears as a single integral expression, and its inductive character leaves us within one parity. For even dimensions n > 2m, Eq. (1) yields nearly free the familiar area formulas as angle excess a + fl +-y - ~r in the spherical case and as angle defect 7r - (a + fl + 3,) in the hyperbolic case of a triangle with angles a, fl, and % These expressions, taken as coefficients in Eq. (1), allow one to proceed to dimension four, and so on. Depending on the combinatorial structure, one can derive closed volume formulae for evendimensional non-Euclidean polytopes by induction. This general reduction principle was discovered by Schl/ifli, who worked it out in different cases; in Chapter 24 entitled "Reduktion der perissosph/irischen Plagioscheme auf artiosph/irische" and in Chapter 26 of [S], he considered the cases of even-dimensional simplices ("plagioschemes') and of even-dimensional orthoschemes. I would like to illustrate the reduction principle with a pretty example. Look at cyclic diagrams E of order n + 3 and of signature (1, n) whose weights are
//
\ \ ?'"-~
//
\\ /
OO
o
o
\
/ ,:X) /
\ \
/
~
\
/
\
/ o
\
Figure 10. A Coxeter polytope P c in H 4 with its graph ~
/
(Pc)
o
(Pc).
c l , . . . , c~+3. These graphs describe n-dimensional hyperbolic polytopes P of finite volume, which, in fact, are (bi)truncated orthoschemes. By means of Schl/ifli's methods, their volumes Fn (G) := con VOln(P), for n = 2m even, are computable by the same formula as for ordinary n-orthoschemes, that is [K1],
l_~ ll
F2ra(G)=~-'~(--1)k(2k) ~--~ k=0
a
~3
(2)
~'~
O~
f-1(0"): = 1, o"
w h e r e a runs through all elliptic subschemes of order 2(n - k) all of whose components are of even order. For example, take the four-dimensional hyperbolic Coxeter p o l y t o p e Pc with graph E having nine finite vertices and one vertex at infinity (Fig. 10). Its elliptic subschemes of orders > 2 are given by al : o
o
and
a3 : o
o
o
0"2 : o
o
o
of order two and of multiplicity two; of order four and of multiplicity one.
Now, the volumes of all elliptic Coxeter diagrams were determined b y Schl/ifli. These results are based on various properties of orthoschemes and Schl/ifli's v o l u m e function f,~ on orthoschemes; one is the multiplicativity fn(0-) = fnl(0-1)"" f,~,(0-r), where a is a linear elliptic scheme of order n + 1 consisting of disjoint c o m p o n e n t s a~ of order ni + 1 > 1 for i = 1 , . . . , r. With this, one finds fl(0"t) = /1(7r/3) = (2/r0(Tr/3) = 2/3, s = 1/2, and f3(0-3) = f1(0-1) 2 = 4/9, and finally, vol4(Pc) = (rc2/12)F4(E) = (rr2/12)[f3(a3) - 2 { f l (0-1)+ f l (r +2] = 7r2/108. For odd dimensions n > 3, deriving v o l u m e formulae for hyperbolic orthoschemes is incomparably more diffi-
Figure 11.
cult. Indeed, in trying to obtain a closed expression for the v o l u m e of a three-dimensional hyperbolic orthoscheme P~(R) : 0 ~1 0 ~2 0 ~3 0 one is confronted with an integral of the form vol3(R) = 1 Y'~,i=l 3 f ~ V~d~i, where the coefficient V/denotes the length of the edge li in R with dihedral angle oq (Fig. 11). The coefficient 11/is expressed in terms of the angles O~1, Ol2, Ot3 b y 1 cos(0 - ~ ) V~ = ~ log cos(0 T ~-7) '
C~i
where
~-7 =
7r
for i = 2 for / = 1,3.
The angle 0 is given by 0 = {~(O~1,OL2,~3)
= arctan
sin2~176
E2]0
COS Otl COS O~3
and the angles c~f are such that vol3(R(c~, c~, c~)) = 0 (take, for example, c~i = 0 for i = 1,2, 3). THEMATHEMATICALINTELLIGENCERVOL.17,NO. 2,1995 27
Figure 12. All ideal regular polyhedra in hyperbolic glass balls.
It is the merit of Lobachevskii to have found an antiderivative for the integral representing vol3(R) by introducing a n e w function which, slightly modified, is nowadays denoted b y JI2(w ) = - f o log [2 sin t[ dt and called the Lobachevskii function. JI2(w ) is highly transcendental and closely related to Euler's Dilogarithm Li2(z) = ~r~--1 (z~/r2) = - J o [ l ~ (1 - t)]/t dt appearing in various branches of mathematics and physics. The volume of a hyperbolic 3-0rthoscheme R is then given
by
by dissection, we can compute the volumes of all regular (all dihedral angles equal) ideal hyperbolic polyhedra (see Fig. 12). For the regular ideal tetrahedron S ~ (7r/3), one has vol3(S~ (7r/3)) = 3JI2(Tr/3 ) "~ 1.01494, and this is the maximal value which the volume on the set of hyperbolic 3-simplices can achieve. The regular ideal octahedron O ~ (7r/4) is of size vol3(Ooo (7r/4)) = 16. vo13(o .:, o o) = 8JIa(~r/4 ) _~ 3.66386, the regular ideal cube or hexahedron H a (7r/3) has volume 48.vo13(o o o 6 o) = 10J12(Tr/6 ) "-" 5.07471, and for the regular ideal dodecahedron Doo (7r/3), we compute
vol3(R) volg(Doo(Tr/3)) = 120. vol3(o 5 o : ~ +
Jla(oL1 q- O) - JIR(OZ1 - ~) q- Jl2 (~ q- oz2 - O) 7F
-
- o)
-- JI2(o~3 - 0) q- 2JI2 ( ~
+ - 0)
(3) .
Now, look at an "ideal" hyperbolic 3-simplex S~, that is, all vertices lie at infinity. Then S ~ is characterized by three dihedral angles a, fl, ~, satisfying the parabolicity relation c~ + fl + "7 = 7r. (This is the angle sum in each of the four congruent Euclidean vertex triangles of S~; see Fig. 8.) By dissection and using Eq. (3), o n e can show that vol3(Sc~) ---- JI2(o~ ) -ff JI2(fl ) q- JI2("/), a formula which was discovered by J. Milnor by direct integration of the volume element over S~. Also 28 THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 2, 1995
o 6 o)
_ 20.58010. The next step is, of course, to look at five-dimensional orthoschemes R c H 5 and to try, having recourse to Lobachevskii's result, to compute their volumes. This problem cannot be settled straightaway, yet there is a general volume formula [K3]. However, this formula involves several dozens of trilogarithms Li3(z) = fo [Li2(t)/t] dt = y~r~176(zr/r 3) in complicated trigonometric arguments related to the five dihedral angles of R. Here, let us confine ourselves to volume computations of Coxeter orthoschemes in H s. Luckily, there are just a few such Coxeter diagrams left which, moreover, belong or are related to the family of 5-orthoschemes of the type
Y~(R) ." 0
with
rxl 0
0:2 0 0~3 O
cos20q
c~I 0
0~2 0
vols(G1) = (1/5)vols(G2) ---- (1/10)vols(G3) = (7/46,080)~(3)
-}- C O S 20~ 2 -~- C O S 2 Of3 ---~1.
For this two-parameter famil~ it is comparatively easy to derive a volume formula [K2]. By means of Schl/ifli's volume differential (1) and of Lobachevskii's result (2) for 3-orthoschemes, we get vol5(R) 1
:~{JI3(oq)q-JI3(ol2)-
7r
~J[3 ( ~ - 0 ~ 3 )
16 JI3 ( 2 + ~ + ~
}
+ JI3 ( 2 - ~ + ~
+3~(3);
T h e V o l u m e Spectra o f H y p e r b o l i c M a n i f o l d s
(4)
here JI3(a; ) = 88Re(Li(e2i~)), for a; E R, is the Trilobachevskii function, and Li3(1) = ~(3) is Riemann's zeta function at 3. Hence, for the remaining three Coxeter orthoschemes in H s, represented by the graphs ~1 : O
O
O
O
O
0t
Y~,2 : o
o
c.
o
o
o,
~3 : o
o
o
o
o
,9,
whose volumes are
and therefore irrational. For odd dimensions n _> 7, the formal structure of hyperbolic polyhedral volumes is completely obscure. Apparently the classical polylogarithms Lik (z) = fo [Lik_l(t)/t] dt = ~r%1 (zr/rk) for k _< [(n + 1)/2] do not suffice anymore to express the volume of hyperbolic n-orthoschemes.
A hyperbolic n-manifold M is an n-dimensional complete, connected Riemannian manifold of constant sectional curvature -1. By the theorem of Hopf-Killing, M can be written as a quotient Hn/F of H n by a discrete group F of hyperbolic isometries acting without fixed points. The size of M = H n/F, as measured by the EulerPoincar6 characteristic x(M), the Betti numbers hi(M), the diameter diam(M), the first eigenvalue ),1 (M) of the Laplacian on M, or the volume voln (M), is a geometricaltopological invariant. This follows either from the definition, or, for n >__3, from the theorem of Mostow-Prasad. So, let us have a look at the set of volumes of hyperbolic n-manifolds, forming the nth volume spectrum Voln = {vol,~(M)[M = H~/F, F discrete, fixed point-free} c
R+.
Figure 13. (a) The 24-cell and (b) the 120-cell. H. S. M. Coxeter, RegularComplexPolytopes, Cambridge: Cambridge University Press ~1974). THE MATHEMATICALINTELLIGENCERVOL.17, NO. 2,1995 29
For even dimensions n = 2ra _> 2, the theorem of Gauss-Bonnet states that
x(M);
vol2m(M) -- (-1) m v~ 2
the spectra Vol2m are therefore discrete subsets of R+. For n = 2, that is, for Riemann surfaces Mg of genus 9 > 1, we k n o w that Vo12 -- 27rN, and that the minimal area 2~r is attained by precisely four nonhomeomorphic surfaces Mg, a sphere with three punctures, a torus with one puncture, the Klein bottle with one puncture, and the projective plane with one handle. For n = 4, the spectrum Voh was recently shown to be (47r2/3)N by J. Ratcliffe and S. Tschantz [RT]. They found a hyperbolic manifold M with x ( M ) = I and with bl (M) > 0: M is obtained from the regular ideal polytope P c ~ with 24 regular ideal octahedra as facets (Coxeter called P the 24-ce11; see Fig. 13a). P can be dissected into 1152 orthoschemes of type o o ~ o o whose volumes, by Eq. (2), can be determined to be 1r2/864. Thus, the volume of P is 47r2/3, which implies x ( M ) = 1 by Gauss-Bonnet. Since bl (M) > 0, M has d-fold coverings for all d E N, giving rise to hyperbolic 4-manifolds of volumes (4~-2/3)d for all d E N. M.W. Davis found a compact hyperbolic 4-manifold which is built u p o n the 120-cell P c H 4 (see Fig. 13b). P can be cut into 14,400 congruent orthoschemes o 5 o o o 5 o of volumes 137r2/5400. Therefore, M is of volume 1047r2/3, that is, of Euler-Poincar6 characteristic x ( M ) = 26. For even dimensions n _ 6, there are no concrete examples of hyperbolic n-manifolds k n o w n together with fundamental region and face identifications from which to determine Voln as in the case of dimensions two and four. Let us turn to the volume spectra Voln for n >_ 3 odd. H.-C. Wang has shown that Voln, for n # 3, is closed and discrete in R+. For n = 3, the situation is completely different. Thurston and Jorgensen proved that Vo13 is a closed, nondiscrete subset of R+ which is well ordered and of order type a;~. The m i n i m u m volume in Vol3 exists, but is still unknown. The smallest volume known, which is about 0.94272, is achieved by the compact hyperbolic 3-manifold which was studied independently by J. Weeks in his Ph.D. thesis and S. V. Matveev and A. T. Fomenko. W. D. N e u m a n n and D. Zagier investigated the structure of Vo13 near the accumulation points vk E Vol3, k > 1, which are the volumes of manifolds with k cusps. Moreover, there are the estimates vk ~ kvOlB(~qoo(~r/3))(with " = " precisely for k -- 1,2) due to C. Adams. For more details and references about Vo13, see [K1]. Finally, let us look at Vols. The only k n o w n concrete example of a hyperbolic 5-manifold M was constructed by J. Ratcliffe and S. Tschantz [RT]. This manifold M is noncompact and has bl (M) > 0. A fundamental polytope
30
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
P c H 5 for M can be dissected into 184,320 copies of the Coxeter orthoscheme o o o o o o with volume 7~(3)/46,080 [see the applications of Eq. (4)]. Therefore M has volume 28((3), and Vo15 contains all numbers 28~(3)d for d E N. In general, every Coxeter polytope P c in H n of finite volume gives rise to a hyperbolic n-manifold; namely, Pc is the fundamental region of a discrete reflection group F acting on H ~, but not without fixed points. Yet, by a lemma of Selberg, one can always find a subgroup F' of finite index in F which acts fixed point-free so that M = H~/F ' is a hyperbolic n-manifold. It is another question, however, to construct explicitly the manifolds whose existence is guaranteed. To seek compact representatives in this geometrical way seems daunting indeed. Acknowledgments
I w o u l d like to thank Konrad Polthier at the Sonderforschungsbereich "Differentialgeometrie und Quantenphysik" of the Technische Universit/it Berlin for producing the very nice pictures presented here. This article is an elaboration of the Colloquium talk I gave at York University, Toronto (Canada), in November 1993. References
[B] M. Berger, Geometry II, Berlin: Springer-Verlag (1987). [K1] R. Kellerhals, The dilogarithm and volumes of hyperbolic polytopes, in Structural Properties of Polylogarithms, Leonard Lewin (ed.), Providence, RI: American Mathematical Society (1991). [K2] R. Kellerhals, On volumes of hyperbolic 5-orthoschemes and the Trilogarithm, Comment. Math. Helv. 67 (1992), 648663. [K3] R. Kellerhals, Volumes in hyperbolic 5-space, Preprint MPI/94-14, accepted for publication by GAFA, 1994. ILl N. I. Lobachevskii, Zwei geometrische Abhandlungen, Leipzig: Teubner (1898). [Le] Leonard Lewin (ed.), Structural Properties of Polylogarithms, Providence, RI: American Mathematical Society (1991). [M] J. Milnor, Hyperbolic geometry: the first ]50 years, Bull. Am. Math. Soc. (2) 6 (1982), 9-24. [RT] J. G. Ratcliffe and S. T. Tschantz, Volumes of hyperbolic manifolds, Research Announcement, 1994. IS] L. Schl/ifli, Theorie der vielfachen Kontinuit/it, Gesammelte Mathematische Abhandlungen, Basel: Birkh/iuser (1950), Vol. 1. IV] E.B. Vinberg, Hyperbolic reflection groups, Russian Math. Surveys 40 (1985), 31-75. [VB] E. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature, in Geometry II, Berlin: Springer-Verlag (1993).
Max-Planck-Institut ffir Mathematik Gottfried-Claren-StraJ3e26 D-53225 Bonn, Germany
Squaring Circles in the Hyperbolic Plane William C. Jagy
The syndicated newspaper column of Marilyn vos Savant was particularly interesting one Sunday in November 1993 [Sa]. Ms. vos Savant announced there that she had no faith whatsoever in the work of Andrew Wiles on Fermat's Last Theorem. In stating her objections to the methodology of Wiles, she wrote that J~nos Bolyai "managed to 'square the circle'--but only by using his own hyperbolic geometry." The word "using" creates the misleading impression that Bolyai used illicit methods to square the circle in the Euclidean plane. What Bolyai did, in fact, was to construct, using the correct intrinsic versions of the compass and straightedge, a square and a circle in the hyperbolic plane with the same area. In this article, I will exhibit all possible such examples (Theorem A). I will also show that the square and circle must be constructed simultaneously: there cannot be a construction that begins with a circle of radius r and produces the correct comer angle rr for the square of equal area (Example B); neither can there be a construction beginning with a that produces the correct r (Example C). Theorem A, discovered independently by the present author, is contained in a 1948 article of Nestorovich [Nel] that has received little attention in English-language publications. That article also has an example similar to those in Example B, but Example C is not considered there. It may be, therefore, that Example C and the interpretation provided by Theorems B and C are new.
this can assume any positive value. Whether we consider constructibility or not, only circles with area _< 2~r (so that cosh r G 2) have a companion square with equal area. The reader may well be more familiar with the theorems of geometry in H 2 than with straightedge-andcompass constructions there. The very simplest constructions used in the Euclidean plane 1r2 are also available in the hyperbolic plane, essentially unchanged. If we imagine a creature in ~.~3 drawing on a flat sheet of paper, our creature can bisect segments, bisect angles, add or subtract segment lengths, add or subtract angles, and draw perpendiculars to lines, either through a point on the line or "dropped" from a point off the line. Differences from ~,,2 begin with the lack of a unique "parallel" to a given line through a given point. For example, it is not generally possible to trisect a line segment [Ma, p. 483]. Rather than "parallel" lines, given a line l and a point P off 1, we may always construct rays ml and m2 (half-
Introduction With the normalization we will be using, the area of a triangle in H 2 is the same as its "defect," that being ~r minus the sum of the three angles (the sum is guaranteed smaller than ~r). It follows that the areas of squares in the hyperbolic plane are bounded above, although the areas of circles are not. Indeed, a convex polygon with n sides has area bounded by (n - 2)~r, and this bound is achieved only by figures with sides of infinite length. In contrast, the circle of radius r has area 2zr(cosh r - 1), and THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 ~) 1995 Springer-Verlag New York 3 1
-1
) I
q
Figure 1.
s
L_
J T m
7
,
Q
f
)
R
l
Figure 2.
P Y
m
T
Figure 3.
32 THEMATHEMATICAL INTELLIGENCERVOL.17,NO.2,1995
l
)
lines), beginning at P, that are "asymptotic" to I, one in each direction (see Fig. 1). These asymptotic rays, which do not intersect I, can be characterized in several ways. First, if we drop the perpendicular from P to a point Q on l, then the angles between ray mj (d = 1,2) and segment PQ are acute (and equal), and any ray through P that makes a smaller angle with segment PQ must of necessity intersect line 1. It is also true that each ray gets "closer and closer" to the line 1: for instance, the distance between a point R on ral and a point S on I, both at distance s from the fixed point P, goes to 0 as s --* ~ . The concept of asymptotic rays leads to the definition of the function 11, which describes a bijection between nonzero lengths t and acute angles p, as illustrated in Figure 1. Then p is called the "angle of parallelism" for the length t, and t could be called the "length of parallelism" for p. The angle must be measured in radians, which are definable in H 2, based on the assignment of ~r/2 radians to the right angle. This defines the monotone decreasing function 1I: for this figure, I I ( t ) = p and I I - l ( p ) = t. J~inos Bolyai provided a construction for the asymptotic ray to a line 1 in a given direction, beginning at a point P off the line (see Fig. 2). We first drop a perpendicular from P to l, arriving at point Q. Next, draw the ray d through P that is perpendicular to segment PQ in the desired direction. Along the line l, in the same direction from Q, choose any point R, and then drop the perpendicular from R back to d, arriving at S. Use the compass to draw a circle around point P with radius equal to the length of the segment QR. The point of intersection of this circle with__fiegmentRS, labeled T, allows us to draw the ray ra = PT, and this ray is asymptotic to 1. It is also necessary to know how to reverse the previous construction: Given an acute angle p, defined by rays m and n through a common point P, construct the ray I that is perpendicular to n and asymptotic to m (see Fig. 3). The illustrated construction is due to Bonola [B, p. 106]. There is the necessity, in this method, of somehow knowing a point W on n that is so far away from P that the ray w through W asymptotic to m makes an acute angle with the segment PW. It is not clear ahead of time how to do this, unless we've already solved the problem for an even smaller angle than p. Knowing some point W sufficiently far away, we draw the ray w, and then drop perpendiculars from P and W to X and Y on rays w and ra. The segments PX and WY will intersect "in the interior" of the infinite triangle formed by segment PW and rays m and w, at a point we label T. Finally, drop the perpendicular from T to the point Q along se__~gmentPW. Extending the segment QT to the ray I = QT gives us the required ray. Bolyai's original work (described in [B, appendix III, pp. 216-226]) gives a lengthy sequence of intermediate constructions to solve the previous problem. Bonola's construction has the virtue of needing no explicit trigonometry for its justification. Martin found a method requiring few steps and no knowledge of any point "sufficiently far away" [Ma, p. 484].
