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.
References
Response to Istvan Hargittai's Imperial Cuboctahedron
1. Hessen and Th~iringen - - Von den Anfa'ngen bis zur Reformation,
Professor Hargittai shows two nice examples of Imperial Cuboctahedra from Japan in The Intelligencer 15, no. 1, pp. 58-59. Readers may be interested to see another specimen, this time from about A.D. 450 in Europe. Two earrings of cuboctahedral shape have been excavated from the tomb of a lady of rank of the East Gothic tribe in Thiiringen, Germany. They are made from gold and a red stone called almadin, most likely by Byzantine craftsmen. The cuboctahedron was presumably known already to Plato. (For details see [2].) I have seen these earrings in the 1992 Hessen-Thiiringen exhibition [1] in Marburg and would like to thank Professor Heinemeyer of the Historische Kommission fiir Hessen for his kind permission to reproduce the photo. The jewels are owned by the Museum fiir Friihgeschichte Thiiringen's in Weimar.
Catalogue of the exhibition in Marburg 1992. (Historische Kommission fiir Hessen, ed.) 2. W. C. Waterhouse, The discovery of the regular solids. Archive for History of Exact Sciences 9 (1972), 212-221.
Benno Artmann Fachbereich Mathematik Technische Hochschule Darmstadt D-64289 Darmstadt Federal Republic of Germany
THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4 (~ 1993Springer-VerlagNew York 3
The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Chandler Davis.
A Residual Category: Some Reflections on the History of Mathematics and its Status I. Grattan-Guinness To the fond memory of Morris Kline
In 1992, the British Society for the History of Mathematics (hereafter BSHM) marked its 21st year of existence with a special issue (21, indeed) of its Newsletter. It contained an introduction by myself (President from 1986 to 1988), followed by a list of the 90-odd meetings held and 450 talks given during those years. The continued occurrence of such activity seems to speak well for interest in the subject; yet speakers and members are well aware that their interest is an isolated one, "too mathematical for historians and too historical for mathematicians. 'd For example, in Britain there is virtually no professional basis for the subject outside the Society itself and the recently constituted Research Group at the Open University (OU). The purpose of this article is to explore this regrettable situation, using Britain as the main (but not the only) example.
became important in this century because of certain needs in quantum mechanics. One would not expect a leading musicologist to inform us that symphonies were introduced into music because of Johannes Brahms or that operas became important because Richard Strauss wrote some, but in the history of mathematics "anything goes," especially standards (see box, "Money for Jam"). In Britain, such attitudes must also have underlain the need for the BSHM to be founded in the first place. Only a year or two before its founding, in 1969, Sir Edward Collingwood had been elected President of the London Mathematical Society (LMS) and had hoped to introduce some historical lectures and meetings into its schedules;
Not for the Mathematicians "Historians are only amateurs," a senior British mathematician confided to foreign historians of mathematics in 1991; "they cannot read a modern mathematics book, whereas any mathematician can read a history." Although about 80 current members of the BSHM are professional mathematicians, such a view has been and is, I suspect, pretty normal in the mathematical community worldwide. But the idea of a typical professional mathematician appraising, say, the various Greek and Arabic editions of Hero of Alexandria is perhaps a slightly optimistic vision of his historical capacities. More typical is a popular lecture given recently by a leading mathematician, in which the public was "informed" that partial differential equations were introduced into mathematics because of electrodynamics, and that complex n u m b e r s - - dating from the 16th century 1See I. Grattan-Guinness,"DoesHistoryof Sciencetreat of the history of science?The caseof mathematics."Historyof Science28 (1990), 147173 (p. 158);the generalconsiderationsdrawn hereare discussedthere in moredetail,and further examplesare given. 4
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~)1993 Springer-Verlag New York
Not for the Educators A p r o m i n e n t place is given by the m e m b e r s h i p of the BSHM to educational questions, thereby suggesting that history of mathematics is gaining a good foothold in the c o m m u n i t y of mathematics education. Further, the founding of the Society helped to launch, in 1972, a s t u d y g r o u p on the "relations between the history a n d p e d a g o g y of mathematics" at the second International Congress on Mathematics Education, due to take place the following August in Exeter. It still flourishes, publishing a w i d e l y circulated Newsletter; indeed, the present BSHM President, John Fauvel, is also currently its Chairman. But such interests are exceptional rather than normal; in general, specialists in mathematical education are little more inclined to take history seriously than are professional mathematicians. As a result, a source of extraordinary educational richness goes a-begging: problems, motives, exercises, rigour at different levels, applications, i n t e r a c t i o n s - - e v e r y t h i n g that goes into learning real mathematics instead of fulfilling educational mores. Of the various associations of mathematics teachers in Britain, contacts for the BSHM with the Mathematical Association have been the strongest, but only once at inter-society level (in 1988).
Elsewhere
but the rebuff from his Council was resounding. 2 As far as I know, the other mathematical societies in Britain have not even m o o t e d such an interest in the first place. An important manifestation of this attitude concerns the neglect of manuscript sources connected with mathematics. Every sketch-scrap of, say, van G o g h will be rightly cherished, but entire Nachliisse of mathematicians will be t h r o w n a w a y for lack of i n t e r e s t - - f o r example, most of Cauchy's in 1937 w h e n the Paris A c a d e m y of Sciences refused to accept it from the family. 3 W h e n the mathematicians take such attitudes, they cannot be surprised (and perhaps not be bothered) w h e n the academic c o m m u n i t y despises their efforts at archival conservation "as a peripheral item with peripheral interest," to quote the Director of Humanities Research Center at Austin (Texas) in 1983 on the transfer there of files of mathematics d e p a r t m e n t s of U.S. universities. 4 21 draw here on personal information and discussions with Coilingwood (who died suddenly in 1970) as well as with others at the time. In recent years, the LMS has been extremely generous in its financial support of BSHMmeetings. 3Information kindly supplied by B. Belhoste (Paris). The library of the Institute of France, located about 100 yards from the Academy offices,holds the NachldsseOfmany important Frenchfigures,including scientists and mathematicians. 4See G. Kolata, Math archive in disarra~ Science210 (1983),940.
Some comparisons with other countries w o u l d be instructive. The current situation in Britain is not an example of its decline, for the subject has always been only an individual effort there (although the quality of some of it, notably that of D.T. Whiteside, is second to none). Whereas the level of historical activity is m o d e s t in e v e r y country relative to the practice of mathematics itself, the difference is greater in Britain, reflecting an especially strong antipathy there. 5 Even the v e r y existence of the BSHM exhibits the status of historical interest within (or rather, without) the mathematical and historical c o m m u nities. Countries showing stronger representation, such as the United States, Italy, France, and Germany, do not have national societies for the subject, probably because it is integrated in a more solid w a y a m o n g the mathematicians and mathematics educators (and the historians of science to some extent). 6 For example, a considerable n u m b e r of people and some dquipes work on the subject in Paris, where Cahiers du sdminaire d'histoire des mathdmatiques is published; meetings on it take place 5 A collective volume about the history of the history of mathematics is currently in preparation, under the editorship of J. W. Dauben. The chapter division is mainly by work done in (not on) different countries, dnd these differenceswill be clearly shown. 6National societies for the history of mathematics have been formed in Canada in 1974 (also covering the philosophy of mathematics, and publishing a Newsletter) and in India in 1978 (with a journal, Ghanita
Bharati). THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4,1993 5
from time to time at the mathematics conference centre at Luminy (Marseille). The corresponding venue in the Federal Republic of Germany, at Oberwolfach, reserves a week for it about once every 2 years, interspersed recently with some small-group workshops; and the German Mathematicians' Association now has a History section, partly because of the absorption of the corresponding society of the former German Democratic Republic, where the subject was very well developed (but is now much destroyed). The change of the Soviet Union into Russia does not seem (yet) to have had comparable effects on the substantial work done especially in Moscow and Leningrad; for example, I understand that the journal Istoriko-matematicheskie Issledovaniya, started in 1948, is trying to continue. Italy is perhaps the most active country of all, with many meetings taking place. The Italian Mathematical Society has sponsored since 1980 a research journal, Bollettino di storia delle scienze mathematiche. Publications now include an excellent new bimonthly magazine on mathematics and its history, Lettera Pristem (the latter word is the acronym of a study group), partly modelled on The Intelligencer. The history of mathematics also appears occasionally in the similar publication Informazione filosofiche. In the United States, special sessions take place regularly at meetings of the AMS and the MAA, and are well attended; and some years ago, the AMS launched a book series (now co-sponsored by the LMS). Indeed, a century ago, the country pioneered the use of history in mathematics education. In Mexico, a group publishes the journal Mathesis, which is especially concerned in the history of foundational questions.
Diagnosis Nevertheless, among the world community of mathematicians, tens of thousands strong, 7 history is usually surplus to requirements. There seem to be several reasons for this normal distaste. One reason is the mistaken inference that, since mathematics shows a continuity of many major concerns (for example, with integers, lines, curves, equations) over the centuries, then it is a cumulative subject, and so only the m o d e m top layer need be learned and the mistakes set aside. However, understanding of mathematical notions has enjoyed great changes--"revolutions," even s _ in
addition to increases and innovations in content; indeed, quite profound unfamiliarity with history is required to achieve ignorance of such features. A second reason follows from this one: that since continuity appears to obtain, the purpose of history to mathematicians (apart from anecdotes, and so on) is to inject the modern versions onto the old texts to find out what the historical figures were really trying to do. (Some things in Lagrange in the late 18th century look like group theory; hence, Lagrange had to be trying to create group theory.) Simple-minded determinism is unavoidable (it happened, therefore it had to happen); the illusion of cumulation is reinforced; the intentions of our predecessors are travestied; and the efforts of historians to reconstruct them are patronised. Of course, modernising old ideas is a perfectly legitimate process, even a good way to do research; but to identify this with the history of those ideas is a profound mistake, and the denigration of the latter a sign of cultural sterility. Another reason concerns quality of problem; as several mathematicians have told me, deep reading of great predecessors has revealed the h u m d r u m nature of their own contributions. Thus, history becomes subversive, at one remove. However, a misunderstanding of the past arises; composers are not discouraged by exposure to Mozart, and mathematicians would do better to recognise the past as an active source of inspiration, in various practical and even technical respects as well as by example. The lack of awareness of history among mathematicians is so marked that quite often, in an interesting mistake, they call it "mathematical history." But that is a quite different subject; in fact, it is part of applied mathematics, including topics such as statistical methods in social history or combinatorial techniques of compensating for fragmentary information. It is quite different from the history of mathematics, although it is applicable in the mathematical history of mathematics. All these factors apply in other sciences as well as other countries: For example, scientists frequently call the history of science "scientific history," which again is quite different, being a part of applied science (and of history also, of course). However, they appear to be particularly marked in mathematics (especially the second reason) and in Britain.
Acknowledgments
This article draws upon information and memories from 7The 1987CombinedMembershipListof the AMS,the MAA,and SIAM, colleague officers and other friends who have nurtured coveringnot onlythe UnitedStatesbut (throughjointmembershipsof the BSHM through its 21 years. It draws upon my introother societies)a number of mathematiciansfor other countries,lists duction to Newsletter no. 21 (October 1992); I am grateful around 43,000persons, with the majorityof them based in institutions to the Committee of the BSHM for permission to reproof highereducation. 8For someconsiderationsof theseissues,see D. Gillies,ed.,Revolutions duce some passages here. in mathematics(Oxford,ClarendonPress, 1992);or my "Scientificrevolutions as convolutions?A scepticalenquiry,"in Amphora. Festschrifl fiir Hans Wussingzu seinem 65. Geburtstag,S. S. Demidov,M. Folkerts, Middlesex University at Enfield Middlesex EN3 4SF, England D. E. Rowe,and C. J. Scriba, eds., (Basel,Birkh/iuser,1992),279-287. 6
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
On Newton's Problem of Minimal Resistance Giuseppe Buttazzo and Bernhard Kawohl
1.
Introduction
In 1685, Sir Isaac Newton studied the motion of bodies through an inviscid and incompressible medium. In his words (from his Principia Mathematica): If in a rare medium, consisting of equal particles freely disposed at equal distances from each other, a globe and a cylinder described on equal diameter move with equal velocities in the direction of the axis of the cylinder, (then) the resistance of the globe will be half as great as that of the cylinder.... I reckon that this proposition will be not without application in the building of ships. The problem of finding a body of minimal resistance to motion in a medium seems to be one of the first problems in the calculus of variations (see, for instance, Goldstine [4]). It can be described roughly as follows. Suppose a body moves with a given constant velocity through a fluid, and suppose that the body covers a prescribed maximal cross section (orthogonal to the velocity vector) at its rear end: Find the shape of the body which renders its resistance minimal. The solution depends on how we define the resistance of a body. The Newtonian pressure law states that the pressure coefficient is proportional to sin 2 ~, with ~ being the inclination of the body contour with respect to the free-stream direction. The sine-squared pressure law can be easily deduced from the assumption that the fluid consists of many independent particles with constant speed and velocity parallel to the stream direction, that the interactions between the body and the particles obey the usual laws governing elastic shocks, and that tanTHE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~1993 Springer-Verlag New York
7
gential friction and other effects can be neglected (see Figure 1). So the resistance of a body depends only on its geometry. Suppose that its front end is described by a function u(x) >_0 defined on the horizontal bottom f~ C R 2. Then it is easy to see that the resistance of the body is proportional to the integral
1 2 dx, F(u) = n 1 + ~Ou[
(1.1)
assuming that the stream direction is vertical downwards. In particular, if the body is a half-sphere of radius ~ - r 2, and an easy calculation R, we have u(x) = v/-R gives the relative resistance Co -
F ( u ) = 0.5
rR 2
as predicted by Newton in 1685. Other bodies with the same value of Co are illustrated in Figure 2. Even though Newton's model is only a crude approximation to real physics, it appears to provide good results in the following situations (see, for instance Funk [3]): for a body in a rarefied gas with low speed, for bodies which move in an ideal gas with high Mach number, and for slender bodies. The literature on Newton's problem and on its consequences for low-speed ballistics and hypersonic aerodynamics is very wide; the interested reader can find problems related to Newton's model in the books by Miele [9] and Hayes and Probstein [5] and references therein. However, as far as we know, most of the literature is concerned with the case of rotationally symmetric bodies, for which the analysis reduces to ordinary (i.e., onedimensional) calculus of variations; in the present article, always within the framework of the Newtonian sinesquared pressure law, we give a first attempt to treat nonsymmetric cases.
g
Figure 2. (a) Half-sphere; (b) Cone; (c) Pyramid. 2.
The Model
Consider a three-dimensional body E whose horizontal bottom is a'subset f~ of R 2, and assume that the upper boundary of E is given by the graph of a function u > 0 defined on fL In other words
E = {(x,y,z): (x,y) e Q, 0 < z < u(x,y)}. Figure 1. Sine-squared pressure law. 8 THEMATHEMATICAL INTELLIGENCERVOL.15,NO.4,1993
As explained in the Introduction, the resistance of the body G is proportional to the integral (1.1) and so
Newton's problem of minimal resistance can be stated as follows: minimize (1.1) over a suitable class of functions u. Now, the integral functional F above is neither convex nor coercive. Therefore, obtaining an existence theorem for minimizers via the usual direct method in the calculus of variations fails. Moreover, if we do not impose any further constraint on the competing functions u, the infimum of the functional in (1.1) turns out to be zero, as is immediately seen by taking, for instance,
Uh ( X ) = h dist(x, 0f~) for every h E N and letting h ~ +cr Therefore, no function u can solve the minimization problem 1 2 dx min f n 1 -4- [Du[
(2.1)
because the integrand is strictly positive on fL Even a constraint of the form 0< u < M
(2.2)
does not suffice. Indeed, a sequence of functions like
uh(x) = M sin:(hlxl) satisfies the constraint (2.2) but is such that lim
h----~q-oo
F(uh) = O,
and again problem (2.1) has no solution. It is, therefore, necessary to restrict the class of admissible functions for (2.1) beyond the constraint (2.2), and we shall consider only convex bodies E, that is, functions u which are concave on f~: min{F(u) : 0 < u < M, u concave on f~}. Other kinds of constraints can be imposed (see, for instance, Miele [9]): Instead of (2.2), we m a y consider a b o u n d on the surface area of E
the resistance due to the first shock, it w o u l d no longer reflect the total resistance of the body. Note that convexity of E is sufficient (but not necessary) to guarantee single collision between particles and body; we shall come back to this observation in the last section of the paper. Other classes of functions u, even if less motivated physically, can be considered from the mathematical point of view; a possible alternative could be the considerably larger class of quasiconcave f u n c t i o n s - - that i s - of functions u whose upper level sets {x E f~ : u(x) > t} are all convex. Note that in the radially symmetric case, a function u = u(Ixl) is quasiconcave if and only if it is decreasing as a function of Ixl. Another intermediate class of admissible functions for which the problem can be studied is the class of superharmonic functions; this will be done in a forthcoming paper [2].
3.
The Radial Case
Before studying general convex bodies, it is instructive to investigate radial shapes; this w a y we shall recover some k n o w n results. The first rigorous proof of existence in the radial case is attributed to Kneser [7], even though the explicit form of the solution was already conjectured by N e w t o n himself (see, for instance, Goldstine [4] or Tonelli [12]). Let f~ be the ball B(O, R), and let us consider only functions of the form u = u([x[) with u nonincreasing as a function of [x[. As described in the previous section, the class of such functions coincides with the class of radially symmetric quasiconcave functions. Then, after integration in polar coordinates, the functional F becomes
r F(u) = 2~r oR 1 + [u'(r)[ 2 dr, so that the resistance minimization problem can be written in the form min
1 + lu'(r)l 2 (3.1)
fn 11 + [Du[2 dx q- Jfof~ udHn-1 < c, or on the volume of E
It is easy to see that the competing functions u m u s t satisfy the conditions u(0) = M and u(R) = 0; in fact, if one of them were violated, the function
jfn u dx G c. From the mathematical point of view, the concavity constraint on u is strong enough to provide an extra compactness which implies the existence of a minimizer; from the physical point of view, a motivation for this constraint is that, thinking of the fluid as of m a n y independent particles, each particle hits the b o d y only once. If E is not convex, it could happen that a particle hits the body more than once. Because F(u) measures only
w
(r) = (1 + c ) ( u ( r ) -
would provide a resistance strictly less than F(u), for some e > 0. Therefore, in (3.1) the conditions u(0) = M and u(R) = 0 can be added. By setting v(t) = u - l ( M - t), problem (3.1) can be rewritten in the more traditional form
f f M vv,3
mint/ (Jo
_~
1 --Za dr:vincreasing, v(O) = O,v(M) = R~. +v (3.2) THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4, 1993 9
The functional in (3.1) is defined even for a general (discontinuous) nonincreasing function as ~0 R
r ,2 dr, l+u a
where u 'a is the absolutely continuous part of the measure u' with respect to Lebesgue measure; analogously, in (3.2) a general increasing function v provides the value
/0 M 1 v+ ,3v~ dt + s
,M]
vv'~,
(3.3)
where v's is the singular part of the measure v' with respect to Lebesgue measure, and the second integral in (3.3) must be interpreted in a particular B V sense. An equivalent simpler expression for (3.3) is
R22 f0
vv"
1 +Vta2 dr.
We refer to Marcellini [8] and Botteron & MarceUini [1] for a rigorous study of minimization problems of this form in the class of functions which m a y have a jump at the origin. It is s h o w n there that the minimization problem (3.1) actually admits a solution which is concave and solves the Euler-Lagrange equation
ru' = C(1 + u'2) 2 on {u' # 0} for a suitable constant C < 0. Therefore, the minimization problems on the classes of concave functions and of quasiconcave functions admit the same solutions. In the radial case, the solution u can actually be explicitly computed, a n d - - a m a z i n g l y e n o u g h - - N e w t o n already gave this solution some 300 years ago; see [4]. Indeed, consider the function
f(t) - -----------~ (1 + t2)
-
Figure 3. Newton's optimal shape for M = R.
+
t4 +
- log t
Vt _> 1;
we can easily verify that f is strictly increasing, so that the following are well defined:
N o t e t h a t lu'(r)l > 1 for allr > r0 and that lu'(r+)l = 1. In fact, as N e w t o n already observed, if u has a small derivative in an interval [a, b], then u can be changed in [a, b] to a function whose slope is first 0 and then -1. This will decrease resistance. In m o d e m terms, we can say the reason is that the convex relaxation f** of the function f ( s ) = 1/(1 + 8 2) which appears in (1.1) detaches from f in the interval (0, 1), so that the solution smells the concavity of f and stays a w a y from it. Denoting by Co the relative resistance
Co = - 2~ fo R 1 +r U'2 dr, we have Co E [0, 1], and some approximate calculations give
ro/R Co
M=R 0.35 0.37
M=2R 0.12 0.16
4RT r0 -- (1 + T2) 2
10
as
--+
For the optimal frustum cone (see Figure 4), an elementary calculation gives
Vr 9 [O, ro] Co-
and, in parametric form,
3 +
as-~- --* + o c
Co ~
With these notations, we get
{ r(t) = ~t(1 +t2i2 u(t) = M - -~ -
M=4R 0.023 0.0049
The optimal radial shape for M = R is shown in Figure 3. Moreover, we obtain ~ 1--6
u(r) = M
M=3R 0.048 0.0082
t2 t4+
THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4, 1993
/ M a s ~ - --* +oo.
) -logt
~-I-~-R-
Vt 9 [1, T].
In particular, for M = R w e get Co = ro/R ,,, 0.38. N o t e that for a slender cone, it is still Co ,,, ( M / R ) -2.
By using Proposition 4.2, the existence statement of Theorem 4.1 follows by the usual direct methods of the calculus of variations. More generally, in this way it can be proved that for every M > 0, the functional
G(u) = s
admits a m i n i m u m point on .AM provided g : 12 x R x R n --* [0, +oo] is a Borel function with g(x,., .) lower selnicontinuous on R x R n for a.e. x E ~. To s h o w that IDul ~ (0, 1) one can generalize Newton's idea. If IDul E (0,1) on a set B of positive measure, we can replace u by the infimum w of all those planes tangent to the graph of u whose slope is outside (0, 1). The proof that F(w) < F(u) can be carried out by using the co-area formula (we refer to [2] for details). This completes the proof of Theorem 4.1. In the case n = 1, ~ = ( - R , R), the solution is explicitly given by
Figure 4. The optimal frustum cone for M ----R.
Summarizing the results which are k n o w n in the radial case, we get: (i) problem (3.1) admits a solution u; (ii) the modulus of the gradient IDul stays outside
(0, 1); (iii) (iv) (v) (vi)
g(x, u, Du) dx
u is concave;
u(R) = 0;
u(x)={M(R-,x,) (R-[x[)AM
ifM>R ifM
For n > 1, however, only these few facts are known about problem (4.1):
the solution is unique; there is always aflat region, that is, there exists r0 > 0 such that u(r) = M for every r E [0, r0]; (vii) u is Lipschitz continuous up to the endpoint.
(i) problem (4.1) admits a solution u; (ii) the m o d u l u s of the gradient IDul stays outside (0, 1); (iii) u is concave and, therefore, locally Lipschitz continuous.
4.
It would be interesting to prove (or disprove) the following facts which were cited above for the radial case:
The Nonradial
Case
Let f~ be a b o u n d e d open convex subset of Rn(n = 2 in the physical case) and let M > 0 be given. Then, according to Section 2, N e w t o n ' s problem of least resistance, for convex bodies having given cross section f~ and given height M, reduces to the minimization problem min
1 + IDu[ 2 dx:O < u < M~ uconcave
}
. (4.1)
Note that every concave function u is locally Lipschitz OO . ~, continuous in f~, so that Du E Lloc(f~,R ) and the integral in (4.1) is well defined. THEOREM 4.1. For every M > 0, problem (4.1) admits at
least one solution, and every solution u has the property that [Dul r (0, 1). The proof of existence relies on the following compactness result (see Marcellini [8]). P R O P O S I T I O N 4.2. For every M > 0 and every p < +c~,
the set .AM =
{~t concave on f~, 0 < u < M }
is compact with respect to the strong topology of W,o~ 1,p( ~ ).
(iv) the solution is zero on 0f~; (v) the solution is unique; (vi) there is always a flat region, that is, an open set ~20 c f~ such that u = M on f~0; (vii) the solution is Lipschitz continuous up to the b o u n d a r y OfL Let us briefly address the question of whether minimizers on symmetric domains f~ are necessarily symmetric. Even in the case w h e n f~ is a disc, the usual methods of symmetrization (see Kawohl [6]) fail. Indeed, it is possible to s h o w that the body of Figure 2(c) has the property that its resistance increases u n d e r Schwarz symmetrization. Nevertheless, we do not k n o w if in the case w h e n f~ is a disc the m i n i m u m in (4.1) is attained on the radial solution of Section 3. Incidentally, among all domains f~ with given area, the disc does not provide the b o d y of least resistance. To s e e this, let f~ be the disc of radius R, let M = R, and consider a long and thin rectangle w = (-E/2, ~/2) x (0, 7rR2/~). Then the function u(x, y) = (E - 2[xDM/~ defined on w has resistance of order e2. In other words, according to Newtonian mechanics, cum grano salis a blade has less resistance than a bullet. T H E M A T H E M A T I C A L INTELLIGENCER VOL. 15, NO. 4,1993
11
Figure 5. Nonconvex bodies with single shock property. If we enlarge the class of admissible functions b y considering bodies having the property that each particle hits the graph of u at most once, even in the radial case w e m a y have minimizers which are not concave. Indeed, the bodies depicted in Figure 5 have M = R and Co "-~ 0.32, and so they have less resistance than every convex b o d y of the same height and radius. The slope of the conical parts is v ~ / 2 . Another class considerably larger than the class of concave functions, in w h i c h the m i n i m u m resistance problem is still meaningful, is the class of quasiconcave functions, that is of functions u whose u p p e r level sets {u _> t} are convex. It is not difficult to see, b y using the coarea formula, that b o u n d e d quasiconcave functions are in BV(f~); therefore, in this framework, the minimization problem (4.1) becomes min
{/1
1 + IDaul 2 dx : 0 < u < M, u q u a s i c o n c a v e
}
,
(4.2) where D~u denotes the absolutely continuous part of the measure Du with respect to the Lebesgue measure. We d o not k n o w if problem (4.2) admits a solution, and if this solution turns out to be actually concave, even if convex bodies s h o u l d be expected to be optimal for the resistance analysis arising from the variational point of view.
Acknowledgments This article was conceived and completed in Oberwolfach; we thank the Mathematisches Forschungsinstitut for providing an excellent research a t m o s p h e r e and Susanne Kr6mker from the IWR Heidelberg for producing the pictures. Financial support came from the project "EUR Homogenization," Contract SC1-CT91-0732 of the p r o g r a m SCIENCE of the Commission of the European 12
THE MATHEMATICAL INTELLIGENCERVOL. 15, NO~ 4,1993
C o m m u n i t i e s (G.B.) as well as t h r o u g h the DFG via SFB 123 and a Heisenberg grant (B.K.)
References 1. B. Botteron and P. Marcellini, A general approach to the existence of minimizers of one-dimensional noncoercive integrals of the calculus of variations, Ann. Inst. H. Poincard Analyse Non Lindaire 8 (1991), 197-223. 2. G. Buttazzo, V. Ferone and B. Kawohl, in preparation. 3. P. Funk, Variationsrechnung und ihreAnwendungen in Physik und Technik, Grundlehren 94 Heidelberg: Springer-Verlag (1962). 4. H. H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century, Heidelberg: SpringerVerlag (1980). 5. W. D. Hayes and R. E Probstein: Hypersonic Flow Theory, New York, Academic Press (1966). 6. B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics No. 1150, Heidelberg: Springer-Verlag (1985). 7. A. Kneser, Ein Beitrag zur Frage nach der zweckmf~igsten Gestalt der GeschoJJspitzen.Arch. Math. Phys. 2 (1902), 267278. 8. P. Marcellini, Nonconvex Integrals of the Calculus of Variations, Proceedingsof Methods of Nonconvex Analysis; Varenna 1989, A. Cellina, ed. Lecture Notes in Mathematics No. 1446, Heidelberg: Springer-Verlag (1990), pp. 16-57. 9. A. Miele, Theory of Optimum Aerodynamic Shapes, New York: Academic Press (1965). 10. I. Newton, Philosophiae Naturalis Principia Mathematica (1686). 11. S. Parma, Problemi di minimo su spazi di funzioni convesse, Tesi di Laurea, Universit~ di Ferrara, Ferrara (1991). 12. L. Tonelli: Fondamenti di Calcolo delle Variazioni, Bologna: Zanichelli (1923).
Dipartimento di Matematica Universita di Pisa 56127 Pisa, Italy Mathematisches Institut Universitiit zu K61n 50923 K61n, Germany (e-mail:
[email protected])
C. N. Yang and Contemporary Mathematics D. Z. Zhang
C. N. Yang, one of the twentieth century's great theoretical physicists, shared the Nobel Prize in physics with T. D. Lee in 1957 for their joint contribution to parity nonconservation. Mathematicians, however, know Yang best for the Yang-Mills theory and the Yang-Baxter equation. After Einstein and Dirac, Yang is perhaps the twentiethcentury physicist who has had the greatest impact on the development of mathematics. I interviewed Dr. Yang in 1991; this article is based on my notes of the interviews together with his published papers and books.
Early Interactions Between Yang and Chern Yang was born in 1922 in Hefei, a mid-sized city in easte m China. His father, K. C. Yang (Yang Ko-Chuen, also known as W. C. Yang), served as a professor of mathematics at Tsinghua (now Qinghua) University in Peking (now Beijing) and later at Fudan University in Shanghai. The elder Yang had received his Ph.D. in number theory from the University of Chicago in 1928 under L. E. Dickson. One of the first to introduce m o d e m mathematics into China, K. C. Yang taught many talented students. Among them, two later became famous: Loo-Keng H w a and S. S. Chem.
have been among the "introducers" in the marriage of the Cherns in Kunming in 1939.
Zhang: As a student of the Department of Physics at Qinghua University in 1938-1942, did you learn mathematics from Chern? Yang: When Chern came back to China to teach in 1937, Qinghua had, because of the Japanese invasion, combined with Beijing University and Nankai University to form the wartime National Southwest Associated University in Kunming. Chern taught at this university for 6 years, 1937-1943. He was a brilliant and popular professor. I was first an undergraduate, later a graduate student, at the same university. I have very fond memories of my student years on that campus and am deeply grateful for the excellent education I received there.
