�
�
�
ANDREI M. RAIGORODSKII
The Borsuk Partition Probem: The Seventieth Anniversary
•
t is indisputable that, in the history of mathematics, some problems are of particular significance: acting as "catalysts," they greatly influence the development of the science. Such problems attract because of the simplicity and lucidity of their statements, fascinating a lot of specialists working in the relevant area of mathematics.
As a result, various new methods and even new theories are elaborated, and new deep and far-reaching questions are proposed. In this survey paper, I shall just speak about one of the most famous "catalyst type" problems-that of Karol Borsuk (Fig. 1).
the study of the problem, but the problem itself has played a considerable part in the making of the science. Now we are ready to state the main question of this ar ticle. First of all, consider an arbitrary bounded point set D lying in the d-dimensional real Euclidean space By the diameter of this set we shall mean the value
Rd.
What is the Borsuk Problem?
The problem I propose to discuss here belongs to "combi natorial geometry." Of course everyone understands the meaning of the words "combinatorial" and "geometry." How ever, the combination of these words is not in common use and appears only in the specialized vocabulary. Thus, I had better discuss this terminology before formulating the exact question of Borsuk Generally speaking, combinatorial geometry is a branch of mathematics dealing with extremal properties of discrete systems of point sets in different (e.g., Euclidean) spaces. One assumes usually that the problems can be expressed in combinatorial terms. As I have said, the Borsuk partition problem belongs to combinatorial geometry. At the same time, I have empha sized that Borsuk's problem is of the catalyst type. Not only has the development of combinatorial geometry stimulated
diam
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
=
SUp
x,yED
X�
y!,
where 1x y denotes, in tum, the standard Euclidean dis tance between vectors x, y E Letf =f(D) be the min imum number such that our set n can be represented as a disjoint union �
Rd.
where we assume that ni c n, that ni n nj = � and that diam ni < diam D, i,j = 1, . . , j, i =1=- j. Of course the last condition on the sets ni is the most important and nontriv ial. So, in words, the quantity j(D) is the minimum number of parts of smaller diameters into which the initial set n can be partitioned. In his paper [ 1], Karol Borsuk raised the fol lowing question: "Ldsst sich jede beschrdnkte Teilmenge E
This work was financially supported by the Russian Foundation for Basic Research (project no. 02-01 -00912).
4
fl
.
Finally, note that Borsuk published his paper in 1933, which means that 2003 marked the seventieth anniversary of the problem. This seems a good moment to cast a glance at the road already passed. Brief Historical Overview
I start by making some general remarks clarifying the na ture of the problem. We can obviously assume from the outset that any 0 c we have to partition is closed, and therefore compact. This means, among other things, that we can take the maximum instead of the supremum in the definition of the diameter. One can also assume without loss of generality that 0 is convex and even of constant width; i.e., the distance between any of its parallel support hyperplanes is always the same (see Fig. 2 and [2], [3] for further details). Finally, we can fix the value of the diame ter of 0: of course . /(d) would not change if we defined this quantity only, say, for those O 's whose diameters are ex actly equal to one. Before proceeding to the "detective story," it is reason able to recall the obvious inequality f(d) 2: d + 1 and to note that it can be derived not only from the consideration of a regular d-simplex but also from the investigation of the corresponding topological properties of a Euclidean sphere: in [4] , Borsuk himself proved that a ball cannot be divided into d parts of smaller diameters.2 The possibility of dividing a ball into d + 1 parts will be discussed in the next section. In 1955 H. Lenz noticed that Borsuk's theo rem about a ball can be transferred onto the case of an ar bitrary convex set of constant width (see [6]). As I said in the previous section, everybody who worked at the problem believed in the correctness of Borsuk's con jecture. At first, such faith received rather firm support: not only was the conjecture proved in the dimensions d ::; 3, but also H. Hadwiger [7] discovered the remarkable fact that any d-dimensional body having a smooth boundary can be parti tioned into some appropriate d + 1 parts. It almost seemed obvious that having obtained this beautiful (and, by the way, quite simple) result, Hadwiger found himself very close to the complete solution of Bor suk's problem. Indeed, it would be natural to approximate somehow an arbitrary (compact, convex) body by a smooth one, to divide the approximating set properly, and to translate the partition onto the initial body. But that's not how it turned out: in spite of all efforts, no good approximation was proposed. Nevertheless, the "catalysis" was in process, and the development of various new methods gained powerful motivation. One such method helps to obtain upper bounds for . /(d) and, in particular, to prove the conjecture in "small" dimensions. It is purely geo metrical, and I shall exhibit it in the next section.
Rd
des raumes R" in (n + 1) Mengen zerlegen, von denen jede einen kleineren /)urchmesser als E hat?" 1 In the notation f(d) = sup f(O), neRd
any n in the supremum being, as before, bounded, this question is whetherf(d) = d + 1 . Note that, by the pigeon hole principle, the properties of a regular d-simplex entail immediately the bound.f(d) 2: d + 1. Although Borsuk was cautious and did not assert any thing definite, his question was quickly transformed into the conjecture that, indeed, one can divide any bounded set 0 c into d + 1 parts of smaller diameters. Moreover, this conjecture received the
Rd
Everybody who worked
at the p roblem bel ieved
name "Borsuk's conjecture."
in the correctness of
This miraculous transfor mation of "Borsuk's question" into "Borsuk's conjecture" suggests that most of the specialists who worked at the problem strongly believed in a positive answer to the question. Actually, nobody believed the contrary! In the numerous attempts to prove the con jecture, many profound and clever ideas have been pro posed, many new and interesting methods have been de veloped, and many fine results have been obtained. However, the quest had an absolutely unexpected and dra matic outcome, and I shall expound this detective story in the next section. In the later sections, I shall discuss the new methods elaborated for the Borsuk partition problem.