There is an "absolute measurement" of length in H 2, an idea apparently d u e to Lambert [B, pp. 44-49]. One m a y associate to a given segment the angle at a vertex of the equilateral triangle with edges congruent to that segment, or some prescribed function of that angle. Further, there are "natural units" of length. One such is Schweikart's constant p, defined b y the equation U(p) = ~r/4. Another is Gauss's constant k, which can be associated with a relationship a m o n g the curves called "horocycles" [Ma, pp. 413-415]. The significance of k in hyperbolic trigonometry is analogous to that of the radius of the sphere in spherical trigonometry. Martin refers to k as the "distance scale." He proves results involving triangles, length, or area for the case k = 1, and then describes the adjustment for other values of k, essentially dividing a n y length by k [Ma, pp. 433-434l. For instance, the comparison between the two constants is sinh (p/k) = 1. I will also restrict to the case k = 1. It should be noted that it will not be possible to construct the length w e are n o w calling I with compass and straightedge; life is like that. A short glossary of trigonometry in ~.~2 is appropriate. This material, with different choices of symbols, is presented in Chapter 32 of [Ma], especially pp. 425 to 433. If we have any triangle (Fig. 4), we find a Law of Sines, along with two distinct versions of the Law of Cosines. The extra law can be said to result from the fact that any two triangles in H 2 that are similar are necessarily congruent. sin a sin fl sin 3` sinh a - sinh b - sinh c' C O S O~ =-
A
C
B
a
Figure 4.
cos a + cos fl cos 3` sin fl sin 3`
For a triangle with a right angle at vertex C, that is, 3' = ~-/2, we find a version of the Pythagorean Theorem and various other facts: cosh c = cosh a cosh b,
cosa-
In discussing the problem of "squaring the circle," Bolyai introduced an angle, which I shall call 8, associated with the radius r of a circle. It should be noted (having arranged k = 1) that the area within a circle in ~.~2 is expressed b y 4~r sinh2(r/2). The angle 0 will be constructed so that tan 0 = 2 sinh(r/2). The result is that the area of the circle is equal to 7r tan 2 0. There are explicit m e t h o d s for beginning with ~ and constructing r [Ma, p. 489] and for beginning with r and constructing ~. As a result, questions of constructibility for r can be rewritten as questions about ~. We exhibit the standard diagram (Fig. 5) illustrating the relationship between 0 and r, which is used in both these constructions.
cosh b cosh c - cosh a sinh b sinh c '
cosh a =
cosh a -
Other Constructions
cos a sin fl' tanh b - tanh c '
cosh c = cot a cot r , sin a -
sinh a sinh c '
tana=
tanh a sinh b"
We have already seen the construction of asymptotic rays, resulting in a type of infinite, "singly asymptotic" right triangle. The trigonometry for these reduces to relations between the finite edge, of length t, and the acute angle II(t): tanh t = cos II(t), cosh t = csc II(t),
\
sinh t = cot II(t),
tP"
,[
~"
II(t)
e - t = tan T
Figure 5. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
33
7
9/
b
b
I
Figure 6. It is necessary to describe the construction of a right triangle with the other two angles a and fl prescribed, and with c~ + fl < ~r/2 (see Fig. 6). First, construct lengths z and t so that II(z) = a and II(t) = ~ r / 2 - f l . As a < 7r/2 - fl, it follows that z > t, so we m a y use the compass to d r a w a right triangle with hypotenuse z and one leg t. Call the other leg of this right triangle length b. Finally, construct the right triangle with one leg of length b and the adjacent angle equal to a. A combination of the various trigonometric relations shows that the angle opposite to the edge of length b is, in fact, equal to ft.
Comparison of Constructibility in ~2 and H 2 Our main tool is an observation that an angle is constructible in ~.~2 if and only if it is constructible in j~,2 [Ma, p. 483]. This follows from comparing trigonometry in H 2 and E 2, as explained below. It is uncertain where the observation was first recorded. Suppose we give the name E to the set of lengths in E 2 that are constructible, beginning with some assigned length denoted 1. The elements of E are thought of as real numbers. By courtesy, the length 0 is a member of E , and if s < 0 and Is[ E E, we agree to say s E E. Recall that one can use the compass and straightedge in ]~-~2 to add or subtract lengths, multiply or divide them, and produce the square root of a given length. By considering intersections of lines and circles, it is s h o w n that the preceding operations characterize E exactly: it is a field and is the smallest subfield of • that is "closed under square roots": if s E E, s > 0, then x/-d c E. As to the hyperbolic plane, it turns out that a length t in H 2 is constructible with compass and straightedge if and only if sinh t E E, or cosh t c E, or tanh t E E, 34 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
these conditions being equivalent. Again, we consider 0 constructible, as also negative t w h e n It[ is constructible. Knowing the correct result, it is not difficult to prove this theorem. The original proof of the theorem is spread over at least four articles. These begin with two by D. D. Mordukha~-Boltovskoi ([M-B1] and [M-B21), which together show both sides of the if-and-only-if statement, but allow the use of a third drawing instrument called the "hypercompass." Later articles by Nestorovich established that all constructions that include the extra "hypercompass" can be performed with just the compass and straightedge. The interested reader m a y consult the m o n o g r a p h of Smogorzhevskii [Sm] or the problem book of Nestorovich [Ne2]. Another proof, discovered without knowledge of [M-B2], appears in the book of Kagan [Ka]. By considering a right triangle with any side of length 1 we find that an angle a is constructible in E 2 if and only if sin a, or cos a, or tan a is in E, these conditions being equivalent. Consider a right triangle in H 2 with one side equal to Schweikart's length p, which satisfies sinh p = 1. Using hyperbolic trigonometry, we find that an angle 7/is constructible in ~.~2 if and only if sin ~/, or cos ~/, or tan ~/is in E, these conditions being equivalent, i.e., A n angle a can be constructed in H 2 if and only if it can be constructed in E 2.
Matching Areas in H 2 We have already mentioned the auxiliary angle 0, with the property that the area of the circle of radius r is ~r tan 2 0. A "square" will be a convex quadrilateral with four equal edges and four equal angles (which must be acute). Let us refer to the corner angle of the square as a. This square can be constructed from eight right
triangles with angles o./2 a n d 7r/4 (Fig. 7). In each triangle, denote b y y / 2 the length of the side opposite the angle 7r/4, so that the e d g e of the resulting square is length y. O n e of the trigonometric relations reads cosh(y/2) = cos(Tr/4) - sin(a/2), f r o m w h i c h follows the r e m a r k (for the square a n d circle of equal area) that cosh y = tan2{(~r cosh r)/4}. Since the area of each triangle is its defect, ~r - 0r/2 + Ir/4 + o'/2) = 7r/4 - o./2, the area of the square is 8(~r/4 - a / 2 ) = 27r - 4o.. O u r m a t c h i n g area p r o b l e m can n o w be written in t e r m s of angles in H 2, that is, 27r - 4 a = Ir tan 2 0. The conditions that the square a n d circle be constructible are therefore expressible in terms of the c o n s t r u c t i b i l i t y - - in ]~-,2 _ _ of angles a a n d 0 that satisfy 27r - 4 a = ~r tan 2 0. Suppose w e give the s y m b o l ~; to the c o m m o n area of the square a n d circle. Since ~; = 21r - 4a, it follows that a; is a constructible angle in ~2. We are considering a; = Tr tan 2 8. As O is a constructible angle, tan O a n d its square are elements of E. If w e write x = tan 2 8, w e h a v e a constructible angle a; and a constructible length x in ]~-,2 such that a; = ~rx. To relate the various symbols, we record w = 2 ~ - - 4 o . = 7rx =Tr tan20 = 4~r sinh 2 ( 2 ) = 2~-(cosh r - 1). We note that 2o. + 7r cosh r = 27r, as well as x + 2 = 2 cosh r. The equation 0; = 7rx can be analyzed u s i n g a f a m o u s result a b o u t transcendental n u m b e r s o v e r Q. For convenience, w e refer to those complex n u m b e r s that are algebraic over Q as s i m p l y "algebraic" a n d assign to t h e m the s y m b o l A. Recall that the algebraic n u m b e r s A c C f o r m a field, and that i E A. One m a y find the following t h e o r e m stated in [Ni, p. 134]: G E L F O N D - S C H N E I D E R T H E O R E M (GS): If ~ and X are nonzero algebraic, ~ ~ 1, and X f~ Q, then any value of ~x is transcendental. R e m a r k s . The value of qax is defined to be e x p ( x log ~), so it is m u l t i v a l u e d like the logarithm. Also, GS applies w h e n X is an algebraic n u m b e r with n o n z e r o i m a g i n a r y part, such as - 2 i . Since one value for i-2~ is e ~, this s h o w s that e ~ is transcendental. Finally, GS prohibits ~ = 1, but allows ~a --- - 1 . The reader will note that E C A, so that E(i) is also a subfield of A. We m a y profitably return to the equation w = 7rx, b e t w e e n a constructible angle w a n d a constructible length x in g2. Since w is constructible, sin w -- sin 7rx a n d cos w = cos 7rx are b o t h in E. It follows that e ~ = cos ~rx + i sin 7rx belongs to E(i) c A. If w e choose l o g ( - 1 ) = 7ri, this m e a n s that ( - 1 ) ~ = exp(x l o g ( - 1 ) ) = exp(iTrx) is in A. O n the other hand, x E E c A, s o t h a t ( - 1 ) ~ ~ A i m p l i e s x i s r a t i o n a l b y G S . F r o m a relationship exhibited earlier, x + 2 = 2 cosh r, w e note that cosh r m u s t also be rational.
r Figure 7. N o w that x E Q, s u p p o s e w e write x as m / n in "lowest terms," that is, m, n E Z, n > 1, a n d gcd(m, n) = 1. Then there m u s t be integers u a n d v such that u m + vn = 1. If w e multiply this t h r o u g h b y 7r/n, w e find umTr/n + wr = 7r/n, or uw + wr = zr/n. A s u a n d v are integers, a n d w is a constructible angle, r / n m u s t also be a constructible angle. This is related to a f a m o u s question, w i t h a f a m o u s answer, s u p p l i e d b y Gauss a n d Wentzel (see [K1]). By placing an angle 27r/n at the center of a circle a n d copying it n times, w e construct, in ~2, a regular p o l y g o n of n sides. The f a m o u s a n s w e r implies that n m u s t h a v e p r i m e factorization n = 2J Fil Fi2 "'" Fi~ 2z
(where the Fin are distinct p r i m e s of the f o r m 1 + 2 ), d -> 0, r > 0. The F~a are often called "Fermat n u m b e r s , " a n d only five are k n o w n to be prime: writing Fz = 1 + 22~, these are F0 = 3, F1 = 5, F2 = 17, F3 = 257, a n d F4 = 65,537. The next one, F5, has a factor of 641, and F6 has a factor of 274,177. In the y e a r 1987, the values F5 t h r o u g h F21 w e r e k n o w n to be c o m p o s i t e [R, pp. 71-74]. In this y e a r 1994, w h e n secret codes are b a s e d on the difficulty t o d a y ' s c o m p u t e r s have factoring a " r a n d o m " n u m b e r w i t h 200 digits, it is sobering to note that even F10 has o v e r 300 digits in decimal notation. Returning to the equations a; = 27r - 4o. a n d w = 7r tan20 = 7rx, w e find that o., the corner angle of the square, satisfies a = (2~ - w)/4, so that o. is a rational multiple of ~, w i t h d e n o m i n a t o r n as described above. We shall reject the "square" with o. = 7r/2; if that m a d e a n y sense in H 2, it w o u l d be a single point, with area 0. We shall allow the square with four infinite edges a n d o. = 0, of area 27r. There is a c o u n t a b l y infinite set of satisfactory angles o., a n d they are dense in the interval f r o m 0 to 7r/2. We h a v e s h o w n that the p r o b l e m of constructing the t w o figures is equivalent to the p r o b l e m of constructing regular p o l y g o n s in ~p2 ( o r , indeed, in H2): THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
35
T H E O R E M A. Suppose a square of corner angle crand a circle of radius r in ~_~2 have the same area a;, so that a; <_ 27r. Then both are constructible if and only if a satisfies these conditions: 0 _< a < 7r/2, and a is an integer multiple of 2~r/n, n a positive integer such that the regular polygon of n sides can be constructed with compass and straightedge in E 2. Note that w e simultaneously p r o d u c e d the square and the circle from auxiliary information, and required that both be constructible with compass and straightedge. The natural question occurs, "What of a m e t h o d that begins with any a or r, and produces the other?" Theorem A is silent on this question; however, there are no such methods. We exhibit two (dense sets of) examples, showing that there is no general m e t h o d in either direction.
Example B. Let m / n be a rational n u m b e r in lowest terms, such that n is not a p o w e r of 2, but has some odd prime factor d. Then 0 = arctan (re~n) is a constructible angle. But a; = 7r tan 2 0 = 7rm2/n 2 cannot be constructible, as that w o u l d imply constructibility for the regular polygon of d 2 sides. If there were a construction that began with r (whence 0) and p r o d u c e d the correct w (whence a), then w h e n e v e r r was constructible, the resulting a w o u l d be the outcome of a long construction. Our family of examples provides constructible r with the corresponding cr nonconstructible, thus precluding the existence of such a method.
that a;/~r is transcendental. As w = Tr tan 2 0, w e have o;/Tr = tan 2 0, and so we k n o w that tan 2 0 is transcendental, finally showing that tan 0 is itself transcendental. Because E c A, this means that, although a = arctan q is constructible, the angle 0 appropriate to a is not. If there were a construction that began with a (whence w) and p r o d u c e d the correct ~ (whence r), then w h e n e v e r we constructed or, the resulting r would have been constructed. Our family of examples precludes the existence of such a method. T H E O R E M C. There can be no general construction in ~.~2 that begins with the corner angle a of a square and produces the radius r of a circle with matching area.
References [BI [C] [G] [Ka]
[KI] [Ma]
T H E O R E M B. There can be no general construction in H 2 that begins with the radius r of a circle and produces the corner angle a of the square with matching area.
[M-B1]
Remark. The article [Nell provides the example
[M-B2]
s i n h ( r / 2 ) = ~ 2V/~-v~. This means that 0 = arctan v / 2 - v~, so r and 0 are constructible. For the corresponding square, however, a = lrv~/4, which is not constructible. The conclusion reached from this example translates as: The class of circulable squares is wider than the class of quadrable circles. For the next example, not contemplated in [Nel], w e quote another t h e o r e m [Ni, p. 41]:
OLMSTED'S T H E O R E M . If T is a rational multiple of ~r, the only possible values of tan T that are rational are O, 1, and -1. Example C. Let q be some rational number, q > 0, q ~ 1. As q E E, w e can certainly construct the angle a = arctan q. Since a # 7r/4, Olmsted's t h e o r e m shows that cr/w is irrational. As cos ~r = 1/V/1 + q2 E E and sin cr = q/x/1 + q2 E E, then e ir c E(i) c A. Choose l o g ( - 1 ) = ~ri, so ( - 1 ) r = e x p ( ( a / ~ r ) l o g ( - 1 ) ) = exp(ai). This time, since ( - 1 ) r is algebraic, we use GS to conclude that a/~r is transcendental. Since ~; = 2~r - 4a, it follows 36
T H E M A T H E M A T I C A L INTELLIGENCER VOL. 17, NO. 2, 1995
[Nell
[Ne2]
[Ni] [R] [Sa] [Sm]
Roberto Bonola, Non-Euclidean Geometry,New York: Dover (1955). H. S. M. Coxeter, Non-Euclidean Geometry, 5th ed., Toronto: University of Toronto Press (1955). Marvin J. Greenberg, Euclidean and Non-Euclidean Geometries, 3rd ed., New York: W. H. Freeman (1993). V. E Kagan, Foundations of Geometry, Part L Moskva: Gosudarstvennoe Izdatel'stvo TekhnikoTeoreticheskoi Literatury (1949). [In Russian.] [Reviewed in MR 12, 731-732.] Israel Kleiner, Mathematical Intelligencer 15 (1993), no. 3, pp. 73-75. [Book review.] George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, New York: Springer-Verlag (1986). D. D. Mordukhai-Boltovskoi, On geometric constructions in Lobachevskii space, in In Memoriam Lobatschevskii, Glavnauka (1927), Vol. 2, pp. 67-82. [In Russian.] D. D. Mordukha~-Boltovskoi, On constructions using algebraic curves in Euclidean and non-Euclidean spaces, Zh. Matematichnogo Tsiklu 1(3) (1934), 15-30. [In Ukrainian.] N. M. Nestorovich, On the quadrature of the circle and circulature of a square in Lobachevskii space (Russian). Dokl. Akad. Nauk SSSR (N.S.) 63 (1948), 613-614. [In Russian.] [Reviewed in MR 10, 562.] N. M. Nestorovich, Geometric Constructions in the LobachevskffPlane, Moskva: Gosudarstvennoe Izdatel'stvo Tekhniko-Teoretichesko1 Literatury (1951). [In Russian.] [Reviewed in MR 13, 969-970.] Ivan Niven, Irrational Numbers, Washington, DC: Mathematical Association of America (1956). Paulo Ribenboim, The Book of Prime Number Records, New York: Springer-Verlag (1988). Marilyn vos Savant, "Ask Marilyn," Parade Magazine, 21 November 1993. A. S. Smogorzhevskii, Geometric Constructions in the Lobachevskff Plane, Moskva: Gosudarstvennoe Izdatel'stvo Tekhniko-Teoreticheskoi Literatury (1951). [In Russian.] [Reviewed in MR 14, 575.]
Department of Mathematics University of California Berkeley, CA 94720 USA
David Gale* For the general philosophy of this section see Vol. 13, no. 1 (1991). Contributors to this column who wish an acknowledgment of their contributions should enclose a self-addressed postcard.
Our lead item for this issue was contributed by Sherman K. Stein.
Packing Tripods Sherman
K. S t e i n
This column is devoted to an unsolved problem so simple to state that it can be told to the proverbial "person-inthe-street." Since it doesn't seem related to known theorems, everyone, specialist or amateur, has an equal crack at it. The puzzle enthusiast, computer programmer, or mathematician may find it a tempting challenge. Moreover, because it hasn't been worked on by many people there is a good chance for a fresh approach. The problem, which concerns placing integers in the cells of a square array, grew out of a geometric question. For a positive integer k consider the tripod formed by a unit cube (the corner) to which arms of length k are attached at three nonopposing faces. The k-tripod consists of 3k + 1 unit cubes. Figure 1 is a perspective view of a 4-tripod. The question is this: What fraction of the volume of space can be filled by nonoverlapping translates of a k-tripod when k is large? Introduce an (x, y, z) coordinate system whose positive parts correspond to the directions of the arms of the tripods. The question leads to this related one: How many nonoverlapping k-tripods can have their corner cubes in the cube of side k, 0 < x, y, z < k? Call this number f ( k ) . It is known that if f ( k ) / k 2 approaches 0 as k increases, then the fraction of space that can be packed by the k-tripods also approaches 0. So we now have the question: Does f ( k ) / k 2 approach 0 as k increases? It is not hard to show that we may assume that each corner cube coincides with one of the k3 unit cubes that make up the k-cube. Identify each of these unit cubes with its vertex that has the largest coordinates, hence with a triplet of integers (x, y, z), 1 < x, y~ z < k. Note
that because the tripods don't overlap, for a given pair (x, y), there is at most one number z such that the triplet (x, y, z) is present. Therefore we can record the presence of a tripod in a packing by entering the number z in the unit cell corresponding to the coordinates (x, p). Because the tripods don't overlap, the resulting entries satisfy the three conditions in the following definition of a "monotonic matrix": Let k be a positive integer. An array of order k consists of k2 empty cells arranged in a k by k square. Place in some of these cells any one of the numbers 1 , 2 , . . . , k, subject to three rules: 1. In each (vertical) column the entries strictly increase in size from bottom to top. 2. In each (horizontal) row the entries strictly increase from left to right. 3. The cells occupied by any specific integer rise as we move from left to right (the "positive slope" condition). Call such an array, with some cells filled in according to the three rules, a monotonic matrix of order k. From now on we deal with monotonic matrices instead of packings by tripods. Then f ( k ) is the maximum number of cells occupied in any monotonic matrix of order k. Clearly, f ( k ) < k 2. Moreover, since you could place the numbers 1 , 2 , . . . , k in order in a single row, f(k) > k. The question is: What happens to the quotient f ( k ) / k 2 as k gets arbitrarily large? Does it have a limit? Is this limit zero?
_/
I
/ /
* Column editor's address: Department of Mathematics,Universityof
California,Berkeley,CA 94720USA.