Zhang: When did you first meet Professor Chern? Yang: I do not remember whether I had met him when he was a graduate student of Qinghua University in Beijing, 1930-1934, where my father was a mathematics professor. But I do remember when and how I first met Mrs. Chem. It was in early October 1929. Her father, Professor Tsen, had been a professor of mathematics in Qinghua University for a number of years, and the Yangs were newcomers that fall. I was 7 years old and was going to elementary school. The Tsens invited us to their house for dinner and that was when I first made the acquaintance of "big sister Tsen." The Tsen and Yang families were very close, and it was a great joy for my parents to THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~)1993 Springer-Verlag New York
13
Yang: Yes, but only after I had left Chicago in 1949. Chern and I, however, have met frequently in Princeton, Chicago, and Berkeley, ever since he came back to the United States in early 1949. Zhang: Did y o u discuss fiber bundles? Yang: N o t until the 1970s. O u r earlier contacts were social. We did discuss mathematicians, but not mathematics. The 1954 Yang-Mills Paper While a graduate student in K u n m i n g and Chicago, Yang was impressed with the fact that gauge invariance determ i n e d all electromagnetic interactions. This was k n o w n from the work in the years 1918-1929 of Weyl, Fock, and London, and through later review papers of Pauli. But b y the 1940s and early 1950s, it played only a minor and technical role in physics. In Chicago, Yang tried to generalize the concept of gauge invariance to nonAbelian groups [the gauge for electromagnetism being the Abelian U(1)]. In analogy with Maxwell's equations, he tried
Fm"
C. N. Yangand his parents. I probably audited several of Professor Chern's courses in mathematics, but a transcript of m y records, which I still have today, shows that I had taken only one course with him, Differential Geometry. That was in the fall of 1940 w h e n I was a junior.
Zhang: This course benefited you, d i d n ' t it? Yang: Of course. But I do not r e m e m b e r that course today very distinctly. Only one thing sticks in m y mind: h o w to prove that e v e r y 2-dimensional surface is conformal to the plane. I k n e w h o w to transform the metric into the form A 2 dU 2 -'I-B 2 dv 2, but for a long time had not been able to make further progress. W h e n Chern told me to use complex variables and write C dz = A du + iB dr, it was like a bolt of lightning which I n e v e r forgot. Zhang: W h e n did y o u arrive in the United States? Yang: In N o v e m b e r 1945. I had h o p e d to s t u d y with Fermi or Wigner after I got to the United States. However, I did not find Fermi at Columbia, w h e r e he had been before 1942. I w e n t to Princeton and discovered to m y deep regret that Wigner would be mostly unavailable to students for the next year. Fortunately, I learned Fermi w o u l d join a n e w l y established Institute at Chicago. That is w h y I went to the University of Chicago for m y Ph.D. Zhang: Chern was a Professor at the University of Chicago for a long time. 14 THEMATHEMATICAL INTELLIGENCER VOL.15,NO.4, 1993
-
OB,
OB~,
Ox~,
Oz.
(*)
which had appeared to him to be a natural generalization of Maxwell's equations. "This led to a mess, and I had to give u p " [1, p. 191. In 1954, as a visiting physicist at Brookhaven National L a b o r a t o r y on Long Island, N e w York, Yang r e t u r n e d once again to the idea of generalizing gauge invariance. His officemate was R. L. Mills, w h o was about to finish his Ph.D. degree at Columbia University. Yang introd u c e d the idea of non-Abelian gauge field to Mills, and they decided to add a quadratic term to the right side of (,). That cleared u p the "mess" and led to a beautiful n e w field theory. They submitted a p a p e r in the s u m m e r of 1954 to the Physical Review, which was published in October of that year as "Conservation of Isotopic Spin and Isotopic Gauge Invariance." [21 Mills wrote later a b o u t this period: During the academic year 1953--1954, Yang was a visitor to Brookhaven National Laboratory ... I was at Brookhaven also .... and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to young physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length .... I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's" [3, p. 495].
Zhang: I read that Mills was in England: In 1954, Yang in the United States and Mills in England constructed a nonlinear version of Maxwell Equations that incorporated a non-Abelian group [4, p. 463].
Yang:That was wrong. Mills was in the United States in 1954. He later did visit England m a n y times, but not in 1954. Zhang: M. E. Mayer said in 1977: A reading of the Yang-Mills paper shows that the geometric meaning of the gauge potentials must have been clear to the authors, since they use the gauge-invariant derivative and the curvature form of the connection, and indeed the basic equations in that paper will coincide with the ones derived from a more geometric approach .... [5, p. 2] Is that correct?
Yang: Totally false. What Mills and I were doing in 1954 was generalizing Maxwell's theory. We knew of no geometrical meaning of Maxwell's theory, and we were not looking in that direction. To a physicist, gauge potential is a concept rooted in our description of the electromagnetic field. Connection is a geometrical concept which I only learned around 1970. That Maxwell's equations have deep geometrical meaning was a surprising revelation to the physicists. Zhang: An interesting question is whether you understood in 1954 the tremendous importance of your original paper on non-Abelian gauge theory. Yang: No. In the 1950s we felt our work was elegant. I realized its importance in the 1960s and its great importance to physics in the 1970s. Its relationship to deep mathematics became clear to me only after 1974. Zhang: As is well known, H. Weyl initiated the idea of Abelian gauge theory. W h y was WeyI's work not mentioned in your paper?
Yang: In the 1940s and 1950s, physicists knew that Weyl had introduced the Abelian gauge idea, but always referred to Pauli's review papers [6, 7]. In fact, I did not read any of Weyl's papers at that time.
Zhang: I read in this beautiful article of yours on Weyl that he had originated the two-component theory of the neutrino. Yang: That is correct. He wrote about this theory in 1929 and pointed out that it did not observe right-left symmetry, and therefore would not be realized in nature. About 30 years later, in 1956--1957, w h e n it was found that right-left symmetry is not strictly observed anyway, Weyl's theory was revived. It is still the correct theory of the neutrino today. By the way, we bought Weyl's house in Princeton 2 years after Weyl died. We lived in it for 9 years: 1957-1966. Zhang: What was Weyl's reaction to the news that his neutrino theory had become reality? Yang:Weyl Unfortunately died 2 years before the great excitement in physics in 1957. Early that year it was announced that right-left s y m m e t r y is not strictly observed, that is, parity is not strictly conserved. Then the Weyl theory was revived. It fitted beautifully the experiments on mu-decay. There followed 6 months of great confusion about beta-decay which is related to the question whether the Weyl neutrino is right-handed or lefthanded. Then in the fall there was the V-A proposal for the structure of beta-decay. In December there was an ingenious experiment which clarified everything, including the finding that the Weyl neutrino is left-handed. Weyl was 37 years older than Yang. They belonged to different academic generations, came from different countries, were in different disciplines. Could it be said that Weyl was a mathematician w h o deeply appreciated physics, and Yang is a physicist w h o deeply appreciates mathematics?
Zhang: Did you meet Weyl in Princeton? Yang"Of course. I shall show you what I said about this matter in m y talk at Zurich in 1985 in celebration of the centenary of Weyl's birth: I had met Weyl in 1949 when I went to the Institute for Advance Study in Princeton as a young "member". I saw him from time to time in the next years, 1949-1955. He was very approachable, but I don't remember having discussed physics and mathematics with him at any time. His continued interest in the idea of gauge fields was not known among the physicists. Neither Oppenheimer nor Pauli ever mentioned it. I suspect they also did not tell Weyl of the 1954 papers of Mills and mine. Had they done that, or had Weyl somehow come across our paper, I imagine he would have been pleased and excited, for we had put together two things that were very close to his heart: gauge invariance and non-Abelian Lie groups. [8, pp. 19-20].
RobertMills THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4, 1993 1 5
Yang-Mills Theory and Geometry After the original paper written by Yang and Mills, a large number of papers were devoted to the quantization and renormalization of gauge theories and to finding solutions of Yang-Mills equations. Relatively few people paid attention to the geometric and topological aspects of gauge theories. Among those who did, however, were S. Mandelstam (1962), E. Lubkin (1963), and H. G. Loos (1967). In addition, R. Hermann published a series of mathematical books for physicists, some of which were on this subject. None of these seemed to have left much impact. I asked Yang about his own experience in realizing the relationship between gauge theory and geometry.
Zhang:Did you study gauge theory continuously after 1954?
Yang:Yes, I did. Although non-Abelian gauge theory was not used in any practical way in physics in the 1950s and 1960s, the elegance of it was more and more appreciated as time went on. For example, in 1964, D. Ivanienko published a collection of Russian translations of 12 articles on gauge theory by Yang and Mills, Lee and Yang, Sakurai, Gell-Mann, and so on. I myself continued to work on various aspects of gauge fields throughout the 1950s, although I did not obtain many useful results. In the late 1960s, I began a new formulation of gauge fields, through the approach of nonintegrable phase factors. It happened that one semester I was teaching general relativity, and I noticed that the formula in gauge theory F , ~ . - cOB, Ox~.
OB~. - - + ir Ox,
- BvB,)
Zhang: I suppose only mathematicians appreciate the mathematical language of today. Yang: I can tell you a relevant story. About 10 years ago, I gave a talk on physics in Seoul, South Korea. I joked, "There exist only two kinds of modern mathematics books: one which you cannot read beyond the first page and one which you cannot read beyond the first sentence." The Mathematical Intelligencer later reprinted this joke of mine. But I suspect many mathematicians themselves agree with me. Zhang:When did
you understand bundle theory?
Yang:In early 1975, I invited Jim Simons to give us a series of luncheon lectures on differential forms and bundle theory. He kindly accepted the invitation, and we learned about de Rham's theorem, differential forms, patching, and so on. That was very useful and allowed us to understand the mathematical meaning of the Aharonov-Bohm experiment and of the Dirac quantization rule of electric and magnetic monopoles. H. S. Tsao and I later also understood the profound and very general Chern-Weil theorem. In retrospect, it was these lectures that taught me the concept of manifold, which I had appreciated only very vaguely.
(1)
Yang-Singer-Atiyah
and the formula in Riemann geometry m
1
m
l
R{; k = -~xd t ik j - ~ x k t ij
(2) are not just similar--they are, in fact, the same if one makes the right identification of symbols! It is hard to describe the thrill I felt at understanding this point.
Zhang:Is that the first time you realized the relation between gauge theory and differential geometry? Yang: I had noticed the similarity between LeviCivita's parallel displacement and nonintegrable phase factors in gauge fields. But the exact relationship was appreciated by me only when I realized that (1) and (2) are the same. With an appreciation of the geometrical meaning of gauge theory, I consulted Jim Simons, a distinguished 16
geometer, who was then the chairman of the Mathematics Department at Stony Brook. He said gauge theory must be related to connections on fiber bundles. I then tried to understand fiber-bundle theory from such books as Steenrod's The Topology of Fibre Bundles, but learned nothing. The language of modern mathematics is too cold and abstract for a physicist.
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
Simons's lectures helped T. T. Wu and Yang to write a famous paper: "Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields" [9]. In this paper, they analyzed the intrinsic meaning of electromagnetism, emphasizing especially its global topological connections. They discussed the mathematical meaning of the Aharonov-Bohm experiment and of the Dirac magnetic monopole. They exhibited a dictionary as presented in Table 1. Half a year later, in the summer of 1976, I. M. Singer of MIT visited Stony Brook and discussed these matters at length with Yang. Singer had been an undergraduate student in physics and a graduate student in mathematics in the 1940s. He wrote in 1985: Thirty years later, I found myself lecturing on gauge theories at Oxford, beginning with the Wu and Yang dictionary and ending with instantons, i.e., self-dual connections. I would be inaccurate to say that after studying mathematics for thirty years, I felt prepared to return to physics. [10, p. 2001.
Table 1. Translation of Terminologies. Gauge field terminology
Bundle terminology
gauge (or global gauge) gauge type gauge potential b~
principal coordinate bundle principal fiber bundle connection on a principal fiber bundle transition function parallel displacement curvature ? connection on a U1(1) bundle connection on a SU2 bundle classification of U1(1) bundle according to first Chern class connection on a trivial U1(1) bundle connection on a nontrivial U1(1) bundle
phase factor ~Qp field strength f ~ source Jff electromagnetism isotopic spin gauge field Dirac's monopole quantization electromagnetism without monopole electromagnetism with monopole Source: Reference 9.
In order to explain the developments of the past decade, Singer reproduced the Wu-Yang dictionary in this paper of 1985. In April-May of 1977, an Oxford-Berkeley-MIT preprint by M. E Atiyah, N. J. Hinchin, and I. M. Singer [11] was circulated. It applied the Atiyah-Singer Index Theorem to the problem of self-dual gauge fields. Thus began the interest of many mathematicians in gauge fields. In 1979, Atiyah published a monograph called Geometry of Yang-Mills Fields [12]. Volume 5 of his Collected Works has the subtitle "Gauge Theories." I found a copy of this volume signed by Atiyah on the bookshelf in Yang's office at Stony Brook. By way of introducing the volume, Atiyah wrote: From 1977 onward my interests moved in the direction of gauge theories and the interactions between geometry and physics. I had for many years had a mild interest in theoretical physics, stimulated on many occasions by lengthy discussions with George Mackey. However, the stimulus in 1977 came from two other sources. On the one hand, Singer told me about the Yang-Mills equations, which through the influence of Yang were just beginning to percolate into mathematical circles. During his stay in Oxford in early 1977, Singer, Hinchin and I took a serious look at the self-duality equations. We found that a simple application of the index theorem gave the formula for the number of instanton parameters .... The other stimulus came from the presence in Oxford of Roger Penrose and his group. [13]
Zhang: Why did you leave a question mark in the middle of your dictionary? Yang: Because mathematicians had not investigated the concept, so familiar and important to physicists, called source, usually denoted by d. It was a key concept in Maxwell's formulation of Coulomb's and of Ampere's laws. In modern mathematical notations, it is *D,f
=J=
source.
The sourceless case satisfies D,f=O which is fulfilled if f = + * f and that is what led both physicists and mathematicians to the study of self-dual gauge fields.
Zhang: This is an extremely interesting story. The study of self-dual gauge fields led later to much beautiful mathematics, including Donaldson's result which won a Fields Medal [see below]. Yang: Yes. The story supplies a m o d e m example of how mathematicians can derive concepts from physics, which was prevalent in earlier centuries, but unfortunately rare now. Zhang: How about ideas in mathematics becoming important for physics? We may recall Einstein was advised to pay attention to tensor analysis. Is that similar to your getting help from Simons? Yang: Einstein's profound depth and stunning insight were such that no mortals should be compared with him in any way. As to the entry of mathematics into general relativity and into gauge theory, the processes were quite different. In the former, Einstein could not formulate his ideas without Riemannian geometry, while in the latter, the equations were written down, but an intrinsic overall understanding of them was later supplied by mathematics. Zhang: There were many scholars who had pointed out earlier that gauge theory is related to bundle theory. Why did their papers not exert the same degree of influence in mathematical circles as your paper? THEMATHEMATICAL INTELL1GENCERVOL.15,NO.4,1993 17
Yang: There may be many factors. The work may have been so formal that physicists were unable to understand what it said. It may have seemed trivial to mathematicians because the physical content may not have been clarified. As to the paper Wu and I wrote in 1975, our discussion of the Aharonov-Bohm experiment and the Dirac monopole helped to draw people's attention to it. Also the dictionary helped. Zhang: Have you had scientific correspondenc e with Singer and Atiyah? Yang: I have met them from time to time, but there has been no research cooperation. Yang-Baxter Equation The other mathematical structure Yang contributed to mathematics is the Yang-Baxter equation which arose from his work on statistics mechanics. In 1967, Yang tried to find the eigenfunctions of a onedimensional fermion gas with delta function interaction [14]. This was a rather difficult problem. He solved it and showed that a crucial identity in the intermediate steps was a matrix equation: A(u)B(u + v)A(v) = B(v)A(u + v)B(u).
(**)
A few years later, R. J. Baxter in his solution of another problem in physics, the 8-vertex model [15], again used equation (**). Both lines of development were later pursued in several centers of research, especially in the USSR, where the largest effort were concentrated. In 1980, Faddeev coined the term "Yang-Baxter relation" or "Yang-Baxter equation" and that has become the generally accepted name today. In the last 6 or 7 years, a number of exciting developments in physics and mathematics have led to the conclusion that the Yang-Baxter equation is a fundamental mathematical structure with connections to various subfields of mathematics, such as knot and braid theory, operator theory, the theory of Hopf Algebra, quantum group theory, the topology of 3-manifolds, the monodromy of differential equations, and so on. There has been an explosion of literature on these subjects [10-12].
Zhang: The YBE (Yang-Baxter equation) is just a simple matrix equation. Why does it have such great importance? Yang: In the simplest situation, the YBE has the form ABA = BAB.
This is the fundamental equation of Artin for the braid group. The braid is, of course, a record of the history of permutations. It is not difficult to understand that the history of permutations is relevant to many problems in mathematics and physics. 18 THEMATHEMATICAL INTELLIGENCERVOL.15,NO.4,1993
R. J. Baxter Looking at the developments of the last 6 or 7 years, I got the feeling that the YBE is the next pervasive algebraic equation after the Jacobi identity
+
+ c;
G =0
The study of the Jacobi identity has, of course, led to the whole of Lie algebra and its relationship to Lie groups.
Zhang: The influences of YBE upon mathematics seems to be stronger than upon physics. Yang: This is true right now. In fact, some physicists think the YBE is pure mathematics. But I think that will change. The YBE is a fundamental structure. Even if a physicist does not like it, h e / s h e may have to use it eventually. In the 1920s, many physicists called group theory "group pest." That attitude persisted well into the 1930s, but disappeared later. Fields Medals of 1986 and 1990 Yang-Mills theory and the Yang-Baxter equation both figure prominently in today's core mathematics. One can see this by the Fields Medals awarded in 1986 and 1990. Simon Donaldson was awarded a Fields Medal at the ICM held in Berkeley in 1986. M. E Atiyah spoke on Donaldson's work:
Together with the important work of Michael Freedman (another Fields Medal winner in 1986), Donaldson's result implied that there exist "exotic" 4-dimensional spaces which are topologically but not differentially equivalent to the standard Euclidean 4-space R4.... Donaldson's results are derived from the Yang-Mills equations of theoretical physics, which are nonlinear generalizations of Maxwell's equations. In the Euclidean case, the solution to the Yang-Mills equations giving the absolute minimum are of special interest and called instantons. [19]. There were four Fields Medalists in 1990: V. Drinfeld, V. F. R. Jones, S. Mori, and E. Witten. The w o r k of three of them was related to Yang-Mills theory a n d / o r the YangBaxter equation. The following quotes are from reports on the Kyoto Conference: We should mention Drinfeld's pioneering work with Manin on the construction of instantons. These are solutions to the Yang-Mills equations which can be thought of as having particle-like properties of localization and size . . . . Drinfeld's interests in physics continued with his investigation of the Yang-Baxter Equation. [20, p. 1210]
that B is wrong, and so forth. Most of the time, it turns out that the original idea of A is totally wrong or irrelevant.
Zhang: In mathematical
circles, too, one has this situ-
ation.
Yang: No, no. It is very different. Mathematical theorems are proved, or supposed to be proved. In theoretical physics, we are pursuing instead a guessing game, a n d guesses are mostly wrong. Zhang: It is, however, necessary to read the newest publications. Yang: Of course. It is important to k n o w what other research workers in one's field are thinking about. But to make real progress, one must face original simple physical problems, not other people's guesses. Zhang: Was
that what y o u were doing with Mills in
1954? Jones opened a whole new direction upon realizing that under certain conditions solutions of the Yang-Baxter equation could be used for constructing invariants of links .... The theory of quantum groups, non-commutative Hopf algebras, was devised by Jimbo and Drinfeld to produce solutions of Yang-Baxter equations. [21, p. 1210] Witten described in these terms the invariant of Donaldson and Floer (extending the earlier ideas of Atiyah) and generalized the Jones knot polynomial to the case of an arbitrary ambient 3-manifold. [22, p. 1214] We note with a m u s e m e n t that there were complaints that the plenary lectures at the ICM-1990 (Kyoto) were heavily slanted t o w a r d the topics of mathematical
physics:
Zhang: What about the Yang-Baxter equation? You were not treating in 1967 a basic important problem in physics. Yang: This is correct. But I was looking at one of the simplest mathematical problems in q u a n t u m mechanics: A fermion system in one dimension with the simplest interaction possible. Zhang: W h y
Everywhere we heard quantum group, quantum group, quantum group! [23]
Mathematics and Physics Zhang: W h y
did y o u r w o r k in physics p r o d u c e such a great impact in mathematics?
Yang: This is, of course, very difficult to answer. Luck is a factor. Beyond that, two points m a y be relevant. First, if one chooses to look into simple problems, one has a bigger chance of coming close to fundamental structures in mathematics. Second, one must have a certain appreciation of the value judgment of mathematics. Zhang: Please say more
Yang:Yes. We asked, "Could w e generalize Maxwell's equations so as to obtain general guiding rules for interactions between particles?"
about the first point.
Yang:Most papers in theoretical physics are p r o d u c e d in the following way: A publishes a p a p e r about his theory. B says he can improve on it. Then C points out
do y o u emphasize "simplest"?
Yang: Because the simpler the problem, the more the analysis is likely to be close to some basic mathematical structure. I can illustrate this with the following observation: If there is a mathematics-based winning strategy in the game of chess, or in Wei-qi (known in the United States by the later Japanese n a m e of "go"), then it must be in Wei-qi, because Wei-qi is a simpler, more basic game. Zhang: Please talk
about the second point.
Yang: M a n y theoretical physicists are, in some ways, antagonistic to mathematics, or at least have a tendency to d o w n p l a y the value of mathematics. I do not agree with these attitudes. I have written: Perhaps because of my father's influence, I appreciate mathematics more. I appreciate the value judgement of the mathematician, and I admire the beauty and power of mathematics: there are ingenuity and intricacy in tactical maneuvers, and breathtaking sweeps in strategic campaigns. And, of course, miracle of miracles, some concepts in mathematics turn out to provide the fundamental structures that govern the physical universe! [1, p. 74] THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
19
Zhang: What were your father's mathematical influences on you?
Mathematics
Physics
Yang: To give one example: I was exposed to the rudiments of group theory by my father when I was a high school student, and I had always been fascinated by the beautiful diagrams in the book by A. Speiser on finite groups that he had on his bookshelf. When I worked on my Bachelor's degree thesis, he suggested that I should learn about group representations from a small book called Modern Algebraic Theory by L. E. Dickson, which presented in a short chapter of 20 pages the essentials of the theory of characters. The elegance and potency of the chapter introduced me to the incredible beauty and power of group theory.
/ I
,z
/
Zhang: It is said that you had been a mathematics teacher, and that your wife was a student of yours. Yang: Yes. I spent the year 1944-1945 teaching mathematics at a high school in Kunming, and she was in one of my classes. But we did not know each other well. Several years later, I accidentally ran into her in Princeton. It was an interesting experience, teaching high school mathematics. But that has nothing to do with my attitude toward mathematics. Zhang: Is it important for a physicist to learn a lot of mathematics? Yang: No. If a physicist learns too much mathematics, he or she is likely to be seduced by the value judgment of mathematices, and may lose his or her physical intuition. I have likened the relationship between physics
and mathematics to a pair of leaves. They share a small common part at the base, but mostly they are separate: They have their own aims and distinctly different value judgements, and they have different traditions. At the fundamental conceptual level they amazingly share some concepts, but even there, the life force of each discipline runs along its own veins (see figure above). [1]
Zhang: For a physicist, experimental results are more important to learn? Yang: This is right. Zhang: Did you have much exchange with mathematicians?
Yang: Some. When T. D. Lee and I were working in 1951 on what was later called the "unit circle theorem," yon Neumann and Selberg had suggested to us to read Inequalities by Hardy, Littlewood, and P61ya, and H. Whitney taught my brother, C. P. Yang, and me in 1965 the topological concept of index. For the method of solving Wiener-Hopf equations, M. Kac referred us to M. G. Krein's long review on this subject. In the 1970s, I collaborated with a mathematics group under C. H. Gu at Fudan University in Shanghai, China. In addition to all these and Simons's lectures, I had benefited from many interactions with A. Borel in Princeton and with my mathematics colleagues in Stony Brook, R. Douglas, M. Gromov, I. Kra, B. Lawson, C. H. Sah, and others. Zhang: Did you have much interaction with Chern?
S. S. Chernand W. Z. Yang,Geneva, 1964. 20 THEMATHEMATICAL INTELLIGENCERVOL.15,NO.4,1993
Yang: As mentioned earlier, I had taken a course on differential geometry with him in my junior year in China and had probably audited some other courses with him. We talked to each other often in 1949 and subsequent years, but we did not go into any real mathematics. I
had heard of the great importance of the Chern class, I think in the 1950s, but did not know what it was. It was only in 1975, w h e n Simons gave a series of talks to us at the Institute for Theoretical Physics at Stony Brook, that I finally u n d e r s t o o d the basic ideas of fiber bundles and connections on fiber bundles. After some struggles, I also finally u n d e r s t o o d the very general Chern-Weil theorem. It is hard to describe the joy I had in u n d e r s t a n d i n g this p r o f o u n d l y beautiful theorem. I w o u l d say the joy even surpassed w h a t I had experienced u p o n learning, in the 1960s, Weyl's powerful m e t h o d of c o m p u t i n g characters for the representations of the classical groups, or u p o n learning the beautiful Peter-Weyl theorem. Why? Perhaps because the C h e r n Weil theorem is more geometrical. But it was not just joy. There was something more, something deeper: After all, what could be more mysterious, what could be more awe-inspiring, than to find that the basic structure of the physical world is intimately tied to deep mathematical concepts, concepts which were developed out of considerations rooted only in logic and in the beauty of form? On this feeling I had written: In 1975, impressed with the fact that gauge fields are connections on fibre bundles, I drove to the house of S. S. Chern in E1 Cerrito, near Berkeley.... I said I found it amazing that gauge theory are exactly connections on fibre bundles, which the mathematicians developed without reference to the physical world. I added "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." He immediately protested: "No, no. These concepts were not dreamed up. They were natural and real." [1, p. 567]
S. S. Chern and C. N. Yang (right) in 1985, w h e n Chern received an honorary DSci at Stony Brook.
References
1. C. N. Yang, Selected Papers, 1945-1980, with Commentary, W. H. Freedman and Company, San Francisco, 1983. 2. C. N. Yang and R. L. Mills, "Conservation of isotopic spin and isotopic gauge invariance," Phys. Rev. 96 (1954), 191195. 3. R. Mills, "Gauge fields," Ann. J. Phys. 57 (1989), 493-507. 4. P. A. Griffith, "Mathematical sciences: A unifying and dynamical resource--Report of the Panel on Mathematic Sciences, initiated by the National Research Council," Notices AMS 33 (1986), 463. 5. M.E. Mayer, Fibre Bundle Techniques in Gauge Theories, Lecture Notes in Physics No. 67, Springer-Verlag, Berlin, 1977, p. 2. 6. W. Pauli, Handbuch der physik, 2nd ed. (Geiger and Scheel, 1933) Vol. 24(1), p. 83. 7. W. Pauli, Reviews of Modern Physics 13, 203 (1941). 8. C.N. Yang, "Herman Weyl's contributions to physics," in Herman Weyl (1885-1955). Springer-Verlag, Berlin, 1985. 9. T.T. Wu and C. N. Yang, "Concept of nonintegrable phase factors and global formulation of gauge fields," Phys. Rev. D 12 (1975), 3845-3857. 10. I.M. Singer, "Some problems in the quantization of gauge theories and string theories," Proc. Symposia in Pure Math. 48 (1988), 198-216. 11. M. E Atiyah, N. J. Hinchin, and I. M. Singer, "Self-duality in four-dimensional Riemann geometry," Proc. Roy. Soc. London Ser. A, 362 (1978), 425--461. 12. M. E Atiyah, Geometryof Yang-Mills Fields,Scuola Normale Superiore, Pisa, 1977. 13. M. E Atiyah, Collected Works, Vol. 5. Gauge Theories. Cambridge University, Press, Cambridge, England, 1988, p. 1. 14. C. N. Yang, "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction," Phys. Rev. Lett. 19 (1967), 1312-1315. 15. R. J. Baxter, "Partition function of the eight-vertex lattice model," Ann. Phys. 70 (1972), 193-228. 16. M. Barber and E Pearce, eds., Yang-Baxter Equations, Conformal Invariance and Integrability in StatisticalMechanics and Field Theory, World Scientific, Singapore, 1990. 17. M. Jimbo, ed., Yang-Baxter Equation in Integrable Systems, World Scientific, Singapore, 1990. 18. C.N. Yang and M. L. Ge, eds., Braid Group, Knot Theory and Statistical Mechanics, World Scientific, Singapore, 1989. 19. M. E Atiyah, "The work of Donaldson," Notices AMS 33 (1986), 900. 20. A. Jaffe and B. Mazur, "Vladimir Drinfeld," Notices AMS 37 (1990), 1210. 21. R. H. Hermann, "Vaughan E R. Jones," Notices AMS 37 (1990), 1211. 22. K. Galwedzki and C. Soule, "Edward Witten," Notices AMS 37 (1990), 1214. 23. Mathematical Intelligencer, vol. 9 (1991), no. 2, 7.
Department of Mathematics East China Normal University Shanghai, 200062 China THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
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The Heavenly Spheres Regained Osmo Pekonen
To my Beatrice
Mathematics can be Hell, Purgatory, or Paradise. Jean Dieudonn6 [6] classified mathematicians into three categories: (1) the great majority, who spend their earthly days idly drawing trivial corollaries from well-known principles; (2) the creative mathematicians, who actually produce new theorems and whose number he estimated at some 150 in France and at some 600 in the United States and in Russia; and (3) the great innovators, of whom only one or two are born per year in the whole world. Ubi sunt qui ante nos in mundo fuere? Touring Hell, Dante Alighieri saw Euclid in the first sphere and Dido in the second s p h e r e - - r e m e m b e r the Phoenician princess who studied the isoperimetric problem in Vergil's Aeneid [10]? What are these spheres in fact? Dante's is a ninefold cosmos: Hell, Purgatory, and Heaven are each subdivided into nine compartments or spheres. The spherical theory goes back to Dionysius the Areopagite, who was converted by St. Paul in the first century A.D. and is believed to have been the first bishop of Athens. Around A.D. 500, a neoplatonist Christian scholar known to posterity as pseudo-Dionysius authored in the bishop's name a treatise on the hierarchy of celestial beings. He declared that they are separated into nine choirs: Angels, Archangels, Principalities, Powers, Virtues, Dominions, Thrones, Cherubim, and Seraphim. This was interpreted by the medieval theologians and by Dante as the existence of nine harmoniously moving, translucent celestial spheres which form the geometry of Heaven. Astronomers did their best to reconcile the prescribed ninefold structure with observations. Beyond the wellestablished spheres of the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn, they posited ad hoc the Firmament and the Crystalline sphere. However, a good number of the theologians who worked on the problem (e.g., Isidore of Seville, the Venerable Bede, Raban Maur, Saint Anselm, Peter Lombard, Michael Scot, William of Auvergne, Saint Bonaventure, and Vincent of Beauvais) proposed the existence of an ultimate immobile tenth sphere above the various mobile 22
ones. This so-called Empyrean sphere or Aqueous Heaven enveloped all the other nine spheres and provided a frame of ref.erence for their movement (Figure 1). Saint Bonaventure described the Empyrean sphere as the ultimate orb "which contains everything and is not contained in anything," omnia continens et a nullo alio contenta. This interesting remark has a modern ring: Obviously, Saint Bonaventure was suggesting that all observed celestial phenomena take place inside an intrinsic manifold which must not be thought of as being imbedded in any ambient space. Pope Urban IV's chaplain, the astronomer Campanus of Novara, categorically stated that "the Empyrean heaven's convex surface has nothing beyond it" [7]. This is reminiscent of Stephen Hawking resorting to Occam's razor to cut out of cosmology all superfluous speculations extending beyond the edge of spacetime [9]. The medieval theologians never seem to have attained universal agreement on the details of the spherical geometry of Heaven, except for the strange numerical fact that there should exist I + 9 celestial spheres. This doctrine prevailed throughout the Middle Ages and, indeed, as to celestial beings, it is professed by the Catholic Church to this day.