Figure 1. THE MATHEMATICALINTELLIGENCER VOL. 17, NO. 2 ~)1995 Springer-Verlag New York 3 7
To get a feel for f(k), consider a few small values of k. Figure 2 illustrates the cases I < k < 5. Clearly f(1) = 1 and f(2) = 2. It takes a little time to show that f(3) = 5 and that f(4) = 8. The case k = 5 was settled by an exhaustive computer search programmed by K. Joy, which showed that f(5) = 11. One of the many solutions he found is displayed in Figure 2. These are the only values of k for which f(k) is known. When k is a square, there is always a monotonic matrix of order k with k 3/2 occupied cells. Figure 3 and the fourth array in Figure 2 illustrate the construction. This suggests writing f(k) in the form f(k) -- k ~(k) and studying the behavior of e(k). At first it was conjectured that
2 [~
3 1
1 3
3
3 4 1 2
3
5 i5
1 1
f(2k + 1) _> 2/(k) + 3k. 4 2
3
3 4 1 2
e(k) approaches 3/2 as k increases. [This would imply that f(k)/k 2 approaches 0.] It is known that e(k) does approach some number. Clearly, this number is no larger than 2. Call it L. It turns out that L is the smallest number that is greater than or equal to all the e(k)'s. With this in mind, consider Figure 4. This monotonic matrix shows that f(7) > 19. Taking logarithms of both sides of the equation shows that e(7) _> (log 19)/log 7 1.513. Thus L is at least 1.513. If you examine Figure 4, you will note that it is based on the monotonic matrix of order 3 shown in Figure 2, with each of the nine blocked-out areas in Figure 4 corresponding to a cell in an array of order 3. The same technique provides the monotonic matrix of order 9 with 28 occupied cells (one more than in Fig. 3), shown in Figure 5. This implies that e(9) _> 1.516, hence that L _> 1.516. The idea behind the construction of Figures 4 and 5 shows that Let k and I be positive integers. By cutting an array of order kl into 12 blocks of size k by k, you can show that
2 4
f(kl) ~ f(k)f(t).
Figure 2.
Also it is clear that f(k + 1) _> f(k) + 1. It is these last two inequalities, together with the fact that f(k) <_ k 2, that imply by a known theorem that e(k) has a limit as k increases. There are several possible "next steps" in determining the behavior of f(k). One is to use a computer to determine some more values of f(k) or at least to find larger values of e(k). D. R. Hickerson has shown (without computer) that f(255) is at least 4638, which implies that L > 1.523. However, there may be a monotonic matrix of a small order that implies that L is even larger. Until someone discovers what happens to f(k)/k 2 as k increases, then, following an honored tradition, I will propose what may be a simpler "practice" problem, also unsolved.
inlll IIIHI IlUl lit'l/I r
12 I1
12] Figure 3.
5 5 8 9 6 7 1234
5 5
i
349 128 34 12 Figure 4.
38
4
8 9 6 7
5
7
1 '2' 3
7
1 6 3
Figure 5.
THE M A T H E M A T I C A L INTELLIGENCER VOL. 17, N O . 2, 1995
4 5~6 1 2 ~3
5
4
1
7 89
7
4
2 7 6
6
5
3 78
9 4 5 6 12 3 Figure 6.
78
9
45 12
6 3
Let J be a fixed positive integer. Let g(d, k) be the maximum number of cells in an array of order k that can be occupied by the numbers 1 , 2 , . . . , d, subject to the three rules given earlier. It is known that for each d, g(d, k)/k approaches a limit as k increases, which will be denoted
c(j).
First of all, g(1, k) = k. (Put l's on the upward sloping diagonal of an array of order k.) Thus c(1) = 1. For j = 2, insert l's and 2's as indicated by Figure 6. This construction, which provides four entries per three columns, shows that c(2) is at least 4/3. It is known that c(2) = 4/3. It is also known that c(3) = 5/3 and that c(4) = 2. Thus, for j = 1, 2, 3, and 4, c(j) = (k + 2)/3. This pattern suggests that c(5) should be 7/3, but all that is known is that it lies somewhere between 16/7 and 5/2. By the way, one could define an n-pod in n-dimensional space. It consists of a corner cube with n arms of length k glued at nonopposite faces. (When n is 2, it looks like the letter L.) In dimensions 1 and 2 it tiles the space. As D. R. Hickerson pointed out, if f ( k ) / k 2 approaches 0, then in all dimensions from 3 on, n-pods pack n-space with a density approaching 0 for large k. So 3 is the critical dimension. Of course, I don't know if f ( k ) / k 2 approaches 0 as k increases. In fact, I don't even know that it has a limit.
References 1. W. Hamaker and S. Stein, IEEE Trans. Inform. Theory 30 (1984), 364-368. 2. S. Stein and S. Szabo, Algebra and Tiling, Mathematical Association of America (1994), Washington, D.C. (Chapter 3) Department of Mathematics, U-C Davis
Some Late-Breaking News (Added in proof by the column editor)
be the origin of the example, a problem in Math. Gaz. 11 (1922) proposed by E. M. Langley. More interesting is the fact that there is actually a small literature stemming from "Langley's problem". In fact all of the results in Machado's article and in the Entertainments column and a great deal more turns out to be already known. I will not attempt to give the complete bibliography (the best easy reference seems to be Math. Gaz. 62 (1978) 174183), but the ultimate story is intriguing. What might be called the generalized Langley problem is that of finding and classifying all "rational quadrangles", that is, all complete quadrangles such that the angle between any two of the six sides is a rational multiple of ~r. This problem was completely solved in 1978 by Paul Monsky, who showed that aside from the obvious examples, (e.g., a rational-angled triangle together with its angle bisectors), the solutions consist of 120 oneparameter families and 1,830 isolated cases. Monsky's manuscript, which ran some 30-odd typed pages, was never published because it turned out that his results had been anticipated 40 years earlier (9 by the Dutch geometer Gerrit Bol in a paper (in Dutch) "Beantwoording van Prijsvraag no 17", Nieuw Arch. f. Wisk. (2) 18 14-68 (1936). By w a y of relating the Bol-Monsky results to some of those described in the Entertainments column, it is not hard to show that the quadrangle problem is equivalent to asking when three or more diagonals of a regular ngon are concurrent. This is illustrated for the original Langley problem in the figure below. It turns out that such concurrences cannot occur for n odd, and, except for obvious cases, can only occur for n divisible by 6. This confirms observations made by Dennis Johnson in his computer search, as reported in last issue's column. Among other interesting results, Bol finds values of n for which 4, 5, 6, and 7 diagonals are concurrent and shows that these are the only possibilities.
I am grateful to Don Chakerian for providing me with some highly relevant information in connection with two of the items treated in last issue's column. Regarding the section "The Dance of the Simson Lines" Chakerian writes, David Kay, College Geometry, Holt, 1969, mentions (p. 248) that the hypocycloid generated by those Sirnson lines was apparently first discovered by Jacob Steiner. Kay gives a development following E. H. Lockwood "Simson's Line and it's Envelope", Math. Gaz. 37 (1953), 124-125. The second communication concerned the section "Configurations with Rational Angles". This was also the subject of last issue's article, "Nineteen Problems on Elementary Geometry" by Armando Machado. I concluded the section by asking for information on the origin of the first problem in Machado's article, which he says "... must be rather well known as it appears repeatedly in mathematical circles". Thanks to Chakerian and also Stan Wagon, I was able to track down what appears to
Correction: In last issue's column I misstated the general result of Jean Brette on paradoxical dissections of triangles. It is not the case that Brette's method allows paradoxical dissections of any right triangle with integral sides. His method does, however, permit one to choose one of the three subtriangles arbitrarily, so that in some cases the "cheating" is not perceptible to the naked eye. THE MATHEMATICAL INTELLIGENCER VOL, 17, NO. 2,1995
39
The Walrus and the M a n d e l b r o t Peter R. Cromwell "The time has come," the Walrus said, "to talk of many things.
"Irrational dimension is a thing that puzzles me.
And yet, it's still self-similar-you've seen it all before.
"We first remove the middle third and then repeat to get four segments (each is one-ninth long), but we're not finished yet. Continue this, and in the end you'll make the Cantor set. "Next, take this line and make a hump, three parts replaced by four, Do this again on each new line, repeat for evermore. Then three of these together m a k e . . . a snowflake that won't thaw! "This snowflake shape will fit inside this circle over here, and yet its edge has boundless length. Have I made myself clear?" "This fractal stuff," the Walrus said, "is just a trifle queer. 40
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 (~ 1995 Springer-Verlag New York
"A fractal shape contains itself although reduced in size. Enlarge, zoom in, it reappears in slightly different guise. These copies can be found in inexhaustible supplies. "But now," the Mandelbrot sat down, "I think we should adjorn. Although I've shown you much today there's still a lot to learn of strange attractors, gasket sets, and how to make a fern." The pictures they had drawn lay on the table in a heap. The Walrus ventured timidly "Are these for me to keep?" The Mandelbrot did not reply for he had gone to sleep.
Tilings of Space by Knotted Tiles Colin C. Adams*
Introduction
The history of tilings of the plane is rich and extensive, dating back to the decoration of walls and floors in early civilization. Mathematicians have been interested in tilings of the plane as well as the generalizations to tilings of 3-space and tilings of higher-dimensional space. Because the tiles utilized in a tiling of the plane by congruent tiles must always be topologically disks, most of the work on filings of higher-dimensional space has focused on tiles that are topologically n-dimensional bails. There has been some interest in tiling 3-space by congruent tiles that are topologically equivalent to unknotted solid tori or unknotted genus-n handlebodies. (See Fig. 1.) In this article, I will examine tilings of Euclidean 3space by congruent tiles, where the tiles will be "knotted" in some sense. In particular, several methods will be described for tiling Euclidean 3-space by congruent tiles, all of which are knotted solid tori. I will then consider exactly what topological shapes can be used to tile 3-space, when all of the tiles are congruent. In the last section, generalizations to higher dimensions will be discussed. All of the filings considered in this paper are monohedral, meaning that every tile is congruent to a single tile, called the prototile. The prototile will always be a compact orientable 3-manifold embedded in 3-space, with one boundary component. It will always be polyhedral, in the sense that its boundary will consist of finitely many faces, each of which is contained in a Euclidean plane. This will avoid consideration of tiles with wildly embedded boundaries. By a tiling, we mean that all of Euclidean
Figure 1. Unknotted solid tori and unknotted handlebodies.
* Supported in part by NSF Grant DMS-9302843 and NSF funds supporting the Regional Geometry Institute at Smith College, July, 1993. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 (~) 1995 Springer-Verlag New York 4 1
Figure 2. (a) A handlebody; (b) a cube-with-holes.
3-space is filled with the union of the tiles, such that pairs of tiles can intersect in common faces, edges, and vertices on their boundaries, but they cannot overlap on their interiors. When I draw a picture with a smooth boundary, it is intended that the reader picture many small flat faces that make up the boundary, and that from a distance, look smooth. We will be looking at several interesting topological objects. A handlebody is a compact orientable 3-manifold obtained by gluing handles onto a ball. The number of handles is called the genus of the handlebody. We say that a handlebody embedded in 3-space is standardly embedded if none of the handles is knotted or tangled with any other. A cube-with-holes is a compact orientable 3manifold of one boundary component that is topologically equivalent to a solid cube with some number of potentially knotted and tangled tunnels drilled out of it. (See Fig. 2.) Note that a standardly embedded handle-. body is a cube-with-holes, it just happens that the tunnels are neither knotted nor tangled. In addition to handlebodies and cubes-with-holes, we will see that there are other compact submanifolds of R 3 with one boundary component. For more background on tilings, particularly tilings of the plane, see Ref. 6. For an elementary introduction to the mathematical theory of knots, see Ref. 1. More advanced material on knots appears in Ref. 10. Two preprints have appeared that were completed independent of this article but over the same time period. In Ref. 11, the author constructs a tiling of space by a tile in the form of a trefoil knot. In Ref. 7, the author demonstrates that any knotted solid torus can be used as a prototile for a monohedral tiling, as I will do here. He goes on to show that a handlebody with handles that are knotted and linked and then with tunnels drilled out, where those tunnels are knotted and linked, can form a prototile for a monohedral tiling. This is slightly weaker than what I prove below; however, it appears that the techniques used in Ref. 7 may be extendable to the more general result.
T i l i n g s b y S o l i d Tori, H a n d l e b o d i e s , and Cubes-with-Holes
Figure 3. A prototile for a tiling by solid tori. 42 THEMATHEMATICAL INTELLIGENCERVOL.17,NO.2,1995
I begin by looking at several methods for tiling space by solid tori, handlebodies, and cubes-with-holes. To construct the first of these, begin with the standard tiling of space by unit cubes. In each cube, drill a square hole from one face through to the opposite face, the hole having width 1/2. Next attach two square "fingers" to two of the remaining opposite faces, each of length, width, and height equal to 1/2. (See Fig. 3.) These tiles fit together to form a wall of one cube's thickness, by having the holes alternate between horizontal and vertical in adjacent cubes. A sequence of such walls can be placed together to fill up all of 3-space.
Figure 4. (a) A handlebody prototile; (b) a cube-with-holes prototile. Sometimes, tilers prefer tilings that have no continuous degrees of freedom, such as occur here when one of the wails can slide along another. In that case, we can just add an indentation to our prototile on one of the two remaining unaffected cube faces, and a bump on the opposite cube face. Note also that we could utilize the same drilling/finger construction in order to tile space with either handlebodies of any genus or cubes-with-holes of any genus and any amount of knotting of the holes, as in Figure 4. Note that although it isn't drawn that way, the second example could easily be made polyhedral. Note also that the tiling by handlebodies could actually be constructed one tile at a time, stacking them up, whereas the tiling by cubes-with-holes could not actually be built by fitting together rigid tiles. A second method for tiling space by solid tori is to take for a tile, a cube with a hollowed-out tunnel as in Figure 5. If we then take a copy of this tile, flip it around and twist it 180 deg, we can fit it to the other tile so that the union of the two tiles exactly fills up the space necessary for two cubes. We can then tile all of three-space with bricks made in this manner. Note that in this tiling, pairs of tiles are nontrivially linked. Note also that by iterating the procedure of hollowing out a tunnel, we can obtain a handlebody of any genus as prototile, such that a pair of tiles again form the equivalent of two cubes. (See Fig. 6.) Interest in tilings by solid tori occurred as early as 1928 [9]. More recently, tilings of 3-space by congruent solid tori and handlebodies, coming out of the theory of regular polytopes, appear in Refs. 3, 5, and 12. See also Ref. 4 for handlebody tilings of 3-space and Ref. 2 for tilings of 4-space by solid tori.
Figure 5. Cube with tunnel.
Casting and Tiling N o w let me describe another approach to tiling with solid tori which will generalize to other more complicated topological objects. If we wish to fill space with unknotted solid tori, then we will, at the very least, need to fill the region surrounding a given solid toms. We can
Figure 6. A prototile for a genus-two handlebody tiling. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
43
utilize the fact that a solid torus can be encased in the interior of a cube, such that the cube decomposes into the solid torus and two balls, as in Figure 7. This is the w a y that one would traditionally cast a bronze solid toms. One w o u l d make a mold from two halves, each of which is topologically a ball, and then pour the molten metal in through a hole to form a solid torus. We can turn this idea into a tiling of 3-space by solid torus tiles. We will take the standard tiling of 3-space by cubes and picture each cube as if it had been decomposed into a solid torus and two balls. Now, we form a prototile by taking one of the solid tori and extending a finger from it out between the juncture of the two balls that surround it to the b o u n d a r y of the cube. We continue to extend the finger up the side of the cube and then attach it to the lower ball in the cube above. We also extend a second finger out between the two bails surrounding our. solid torus and then d o w n to the upper ball of the cube below. Since the fingers and balls are merely unsightly protrusions on the solid torus, and have no topological impact on the prototile, the prototile is still a solid torus. This prototile, which is depicted in Figure 8, can then be used to tile all of space. (Although the prototile is not polyhedral in the figure, it can easily be m a d e polyhedral, approximating it by an object with a finite n u m b e r of planar faces and the same topology.) Figure 7. Casting a solid torus.
Figure 8. Prototile for a solid torus tiling.
44 THEMATHEMATICAL INTELLIGENCER VOL.17,NO.2,1995
Well, why not use the same idea to tile all of space with ponent. No shaded subsurface of a given boundary comtiles that are knotted? We could just encase a knotted ponent of this submanifold can be identified to a shaded solid torus inside a cube such that the remainder of the subsurface of any other boundary component, since we cube decomposes into two balls, and attach the knotted would have to pass through the balls themselves to do solid torus to two balls in the adjacent cubes via fingers. the identification. Therefore, after the rest of the idenOf course, this brings up the question, probably de- tifications are completed, there remains more than one bated by metalworkers over beers, "Can you cast a boundary component, a contradiction. Thus, all of the shaded regions are disks. Gluing toknot?" That is to say, can you decompose a cube into a knotted solid torus and two balls, such that any two of gether a pair of them, one from the surface of each ball, we obtain a ball B in S 3. Hence, the closure of the comthe three of them only intersect in their boundaries? In fact, we can ask the more general question: Exactly plementary region is also a ball. Disks on the boundwhat topological objects can we cast using a mold with ary of B are identified to one another via isotopy of the two halves that are topologically balls? boundary of B through S 3 - int(B). This is the same as In addition to an unknotted solid toms, we can cer- gluing handles on B or, equivalently, drilling tunnels in tainly cast a ball. Using the same method we used for S 3 - int(B). Hence W, which is the closure of the comcasting an unknotted solid toms, we can also cast a stan- plement of B1 U B2, is obtained by drilling tunnels in a dardly embedded handlebody of any genus. In fact, we ball and is therefore a cube-with-holes, or in the case that can also cast any cube-with-holes, the top ball filling the no tunnels are drilled, a ball. 9 So, in particular, because a knotted solid torus is not top half of the cube and the top half of each of the holes, while the bottom ball fills the bottom half of the cube topologically equivalent to a cube-with-holes (proof of and the bottom half of the holes. It turns out that these this utilizes the fundamental group), we cannot cast a examples include all of the possible objects that can be knotted solid torus with a mold made from two balls. As cast. interesting as that is for the metalworkers, it will have THEOREM 1. If a cube is decomposed into three polyhedral no bearing on whether or not we can tile with knotted pieces, two of which are balls, and the third of which is con- solid tori, as we can just use three balls, rather than two, tained in the interior of the cube, then the third must be a ball or to fill up the space in the cube surrounding the knotted a cube-with-holes (where a standardly embedded handlebodt/ solid torus. For instance, in Figure 9 we show a decomcounts as a cube-with-holes). position of the cube into four pieces, one of which is a Proof. Say we want to cast W. So we want to decompose trefoil knotted solid toms and the other three of which a cube into three pieces, W, B1, and B2, where B~ and are topologically balls. B2 are topologically balls and W is contained entirely in the interior of the cube. The fact that M is a manifold with polyhedral boundary implies that B1 and B2 intersect one another in subsurfaces on their boundaries. In particular, a component of the intersection cannot be a single vertex or single edge, for instance. Note that each of B1 and B2 must intersect the boundary of the cube in a single disk if they are both to be balls. For convenience, we will glue a solid ball to the boundary of the cube, adding the ball to B~, so that now B1, //2, and W fill all of the 3-sphere, and B 1 U B2 has only a single boundary component, namely, cgW. Also for convenience, we think of S 3 a s the one-point compactification of R 3 and we let the point {cx~} be a point within W. Then B1 U B2 is a submanifold of//3. If we cut open the identified subsurfaces of the two balls, one pair at a time, until we are left with two balls, at each stage the resulting manifold remains in//3. Hence, we can picture the entire process of constructing the manifold B1 U/~2 a s occurring in R 3, by simply identifying subsurfaces of the boundaries of two bails i n / / 3 Let's shade those regions on the surfaces of the two balls that are to be identified. Suppose that there is a shaded region on one of the ball boundaries that is not a disk. Identifying this region to the corresponding region on the boundary of the other ball results in a connected Figure 9. D e c o m p o s i n g a cube into a knotted solid torus and submanifold of//3 with more than one boundary corn- three balls. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
45
Figure 10. A prototile for a tiling by trefoil knotted solid tori.