THE MATHEMATICALINTELLIGENCER VOL. 15, NO. 4 (~)1993 Springer-Verlag New York
On August 6, 1986, the Pope delivered in Rome a sermon on celestial beings in which he duly mentioned the nine heavenly choirs of Dionysius the Areopagite. For whatever r e a s o n - - m a y b e one of the heavenly host has a direct line to UPI - - international news agencies seized on this particular item from the Pope's homily. This odd tidbit of news happened to reach me on August 7 at about 8:00 A.M. while I was sipping my morning coffee in a campus dormitory in Berkeley-- I was attending the International Congress of Mathematicians. At 9:30 A.M., I secured a seat in the auditorium of Zellerbach Hall, where Edward Witten was about to deliver a plenary address dealing with superstring theory [16]. I had never seen him before, and I knew almost nothing about superstrings. Imagine my shock when I saw him produce a slide featuring the Lorentzian signature of his spacetime:
(-+++++++++). The strange numerical coincidence between the 1 + 9 celestial choirs and the 1 + 9 spacetime dimensions heralded the same morning by leading figures from the worlds of Faith and Science really startled me. "Good heavens, we are all going back to the Middle Ages," I said to m y s e l f - - a n d ever since, when alone at night, I read my Dante more and Bourbaki less. I have even checked the original text of the Pope's sermon which appears in the edition of L'Osservatore Romano dated August 7, 1986. "Contemplation of superstrings may evolve into an activity.., to be conducted at schools of divinity by future equivalents of medieval theologians," Sheldon Glashow prophesied not long ago. In his opinion, Witten's theory is not physics at all, but merely mathematical "smoke and mirrors." "Nonbelievers worry that Witten may be a Pied Piper, leading his followers away from reality and into a phantasmagoria of pure mathematics," John Horgan commented [11]. But who are these people after all to call Pure Mathematics a "phantasmagoria"? Come on, Ed Witten is our man; we awarded him the Fields medal and we intend to follow him wherever he goes--just as Dante followed Vergil. All the way to 10 dimensions! How can one possibly fit Dante's celestial spheres into Witten's Universe? Well, we can imagine the spheres of various dimensions up to the 10-dimensional Empyrean sphere and listen to their music. The Pythagoreans set the basis of the Western conception of music by plucking a single string, which can be taken to be a 1-sphere as far as the spectrum (i.e., the set of eigenvalues of the Laplacian on functions) is concerned. The Pythagoreans discovered that the basic musical intervals of Greek string instruments had the proportions I : 2 (octave), 3 : 2 (fifth), and 4 : 3 (fourth). They summarized this cosmic rule in the tetraktys, or the Holy Decad, which they expressed by the formula 1+2+3+4=10.
Figure 1. The I + 9 heavenly spheres as illustrated in Apian's Cosmographia (1553). Alfonso X (1221-1284), king of Castile and Leon, is said to have remarked, after the Ptolemaic system had been explained to him, "If the Lord Almighty had consulted me before embarking upon Creation, I should have recommended something simpler." Reprinted from E. R. Harrison, Cosmology: The Science of the Universe, Cambridge, England: Cambridge University Press (1981), p. 78.
Thanks to Dennis DeTurck, the music of some higherdimensional spheres is now available as a sound sheet [8]. As a partial answer to Mark Kac's classical question "Can you hear the shape of a drum?," it is known that the spectrum of the n-sphere characterizes the standard sphere for n ~ 6. For n > 6, the question has not been settled. The spherical harmonics of the 7-sphere have recently been investigated by superstring theorists [2]. Indeed, various n-spheres do appear in superstring theory in several mysterious ways. For starters, a closed bosonic string itself is topologically nothing but a 1sphere. To include fermions in the theory, we convert the string into a superstring by introducing anticommuting coordinates. Heterotic superstring music was first performed by the "Princeton string quartet" (David Gross, Jeffrey A. Harvey, Emil Martinec, and Ryan Rohm). The reparametrization space of the superstring is the superdiffeomorphism group of the 1-sphere modulo the subgroup of rotations. This moduli space .&I carries the structure of an infinite-dimensional super-K/ihler manifold whose super-Ricci curvature can be computed: Its nonvanishing is interpreted as an anomaly of the reparametrization invariance. To cancel this anomaly, a "vacuum bundle" is also introduced over d~. In an ambient flat Minkowski spacetime of dimension D, the total curvature is then found to be proportional to the number D - 10. Accordingly, there is no reparametrization anomaly if and only if D = 10 [15]. Thus, plucking a superstring also leads to the Holy Decad! THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993 2 3
F i g u r e 2. T h e 1 - s p h e r e is p a r a l l e l i z a b l e . I l l u s t r a t i o n b y G u s t a v e Dor~, b a s e d o n a p a s s a g e f r o m D a n t e ' s 2 4 THEMATHEMATICALINTELLIGENCERVOL.15,NO. 4, 1993
Divine Comedy.
The gauge group of superstring theory is believed to be Es x Es. This implies some relationship with both the 7-sphere and the 10-sphere. Indeed, the root lattice of the exceptional Lie algebra Es is generated by the Cayley integers lying on the 7-sphere. Amazingly, the dimension of E8 x Es, or 496, happens to be one of the perfect numbers which were divine for the Pythagoreans. It also happens to be half the number of components of the orientation-preserving group of diffeomorphisms of the 10-sphere [14]. Michio Kaku's view on string theory, as expressed in [12], is definitely Pythagorean: The myriad particles found in nature can be viewed as the vibrations of a string, in much the same way that the notes found in music can be explained as the modes of a vibrating string. Pursuing this analogy, the basic particles of our world correspond to the musical notes of the superstring, the laws of physics correspond to the harmonies that these notes obey, and the universe itself corresponds to a symphony of superstrings. In his illustrations for La Divina Commedia, Gustave Dot6 depicted the heavenly spheres as 2-spheres around which myriad angels whirl in eternal bliss. However, his beatific vision led him into a major difficulty of topological nature: a dynamical system on a 2-sphere always exhibits a singularity. Indeed, whenever I hear the word "singularity," one of Gustave Dor6's plates comes to my mind (see cover). The artist could not have avoided drawing a singularity because of the well-known Hairy Ball Theorem. Undoubtedly, he was aware of the problem. In another plate (Figure 2), he drew the angels flying in circles, thereby avoiding a singularity (for the reason, as we express it, that the 1-sphere is parallelizable). Gustave Dor6 might have circumvented the Hairy Ball Theorem by contenting himself with a discrete set of angels on the 2-spheres rather than a continuously whirling flow. Even so, for the sake of symmetry, he should have distributed a large number N of angels at approximately equal distances on the 2-sphere, and this is a very difficult problem, except for the special cases N = 4, 6, 8, 12, 20 which were so prominent in the cosmology of Pythagoras, Plato, and Kepler. An efficient method of distributing N angels at approximately equal intervals on a 2sphere, with an asymptotic formula for the error term when N tends to infinity, was discovered by Lubotzsky, Phillips, and Sarnak in 1987. Their construction depends crucially on the Ramanujan-Weil Conjectures that were established by Deligne [4]. Let me now fly to the sixth sphere, that of Jupiter, to illustrate the above discussion. First of all, the Red Spot in the Jovian atmosphere is a famous instance of the Hairy Ball Theorem in action. The Poincar6--Hopf Theorem then guarantees the existence of at least one other point of turbulence (although it may be less visible than the Red Spot because of a fainter color). On the other hand, the 12 major moons of Jupiter could easily be rearranged at equal distances at positions corresponding to the vertices of a regular icosahedron concen-
tric with the planet. Suppose now that the moons were as big as Jupiter itself and tangent to it. Interestingly, the 12 moons would not then be tangent to each other and could be moved freely. Indeed, any permutation of the 12 satellites could be achieved by rolling them around Jupiter. Thus, there are infinitely many ways to arrange 12 billiard balls of equal size around a central one. This fact gave rise to a famous debate between Isaac Newton and David Gregory in 1694. Gregory suggested that a 13th sphere of equal size could be added as a tangential outer sphere. Newton objected, and he was right. Some arguments supporting his view were advanced in the 19th century, but it was not until 1953 that Schfitte and van der Waerden supplied a rigorous proof. The optimal tangency number 12 is fondly called the kissing number of 2-spheres. The kissing number for nspheres is known in only three other cases: It is 6 for 1-spheres, 240 for 7-spheres, and 196560 for 23-spheres. In particular, the kissing number 240 is realized by 7spheres centered at the lattice points of Gosset's lattice E8 in 8-dimensional octonion space [5]. Of course, no one should be misled into believing that the 2-sphere is an ugly sphere because of the Hairy Ball Theorem. On the contrary, it is the unique complex sphere: the Riemann Sphere. In higher dimensions, none but the 6-sphere carries an almost complex structure. Also, the theory of spherical triangles is a marvel which is not easily generalizable for higher-dimensional spheres. The area of a spherical triangle with angles c~, fl, and "/is given by the simple formula c~ + fl + "y - ~r; hence, if its angles are rational multiples of 7r, so is its area. Yet Cheeger and Simons [3] have conjectured that for n > 3, there exist simplices in the n-sphere all of whose dihedral angles are rational multiples of ~r, but whose volume is not a rational multiple of 7r. H o w about flights of angels in higher-dimensional spheres? The Hairy Ball Theorem extends to any even dimension to confirm the presence of singularities in any continuous global vector field. On the other hand, it is a very easy exercise to produce at least one smooth nowhere vanishing tangent vector field for any odddimensional n-sphere. Frank Adams used K-theory to compute the precise number of everywhere linearly independent vector fields on the n-sphere. For the remaining celestial spheres, his formula yields 3 for n = 3, 1 for n = 5, 7 for n = 7, and 1 for n = 9 [13]. Hence, the 3-sphere and the 7-sphere are parallelizable: They admit a global framing. This is not surprising for the 3-sphere, for all orientable 3-manifolds are known to be parallelizable. Moreover, the 3-sphere can be identified with the Lie group SU(2) or the multiplicative group of unit quaternions, and all Lie groups are parallelizable. Apart from the 1-sphere, the 3-sphere is the only sphere which is also a Lie group; a fact which was already known to ]~he Cartan. The 7-sphere is parallelized by octonion multiplication, but as the set of octonions of unit norm, it falls short of being a multiplicative THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993
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group; indeed, A r m a n d Borel identified the 7-sphere as the h o m o g e n o u s space Spin(7)/G2. The 3-sphere and the 7-sphere thus seem to occupy places of h o n o r a m o n g the celestial spheres. Both of them can also be v i e w e d as H o p f fibrations: the 3-sphere fibers as a 1-sphere b u n d l e over the 2-sphere, and the 7-sphere fibers as a 3-sphere b u n d l e over the 4-sphere. The former fibration describes the magnetic monopole; sooner or later the latter is also b o u n d to harbor some physical meaning for superstring theory. The H o p f fibration of the 3-sphere has been illustrated in [1]. In 1959, Stephen Smale p r o v e d that for n = I and n = 2 the diffeomorphism g r o u p of the n-sphere has the homotopy type of O(n + 1). Allen Hatcher e x t e n d e d this result to the case n = 3 in 1983. For n = 4, the assertion is unsettled, and for n ~ 5, it is definitely false. The n u m b e r of distinct smooth structures on the 4-sphere is unknown. The orientation-preserving diffeomorphism g r o u p of the 5-sphere is k n o w n to be connected. John Milnor discovered in 1958 that the orientation-preserving diffeomorphism group of the 6-sphere breaks into 28 components. The orientation-preserving diffeomorphism groups of the remaining celestial spheres have c o m p o n e n t s that n u m b e r as follows: 2 for the 7-sphere, 8 for the 8-sphere, 6 for the Crystalline 9-sphere, and as m a n y as 992 for the Empyrean 10-sphere [14]. As a Finn, I find it amusing that the ancient mythologies of N o r t h e r n Europe also concur with Dante and Witten on the 1 + 9 structure of the Universe. To cite but two examples: In the Icelandic sagas there is a mythological p o e m called Vfguspa dating from about A.D. 1000 which quotes the existence of nine worlds; in the Finnish national epic, the Kalevala, there is a famous scene where a shaman m o t h e r m e t a m o r p h o s e s into a bee and flies to the ninth celestial sphere to seek r e m e d y for her badly w o u n d e d son Lemmink/iinen. In both myths, the Tree of Life plays the role of the unifying tenth object. Indeed, for the shaman, it is possible to climb the Tree of Life to reach the higher spheres unseen b y mortal eyes. Pythagoras must have been some kind of shaman, and the same characterization might well a p p l y to W i t t e n - unless he is a Martian as others have suggested [11]. Let me give the final w o r d to Dante w h o punctuated his journey t h r o u g h the heavenly spheres with the following strains: Qual ~ 'l geom~tra che tutto s' affige per misurar lo cerchio, e non ritrova, pensando, quel principio ond'elli indige, tal era io a quella vista nova: veder voleva come si convenne l'imago al cerchio e come vi s'indova; ma non eran da ci6 le proprie penne: se non chela mia mente fu percossa da un fulgore in che sua voglia venne. A l'alta fantasia qui manc6 possa; ma gi?~volgeva il mio disio e "I velle, sf come rota ch'igualmente ~ mossa, l'amor che move il sole e l'altre stelle. 26
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
An English translation is: As one, Who versed in geometric lore, would fain Measure the circle; and, though pondering long And deeply, that beginning, which he needs, Finds not: e'en such was I, intent to scan The novel wonder, and trace out the form, How to the circle fitted, and therein How placed: but the flight was not for my wing; Had not a flash darted athwart my mind, And, in the spleen, unfolded what it sought. Here vigour fail'd the towering fantasy: But yet the will roll'd onward, like a wheel In even motion, by the love impell'd, That moves the sun in heaven and all the stars.
References 1. M. Berger, Geometry I-II, Berlin-Heidelberg-New York: Springer-Verlag (1987). 2. L. Castellani, R. D'Auria, and P. Fr6, Supergravity and superstrings: a geometric perspective I-II, Singapore: World Scientific (1991). 3. J. Cheeger and J. Simons, Differential Characters and Geometric Invariants, Lecture Notes in Mathematics No. 1167, Berlin-Heidelberg-New York: Springer-Verlag (1985). 4. Y. Colin de Verdi6re, Distribution de points sur une sph6re (d'apr6s Lubotzsky, Phillips et Sarnak), S6minaire Bourbaki, expos6 703, Astdrisque 177-178 (1989), 83-93. 5. J. H. Conway, and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Berlin-Heidelberg-New York: SpringerVerlag (1988). 6. J. Dieudonn6, Pour l'honneur de l'esprit humain, Paris: Hachette (1987). 7. P. Duhem, Medieval Cosmology: Theories of Infinity, Place, Time, Void, and the Plurality of Worlds, Chicago-London: The University of Chicago Press (1985). 8. C.S. Gordon, When you can't hear the shape of a manifold, Mathematical Intelligencer 11(3) (1989), 39-47. 9. S. W. Hawking, The edge of spacetime, American Scientist 72 (1984), 355-359. 10. S. Hildebrandt and A. Tromba, Mathematics and Optimal Form, New York: Scientific American Books (1985). 11. J. Horgan, Edward Witten, the Pied Piper of superstrings, Scientific American (November 1991), 18-20. 12. M. Kaku, Strings, Conformal Fields, and Topology, BerlinHeidelberg-New York: Springer-Verlag (1991). 13. M. Karoubi, K-theory. An Introduction, Berlin-HeidelbergNew York: Springer-Verlag (1978). 14. J. Milnor, Remarks on infinite dimensional Lie groups, in Relativily, Groups, and Topology II, Les Houches, Session XL, B. J. DeWitt and R. Stora, eds., Amsterdam: NorthHolland (1986), pp. 1007-1057. 15. P. Oh and P. Ramond, Curvature of super Diff(S1)/S 1, Physics Letters B 195 (1987), 130-134. 16. E. Witten, Physics and geometry, in Proceedings of the International Congress of Mathematicians, Berkeley, A. M. Gleason, ed., Providence, RI: American Mathematical Society (1987), pp. 267-303. Department of Mathematics University of Jyviiskylii SF-40351 Jyviiskylfi, Finland (e-maih
[email protected])
Dealing with the Political Past of East German Mathematics Reinhard Siegmund-Schultze
The current difficult situation of East German mathematics moves me to look back at the first third of the history of mathematics in the German Democratic Republic (GDR). I will bring in some sources I which became accessible only recently. I will try to put the development of mathematics in the GDR against the background of East Germany's striving for international recognition and its deepening demarcation from the capitalist West. East German mathematicians sometimes found their interests conflicting with the "leaders of the party and the state" but sometimes coinciding, leading to political adaptation. The two German mathematical cultures gradually diverged at least since the end of the 1950s. No assessment of the (undoubtedly considerable) scientific achievements of East German mathematicians will be given in this article.
serve to survive. Political indoctrination of students and scholars, overcentralization, restriction of international communication, and a lack of exchange of personnel between universities which promoted personal and political nepotism are instances of untenable and counterproductive conditions within the East German system.3 On the other hand, the one-sidedness of the power constellation following the victory of the West i n the Cold War is not without problems. It results in proclaiming the West German science and university system as a n o r m - - numerous though its critics are in Germany and
3 In this respect, a report of the applied mathematician of Dresden, Kurt Reinschke, accepted as a resolution of the "Union of Democratic Scientists" (VdWi) on June 16, 1990, is, despite its emotional attitude, basically correct [2].
The A n x i o u s R e e x a m i n a t i o n The political "turn" of October 1989 in the former German Democratic Republic (GDR) had dramatic consequences for its scholars, among them mathematicians.2 The diverging development of the East and West German systems in the past 40 years had led to a real incompatibility of structures, and at the same time to a deep psychological divide between East and West German scholars. It goes without saying that much of the East German science system was not fit for survival and did not de-
1 These are the files of the former "Department of Science" ("Abteilung Wissenschaft') within the Zentralkomitee of the Socialist Unity Party (SED), which are located in Berlin (SAPMO, Zentrales Parteiarchiv), and the papers of the two leading East Berlin mathematicians Kurt Schr6der (NLKuSch) and Karl Schr6ter (NLKaSch) at the Archives of the former East German Academy of Sciences in Berlin. I have to thank the staff of these archives as well as the staff of the Library of Congress (Washington), the Courant Institute (New York), and the Harvard University Archives for support and permission to quote. Special thanks to Chandler Davis who kindly corrected my English. 2 Many of these problems were discussed at the first all-German convention of mathematicians after the reunification, Bielefeld, September 1991 [1]. THEMATHEMATICALINTELLIGENCERVOL,15, NO. 4 Q 1993Springer-VerlagNewYork 27
abroad 4 - a n d in mechanically t r a n s f o r m i n g the East G e r m a n system to follow West G e r m a n patterns. N e e d less to say, a l m o s t all decisive positions in the process of transforming East G e r m a n science, in particular, the so-called Science Council, are occupied b y West G e r m a n scholars. This has serious consequences for the h u m a n beings involved. East G e r m a n scholars h a v e to a p p l y on the basis of West G e r m a n reviews for their o w n f o r m e r positions, which m a y be advertised in all Germany. In s o m e East G e r m a n states, especially in MecklenburgV o r p o m m e r n (Mecklenburg-Pomerania) a n d Sachsen (Saxony), it is clear f r o m the outset that only a b o u t onehalf of the original positions will be filled .5 The two Germ a n university s y s t e m s had a v e r y different facultystudent ratio: In the East, there were m u c h fewer students per teacher, to the benefit of both. Younger scholars generally h a d p e r m a n e n t positions in the GDR. In general, their career prospects are r e d u c e d in the unified G e r m a n y (unless they are exceptionally talented): they do not satisfy West G e r m a n s t a n d a r d s in curricula vitae ( m e m b e r s h i p in political organizations, lack of international experience), and the old-boys n e t w o r k of their teachers is n o w ineffectual. Moreover, the East Germ a n science s y s t e m is threatened b y the e m i g r a t i o n of its outstanding mathematicians, as long as the living conditions in the East are m u c h worse than in the western part of Germany. Another incompatibility of the two s y s t e m s was the strong e m p h a s i s on research institutes of the f o r m e r East G e r m a n A c a d e m y of Sciences, following the Soviet example. A m o n g these institutes w a s the w e l l - k n o w n East Berlin Karl Weierstrat~ Institute for mathematics, with about 150 p e r m a n e n t collaborators. A l t h o u g h the prospects for f o r m e r a c a d e m y w o r k e r s in s o m e other sciences and the h u m a n i t i e s are bleak, m a t h e m a t i c i a n s of the A c a d e m y did rather well so far in the process of reconstruction. Meanwhile, an Institute for Applied Analysis and Stochastics with about 80 w o r k e r s has been f o u n d e d in Berlin, a n d s o m e other w o r k i n g g r o u p s of the Weierstrat~ Institute are t e m p o r a r i l y m a i n t a i n e d b y the Max Planck Society. A m o n g the reasons for this are the internationally recognized a c h i e v e m e n t s of the Weierstrat~ Institute 6 (reflected in the " e v a l u a t i o n " b y the Science Council) and the apolitical character of m a t h e m a t i c a l results. As for East G e r m a n universities, m a t h e m a t i c s m a y again fare relatively well just because in m a t h e m a t i c s it should be
4 These problems are discussed in detail in [3]. 5 In mathematics, at this moment the situation for former professors of the six East German universities is much more promising in Berlin, Greifswald, and Jena than in Leipzig, Rostock, and Halle. Even less clear is the situation of the younger generation. 6 It seems to have been a technology gap in the former Federal Republic in the mathematical theory of circuit design which made the results of the Weierstrat~ Institute especially impressive to the Science Council. 28
THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4, 1993
m u c h h a r d e r than in s o m e h u m a n i t i e s to fill the v a c a t e d positions with second-rate West G e r m a n scholars. If, as it seems so far, m a t h e m a t i c s departs s o m e w h a t f r o m the overall pattern of a d a p t a t i o n to the West Germ a n system, this exceptional position could tend to restore the ideal of "apolitical" science, p r o m o t i n g selfd e c e p t i o n of East G e r m a n m a t h e m a t i c i a n s on their role in the history of East Germany. I h a v e b e e n describing consequences of G e r m a n unification which affect all m a t h e m a t i c i a n s of East G e r m a n y , w h e t h e r s y m p a t h i z e r s or a d v e r s a r i e s of the old system. 7 In mathematics, however, as in other sciences, the struggle for n e w careers is tied to settling political a n d ideological scores with the defeated regime. N o w , it is just, of course, to p u t those scientists last w h o u s e d political m e a n s in East G e r m a n y to h a r m others for selfish reasons. 8 It w o u l d s e e m unwarrantable, however, to blacklist entire g r o u p s of citizens on the g r o u n d s of formal political criteria o v e r r i d i n g the principle of the p r e s u m p t i o n of innocence. A m o n g n u m e r o u s difficult j u d g m e n t s which arise in the evaluation of political behavior of scholars in the GDR, let m e just m e n t i o n the following: H o w is one to distinguish deliberate political n o n c o o p e r a t i o n with the r e g i m e f r o m an essentially apolitical attitude and f r o m lack of willingness to take on t i m e - c o n s u m i n g administrative work? In this process of settling political a n d ideological scores, w h i c h unfortunately greatly overlaps w i t h personal interests, historical a r g u m e n t a t i o n s are e v i d e n t l y critical. A p p a r e n t l y o w i n g to ignorance of history, d o u b t ful analogies are used. S o m e formerly discriminated G D R m a t h e m a t i c i a n s [4] as well as s o m e West Germ a n m a t h e m a t i c i a n s c o m e f o r w a r d with rather superficial c o m p a r i s o n s b e t w e e n the situation of m a t h e m a t i c s in N a z i G e r m a n y and in the GDR. 9 This contributes to m o r e general ideological c a m p a i g n s , which try to shift all responsibility for undesirable d e v e l o p m e n t s in Ger-
7 A cap on pensions of 2,010 Deutsche Mark has been introduced. Retired colleagues in West Germany often get three times as much. 8 A particularly grave instance of an East German mathematician abusing his power to the disadvantage of the woman mathematician B. Schultz-Heinecke is reported by the Munich mathematician W. Heise in a circular letter to members of the German Mathematicians Association of December 1990. At the former Weierstral~ Institute of the Academy, reports by H. Koch and E. W. Zink were given concerning the blocking of the careers of Th. Zink and E. Kirchberg. 91 am not opposed in principle to such comparisons. I just think they require a larger historical context. This includes the relation of "reason and power" in the German "authority state" ("Untertanenstaat') over several centuries. A broad examination of this kind will look critically at anticommunist policies in the West as well. Moreover,one has to take into account the fundamental social differences between the Nazi and GDR systems, the incomparable criminal character of the Hitler regime, and the much more supportive attitude of the Marxist ideology to basic science, which sometimes even amounted to uncritical scientism. As to the "coordination" of mathematics in Nazi Germany, see [5]. For a comparative discussion of the political systems of Nazi Germany and the GDR, see [6] and [7].
man history to East Germany. Mysticizing the Stasi, the East German secret service, contributes to a mostly unintended downplaying of Nazi crimes [7]. West Germans divert attention from blots on their own history, for example, the infamous career-bans (Berufsverbote) of the 1970s. 1~Some West Germans, who did not witness either the Nazi regime or the GDR system, are utterly unable to appreciate their social constraints. On the other hand, the overwhelming majority of loyal GDR-citizens (this author was one of them), feeling themselves treated unjustly and with ignorance, may be tempted by an inappropriate and uncritical "GDRnostalgia." It is time for the historian to introduce some calm. I have neither the space nor the historical distance to give the full story. I will try to outline the first third of the political history of mathematics in the GDR, to name some problems, and to point to some sources for a thorough discussion of these problems.
How Mathematics Was Organized in the GDR After the erection of the Berlin Wall in August 1961, the situation of mathematics in the GDR changed considerably. There was an increase in international isolation, which was partly caused by the West) 1 Bureaucratization and ossification were growing, the infamous Third University Reform of 1968 being a turning point which was hardly welcomed by anybody at least of the older generation of scholars. 12 Gradually, a younger generation of scholars, educated in the GDR and bound to severe party discipline, took over all decisive positions of the science system. The erection of the Berlin Wall, which provided shortand medium-term economic relief, turned out to be a long-term political trap for the East even with regard to science. 13 10 The consequences of the Berufsverbote for academic life in the Federal Republic are described in a 48-page unpublished German report by the mathematician B. Booi~ (Bielefeld/Roskilde), 1978. The title translates to "Academic life in West Germany: 1.3 million 'examinations' and 4000 career-bans could not suppress freedom of opinion." 11 In a letter to H. Brown (National Academy of Science) of November 4, 1965, the New York mathematician Richard Courant, who had been expelled from Nazi Germany, reported on his participation in the East Berlin celebration of Weierstra~'s 150th birthday (RCP1). Courant wrote that the isolation of East Germany stemming from the erection of the Wall was considerably exacerbated by the policy of the Allied "Travel Office" in West Berlin. This office even prevented a trip of the East German Nobel prize-winner Gustav Hertz to Kopenhagen, which Hertz planned to undertake as a representative of the East German Academy. 12 Immediately after the "Mathematikbeschlul~" of 1962, to be mentioned below, there came an initiative of the East German Ministry for Higher Education (Staatsekretariat). Deliberately putting up with resistance of the competent mathematicians, the formation of "leading institutes" was decreed by the ministry; more abstract fields, such as number theory, were by and large banned from the university system (SAPMO1, fol. 97-213).
My remarks here will deal mostly with the period prior to 1961. The starting situation of mathematics in Germany after World War II is known, albeit only roughly. A considerable part of the leading German mathematicians as of 1933 (Weyl, Artin, Siegel, Courant, Schur, Landau, von Mises, Dehn, etc.) had been expelled. Only very few of them (Siegel, Artin, Hamburger) returned after the war. Although most of the mathematicians persecuted under the Nazis were able to emigrate, some of them were murdered or driven to suicide (O. Blumenthal, F. Hartogs, E Hausdorff, R. Remak, A. Tauber, and others). Some very promising young mathematicians (O. Teichm/.iller, G. Gentzen) lost their lives during the war or in its aftermath. The generation of German mathematicians educated between 1933 and 1945 is very weak, at least in numbers. This was partly a result of the decline in enrollments in mathematics in this period. During the last months of the war, numerous mathematicians fled the eastern part of Germany, which presumably was to be occupied by Soviet forces. Important libraries were moved to the West as well. A particular point of attraction was Oberwolfach in the Black Forest in the southwest of Germany, where in September 1944 an Imperial Institute of Mathematics (Reichsinstitut f~ir Mathematik) had been founded while the Nazis were still in power. The occupation of Germany by the Allies entailed a second wave of emigration, primarily to the United States (E. Hopf, A. Busemann, etc.), a n d - - t o a smaller degree and as part of the reparations of the E a s t - - t o the Soviet Union (H. Reichardt). The mathematicians remaining in Germany (H. Behnke, F. K. Schmidt, H. L. Schmid, E. Schmidt, E. Kamke, etc.) made many efforts to rescue the existing potential. This led to early EastWest German contacts, especially in connection with the reappearance of German mathematical journals and the Zentralblatt fiir MathematikJ 4 The so-called denazification after the war led, of course, to some problems in reviving German mathematical research and teaching. In the first years after the war, former Nazi party members could not get university jobs--which were more essential for mathematicians than in some other, more applied sciences. After the handing over of denazification to German authorities, almost all university teachers returned to their positions. (An exception was Ludwig Bieberbach, who had compromised himself too much as a former ardent Nazi.) This applied also to East Germany, which, in addition,
13 In the letter to H. Brown (RCP1) mentioned above, Courant called the Wall a "repugnant symbol (notwithstanding its possible justification as the brutal way of stopping the previous mass defection of skilled technicians and intellectuals)." 14 In November 1947, the mathematician H. Behnke of Miinster (West Germany) took a trip to East Berlin, following an invitation of the Academy (OVP). In East Berlin, the reappearance of the Zentralblatt was prepared, which worked as an all-German enterprise until 1977. THEMATHEMATICALINTELLIGENCERVOL.15,NO. 4,1993 2 9
faced losses of p e r s o n n e l in the last m o n t h s of the w a r a n d a s t e a d y loss of m a n p o w e r thereafter. Offsetting this d i s a d v a n t a g e w a s the m u c h m o r e systematic s u p p o r t of science a n d culture in the Soviet Occ u p a t i o n Z o n e (SBZ). In the three w e s t e r n zones, cultural a u t h o r i t y w a s g r a d u a l l y restored to the states as before 1933. Bureaucratic w a s t a g e in this process w a s considerable. In the SBZ, h o w e v e r , a centrally directed university and academy system was introduced, which w a s controlled b y the Soviet Military A d m i n i s t r a t i o n (SMAD) in B e r l i n - K a r l s h o r s t until the f o u n d i n g of the G D R in 1949. P e r h a p s , relatively fewer of the m a t h e m a t i cians in East G e r m a n y than in West G e r m a n y h a d c o m p r o m i s e d t h e m s e l v e s b y collaboration w i t h the Nazis. ( A n t i - c o m m u n i s m a n d anti-Sovietism, w h i c h s h a p e d the C o l d War on the Western side, h a d b e e n constituents of the N a z i i d e o l o g y as well.) H o w e v e r , e v e n political implication g o i n g b e y o n d formal ties to N a z i o r g a n i z a tions w a s not a n a b s o l u t e obstacle to a career in East G e r m a n y . This a p p l i e s especially to e x c e p t i o n a l m a t h e maticians, w h o s e c o m p e t e n c e East G e r m a n authorities w e r e eager to secure. 15 E v e n s o m e politically c o n t a m i nated, s e c o n d - r a t e m a t h e m a t i c i a n s g o t their chance, prov i d e d t h e y s i d e d o s t e n t a t i o u s l y w i t h the n e w political system. 16 Since the m a j o r i t y of G e r m a n scientists a n d
15Particularly grotesque was the appointment of the politically incriminated, renowned number theorist H. Hasse in East Berlin. The communist German science functionaries recommended Hasse in gradually more emphatic letters to the SMAD. The last of these letters of 1948 even says "Moreover, it is likely that Hasse will promote the democratic education of students." (AHUB UK H 134.) 16Max Draeger, author of the article "Mathematik und Rasse" of 1941 in Bieberbach's journal Deutsche Mathematik, became full professor in Potsdam. It is not clear from the files (AHUB, Institute ffir Mathematische Logik, allgemeiner Schriftverkehr 1956/57, Nr. 212), though, whether his Nazi activities were known to the East German officials. At any rate, Draeger had become a member of the SED and was able to play a political role even some time after a protest of the logician Karl Schr6ter.
Berlin University in 1946. 30 THEMATHEMATICALINTELLIGENCER VOL.15,NO.4, 1993
m a t h e m a t i c i a n s did not e x p o s e t h e m s e l v e s politically d u r i n g the N a z i years [8], t h e y d i d not n e e d to be exc l u d e d b y the n e w system; the m o r e so, b e c a u s e the p a r t y w a s v e r y m u c h inclined to p r e s u m e science to be "in itself h u m a n e a n d progressive." L e a d i n g m a t h e m a t i c i a n s of the first y e a r s of the G D R , s u c h as O. H. Keller, N. J. L e h m a n n , H. L. S c h m i d , E. S c h m i d t , K. Schr6der, K. Schr6ter, H. Schubert, a n d H. Willers, w e r e n o t p u t u n d e r p r e s s u r e to join the "Socialist U n i t y P a r t y " (SED). 17 N o d o u b t s o m e of these m a t h e m a t i c i a n s r e c o n s i d e r e d their p r e v i o u s political p o sitions, d e p e n d i n g on the s u p p o r t t h e y g o t for their disciplines u n d e r the n e w system. In addition, s o m e p o litical e v e n t s in the West, especially the witch h u n t for dissidents in the M c C a r t h y era of the U n i t e d States [10] a n d the political restoration a n d remilitarization in the F e d e r a l Republic, repelled East G e r m a n m a t h e m a t i c i a n s a n d w e r e u s e d b y the eastern p r o p a g a n d a to conjure u p m e m o r i e s of the N a z i past. 18 In s o m e cases, h o w e v e r , p o litical doctrines (e.g., the religious intolerance of the first d e c a d e s , w h i c h inevitably a n t a g o n i z e d d e v o u t m e n s u c h as O. H. Keller of Halle) m a d e a r e a s o n a b l e collaboration of scientists a n d the state impossible. A l t h o u g h g o v e r n m e n t s u p p o r t for m a t h e m a t i c s in g e n e r a l w a s s t r o n g in East G e r m a n y , as will be s h o w n later, it w a s a l w a y s a m b i g u o u s a n d regionally u n e v e n (with clear preference g i v e n to Berlin). F r o m the outset, political c o n t r o l w a s felt e v e n in m a t h e m a t i c s . Working-class c h i l d r e n got preferential access to the universities. As e a r l y as 1949, H a s s e called for a "talents clause" (Begabtenklausel), g i v i n g the ext r a o r d i n a r i l y gifted easier access to the u n i v e r s i t y reg a r d l e s s of political prerequisites (AHUB, Math.-Nat. D e k a n a t , Nr.1, fol. 91/92). S o m e elder m a t h e m a t i c i a n s
17There was a clear parallel in this respect in Nazi Germany. Kurt Schr6der, for instance, joined the Nazi party in 1940 to get a position at the university, whereas established scientists were not compelled to do so [9, p. 61]. In the GDR, however, now being a prominent scientist himself, Schr6der got the highest awards and positions (among them the rectorate of Humboldt University) without party membership. Younger scholars, on the other hand, certainly promoted their careers by joining the party if they did not do it anyway for honest political convictions. is Richard von Mises's declining to become a member of the Berlin Academy was explained as a result of the suppression of free opinion in the United States, especially because von Mises's letter of September 1950 can be read this way (NLKusch 508). It does not come as a surprise that von Mises took a much clearer stand against the East Berlin Academy when explaining his refusal to American colleagues (RvMP). Von Mises (as well as John von Neumann) had been informed by the geneticist H. J. Muller of the political control over the Academy, especially in the case of the expelled geneticist H. Nachtsheim [11] and in a servile telegram of the Academy on the occasion of Stalin's seventieth birthday. The persecution of MIT professor D. J. Struik during the McCarthy era became an issue of East German propaganda as well. In a resolution of the presiding committee of the Berlin Academy of May 6, 1954, it is stated with regret "that Prof. Struik faces troubles in the United States which unequivocally remind us in Germany of similar events during the Hitler era." (AdWL 159, addendum IV, fol. 39.)
Humboldt University in Berlin.
saw with regret the gradual curtailment of academic self-government. This problem was discussed in 1950 at H u m b o l d t University on the occasion of the reduction of pure mathematical lectures in favor of pedagogical and philosophical lectures (AHUB Rektorat 165, foi.149). The mathematician Rudolf Kochend6rffer, w h o as a dean in Rostock was quite willing to cooperate with the government (he even was temporarily a m e m b e r of the SED), dared in 1951 to dispute the interpretation of dogmas such as the "experiences of the Soviet Union" or the "superiority of socialist democracy." Discouraged, he finally left the GDR for the Federal Republic [12]. Especially in connection with strong secrecy regulations, the bureaucratization of the East G e r m a n science and university system was a very early obstacle to international communication. 19 19Typical of this is an instruction given by the East German Ministry for Higher Education on June 8, 1953,demanding an increase in the number of guest lecturers from West Germany. At the same time, however, this instruction calls for numerous bureaucratic and political precautions, among them "a short review of the scientificand political activity of the scientist to be invited" (AHUB,Math.-Nat. Fak. 14, fol. 55).
On the other hand, however, there was the foundation of institutes such as the Research Institute for Mathematics at the East G e r m a n A c a d e m y in 1946 and the Institute for Mechanical C o m p u t i n g Techniques in Dresden in 1956. A systematic p r o g r a m of translating Soviet textbooks [13] as well as visits of Soviet guest lecturers to East G e r m a n y had a considerable impact on mathematics in the GDR. The obvious importance of Soviet mathematics could not fail to influence even the political positions at least of y o u n g e r mathematicians. The promotion of mathematical logic in the 1950s encouraged the leading East G e r m a n logiciar~ Karl Schr6ter (19051977) to initiate a "call for a competition in h o n o r of the Fifth Party Congress," although Schr6ter himself was not a m e m b e r of the SED (NLKaSch, 138). This is to say there were sources in mathematics itself to p r o m o t e a certain GDR patriotism even on the part of some elder mathematicians. State restriction of international communication, referred to above, affected y o u n g e r mathematicians rather than the leading ones. Kurt Schr6der called for the participation of y o u n g e r mathematicians at a conference of the Gesellschaft f~ir A n g e w a n d t e Mathematik u n d THEMATHEMATICAL INTELLIGENCERVOL.15,NO.4,1993 31
A ceremony of the Young Pioneers on the cover of a mathematics textbook (Source: Mathematik-Lehrbuch fiir die Klasse 3, Berlin (Ost): Deutscher Verlag der Wissenschaften (1975).
Mechanik (GAMM) in H a m b u r g in 1957, but his objections were rejected with a typical authoritarian argument: allegedly the "reputation of the A c a d e m y " w o u l d suffer if younger, scientifically less p r o m i n e n t mathematicians took part (AdWL 708, part 1, fol. 32). The traveling privileges of East Berlin mathematicians, especially of some members of the Academy, u n d o u b t e d l y contributed to their political adaptation. At times, the high salaries for elder scientists, w h o m the political leaders wanted to retain in East G e r m a n y at all costs, antagonized y o u n g e r scholars (SAPMO2, fol. 36). Although the SED leaders handled carefully the prominent mathematicians w h o had been educated scientifically and politically before the existence of the GDR, they m a d e d e t e r m i n e d efforts to raise an intelligentsia of their own, which was "faithfully d e v o t e d " (treu ergeben) to the party. Victims of Fascism, in particular communist emigrants to the Soviet Union, were the most influential a m o n g party functionaries. It would be gratuitous and unhistorical to d o u b t the honest aspirations of m a n y of these functionaries. T h e y intended to build a totally new Germany, unable b y its social structure to repeat crimes 32
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993
like the Nazis'. Indisputably they had resisted the Hitler r e g i m e - - u n l i k e most of the G e r m a n people. However, these p a r t y functionaries inferred the right to decide for the p e o p l e even in the future a n d to realize the "historical laws" without democratic l e g i t i m a t i o n - - i f necessary, b y dictatorial means. This included the "political education" of the people, which had to be p e r f o r m e d in the sciences as well. 2~ The "Department of Science" ("Abteilung Wissenschaft') within the Central Committee of the SED, which was the administrative center of all East G e r m a n science, considered it an obstacle to political education in the field of mathematics that in 1958 only 5 professors of mathematics (out of a b o u t 40) were m e m b e r s of the SED (SAPMO2, fol. 51). In the same report, however, the encouraging fact was mentioned that 27 out of 121 y o u n g e r mathematicians were party members. Already in the early 1950s, the Department of Science had consulted y o u n g and scientifically promising party m e m b e r s on a regular basis. More than "socialist education" was on the agenda of these consultations. As a matter of fact, this collaboration with y o u n g party m e m b e r s played a role in the foundation of the computing institute in Dresden in 1956 and in the process of revitalization of probability and statistics. M a n y of the y o u n g scientists involved were convinced of a close link b e t w e e n their "progressive Weltanschauung" and their progressive u n d e r s t a n d i n g of mathematics. 2t Thus, it w o u l d be simplistic to accuse these y o u n g scientists of dishonestly taking advantage of the political situation. The convictions of these y o u n g party m e m b e r s were also m u c h shaped by the authority of Soviet mathematics and b y the influence of Soviet guest lecturers. (But not in the sense of the short-lived "idealism debate" in m a t h e m a t ics, the aftereffects of which reached the GDR in some articles of the beginning 1950s. 22) To charge the D e p a r t m e n t of Science with "hostility to science" is equally dubious, at least in mathematics. In the files of this department, which are n o w accessible, m a n y statements can be found against shortsighted utilitarian policies in mathematics. It was also in the office of the D e p a r t m e n t of Science that the 20Also school books of mathematics, however good they might have been as to mathematical substance, reflected this educational goal. This includes assignments fostering attachment to the Soviet Union, to the "Young Pioneers" organization, or preparing for the military service. 21Typicalis the followingjudgment which was made by some of these young mathematicians on a consultation with the "Department of Science" in February 1956: "There is an Academy institute for statistics which is very weak politically.It is headed by Professor Lorenz. Lorenz is no theorist, is opposed to theory, applies obsolete methods." (SAPMO4, fol. 68.) In this quotation, the scientificassessment, which was certainly not unjustified, was put into intimate connection with the political judgment. 22Such a late-corneris [14].However, [15]is rather a defenseof Hilbert's axiomatics against attacks of vulgarized "Marxist" philosophy.On the problem of political interferenceinto mathematics in the Soviet Union, see [16].
Mathematics-Decision (Mathematikbeschlut~) of D e c e m b e r 1962 w a s p r e p a r e d in its basic features. 23 This decision b r o u g h t a m o n g other things a r e f o r m of m a t h e matical high-school education and the internationally successful o l y m p i a d of y o u n g mathematicians. 24 The Mathematikbeschlut~ w a s cheerfully w e l c o m e d b y m a n y mathematicians of the older generation too because it p r o m i s e d to considerably i m p r o v e the social standing of their science in the GDR. A n o t h e r historical fact m a d e it easier for s o m e older m a t h e m a t i c i a n s to c o m p l y w i t h the East Germ a n science policy of the 1950s. In spite of all rhetoric of differentiation f r o m the West a n d the Federal Republic, the G D R m a i n t a i n e d the goal of national unity. This coincided with Soviet policies of that time. (Think of Stalin's p r o p o s a l to f o r m a neutral, unified G e r m a n y , which the West G e r m a n chancellor, K o n r a d Adenauer, hastened to reject.) As late as 1955, in guidelines given b y the D e p a r t m e n t of Science, one reads, "All c o m r a d e s - m a t h e m a t i c i a n s o u g h t to be m e m b e r s of the G e r m a n Mathematical Association immediately." (SAPMO2, fol. 12.) 23Preparation, text, and consequences of this obviously unpublished resolution in SAPMO1. 24The organization of the olympiads, again, followed the example of the Soviet Union [18]. Pupils were involved as early as the age of 11, whereas in West Germany, a comparable competition did not start until the age of 17. Today,this causes trouble for the future of olympiads in the unified Germany [17[. In a letter of November 15, 1965, to M. E. Rose of the National Science Foundation, R. Courant wrote of East Berlin: "My impression was confirmed that secondary education in mathematics is on a very high level, apparently better than in West Germany and strongly influenced by the Soviet models. This can also be seen from the excellent mathematics textbooks in East Germany."
First issue of the popular mathematics journal for schoolchildren (1967) which contributed to the East German success in Olympiads.
Military education for third graders (Source: M a t h e m a t i k - Lehrbuch fiir die Klasse 3, Berlin (OstA: Deutscher verlag tier Wissenschaften (1987), p. 40). THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
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At the Euler celebration, East Berlin, 1957. From left to right: R. Nevanlinna, W. Hartke (Academy President), K. Reidemeister (West Germany).
In the "apolitical" field of mathematics, it was easier to pursue an all-German policy. Moreover, the buildings of the University of Berlin and of the Academy, founded by Leibniz, were located entirely on the territory of East Berlin--further incentive for both German sides to keep communication going in the name of historical tradition. Mathematicians such as the East German Erhard Schmidt (1876-1959) considered those institutions as well as the German Mathematical Association (DMV) as "clamps" which could hold the German culture together [19, p. 9]. On this point, the convictions of many East Germans coincided with those of West Germans such as E. Kamke (1890-1961). The GDR leaders emphasized the idea of national unity by founding in 1949 the well-endowed Nationalpreis and awarding it to East and West German scientists. 25 Paradoxically, political confrontation between East and West Berlin did not destroy East-West German communication. As a matter of fact, the foundation in 1948 of the Free University in West Berlin, in connection with the Berlin Blockade, was considered by many West Germans as an unpleasant threat to the collaboration with East Germans which was well under way at that time. (To be sure, resentments against the antitraditional constitution of the Free University, allowing student participation from the beginning, may have contributed to these 25 In this connection, once again, the East German government acted very pragmatically and unideologically. This was demonstrated by the awarding of the 1954 Nationalpreis to W. Blaschke, who had unduly collaborated with the Nazis. In hindsight, Blaschke commented ironically on the paradox that "curiously enough a communist state wanted me to become a millionaire" [20, p. 3].
34 THEMATHEMATICALINTELLIGENCERVOL.15,NO. 4, 1993
feelings.) Another historical accident was more specific to mathematics: The former leading Nazi among mathematicians, Ludwig Bieberbach, was living in West Berlin. Repeatedly, rumors were spread to the effect that Bieberbach would get a position at the Free University. This could not fail to infuriate victims of the Nazi regime such as the West German Kamke. 26 Until the end of the 1950s, East German mathematicians such as Kurt Schr6der stayed in the presiding committee of the DMV. The DMV remained the mathematical association for all Germans until 1962. On the other hand, the overall political confrontation between West and East was not without impact on mathematics policies. Partly as a response to the remilitarization in the Federal Republic, the Department of Science tried to promote the applications of mathematics to the military and to use the experiences of the older generation of mathematicians from World War II (SAPMO2, fol. 17). Finally, up to the end of the 1950s, the aspiration for international recognition of the GDR had priority. It was primarily for this reason that the Department of Science promoted the foundation of a separate East German mathematical society. In the files of the department is a 26 A correspondence between E. Kamke and A. Dinghas of December 1950 testifies to these frictions between West German and West Berlin mathematicians (NLKuSch 256). Dinghas had left the University of East Berlin in 1948 and became a professor of the Free University, at which East Berlin mathematicians like E. Schmidt took offense. Kamke in his letter criticized Dinghas's concern for Bieberbach. Dinghas, however, complained about the participation of West German mathematicians in the celebration of the 250th anniversary of, as he called it, "the Academy of Sciences of the Eastern Zone."
first note of February 1959, calling for the foundation of such an association to "completely exploit all research capacities," to "enrich scientific life," and to "build socialist consciousness" (SAPMO2, fol. 69). In the foundation of the Mathematical Society of the GDR (MGDDR), the scientifically renowned Berlin aerodynamicist Kurt Schr6der (1909-1978) took a prominent part. It would be unhistorical, however, to denounce Schr6der as merely a henchman of the SED. By the end of the 1950s, in the sessions of the presiding committee of the DMV one feels a gradual divergence of interests of East and West German mathematicians. This became particularly striking in 1959, when West German mathematicians discussed founding a research institute following the example of the IAS in Princeton. A substantial argument in favor of this plan was the West German mathematicians' heavy teaching load. East German mathematicians, of course, had no such problem. Moreover, Schr6der pointed to the existence of a research institute in East Berlin when the matter was discussed at a convention of the DMV leaders in M~inster 1959 (NLKuSch 256). Even if he had wanted to, Schr6der could not possibly have resisted the aspirations of the GDR leaders for international recognition. As a matter of fact, participation in international conferences was totally dependent on political decisions. The International Congress of Mathematicians in Edinburgh in 1958 proved to be a turning point in East-West German mathematical relations. For the first time, GDR mathematicians enrolled separately as members of a "National Committee of the GDR." (One of the more immediate reasons for this was the impending vote on the admission of Taiwan to the International Mathematical Union.) Increasing political demarcation of both German states resulted in a decline in numbers of West Germans attending DMV meetings on the territory of the GDR. A memoir of the Department of Science of March 1962 reports that junior West German mathematicians "refrain from all actions which might endanger studies in the United States, in particular, visits to our republic" (SAPMO2, fol. 165.) The foundation of an independent East German mathematical society has also to be seen in the context of the imminent "Mathematikbeschlut~" of December 1962 with its consequences for school and university instruction in mathematics. Moreover, there were some signs of bureaucratic ossification in the West German university system of that time, which reinforced the belief of leading East German mathematicians that they had the more progressive system, more favorable to mathematics. 27 The Mathematical Society of the GDR (MGDDR) was finally founded in June 1962. There were some undeniable advantages over the DMV: contact to applied mathematics, strong involvement of schoolteachers. These advantages were mentioned and recognized even after the political events in 1989, on the occasion of the unification
of the two mathematical societies. 28 Nevertheless, a majority of leading East German mathematicians in 1962 agreed only reluctantly to the foundation of the MGDDR. 29 This has to be seen against the background of the new political conditions. The Berlin Wall had been erected in 1961; the foundation of the MGDDR must have appeared as an aggravation of the East-West German divide and an impediment to international communication. Still, East German mathematicians had no choice but to make the best of the new circumstances. Even a declared political dissident like the mathematician from Halle, O. H. Keller, had to bow to the pressure of the East German ministry of education (Staatssekretariat f/Jr Hochschulwesen). In a letter to the DMV chairman W. Haack, Keller stood up for the foundation of the GDR National Committee for Mathematics and maintained that "the National Committee's attitude is far from unduly linking politics and science" (AdWL 531, fol. 7). The former GDR representative in the DMV, Kurt Schr6der, had to accept an editorial distortion of a talk he gave in 1969: the published version [21] contained a strong political attack on the DMV which was not in the much more comprehensive manuscript. 3~ One sees from these examples, as from the whole prehistory of the foundation of the Mathematical Society of the GDR, the priority of political conditions, and, at the same time, the pressure on the mathematicians of their social bonds and the concomitant material and moral commitments. One may wonder whether West German mathematicians would have acted otherwise than the leading East Germans around 1960 if it had been their country which was still striving for international recognition.
27 In a indirect report to the Department of Science, a meeting of the East German H. Grell and the American M. H. Stone of September 1961 in Prague is mentioned. According to this source, which has to be judged cautiously (SAPMO5, fol. 14/15), Stone called for a gradual establishment of equal rights for GDR mathematics in all international organizations. Stone is reported to have had the impression of a decline in West German mathematics. About the same time, R. Courant, in a review that disappointed West German mathematicians, opposed the foundation of a research institute for mathematics. Courant criticized harshly a bureaucratic ossification in West German science. He was especially critical of the insecure position of young West German mathematicians and of tendencies towards a separation of mathematics from its applied fields. On the West Germans' lack of sensibility to their political past, Courant commented "It was the Nazis and the compromising scientists who ruined mathematics in Germany almost beyond hope. It is infuriating that this fundamental fact is not mentioned in an application which is concerned with the role of science in Germany on an international scale" (RCP2). 2s The first-mentioned advantage is conceded in [4, p. 59]. 29 This reluctance is also reported in an official memoir of the Department of Science of early 1962: "Almost all of the professors stick to the illusion of an all- German unified science" (SAPMO2, fol. 155). 3o NLKuSch 160. This detail, typical of the political constraints in the GDR, was certainly unknown to Kiihnau when he [4, p. 58/59] polemicized against [21]. THEMATHEMATICALINTELLIGENCERVOL.15,NO.4, 1993 35
Conclusions
References
Incomplete though my remarks may be, I hope I have been able to show the following: Looking at the early political history of mathematics in the GDR, it is pointless to differentiate between "good mathematicians" and "bad politicians." Mathematicians sometimes quite unexpectedly find that circumstances compel them to act politically to make mathematics. Their moral judgments, therefore, have always to consider both domains, mathematics and society as a whole. A reliable picture of the history of mathematics in East Germany has to combine domestic political constraints, international relations, and, last but not least, the individual and collective interests of mathematicians. The political history of mathematics in the G D R - yet to be w r i t t e n - - w i l l be a new occasion to discuss the political and social responsibility of mathematicians in the modern society.
1. W. Scharlau, Ansprache des Vorsitzenden der Deutschen Mathematiker-Vereinigung Prof. Dr. Winfried Scharlau; Mitteilungen der DMV 1991, Heft 4, 135-141. 2. K. Reinschke, Vergangenheitsbew/iltigung an den Hochschulen und der Akademie der Wissenschaften der DDR, MUT 277 (1990), 33-37. 3. "Bald knallt's," Der Spiegel 1991, Heft 30, 36-59. 4. R. K/ihnau, Zur Situation der Mathematik und der Mathematiker in der ehemaligen DDR, Mitteilungen der DMV 1992, Heft 2, 57-63. 5. H. Mehrtens, The Gleichschaltung of mathematical societies in Nazi Germany, Mathematical Intelligencer 11 (1989), No. 3, 48-60. 6. E. J/ickel, Die doppelte Vergangenheit, Der Spiegel 1991, Heft 52, 39-43. 7. H. Obenaus, Stasi k o m m t - - Nazi geht? , Die Zeit1992, Heft 32, Uberseeausgabe, S. 16 8. R. Siegmund-Schultze, Mathematics and ideology in Fascist Germany, in World View and Scientific Discipline Formation, W. R. Woodward and R. S. Cohen, eds., Dordrecht/Boston: Kluwer (1991), pp. 89-95. 9. R. Siegmund-Schultze, Zur Sozialgeschichte der Mathematik an der Berliner Universit/it im Faschismus, N T M - Schriftenreihe 26 (1989), Heft 1, 49-68. 10. Ch. Davis, The Purge, in A Century of Mathematics in America, Part I, P. Duren, ed., Providence RI: American Mathematical Society (1988), pp. 413-428. 11. H. Nachtsheim, For a new Academy, Science 113 (1951), 30-31. 12. Geschichte der Universitfit Rostock1419-1969, Band lI (19451969), Rostock, 1969, pp. 116-118. 13. L. Boll, Die Bedeutung der russischen Obersetzungen in der Mathematik der DDR, Mitteilungen Math. Gesellschafi DDR (1977) H. 2/3. 14. A. D. Aleksandrov, Uber den Idealismus in der Mathematik; Forum, Wissenschaftliche Beilage, 29 November 1952, S. 3-16, and 6 Dezember 1952, S. 3-7. 15. B. W. Gnedenko and L. Kaloujnine, Uber den Kampf zwischen dem Materialismus und dem Idealismus in der Mathematik, Wissenschaftliche Zeitschrift der TH Dresden 3 (1953/54), Heft 5, 631-638. 16. A. Shields, Years Ago, Mathematical Intelligencer 10 (1988), No. 3, 7-11; 11 (1989), No. 2, 5-8. 17. D. Rehbinder, Quo vadis, Mathe-Olympiade?, Neues Deutschland January 28, 1993, p. 10. 18. A. M. Vershik, O. Ya. Viro, and L. A. Boku{,To Guard the Future of Soviet Mathematics, Mathematical Intelligencer 14 (1992), No. 1, 12-15. 19. Ansprachen anl~it~lich der Feier des 75. Geburtstages von Erhard Schmidt durch seine Fachgenossen (mit Erwiderungen Schmidts); Berlin, printed manuscript 1951. 20. W. Blaschke, Reden und Reisen eines Geometers, Berlin (Ost) 1957. 21. K. Schr6der, Zwanzig Jahre Entwicklung der Mathematik in der Deutschen Demokratischen Republik, Mitteilungen Math. Gesellsch. DDR (1969), Heft 1/2, 17=27.
Sources and Abbreviations AdW AdWL
AHUB fol. SAPMO
SAPMO1 SAPMO2 SAPMO3 SAPMO4 SAPMO5 NLKaSch NLKuSch OVP
RCP
RCP1 RCP2 RvMP
36
Akademie der Wissenschaften der DDR Files AdW-Leitung, Naturwissenschaftliche Einrichtungen, Archiv der AdW Berlin Archiv Humboldt-Universit/it Berlin folio Stiftung Archiv der Parteien und Massenorganisationen der DDR im Bundesarchiv, Zentrales Parteiarchiv, Berlin SAPMO ZPA IV 2/904/281 SAPMO ZPA IV 2/904/280 SAPMO ZPA IV 2/904/283 SAPMO ZPA IV 2/904/285 SAPMO ZPA IV 2/904/284 Nachlat~ Karl Schr6ter, Archiv AdW Berlin Nachlat~ Kurt Schr6der, Archiv AdW Berlin Oswald Veblen Papers, Library of Congress Washington, here: container 4, folder: Courant 1939-1948, 5-page report Richard Courant Papers, Courant Institute, New York University, Library RCP, file: "Deutsche Akademie" RCP, file: trip to Europe 1960 Richard von Mises Papers, Harvard University Archives, Cambridge, Massachusetts, here: 4574.5, Box 9, folder: Feb. 1, 1950-Sept. 1950
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993
Kastanienallee 12 Berlin, 10435 Federal Republic of Germany
A Geometrization of Lebesgue's Space-Filling Curve Hans Sagan
1. Introductory Remarks G. Cantor demonstrated in 1878 that any two finitedimensional smooth manifolds, no matter what their respective dimensions, have the same cardinality [1]. This is true, in particular, for the interval I = [0, 1] and the square Q -- [0, 1] • [0, 1], meaning that there exists a bijective map from I onto Q. The question arose almost immediately whether or not such a mapping can possibly be continuous. E. Netto put an end to such speculation by showing in 1879 that such a bijective mapping is, by necessity, discontinuous [8]. Is it then, it was asked, still possible to obtain a continuous surjective mapping if the condition of bijectivity is dropped? G. Peano settled this question once and for all in 1890 by constructing the first "space-filling curve" [10]. (A "space-filling curve" is a continuous map from I to E n (n > 2) whose image has positive n-dimensional Jordan content [11].) Other examples followed: [3, 6, 14]. However, this was not the end of it because the following related question arose: Whereas the interval I cannot be mapped continuously and bijectively onto an n-dimensional set of positive Jordan content (such as Q in E2), can it be done with an image set of positive outer measure? In other words, are there Jordan curves (continuous injective maps from I into E n) with positive n-dimensional outer measure? There are indeed, as W. F. Osgood showed in 1903 [9] when he constructed a oneparameter family of such curves. In fact, Osgood's curves are Lebesgue measurable. It is reasonable to assume that word of the Lebesgue measure had not reached Harvard by Thanksgiving of 1902 when Osgood submitted his paper for publication, because Lebesgue's pivotal thesis only appeared that year in the Annali di Matematica pura et applicata and there was no airmail. One does wonder, however, w h y Osgood did not make use of the Borel measure, which would have given him a stronger result. The limiting arc of Osgood's family is Peano's spacefilling curve. This is not a coincidence. Peano's ingenious result undoubtedly inspired Osgood's construction. Jordan curves with positive Lebesgue measure are now called Osgood curves.