Figure 11. Can this be the topological shape of a prototile? 46
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
The one funny-looking ball, which resembles a divining stick, is simply filling the vertical space between the crossings. The "arms" of that ball run along the knot from crossing to crossing, so that the ball will be connected. Once this ball is added to the knotted solid torus, a standardly embedded handlebody results, and we can fill the rest of the cube with two balls, as we have done before. We can then form a prototile as in Figure 10. This prototile can easily be made polyhedral. This procedure will work for any knot whatsoever. We use a ball shaped like a "tree" that reaches in and fills the space between the strands of the knot at each crossing. Hence, we can tile space with congruent tiles, a prototile for which is topologically a solid torus, knotted into the shape of any knot whatsoever. What about more general topological objects? Can we tile with them? So far, we have seen tilings by solid tori, knotted in any way we want. We have seen tilings by "unknotted" handlebodies. We have seen tilings by cubeswith-holes. What else could there be? Given any polyhedral embedding of a surface in 3space, it bounds a compact 3-manifold that is a potential topological shape for a tiling. So the question is, can we tile with prototiles that are topologically equivalent to such a shape? Figure 11 shows a nasty-looking example. (This too could easily be made polyhedral.)
Figure 12. Triangulate the 3-manifold Q. Surprisingly enough, the answer is always yes, we can tile with tiles that are topologically of any polyhedral shape possible within 3-space. We will utilize the casting idea one more time. In particular, we will see that any compact 3-dimensional submanifold M of R 3 with one boundary component can be encased in a cube, where the remainder of the cube is made up of four balls. How? Any compact submanifold M of R 3 is bounded, so just take a cube C that is big enough to contain M Figure 13. A handlebody decomposes into two balls. entirely in its interior. Now we will decompose Q = But we are interested in decomposing the manifold Q C - int(M) into four balls. into four balls. Each of the two handlebodies can easily First, use the fact that we can triangulate the 3be split into two balls as in Figure 13. manifold Q. That is to say, we can cut Q up into finitely Thus, we have decomposed the cube C into five pieces, many tetrahedra, so that any two tetrahedra either do one of which is M and four of which are balls, denoted not intersect, or they intersect in a vertex, an edge, or a face. The boundaries of Q are decomposed into faces of B1, B2, B3, and B4. Since each of/-/1 and/-/2 touches both boundary comtetrahedra. (See Fig. 12.) Although any 3-manifold can be ponents of Q, we can make sure that each of our four bails triangulated, as proved by Edwin Moise in 1952 [8], we touches the boundary of the cube, by snaking fingers out, do not need the power of that theorem in our situation. if necessary. Since they are balls, each must touch the We just want to triangulate a compact submanifold of R 3 boundary of the cube in a set of disks. with a polyhedral boundary consisting of finitely many We will deform the balls so that the top face of the cube faces. To obtain our triangulation, we simply cut Q up is entirely contained in a disk within the boundary of/~1 into pieces using the set of all planes that contain three or and the bottom face of the cube is entirely contained in a more of the vertices on the boundary of Q. Each of these disk within the boundary of B2. We can do this because, pieces will be a convex polyhedral ball. For each such after all, this is topology, and everything is made of amazpiece, we can add edges to the boundary to triangulate ingly deformable rubber. We Will make sure that at least it, so that pieces which abut one another in Q have the one point on the boundary of B4 is contained in one of same edges added. We can then pick a point in the intethe remaining faces that is opposite a face containing a rior of each piece and run straight edges from it to the boundary vertices. This yields a decomposition of each point from the boundary of/~3Now, we are ready to form our prototile. Via integer piece into tetrahedra and a decomposition of Q into a translation in the x, y, and z directions, we first tile all of finite set of tetrahedra. space with copies of our cube. In our initial cube C, we The 1-skeleton of the triangulation is the union of all of will run a finger from M to the boundary of the cube. the vertices and edges. Let H1 be a neighborhood of the 1We do this by starting at a point of 0M where it touches skeleton in Q. Note that H1 is simply a handlebody. Note one of the balls B1. Then we run a small finger along the also that Q - int(H1) is also a handlebody. This is because boundary of B1, absorbing a small amount of material it is just a regular neighborhood of a graph that is dual from B1, until we reach the surface of the cube. to the triangulation, with vertices at the center of each We repeat the procedure, forming a second finger that tetrahedron and edges passing through faces shared by runs from 0M to the boundary of the cube, where this two tetrahedra. Let/-/2 be this second handlebody. This technique can be used to decompose any compact new finger runs along the boundary of B2 and never 3-manifold into two handlebodies. It is called a Heegaard touches the first finger. We repeat this procedure twice more until we have four fingers attached to M that run splitting of the manifold. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
47
Figure 14. One of four trefoil knots making up a cube.
from M out to the boundary of the cube and that do not touch one another. We run them around the boundary of the cube, if necessary, so that one finger ends on the top face, one finger ends on the bottom face, and two fingers end on the two opposite faces containing points of B3 and/?4, respectively, again making sure in the process that none of the fingers touches any other. We now attach the finger ending on the top face to the copy of/?2 in the cube directly above. We attach the finger ending on the bottom face to the copy of B1 in the cube directly below. The remaining two fingers are attached to the copies of B4 and B3 in the adjacent cubes. Our precaution that the top and bottom faces are entirely contained in disks contained in B1 and B2 ensures that none of the four balls that our fingers are attached to will touch any other and des'troy our topology. Hence, this prototile is topologically equivalent to M. By design, this prototile does not intersect any of its translates in their interiors. Moreover, for any cube in the cubical tiling, each of the four balls and the copy of M that together make it up is itself a part of an integer translate of this tile. Hence, all of 3-space is filled with these tiles. Thus, we see that we can tile Euclidean 3-space with any topological shape of one boundary component in 3space that is polyhedral. Knots in Cubes
The tilings that we have looked at so far have succeeded resoundingly in answering the question whether or not space can be tiled with tiles of any potential topological 48
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
type. However, they lack a certain aesthetic appeal. Here we are using contorted objects to tile space, and yet the tiles are not in any way knotted with one another. In this section, we will look at an alternative method for tiling with knotted solid tori. Here, we will see that a solid cube can be decomposed into a set of congruent knotted tiles and, therefore, when 3-space is tiled with copies of the cube, we obtain a tiling of 3-space by knotted solid tori. We begin with a decomposition of a single cube into four congruent solid tori tiles, each knotted as a trefoil knot. The four knotted solid tori will be permuted cyclically when we rotate the cube a quarter-turn about an edge that runs through the centers of two opposite faces. In Figure 14, we see the basic shape for a single one of the trefoil knots within the cube. The actual polyhedral prototile has various thickenings and protrusions not depicted in this figure. The easiest way to view the decomposition of the cube into four tiles is to look at horizontal slices of the cube. In this example, all of the faces of the tiles have been chosen to be parallel to the three coordinate planes. In Figure 15, we see various horizontal slices of the cube, starting at the top and then viewing subsequent slices at the levels where a change occurs in the pattern of the slice. Note that the four knots in the cube are linked with one another. Two adjacent knots are linked as in Figure 16a, and two opposite knots are linked as in Figure 16b. Translating such a decomposed cube around space by integer translations in the x, y, and z directions yields a tiling of space by trefoil knotted solid tori. Although groups of four trefoils are linked, the trefoils in a given cube are not linked with the trefoils in neighboring cubes. If we desire a tiling where the tiles from adjacent cubes are tangled with one another, so that the tiling cannot be "pulled apart," we can modify the given tiling. The four knots in a given cube can be linked with the four knots in the cube above, by taking as our prototile the handlebody depicted in Figure 17a. Each of the handles consists only of material from a single one of the knots. This allows us to form vertical chains of cubes. Now, on the four vertical sides of the cube, we put in two handles as in Figure 17b. Then, if we flip over every other vertical chain of cubes, we can fit them all together to fill space so that every knot is linked with four knots from three other cubes, in addition to being linked with the three knots in its own cube. The decomposition of the cube into four trefoil-knotted solid tori brings up the question whether one can decompose the cube into three trefoil-knotted solid tori. In fact, one can. The cube has a threefold symmetry about a diagonal. One can draw a sequence of slices of the cube perpendicular to this axis, colored with three different colors such that each slice has a threefold symmetry with respect to the perpendicular axis, as we did in Figure 15 for the fourfold symmetry. Each color corresponds to a trefoil-knotted solid torus, and the three trefoil knots that
Figure 15. Decomposition of a cube into four trefoil knots.
m a k e u p the cube are p e r m u t e d b y the threefold s y m m e try. This figure is not included. Can w e do better again? Can the cube be d e c o m p o s e d into t w o trefoil-knotted solid tori? Here is w h e r e o u r luck r u n s out. In fact, w e have. L E M M A 1. A topological ball cannot be decomposed into two solid tori where one or both are nontrivially knotted.
Proof. Let B be the topological ball, a n d let V be a nontrivially knotted solid t o m s in a d e c o m p o s i t i o n of B into two solid tori. Then U = cl (B - V) is the other solid torus. N o t e that 0 B N 0V is n o n e m p t ~ in order that a U be connected. S u p p o s e first that there exists a nontrivial curve 7 on cgV that is also in 0B. If 7 is a longitude of V, then w e can p u s h the core curve of the knotted solid torus onto OB. H o w e v e r , e v e r y knot on a sphere is trivial, conTHE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
49
is a cable knot on the core curve of the solid torus V. However, a cable knot on a nontrivial knot is always itself a nontrivial knot (this is proved using the Seifert-Van Kampen theorem applied to the fundamental groups [see Ref. 10 (p. 113)], contradicting the fact that "~ lies on the spherical b o u n d a r y of B. Thus, there are no nontrivial curves on 0V that are in COB,and therefore OV n cOB consists only of planar surfaces. Since cOU is connected, cOV n cOB consists only of disks. If there is more than one such disk, cOU will have genus greater than 1. Hence, cOV intersects cOB in a single disk. Cutting V away from cOB along this disk would result in a knotted solid torus floating freely in the ball. In particular, this means that U is the complement of a knotted solid torus in a ball with a tunnel drilled from the missing knotted solid torus to the spherical boundary. Such a manifold is a cube-with-knotted-hole rather than a solid torus, as can be seen from the f u n d a m e n t a l group. 9
Generalizations and Questions Figure 16. (a) Linking of adjacent trefoils; (b) linking of opposite trefoils. tradicting the nontrivial knotting of V. If 3' is a meridian of V, then there is a meridianal disk of V with b o u n d a r y on OB. This meridianal disk separates the ball into two components. However, a meridianal disk is punctured once by the core curve of V. Once the core curve leaves one component and enters the other, it has no w a y to return, contradicting the fact that the core curve is a closed curve.
Hence, "y must be a so-called (p, q) curve on the solid torus V, wrapping 19times meridianally and q times longitudinally, with both/9 and q nonzero. This means 7
First, note that the constructions of knotted tilings above are not restricted to tilings of Euclidean 3-space. They would work just as well in spherical 3-space or hyperbolic 3-space, for instance. However, instead of utilizing the standard cubical tiling of Euclidean 3-space, we can use the dodecahedral tiling of spherical space or the dodecahedral tiling of hyperbolic space. For instance, rotating around a vertex of the dodecahedron gives a threefold s y m m e t r y of the dodecahedron. Just as the cube could be decomposed into three trefoil knots, so the dodecahedron can be decomposed into three trefoil knots, such that they are permuted by the symmetry. The dodecahedral tilings of either spher-
Figure 17. (a) Adding handles to the top and bottom of the cube; (b) adding handles to the sides of the cube. 50
THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 2,1995
ical 3-space or hyperbolic 3-space can then be decom- 2. Given any particular knot K, can a cube always be decomposed into four congruent knotted solid tori, posed into trefoil-knotted solid tori tilings of those two each knotted as K? How about three? spaces9 The previous construction for tilings of Euclidean 3- 3. Are there more "natural" decompositions of the cube into a set of congruent knotted solid tori? Are there space by arbitrary polyhedral submanifolds of R 3 with "natural" decompositions of the other regular solids one boundary component extends to similar tilings of into congruent knotted solid tori? Euclidean n-space. Given any polyhedral submanifold M with one boundary component in R '~, we can em- 4. Can one decompose the cube into a set of possibly incongruent knotted solid tori, but where each solid bed it in the interior of an n-dimensional solid cube C. torus has a constant "thickness"? Just as in the 3-dimensional case, the compact polyhedral n-manifold Q = C - int(M) can be triangulated into 5. Are there any molecular structures that exhibit "spacefilling knottedness"? Could such molecules be synthefinitely many n-simplices. sized? We can see that the n-manifold Q can be decomposed into n + 1 n-dimensional bails. A neighborhood of the 6. What about higher-dimensional analogues of the decompositions of cubes into knotted solid tori? Can union of the 0- and 1-simplices is an n-dimensional hanthe four-dimensional hypercube be decomposed into dlebody that can be decomposed into two n-balls, just as thickened knotted 2-spheres? for three-dimensional handlebodies. Remove the interior of this first handlebody from Q. A neighborhood of the A c k n o w l e d g m e n t s union of what remains of the 2-simplices and 3-simplices is again a handlebody that can be decomposed into two Thanks to the Regional Geometry Institute at Smith Coln-balls. Remove the interior of this handlebody from Q. lege in the summer of 1993, where these questions came For each subsequent even i, with 0 < i < n - 1, the union up, and thanks to John Conway, Branko Griinbaum, of what remains of the i-simplices and i + 1-simplices is a Frank Morgan, Egon Schulte, and Jeff Weeks for stimuhandlebody of dimension n that can be split into two n- lating and helpful conversations and comments. Thanks balls. Remove the interior of that handlebody and move also to Krystal Williams and Pier Gustafson for help with on to the next larger even i. If n is odd, we are done, hav- the illustrations. ing decomposed Q into n + 1 n-balls. If Q is even, we still A d d e d in Proof. A recent preprint of Seungsang Oh have the remains of the n-simplices with which to deal. answers affirmatively both questions posed in No. 2 However, we can steal a little material from the two balls above. that make up the previous handlebody in order to con- R e f e r e n c e s nect each of these n-simplices into a single ball. Hence, we have decomposed Q into n + 1 n-balls. 1. C. Adams, The Knot Book, New York: W9 H. Freeman and We can tile n-dimensional space with n-dimensional Co., 1994. 2. T. Banchoff,Torus decompositions of regular polytopes in unit hypercubes, such that each hypercube decomposes 4-space, in Shaping Space--A PolyhedralApproach, edited by into a copy of M and n + 1 n-balls. Starting with a parM. Senechal and G. Fleck, Boston: Birkh/iuser, 1988. ticular copy of M, we extend n + 1 fingers out through 3. H.S.M. Coxeter and G. C. Shephard, Regular 3-complexes its cube to the boundary of the cube, having them end with toroidal cells, J. Comb. Theory B 22 (1977), 131-138. on n + 1 different faces. Note that a hypercube has 2n 4. H. E. Debrunner, Tiling three-space with handlebodies, Studia Scientiarum Mathematicarum Hungarica 21 (1986), faces. We then attach the fingers to balls in the adjacent 201-202. hypercubes, such that no two of the different balls that 5. V. Frei,O osmnactem Hilbertove problemy, PokrokyMatemwe attach the fingers to are identified by integer translaatiky, Fysiky Astronomie 20 (1975), 260-268 (in Czech). tions in the various coordinate directions. We can again 6. B. Grunbaum and G. C. Shephard, Tilings and Patterns,New arrange the intersections of the n + 1 bails with the faces York, W. H. Freeman and Co., 1987. 7. W. Kuperberg, Knotted lattice-like space fillers, Discrete of the hypercubes to ensure that the n + 1 balls that we Comput. Geom. (in press). are attaching the fingers to do not intersect one another 8. E. E. Moise, Affine structures in 3-manifolds V. The Triand destroy the topology. The result is a prototile for a angulation theorem and Hauptvermutung, Ann. Math. 56 tiling of Euclidean n-space by tiles that are each isotopic (1952), 96-114. to the polyhedral submanifold M. 9. K. Reinhardt, Zur Zerlegung der euklidischen Raume in kongruente Polytope, S.-Ber. Preuss. Akad. Wiss. Berlin What questions remain? Here are a few. (1928), 150-155. 1. We might try to subdivide the cube into n 3 smaller 9 10. D. Rolfsen, Knots and Links, Publish or Perish Press, Berkeley 1976. cubes, n along each edge of the cube, so that we can 11. R Schmitt, A spacefiUingcloverleaf knot, preprint. construct our example of the decomposition of the 12. E. Schulte, Space fillers of higher genus, J. Comb. Theory A cube into four symmetrically placed trefoil knots by 68, no. 2 (1994), 438-453. simply coloring each of the small cubes one of four Department of Mathematics colors, one color for each knot. What is the least n for Williams College Williamstown, MA 01267 USA which this can be done? THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995 5 1
Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions--not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous ini-
tials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? 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 the Mathematical Tourist Editor, Ian Stewart.
Octagons Abound Istv in Hargittai A recent article [The Mathematical Intelligencer, vol. 16 (1994), no. 2, 18-24] on octagons in Renaissance architecture prompted me to communicate a few examples of octagons in design and decoration. A beautiful three-dimensional example of octagonal architecture is Castel del Monte in Apulia, southern Italy (Fig. 1), built in the 13th century for nonmilitary purposes on the top of a hill. The outer shape is an octagon, as is the inner courtyard. Even the eight small towers have octagonal symmetry. Tilings provide characteristic two-dimensional examples. As is known, the regular octagon cannot tile the
surface without gaps or overlaps, and, of the regular polygons, only the equilateral triangle, the square, and the regular hexagon can, as is illustrated in Figure 2. Nonetheless, regular as well as nonregular octagons are often used for tiling together with squares. Examples are shown in Figures 3-6.
Budapest Technical University Szt. Gelldrt 4 H-1521 Budapest, Hungary e-maih
[email protected]
Figure 1. Castel del Monte on Italian stamp. * Column Editor's address: Mathematics Institute, University of Warwick,Coventry,CV4 7ALEngland. 52
Figure 2. Planar networks of regular polygons with up to eightfold symmetry.
THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 2 (~)1995Springer-VerlagNew York
Figure 3. Octagons and squares after Victor Vasarely.
Figure 4. Hungarian embroidery.
Figure 5. Pavement on the campus of Northwestern Univex'sity, Evanston, Illinois.
Figure 6. Detail of Tintoletto's El lavatorie (@ Prado Museum, Madrid. Rights reserved.). THs M~TFIEMATIC&L'INT~LLIGs ICE'~,V'OL.17~~O. 2, [s
53
Penrose Tiling in Northfield, Minnesota Brian J. Loe Northfield, Minnesota proclaims itself to be a town of "Cows, Colleges, and Contentment." This attitude of rural contentment leads many to draw comparisons of Northfield to Lake Wobegon, the fictional Minnesota town of Garrison Keillor's A Prairie Home Companion, which is described as "the little town that time forgot, that the decades could not improve." However, at Carleton College, the Department of Mathematics and Computer Science, having been housed in the Goodsell Observatory since it was erected in 1887, felt that after ten decades there was room to improve. In September of 1993, Carleton College opened the doors of its Center for Mathematics and Computing.
While the new building provides contemporary computing and teaching facilities which are common to many college and university campuses, it also has one unique feature: the floor of the atrium of the Center is laid with a Penrose tiling. Roger Penrose's kites and darts were popularized in the January, 1977 issue of Scientific American by Martin Gardner [11. The Carleton tiling is a central portion of the infinite cartwheel, like the pattern which appeared on the cover of Scientific American. The aptness of the cartwheel design is derived from the special place that cartwheels hold in proving fundamental results about tilings by Penrose kites and darts, e.g., local isomorphism (consult [2] for details). The tiles were cut from standard 8-inch square ceramic tiles of four different colors. The tiling contains 650 tiles and is approximately fifteen feet in diameter. The tiling was installed with the consent of Roger Penrose who was a guest of the college, and featured speaker, for the formal dedication of the building April 11, 1994.