In 1917, K. Knopp constructed another family of Osgood curves with Sierpifiski's space-filling curve as its limit [4, 13, 14]. Apparently unaware of this earlier w o r k T. Lance and E. Thomas, in a recent note, developed the same idea, albeit by a different approach [5]. It is the purpose of this note to put Osgood's and Knopp's approach in juxtaposition to the one by Lance and Thomas and to point out how the latter narrowly missed obtaining Lebesgue's space-filling curve as the limit of their family of Osgood curves and therewith the opportunity to put the first geometric generating process of Lebesgue's space-filling
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~)1993 Springer-Verlag New York
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curve on record. Geometric generations of the spacefilling curves by Peano,Hilbert, and Sierpifiski have been known as long as the curves themselves in the cases of Hilbert's and Sierpifiski's curves and almost as long in the case of Peano's curve [3, 7, 12, 13]. Besides being of interest by themselves, such geometric generations lead to uniformly convergent sequences of approximating polygons, which, in turn, lead to simple proofs of the continuity of the map. In the case of the Lebesgue curve, in particular, such a proof turns out to be much simpler than the conventional proof that is based on the structure of the Cantor set (such as the one in [2]).
2.
The Curves by O s g o o d and Knopp
Osgood's construction of a family of Jordan curves with positive Lebesgue measure consists in the successive removal of grate-shaped regions from squares, starting out with the unit square and proceeding as indicated in Fig-
Figure 1. Osgood's construction.
Figure 2. Construction of Peano's space-filling curve. 38 THEMATHEMATICAL INTELLIGENCERVOL.15,NO,4,1993
ure 1 for the first two steps. The shaded squares are what is left after each step. Starting with square $1, they are then connected by "joins" as indicated by the bold line segments in Figure 1. The dimensions of the grateshaped regions can be chosen so that the sum of their areas tends to some positive ,k < 1. If An denotes the point set consisting of the 9 n squares and 9'~- 1 joins that are obtained at the nth step and d(An) its Jordan content, then the set C = N~_lAn that is obtained after infinitely many steps has Lebesgue measure #(C) = l i m n ~ d(An) = 1 - ,~ > 0. It represents a Jordan curve [9] that may be parametrized as follows: Dividing the interval I into 17 congruent subintervals and excluding the even-numbered subintervals without beginning and endpoint, and repeating the process for each of the remaining 9 closed subintervals, and then again for each of the remaining 81 closed subintervals, and then again and again, ad infinitum, generates a Cantor-type discontinuum F17. At the first step, the remaining 9 closed
Figure 3. Knopp's construction of Osgood curves.
intervals are mapped into the squares S1, 82 . . . . . $9 and the excluded 8 open intervals are mapped linearly onto the joins (without beginning points and endpoints) from Si to $2, $2 to $3 . . . . . $8 to $9, and the process is repeated ad infinitum. The complement of P17 is mapped bijectively onto the set of all joins, and r17 is mapped onto N~=I Qn, where Qn represents the 9n squares of the nth iteration. (Osgood mapped the excluded even-numbered closed subintervals onto the joins with beginning points and endpoints and the remaining odd-numbered open intervals into the squares. We have deviated from his construction with a view toward what we are going to do in Section 3.) In the limit with & ~ 0, Peano's space-filling curve is obtained. The first two steps in the construction of Peano's curve are illustrated in Figure 2 where the bold polygonal lines indicate the order in which the squares have to be lined up. Joining the beginning points of these polygonal lines to (0, 0) and their endpoints to (1, 1) by straight line segments yields the approximating polygons we mentioned in Section 1. (Another notion of approximating polygons may be found in [7, 12, 13].) Compare Figure 2 with Figure 1. Since consecutive squares now have an edge in common, the injectivity of the mapping is lost. This is to be expected because manifolds of different dimensions cannot be homeomorphic. Knopp's construction of a family of Osgood curves consists in the successive removal of triangular regions
from an initial triangle as indicated for the first four steps in Figure 3, where the shaded triangles are the ones that are left after each step. (The initial triangle T need not be a right isosceles triangle.) At the first step, we remove a triangle of area rim(T), where re(T) is the area of the initial triangle T and where rl C (0~ 1), to be left with two triangles To, T1 of a combined area re(T)(1 - rl). From To and T1, we remove triangles of area r2m(To) and r2ra(T1) for some r2 E (0, 1) to be left with four triangles Too, To1, Tlo, and Tu of a combined area re(T)(1 - rl)(1 - r2), etc. In the limit, we obtain a point set C of Lebesgue measure #(C) = re(T) II~_ 1(1 - rk). If we choose the rk such that E~~ rk converges, then #(C) is positive. If, at each step, the triangles that are to be removed are placed judiciously, then all dimensions of the remaining triangles tend to zero and the remaining triangles shrink into points. If the interval [0, 89 (all numbers 020(22(23...) is mapped into To and [89 1] (all numbers 021(22(23...) into T, so that [0, 88 (all numbers 0200(23(24...) is mapped into Too, [88 89 (all numbers 0201(23(24. . .) into To1, [89 3] (all numbers 0210(23(24...) into T1o, and [3, 1] (all numbers 0~11(23a4..-) into T11, etc., with the points common to two adjacent intervals being mapped into the vertices common to the corresponding adjacent triangles, we see that the mapping from [0, 1] to C is bijective and continuous and C is an Osgood curve of Lebesgue measure #(C) > 0. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
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If we choose, for example, as initial triangle a right isosceles triangle of base 2 [and hence, m(T) = 1] and rk = 1/4k 2, k = 1,2, 3,..., we obtain from Weierstrass's factorization theorem that the corresponding Osgood curve has Lebesgue measure 2/Ir. Setting r k = r / k 2 instead and taking the limit as r --* 0, a space-filling curve, namely, the Sierpifiski curve, is obtained because the combined area of the removed triangles shrinks to zero (see also [13]). Since adjoining triangles are then not only joined at vertices but also along edges, the injectivity of the mapping is lost. (See also Fig. 4, where the bold polygons indicate the order in which the triangles have to be taken, and compare Figures 3 and 4. These polygonal lines, when extended by line segments to entry point and exit point, represent approximating polygons that converge uniformly to the Sierpifiski curve.)
3. The L a n c e - T h o m a s Curve and the Lebesgue Curve Instead of removing grates from squares or triangles, Lance and Thomas, by contrast, remove cross-shaped regions from squares as we have indicated for the first two steps in Figure 5 [5]. They connect the remaining squares by joins as indicated in Figure 5 by bold line segments. As in the preceding cases, the dimensions of the regions that are to be removed may be chosen so that the sum of their areas tends to some positive ,~ < 1. An Osgood curve of Lebesgue measure 1 - ,~ is obtained.
Figure 4. Generating the Sierpifiski curve. 40 THEMATHEMATICAL INTELLIGENCERVOL.15,NO.4,1993
Its parametrization may be accomplished as in the case of Osgood's original example, using instead of a Cantortype discontinuum the Cantor set itself, mapping the excluded (open) middle thirds linearly onto the joins and the remaining closed intervals into the squares. Specifically, at the first step, the closed intervals [0, 1/9], [2/9, 1/3[, [2/3, 7/9], [8/9, 1] are mapped into the squares S1 to $4 with the endpoints going into the appropriate corners and the open intervals (1/9, 2/9), (1/3, 2/3), (7/9, 8/9) linearly onto the joins (without beginning points and endpoints) from S1 to $2, 82 to $3, and 83 to 84. The process is continued ad i n f i n i t u m to obtain a continuous bijective map from I onto n~= 1 An, where An denotes the set consisting of the 4 n squares and 4 n - 1 joins that are obtained at the nth iteration. As in the case of Osgood's example, Nn~ An is Lebesgue measurable with measure limn--,~ J ( A n ) = 1 - )~ > O. While every part of Knopp's Osgood curve is again an Osgood curve, this is not the case for the examples by Osgood and Lance and Thomas because of the presence of joins. Because of this, Knopp leveled in [4], p. 109, footnote 2, some justifiable criticism at Osgood's construction. (In the same footnote, he dismisses an attempt by Sierpi~ski in [15] to construct a curve without this shortcoming as too complicated.) This criticism applies to the construction by Lance and Thomas as well, and it could be viewed as a throwback to Osgood's original attempt, were it not for the fact that a slight modification of their construction yields Lebesgue's space-filling curve as the limit.
Figure 5. Generating the Lance-Thomas curve.
Lebesgue's space-filling curve is defined on the Cantor set F = {032a12a22a3... : a j = 0 or 1} by
(1) x=O2ala3a5...,Y=O2a2a4a6
....
and on the complement F c = [0,1]\F of the Cantor set by linear interpolation (see [6] or [11]). If one replaces the joins in Figure 5 by the ones in Figure 6 and starts with the lower left corner, one obtains Lebesgue's space-filling curve as the limiting arc as ,~ --* 0. This may be seen as follows: By our construction, the interval [0, 1/9] (all numbers of the form 0 3 0 0 a 3 a 4 a s . . . ) is mapped into $1 in Figure 7 and the points of $1 have coordinates (020b25354... ~020c2c3c4...). The interval [2/9, 1/3] (all numbers 0302a3a4...), is mapped into $2 with points (0~0b2b364..., 021c2c3c4...), the interval [2/3, 7/9] (all numbers 0 3 2 0 a 3 a 4 a 5 . . . ) into $3 with (O~1626364...,020c2c3c4...) and [8/9, 1] (all numbers 0 3 2 2 a 3 a 4 a 5 . . .) into S4 with ( 0 2 1 b 2 b 3 b 4 . . . , 021r162162 . .). The process is to be repeated with the intervals [0, 1/81], [2/81, 1/27] . . . . . [80/81, 1] (all numbers of the type 030000a5asa7...
~0 3 0 0 0 2 a s a 6 a 7 . . . , . . . ,
032222a5asa7...)
and the squares S~j within each of the squares Si (i, d = 1,2,3,4) with points (0~00b364b5.... 0200c3c4c5...), ( 0200b3b4b5 . . . , 0 2 0 1 c 3 c 4 c 5 . . . ), . . . , (0211b3b4b5..., 0~11c3c4c5...), etc. This process, continued a d i n f i n i t u m , demonstrates that the mapping satisfies (1). Because = 1/3 = 0~02 is mapped by (1) into the point (1/2, 1) and t = 2/3 = 032 into (1/2, 0), the limiting position of the join from the exit point of the second square in Figure
Figure 6. Recursion operator leading to Lebesgue's curve.
6 to the third square, as & ~ 0, represents the linear interpolation on (1/3, 2/3) C FC; because t = 1/9 = 03002 is mapped into (1/2, 1/2) and t = 2/9 = 0302 into (0, 1/2), the limiting position of the join from the first square to the second square represents the linear interpolation on (1/9, 2/9) c FC; because t = 7/9 = 03202 has the image (1, 1/2) and t = 8/9 = 0322 the image (1/2, 1/2), the limiting position of the join from the third square to the fourth square represents the linear interpolation on (7/9, 8/9) c pc, etc. (See also Fig. 6.) Repeating this argument a d i n f i n i t u m reveals the limiting positions of the joins to THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993
41
represent the linear interpolation on F c as called for by Lebesgue's definition. We can now utilize this geometric generation of the Lebesgue curve to construct approximating polygons as follows: In each square, we join entry point and exit point by a straight line (diagonal) and leave the joins as they are as we have indicated in Figure 8 for the first two steps. (In our illustration, we have rounded off some corners to prevent the polygon from bumping into itself and obscuring its progression.) These polygons are approximating polygons in the conventional sense because within each square, the distance from the polygon to the Lebesgue curve is bounded above by the length of the diagonal of the square, namely, by 21/2-n, and the polygons coincide with the Lebesgue curve along the joins.
82 81
84
Hence, they form a sequence that converges uniformly to the Lebesgue curve, the continuity of which is thus established. In conclusion, let us note that Osgood's limiting curve, namely, the Peano curve, and Knopp's limiting curve, namely, the Sierpifiski curve, are nowhere differentiable, whereas the limiting curve of the Lance-Thomas family is, just as the Lebesgue curve, differentiable a.e.
Acknowledgment Dionne R. Wilson prepared the illustrations for this article by scanning and enhancing the author's handdrawn sketches, using Adobe Illustrator on an Apple Macintosh.
~22
824
S42
S21
~23
841
S14
83
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844
34
Figure 7. Image of the Cantor set.
//// //// Figure 8. Approximating Polygons for Lebesgue's space-filling curve. 42
THE MATHEMATICAL [NTELLIGENCER VOL. 15, NO. 4, 1993
References 1. G. Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, CrelleJ. 84 (1878), 242-258. 2. A. Devinatz, Advanced Calculus, New York: Holt, Rinehart, Winston (1968), 253. 3. D. Hilbert, Ueber die stetige Abbildung einer Linie auf ein Flaechenstueck, Math. Ann. 38 (1891), 459-460. 4. K. Knopp, Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und von Koch, Arch. Math. Phys. 26 (1917), 103-115. 5. T. Lance and E. Thomas, Arcs with positive measure and a space-filling curve, Amer. Math. Monthly 98 (1991), 124127. 6. H. Lebesgue, Leconssur l'Intdgration et la Recherchedes Fonctions Primitives, Paris: Gauthier-Villars (1904), 44-45. 7. E. H. Moore, On certain crinkly curves, Trans. Amer. Math. Soc. 1 (1900), 72-90. 8. E. Netto, Beitrag zur Mannigfaltigkeitslehre, Crelle J. 86 (1879), 263-268. 9. W. E Osgood, A Jordan curve of positive area, Trans. Amer. Math. Soc. 4 (1903), 107-112. 10. G. Peano, Sur une courbe qui remplit toute une aire plane, Math. Ann. 36 (1890), 157-160. 11. H. Sagan, Some reflections on the emergence of spacefilling curves, Franklin ]. 328 (1991), 419-430. 12. H. Sagan, On the geometrization of the Peano curve and the arithmetization of the Hilbert curve, Int. J. Math. Educ. Sci. Technol. 23 (1992), 403-411. 13. H. Sagan, Approximating polygons for the SierpifiskiKnopp space-filling curve, Bull. Acad. Sci. Polon. 40 (1992), 19-29. 14. W. Sierpifiski, Sur une nouveUe courbe continue qui remplit tout une aire plane, Bull. Acad. Cracovie, (Sci. Mat. Nat. Serie A) (1912), 462-478. 15. W. Sierpifiski, Sur une courbe non quarrable, Bull. Acad. Cracovie (Sci. Mat. Nat. Serie A) (1913), 254-263.
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43
Jeremy J. Gray*
Jakob Nielsen (1890-1959) Vagn Lundsgaard Hansen Jakob Nielsen stands out as one of the greatest Danish mathematicians of all times, and his papers are often quoted in the literature. It was, therefore, appropriate that Danish mathematicians wished to celebrate the centenary of his birth. One such celebration among others took place at a meeting of the Danish Mathematical Society on May 31, 1991. In the period 1925--1951, Jakob Nielsen was professor at the Technical University of Denmark, spending the main part of his active life as researcher and teacher there. When the Mathematical Institute decided on its logo in 1986, it was an obvious idea to embed it into the hyperbolic plane which played a central role in many of Nielsen's investigations (see illustration).
Jakob Nielsen won international recognition for his pioneering work in group theory and the topology of surface transformations. His collected mathematical papers were published in two volumes by Birkh/iuser in 1986, with many of his long memoirs in translations from German to English by John Stillwell. * Column Editor's address: Faculty of Mathematics, T h e O p e n U n i v e r sity, Milton Keynes, MK7 6AA, England.
Logo of the Mathematical Institute, The Technical University of Denmark, Lyngby, Copenhagen: Petersen graph in the hyperbolic plane. 44 THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4 (D1993Springer-VerlagNew York
This article about Jakob Nielsen consists of two parts. The first part contains a short description of his exciting life. For most of the biographical material, m y source is the memorial paper by Werner Fenchel in Acta Mathematica 103 (1960), vii-xix, reprinted in Nielsen's collected works. The second part contains a description of Jakob Nielsen's mathematical research. 1 In addition to Nielsen's own papers, I have, in particular, made use of a series of essays at the end of the second volume of his collected works, in which eminent mathematicians have described his research and its importance for later developments. The author is grateful to Thoger Bang, Robert E Brown, David Chillingworth, Robert E. Greene, Jane Gilman, Paul Hjorth, and Heiner Zieschang for valuable help in connection with this article.
Biography of Jakob Nielsen Childhood and Years of Study. Jakob Nielsen wasborn in the small village of Mjels on the island of Als in North Schleswig (the southern part of Denmark) on October 15, 1890. His father owned a small farm in Mjels, and here he grew up in quiet and peaceful surroundings as the youngest of four children. North Schleswig (called Sonderjylland in Denmark) was ruled by Germany in the period 1864-1920. Like many people of Danish origin from this region, Jakob Nielsen felt a strong association with his birthplace throughout his life. At the age of 3 he lost his mother. In the following years, an aunt, who was a teacher at Rendsborg, followed his progress closely. She noticed early on that he was an unusually gifted child, and in the year 1900 he moved to live with her in Rendsborg, as this town offered a far better educational system. Here he attended the so-called Realgymnasium, where the teaching of Latin carried considerable weight in the curriculum. Throughout his life he retained a deep passion for Roman poetry. After a few years, the relations between him and his aunt deteriorated because they were both rather uncompromising characters, and at the age of 14 he left her home. For the remainder of his schooldays, and later during all his years of study, he earned his living by tutoring pupils in a variety of subjects--even Norwegian, as he once said. In December 1907, he was expelled from the Realgymnasium because he and a few of his schoolmates had founded a pupils' club which, though quite harmless, was against the rules. Full of confidence, however, he continued studies on his own and matriculated at the University of Kiel in the spring of 1908, also obtaining his school-leaving certificate privately at Flensborg in the autumn of 1909. 1 The mathematical figures are taken from the author's article (in Danish) "Om lakob Nielsen (1890-1959)/' Nordisk Matematisk Tidskrifl (normat), 40 (1992), pp. 63-74, with the kind permission of Scandinavian University Press (Oslo-Stockholm).
Jakob Nielsen spent all his years of study in Kiel, with the exception of the summer of 1910, which he spent at the University of Berlin. In the first years of study, he attended lectures in mathematics, physics, chemistry, geology, biology, literature, and philosophy. Only slowly did mathematics begin to play the central role in his studies, but philosophy also was a subject close to his heart. Among his teachers in Kiel, Jakob Nielsen valued in particular the mathematician Georg Landsberg, known among other things for his work on algebraic functions. It was Landsberg who encouraged Nielsen to study the mathematical problems underlying his doctoral dissertation of 1913. In a short autobiography appended to the thesis, he expresses his devotion to Landsberg, who had died shortly before. Of the utmost importance to Nielsen for his start as a research mathematician, however, was Max Dehn (1878-1952), who was attached to the university in Kiel at the end of the year 1911. Dehn was already considered a very eminent mathematician, and through him Nielsen came into contact with the most recent advances in topology and group theory. It was precisely to these fields that Nielsen devoted most of his research. In itself, Nielsen's thesis was not epoch-making. In it, however, y o u find the roots of his pioneering work on surface transformations, marked in particular by four long memoirs in Acta Mathematica (1927,1929,1932,1942) and a memoir in Meddelelser fra det Danske Videnskabernes Selskab from 1944. In addition, the group-theoretical papers of Nielsen, including a very important paper in Danish from 1921 on free groups, have roots going back to his years of study in Kiel, and, in particular, to the inspiration from Dehn, with w h o m he developed a lifelong friendship.
Military Service and the First Years after the War. Immediately upon receiving his Ph.D. in 1913, Jakob Nielsen was drafted to do military service in the German marines and was assigned to coastal defence. After the outbreak of the war in 1914, he was sent first to Belgium and then, in April 1915, to Constantinople as one of the German officers assigned to advise the Turkish government on the defence of the entrance to the Black Sea through the Bosporus and the Dardanelles. At the end of the First World War in 1918, he finished his military service. During these years there was, of course, not much time for mathematics, but somehow Jakob Nielsen found time to write a couple of important short group-theoretical papers published in Mathematische Annalen and a paper on a subject from ballistics. The home journey from Turkey went through Russia and Poland in November 1918, and during this journey he kept a diary, which was published in the Danish newspaper Politiken on the tenth anniversary of Armistice Day. The whole period made a strong impression on him and probably contributed to his complete openmindedness throughout his life towards people with a background different from his own. Professor Th~ger THEMATHEMATICALINTELLIGENCERVOL.15,NO.4,1993 45
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Bang has pointed out to me some poems by an anonymous writer. One of these poems is written in 1917 by a soldier on leave from the front. We n o w k n o w that the author was Jakob Nielsen. Its serious words m a d e a strong impression in the shadows of the war in Denmark and it was set to music by the composer Max Springer (see above). An English translation of the first verse and the final verse could go as follows: 1. Farewell, my home in the quiet village. Thank you father and mother for my stay. Now duty calls me away to savage, will peace let me return some day? If this the last farewell should be, thanks for everything to thee. 4. As off I go, away again. I fully know what I am leaving: my bride who became my trusted friend, my brother, sister, and parents giving. I love you more than words can tell, farewell, farewell! 46
T H E M A T H E M A T I C A L 1 N T E L L I G E N C E R V O L . 15, N O . 4, 1993
In the spring of 1919, Jakob Nielsen married the German medical doctor Carola von Pieverling, w h o m he had met in H a m b u r g shortly after the war. They lived a h a p p y family life together and had one son and two daughters. Nielsen wrote m a n y other poems, but apparently he never told his wife about the above poem. In the summer term 1919, Jakob Nielsen stayed at G6ttingen, which was at that time the undisputed centre for mathematical research in the world. Here, he was especially attracted by the algebraist Erik Hecke (18871947), and w h e n Hecke the same year received nomination to a chair in mathematics at the recently established university in Hamburg, Jakob Nielsen followed as his assistant with the title of Privatdozent. Already in 1920, however, Nielsen himself was named to a professorship at the Technical University of Breslau. Here he could res u m e close contact with Max Dehn, who had been a professor at the University of Breslau for some years. In t w o inaugural lectures in Breslau in 1921, Jakob Nielsen formulated clearly that circle of problems concerning surface transformations upon which he was so strongly engaged for the rest of his life. Handwritten notes from these lectures were translated into English and published for the first time in 1986 in connection with the publication of his collected mathematical papers.
Researcher and Teacher. His stay in Breslau was not to last long, for, at the reunion of North Schleswig with Denmark in 1920, Jakob Nielsen opted for Denmark, and in 1921 he took over the vacant position as lecturer in mathematics at the Royal Veterinary and Agricultural University in Copenhagen. In 1925, he succeeded Christian Juel as professor of theoretical mechanics at the Technical University. For some years, Jakob Nielsen based his teaching of mechanics on Juel's textbook, but gradually it became clear that a revision was needed. Nielsen plunged into this work with great energy, and the n e w textbook in theoretical mechanics was published in two volumes in 1933-1934. The book was to a large extent original pedagogical work on an advanced level for its time, and in his exposition, Nielsen made strong use of relatively new mathematical tools such as vectors and matrices. The text is not easy, and Jakob Nielsen's lectures were rather demanding. He was, however, well k n o w n for his ability to express himself with great clarity and intensity. Jakob Nielsen allowed the students to bring their books for the written examinations in theoretical mec h a n i c s - - a novelty at the time. A student once asked Jakob Nielsen several times whether this was true. Eventually, Nielsen got irritated and said, "Yes, you can bring all the books you like, even the collected works of Shakespeare." Finally, the examination came, and the student did rather poorly. I imagine that Nielsen laughed w h e n he read the last sentence in the essay: "The rest is sil e n c e - Hamlet, act V, scene II."
In 1935, the textbook was translated into German by Werner Fenchel; it was reprinted by Springer-Verlag in 1985. In the beginning of the 1940s, teaching of aerodynamics was introduced at the Technical University of Denmark and put into the hands of Jakob Nielsen. The more theoretical parts of the lecture notes in this connection were published in 1952 as the third volume of his textbook on theoretical mechanics. The book is remarkable for its clear distinction between the empirical foundation and the mathematical theory. As soon as his work with the first edition of the textbook was completed, Jakob Nielsen returned to topology and group theory. He often lectured on these subjects to small groups of interested younger mathematicians at the University of Copenhagen. Of particular importance is a series of lectures in the year 1938-1939 on discontinuous groups of isometries in the hyperbolic plane. Inspired by these lectures, Svend Lauritzen wrote his thesis: "En Indledning til en gruppeteoretisk Behandling af de ikke orienterbare Flader." It quickly proved desirable to take up studies of such groups in their full generality, and Jakob Nielsen began with Werner Fenchel (1905-1988) to prepare a manuscript for a monograph on this subject. Even though he was heavily engaged in this project, Nielsen could only devote a limited part of his time to it because after the Second World War he became more and more involved in international work, in particular in UNESCO, where he was a member of the executive board from 1952 to 1958. In this context also, he was highly esteemed for his personal integrity. After the death of Harald Bohr in 1951, Jakob Nielsen was nominated as his successor as professor of mathematics at the University of Copenhagen. Already in 1955 he resigned from the chair, however, because he felt that he could no longer carry out his work as a professor fully due to his many international obligations. A first version of the manuscript just mentioned was completed, but both Jakob Nielsen and Werner Fenchel felt that it needed a thorough revision. The revision was not finished when Nielsen died in 1959, and later the original of the manuscript was stolen from a parked car. Various copies have, however, circulated among specialists, and in several cases, other mathematicians have found alternative proofs of the most important results in the manuscript. Major parts of the theory, now known as the Fenchel-Nielsen theory, have therefore gradually come to be known among the researchers in the field. Furthermore, it should be mentioned that in his last years Werner Fenchel almost completed the revision of the manuscript, and with further work, particularly by Professor Siebeneicher in Bielefeld, it may be published in the near future. In all the areas in which Nielsen worked, he left strong evidence of an unusual personality. He was elected member of the Royal Danish Academy of Sciences and Letters in 1926, and in his last years, he lived in the Academy's
Jakob Nielsen in 1946.
honorary residence close to the castle of Hamlet in the town of Elsinore. In January 1959, Jakob Nielsen was stricken by the illness which led to his death on August 3, 1959. Memories of Jakob Nielsen as a person are still alive, and his mathematics will never be forgotten.
The Mathematics of Jakob N i e l s e n On the Work of Jakob Nielsen in Group Theory. When Nielsen began his investigations of groups, combinatorial group theory was just beginning, with emphasis on finding descriptions of groups by generators and relations. For that purpose, free groups play a decisive role. To define the notion, let { x l , . . . ,xn} be an alphabet with n letters. In this alphabet, we can make words from the letters and associated formal inverse letters, denoted by x l l , . . . , Xn 1. As an example, is a word in an alphabet with at least four letters. We now define the product of any two of these words by juxtaposing them and then cancelling factors of the form x~x~1 and x~-1xi, until no further cancellations are possible. By this cancellation procedure, we arrive perhaps
X2xllx4x2x2x41
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993 4 7
at the empty word, that is, the word without any letters. If we denote the empty word by 1, and let it act as neutral element for the product, we have defined the free group on the generators x l , . . . , x,~. For this construction to work, there is no need to assume that the alphabet is finite, but if it is, we say that the free group is finitely generated. In a very important paper in Danish published in Matematisk Tidsskrift in 1921, Nielsen proved that every subgroup of a free group is itself free. (This fundamental paper is included in his collected works in an English translation by Anne W. Neumann, which was first published in The Mathematical Scientist 60 years after the original paper.) Nielsen assumed the free group to be finitely generated, but 5 years later, Otto Schreier proved that this assumption is not necessary, so that the result is true in complete generality. The theorem is n o w known in the mathematical literature as the Nielsen-Schreier theorem. The theorem is of fundamental importance when dealing with the relations in a group. Though this result is extremely significant, the main goal of Nielsen was, however, to describe the automorphism group of a free group, that is, the group of isomorphisms of the free group onto itself. For that purpose, he introduced some basic automorphisms, now known as Nielsen transformations. In this context, a paper in Mathematische Annaten from 1918 should be mentioned, in which it is shown that the automorphism group of the free group on n generators is generated by n + 1 automorphisms. The corresponding relations for the automorphism group are determined in a paper which is very difficult to read, published in Mathematische Annalen in 1924. As an example, Nielsen gives in his paper from 1918 the following system of five automorphisms (Nielsen transformations) that generates the automorphism group of the free group on four generators X l ~X2~ X3~ X4:
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Figure 1. Homeomorphism T:~o --~ ~oon a surface ~o of genus 2.
48
THE MATHEMATICALLNTELLIGENCERVOL. 15, NO. 4, 1993
The theory of automorphisms of free groups has only recently approached a definitive stage with the works of Culler and Vogtman [1] and Bestvina and Handel [2]. O n the Work of Jakob N i e l s e n in Topology. In topology, as is well known, one investigates the qualitative characteristics of a geometrical object: Is it connected? Does it have edges? Holes? Quantitative properties associated with measurements of lengths on the object stay in the background. An important method in the investigation of a geometrical object is to study its degree of symmetry. If you are interested only in the topological properties, you need also to consider "qualitative symmetries," where certain distortions are allowed. Nielsen's investigations deal with topological transformations ("qualitative symmetries") of surfaces. I have already mentioned the four long memoirs in Acta Mathematica and the memoir in Meddelelser fra det Danske Videnskabernes Selskab. When Nielsen began his studies of transformations (homeomorphisms) of surfaces, topology was still a field at its formative stages with its roots particularly in work by the French mathematician Poincar6 at the end of the previous century. Concerning the study of manifolds, the subject had not yet come very far, but it was known that one can realise every closed, orientable surface in space by adding handles to a sphere; the number of handles is called the genus of the surface. As Nielsen writes in his first long memoir in Acta Mathematica from 1927, "the 2-dimensional manifolds (i.e. the surfaces) have thereby prematurely offered themselves for deeper study," and he gives almost no further motivation to embark on his detailed study of surface transformations. Let ~ denote a closed, orientable surface of genus p _> 1, and let AJ(~) denote the group of isotopy classes of orientation-preserving homeomorphisms of 9~onto itself; compare Figure 1. Two homeomorphisms belong in other words to the same mapping class if they can be deformed into each other through homeomorphisms. By a result of Baer from 1928, it is sufficient to require that the two homeomorphisms be homotopic. The group .M(~) is called the mapping class group of ~. Already in his the-
sis from 1913, Nielsen had proved that the mapping class group of the torus (a surface of genus 1) is nothing else but the unimodular group SG(2, Z) of integral 2 x 2matrices with determinant 1. The group SL(2, Z) is called the elliptic modular group and it is closely associated with the theory of doubly periodic algebraic functions. Now consider a closed, orientable surface of genus p >_ 2. As the keystone in his investigations of surface transformations, Jakob Nielsen in 1942 succeeded in proving for a surface of genus p ~ 2 that if the nth iterate of a homeomorphism of the surface can be deformed into the identity homeomorphism, then the homeomorphism itself can be deformed into another homeomorphism for which the nth iterate is exactly the identity homeomorphism. This main result on surface transformations can be given the following formulation: Every cyclic subgroup of .M(~) can be represented by a cyclic subgroup of homeomorphisms of ~. In this formulation, the problem can be generalised. In 1948, Fenchel proved that every finite solvable subgroup of ; ~ (~) can be represented by a subgroup of homeomorphisms of ~. In 1981, Kerckhoff proved this result in complete generality for an arbitrary finite subgroup of .M (~), thereby solving what by then had become known as the Nielsen realization problem. Heiner Zieschang pointed out in 1976 that one of the arguments by Nielsen in his memoir of 1942 is not correct in that his proof does not cover all cases. The first complete proof of Nielsen's theorem is, therefore, contained in Fenchel's paper of 1948, where other methods are used. Nielsen's general description of surface transformations in terms of primitive homeomorphisms, which is perhaps even more important and to which we shall later return, is completely correct, however.