References 1. Martin Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Scientific American, January 1977, pp. 110-121. 2. B. Griinbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987, pp. 520-582. Brian J. Loe 7102 135th Street Apple Valley, MN 55124, USA
54
THE MATHEMATICALINTELLIGENCERVOL. 17, NO. 2 (~ 1995 Springer-VerlagNew York
Infinitesimals and the Continuum John L. Bell
The belief that a continuum can be "composed of" or "synthesized from" points has been frequently challenged, as witness the following quotations: Aristotle: "... no continuum can be made up out of indivisibles, as for instance a line out of points, granting that the line is continuous and the point indivisible." ([1], Book 6, Chap. 1) Leibniz: "A point may not be a constituent part of a line." ([11], p. 109) Kant: "Space and time are quanta continua.., points and instants mere positions.., and out of mere positions viewed as constituents capable of being given prior to space and time neither space nor time can be constructed." ([6], p. 204) Weyl: "Exact time- or space-points are not the ultimate, underlying, atomic elements of the duration or extension given to us in experience." ([12], p. 94) Brouwer. "The linear continuum is not exhaustible by the interposition of new units and can therefore never be thought of as a mere collection of units." ([4], p. 80) Ren6 Thorn: "... a true continuum has no points." ([5], p. 102) These views are in striking contrast with the generally accepted set-theoretical account of mathematics according to which all mathematical entities are discrete: on this account there is, in particular, no "true" continuum. Closely associated with the concept of continuum is that of infinitesimal, which is, roughly speaking, what remains after a continuum has been subjected to a mathematically or metaphysically exhaustive analysis. An infinitesimal may be regarded as a continuum "viewed in the small." On the set-theoretical or discrete account, infinitesimals are just points (or singletons); however, if continua are truly continuous and do not have points as parts, then an infinitesimal, as a part of the continuum from which it is extracted, cannot be a point. Let us call such infinitesimals nonpunctual or continuous. (Nonpunctual) infinitesimals have a long and fascinating history. They first show up in the mathematics of the Greek mathematician-philosopher Democritus (himself an atomist!), only to be banished by Eudoxus (c. 350 B.C.)
from what was to become official "Euclidean" mathematics. Taking the somewhat obscure form of "indivisibles," they reappear in the mathematics of the late middle ages and were systematically exploited during the 16th and 17th centuries by Kepler, Cavalieri, and others in determining areas and volumes of curvilinear figures. As "linelets" and "timelets" they play an essential role in Barrow's "method for finding tangents by calculation," which appears in his Lectiones Geometricae of 1670. As "evanescent quantities" they were instrumental in Newton's development of the calculus, and, as "inassignable quantities," in Leibniz's. De l'Hospital, the author of one of the first textbooks on the (infinitesimal!) calculus in 1696, invoked the concept in laying down the principle that "a curved line may be regarded as an infinite assemblage of infinitesimally small straight lines." Memorably derided by Berkeley as "ghosts of departed quantities" and condemned by Bertrand Russell as "unnecessary,
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 (~) 1995 Springer-Verlag New York
55
erroneous, and self-contradictory," they were believed to be finally suppressed through the set-theorization of mathematics achieved in this century. In fact, the suppression of infinitesimals within "respectable" mathematics did not eliminate them altogether but, instead, drove them u n d e r g r o u n d . Physicists and engineers, for example, never a b a n d o n e d their heuristic use for deriving (correct) results in the application of the calculus to physical problems. A n d even differential geometers as reputable as Lie and Cartan did not disdain to use them in formulating concepts which would later be p u t on a "rigorous" footing. One of the greatest champions of the concept of (continuous) infinitesimal was Charles Sanders Peirce. He saw the concept of the continuous (as did Brouwer and Weyl and as does Ren6 Thom) as arising from the subjective grasp of the flow of time, and the subjective "now" as a continuous infinitesimal. Here are some quotations. It is difficult to explain the fact of memory and our apparently perceiving the flow of time, unless we suppose immediate consciousness to extend beyond a single instant. Yet if we make such a supposition we fall into grave difficulties, unless we suppose the time of which we are immediately conscious to be strictly infinitesimal. ([9], p. 124) We are conscious only of the present time, which is an instant if there be any such thing as an instant. But in the present we are conscious of the flow of time. There is no flow in an instant. Hence the present is not an instant. ([9], p. 127) ... The fact that the continuity of space and time is a natural belief is perhaps evidence that it is true. Better evidence is that it explains the personal identity of consciousness in time, which is almost if not quite incomprehensible otherwise ([9], p. 62) This continuum does not consist of indivisibles, or points, or instants, and does not contain any except insofar as its continuity is ruptured. ([9], p. 925) It is singular that nobody objects to x/Z1 as involving any contradiction, nor, since Cantor, are infinitely great quantities objected to, but still the antique prejudice against infinitely small quantities remains. ([9], p. 123) Recently, thanks to developments in category theory and mathematical logic, it has become possible to construct a consistent f r a m e w o r k within which both "true" continua and continuous infinitesimals can be accommodated. This f r a m e w o r k is the so-called synthetic differential geometry (SDG)J It is a theory of the smoothly continuous world: in it all functions or correlations between mathematical objects are smooth, thus realizing Leibniz's doctrine of continuity, natura non facit saltus. It is interesting to note that the idea of a f r a m e w o r k of this kind was anticipated b y H e r m a n n Weyl in 1940:
1 See [7] or [8]. The idea of basing s u c h a n a p p r o a c h to differential g e o m e t r y on category theory is d u e originally to F. W. Lawvere.
56 THE MATHEMATICALINTELLIGENCERVOL.17, NO. 2, 1995
A natural way to take into account the nature of a continuum which, following Anaxagoras, defies "chopping off its parts from one another, as it were, with a hatchet", would be by limiting oneself to continuous functions. ([13], p. 294) The pervasive nature of continuity within SDG forces
a change of logic: from classical to intuitionistic--in which the law of excluded middle fails and so in which there m u s t be more than two "truth values" (nonbivalence). H o w does this come about? If excluded middle held in SDG, then each real n u m ber w o u l d be either = 0 or # 0, and so the correlation 0 H 0, x H 1 for x # 0 w o u l d define a map from the real line to 2, which is clearly discontinuous. To see that logic cannot be bivalent in SDG, let f~ be its set of "truth values." Then as in ordinary set theory, for any object X, correlations X --* f~ are in bijective correspondence with parts of X. If X is a connected c o n t i n u u m (e.g., the real line), it presumably does have p r o p e r n o n e m p t y parts but certainly no continuous nonconstant maps to the two element set {true, false}. It follows that f~ ~ {true, false} in SDG. In (many models of) SDG, any classical space X (e.g., R, R n) has a counterpart X* which is indecomposable w h e n e v e r X is connected. (A space X is indecomposable if no p r o p e r n o n e m p t y part U is detachable in the sense that there is a part V such that U U V = X, U N V = 03 Thus the connected continua of SDG are true continua in something like the Anaxagoran sense. Even more remarkably, perhaps, SDG embodies a fruitful concept of continuous infinitesimal-- that of an infinitesimal tangent vector. A tangent vector to a curve C at a point p on it is a short (nondegenerate) straight line segment & around/9 pointing along C. In SDG we m a y take & to be a part of C: in SDG, therefore, curves are " c o m p o s e d " of infinitesimally small straight lines in something like de l'Hospital's sense. Since a curve is a continuous m a p f with d o m a i n a part of the real line, it turns out that we can take & to be the image u n d e r f of the intersection D of a circle with a tangent: in particular, the intersection of the circle x 2 + (y - 1) 2 = 1 with its tangent, the line x = 0. This choice makes D that part of the real line consisting of the points x for which X 2 = 0: the square-zero infinitesimals. Notice that, in SDG, D must be nondegenerate, that is, not identical with (0}! In SDG, D is subject to the principle of infinitesimal linearity. This m a y be paraphrased b y saying that D remains straight and unbroken u n d e r any m a p or that it is too small to bend or break (but larger than zero!); that is, D can be subjected solely to translations and rotations; it is, in other words, a pure synthesis of location and direction. 2 These facts enable differential geometry [8] and the calculus [2] to be developed within SDG in a direct in-
2 D m a y also be regarded as a geometric representation of the present (as o p p o s e d to the pointlike instant).
specious
tuitive manner. For example, a tangent vector at a point x of a space ("manifold") X is an infinitesimal straight path on X passing through x, that is, a m a p p : D --* X with p(0) = x. So the tangent bundle of X is just X D, the space of all m a p s 3 D --* X, and a vector field on X any m a p X --* X D w h o s e composite with the base point m a p p H p(O) : X D --* X is the identity on X. This is just the beginning of the remarkable conceptual simplification of differential g e o m e t r y m a d e possible b y SDG. Since, as we have observed, the law of excluded middle fails in models of SDG, it follows that an assertion is, in general, no longer implied by its double negative. This fact is exploited [10] in a remarkable w a y in certain models of SDG, where infinitesimal real n u m b e r s can be equivalently identified either as nilpotent elements or as elements not unequal to or indistiguishable from O. In this situation, we can say that two points are infinitesimally close if they are not unequal, or indistinguishable; the assertion that all m a p s are continuous then reduces to the purely logical fact that any m a p automatically preserves the relation of indistinguishability. Here we have a remarkable example of a reduction of topology to logic. The infinitesimals of SDG are to be contrasted with those of A b r a h a m Robinson's nonstandard analysis (NSA) (see, e.g., [3], Chap. 11). In Robinson's approach, the field R of (standard) real numbers is "enlarged" to a field R* containing "infinitely large" elements in such a way as to preserve the usual algebraic properties of the real n u m ber field. In particular, every nonzero element of R" is (multiplicatively) invertible; the infinitesimals of NSA are obtained as the inverses of the infinitely large elements of R*. By contrast, the nilpotent infinitesimals of SDG do not (of course) possess multiplicative inverses and so cannot be obtained in this way. 4 The infinitesimals and infinities of NSA m a y be regarded as "ideal" elements playing m u c h the same role with respect to the standard real n u m b e r system as the "ideal" points and lines "at infinity" of classical projective geometry play with respect to the standard Euclidean plane. In neither case does the use of these ideal elements add to the (classically) provable facts about the standard elements. NSA is, accordingly, conservative, in that R" possesses no mathematical features not already possessed b y R. This is to be contrasted with SDG w h o s e version of the real n u m b e r system differs essentially from its standard counterpart, not being a field. The enlarged system R* of NSA m a y be regarded as just the standard system R viewed through n e w mathematical "spectacles" w h o s e resolving p o w e r is sufficient 3 This implies that, in SDG, the tangent space to any manifold M at any point on it may be identified with a part of M. In other words, in SDG, just as every curve is "infinitesimallylinear," so every manifold is "infinitesimallyflat." 4 It should be noted, however, that models of SDG incorporating "Robinsonian"infinitesimalsin additionto nonpunctual ones have been constructed: see [8].
to reveal the presence of ideal elements a m o n g the standard ones. (R* is, in particular, still a discrete structure c o m p o s e d of "distinguishable" elements.) In view of the conservative nature of the enlargement, these ideal elements are infinitesimal or infinite not in an absolute sense, but only in relation to the standard elements; that is, speaking metaphorically, an "observer" situated within a model of NSA w o u l d be unable to detect the presence of infinitesimals or infinities in R*: this is because, within any such model, R* satisfies the usual axioms for the real n u m b e r field which of course excludes infinitesimals (Archimedean property). By contrast, the nilpotency of the infinitesimals of SDG is an absolute property which is perfectly "detectable" within a model of SDG. I conclude with a final quotation from Peirce which reveals that, even before Brouwer, he was aware that a faithful account of the truly continuous will involve jettisoning the law of excluded middle: Now if we are to accept the common idea of continuity.., we must either say that a continuous line contains no points or we must say that the principle of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual.., but places being mere possibilities without actual existence are not individuals. ([9], p. xvi: the quotation is from a note written in 1903.) A remarkable insight, indeed!
References 1. Aristotle, Physics, Vol II, Cambridge, MA: Harvard University Press (1980). 2. J. L. Bell, Infinitesimals, Synthese 75 (1988), 285-315. 3. J. L. Bell and M. Machover, A Course in Mathematical Logic, Amsterdam: North-Holland (1977). 4. L. E. J. Brouwer, Intuitionism and formalism, reprinted in P. Benacerraf and H. Putnam (eds), Philosophy of Mathematics, Selected Readings, Oxford: Blackwell (1964). 5. C. Cascuberta and M. Castellet (eds), Mathematical Re6. 7. 8. 9. 10. 11. 12. 13.
search Today and Tomorrow: Viewpoints of Six Fields Medallists, Berlin: Springer-Verlag (1992). I. Kant, Critique of Pure Reason, New York: Macmillan (1964). C. McLarty, Elementary Categories, Elementary Toposes, Oxford: Oxford University Press (1992). I. Moerdijk and G. E. Reyes, Models for Smooth Infinitesimal Analysis, New York: Springer-Verlag (1991). C. S. Peirce, in The New Elements of Mathematics, Vol. III (Carolyn Eisele, ed.) Atlantic Highlands, New Jersey: Humanities Press (1976). J. Penon, Infinit6simaux et intuitionnisme, Cahiers Topoi. Gdomd. Diffdrent. XXII (1) (1981), 67-71. N. Rescher, The Philosophy of Leibniz, Englewood Cliffs, NJ: Prentice-Hall (1967). H. Weyl, The Continuum: A Critical Examination of the Foundation of Analysis (S. Pollard and T. Bole, transl.), Philadelphia: Thomas Jefferson University Press (1987). H. Weyl, The ghost of modality, in Philosophical Essays in Memory of Edmund Husserl, Cambridge, MA: Harvard University Press (1940).
Department of Philosophy University of Western Ontario London, Ontario N6A 3K7 Canada THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
57
Eisenstein's Footnote John Stillwell
Introduction Solution of the general quintic equation x 5 + a l x 4 --i-a2 x3 -~ a3 x2 q- a4x if- a5 ~- 0
(1)
was once the Holy Grail of algebra. After the general cubic and quartic equations were solved early in the 16th century, nearly three centuries of effort were expended unsuccessfully on the quintic. Several famous mathematicians actually produced faulty "solutions" in their youth. Among them were Abel and Galois, who of course later proved the impossibility of any solution of the quintic by radicals, that is, by any formula expressing the roots in terms of the coefficients al,. 99 as and a finite number of operations + , - , x, + and nth roots. Abel's impossibility proof [18261 was first, but Galois [18311 clarified the situation enormously by introducing what we now call the Galois group. He showed that the group of the general nth degree equation is the symmetric group Sn and that solvability of an equation by radicals is reflected in a group-theoretic "solvability" property, possessed by $3 and $4 but not by Ss. This explains why the general cubic and quartic are solvable by radicals and the general quintic is not. The group theory may be found in almost any modern algebra text. It is vital to an understanding of solvability by radicals. However, I believe it is not very helpful in understanding how the quintic can be solved by transcendental means, such as infinite series. In particular, I believe that the classic book on the subject, Klein's Lec-
were noticed by Galois himself and developed into a transcendental solution of the quintic by Hermite [1858]. The connections became even richer when Klein related them to the geometry of the icosahedron and its symmetry group As, which is the subgroup of Ss obstructing solution of the quintic by radicals. But is it necessary to know all this to solve the quintic? Not at all. Klein doesn't tell us, but the quintic was first solved by Eisenstein [1844], using quite elementary methods. Eisenstein's solution is so simple he mentioned it only in a footnote, without explanation or proof!
tures on the Icosahedron and the Solution of Equations of Fifth Degree [1884], makes the quintic look more complicated
than it really is. It is true that the quintic has fascinating connections with Klein's pet topic of elliptic modular functions. They 58
THE MATHEMATICALINTELLIGENCERVOL, 17, NO. 2 (~)1995SpringeroVerlagNew York
Of course some explanation of Eisenstein's solution is necessary, but not a whole book. A little linear algebra, divisibility theory of polynomials, and a trick with Laurent series is about all. The solution boils down to the formula y(x)=x-xS+
lO~w.9- 15 914~t.3 + 20.19 918x4( . . . . ,
where yS+y=x.
where xi is a root of (1), by definition of (3). Conversely, if x~ is a root of (1), then it is also a root of Yi -- box4 - hi x3 - b2 x2 - b3x - b4 -~ O.
hence x - x~ is a common divisor of the polynomials x s + a l x 4 + a2x 3 + a3x 2 + a4x + as and Yi - box 4 - b l x 3 b2x 2 - b3x - b4. In fact, it is t h e i r greatest common divisor (up to constant factors) because another divisor x - xj would imply Yi = box4 + " "
In the pages that follow I shall explain why Eisenstein's formula solves the general quintic, and how it may be derived. I am essentially filling in the elementary details of the paper by Patterson [1990], which paints a backdrop for Eisenstein analogous to the one Klein painted for Galois and Hermite. Readers looking for the bigger picture may then proceed to Patterson, or perhaps to Serre [1980], which is in some ways a modernization of Klein.
Tschirnhaus Transformations Ehrenfried Walter von Tschirnhaus (1651-1708) was a friend of Leibniz and perhaps the first man to believe he had solved the general quintic equation. In his work [1683] he published a method for transforming equations so that certain terms vanish, now known as the Tschirnhaus transformation. The method does indeed solve the general cubic and quartic equations, but not the quintic, as not enough terms vanish. Nevertheless, it does give the greatest possible simplification, so it is worth seeing how a quintic is related to its Tschirnhaus transform. A Tschimhaus transformation of the quintic (1) is the change of variable to y = box 4 + b l x 3 + b2x 2 q- 5328q- b4.
(2)
Eliminating x between (1) and (2) (see next section for details) gives a quintic equation for y, y5 + A l y 4 + A2y3 + A3y2 + A 4 y + A5 = 0
(3)
which is equivalent to (1) in the following sense.
TSCHIRNHAUS TRANSFORMATION THEOREM. The roots yi of (3) are the rational functions
= box 4 + b,x
+ b2x + b328 + b4
of bo, . . . , 54 and the roots xi of(l). If(3) has no repeated roots, then each xi is a rational f u n c t i o n of bo, . . . , b4, a l , . . . ,as, and y~. Proof. Each root y~ of (3) is of the form yi = box 4 § bl x3 + b2x2i -4- b3xi q- b4,
+ b4 = box 4 + " "
+ b4 = y j ,
contrary to the assumption that the roots y~, yj of (3) are distinct. But the greatest common divisor of two polynomials can be found by the euclidean algorithm, which involves rational operations. Hence x~ is a rational function of b0, 9 9 b4, a l , . . . , as, and y~. [] It is doubtful that Tschirnhaus had a clear formulation and proof of this theorem, though the necessary ingredients were available in his time. The correspondence between roots xi and divisors x - xi was given by Descartes [16371, and the euclidean algorithm for the greatest common divisor of two polynomials w a s given by Stevin [15851.
Eisenstein's solution is so simple he mentioned it only in a footnote, without explanation or proof!
The Tschirnhaus theorem reduces the solution of (1) by radicals to the solution of (3) by radicals, provided the transformation (2) is chosen so that (3) has no repeated roots and the coefficients bj of (2) can be expressed as radicals in the a~. Tschirnhaus believed he could choose the bj so that (3) becomes y5 + As = 0, which indeed has no repeated roots and is solvable by radicals. However, Leibniz pointed out that finding by so that A1 -- A2 -- A3 -- A4 = 0 requires the solution of a quintic equation, so the argument is circular. The best that can be done is to make A1 -- A2 --- A3 -- 0, for which it suffices to solve quadratic and cubic equations in the bj. This was first achieved by Bring [17861, but it went unnoticed until rediscovered by Jerrard [1834].
The Bring-Jerrard Reduction The elimination of x between x 5 + al x4 + a2x 3 -b a3 x2 + a4x --b a5 = 0
(4)
THE MATHEMATICAL 1NTELLIGENCER VOL. 17, NO. 2, 1995 ~ 9
and y = box4 q- bl x3 q- b2x2 q- b3x q- b4
(5)
y5 + Aly4 + A2y3 + A3y2 + A4y + A5 = 0
(6)
to produce
has to be m a n a g e d so A1, A2, and A3 can be seen to be homogeneous polynomials in the by, of degrees 1, 2, and 3, respectively. All proofs of this I have seen use Newton's formulas [1665-66] for the sums of powers of the roots, but it seems easier to use linear algebra the w a y Dedekind did in his study [1877] of algebraic integers. Linear algebra also helps with the next stage of the proof, getting A1 = A2 = A3 = 0. BRING-JERRARD equation
and since each term is of total degree 5 in the bJk) and y, the coefficients A1, A2, and A3 are homogeneous of respective degress 1, 2, and 3 in the b~k), and hence in the by. We can, therefore, make A1 = A2 = A3 = 0 by finding bo, 999 b4 satisfying
A, = Z
cjbj = 0,
(7)
A2 = E
Djkbdbk = O,
(8)
A3 = E
Ejktbybkbz = O,
(9)
where the coefficients Cj, Djk, and Ejkl are k n o w n polynomials in the ai. By eliminating b0, (8) and (9) reduce to equations
THEOREM. The general quintic 4
E x 5 + a l x 4 + a2 x3 -t- a3 x2 + a4x + a5 = 0
DIJk b j b k = O '
j,k=l 4
is equivalent to an equation
E ElYk'bjbkbl = O, j,k,t=l
yS + y + A =O,
and by the diagonalization process for quadratic forms (Lang [1965], p. 359), we can rewrite the former as
where A is a radical expression in al , . . . , a5. Proof. Multiplying (5) by x, x 2, x 3, and x 4 and using (4) to rewrite x 5, x6,.., as linear combinations of 1, x, x 2, x 3, and x 4, we get
4
,, I q- Di2b ,, 2 q- Di3b3 ,, ,, 2 = O, Dilb q- Di4b4)
E( /=1
where the Dij are (at worst) square roots of rational functions of the D}k. This equation can be satisfied by choosing the bj to satisfy two linear equations:
y = box 4 q- blx 3 q- b2x 2 q- b3x q- b4, x y = b~ox4 + b~x3 -I-''" -I- b~4,
l/
x4y = b~o"X4 + ~1111 01 x 3 + . . . + bIIII 4 ,
II
II
II
Dnbl + D12b2 + D13b3+ D14b4 = i(D~tlbl + D22b2 " ~- D"23b3 + D24b4), "
where the/)3(,k) are linear combinations of the bj whose coefficients are polynomials in the ai. This means that the homogeneous linear equations
(10a)
D~1 bl + D~2b2 + D~3b3 q- D~4b4 = i(DI411bl + DI412b2~-- D"b43 3 + D~b4).