/
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b r
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a
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The G e o m e t r y o n a Surface of G e n u s 1. In the following two sections we shall, using Nielsen's own notation, describe the tools behind his methods for studying surface transformations. These are now the standard tools in the area and the point at which students seeking to work in the field begin their graduate study. Consider a closed, orientable surface ~ of genus p = 1, that is, a toms. By cutting along the two canonical curves a and b on the torus shown in Figure 2, we see that the torus can be identified with a square in which opposite edges are glued pairwise together; compare Figure 3. In the euclidean plane ~2, we now consider the two parallel translations that translate vectors in I1~2 by (0, 1), respectively (1, 0). Due to their close relation to the canonical curves, we also denote these parallel translations by a, respectively b; compare Figure 4. The two parallel translations a and b generate the free abelian group of rank 2, that is, Z 2 = Z | Z. This group corresponds to the integral lattice in I~2, and we find that the torus can be constructed as the orbit space ~2/Z2, also called the modular surface, for the action of Z 2 on I~2. It follows that the torus has ~2 as its universal covering space and Z2 as its fundamental group. We say that the unit square in 1t~2 is a
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THEMATHEMATICAL INTELL[GENCERVOL15,NO.4,1993 49
Figure 5. Surface ~oof genus 2.
Cut the surface 9~ along the 2p canonical curves
al, bl, a2, b2,..., ap, bp, and fold it out into a regular 4pgon as s h o w n in Figures 5 and 6. If the edges in the regular 4p-gon are glued together in pairs as indicated in Figure 6, w e get back the surface ~. A particularly nice description of these identifications can be found in Hilbert and Cohn-Vossen's Anschauliche Geometrie, w The s u m of the angles in a regular @-gon in the euclidean plane is given b y 1@-2,, ~ - - - ~ p zrr = (2p - 1)21r _> 6~r f o r p > 2.
Figure 6. Surface ~o obtained by identification.
fundamental polygon for the torus. The construction shows that we can tessellate the euclidean plane with parallel translates of the fundamental polygon for the torus. The geometry on the torus is, in other words, euclidean. The G e o m e t r y o n a Surface of Higher Genus. Next we consider a closed, orientable surface ~ of genus p > 2. The considerations by Nielsen in his Acta memoirs are to a large extent concentrated on this case, because the other cases either were already well k n o w n or easy; the case of surface transformations on the torus he had, for instance, already completely covered in his doctoral dissertation. As we shall see, the g e o m e t r y on ~ is in this case hyperbolic, and therefore Nielsen's investigations are closely tied to the hyperbolic plane. Figures in the following are carried out for p = 2. 50
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
For p > 2, the euclidean plane cannot be tessellated with congruent regular 4p-gons such that { appears as the m o d u l a r surface for a g r o u p of parallel translations preserving the tessellation. For if such a tessellation existed, then 4p regular @-gons w o u l d meet in each corner, because a n e i g h b o u r h o o d of the base point for the 2p canonical curves on { w o u l d correspond to a neighb o u r h o o d of each c o m e r in the tessellation. But the s u m of the angles in a c o m e r for such a tessellation is given by ( 2 p - 1)2Tr which is >27r. At the beginning of this century, Poincar4 s h o w e d that the hyperbolic plane 9 can be used, because one can construct a fundamental @ - p o l y g o n for ~ in this plane in which the sum of the angles is 2rr for every p > 2. We shall n o w describe this construction in some detail. Let C denote the complex n u m b e r plane with the complex variable z. By IzI we d e n o t e the m o d u l u s of z. Let E : {z E CI Izl = 1} and ~5 = {z E C I IzI < 1} denote the unit circle, respectively the interior of the unit circle in C.
In a memoir of 1887, Poincar6 described w h a t is probably the most c o m m o n l y used model of the hyperbolic plane. We take as lines those circular arcs in 9 that are orthogonal to E, and as points, the usual points in ~. One can equip 9 with a metric (the logarithm of a suitable cross-ratio of euclidean distances), such that the size of an object m e a s u r e d b y this so-called hyperbolic length tends to infinity w h e n the object is m o v e d towards the b o u n d a r y E of ~. Angles in 9 are the usual euclidean angles. The orientation-preserving isometries in 9 are exactly the linear fractional transformations (M6bius transformations) of the form
az+b bz +-d
z ~ -
with a~ - bb > 0
for complex n u m b e r s a, b. We can normalize so that a~ bb = 1 if we wish. In the following, we only consider orientationpreserving isometries and homeomorphisms. The isometries in 9 can be divided into rotations, limit rotations, and translations. Regarded as a transformation in C, every M6bius transformation has two fixed points, which m a y coincide. If one of the fixed points lies in the interior of 9 and one in the exterior of ~, we get a rotation in ~. If the two fixed points coincide at a point of E, we get a limit rotation. A translation is an isometry in 9 for which the two fixed points are different and lie on E. Hence, a translation will m a p a line in ~ - - c a l l e d the axis for the translation - - into itself, with the points on the axis being p u s h e d from one of the fixed points towards the other; compare Figure 7. In this article, we shall consider only translations, also called hyperbolic transformations. In the hyperbolic plane ~, one can construct a fundamental 4p-gon with angle sum 2~r, which is symmetric with respect to the centre of q~. [Proof'. A v e r y small regular 4p-gon a r o u n d the centre of 9 is approximately euclidean and has, therefore, approximately the angle s u m (2p - 1)2~r, whereas a regular 4p-gon with its corners on E has the angle sum 0. Because the sum of the angles varies continuously with the distances of the corners from the centre of ~, there exists a regular 4pgon in which the corners have a suitable distance from the centre of ~, such that the angle s u m is 2~r.] Associated with such a regular 4p-gon in 9 w e can define 2p uniquely d e t e r m i n e d hyperbolic transformations al, bl, a2, b2,..., ap, bp, which m a p pairs of edges in the polygon to be identified into each other with the lines for the edges in b e t w e e n as the axes, respectively, such that bl H bl, al ~ a l , . . . , bp ~ bp, ap ~ ap, a s s h o w n in Figure 8. Let F be the g r o u p of hyperbolic transformations generated by a l , bl~ a2, b 2 , . . . ~alas bp. The hyperbolic transformations in F transfer the fundamental 4p-gon into congruent 4p-gons, which in the limit tessellate the hyperbolic plane ~. The g r o u p F acts freely and discontin-
Figure 7. A hyperbolic transformation.
Figure 8. Hyperbolic transformations implemented by identification of edges.
uously on ~. The orbit space ~ / F , also called the modular surface for the action of F on ~, can be identified with the fundamental 4p-gon with the edges pairwise glued together as indicated in Figure 8, that is, with the original surface ~ of genus p > 2. The projection m a p --* ~ / F = ~ is the universal covering m a p of ~, and the g r o u p F is therefore the fundamental g r o u p of ~. THEMATHEMATICAL INTELLIGENCER VOL.15,NO.4, 1993 51
In Figure 9, w e indicate the tessellation of the hyperbolic plane with congruent 8-gons. O n Nielsen Fixed Point Theory. In this section, we shall briefly discuss Nielsen's investigations of surface transformations (homeomorphisms) of the orientable closed surface ~ of genus p _> 2 onto itself. In particular, we shall mention the investigations which have led to the d e v e l o p m e n t of a theory n o w k n o w n as Nielsen
fixed point theory. Every h o m e o m o r p h i s m ~- : ~ ~ ~ can (in m a n y ways) be lifted to a h o m e o m o r p h i s m t : 9 ~ ~, 4p
t
l
ffp
l T
By a clever a r g u m e n t Nielsen shows that t : 9 --* can be extended to the closed unit disc r so that t defines a h o m e o m o r p h i s m tiE : E --* E of the unit circle E onto itself. It is by a close examination of the h o m e o m o r p h i s m t IE that Nielsen gets his results. In the first Acta memoir, it is shown to begin with that every a u t o m o r p h i s m of the fundamental g r o u p of ~, that is, of the group F, can be realised by a h o m e o m o r p h i s m of ~ onto itself. This theorem is due to Dehn, but the first proof of it in print is in Nielsen's memoir. Nielsen later always gave full credit to Dehn, and the t h e o r e m is n o w k n o w n as the D e h n - N i e l s e n theorem. The fixed points for 7- : ~ ~ ~, that is, points x E
Figure 9. Tessellation of the hyperbolic plane. 52 THEMATHEMATICAL INTELLIGENCER VOL.15,NO.4, 1993
for which T(X) = X, play a decisive role. A fixed point ~: E 9 for a lift t : 9 -* 9 of 7 : ~ ---* ~, that is, t(:~) = 5:, is projected onto a fixed point x E ~o for 7-, that is, ~-(x) = x. Two lifts t, t' : 9 ~ 9 of the same h o m e o m o r p h i s m ~- : ~ ~ ~ have the same fixed point projections onto ~ if and only if they are conjugate u n d e r F, that is, t' = T t T - 1 for a hyperbolic transformation T E F. Every fixed point x E ~ for 7- : ~ ~ ~ is the projection of a fixed point ~ E for some lift t : 9 ---* 9 of ~-. [Proof:. If t is a lift, then t T is a lift for all T E F.] The collection of fixed points for T, which are the projections of all the fixed points for the lifts t of T in a conjugacy class of lifts, is called a fixed point class. The index of an (isolated) fixed point x of ~measures the twisting of T about x and can be identified with the winding number, as in complex analysis, of 1 - % w h e r e 1 denotes the identity map. After deforming the m a p T to have only finitely m a n y fixed points, the index of a fixed point class is defined to be the s u m of the indices of its members. The n u m b e r of fixed point classes with index ~ 0 is n o w called the Nielsen number of T and is d e n o t e d b y NO- ). It was this n u m b e r that Nielsen tried to determine. Clearly, the Nielsen n u m b e r N(T) provides a lower b o u n d for the n u m b e r of fixed points of r. It can p r o v e d that two fixed points x~, x2 E ~ for T : ~ ~ ~ belong to the same fixed point class if and only if xl and x2 can be connected b y a curve C in q0 such that C is homotopic to ~-(C) b y a h o m o t o p y keeping xl and x2 fixed. The Nielsen n u m b e r N(~-) is again the n u m b e r of "essential" fixed point classes, that is, those with index # 0. In this formulation, the notion of fixed point classes and index for these can be generalised to m a p p i n g s f : X --* X b e t w e e n more general types of spaces than surfaces, for example, p o l y h e d r a and manifolds. There is an extensive literature on the subject. H o m o t o p i c maps f~ g : X --* X have the same Nielsen number, that is, N ( f ) = N(g). The following question about the Nielsen n u m b e r N ( f ) for a m a p f : X --, X is therefore interesting: Does there exist a m a p g : X ---+X h o m o t o p i c to f : X ~ X, such that the n u m b e r of fixed points for g is exactly N(g) = N ( f ) ? In other words: Can the Nielsen n u m b e r be realised? In 1942, it was p r o v e d b y Wecken that for a fairly large class of finite polyhedra, containing a m o n g others all triangulable manifolds of dimension ~ 3, such a minimality theorem holds. Nielsen conjectured in his 1927 m e m o i r that Nielsen n u m b e r s of maps of surfaces can be realised. For homeomorphisms, the answer is correct, as Nielsen himself p r o v e d in part (see also [3]), t h o u g h the proof was completed only recently [4]. H o w e v e r , for continuous maps, in [5, 6], Boju Jiang p r o d u c e d examples which show that it is not always possible to realise the Nielsen n u m b e r on surfaces. In fact, for any surface of negative Euler characteristic, Jiang has recently p r o v e d that there is a m a p such that the gap between its Nielsen n u m b e r and the m i n i m u m n u m b e r of fixed points of all maps h o m o t o p i c to it is arbitrarily large [7]. This w o u l d most certainly have come as a surprise to Jakob Nielsen.
As a witness to the continued strong interest in Nielsen fixed point theory, a conference was organised by the American Mathematical Society in June 1992. The conference proceedings will be published in the society's series Contemporary Mathematics. It contains several survey papers in Nielsen fixed point theory; in particular, a paper by R. E Brown [8] discusses the realisation problem mentioned above.
The Synthesis of Nielsen's Work on Surface Transformations. As the synthesis of his work on homeomorphisms of a closed, orientable surface ~ of genus p _> 2 - - f o r obvious reasons also called a hyperbolic surface--Jakob Nielsen gained the deep insight that up to isotopy, and possibly after a finite iteration, every homeomorphism of a hyperbolic surface can be written as a composition of certain primitive homeomorphisms defined essentially on disjoint subsurfaces. The work of Nielsen was based, as indicated above, on a thorough analysis of the fixed point sets for the homeomorphisms. Nielsen found that the primitive homeomorphisms of a hyperbolic surface were of two types, one type consisting only of periodic homeomorphisms. The second type of primitive homeomorphisms was not clearly identified before the end of the 1970s, where, by completely different methods, William Thurston found that they are nonperiodic and that they preserve a pair of transverse measured foliations by geodesic lines. They are now called pseudo-Anosov homeomorphisms. These homeomorphisms are very important both in the theory of three-dimensional manifolds and in the s t u d y of iterations of mappings. The study of the connections between the work of Nielsen and the work of Thurston has been the subject of several papers, among which we mention the papers by Gilman [9], Miller [10], and Handel [11]. See also the paper by Thurston [3] and the book by Bleiler and Casson [12].
References 1. M. Culler and K. Vogtman, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91-119. 2. M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. Math. 135 (1992), 1-51. 3. W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 417-431. 4. B. Jiang and J. Guo, Fixed points of surface diffeomorphisms, Pacific ]. Math. (to appear). 5. B. Jiang, Fixed points and braids I, Invent. Math. 75 (1984), 69-74.
6. B. Jiang, Fixed points and braids II, Math. Ann. 272 (1985), 249-256. 7. B. Jiang, Commutativity and Wecken properties for fixed points of surfaces and 3-manifolds, Topology Appl. (to appear). 8. R. E Brown, Wecken properties for manifolds, in Nielsen Theory and Dynamical Systems, C. McCord, ed., Providence, RI: American Mathematical Society, (to appear). 9. J. Gilman, On the Nielsen type and the classification of the mapping-class group, Advan. Math. 40 (1981), 68-96. 10. R. T. Miller, Geodesic laminations from Nielsen's viewpoint, Advan. Math. 45 (1982), 189-212. 11. M. Handel, New proofs of some results of Nielsen, Advan. Math. 56 (1985), 173-191. 12. S. Bleiler and A. Casson, Automorphisms of Surfaces after Nielsen and Thurston, London Math. Soc. Student Texts Vol. 9, Cambridge: Cambridge University Press (1988).
Bibliography Joan S. Birman, Braids, Links and Mapping Class Groups, Ann. Math. Studies vol. 82, Princeton: Princeton University Press (1975). Joan S. Birman, Mapping class groups of surfaces, in Braids, J. S. Birman and A. Libgober, eds., Providence RI: American Mathematical Society (1988). Contemporary Math. Vol. 78, 1988, pp. 13-43. Robert E Brown, The Lefschetz Fixed Point Theorem, Glenview, IL: Scott-Foresman (1971). Bruce Chandler and Wilhelm Magnus, History of Combinatorial Group Theory. A Case Study of the History of Ideas, New York: Springer-Verlag (1982). A. Fathi, E Laudenbach, and V. Po6naru, eds., Travaux de Thurston sur les surfaces, S6minaire Orsay, Astdrisque vol. 66--67, Societ6 Math6matique de France (1979). Werner Fenchel, Elementary Geometry in Hyperbolic Space, De Gruyter Studies in Mathematics Vol. 11, Berlin: Walter de Gruyter (1989). Jane Gilman, Review of Collected Mathematical Papers of Jakob Nielsen, Bull. Amer. Math. Soc. 21 (1989), 125-129. D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie, Berlin: Springer-Verlag (1932). B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemporary Math. Vol. 14, Providence, Rh American Mathematical Society (1983). T. Kiang, Theory of Fixed Point Classes, Beijing: Scientific Press (1979). Translation of the revised 2nd Chinese edition, Springer-Verlag, 1989. Jakob Nielsen, Collected Mathematical Papers, Vagn Lundsgaard Hansen, ed., Volume I (1913-1932), Volume 2 (1932-1955), Boston: Birkh/iuser (1986). John Stillwell, Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, Vol. 72, New York: Springer-Verlag (1980). Heiner Zieschang, Finite Groups of Mapping Classes of Surfaces, Lecture Notes in Mathematics No. 875, Berlin: SpringerVerlag (1981).
Mathematical Institute Technical University of Denmark DK-2800 Lyngby, Denmark
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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 card 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.
The Jubilee Maze Klaus Treitz
Labyrinths have been the object of great fascination for thousands of years. One reason for their long-lasting popularity is that they may be a metaphor for the complexity of life with its roundabout ways, blind alleys, and alternations of failure and success. But surely there is also quite a different reason for the continuing interest in labyrinths: their challenging mathematical character. The oldest form, which is spread all over the world, is the Cretan type. It is found on many Cretan coins, in many stone structures in Sweden, and in wall carvings. The adjective Cretan calls to mind the legend of Theseus and the Minotaur. The Minotaur was a creature with the head of a bull and the body of a man. He was caged in a complicated building on the isle of Crete in the Mediterranean Sea. Annually, he demanded seven Greek maidens and youths for sacrificial food. Theseus, a prince of Athens, found and conquered the Minotaur. He himself escaped the Labyrinth with the aid of the clue of thread provided by Ariadne. If the prison of the Minotaur was as intricate as the legend says, the preserved pictures must be only ideograms for it (Fig. 1). Nevertheless, they represent the classic type of labyrinth. This type is called "unicursal," having no branches. A similar sort of labyrinth is the Chartres type. In the Middle Ages, it was used in the pavements of important churches and was traced by pilgrims fulfilling vows. These labyrinths also belong to the unicursal type. A c * Column Editor's address: Mathematics Institute, Universityof Warwick, Coventry,CV4 7AL England. 54
Figure 1. Graffito on a pillar at Pompeii (about A.D. 79): "Here lives Minotaur." (From alpha, Mathematische Schiilerzeitschrifl, Berlin 17 (1983), 4.
cording to Christian philosophy, the journey through life may be difficult and troublesome, but in any case can lead to the goal. Theseus became an allegory for Christ (Fig. 2).
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~ 1993 Springer-Verlag New York
Figure 3. Maze at Hampton Court Palace. Photograph by G. Gerster, 1990. ([2], p. 92)
Figure 2. Cathedral of Chartres. (From Notre-Dame de Chartres, page 13, 15 ANNEE, Chartres 1984.) Photograph by B. Deffontaines, 1983.
Then a striking change takes place: With the Reformation and the beginning of modern times, a new type of labyrinth replaced the unicursal. Human life no longer appeared as well programmed as it did in medieval times. Garden labyrinths appeared in which it is really possible to find the way to the center and the way out only by trial and error ("Irr-g~rten"). The walls consist of hedges. Although the old unicursal labyrinths are of some mathematical interest (it is possible to look for a classification, or to ask about methods of construction), the mazes are, of course, more interesting. Questions about solving algorithms which they raise connect to modern and important questions of graph theory and informatics (computer science). For the mathematician on vacation, seeking recreation and inspiration in visits to mazes, the place to go is England. Many castles have not only an old history and spirit of their own, but also a marvelous garden with a maze [1]. For instance, Leeds Castle, halfway between Dover and London: Its maze has a subterranean exit
through a romantic grotto. Thirty kilometers away is Hever Castle, the childhood home of Anne Boleyn, often visited by Henry VIII. In the garden, you find a pretty maze. On the Thames not far from London lies the royal palace of Hampton Court. In the gardens, you can walk through the oldest and most famous maze of all, known as "Wilderness" since the 16th century (Figs. 3 and 5). Models of it are favored for tests of new labyrinthsolving computer mice (and for real mice in ethology). On the top of Catherine's Hill near Winchester is a turf labyrinth cut into the ground. It dates back to the dawn of history. The labyrinths of Rocky Valley date from between the 18th and 14th century B.C. You reach them from Tintagel after an hour's walk along the breathtaking Cornish coast. It is a unique experience to follow, with your fingertips, the geometric lines, which someone carved into the stone nearly 4000 years ago (Fig. 4).
Figure 4. Mazes at Rocky Valley near Tintagel/Cornwall (with my wife). Photograph by K. Treitz, 1992. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993 5 5
M e t h o d s of M a z e S o l v i n g
Figure 5. Layout of the Hampton Court Maze.
It is fun to walk about in a maze, and usually in 15 or 20 minutes you will reach the center. But as a mathematician you will feel challenged to find it systematically. Usually you do not have a map and can gather only local information about the structure of the maze; then, in many cases, the rule of right-hand-on-right-wall will work. In a simple maze, you will then go along all the alleys twice, including the blind alleys. You will traverse the center and in the end you will return to the entrance. But if the maze is more complicated, this rule will fail. If the center is--like the asterisk in Figure 5--located between walls of islands (between walls that are not connected to the remaining walls), then you will only be led through some parts of the maze and back to the exit, without having reached the center. Moreover, if you make a mistake and cross over to the wall of an island, you would be condemned to walking around it for the rest of the day without finding the way out. There are better rules known, which allow you to thread any maze: the methods of Tarry and Tremaux. Let us represent the maze as a graph (Fig. 6). The junctions become the nodes and the paths the edges; we get a finite, connected graph. Then the method of Tremaux consists of the following rules ([2], p. 74). 1. No edge may be traversed more than twice. 2. When you come to a n e w knot, take any edge you like. 3. When you come along a new edge to an old knot, return along the edge by which you came. 4. When you come along an old edge to an old knot, take a new edge if possible, otherwise an old.
The Jubilee Maze Figure 6. Tremaux's algorithm.
Figure 7. Center of Jubilee Maze. Photograph by K. Treitz, 1992. 56
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
There are about 80 mazes in England, but one of the most attractive for the mathematical tourist is surely the Jubilee Maze at Symmonds Yat. It is situated 40 km north of Bristol on the border with Wales, in the beautiful valley of the river Wye. The maze has the octagonal form of a Labyrinth of Love, popular around 1600. It was set out by two young people, the brothers Edward and Lindsay Heyes, to celebrate the Silver Jubilee of Queen Elizabeth II in 1977. The cypress hedges are 2 m high. In the center stands a graceful stone pavillion (Fig. 7). The Heyes are there all day and are conspicuous in their Victorian white flannels, blazers, and straw hats. The second attraction in the Jubilee Park is a museum. In a very appealing exhibition "Mystery and Mazes," the history and the mathematics of different labyrinth types are presented. This is a "hands-on" museum. Visitors are invited to do several experiments and to construct various forms of labyrinth. The third attraction of the Jubilee Park is the show "Mazes and Micros," where you can see maze-solving
computers at work. The highlight of this exhibition is a maze-solving robot (Fig. 8). It uses an algorithm invented by N. Lee at the Bell Corporation ([3]). Adults and children are invited to build a maze from small boards on a table divided into squares. Then the technical function of the mouse is explained. It feels the walls around its present position with infrared sensors. The location of the walls is recorded in the "memorymap" in a Random-Access Memory (RAM) which can be updated. The maze-solving algorithm is stored in a Read-Only Memory (ROM). The program is carried out by a microprocessor, which controls the step-motors. At the beginning, the micromouse knows from its program that it points "South" in the Northeast corner (Fig. 9). Diagonal moves are not allowed. Finding the Way. The robot uses a readily understandable and efficient method. Imagine that the squares of the maze are like steps of a staircase: Each square is assigned a number which is the number of steps from that point to the goal . . . . One step for each square. If there is a path from start to goal and if the robot has a map of this path in its memory, it has simply to step d o w n the steps following this map. Updating the Step-Map. But how can the robot get a map? To begin with, it has a step-map in its RAM without any walls. Running through the maze, it checks the edges of ceils for walls. When a wall is found, the robot waits and the step-numbers of all the squares are recalculated to update the memory-map: This is done by testing all of the 25 squares in turn against squares accessible from them. If no step down is accessible from a square, then the robot's microprocessor adds one to the step-number for that square, until the step-number exceeds that of an adjacent square. For this operation, it needs only a fraction of a second. Then also the position of the discovered wall is recorded in the RAM-map: South, West, North, East of the square the robot is in at the time. Then it will go on to the next cell. Figure 9b shows the state of the step-map after the robot has reached the square (*) and the updating is finished. The method always results in a shortest route; and it is possible for the robot to skip over the investigation of whole parts of the maze. For instance, in our example it will never be concerned with the wall at (+); it will not even detect it! This algorithm is important for transport problems, for layouts of electronic circuits, and for pattern recognition. Mr. Heyes is mathematically knowledgeable and will also answer specialized questions. If you are accompanied by your family, it will be pleasant to find them sharing enthusiasm for your science. It should be mentioned that there is also an amazing puzzle shop, a pleasant restaurant, and a picnic area at the Jubilee Park.
Figure 8. Robot in the exhibition "Mazes and Micros." Photograph by K. Treitz, 1992.
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Figure 9. (a) Robot at start position. (b) Step-map after the eighth step of the robot.
References
1. Fisher, Adrian, and Gerster, Georg, The Art of the Maze, Weidenfeld and Nicolson, London 1990. 2. Hemmerling, A., "Labyrinth-Problem," alpha, Mathem. Schiller- zeitschrift, Berlin 17 (1983), pp. 73-75. 3. Lee, C.Y., "An Algorithm for Path Connections and its Applications," I.R.E. Transactions on Electronic Computers, Sept. 1961. Gymnasium Rheinfelden 7888 Rheinfelden Federal Republic of Germany THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
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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.
In this issue we welcome guest author Donald J. Newman who contributed the following lead item.
Thought Less Mathematics Donald J. Newman One type of problem that we all "teethed on" in our mathematical youth was the so-called weighing problem. We learned therein the valuable lesson of "branching" procedures: if this and this happens then we do that and that, but if it does not happen then instead we do such and such. These "branching" procedures emerged as a fundamental method in weighing problems, perhaps the right method for problems in general. Our minds were even tempted to go further. This might be the right path to follow for mathematics in general. And if it is right for mathematics, then it might be the right w a y to think altogether! WOW. In his recent article in the New Yorker magazine, Jeremy Bernstein pointed with admiration to the use of this branching reasoning as an index of real mathematical talent. The example he chose was the famous 12-coin problem, and the solver he pointed to was the then Harvard undergraduate Charles (Ariel) Zemach. So, as we already said, this branching reasoning appeared to be fundamental and near-universal.
Our purpose, however, is to deflate this notion! We illustrate with the 12-coin problem itself and a few other examples, and hope to convince the reader that this branching reasoning is perhaps never needed. Any time there is a solution using branching, there is another one which does not use it (and is, as a result, cleaner and simpler).
The 12-Coin Problem Twelve identical-looking coins are given, and we are told that one of them has a different weight from the other 11. The problem is to determine which coin it is and whether it is heavier or lighter, in only three weighings of these coins on a balance scale. Note first that even to describe a solution after one has found it would seem to require branching. The outcome of a weighing is that the left tray of the balance goes down or up or remains level. We encode these outcomes by 1, - 1 , and 0, respectively. Each weighing involves picking a pair of subsets L and R of the same cardinality and putting them on the left and right tray, respectively. The "flow diagram" for a weighing procedure would look like this:
* C o l u m n e d i t o r ' s a d d r e s s : D e p a r t m e n t of Mathematics, University of California, Berkeley, C A 94720 USA.
58
THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4 (~)1993Springer-VerlagNew York
t,efl
if...,t,e0 F
,/0 \_,
where for each L and R one would have to list the elements of the appropriate subsets. Compare this to the following nonbranching instructions. Let the coins be labeled A, B , . . . , L. Then do ABCD
EFGH I
I
0 A H I K
CD
EJ
I
0 CG
I L
ADFK
I
I
0 One easily verifies that this works. Namely, if A is the phony coin and it is heavy, then the outcome will be 1, 1, - 1 , and if it is light, the outcome will be - 1 , - 1 , 1. If B is the culprit and is heavy, then the outcome will be 1, 0, 0, and if it is l i g h t - - b u t wait a minute. We do not have to go through all this. All we have to do is notice that no two coins have the same or opposite "itineraries," that is, no two coins are always on the same tray or always on opposite trays. Therefore, for each of the 24 possible states (that is, which coin is counterfeit and whether it is heavy or light), there will be a different outcome; so given the outcome, we will know the state; for example, if we know the outcome is - 1 , 0, 1, then the coin G must be heavy. Note, by the way, that it makes no difference in which order the weighings are performed. Also note that we can solve a slightly harder problem in which we allow the possibility that none of the coins is counterfeit, which will be true if a n d only if the outcome is 0, 0, 0. But the question n o w is what sort of ingenuity was required to find these three weighings. The answer, none. We let the solution give itself. The real message we wish to impart then is that the old, complicated, clever solution was a waste of effort, a wrong attitude! The new solution reasons backward from the 27 outcomes to the 24 counterfeit possibilities. Here's how it goes. First, we make a list of 12 different outcome vectors, such that no outcome and its negative is on the list. A simple w a y to do this is to list the lexicographically positive vectors in lexicographic order, as in the columns of the following table. A
B
0 0 1
0 1 -1
C 0 1 0
D
E
F
G
H
0 1 1 1 1 1 -1 -1 -1 0 1 -1 0 1 -1
I
J 1 0 0
K 1 1 0 1 1 -1
L 1 1 0
N o w for the procedure to work we must have the same number of l's and - l ' s in each row. The bottom row is all right as it stands. Reversing the sign of column C fixes
up the middle row, and reversing columns F, H, J, and L takes care of the top row; so we have A
B
C
0 0 1
0 0 1 -1 -1 0
D
E
F
G
H
0 1 -1 1 -1 1 -1 1 -1 0 1 -1 0 1 1
I
J
K
L
1 -1 1 -1 0 0 1 -1 0 -1 -1 0
N o w each row of the table corresponds to a weighing. Namely, put the + l ' s on the left and - l ' s on the r i g h t - and there it is. Perform the weighings, write d o w n the outcome, and read off the guilty coin from the table. (The capital letters of the tables differ by a permutation from the ones given earlier, but this clearly makes no difference.) In some cases, non-branching solutions can be found easily. I have picked a n u m b e r between 1 and 8 a n d you must guess it by asking three yes-no questions. Of course, everyone uses the branching or "interactive" strategy of successive bisecting, but one need not do this. W h y not just ask in advance these three questions, in any order. "Is the number in the set {1, 2, 3, 4}? in the set {1, 2, 5, 6}? in the set {1, 3, 5, 7}?" Of course, this would not w o r k if the allowable question had to be of the form "Is the n u m b e r greater than x?" The example shows what is going on generally. There is a set of possible states and one wants to learn the true state. Every question, or weighing, or "experiment" gives a partition of this set. One defines the intersection of k partitions in the obvious w a y as the partition formed by all intersections of sets of the k partitions. Then the true state can be learned in n experiments without branching if and only if one can find n partitions whose intersection is the partition by singletons. For our next examples we have chosen two wellk n o w n problems involving four coins, each of which is a good or counterfeit one. N o w we do not k n o w h o w m a n y of each there are, and we are required to find exactly which are which.