(10b)
(/)4 -- y)Xo q- b3Xl + ' " q- boX4 = O, b~4Xo + (b~3- p)Xl + " " "1- bloX4 ---=O,
bl4'"Xo + b3pip, X1 q - " " q- (blom
-- y)X4
have the nonzero solution X0 = 1, and therefore their determinant
Xl
~
=
x,
0
. . . , X4
~
x 4,
The latter equations enable us to express two of the bj, say bl and b2, as linear combinations of the other two, b3 and/)4. Substituting these combinations in (9), we get a cubic equation for b3/b4. Hence, A1 = A2 - A3 = 0 canbe satisfied by values of the by which are radical expressions in the a~. The result is an equation y5 + A4y + A5 = 0,
b4-y
b3
...
=0. by"
...
b
"-y
The determinant is a quintic polynomial in y, y5 + Aly4 + A2y3 + A3y2 + A4y + As,
60
(11)
bo
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
where A4 and A5 are radical expressions in the ai. Finally, we get y5 + y + A = 0 (12) by replacing y by A41/4y, w h i c h is still within the scope of the Tschirnhaus transformation (5), and A = A45/4A5 is a radical expression in al, 999 a5.
It follows from the Tschirnhaus theorem that (12) is equivalent to the general quintic (4), provided (12) has no repeated roots. This is indeed the case. Any repeated root of (12) would also be a root of the derivative equation
roots can be viewed as functions of a single variable A. To emphasise the dependence of y on one variable, we shall now rename this variable - x and consider the problem of solving
5y4 § 1 = 0.
yS+y=x by expressing y as power series in x. As mentioned above,
However, the roots of this are the fourth roots of - 1 / 5 , none of which satisfies
yS + y + A =O, since A is an indeterminate, and hence unequal to a number. [For a quintic (4) with numerical coefficients a l , . . . , a s , the same argument shows that it either is equivalent to (12) or has a root ~ - / 5 , in which case it reduces to a quartic and is therefore solvable by radicals.] [] Bring's reduction appeared in an obscure Swedish publication and went unnoticed until rediscovered by Jerrard and used by Hermite [1858] in his solution of the quintic by elliptic modular functions (see Harley [1864], which includes the relevant part of Bring's paper). Hermite praised Jerrard's result as "the most important in the algebraic theory of fifth degree equations since Abel showed it is impossible to solve them by radicals." Indeed, until the impossibility of solution by radicals is
E i s e n s t e i n may well have considered this solution to be child's p l a y , because i t is based on a method he d i s c o v e r e d f o r h i m s e l f a t age 1 4 . . . known, the Bring-Jerrard theorem looks like a "near miss" rather than the best possible result of its type. Ironically, Jerrard refused to believe it was best possible and claimed, like Tschirnhaus, that he could go all the way and solve the quintic by radicals. He rejected Abel's proof and clung to his own despite resolution of the dispute in Abel's favour by Hamilton [1837, 1837']. This is where Eisenstein enters the story. In 1843 he travelled to Dublin and met Hamilton, who gave Eisenstein a copy of his work on Abel's theory (it should be remembered that Galois' theory was still unpublished and unknown at this time). Presumably this was what inspired Eisenstein to make his own contribution to the theory of equations. The little paper containing his footnote on the quintic is dated 1 January 1844. It presents uniform formulas for the solution of cubic and quartic equations by radicals, adding the infinite series solution of yS + y + A = 0 only as an afterthought. Eisenstein may well have considered this solution to be child's play, because it is based on a method he discovered for himself at age 14-- see below.
Eisenstein does not explain how he did this, but a clue is given in his posthumous Autobiographie ([1895], p. 895), where he stresses the importance "of basing the whole theory of inversion of series, as done for example by Jacobi, on the fact that the term 1/x cannot appear in the derivative of a rational function." (This method is used in another remarkable paper, [1844'], where he finds power series for functions like x x. He says this was his first mathematical research, done in his fifteenth year.) Many functions, including all the rational functions, can be expanded as power series with a finite number of negative powers, called Laurent series. In any Laurent series f(x), the coefficient of x -1, called the residue of f or res(f), plays a key role. In particular, residues help us to find a series y(x) : ClX§
x3 §
x2§
such that
yS + y = x , because
y'(x) = cl + 2C2X § 3C3x2 q-"" and therefore
cj = res(y'(x)/jxJ). The fact that x -1 is absent from the derivative of any Laurent series (since x -1 is not the derivative of any power of x) yields the following lemma, which makes a crucial simplification in the calculation of residues involving
y'(x). LEMMA. If g(x) = Ei~=N b~xi and y(x) = Ei~l ci xi, then res(g(y(x))y'(x)) = res(g(x)).
Proof. Since g(x)
= ~i~176
b~xi, (2O
=
b y(x) i=N
and therefore
f(x) = g(y(x))y'(x) = ~
biy(x)iy'(x)
i=N
= b-1 y(x)
+
b~ =
i+ 1
Y(X)i+l'
The M e t h o d of Residues
Since the power series expansion of each derivative d [y(z) i+1]/dz contains no z -1 term, we are left with
The great advantage of the equation y5 + y + A = 0 over the general quintic, apart from its simplicity, is that the
res(f) = res (b-i y'(x) / THE MATHEMATICALINTELLIGENCERVOL.17, NO. 2, 1995 61
res
['b ~,
in p o w e r s of x as a special case. Of course, in 1758 it was not k n o w n that the general quintic equation could be reduced to this form.
cl -4- 2c2x -4- 3c3x -4- 999 - - 1 C l Z -4- C2 x 2 -4- C3x3 -4-''' ]
( b - 1 Cl § 2C2x § 3c3x2 q- ' ' ' ) -x cl + c 2 x + c - ~ + : " cl + 2c2x + .." = b-l, since ~ I as x --* 0
~- r e s
References
C1 -4- C2X -4- "' "
= res(g).
[]
To compute the coefficients c3 = r e s ( y ' ( x ) / j x j) we n o w express 1 / j x J as a function g(y) of y, and apply the lemma. EISENSTEIN'S T H E O R E M . If y5 + y = x, then y has a power series expansion x 17
Y=X-x5+lO~..
-15"14~. 3+201918.
4, . . . .
"
Proof. Since x = y5 + y = y(1 + y4), we have x - j = y-J(1 -4- y4)-j and therefore
cj = r e s ( y ' ( x ) / j x j) = res(y - j (1 + y 4 ) - j y ' ( x ) / j ) = res(g(y(x))y'(x)), where g(x) = x - j (1 + x4) - j / j . And b y the l e m m a
res(g(y(x))y'(x)) = res(g(x)), cj = res(x-J(1 + x 4 ) - j / j ) .
so
Now (1 + x4) - j = 1 - j x 4 § ( - J ) ( - J - 1) x8 . . . . 2~ + (-j)(-j1)...(-ji + 1) x4~ + " b y the binomial theorem. It follows that cj = res(x-d(1 + x 4 ) - j / j ) = coefficient o f x j-1 in (1 + x 4 ) - j / j , which is ( - 1 ) i ( j + 1 ) . . . ( j + i - 1)/i! if j - 1 = 4i and zero otherwise. Thus only the coefficients of x~ xh~ x9~ x13~ 9 : x 4 i + l ~ 9 9 are nonzero, and they are 10 1 , - 1 , 2!'
14.15 3~'""
(_1) ~(4i -4- 2)(4i + 3 ) - - . (5i) i! "'"
as found by Eisenstein.
[]
Added in Proof I have recently learned that an infinite series solution of the equation x m + px = q was given b y Lambert [1758]. Lambert's series, in p o w e r of q/p, includes Eisenstein's solution of
yS+y=x
62
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO- 2, 1995
Abel, N.H. [1826] D6monstration de l'impossibilit6 de la r6solution alg6brique des 6quations g6n6rales qui passent le quatri6me degr6. J. Reine Angew. Math. 1, 65--84. Oeuvres Completes 1, 66-87. Bring, E.S. [1786] Meletemata quaedam mathematica circa transformationem aequationum algebraicarum. Lund University, Promotionschrift. Dedekind, R. [1877] Sur la th6orie des nombres entiers alg6briques, IIL Bull. Sci. Math. Astron., ser. 2, 1, 144--164. Descartes, R. [1637] La Gdomdtrie. English translation: The Geometry, Dover, New York, 1954. Eisenstein, F. G.M. [1844] Allgemeine Aufl6sung der Gleichungen von den ersten vier Graden. J. Reine Angew. Math. 27, 81-83. Mathematische Werke I, 7-9. [1844'] Entwicklung von a ~~ . J. Reine Angew. Math. 28, 49-52. Mathematische Werke I, 122-125. [1895] Eine Autobiographie von Gotthold Eisenstein. Mathematische Werke II, 879-904. Galois, E. [1831] M6moire sur les conditions de r4solubilit6 des 6quations par radicaux. F:critsMdm. Math., 43-71. Hamilton, W. R. [1837] Inquiry into the validity of a method recently proposed by George B. Jerrard, Esq., for transforming and resolving equations of elevated degree. British Assoc. Report, 295-348. Mathematical Papers III, 481-516. [1837'] On the argument of Abel, respecting the possibility of expressing a root of any general equation above the fourth degree, by any finite combination of radicals and rational functions. Trans. Roy. Irish Acad. 17, 171-259. Mathematical Papers III, 517-569. Harley, R. [1864] A contribution to the history of the problem of the reduction of the general equation of fifth degree to a trinomial form. Quart. J. Pure Appl. Math. 6, 38--47. Hermite, C. [1858] Sur la r6solution de l'6quation du cinqui6me degr6. Comptes Rend. Acad. Sci. Paris 46, 508-515. Oeuvres, Vol. 2, 5-12. Jerrard, G.B. [1834] Mathematical Researches, Vol. 2, William Strong, Bristol. Klein, E [1884] Vorlesungen iiber das Icosaeder und die Auflhsung der Gleichungen vom fiinflen Grade. English Translation: The Icosahedron and the Solution of Equations of the Fifth Degree. Dover, New York, 1956. Lambert, J. H. [1758] Observationes variae in Mathesin puram. Acta Helvetica, vol. 3, 128-168, w Opera Mathematica, vol. 1, 16-51. Orell Ffissli Verlag, Ziirich 1946. Lang, S. [1965] Algebra. Addison-Wesley, Reading, Mass. Newton, J. [1665--66] Researches in the theory of equations. Math. Papers 1,517-539. Patterson, S.J. [1990] Eisenstein and the quintic equation. Historia Math. 17, 132-140. Serre, J.-P. [1980] Extensions Icosa6driques. Sere. Th. Nombres de Bordeaux, Annde 1979-1980, exposd 19. Collected Papers III, 550-554. Stevin, S. [1585] L'arithmdtique. Abridgement in: Principal Works of Simon Stevin, Vol. liB, pp. 477-708. C. V. Swets & Zeitlinger, Amsterdam, 1958 Tschirnhaus, E.W. [1683] Methodus auferendi omnes terminos intermedios ex data aequatione. Acta Eruditorum IL
Mathematics Department Monash University Clayton, Victoria 3168 Australia
In Memory of My Friend Wilhelm Magnus Abe Shenitzer
I first met Wilhelm Magnus in 1950, w h e n I came to the N e w York University graduate school and enrolled in Magnus's course in algebra. I took an immediate liking to this polite, s o m e w h a t shy, and strikingly intelligent man. I was older than most of the students in the class and I approached him without hesitation. The fact that I was a good student helped, and I found myself talking to Magnus about nonmathematical matters as well as mathematical issues. One day he said to me: "You've done more for me than any person can d o for another." H e was visibly m o v e d and I was utterly perplexed. "Yes," he said, " y o u are a Jew w h o was in G e r m a n concentration camps and I am a German." "But I deal with individual people, not with labels" was m y response. This was the beginning of our friendship. Magnus was an extremely "practical" person in a special sense of the word. Quite early in m y student career at NYU he suggested that I should start working on a doctoral problem. Shortly thereafter, he said that it was time for me to prepare for the preliminary examination. I said, "This doesn't make m u c h sense. The prelims are supposed to determine a student's ability to w o r k on a doctoral thesis and y o u seem to think that I can do this. W h y waste time? .... You are right," said Magnus; " n o w start working on the prelims." W h e n I got m y degree, I began to agonize over the choice of a job. I had an offer from Bell Laboratories and a teaching offer from Ohio State University. W h e n I asked Magnus's advice he said, "You must decide. All I want y o u to keep in mind is that there is nothing irreversible about y o u r choice." Instead of examinations, Magnus assigned to students in his algebra course 10 substantial problems. H e checkm a r k e d correctly solved problems and returned the others without comment. H e gave out only As and Bs. It was not difficult to figure out that these were really "pass" and "fail" grades. The advantage of this system was that serious students w o r k e d their heads off to avoid the embarrassment of a B. Magnus was extremely witty. I was in his office w h e n a student came in and asked him to sign a form. W h e n the student began to explain to him what the form was about, Magnus interrupted him: "Just show me where to sign. And if I hang for this it'll be on y o u r conscience."
At the beginning of m y teaching career I had serious doubts about m y usefulness as a teacher. I said to Magnus, "What am I being paid for? .... You'll find out," he replied. "In the meantime, d o n ' t tell y o u r dean to cut y o u r salary. We'll all suffer if y o u do." W h e n told that the nuns in a certain religious school told the girls to wear long-sleeved blouses, Magnus was visibly angry and said, "They should be reminded that God created A d a m and Eve naked." To say of someone that he or she is a great teacher is to resort to "generalized praise." It is important to realize that in all areas, one can function as a creator or as a critic. Magnus was a creative mathematician and, as he told me, he liked best to w o r k with gifted doctoral students. On the other hand, he was far too intelligent not to function occasionally as a critic w h o sheds light on a whole area with a single aphoristic remark. When, as a rank beginner, I asked him what m a d e groups of automorphisms important, he replied, "They are the algebraic counterpart of symmetries in geometry." H e began his first lecture in a course on g e o m e t r y with the remark: "The fundamental difference between Hilbert and Euclid is that Hilbert realized that y o u can s t u d y form without substance."
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO- 2 (~)1995 Springer-Verlag New York 6 3
knew few bounds. "Yes, b u t . . . " was his response to utter nonsense. When I couldn't do a thing with my first doctoral problem, Magnus told me that he had just read a paper which made him think that he, Magnus, didn't understand the problem. Outwardly, he had the patience of a saint. This is all the more remarkable if one knows that he didn't suffer fools gladly. But for his encouragement many of his doctoral students would have given up; I know that I, for one, would have given up. When I read Psalm 145 and get to the lines The Lord supports all who stumble and makes all who are bent stand straight, I think not of the Lord, who is beyond human comprehension, but of my friend Wilhelm Magnus. The last memorable conversation I had with Magnus involved poetry. I saw a striking motto taken from a poem by Heine. I picked up the phone and asked Magnus which poem the quoted lines came from. "Oh," said Magnus, "it's one of the Lazarus poems. I remember it perfectly well. I'll write it down and send it to you." He did. Here is the poem.
Wilhelm Magnus When he retired he intended to function as a critic. He wanted to write an intellectual account of mathematics. He sent me a splendid article and asked my wife and me to tell him whether we liked it. He said he hoped that we would find the time to reply w i t h i n . . , a year. I called him a few days later, told him how much we liked the article, and asked him when he would write the next installment. This brings me to Magnus's intellectual eminence. He was deeply versed in history, philosophy, and literature, but he had a special passion for poetry. His learning was an integral part of his mind. He was the epitome of a cultivated person. He once told me that if he got tired of algebra, then he could always teach a course on Plato. I was present when the philosopher Hans Jonas, the mathematician Fritz John, and Magnus got together to listen to Jonas's report on a conference on gnosticism which he had just attended. There ensued an animated discussion by three people who seemed equally at home in history and in philosophy. An outsider would have found it hard to believe that two of the three participants were eminent mathematicians. When, at Magnus's suggestion, I translated Wussing's Genesis of the Group Concept, I got stuck on a lengthy note involving some of Kant's views. Magnus clarified the issue for me without a moment's hesitation. There is a Jewish legend to the effect that the continued existence of the world is due to 3 6 - - in Hebrew, "lahmed vahv" - - j u s t people. Lipman Bers called Magnus one of the "lahmed vahv." Indeed, Magnus was the essence of politeness and consideration, and his readiness to help 64
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
Lass die heilgen Parabolen, Lass die frommen Hypothesen-Suche die verdammten Fragen Ohne Umschweif uns zu 16sen. Warum schleppt sich blutend, elend, Unter Kreuzlast der Gerechte, W/ihrend glficklich als ein Sieger Trabt auf hohem Ross der Schlechte? Woran liegt die Schuld? Ist etwa Unser Herr nicht ganz allm/ichtig? Oder treibt er selbst den Unfug? Ach, das w/ire niedertr/ichtig. Also fragen wir best/indig, Bis man uns mit einer Handvoll Erde endlich stopft die M/iuler-Aber ist das eine Antwort? Leave the holy parables, leave the pious hypotheses-try to solve for ourselves these damned questions without beating around the bush. Why does the just man drag himself along, bleeding and wretched, beneath the weight of the cross, while the bad man trots along on a high horse, happy, as a victor? Where does the fault lie? Is Our Lord perhaps not quite omnipotent? Or is he himself responsible for this [mischief]? Ah, that would be base. So we constantly ask until someone eventually stops up our mouths with a handful of e a r t h - - b u t is that an answer? (Prose translation by Peter Branscombe, from Selected Verse by Heinrich Heine, in Penguin Books edition, 1968. Used by permission.)
Department of Mathematics and Statistics York University North York, Ontario M3J 1P3, Canada
How Joe Gillis Discovered Combinatorial Special Function Theory Doron Zeilberger
Girsa d'yankuta [la mishtakcha]1 (Talmud, Tractate Shabbat 22b) November 19,1993: When I checked my mail this morning, I was shocked to learn that Joe Gillis died last night in his sleep. He was 82 years old. Only last May, during my last visit to Israel, he discussed his research plans with me. He was hoping to generalize results on Hausdorff dimension and what are now called fractal sets, which he had found back in the mid-thirties. Joe had a great influence on my mathematical development, as he did on many generations of Israeli mathematicians, who got their first taste of advanced mathematics through Gilyonot Lematematika, an outstanding mathematical magazine, in Hebrew, "for highschool students and amateurs." This magazine, which Joe edited for many years, had an extensive problem section. The problems periodically challenged the best and brightest among us. Joe also initiated the Israeli Math Olympiad and was the coach of the Israeli I.M.O. team for a long time. However, Joe's influence on mathematics at large was far greater than that; in particular on my own two specialties: combinatorics and special functions. Gillis, in his paper with Even [1], initiated the marriage of these hitherto unrelated subjects, from which was soon born the flourishing new field of combinatorial special functions (e.g., [4], [5]). I feel that the story of how this union came to be, narrated to me years ago by Joe himself, must be recorded for posterity, as it testifies not only to Joe's genius but to the genius of the human spirit.
1928: Joe spent his last year in high school preparing for the competition for the coveted scholarship to Trinity College, Cambridge, which he subsequently won. The textbook he studied was Chrystal's famous Algebra [2]. One of the problems discussed there particularly appealed to Joe. It was the classical derangements problem: In how many ways can one stuff n different letters in the corresponding n envelopes, in such a way that no letter gets sent to the right address? The well-known answer, given in Chrystal's text, is that this number, D(n), equals n!(1/0! - 1/1! + 1/2! - 1/3! + . . . + (-1)~/n!) = [n!/e].
1 Aramaic: The lesson of infancy is not forgotten. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 (~) 1995 Springer'Verlag New Y~
65
tially by his desire to contribute to the welfare of the then young state of Israel, to which he immigrated in the late forties), he encountered a "practical" problem. In the course of trying to solve a certain differential equation, he needed to compute the following "linearization coefficients" for the Laguerre polynomials:
New!
IN SEARCH OF INFINITY by N. Ya Vilenkin, member of Soviet Academy of Sciences Translated from Russian by Abe Shenitzer, York University, Toronto The concept of infinity has been for hundreds of years one of the most fascinating and elusive ideas to tantalize the minds of scholars and lay people alike. The theory of infinite sets lies at the heart of much of mathematics, yet has produced a series of paradoxes that have led many scholars to doubt the soundness of its foundations. The author of this book presents a popular level account of the roads explored by human thought in attempts to understand the idea of the infinite in mathematics and physics. 1995 Approx.170 pp., 36 Illus. Hardcover $24.50(tent.) ISBN0-8176-38190
~7(nl, n2, n3)
= (--1) (n'+n2+n3) f Ln, (x)Ln2(x)Ln3(X)e -x dx. d
Once again, he was unable to find a "closed form" expression. However, he and George Weiss [3] obtained recurrence relations for the E(nl, n2~ riB), which enabled one to compile a table of these for any specified range of the arguments, obviating the need to integrate every time anew. 1975: A decade and a half later, as he was browsing through his old paper [3], Joe made a connection. He had seen these recurrence relations for Eq. (2), established 15 years earlier, much earlier than that! They were identical (up to some trivial change of notation) to the recurrences he established for D(nl, n2~ n 3 ) almost half a century before, during his last year of high school. Matching the obvious initial conditions at n3 = 0, for which E coincides with Eq. (1) (due to the orthogonality of the Laguerre polynomials), it followed that [1]
ELLIPTIC BOUNDARY PROBLEMS FOR DIRAC OPERATORS by B. Booss-Bavnbek, Roskilde University, Denmark & K.P. Wojciechowski, IUPUI, Indianapolis This book focuses on elliptic differential operators of first order as, for example the Caochy-Riemann operator, the signature operator, the Dirac operator over spin manifolds, and other generalized Dirac operators. It also deals with related, global, boundary problems like the Atiyah-PatodiSinger condition, the Calderon projection, and other analytically or geometrically defined projections. 1993 307pp. Hardcover $49.50 ISBN0-8176-3681-1 Mathematics:TheoryandApplications FOR ORDERS AND INFORMATION CALL: 1 800 7774643 or Fax (617) 876-1272 WRITE: Birkh~user,MarketingDept.,675 MassachusettsAve., Cambridge,MA 02139-3309
D(nl, Joe, being the bright youngster that he was, started to wonder what happens if there are multiple letters addressed to each address. In other words, what can one say about D ( n l , . . . , nk), the number of ways of stuffing nl letters addressed to 1 , . . . , nk letters addressed to k, into the corresponding nl + . - . + nk envelopes, in such a way that no letter gets to the right destination? Of course, he realized that D(Itl,
n 2 ) ---- 6nl, n2
n2~ riB) =
E(nl,
n2~ riB).
(2)
Thus began the beautiful field of combinatorial special function theory. (The referee pointed out that the connection between combinatorics and function theory goes back to Euler, Gauss, and Jacobi. However, the connection between combinatorics and the classical special functions of mathematical physics was first made, as far as I am aware of, by Joe Gillis. I wish to thank the referee for many valuable comments.)
(1)
but was unable to find a "closed form" expression even for the case k = 3. Failing this, he went on to establish recurrence relations that enabled him to compile a table for J ( n l ~ n2~ r~3) , for small (and not so small) values of the arguments, starting from the obvious "initial conditions" (1). Having accomplished this and realizing that there probably is no reasonable formula for D(nl, n2, n3), he went on to "bigger and better" things, or so it seemed then. After completing his undergraduate studies with distinction, he went on to write a brilliant dissertation under Besicovitch, was one of the first collaborators of Erd6s, was stationed in Bletchley Park, made important contributions to fluid dynamics; and so on, but this is a different story. 1960: About one-third century later, and long after he "changed fields" to "applied" mathematics (spurred ini66 THE MATHEMATICALINTELLIGENCERVOL. 17, NO~2, 1995
(2)
References 1. S. Even and J. Gillis, Derangements and Laguerre polynomials, Proc. Cambridge Phil. Soc. 79 (1976), 135-143. 2. G. Chrystal, Algebra, Chelsea, New York, Vol. 2., 1964. 3. J. Gillis and G. Weiss, Products of Laguerre polyn6mials, M.T.A.C. (now Math. Comp.), 14 (1960), 60-63. 4. D. Foata, Combinatoire de identiti6s sur le polyn6mes orthogonaux, Proc. Internat. Congress of Mathematicans [Warsaw, Aug. 16-24, 1983], Warsaw, 1983. 5. J. Zeng, Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials, Proc. London Math. Soc. (3) 65 (1992), 1-22.
Department of Mathematics Temple University Philadelphia, PA 19122 USA e-maih
[email protected]
Jet Wimp*
Arcadia A Play by Tom Stoppard
Theatre Royal, Haymarket, London Vivian Beaumont Theater, New York (March 2 - April 30)
Reviewed by Mary W. Gray After a period of time enmeshed in the politics of the Cold War, prolific dramatist Tom Stoppard has returned to less overtly political themes. At their best his plays are both deeply engaging and entertaining, and many believe Arcadia is his best. His skill with language and his gift for conjuring up characters and situations make for exciting and memorable theater. Perceptions of Hamlet were altered permanently for many who saw Stoppard's Rosencrantz and Guildenstern Are Dead. Logical positivism for a generation of theatre-goers is explicated by Jumpers. Travesties, Stoppard's exploitation of the juxtaposition of Lenin, James Joyce, and Dadaist Tristan Tzara in World War I Zurich, weaves an elaborate plot, coloring forever the view its audiences have of momentous events in history. His latest effort, Arcadia, may do the same not only for the audience's perception of Lord Byron and the early nineteenth-century British literary scene, but also for their view of mathematics. "The Coverly set .... See? In an ocean of ashes, islands of order. Patterns making themselves out of nothing. I can't show you how deep it goes. Each picture is a detail of the previous one, blown up. And so on. For ever. Pretty nice, eh?" So we learn, according to Tom Stoppard, that Mandelbrot was anticipated in 1809 by 13-year-old Thomasina Coverly. For most theatre-goers, the plot of Arcadia centers on whether or not Lord Byron was forced to leave *Column Editor's address: Department of Mathematics, Drexel Universi~, Philadelphia, PA 19104USA.
England in 1809 because he killed a jealous husband in a duel. But more interesting to mathematicians is the subplot concerning Thomasina's discovery of recursion and its application by Valentine, a twentieth-century Coverly, to the question of the grouse population at Sidley Park, the family estate. Whereas our concern may be with the construction of the appropriate algorithm, the significance to others of the game book is that Lord Byron's presence at Sidley Park is established by its notation that the poet was responsible for the hare among one day's bag of fourteen grouse and one hare. Arcadia opened in London in 1993 two months before the original announcement of the proof of Fermat's Last
This illustration, produced by Alun Lloyd of Oxford, accompanies the essay "From Newton to Chaos" in the playbill for Arcadia.
THE MATHEMATICALINTELLIGENCER VOL. 17, NO. 2 (~)1995 Springer-Verlag New York 6 7
Theorem; but hearing now that "Fermat wrote that he had discovered a wonderful proof of his theorem but, the margin being too narrow for his purpose, did not have room to write it down. The note was found after his death, and from that day to this..." gives one a little frisson. Thomasina is exhorted by her tutor: "My lady, take Fermat into the music room. There will be an extra spoonful of jam if you find his proof." He is met by the reply, "There is no proof, Septimus. The thing that is perfectly obvious is that the note in the margin was a joke to make you all mad." Here we c a n - - a t least we hope we c a n - - e n j o y an insider's chuckle. Thomasina, a feminist at heart, expresses a preference for another mathematician, Queen Dido, over Cleopatra, whom she characterizes as making "such noodles of our sex." In spite of her expressed scorn for the debilitating effects of love on some women, Thomasina is much taken by Lord B y r o n - - who did ultimately marry someone with an interest in mathematics, an interest passed on to their daughter Ada, Countess Lovelace. Although the particle physics and labyrinthine plot of Hapgood, a previous Stoppard effort, were generally not well received by audiences, viewers seem to take the mathematics of Arcadia in stride. There were a few gasps at the recitation of Fermat's theorem itself. During the National Theatre production, sales of the script set an all-time record, as theatre-goers rushed to see whether they might get a better understanding from the printed word of just what an iterated algorithm might be. A recent article in the London Times asked, "Why doesn't maths have mass appeal?" Its answer is that we have insufficient glamour and romance in our lives and work, at least as we present them publicly. Mathematicians are not as lacking in interest--even to nonmathematicians--as the article suggests. E.T. Bell, in the preface to the notorious Men of Mathematics, is at some pains to describe mathematicians as very ordinary folk, but in the book itself he presents his Men (and one woman) in a lively, even romanticized way. I have always been a little embarrassed to confess the impression made on me by Bell's effort. I was greatly heartened to learn that Julia Robinson, the first woman mathematician elected to the U.S. National Academy of Science, was also inspired by the book's revelation that people really do mathematics. We need more writers who show that mathematicians are not so dull and who convey the excitement in what we do. Stoppard has Valentine exclaim over Thomasina's discoveries: "It makes me so happy. To be at the beginning again, knowing almost nothing ... a door like this has cracked open five or six times since we got up on our hind legs. It's the best possible time to be alive, when almost everything you knew is wrong." Perhaps a publicist like Stoppard is just what the discipline needs. Arcadia is billed as an exploration of the differences between classical and romantic i m a g i n a t i o n - - o n the sur68
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
face in landscape architecture, but more interestingly in mathematics. In fact, Stoppard's idea for the play began to take shape when he read James Gleick's Chaos and saw it as the antithesis to classical mathematics. Thomasina's untimely demise in a fire the night before her seventeenth b i r t h d a y - - a s well as her lack of a c o m p u t e r - - c u t short her development of chaos theory as well as her ideas on non-Euclidean geometry. The script is also filled with references to entropy, the second law of thermodynamics, the Newcomen atmospheric engine, and population b i o l o g y - - i n fact, an interview in The Times on the day Arcadia opened was accompanied by "Tom Stoppard's Science: A Playgoer's Guide." Arcadia is not the first Stoppard work to contain mathematical allusions. The double agent physicist in Hapgood casually refers to the Koenigsberger bridge problem. In Jumpers we hear of Zeno's paradox (Thumper the rabbit is killed by the arrow that theoretically never left the bow of the logician protagonist), and of Cantor's work on the infinite. Catastrophe theory is used as a metaphor in Professional Foul. In Every Good Boy Deserves Favour the young boy encounters a geometry lesson: "A circle is the longest distance to the same point!... A plane area bordered by high walls is a prison not a hospital." Later in the play the ravings of the prisoner Ivanov center on his version of Euclid's axioms. Rosencrantz and Guildenstern Are Dead opens with a discussion of probability instigated by the phenomenon of 85 heads appearing in a row as the title characters toss a coin. Subsequently a dead-certain wager hinging on multiples of two being even numbers is proposed. Squaring the Circle is the metaphor for the attempt "to put together two ideas which wouldn't fit, the idea of freedom as it is understood in the West, and the idea of socialism as it is understood in the Soviet empire. The attempt failed because it was impossible, in the same sense as it is impossible in geometry to turn a circle into a square with the same a r e a - - n o t because no one has found out how to do it, but because there is no way in which it can be done." In Arcadia Stoppard has done a great job with a fictional mathematician, although the play as a whole does not quite measure up to the portrayal of Turing in Breaking the Code. Maybe Stoppard might consider applying his great skill to give a new twist to the life and work of Kovalevskaya. Arcadia, having opened in London's Royal National Theatre, is now playing at the Vivian Beaumont Theater in New York and will be produced in other cities in the future. The current production is terrific in all respects. In particular, what a joy to see full houses enjoying a popularization of mathematics!
Department of Mathematics and Statistics American University Washington, DC 20016, USA
Memorabilia Mathematica: The Philomath's Quotation Book by Robert Edouard Moritz Washington DC: The Mathematical Association of America, 1993. vii + 410 pp. US $24.00, ISBN 0-88385-321-3
Out of the Mouths of Mathematicians: A Quotation Book for Philomaths by Rosemary Schmalz Washington DC: The Mathematical Association of America, 1993. x + 294 pp. US $29.00, ISBN 0-88385-509-7
Reviewed by Donald M. Davis The first of these books consists of quotations about mathematics by famous people, including nonmathematicians as well as mathematicians, made prior to 1914 (the present book is a reissue), whereas the second is a similar compilation of post-1890 quotations by mathematicians. Each book is broken down into about 20 chapters. My favorites are the ones containing anecdotes about mathematicians. Some other chapters in the earlier work are "The Value of Mathematics," "Mathematics as a Fine Art," "Mathematics and Science," and "The Fundamental Concepts of Time and Space." Some others in the work by Schmalz are "Particular Disciplines in Mathematics," "The Love of Mathematics," "Mathematics and the Computer," and "Mathematics Education." In her preface, Schmalz describes how her book can be useful to researchers, writers, teachers, and aspiring mathematicians. These books would have been invaluable to me in writing The Nature and Power of Mathematics [1], but the newer book didn't exist and the older one was unknown to me. My book, which weaves some history and anecdotes into its mathematics, contains quite a few quotes in common with Memorabilia, but the wellorganized and documented compilations under review would have made my job easier and improved my product. I envision using these volumes extensively in my teaching. I may offer a quote a day to the students in my course, "Introduction to Mathematical Thought," which uses The Nature and Power of Mathematics for liberal arts students at Lehigh University. One which I will certainly use, and wish I had known when writing my book, is the following from Bertrand Russell: It has gradually appeared, by the increase of non-Euclidean systems, that Geometry throws no more light upon the nature of space than Arithmetic throws upon the population of the United States. Reading every quotation, which I felt I should do as a reviewer, was somewhat tedious, but I think many math students, both undergraduate and graduate, would enjoy reading selected portions. They should find it quite inspiring to read such statements as the following, made in 1890 in German by J. E Herbart:
Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself will some day find it employed against himself. Or this, by the American H. S. White in 1905: Mathematics is a science continually expanding; and its growth, unlike some political and industrial events, is attended by universal acclamation. In our current environment of shrinking external support for mathematics, this sounds pretty good. I was unfamiliar with the word "philomath," used in the title of both books. My colleagues pointed out that "philo" means "lover," which yields an obvious definition, but Webster's Unabridged Dictionary lists "lover of learning" as the principal definition. This universality of mathematics can be bolstered by numerous quotations from these books. Two by Lord Kelvin cited in Memorabilia describe mathematics as "the only true metaphysics" and as "merely the etherealization of common sense." The modern book is particularly rich in anecdotes. There are the famous absentmindedness stories about Isaac Newton (Moritz, p. 169), David Hilbert (Schmalz, p. 226), and Norbert Wiener (Schmalz, p. 233). There are great descriptions by Gian-Carlo Rota of the lecture styles of Alonzo Church and William Feller. Church had a clean-blackboard fetish, and his lectures imparted significant learning, despite being verbatim from available text. Feller's style was just the opposite; the expression "proof by intimidation" was coined to describe his method. Ernst G. Straus relates the following anecdote about Albert Einstein. We had finished the preparation of a paper and we were looking for a paper clip. After opening a lot of drawers we finally found one which turned out to be too badly bent for use. So we were looking for a tool to straighten it. Opening a lot more drawers we came on a box of unused paper clips, Einstein immediately starting to shape one of them into a tool to straighten the bent one. When I asked him what he was doing, he said, "When I am set on a goal, it becomes difficult to deflect me." Another tale that I found amusing concerned Lord Kelvin (William Thomson). He left a note on a lecture room door stating, "Professor Thomson will not meet his classes today." His students thought they were very clever when they erased the "c" from this note, but chagrined the next day to find that the professor had proceeded to also erase the letter 'T' from the word "lasses." These books are predominantly filled with more serious material, and a few themes seem to recur. Both genius and hard work are cited as being important for success in mathematics, the latter being mentioned much more frequently. Tales of the genius of Leonhard Euler (Moritz, p. 154) and John yon Neumann (Schmalz, p. 232) are particularly memorable. In her "Moments of Mathematical Insight" chapter, Schmalz relates many incidents similar to the following account of Henri Poincar6. THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
69
For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours. Ideas come w h e n stepping onto a bus (Poincar6), attending the theatre (Wiener), walking u p a mountain (J. E. Littlewood), sitting at the shore (P. S. Aleksandrov), or walking in the rain (Littlewood), but only after a long struggle of intensive work. We learn that the great N e w t o n could not admit that there was any difference between him and other men, except in the possession of such habits as ... perseverance and vigilance. When he was asked how he made his discoveries, he answered, "by always thinking about them." There are m a n y quotations highlighting the importance of discoveries and the greatness of their discoverers, but N e w t o n and his law of gravitation recur most often. We have, from D. Brewster's 1831 biography, The name of Sir Isaac Newton has by general consent been placed at the head of those great men who have been the ornaments of their species. and, from W. Whewell's History of the Inductive Sciences, The law of gravitation is indisputably and incomparably the greatest scientific discovery ever made. We even have the following from N e w t o n ' s competitor, G. W. Leibnitz: Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half. C. E Gauss also gets a lot of attention. We learn that he wished that "+1~ - 1 and ~ had been called direct, inverse, and lateral units instead of positive, negative, and imaginary," and that he disliked the notation sin 2 G feeling that this suggests sin(sin 0). In 1856, W. v. W. Sartorius wrote this about Gauss: We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically. Schmalz's chapter, "The Creative Process in Mathematics," contains m a n y quotations about the dual roles of intuition and rigor in mathematics. Carl Allendoerfer wrote in 1962, The products of this intuitive discovery are frequently wrong, usually unorganized, and always speculative. And 70
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
so there follows the task of sorting them out, weaving them into a proper theory, and proving them on the basis of a set of axioms. In 1968 Paul Halmos wrote, The deductive stage, writing the result down, and writing down its rigorous proof are relatively trivial once the real insight arrives; it is more like the draftsman's work, not the architect's. I k n o w this distinction well, having been the draftsman for the architect Mark M a h o w a l d in 28 joint papers in algebraic topology. Michael Atiyah writes, "The importance of a part of mathematics is something one can judge roughly b y the a m o u n t of interaction it has with other parts of the subject." This theme recurs frequently, for example in a statem e n t b y Einstein (Schmalz, p. 52). M a n y writers t h r o u g h o u t the centuries, b e g i n n i n g with Plato, assert that learning mathematics is difficult. Students reading this can see that they are not alone; teachers can obtain from these readings a tolerance for the difficulties faced by students. I close b y mentioning a few differences between these books. Moritz's book contains m a n y quotations from such nonmathematicians as I m m a n u e l Kant, John Locke, J. S. Mill, and William Wordsworth. For me, these were the least interesting part of the book, but I a m sure that m a n y w o u l d find t h e m valuable. Moritz organizes quotes b y subtopic within a chapter, whereas Schmalz organizes them chronologically. Both arrange the anecdotes about mathematicians alphabetically. Moritz's m e t h o d of arrangement was particularly nice w h e n one quote w o u l d rebut another, such as J. J. Sylvester's rebuttal (p. 26) of Thomas Huxley's statement that "mathematics k n o w s nothing of observation, nothing of experiment, nothing of induction, nothing of causation." Moritz's b o o k true to its time, contains virtually no quotations by or about women. It contains m a n y quotations that would n o w a d a y s be considered offensive, such as the following statement of Plato, quoted, interestingly, b y Sophie Germain: He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side. Schmalz makes u p for this with m a n y quotations b y w o m e n about h o w they were led into mathematics, and h o w they achieved success. H e r book is particularly reco m m e n d e d for aspiring female mathematicians.
Reference 1. D.M. Davis, The Nature and Power of Mathematics, Princeton, NJ: Princeton University Press (1993).
Department of Mathematics Lehigh University Bethlehem, PA 18015 USA
The Joy of Sets by Keith Devlin New York: Springer-Verlag, 1993. vii + 192 pp. US $29.95, ISBN 0-387-94094-4
Reviewed by J. Donald Monk From the preface of the book: This book provides an account of those parts of contemporary set theory of direct relevance to other areas of pure mathematics. The intended reader is either an advanced-level mathematics undergraduate, a beginning graduate student in mathematics, or an accomplished mathematician who desires or needs some familiarity with modern set theory. This review is divided into four parts: (1) Comments on the extent to which these goals are met, including an outline of the book. (For this survey, I assume that certain defects in the book can be rectified, either in future printings or by an instructor using the book in a course.) (2) Details on the most unusual feature of the b o o k - the inclusion of a treatment of the antifoundation axiom. (3) A comparison of Devlin's approach with other recent books at about this level. (4) What I consider to be important shortcomings of the book.