Three W e i g h i n g s o n a True Scale Our "scale" will tell us, for any chosen subset, exactly h o w m a n y of them are good coins (not which ones but h o w m a n y of them). Our solution, again unbranched, is obtained by taking as our first subset coins 1 and 2; second subset, coins 2 and 3; third subset, coins 1, 3, and 4. So these are our three "weighings" and the m e t h o d of determination from the three answers, call them a, b, and c, is quite charming. We first add these three and get a + b + c, which counts coin 1 twice, coin 2 twice, coin 3 twice, but coin 4 only once. Thus, by reducing a + b + c m o d u l o 2 we obtain the nature of coin 4 itself. Then subtracting this off from c we obtain the balanced system for coins 1 and 2, coins 2 and 3, coins 1 and 3. This then determines the nature of the first three coins. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
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Seven Q u e s t i o n s to a Liar Back again to our four coins, but this time we may ask any yes-or-no questions about them. The person we are asking, however, is permitted to lie in response to (at most) one of these questions. As a result we must ask more than the obvious four questions we would need with a truth-teller. Indeed the correct number of questions needed, using branching, from this liar is seven. Once more our purpose is to achieve this same determination with seven nonbranching questions. Our first four questions are simply: Is 1 a good coin? Is 2 a good coin? Is 3 a good coin? Is 4 a good coin? The next three questions may seem to use branching, but, in fact, the questions do not though the answers do. Question 5. Were your answers to questions 1, 2, and 3 all correct? Question 6: Were your answers to questions 2, 3, and 4 all correct? Question 7: Were your answers to questions 1, 2, and 4 all correct? The determinations are obtained then as follows: If at most one of questions 5, 6, and 7 were answered NO, then all four of the first answers were true and the coins are determined. If answers to 5, 6, and 7 were YES, NO, NO, then question 4 was lied to, so negate that one and obtain the correct determination. Similarly, if the answers are NO, YES,NO, just negate the answer in question 1. If the answers are NO, NO, YES, negate answer 2. And, finally, if they are NO, NO, NO, then negate answer 3.
D .J.N. Temple University
More on Squaring Squares and Rectangles Squaring a square or rectangle means tiling (partitioning) the rectangle or square as a union of subsquares. For rectangles with commensurable sides, there are the trivial tilings [letting h be the height and w the width, if h = (p/q)w, then tile with a p-by-q array of squares of size h/p = w/q]. Of interest, however, are perfect tilings where no two squares are the same size. A second natural restriction is to require that no subset of squares form a subrectangle. Such tilings are called simple. Here is a brief and very incomplete chronology of some of the work on this problem. 1903. Dehn proves that if a rectangle is tiled by squares, the sizes of all squares must be commensurable. (By the size of a square we mean the length its sides.) 1925. Moron finds a perfect tiling of a rectangle by nine squares. (We call the number of squares the order of the tiling.) 1939. R. Sprague publishes the first example of a squared square. It has order 55. 60
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1940. Brooks, Smith, Stone, and Tutte prove that no perfect tiling of a rectangle can have order less than 9, and there are exactly two tilings of this order. 1948. Willcocks finds a perfect (but not simple) squared square of order 24. 1960. C. J. Bouwkamp and associates find 4094 simple rectangle tilings (3663 perfect, 431 imperfect) of order less than 16, including 2609 perfect ones of order 15. 1962. A. W. J. Duijvestijn proves that there are no simple perfect squared squares of order less than 21. 1978. Duijvestijn finds a simple perfect squared square of order 21 and shows it is the only one of that order. 1992. Bouwkamp and Duijvestijn publish an illustrated "catalogue" of all simple perfect squared squares (up to obvious symmetries) of orders 21 through 25, containing 207 tilings, one of order 21, 8 of order 22, 12 of order 23, 26 of order 24, and 160 of order 25. If one takes the square being tiled to be the unit square, then, from Dehn's result, the size of all the subsquares will be rational, so one can normalize by a uniform stretching making all sides integers which are relatively prime. The size of the big square is then called the "reduced size" of the tiling. Of the 160 tilings of order 25, the smallest has reduced size 147, the largest 661. Even 30 years ago one knew more than 2600 perfect tilings of rectangles by 15 squares, so one would expect the number of tilings of rectangles of order 25 to be very large indeed, especially if one does not require the tilings to be simple or perfect. Bouwkamp says that there are about 5,000,000 perfect simple squared rectangles to every such square (for order greater than 20)! Can one even be certain that as the order gets larger the number of tilings remains finite? The answer is yes, but the proof does not seem to be completely obvious. However, it might fit nicely as a nonroutine application in an undergraduate course in linear algebra. Further, the proof gives, as a by-product, Dehn's theorem. Extend the horizontal sides of all the tiles. The regions between consecutive lines will be called strips. In this figure, there are nine squares and five strips. If there are m strips and n squares, we construct the m-by-n
2
3
5
""4 6 8
The original example of Moron.
horizontal intersection matrix A of the tiling b y the rule: aij is 1 if strip i meets the interior of square j and is 0 otherwise. The intersection matrix for the figure shown is 1 2 3 4 5 6 7 8 9 1 1 1 0 0
1 0 0 0 0
1 1 0 0 0
0 1 0 0 0
0 1 1 0 0
0 0 1 1 0
0 0 0 1 1
0 0 0 1 0
We rise out of disorder into order, and the poems that I make are little bits of order. If I make a basket or a piece of pottery and a vase or something .... If you suffer any sense of confusion in life the best thing you can do is make little forms, blow cigaret smoke rings (even those have form, you know) .... To this list a mathematician might add, "or prove a theorem."
0 0 0 0 1
Let x (Xl,X2,...,Xn) be the vector w h o s e components are the sizes of the n tiles, and let y = (yl, Y2,.-., Y,~) be the heights of the m strips. Let 1 be the m-vector all of w h o s e entries are l's. =
Correction: In the column on "the industrious ant" in the Spring issue, the proof that the Ant's trajectory is always u n b o u n d e d should have been attributed to Buminovich and Troubetzkoy rather than C o h e n and Kong.
T H E O R E M (for Sandra). The vectors x and y are determined, up to multiplication by a constant, by the matrix A,
and are the unique solutions of the equations ~a~Wy -----1
and
x = ATy.
(1)
The height, hence width, of each square is the sum of the heights of the strips which meet it, so b y the definition of A, x = ATy. Also, the sum of the widths of all squares which meet a given strip must be the width w of the rectangle, which we m a y take to be 1; so, again by the definition of A, w e must have Ax = AATy = 1. It remains to s h o w that AA T is nonsingular so that y and x are unique. We s h o w first that A has rank m. To see this, note that for a n y two consecutive strips, say, k and k + 1 reading d o w n w a r d , there is a square that intersects the first but not the second. The corresponding column of A will have a 1 in row k and O's in all subsequent rows. Choosing such columns for all k (columns 2, 3, and 5-7 in the earlier exampIe), we get an u p p e r triangular square submatrix with l's on the diagonal which is, therefore, nonsingular. It is then a familiar fact that AA T is nonsingular. (This is, indeed, exactly the result used in obtaining the formula for least squares approximations; see, for example, Strang, Linear Algebra and its Applications, p. 102. Here it comes u p in quite a different context, although this one also involves squares!) The finiteness r o w follows; Given the n u m b e r of tiles, there are only finitely m a n y possible incidence matrices, and given the sizes of the squares there are only a finite n u m b e r of possible tilings. Dehn's Theorem follows: the matrix A is rational, so the unique solution of (1) must be rational Because the T h e o r e m is proposed for a linear algebra text, the next thing w o u l d be,
Exercise: Use (1) to find the sizes of the nine squares in Moron's example.
The Two Cultures The following is from a TV interview with the poet Robert Frost. THE MATHEMATICAL INTELLIGENCERVOL. 15, NO. 4, 1993
61
Jet Wimp*
A Basic Course in Algebraic Topology by William Massey New York: Springer-Verlag, 1991. xvi + 428 pp. US $65, ISBN 0-387-97430-X.
Reviewed by Peter Hilton One of the problems encountered in a systematic exposition of algebraic topology is deciding on a suitable category of spaces to be studied. If the category chosen is too narrow and restricted, the theorems are not likely to be applicable in other parts of mathematics. On the other hand, if the category chosen is too broad and inclusive, many of the theorems one desires to prove will become very difficult or false.... We have preferred to emphasize CW-complexes rather than simplicial complexes .... Another choice occurs in the actual definition of singular homology groups. Should one use singular simplices or singular cubes? From a strictly logical point of view it does not matter because the resulting homology and cohomology theories are isomorphic in all respects. From a pedagogical point of view, it does make a difference, however. So, stating his case with typical lucidity, Bill Massey makes his choices: CW-complexes and cubical singular homology theory. The result is, predictably, an excellent book. As the author admits, however, other choices are possible. I cannot resist pointing out one, taken by Shaun Wylie and myself 40 years ago, namely, one may adopt the principle: When a fundamental alternative presents itself, treat both possibilities, but in a carefully selected order.
(,)
* C o l u m n Editor's a d d r e s s : D e p a r t m e n t of M a t h e m a t i c s , Drexel University, Philadelphia, PA 19104 USA.
62
The reader must be assumed - - this is surely not too opt i m i s t i c ! - to grow in maturity as he or she studies the text, and thus to become, at some stage, ready for a more sophisticated concept or treatment. Accordingly, in [1] the authors started with simplicial complexes and simplicial homology theory, confining themselves almost exclusively in Part I to the algebraic topology of polyhedra. They moved on to the singular theory for general topological spaces only in Part II. Besides, they began their discussion of singular theory by using simplicial singular theory and only proceeded to cubical singular theory when the payoff was unmistakable, namely (as very explicitly and compellingly observed by Massey), in the study of topological products and of the cohomology ring. I cannot entirely agree with Bill Massey that it does not matter "from a strictly logical point of view" whether one uses singular simplices or singular cubes. For if one uses singular cubes, one is compelled to factor out the degenerate cubes (however defined) to get the "right" homology groups, whereas, in the simplicial theory, one has a choice whether to do so or not. Can the theories be truly "isomorphic" with this substantial difference between them? Of course, if we simply mean that there is a natural isomorphism between the homology groups based on singular cubes and the homology groups based on singular simplexes, then the theories are equivalent, but, to most of us, a claim of "isomorphism" is a stronger claim than this. It would be hard to sustain the view that the formula for the boundary of a cube is exactly like the formula for the boundary of a simplex. The simplicity of the latter springs from the fact that a simplex is entirely and most naturally specified by the set of its vertices, an orientation of the simplex by an ordering of its vertices, and a face of a simplex by a subset of its vertices. A cube has, compared with a simplex, the massive disadvantage that a subset of its vertices may not span a face, indeed,
THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 4 (~)1993Springer-VerlagNew "~ork
usually will not. Is it just special pleading on my part to find here a perfect opportunity for the implementation of the principle (,) enunciated earlier, that is, for the prior presentation of the simplicial theory, followed by the presentation of the cubical theory precisely when it is most useful? It is relevant here to recall that Serre, in his sensational thesis introducing spectral sequence theory into the study of the homology and cohomology of fiber spaces, exploited the cubical theory to wonderful effect; but Eilenberg just as effectively used the simplicial theory to lay the foundations of obstruction theory. It will be very interesting to observe the success of Massey's experiment in avoiding the simplicial theory altogether. I believe that it will create casualties. But I believe the survivors will be well prepared to proceed further toward the frontiers of algebraic topology (though, at the pace at which these are currently receding, some may never reach them). Massey will have not only his vast experience but also the corroboration of many experts to support his choice of strategy. I recall Serre's comment when I told him that Wylie and I had decided to highlight the simplicial theory of polyhedra: "There are two important homology theories, the singular theory and the Cech theory. You have chosen their intersection." We were unrepentant, believing that the conceptual simplicity of the simplicial theory of polyhedra would be a help to the beginning student of algebraic topology. Also, we were convinced there would be a significant computational advantage with regard both to the homology groups, because these would be based on finitely generated chain complexes for compact polyhedra, and to the fundamental group, because a system of generators and relations could be read off the 2-skeleton of a simplicial complex. I find some implicit support for this point of view in the text under review. Massey begins the calculation of the fundamental group of the circle in Chapter II (p. 49), but only completes it in Chapter V (p. 126). Again, he p r e s e n t s - - v e r y attractively--the homology theory of CW-complexes but specializes to regular CW-complexes (we used to call these Steenrod cell complexes) and even "almost simplicial complexes" when he wants to make computations. Here again, in considering, from an expository point of view, the relative merits of simplicial complexes and CW-complexes, there is a good opportunity to apply the principle (,) of treating first one and then the other because both have merit, but there is a distinct difference in level of sophistication between them. Simplicial complexes, like simplexes themselves, are conceptually very simple and natural. An n-simplex is simply the convex hull of a set of (n + 1) independent points or vertices (in some high-dimensional Euclidean space) and, thus, the natural generalization of a triangle (n = 2). Moreover, as already pointed out, a face is just the convex hull of a subset of the vertices, and the boundary of an oriented n-simplex is the sum of its (n - 1)-faces, appropriately oriented. On the other hand, a (closed) n-cell in a White-
head CW-complex X is the continuous image of a closed n-ball B n under a map f which is a homeomorphism on the complement of the boundary of the ball and sends the boundary into the (n - 1)-skeleton X n-1. This is a much more sophisticated notion, as I am sure many of m y appalled readers will agree. There is a tremendous advantage in using CW-complexes, of c o u r s e - - a s Henry Whitehead understood and as Bill Massey sets in unmistakable e v i d e n c e - - o n e uses far fewer cells in giving a space a cellular structure than in giving it a simplicial structure. For example, the n-sphere S n requires only a 0-cell e~ and an n-cell en (so that e~ is the "boundary" of en), whereas a simplicial triangulation of S n, say as the surface of a (n + 1)-dimensional ball, would require
Massey's experiment in avoiding the simplicial theory altogether will create casualties. 2 n+2 -- 2 simplexes. However, calculation of the homology and cohomology groups of a compact polyhedron by means of a simplicial structure on the polyhedron is entirely algorithmic once the structure has been chosen, whereas there may be some highly nontrivial difficulties in trying to exploit a CW-structure to calculate these groups. It is not surprising that Massey prudently restricts attention to "almost simplicial complexes" when it comes to actually making calculations.* There are, to be sure, other areas of choice when one writes "A Basic Course" in so vast and deep a subject as "Algebraic Topology." What should one prove? When should one refer to published work? I believe that a basic course should be as self-contained as possible, even at the cost of extra length; Massey sometimes obliges the reader to consult other publications and sometimes reproduces what is readily available elsewhere (e.g., his treatment of the Hopf Invariant). H o w general should one be? And, if one plans to be general, should one first deal with somewhat special, but easier, cases? My answer to the second question is (of course!) "Yes, provided the special cases occur sufficiently often." This is not Massey's strategy. Thus, in his very valuable chapter on "Duality Theorems for the Homology of Manifolds," we find (p. 351), "Let M be an arbitrary n-dimensional manifold; we emphasize[my italics] that M need not be compact or connected; in fact, we do not even need to assume that M is paracompact! [his exclamation mark]." Again (p. 360): "Let M be an n-dimensional manifold with orientation #; we stress [my italics] that we do not need to assume that M is compact, connected, or even paracompact. Also, we do not need to make any hypothesis of * I have quoted Massey as declaring that he has preferred "to emphasize [my emphasis] CW-complexes rather than simplicial complexes." In fact, simplicial complexes are not mentioned at all in the text! They appear shamefacedly in the index with the instruction "See almost simplicial complex"!
THEMATHEMATICAL INTELLIGENCERVOL.15,NO.4,1993 63
triangulability or differentiability." As a result, Massey must introduce (p. 358) cohomology with compact supports to state Poincar6 duality, and Poincar6 duality for compact (=closed) manifolds appears only on p. 365 as an "application.' These choices concern matters of taste and style. What is not in dispute is that Massey has produced a "basic course" that contains all the important ideas of homology t h e o r y - - except obstruction theory and the theory of spectral sequences-- treated with meticulous care and thoroughness. He has, indeed, refuted the false view that "algebraic topology is a subject that is esoteric, specialized, and disjoint from the general sweep of mathematicaI thought." He has achieved readability with no concessions to superficiality; and, in providing historical notes after most chapters (why are there none for Chapters IX and X?), he has added greatly to the reader's interest and potential understanding. I must add that it is particularly regrettable that a text written with such care for the needs of the reader should be marred by such a plethora of typographical errors. (Where was Bill when the proofs were being read?) To mention but three, each remarkably conspicuous: On p. 4, line 6, somebody (the copy-editor?) has inserted a mischievous comma after "However" in the clause "However we define the M6bius strip," with a devastating effect on the intended meaning. On p. 42, line -3, "homomorphism" should be "homeomorphism," and, on line -2, "homeomorphism" should be "isomorphism" - two errors calculated to sow great confusion in the ranks of inexperienced graduate students. On p. 308, the following strange couplet appears:
The Apprenticeship of a Mathematician by Andr6 Weil translated from the French by Jennifer Gage Birkh/iuser Verlag, Basel (1992) 197 pp., US $29.50 ISBN 3-7643-2650-6 and 0-8176-2650-6
Reviewed by Lawrence Zalcman
1. Hilton, Peter, and Shaun Wylie, Homology Theory, Cambridge University Press 1950 (reprinted 1965).
The Apprenticeship of a Mathematician (Souvenirs d'apprentissage in the original French) is Andr6 Weil's memoir of his first 40 years. Autobiographies of great mathematicians are in sufficiently short supply that the appearance of this volume will inevitably evoke comparison with other examples, most notably the books by Norbert Wiener [6], [7] and Stan Ulam [4]. Wiener, Ulam, and Well were very different sorts of personalities and very different sorts of mathematicians as well; but the similarities between them seem no less striking than the differences. All three were born into highly assimilated, intellectual Jewish families of the upper middle class. Both Wiener and Weil remained in the dark about their Jewish origins until their early teens. 1 (Presumably such a feat would have been impossible in Ulam's Poland.) All three married around the age of 31 and enjoyed extraordinarily happy marriages. Coincidence? No doubt, but it is amusing to speculate that something more might be involved as well. Well had an extremely precocious youth, though he was perhaps not a prodigy in quite the same sense as Wiener. His passions were mathematics and classical languages: Latin, Greek (which he taught himself when he was 11), and Sanskrit. Already at the age of 15 he had made the acquaintance of Hadamard and Sylvain L4vi, the distinguished orientalist. These men became, and remained, his intellectual heroes. At 16 he entered the Ecole Normale and began attending Hadamard's legendary seminar at the Coll~ge de France. At the Ecole Normale, Weil made friendships that were to last a lifetime: Henri Cartan, Chevalley, Delsarte. He also found time to attend several lecture courses on Sanskrit grammar and literature and IndoEuropean linguistics at the Sorbonne and the Coll6ge de France. Hadamard remained a major influence: "The bibli [library at the Ecole Normale] and Hadamard's seminar ... are what made a mathematician out of me" (p. 40) a as well as the standard against whom he measured others and himself. Thus, Vito Volterra, "It]hough probably a less universal mathematician than Hadamard ... was an admirable man in all respects" (p. 47), while Erhard Schmidt's "keen mind was nearly comparable to H a d a m a r d ' s - - a n d to say this is no mean compliment" (p. 53). Weil himself conceived
Department of Mathematical Sciences State University of New York at Binghamton PO Box 6000 Binghamton, NY 13902-6000 USA
1Wiener's discovery that he was Jewish came as a great shock and played a major role in his development; he devotes an entire chapter of [6] to it. Well,on the other hand, "certainlydid not considerit of any importance" (p. 42).
will all the usual properties. For all the usual properties: seeming, perhaps, to come from an Ezra Pound c a n t o - but all that the author intends is "with all the usual properties." (There is some strange numerology in this book, too. I supply the observation that Graduate Text 127 is just a little more than Texts 56 + 70. Springer-Verlag supplied the mysterious information "With 57 illustrations in 91 parts,"--are they thereby acknowledging a gruesome act of vandalism?) But let not the final note of this review be critical. Bill Massey has performed many splendid services in his career as researcher, teacher, and writer; this is one of the best. We are in his debt.
Reference
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the ambition of becoming, like Hadamard, a "universal" mathematician: the way I expressed it was that I wished to know more than non-specialists and less than specialists about every mathematical topic. Naturally, I did not achieve either goal. (p. 55) On the other hand, he could ignore the advice of the master when such practical matters as presenting a thesis were at issue (p. 57). Even before obtaining his doctorate, Weil travelled extensively in Italy and Germany. In Rome, he became a familiar at the home of Volterra, with whose son Edoardo (later an extremely distinguished jurist) he formed a close friendship. Weil's stay in Germany the following year provides the background for some of the most incisive verbal snapshots in the book. Here, for instance, is Erhard Schmidt: Erhard Schmidt received me like the true aristocrat he was. On his mantelpiece was a marble bust of him. It resembled those of the ancient Roman patricians, and he resembled his bust. (p. 53) By contrast, Hartogs (whose work Weil greatly admired) was a "timid, self-effacing man with the look of a harmless rodent" (p. 56). In Frankfurt, Weil found an extremely congenial and close-knit group of mathematicians, led by Max Dehn. There he also met Carl Ludwig Siegel, "already a legend," and their long mutual admiration began. But it was Dehn who made the deepest impression. Max Dehn ...--like Socrates as we picture him from the accounts of his disciples-- possessed a radiance that makes one naturally bow down before [his] memory: a quality, both intellectual and moral, that is perhaps best conveyed by the word "wisdom"; for holiness is another thing altogether. In comparison with the wise man, the saint is perhaps just a specialist-- a specialist in holiness; whereas the wise man has no specialty .... for such a man, truth is all one, and mathematics is but one of the mirrors in which it is reflected-- perhaps more purely than it is elsewhere. (p. 52) Not all the portraits are so flattering. Weil does not seem to have gotten on well with administrators (pp. 83, 111) or other authority figures (pp. 143, 155, 158); and Courant, who was in charge at G6ttingen, was no exception. "It has sometimes occurred to me," confesses Weft, "that God, in His wisdom, one day came to repent for not having had Courant born in America, and He sent Hitler into the world expressly to rectify this error" (p. 50). And very occasionally, as in his description of Paley (from a slightly later period), Weil's comments on others contain a flash of self-revelation. Our conversation turned to a comparison of our approaches to work. At first, we seemed to be on completely different wavelengths. Finally, it became apparent to me that he worked fruitfully only when competing with others: having the rest of the pack at his side spurred him to greater efforts as he tried to surpass them. In contrast, my style was to seek out topics that I felt exposed me to no competition whatsoever, leaving me free to reflect undisturbed for years. (p. 94)
One expects mathematicians to rub elbows with other mathematicians, but it is often from their interactions with non-mathematicians that we learn most about them. Ulam's work at Los Alamos brought him into contact with physicists of the caliber of Bethe, Fermi, Feynman, Gamow, and Teller. Weil's trajectory crossed those of people no less eminent, but from fields very far removed from the exact sciences: Georges Dum6zil, the doyen of Indo-European studies; the great classicist Wilamowitz, whose very last lecture Weil attended in Berlin; Paul Val6ry; W.H. Auden (who vetted the English of the introduction to Weil's Colloquium volume on algebraic geometry); and Claude L6vi-Strauss, who The saint is a specialist in holiness; the wise m a n has no specialty. managed to coax Weil into the only applied mathematics of his career. Weil's peregrinations once brought him into the same room with Hitler (p. 95); at roughly the same time, Trotsky was sleeping in his bed in Paris (p. 97). Weft obtained his doctorate in 1928, with Picard chairing the thesis committee and Garnier submitting the official report. He did his compulsory military service and then spent two years in I n d i a - - Sylvain L6vi had helped him obtain a position at Aligarh Muslim University where he became fast friends with Zakir Husain (who was later to be president of India), discussed politics with Nehru, and had tea with Gandhi. He also proved a result that garnered him the most flattering praise of his entire career (pp. 91-92). Back in France, Weft was present at the birth of Bourbaki, whose history he recounts in loving detail. The reader will search this review in vain for the secrets of Bourbaki divulged in this volume. To learn Bourbaki's nationality (p. 102), or his daughter's name (p. 133), or whether he has a big white beard (p. 102), or even the proper adjectival form of Bourbaki (p. 149), you will simply have to read the book (or, at least, glance at the pages indicated). Photographs from the Bourbaki "congresses" of 1936, 1938, and 1951 should enable math trivia fans to reconstruct the composition of Bourbaki at various stages of its corporate identity. Some mathematicians, like Beurling and Turing, spent World War II breaking enemy codes. Others, like von Neumann and Ulam, helped build the atom bomb. Weil spent the war, at least several months of it, in jail. Trapped in Finland at the onset of hostilities with Russia, he was imprisoned by the Finns as a spy and escaped being shot only through the intervention of Nevanhnna (p. 134). Repatriated to France, he was imprisoned by the French as a deserter and again narrowly escaped being s h o t - this time by the wardens of the Rouen prison, who were reluctant to show their flight before the advancing Germans with the extra burden of prisoners (p. 152), THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
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Actually, the nearly three months Weil spent in the military prison at Rouen were a m o n g the most mathematically p r o d u c t i v e periods of his life. H e r e is h o w he describes them in a letter to his wife. My mathematics work is proceeding beyond my wildest hopes, and I am even a bit worried--if it's only in prison that I work so well, will I have to arrange to spend two or three months locked up every year? In the meantime, I am contemplating writing a report to the proper authorities, as follows: "To the Director of Scientific Research: Having recently been in a position to discover through personal experience the considerable advantages afforded to pure and disinterested research by a stay in the establishments of the Penitentiary System, I take the liberty of, etc. etc." [...] As for my work, it is going so well that today I am sending Papa Cartan a note for the Comptes-Rendus. I have never written, perhaps never even seen, a note in the ComptesRendus in which so many results are compressed into such a small space. I am very pleased with it, and especially because of where it was written (it must be a first in the history of mathematics) and because it is a fine way of letting all my mathematical friends around the world know that I exist. And I am thrilled by the beauty of my theorems .... (pp. 146-147). After additional adventures, the Weils finally managed to reach the United States. An earlier visit to the Institute for A d v a n c e d Study in 1936/7 had left Andr6 "quite pleased with [his] stay in Princeton, but with no particular wish to come back" (p. 117). N o w he was to learn that "Americans, w h o so w a r m l y welcome those w h o do not need them, are m u c h less hospitable to those w h o h a p p e n to be at their mercy" (p. 128). It was not an easy lesson. The low point came during 1942-44. The m a n who, just five years earlier, had considered himself "not unworthy" of succeeding H a d a m a r d at the Coll6ge de France (p. 120) n o w f o u n d himself an assistant professor at Lehigh University in Bethlehem, Pennsylvania. Even this position he obtained only after haggling; the original offer was for an instructorship. The m e m o r y of these years still rankles; and, even now, Weil cannot bring himself to mention Lehigh b y name. The institution to which I belonged (a word that all too accurately describes my relationship with my employer) was graced with the noble title of "university"; but in fact, it was only a second-rate engineering school attached to Bethlehem Steel. The only thing expected of me and my colleagues-who were totally ignorant so far as mathematics w e n t - - was to serve up predigested formulae from stupid textbooks and to keep the cogs of this diploma factory turning smoothly. (p. 180) During this period, Weil must often have t h o u g h t of Herm a n n Weyl's offer (p. 178) to use his influence to have him put in prison a g a i n - - a n d wished that it had not been m a d e in jest. Weil's attempts to extricate himself p r o v e d unavailing. Lefschetz seems to have played some role in frustrating these efforts, and Well has never forgiven him for it. He doesn't miss a chance to offer a disparaging c o m m e n t on 66
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Lefschetz, placed, if possible, conveniently in the m o u t h of authority (pp. 49, 154). H e even accuses him of antiSemitism: Lefschetz was reputed to be among those Jews who, to avoid any accusation of favoring their co-religionists, go to the other extreme and display patently anti-Semitic behavior. (p. 183). This last charge has been heard before but still s o u n d s a bit strange, in view of the extremely problematic relationship to Judaism of Weil's o w n sister, the noted writer Simone Weil. 2 Stranger yet is Well's coyness concerning Pontrjagin's virulent anti-Semitism (pp. 106-107); here, at least the translator has had the sense to add a footnote m a k i n g things plain. Strangest of all, however, a n d really grotesque b e y o n d words, is Well's choice of Shafarevich, of all people, to cite with a p p r o v a l concerning the possibility of resisting anti-Semitic legislation (p. 171). 3 All in all, anti-Semitism is one subject on which Well seems surprisingly ill-informed. Take, for instance, his characterization (p. 42) of anti-Semitism as the "complem e n t a r y concept" to Judaism, w h a t e v e r that m a y mean; replace "anti-Semitism" b y its s y n o n y m "Jew-hatred," and it becomes evident that this is complete nonsense. Similarly, Weil's view that (gentile) anti-Semitism is the result of e n v y of Jewish success (pp. 80-81) ignores a two-thousand-year legacy of theological, economic, ethnic, racial, and political hostility to Jews. Coming at the end of the Twentieth Century, such a statement is little short of astonishing. On the other hand, Well's grasp of the practical aspects of anti-Semitism was sufficiently sure for him to have urged, "Make sure the lawyer isn't n a m e d Levy or Cohen" (p. 140) w h e n he n e e d e d an advocate to defend him before the military tribunal. One w o n d e r s what he might h a v e m a d e of such a sentence had it emerged from the m o u t h of Lefschetz! Eventually, through H e r m a n n Weyl's intervention, Weil received a G u g g e n h e i m Fellowship and so was delivered from Lehigh. H e also m a n a g e d to obtain a position at the University of Sao Paulo, which he was to o c c u p y until 1947, w h e n he settled p e r m a n e n t l y in the United States, first at the University of Chicago and later
2 For which, see Thomas R. Niven's study, Simone Weil: Portrait of a Self-Exiled Jew (Chapel Hill: University of North Carolina Press, 1991), reviewed by George Steiner in the New Yorker (March 2, 1992). In his review of the same book in the JerusalemReport(February6,1992),Hillel Halkin opines that Simone Weil was "arguably the most self-hating Jew since Paul of Tarsus." Undoubtedly, this is a wild exaggeration. However, if you replace "Paul of Tarsus" by "Otto Weininger," the statement would not be too far off the mark and might even be literally true. 3 I.R. Shafarevich,a distinguished Russian mathematician, was a leading dissident during the Sovietperiod and gained many admirers both for his mathematics and for his brave behavior during difficult times. In the last decade, however, he has emerged as intellectual godfather of a resurgent popular anti-Semitism in Russia. See The Mathematical Intelligencer, vol. 12, no. 3, p. 4; vol. 14, no. 1, pp. 61-62; vol. 15, no. 1, pp. 4-5; and sources cited there. -- Editor
at the Institute for A d v a n c e d Study. But this is going outside the time-frame of the book. Weil's narrative ends, rather abruptly, as he puts into Curacao on his w a y back to Rio; it is August 7, 1945, and a m u s h r o o m cloud floats over Hiroshima. The Apprenticeship of a Mathematician contains a great deal more than the d r y details of Andr6 Weil's life. The more includes some of Weil's poetry, including a mathematical sonnet (pp. 118-119); two proverbs (pp. 135,151), only one of which is appropriate for publication in a family newspaper; a witty Latin saying, attributed to Gauss, comparing the pleasures of mathematical invention and sex (p. 91); a pragmatic refutation of Kant's categorical imperative (p. 124); Weil's views on the p r o p e r way to teach fractions in elementary school (p. 150); and seven photographs of Andr6 (five of them with Simone) between the ages of five and sixteen. Here and there, as on p. 99, the reader m a y even glean some insight into w h a t m a d e his sister tick. What is missing, and sorely missed, is any real discussion of the mathematics. Most readers of this b o o k are likely to be professional mathematicians. In view of the author's not inconsiderable talents as an expositor, as well as a creator, of mathematics, it is a pity that he did not make the effort to share with his readers w h a t has been most precious to him: his mathematics. Mark Kac did it in his a u t o b i o g r a p h y [2] and almost m a n a g e d to carry it off. In writing an autobiography, the temptation to put one's brand on some "big ones that got a w a y " seems well-nigh irresistible. Wiener succumbs to it w h e n he recounts four times [6, pp. 224, 281], [7, pp. 60, 93] (cf. [4, p. 137]) h o w he invented Banach spaces, i n d e p e n d e n t l y of, and almost simultaneously with, Banach; that was his first and last contribution to the subject. Weil has a similar tale to tell about the ergodic theorem. In his seminar, [Hadamard] often emphasized what was then called the "ergodic hypothesis." On this topic, he had never gone beyond Poincar6 and Boltzmann. Even before leaving France, I had thought of applying von Neumann's recent work on unitary operators in Hilbert spaces to these problems. When I spoke to von Neumann about it in 1931, it seemed to me that the idea had not yet occurred to him, and he expressed interest in it. I thought it was a big step forward when I conjectured the truth of what is called the ergodic theorem in the L 2 s e n s e . When I spoke to Elie Cartan about this, he objected that this finding was too general and imprecise to be really useful in the study of differential equations, and I ended up being convinced that he was right. (p. 90) This just goes to s h o w that, as Bertrand Russell was w o n t to say, "You should not have too m u c h respect for distinguished men." Jennifer Gage's translation is, on the whole, smooth, if not elegant. Occasional infelicities are m o r e than compensated by the frequent translator's notes, most of which deal with aspects of French culture unlikely to be familiar to the English reader. When it comes to other languages, however, the reader is on his own. Many, per-
haps most, will identify the Italian of p. 191 as adapted from the first line of the Inferno; one imagines that far fewer will recognize the Latin of p. 11 as (a slight m o d ification of) Catullus viii.8. The translator's prudishness (the author is anything but prudish) m a y occasionally bring a smile to the reader's lips: twice (on p. 114 and again on p. 119) she rendersfoutu as " f . . . ed." There are the usual misprints one has come to expect and learned to endure; the first appears on the back of the title page, which gives the original French title as Souvernirs d'apprentissage. Such an inauspicious beginning prepares us for "consitituted" (p. 58, 1. - 1), "solliciting" (p. 60, 1.10), "offical" (p. 83, 1.-10) and "attribrute" (p. 113, 1. - 6). M y dictionary lists half-a-dozen acceptable ways to spell "parakeet" (and the OED lists two d o z e n more), but "parrokeet" (pp. 68, 70, 151) is not one of them. Someone also needs to teach Birkh/iuser's c o p y editor h o w to handle inner quotes (p. 140, 11. 9-11) and the p r o p e r use of the word "transpire" (pp. 138, 143). Readers of The Apprenticeship of a Mathematician will u n d o u b t e d l y want to learn more about Professor Weil's views on mathematics and mathematicians. A g o o d place to start is with his article "The Future of Mathematics" [5], written in 1946 and published (in an authorized translation) in the American Mathematical Monthly in 1950. Only Weil could have written it; and, even today, it richly repays careful reading. Weil found evidence for the "robust vitality" of mathematics in the continuing s u p p l y of great problems with which it is endowed. But, if logic is the hygiene of the mathematician, it is not his source of food; the great problems furnish the daily bread on which he thrives. "A branch of science is full of life," said Hilbert, "as long as it offers an abundance of problems; a lack of problems is a sign of death." They are certainly not lacking in our mathematics; and the present time might not be ill chosen for drawing up a list, as Hilbert did in the famous lecture from which we have just quoted. Even among those of Hilbert, there are still several which stand out as distant, although not inaccessible, goals which will continue to suggest research for perhaps more than a generation; an example is furnished by his fifth problem, on Lie groups. H e was m u c h less sanguine about the prospects of American mathematicians for participating in these developments. [L]et us merely indicate how the future mathematician is formed in this country which produces more "mathematicians" than perhaps all the rest of the world. In the most favorable cases, one sees a student who, towards the end of his stay at the University, has three or four years at his disposal in which to acquire at the same time the knowledge, the method of work and the elementary intellectual apprenticeship, for which nothing that he has experienced before has in the least prepared him. His only way out under these circumstances is to seek his salvation in the most narrow specialization; in this way, he can often, if he is intelligent and has good guidance, do useful work. Beyond this, he runs the great risk of not being able to survive the stupefying effects of the purely mechanical teaching which he will have to inflict on others, in order to earn his living, after having undergone it himself for too long a time. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4,1993
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Within two years of the appearance of these lines Hilbert's fifth problem had been s o l v e d - - b y three American-born, American-trained mathematicians [1], [3]. "A well-written Life," said Carlyle, "is almost as rare as a well-spent one." The Apprenticeship of a Mathematician is the well-written account, volumen non illepidum neque invenustum, of (half) a life well-spent. Dare we hope for a sequel?