The Content of the Book The book can be divided into four unequal parts: (1) Naive set theory (nonaxiomatic), which is Chapter 1, 28 pp. The development does not go very far: a little of the customary notation for relations and functions, and a discussion of ordinals and their relationship to well-orderings, with no treatment of the arithmetic of ordinals. (2) Introduction of the axioms of ZFC set theory, in Chapter 2, 37 pp. The naive set theory of Chapter 1 provides an intuitive foundation for building a cumulative hierarchy of sets that motivates the usual axioms. The author proves the important recursion principle in detail, using a careful treatment involving classes, but he provides sufficient indications that in dealing with "classes" one is really working with formulas of the set-theoretic language. (3) A development of axiomatic set theory, in Chapter 3, 35 pp. Here the author skips over the elementary set theory of Chapter 1, assuming that it is sufficiently clear that it can be developed in the axiomatic framework. Evidently impatient with elementary things, the author even leaves to the reader the proof that the set of non-negative integers, w, exists (this is not completely trivial, as he takes the axiom of infinity in an unusual form). However, given the basics, the author provides a development of standard topics of set theory with full proofs: arithmetic of ordinal numbers, cardi-
nals and cardinal arithmetic (through the theorem that ~cf~ > ~ and a discussion of inaccessible cardinals), equivalents of the axiom of choice. (4) Advanced topics, Chapters 4-7, 84 pp. Chapter 4 deals with some more advanced topics in standard set theory: the Borel hierarchy, closed unbounded sets and stationary sets (including Fodor's theorem), trees, extensions of Lebesgue measure, and a result about GCH. The treatment in this chapter is complete with proofs for the results mentioned. Chapter 5 is about the Axiom of Constructibility, of which the author is a world expert. He provides an outline of the consistency of this axiom relative to ZF, but without complete details (of course). Chapter 6 sketches the ideas behind Boolean-valued consistency proofs, including the nonprovability of the Continuum Hypothesis (CH). Chapter 7 concerns an unusual topic, set theory in which the axiom of foundation is negated. It is an exposition, with motivation and complete proofs, of the approach of P. Aczel to this topic. That is an outline of the book. How it meets the goals mentioned in the preface depends on one's philosophy about learning the material. Traditionally, mathematics books contain a thorough development of the topics they deal with, which enables a reader to start on page I and work through to the end, absorbing the material to the best of his/her ability. That is not the philosophy of this book. Rather, the author seeks to indicate how set theory can be developed rigorously, and he devotes more space to motivation of the ideas involved and exposition of a spectrum of topics.
The Antifoundation Axiom Traditional set theory (as treated in all but Chapter 7 of the book) builds on the intuitive idea of cumulatively forming new sets by starting with the empty set, repeatedly taking the set of all subsets of sets constructed at some state, and taking unions of previous stages at a limit stage. At each nonlimit stage, new sets are introduced, but old sets are left a l o n e - - no new members are added to them. These intuitive principles give rise to an axiom that has been called into question on occasion: the foundation axiom. It says that any nonempty set A has a member a such that a n A is empty. This implies, for example, that there is no set a such that a is the only member of a. It is possible to develop most of mathematics without worrying about this axiom, as the sets ordinarily encountered in mathematics are anyway not this "irregular" kind of set. In foundational studies, negating the axiom of foundation has had some uses; see, e.g., Ref. 1. Set theory with axioms contradicting the axiom of foundation has not had much use yet, however, The rather extensive treatment of this topic, although intrinTHE MATHEMATICAL 1NTELLIGENCER VOL. 17, NO. 2, 1995
71
sically interesting, conflicts with the statement made in the preface and quoted at the beginning of this review. Devlin gives an in-depth treatment, including a lengthy informal motivational discussion, of P. Aczel's antifoundation axiom (AFA); see Ref. 2. This is not the only possible such axiom, and it is inconsistent with the version implicitly used in Ref. 1. The idea is that one associates with each set a a directed graph (possibly with loops), by letting the nodes be the members of a, with an arrow from b to c exactly w h e n b C c. The AFA is, roughly speaking, that for any directed graph there is exactly one set corresponding to it. Taking a graph with only one node and a loop at that node gives the consequence that there is exactly one set a such that a = {a}. The major result that Devlin proves is that Z F C - + AFA is consistent relative to ZFC-, where the latter is ZFC without the axiom of foundation.
Comparisons There are m a n y books at roughly the level of Devlin's book. I am not going to attempt a simultaneous review of these, but I w a n t to compare them with Devlin's concerning certain key points. I consider Devlin [3] (the first edition of this book), Enderton [4], Kunen [5], Roitman [6], and Vaught [7]. The book under review is a new edition of [3]. Although the two editions appear to be the same size, the n e w edition is considerably bigger, for the first edition was a camera-ready book with extra spacing between lines. It appears that the main revision of the new edition was the addition of the chapter on the Antifoundation Axiom. Enderton's work [4] is considerably easier than Devlin's book. Enderton develops set theory thoroughly and with no gaps left for the reader, as far as the book g o e s - - w h i c h is not as far as Chapter 3 of Devlin. (An unusual feature of Enderton's book is its dual treatment of naive and axiomatic set theory: one simply omits material with a vertical bar in the margin if one wants a nonaxiomatic approach.) On the other hand, Kunen [5] is at a much higher level but still is accessible to the audience mentioned at the beginning of this review (although an advanced-level undergraduate mathematics major will find it hard sledding). Kunen does everything rigorously, and covers all the topics of Devlin's book m u c h more thoroughly, except that there is no treatment of an antifoundation axiom. The level of Roitman [6] seems about the same as Devlin's book; the main part of it is more-or-less standard axiomatic set theory, and there is a brief section on constructibility. A useful feature of this book not found in Devlin's book is some consistency results of an easier nature than the deep ones involving constructibility and Boolean-valued models. Vaught [7] does m u c h more in naive set theory than Devlin or the other books before turning to axiomatic set theory. From that point the book is rigorous, more so than the others, except Kunen. This author also proves some simple consistency results. 72
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
Shortcomings Devlin's book has mathematical errors, typographical errors, and arguable points. Presumably the things of the first two kinds can be easily corrected in a further printing. Items of the third k i n d - - a r g u a b l e p o i n t s - - c o u l d be taken care of by a few disclaimers. For all three shortcomings I have to go into some detail.
Mathematical Errors. I list only ones that might genuinely bother a reader. Page 8. It is not clear what the intersection of the empty set should b e - - presumably the collection of all sets, but if so, some words should be said about this and its associated paradoxes. Page 25. The definition of a Boolean algebra is defective on two counts. First, the author allows the e m p t y set to be a Boolean algebra. More seriously, the axioms are not sufficient: one could set - b = b for any element b in a distributive lattice and the axioms w o u l d hold. Page 27, definition of an ideal. Condition (b)' is clearly not correct. Page 28, D. This is clearly not right for X the ordering of the reals. Page 46. The reference to the reviewer's book Monk [8] is in error: Devlin claims that Monk treats the BernaysG6del system in this book; but Monk uses the Kelley-Morse system, and this system is stronger than ZFC w h e n restricted to sets alone (see Ref. 9). Page 100, E. The author states that wl + 1 is not locally compact. It is well k n o w n and easy to show that it is, and is even compact. Pages 114-115. One needs the condition on the measure # that #({c~}) = 0 for all c~.
Typographical Errors. The book has many, but there is no point in making an extensive list. One major typographical error should be mentioned, though (and I hope it really is a typographical error!). Throughout the book where most mathematicians use f r a for the restriction of a function f to a subset a of its domain, Devlin's book simply has f a, that is, f followed by a big space, and then a.
Arguable Points. As one might expect, one can quarrel with some statements m a d e in those parts of the book concerned with motivation rather than just mathematics. Here are some that bothered me: In motivating the customary cumulative hierarchy of sets, Devlin makes the argument that one should start with the empty set because there should be " . . . no unnecessary and restrictive assumptions" (p. 36). However, this seems to be an argument for taking as a starting point a fixed but arbitrary system of atoms (Urelemente). On page 85, the author says, "In fact, infinite cardinal arithmetic is essentially trivial . . . . " This needs a disclaimer to the effect
that this is only true for a d d i n g or multiplying two infinite cardinals. W h e n one turns to exponentiation, there are difficult problems and theorems, e.g., there are the recent results of Shelah using pcf theory (see Ref. 10). The treatment of Boolean-valued models in Chapter 6 is technically correct, but the author seems to imply that this is the w a y i n d e p e n d e n c e proofs usually go: pick out a suitable complete Boolean algebra and form the associated model to establish independence. The Booleanalgebraic viewpoint is usually hidden, as one usually works directly with certain partially ordered sets. To this extent, this chapter is misleading. It is true that there is a routine translation process from the partially ordered set (forcing) machinery to the Boolean-valued machinery (see Ref. 5). The chapter on the Antifoundation Axiom has a couple of defects, in m y opinion. The difference between the motivational part of the chapter and the rigorous part is not sharp enough, and one can get confused. The author implies that the AFA given is the correct antifoundation axiom, but this is questionable. References
1. E. Specker, Zur Axiomatik der Mengenlehre (Fundierungsaxiom und Auswahlaxiom), Z. Math. Logik Grund. Math. 3 (1957), 173-210. 2. P. Aczel, Non-Well-Founded Sets, CSLI Lecture Notes Vol. 14, Stanford, CA: CSLI Publications (1988). 3. K. Devlin, Fundamentals of Contemporary Set Theory, New York: Springer-Verlag (1979). 4. H. Enderton, Elements of Set Theory, New York, Academic Press (1977). 5. K. Kunen, Set Theory, Amsterdam: North-Holland (1980). 6. J. Roitman, Introduction to Modern Set Theory, New York, Wiley (1990). 7. R.L. Vaught, Set Theory,an Introduction, Boston: Birkh/iuser (1985). 8. J.D. Monk, Introduction to Set Theory, New York: McGrawHill (1969). 9. A. Levy, BasicSet Theory, New York: Springer-Verlag (1979). 10. M. Burke and M. Magidor, Shelah's pcf theory and its applications, Ann. Pure Appl. Logic 50 (1990), 207-254. 11. K. Devlin, The Joy of Sets, New York, Springer-Verlag (1993). Department of Mathematics University of Colorado Boulder, CO 80309-0395 USA A Century of Mathematics e d i t e d b y John E w i n g
Washington DC: The Mathematical Association of America, 1994. xii + 323 pp. US $39.50, ISBN 0-88385-457-0
Reviewed by Underwood Dudley H a v e y o u ever noticed h o w children accept the world without question? As far as they are concerned, everything is as it ought to be, always has been, and always will be. They touchingly m e m o r i z e the quadratic formula without question, because memorizing the quadratic formula is something that e v e r y o n e has to do. This attitude
persists a m o n g those children w h o grow u p to be students of mathematics. They learn about groups and rings and ideals because they clearly are part of mathematics and, of course, always have been. Can it be that some of this childishness remains even in m a t u r e and wise professional mathematicians? Maybe so, m a y b e not. Take the following test to see. The book u n d e r review contains material from the first century of the American Mathematical Monthly. Here are nine excerpts from it, where 8 excerpt i was published in the year 19nimi, with {n i}i=0 a p e r m u t a t i o n of {i}~=0. Your task is to estimate ni with ~ , i = 0, 1 , . . . , 8. Or, in language y o u m a y even prefer, the excerpts were published one in each decade from the 1900s to the 1980s and you are to try to match the excerpt with its decade. 0. A ton of ore contains an almost infinitesimal amount of gold, yet its extraction proves worthwhile. So if only a microscopic part of pure mathematics proves useful, its production would be justified. Any number of instances of this come to mind, starting with the investigations of the properties of the conic sections by the Greeks and their application many years later to the orbits of the planets. Gauss' investigations in number theory led him to the study of complex numbers. This is the beginning of abstract algebra, which has proved so useful for theoretical physics and applied mathematics. 1 . . . . students always have to be taught what they should
have learned in the previous course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in the calculus course, either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. 2. For example, if you are going to explain to an average class how to find the distance from a point to a plane, you should first find the distance from (2, - 3 , 1 ) to x - 2y - 4z + 7 = 0. After that, the general procedure will be almost obvious. Textbooks used to be written that way. It is a good general principle that, if you have made your presentation twice as concrete as you think you should, you have made it at most half as concrete as you ought to. Remember that you have been associating with mathematicians for years and years. By this time you probably not only think like a mathematician but imagine that everybody thinks like a mathematician. Any nonmathematician can tell you differently. 3. But what of the teaching of college mathematics and the preparation of teachers for work in colleges? Where are the associations devoted to the improvement of teaching mathematics to freshmen and sophomores? What universities are offering courses in education especially intended for the preparation of teachers for college? 4. A good instructor does very little talking himself. He will explain just enough to keep his students from getting too discouraged. Students gain strength, power and skill in mathematics by doing it themselves. They can no more learn mathematics by listening to someone explain it than they can learn to play baseball by sitting in the bleachers. It is for THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2,1995
73
this reason that the lecture method is a poor method to teach mathematics. 5. . . . much of the teaching to which freshmen are subjected in their mathematics is sadly in need of reform. Some of it is unbelievably petty and pedantic; much of it is mechanical and uninteresting, as if it were a necessary evil; and not a little of it is unimaginative and uninspiring... Under such circumstances, are students to be blamed if they voice distaste, disgust, or abhorrence for mathematics? I hardly think so. Let us then, in the light of honest self-criticism, make a determined and forthright effort to improve the teaching of freshman mathematics through better selection and organization of content as well as better craftsmanship in teaching. 6. That the calculus is regarded as dry and uninteresting by many students, and that its value is occasionally doubted, is the strongest proof possible that its significance is not grasped. Here the connections with reality are so easy and so abundant that it is actually a skillful feat to conceal the fact. Yet it is done. I know students whose whole conception of integration is the formulistic solution of integrals of set expressions by devices whose complexity you well know. 7. Some students arrive at a creative apex of their field by different roads, but others are lost because at their school ordinarily only one road is available, a certain agreed upon road which simplifies the life of the educator, but not of the student. To be sure, brilliance and depth do not lack with a large number of those who do not make the mark, but they just don't have the knack of walking in the footsteps of their elders or in absorbing themselves along traditional lines. 8. There was a time when you could make a good beginning by a reference to a liberal education. We have to recognize now that the thing formerly called by that name no longer exists. There is no body of knowledge or type of intellectual experience which is the common possession of persons of academic training, a bond of fraternity and medium of understanding between one educated man and another. N o w let us see h o w well y o u did. Calculate the Spearm a n rank correlation coefficient b y looking u p ni in the table at the end and computing r 1 ~ s hi) 2. If r < 0, y o u r sense of history could do with some work. But even if r > 0, y o u r result m a y have been owing to chance. In fact, to have a statistically significant value of r (one-sided, at the 5% level), y o u m u s t have r > .6. Of course, the test was not fair since I chose excerpts that for the most part dealt with timeless topics in mathematics education. So, if y o u failed, you are not disgraced. Since readers of The Intelligencer have a better u n d e r s t a n d i n g of the flow of mathematics than the general mathematical public, y o u m a y even have passed. Pass or fail, y o u would benefit from reading this splendid book. T h o u g h some mathematical things remain the same, others change, and it is nice to k n o w which is which. This book will tell you, and in the best possible way. It contains primary sources, selected with amazing skill b y John Ewing. W h e n y o u read B. E Finkel reminiscing in 1931 about the early days of the Monthly (p. 79), y o u can see and feel, a little, w h a t life was like in Kidder, Missouri in 1895. The table of the n u m b e r of Ph.D. degrees in mathematics granted u p to 1934 (p. 1 0 2 ) - - a n average of 26 =
74
-
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2, 1995
per year in 1 9 2 0 - 1 9 2 4 - - l e t s y o u see that higher education, in mathematics and in general, has not always been w h a t it is now. You can read a statement that t o d a y could hardly even be thought, m u c h less written: "The function of the university as a teaching institution is p u r e l y i n c i d e n t a l . " - - C . C. MacDuffee, 1948, (p. 187). You can consider the syllabus (p. 154) for the course in War Mathematics (Topic 7: Safety Areas and Dead Zones) and its implications. You can see w h a t has happened, t h r o u g h the eyes of those who were there w h e n it happened. This is a rich and fascinating book. It has everything, and everything that it has is delightful, curious, enlightening, engrossing, interesting, informative, f u n n y (when an author writes "X.'s proof is ingenious" what is m e a n t is "I can u n d e r s t a n d X.'s proof." (p. 255)), stirring, poignant (see the picture on p. 40), or some combination of the preceding. It has Ivan Niven proving that 7r is transcendental, D. H. L e h m e r p r o u d l y e x p l a i n i n g his c o m p u t e r - - the one with gears, with teeth, that w e n t r o u n d and r o u n d - - a n advertisement for a slide rule (p. 172), K. O. May complaining in 1970 that book prices are too high (p. 284), the a n n o u n c e m e n t of the first Fields Medals (p. 98), the longest title (all thirty-three w o r d s of it) ever borne b y an article in the Monthly (p. 82), M o r g a n Ward proving that all real n u m b e r s are uninteresting (p. 167), Andr6 Weil on the future of m a t h e m a t i c s , . . , the list could be extended on and on. It is too bad that the book is not twice as long as it is, or three times as long. Besides excerpts from articles, there are m a n y pictures, n u m e r o u s small items (1933 appointments to instructorships of mathematics included Hassler Whitney, E. J. McShane, A. W. Tucker, and G. B. Price (p. 97); the 1943 investments of the Association included two Pennsylvania Railroad 3 3/4% bonds, maturing in 1970 (p. 1 6 4 ) - - b u t I m u s t stop giving in to the temptation to cite example after example), and introductions b y the editor to each of the nine chronological sections, to connect the history of mathematics with history in general. Reading this book will give you not just a sample of what has a p p e a r e d in the Monthly, but a view of mathematics and mathematicians in the United States during the past h u n d r e d years. It will give y o u perspective. It will instruct and entertain. It will enlarge y o u r world. What more could y o u ask from a history book, or from any book? Here are the answers to the test: i
0
1
2
3
4
5
6
7
8
n~
7
5
8
0
4
3
1
6
2
Excerpt i can be found on page 273, 229, 304, 16, 178, 112, 32, 258, 52, i = 0, 1 , . . . , 8. This review w o u l d not be complete without noting that the last misprint in the book occurs on page 307, line 4 - , where "content" should be "contend".
Department of Mathematics DePauw University Greencastle, IN 46135 USA
Robin Wilson* Jurij Vega Toma
Slovenian Mathematician Pisanski and Robin Wilson
Slovenia differs from other European countries in two respects--it is very young and very small. It gained independence in 1991, and has only 2 million inhabitants. However, the Slovenian language, traditions, and culture are much older, and can be traced back through several centuries. In particular, the first book in the Slovenian language was printed in 1551. The Slovenian mathematician Jurij Vega was born in 1754 in Zagorica, near Ljubljana. At that time, the territory of Slovenia was part of the Austrian Habsburg monarchy. A poor village orphan (his father died when he was six), Vega received his education in Ljubljana. Later he travelled to Vienna, where he became a professor at the artillery school. He completed a four-volume textbook, Vorlesungen fiber die Mathematik, and a popular series of logarithm tables of various sizes in almost 300 editions in many languages. In 1794 he published his greatest work, Thesaurus logarithmorum completus. Vega was the first to calculate the value of 7r to 140 decimal p l a c e s - - a record that held for over half a century. Jurij Vega took part in three wars (against the Turks at Belgrade, against Prussia, and against the French Revolutionary Army along the Rhine), and his military achievements and work in mathematics and ballistics caused
him to attain the rank of Lieutenant Colonel; the title of hereditary baron was conferred on him in 1800. His death in Vienna in 1802 was mysterious: he was reported missing, and a few days later his body was found in the Danube. In October 1994, The Society of Mathematicians, Physicists, and Astronomers of Slovenia held its first Congress. This meeting was dedicated to Jurij Vega, and there was a special exhibition featuring his life and work. The Republic of Slovenia honoured him by issuing a 50-tolar banknote, and a stamp was issued to commemorate the 240th anniversary of his birth and the 200th anniversary of the Thesaurus.
The Slovenian Coat of Arms
Jurij Vega
*Column editor's address: Facultyof Mathematics, The Open University,Milton Keynes,MK76AA,England. 76
THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 (~)1995 Springer-Verlag New York