References 1. Andrew M. Gleason, Groups without small subgroups, Ann. of Math. (2) 56 (1952), 193-212. 2. Mark Kac, Enigmas of Chance, New York: Harper & Row (1985). 3. Deane Montgomery and Leo Zippin, Small subgroups of finite-dimensional groups, Ann. of Math. (2) 56 (1952), 213241. 4. S.M. Ulam, Adventures of a Mathematician, Berkeley: University of California (1991). 5. Andr6 Weil, The future of mathematics, Amer. Math. Monthly 57 (1950), 295-306. 6. Norbert Wiener, Ex-Prodigy, Cambridge: M.I.T. (1964). 7. Norbert Wiener, I Am a Mathematician, Cambridge: M.I.T. (1964).
Department of Mathematics and Computer Science Bar-Ilan University 52900 Ramat-Gan, Israel
The Illusion of Reality by Howard Resnikoff New York: Springer-Verlag, 1989. viii + 339 pp. US $54.00, ISBN 0-387-96398-7
The Geometry of Vision edited by Robert Melter, Azriel Rosenfeld, and Prabir Bhattacharya Providence, Rh American Mathematical Society, 1991. viii + 237 pp. US $90.00, ISBN 0-8218-5125-X
Visual Structures and Integrated Functions edited by Michael Arbib and Ji~rg-PeterEwert New York: Springer-Verlag, 1991. xii + 441 pp. US $49.00, ISBN 0-387-54241-8
Reviewed by Shimon Edelman The first steps of a person who embarks on a study of visual perception are difficult for many of the same reasons that we find it difficult to imagine how our mother tongue sounds to someone who does not understand it, and what it would be like for someone to learn it. We are 68
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so loath to acknowledge the difficulty of learning our own language that in speaking to a foreigner we tend to raise our voice, as if unconsciously assuming that we are not understood merely because we are not heard clearly. In this case, as well as in thinking about vision, the first obstacle to be overcome is the conviction that what seems so easy and natural cannot be too complicated. Such misconceptions regarding everyday linguistics can be easily cured by trying to learn a foreign language. In the study of vision, a good way to appreciate the difficulties involved in seeing is to try to program a computer to see. It becomes apparent that making a computer see is harder than making it solve other informationprocessing tasks, some of which were once thought to epitomize intelligence (for example, playing chess at the level of a grand master). This realization is apt to bring with it a profound sense of wonder at the performance of our visual systems. Our eyes provide us with a stable and reliable impression of the surrounding world, and do so in the face of the ever-changing illumination and vantage point, and despite the limitations of the available information-processing units, each of which is slow and unreliable. As Marvin Minsky once remarked, the reliability of our visual systems is constantly tested throughout our lifetimes. The degree to which they stand up is manifest in the extreme rarity of situations in which silly behavior, such as an attempt to walk through a window, stems from a failure of vision. It is ironic that vision, so seldom fooled in real-world situations, is so easily fooled by specially contrived stimulation such as that supplied by the well-known visual illusions. This credulity of vision prompts one to consider the possibility that the exquisite tapestry it weaves is nothing but an illusion of reality, maintained, perhaps, for the sake of inner peace, and sufficiently reliable for normal behavior, but surprisingly amenable to manipulation. If true, this possibility could have serious implications for understanding the way vision and other perceptual faculties work. Most importantly, it could mean that the representation of the world built by the perceptual systems is not nearly so complete and veridical as the current central dogma of cognitive science would have it. Contrary to my expectations when I first glanced at the title of The Illusion of Reality, Howard Resnikoff's book does not go so far as to question the basic tenets of the state-of-the-art theories of vision. The book is aimed at developing a unifying approach to several problems of measurement and computation. It starts by considering simple problems, such as measuring the length of a rod with a yardstick or calculating successive digits in the decimal expansion of v~, and progresses over the course of four chapters to the considerably more complicated problem of understanding how the measurements performed by the human visual system on its input can produce many of the known visual illusions. The reader is well-prepared for (or, at least, forewarned of) the eclectic style of the book by the introductory chapter in which
Resnikoff sets out his framework for information science that brings together the disciplines of thermodynamics, psychophysics (or the study of sensory information processing), communication engineering, and computability. The sheer courage of attempting to review the history of all those disciplines together in 20 pages, which Resnikoff does in search of common threads that are to bind them into an integrated information science, is admirable and should mitigate any complaints of omissions or bias. Nevertheless, fair play requires that at least some of the reviewed material be put into a proper perspective. It happens that heuristics (see Judea Pearl's book bearing this title [1]) is a well-established branch of computer science and not merely "a term to name neatly an essential process which is still largely a mystery," as Resnikoff would have it. Similar instances of missing essential references occur in other parts of the book. Chapters 2 and 3 are devoted to the mathematics and the physics of information measurement, respectively. A nonphysicist, whose last encounter with Heisenberg's uncertainty principle probably occurred in undergraduate physics, will surely appreciate Resnikoff's lucid treatment of uncertainty in measurement. Chapter 4 rounds off the programmatic first half of the book with an overview of principles of information-processing systems: hierarchy, optimization of information, signals and modulation, and sampling. Having outlined his framework for information science in the first half of the book, Resnikoff spends the
...making a computer see is harder than making it solve other information-processing tasks, some of which were once thought to epitomize intelligence... last two chapters applying the newly expounded general principles to an analysis of biological information processing. Most of this part of the book deals with vision; it is here that I felt the most uneasy about Resnikoff's presentation. Some of the biological details found in Chapter 5 are wrong. Retinal rod cells are not insensitive to the spectral composition of light. In fact, their spectral selectivity is similar to that of the cones. The reduced color perception in low-light (scotopic) conditions when only the rods are active is due to there being only one type of rod (compared to three types of cones whose differing peak spectral tuning make the perception of color possible, despite poor selectivity of each individual cone type). Another error appears in the discussion of eye movements, where saccades (fast voluntary "jumps" of the fixation point) are confused with tremor (small irregular involuntary oscillations of the eye). Some of the details are simply irrelevant: What does the chem-
ical structure of rhodopsin have to do with the topic of the book? Perhaps a more problematic thing about Resnikoff's attempt to explain perception by appealing to the principles of information theory is that so little understanding seems to be gained as a result. For example, the explanations of various visual illusions offered in Chapter 5, clearly designed to be the culmination of the b o o k are disturbingly circular: It remains to ask why the vision system systematically distorts angular measurements. In the cases discussed, it appears that the quantity of information obtained from an observation tends to be reduced by the illusory effect. This can be interpreted as a mechanism for omitting information that was present in the original scene... [p. 230] I feel that such a statement adds nothing to the neurophysiological account of the illusions, offered immediately prior to it in Chapter 5. Unless a good computational reason is found for the omission of information, it is better to assume that it is a by-product of the neural implementation of some other, intrinsically significant, perceptual function. The above example is symptomatic of more general problems that beset attempts to apply information theory to perception. The initial flood of publications that followed the discovery of information theory by psychologists in the fifties dried up a decade later, with review papers bearing titles such as "Information theory and figure perception: the metaphor that failed" [2]. The abuse of information theory in popularizing genetics has been pointed out recently in the Mathematical Intelligencer by Jack Cohen and Ian Stewart [3]. Space limitations prevent me from reproducing here the issues raised by these and other critics of the informationtheoretic approach. The main argument, however, can be stated in just one sentence: There are certain formal prerequisites for a valid application of information-theoretic tools. Two of these are the knowledge of the representations involved in a communication or computation process and the knowledge of prior probabilities of the various events in the system. Resnikoff's treatment of perception, for example, presupposes that its final product is a representation that is, in a sense, a reconstruction of the world. For an entertaining and provocative discussion of w h y this may not be true, I refer the reader to a recent book by Daniel Dennett [4]. If the thesis of Resnikoff's book is taken to be a unified approach to vision and other information-processing activities, The Geometry of Vision, edited by Melter, Rosenfeld, and Bhattacharya, can be considered its Hegelian antithesis, in form if not in content. This book presents a fragmented picture of the state of the art of digital geometry, one of the many computational aspects of vision. The fragmentation is due to the absence of any form of editorial integration of the 14 papers included in the collection, which were presented at a special session of a meeting of the American Mathematical Society. For example, the reader will find there a paper that advocates a general approach to shape representation THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993
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based on Minkowski operators of addition and decomposition, and, three chapters later, another paper that proves certain problems of Minkowski decomposition to be intractable (NP-complete). The collection includes two reviews (on digital metrics and on applications of computational geometry in computer vision) and a diverse set of papers on digital straight lines, properties of polygons, and graph connectivity. The reviews (A Survey of Digital Metrics by Melter and Computational Geometry and Computer Vision by Toussaint) are informative and include ample bibliography. Of the other papers, I found two to be especially good reading. Self-Similarity Properties of Digitized Straight Lines by Bruckstein presents several prop-
Considering both perception and action within the same framework makes it easier to accept the possibility that complex behavior need not rely on intricate representations of the world... erties of chain codes (which are discrete representations of contours in binary images) that express the invariance of the digital straightness property over different possible regular subgrids embedded in the integer lattice. Bruckstein's treatment is both comprehensive and definitive: He shows that his approach covers all possible types of chain-code self-similarity. The other paper, Galleries and Light Matchings: Fat Cooperative Guards by Czyzowicz, Rival, and Urrutia, gives a good example of a typical problem in computational geometry ("how many guards are necessary, and how many are sufficient, to patrol the n paintings of an art gallery?"), and includes an elegant solution, based on a reduction to graph coloring. For those who cannot wait until they get a copy of the book to have the answer, here it is: F3n + 41 guards are sometimes necessary and always sufficient. If you take dialectics seriously, you probably half anticipate my labeling the third book, Visual Structures and Integrated Functions edited by Michael Arbib and J6rgPeter Ewert, as a synthesis of a wide variety of topics in biological information processing. Indeed, this highly informative collection provides no less, illustrating how the notion of the nervous system as an information processor can be given explanatory power by considering its various functions in the wider context of the organism's behavior. Considering both perception and action within the same framework makes it easier to accept the possibility that complex behavior need not rely on intricate representations of the world, an idea that many students of vision still reject out of hand. An example of the integrated approach, and one of the central topics of the book, is a set of computational models which together comprise Rana computatrix, the 70 THEMATHEMATICAL INTELLIGENCER VOL.15,NO.4,1993
"frog that computes." A survey of the development of Rana computatrix over the last decade, by Michael Arbib, opens the collection. Arbib's article provides enough of a background even for a newcomer to the field to enjoy the book, and ends with the bold claim that "... the roots of our intelligence in visuomotor coordination point the way to a theory of higher mental functions based on this new paradigm. Ex Rana computatrix ad omnia." The new paradigm to which Arbib refers is the integrated use of schema theory and neural modeling. The concept of a schema in various disciplines of cognitive science (such as neurobiology or artificial intelligence) usually denotes a behavior or a mechanism that subserves a well-defined and limited function (perceptual or motor) and is activated by a certain pattern of values of cognitive variables. Prominent examples of theories of mind that rely on schemalike building blocks are Minsky's The Society of Mind [5] and the work by Dennett mentioned above [4]. The second article in the collection, A Prospectus for the Fruitful Interaction Between Neuroethology and Neural Engineering by J6rg-Peter Ewert, complements Arbib's opening overview by considering several concrete examples of behavioral tasks, the neural mechanisms that support these tasks (in real frogs), and the questions they raise for computational modelers. The rest of the book is divided into five sections (From the Retina to the Brain; Approach and Avoidance; Generating Motor Trajectories; From Tectum to Forebrain; Development, Modulation, Learning and Habituation) and contains a satisfyingly high proportion of interesting articles. It should be noted that both the range of organisms (from toads via zebra finches to macaque monkeys) and the range of behaviors (from fly-catching via bird-song to visual face recognition) that appear in the book are wide enough to vindicate at least to some extent Arbib's claim of the generality of his approach, expressed in the Preface. The frog may well one day turn into a prince and help us see through the illusion of reality.
References
1. J. Pearl, Heuristics, Reading, MA: Addison-Wesley (1984). 2. R.T.Green and M.C. Courtis, Information theory and figure perception: the metaphor that failed, Acta Psychologica 25 (1966), 12-36. 3. J. Cohen and I. Stewart, The information in your hand, Mathematical Intelligencer 13 (1991), 12-15. 4. D. Dennett, Consciousness Explained, Boston: Little, Brown (1991). 5. M. Minsky, The Society of Mind, New York:Simon and Schuster (1985).
Department of Applied Mathematics and Computer Science The Weizmann Institute of Science Rehovot, 76100 Israel
The Problems of Mathematics, Second Edition by Ian Stewart Oxford: Oxford University Press, 1992. x + 347 pp. US $16.95, ISBN 019-286-1484
Reviewed by David M. Bressoud The first edition of The Problems of Mathematics appeared in 1987 and was reviewed in the Mathematical Intelligencer, vol. 13 (1991), no. 3, 81-83. Ian Stewart is sufficiently visible to the public as a spokesperson for the mathematical community that a new edition warrants a new review. This edition is truly new. Stewart has added three chapters, updated many others, and replaced his contentious conclusion that pure and applied mathematics are at loggerheads with an enlarged vision of 21stcentury mathematics. The style is light, and Stewart paints his mathematics with broad strokes. The Foreword to the second edition is the succinct but less than useful quotation from Yogi Berra: "It's ddj~ vu all over again." He is being cute. The material one would expect to find in the Foreword is hidden amidst his "Interview to the second edition." In the text proper, Stewart's carelessness with dates and facts makes one suspicious of the details he supplies. Stewart credits Copernicus with postulating a heliocentric universe in 1473. However, that was the year of Copernicus's birth. He states that Dirichlet and Legendre proved the n = 5 case of Fermat's Last Theorem in 1828 and 1830, respectively. Dirichlet made important progress on this problem in 1825. Legendre used Dirichlet's result to complete a proof that was published in the same year. These are minor points that should not detract from an appreciation of what Stewart has accomplished in this book. He describes honest mathematics, and he does so with sureness and simplicity. He can convey the essence of what is interesting and important about a mathematical topic without getting lost in the definitions and the details. His purpose in this book is to communicate to the interested lay public a sense of what mathematics is truly about. He succeeds admirably. The fresh material in this edition comes from many areas of mathematics, and even professional mathematicians should find much that is new. It includes Noam Elkies's discovery that there are infinitely many sums of three fourth powers that are each a perfect fourth power. The smallest of Elkies's examples is 26824404 + 153656394 + 187967604 = 206156734. (Stewart incorrectly states the third integer to be 187960.) Euler had conjectured that no nth power could be expressed as a sum of (n - 1) nth powers. L. J. Lander and T. R. Parkin disproved this conjecture for n = 5 in 1966: 275 + 845 + 1105 + 1335 = 1445.
The section on Fermat's Last Theorem ends with a discussion of the A B C conjecture of R. C. Mason, P. Vojta, J. Oesterl6, and others. Formulated over the field of rational numbers, the conjecture reads as follows: if A, B, and C are relatively prime integers for which A + B = C, and if r is the product of the distinct prime divisors of ABC, then for any positive e there is a constant c, depending only on e, for which the absolute values of A, B, and C are each less than cr ~. A proof of this conjecture would imply that Fermat's Last Theorem is true for all sufficiently large exponents. Stewart misses Elkies's 1991 announcement that the A B C conjecture over an arbitrary number field K implies the Mordell conjecture for that number field ( " A B C implies Mordell," International Mathematics Research Notices, 1(7) (1991), 99-109). Although Faltings and others have proven the Mordell conjecture and the A B C conjecture is still open, this approach offers the possibility of a direct proof with actual bounds on the height of the rational points of a curve of genus > 2 - - a substantial improvement over current knowledge that the number of rational points is finite. As Don Zagier has observed, "Mordell is as easy as ABC." The first new chapter deals with Wu-Li Hsiang's proof of the Kepler conjecture that the obvious dense packing of identical spheres is, in fact, the densest possible packing. As this book went to press, and even still as this review is being written, the status of Hsiang's proof is unclear. Mathematics may deal in absolute truths, but recognizing them is an uncertain process. Like many proofs, Hsiang's is long and technically complicated. There are passages that are not right but that may be correctable. The community as a whole must suspend its appraisal of the validity of Hsiang's proof until the expert witnesses have passed judgment. Stewart also discusses kissing numbers, connections with crystal structure, and the insight that walking on a beach of damp sand does not compress but rather loosens the packing of the sand. See figure. The second new chapter concerns "Squaring the unsquarable." The problem is one Alfred Tarski posed in 1925: to dissect a circular disc into finitely many pieces that can be reassembled to form a square. After a discussion of related problems, Stewart expounds on the nature of Miklog Laczkovich's solution. The solution arose from Laczkovich's disproof of a conjecture made by Richard Gardner in 1988 that would have shown Tarski's problem to have no solution. The third chapter of new material is the second chapter devoted to knots. Stewart explores the Jones polynomials, their antecedents, their descendents, the Tait conjectures, and their solutions. He introduces links to biology and mentions ties to quantum field theory. It is the last chapter that I find most intriguing: Stewart's vision of the future of mathematics. He speaks of two trends that will increase in importance as we move into the next century: the influence of the computer and the interplay between mathematics and the other sciTHE MATHEMATICAL INTELLIGENCER VOC. 15, NO. 4, 1993
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Different ways to stack layers. (a) leads to a non-lattice packing in which spheres in the third layer are directly above those in the first; (b) leads to a lattice packing. Both packings have the same density.
ences. As his book makes clear in example after example, mathematics today is a vibrant subject. It draws much of its vitality from cross-pollination with other fields. The traditional bastions of pure mathematics, Stewart claims, are being refreshed with new ideas and applications that arrive from unexpected connections. He is correct, and it is a trend that is going to accelerate. More and more lines of interdisciplinary communication are opening up, and the problems that emerge are too interesting to ignore. However, Stewart understates his case for computers as a critical influence in mathematics. He speaks of experimental capabilities and argues with a straw opponent who sees the increasing use of computers as an attempt to equate mathematics with mere computation. As Stewart points out, computers are no longer just calculating prodigies. Mathematics is the search for structure, and computers are becoming increasingly adept at identifying such structure. He sees a role for the publication of experimental results, though my guess is that this type of mathematical communication will be the first to pass to a purely electronic format, one where the terms "publication" and "journal" cease to have clear meanings. I believe that the computer will have as profound an effect on the nature of mathematics as the telescope did on astronomy. The computer is already changing the kinds of questions we ask. As its capabilities improve and we learn to place more reliance on it, it will also change the kinds of answers we accept. Many mathematicians still view the Appel and Haken proof of the four-color theorem with m i s t r u s t - - a t best with a longing for in72
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dependent confirmation. The situation is reminiscent of the controversy over Galileo's sighting of spots on the sun and mountains on the moon. Many astronomers of the era argued that one could not accept anything that had only been seen through a telescope, an instrument that might be subject to optical aberrations. The existence of the computer is giving impetus to the discovery of algorithms that generate proofs. I can still hear the echoes of the collective sigh of relief that greeted the announcement in 1970 that there is no general algorithm to test for integer solutions to polynomial Diophantine equations; Hilbert's tenth problem has no solution. Yet, as I look at m y own field, I see that creating algorithms that generate proofs constitutes some of the most important mathematics being done. The allpurpose proof machine may be dead, but tightly targeted machines are thriving. One example lies in the W-Z pairs of Herb Wilf and Doron Zeilberger that can be used to prove identities for hypergeometric series. Zeilberger has been using Shalosh B. Ekhad as the nom de plume for his computer. Zeilberger has also discovered an algorithm, recently improved by Frank Garvan and Gaston Gonnet, for proving the Macdonald constantterm conjectures; see the Appendix. This is mathematics shaped by the computer. It is exciting. Its directions are unpredictable. The era of mathematical navel-gazing is o v e r - - i f it ever existed. We are weaving a new creation from traditional mathematics, the power of the computer, and the reestablished ties to other disciplines. In this creative endeavor, we need fluent communicators.
As Stewart concludes, A sales job remains. The world will not beat a path to the mathematical door unless it knows about the improved mathematical mousetrap. Contrary to what many mathematicians appear to assume, the world is not full of people scouring technical journals for a new piece of mathematics that they can put to use in their area. They need to be told what's available, in their language, and in the context of their concerns. Ian Stewart is one of these translators. We need more of them.
APPENDIX: A n Algorithmic Approach to the Macdonald Constant-Term Identities In 1982, Ian Macdonald [1] published his constant-term conjectures, defined for any root system R:
[x~ I-[ Z ( 1 -
1 - x- q
aER + r=l
= a certain explicit product. Here, Ix~ signifies the constant term in the Laurent polynomial in the x+% The k~ are arbitrary positive integers that depend only on the length of a, and u~ and e~ are constant integers determined by the associated affine root system. As an example, if R = An-I = {el - ed I1 <_ i r j < n} and we use xi/xj for x r162 then Macdonald's conjecture becomes [x~
I-[
k (
~-2 1 l<~<j<_n~=1
first proof for the Macdonald identity associated with the affine root system S(G2) v. In 1992, Frank Garvan and Gaston Gonnet [3] published a modification of Zeilberger's algorithm that generates linearly independent equations. In effect, they proved that some explicit product must exist on the right-hand side, though, unless one implements the algorithm, it is not apparent that it must be Macdonald's product. Garvan and Gonnet were able to verify Macdonald's conjecture for S(F4) and S(F4) v. Now the conjecture remains unresolved only for E6, ET, and Es.
References 1. I. G. Macdonald, Some conjectures for root systems and finite reflection groups, SIAM J. Math. Anal. 13 (1982), 9881007. 2. Doron Zeilberger, A unified approach to Macdonald's rootsystem conjectures, SIAM J. Math. Anal. 19 (1988), 987-1011. 3. Frank Garvan and Gaston Gonnet, A proof of the two parameter q-cases of the Macdonald-Morris constant term root system conjecture for S(F4) and S(F4) v via Zeilberger's method, J. Symbolic Comput. 14 (1992), 141-177.
Department of Mathematics Pennsylvania State University University Park, PA 16802 USA
q~_lXi~( qrXj ~ 1-xj ] xi /
(1 -- q)(1 -- q2)... (1 qnk) [(1 - q)(1 - q2)... (1 - qk)]n" -
Researchers have applied a vast array of machinery to these conjectures: multidimensional beta integrals, the combinatorics of tournament counting, Lie algebra cohomology, hypergeometric series in SU(N), Schur functions, and representation theory. Slowly, root system by root system, proofs have emerged. In particular, these identities have been verified for the infinite families of root systems: An, B,~, Cn, BCn, and D,~. The exceptional root systems have proven harder. In 1987, Doron Zeilberger [2] announced an algorithm "that systematically handles the Macdonald conjectures for any given, fixed root system, provided there are sufficient computer resources, and, for the time being, some luck." His algorithm generates seemingly random linear equations. If a system of linear equations of full rank can be found, then its solution describes the explicit product of the right-hand side of the first equation above. The computations quickly become daunting. For the root system Es, the algorithm generates linear equations in 212~ variables. Nevertheless, this approach did yield the THE MATHEMATICAL 1NTELLIGENCER VOL. 15, NO. 4,1993
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Robin Wilson*
Islamic Mathematics and Astronomy I The period 750-1400 was an important time for the development of mathematics. United b y their n e w religion, Islamic scholars siezed on Greek and H i n d u culture and m a d e it their own. Words like algebra and algorithm are Arabic words dating from this t i m e - - i n fact, from the Persian mathematician and astronomer al-Khwarizimi (c.780-c.850), w h o wrote an influential w o r k containing the word al-jabr in its title. Other Islamic scholars of the period included al-Farabi (c.878-c.950), a brilliant Islamic philosopher w h o wrote an important c o m m e n t a r y on Euclid's Elements, and al-Biruni (973-1048), an outstanding intellectual figure w h o helped to m a k e the H i n d u positional m e t h o d of counting (units, tens, h u n d r e d s . . . . ) familiar to the Islamic world, and w h o showed h o w to inscribe a 9-gon in a circle.
The most celebrated of all the Arabian philosopherscientists was ibn $ina (980-1037), k n o w n in the West as A v i c e n n a , who applied results in mathematics to problems in physics and astronomy, and w h o made a trans- 9 lation of the Elements. Also influential, especially w h e n his writings were translated into Latin during the Renaissance, was i b n a l - H a i t h a m (965-1039), k n o w n in the West as Alhazen, a geometer w h o s e main contributions were in optics; even now, one sometimes hears of A1hazen's problem, which asks for the point on a circular mirror at which light from a point source is reflected into the eye of an observer.
* Column editor's address: Faculty of Mathematics, The Open University,Milton Keynes, MK7 6AA England. 74
THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~) 1993 Springer-Verlag New York