Borsu k's conjecture .
'Can every bounded set E in the space Rn be divided into (n
+
1) parts so that each of these parts has diameter smaller than that of E?
21n fact, the same was also done independently by L.A. Lyusternik and L.G. Shnirel'man even a little bit earlier (see [5]).
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26, NUMBER 3, 2004
5
Moreover, it appeared plausible that the case of the poly topes would be next (by simplicity) after that of the smooth bodies. In their book Results and Problems in Combina torial Geometry [8], V. G. Boltyanskil and I. Ts. Gokhberg posed the following problem: "Prove that every d-dimen sional polytope . . . can be divided into d + 1 parts of smaller diameter. " Another illustration of how sure peo ple were of a future positive answer. We shall come back to the polytopes later; now I would like to trace another part of the story. It was absolutely rea sonable to try to obtain upper bounds for the value f(d). Here the sequence of results was roughly as follows: •
• •
:s
2d - l + 1 (M. Lassak [9]). :s (v2 + o(1))d (C. A. Rogers [ 10]). f(d) :s + o( I ) d (0. Schramm [ 1 1], and J. Bourgainf(d) f(d)
(H
)
J. Lindenstrauss [ 12]).
Note that, in small dimensions, the result of Lassak is bet ter than those of Rogers and Schramm, and Bourgain Lindenstrauss. Certainly the above-mentioned results are very disap pointing: the gap between an exponent and the desired lin ear function is too large! This might have suggested to spe cialists that something could be wrong with the conjecture. However, only P. Erdos [ 13], C. A. Rogers [ 14], and D. Lar man [ 15] (as far as we know) dared explicitly express skep ticism. Thus, Rogers tried to disprove Borsuk's conjecture, but arrived only at the positive assertion that every d-di mensional set which is invariant under the action of the group of congruences leav ing invariant a regular d simplex can be partitioned into d + 1 smaller subsets. In his article [ 14], he wrote: "The results in this note were obtained in an un successful attempt to disprove Borsuk's conjecture. IfIfelt that the work threw much light on the conjecture, I should be looking at it in this light, rather than putting pen to paper. " In fact, the result of Rogers is very interesting and general. It treats of point sets having rich groups of sym metries. So the disappointment was great: although knowledge about the problem was being constantly enriched by new deep and nontrivial data, the case deadlocked, and nobody could be sure that things were coming to a head. Anyway, nobody could predict the real outcome, and it is not "far out" to say that even those who had some doubts about the correctness of Borsuk's conjecture would prefer to bet on its final confirmation. Larman recently told me (of course I cite him not word for word), ''If the conjecture was true, we would have an absolutely independent definition of the dimension. " You can already guess what eventually happened: the conjecture was dramatically disproved. In 1993, sixty years after the publication of Borsuk's paper, J. Kahn and G. Kalai [ 16] constructed a . . . 2014-dimensional counterexample. The approach Kahn and Kalai used was quite nontrivial and
yet amazingly simple. What was even more astonishing and unexpected, their approach was based on the considera tion of a polytope. So it turned out that the Boltyanski'i Gohberg problem cited above was as complicated as the general one. At the same time, Kahn and Kalai's approach enabled its authors to establish the lower bound f(d) � ( 1.203 . . . + o(1))v'd , which reduced substantially the gap between former linear and exponential estimates. The new estimates were much less disappointing, although the gap between an exponential function and a subexponential one still remained large enough. In a way, the Borsuk partition problem was solved. How ever, it certainly was not solved completely. First, one should be interested in understanding what happens in the dimensions dE [4, 2013]. Second, the bounds ( 1.203 . . . + ' o(1))v d :sf(d) :s + o(I))d are not tight. So a new era dawned, and, during the last ten years, specialists' energies have been directed toward reducing the dimension of a counterexample and demolishing the gap between the up per and lower bounds for f(d). The improvements to the results of Kahn and Kalai look like consecutive records in a competition, and the best of them works for d � 298 (see [ 17]). As for estimating f(d) when d tends to infinity, only one slight strengthening of Kahn and Kalai's bound has been ' obtained: f(d) � (� + o(1))v2v'd = (1.2255 . . . + o(1))v d . This result is mine [18]. I have already observed that it was surprising that the counterexample constructed by Kahn and Kalai was based on the consideration of a polytope. Moreover, the body be longs to the very special class of the so-called (0,1) polytopes, which means that its vertices are vectors whose coordinates can be only 0 or 1 . In fact, almost all the counterexamples I have just cited were produced with the help of (0, 1)-poly topes, and the best known lower estimate for the quantity f(d) follows from the investigation of some combinatorial properties of the set of vertices of a (0, 1,- 1 )-polytope (cross polytope). So, in the last few years, the natural question has been studied intensively: what can be said in general about "optimal" partitions of polytopes whose vertices have pre scribed arithmetic structures of their coordinate sets?" In the case of small dimensions, this question was attacked by G. M. Ziegler and his research group [ 19]: they succeeded in proving "Borsuk's conjecture" for (0, 1)-polytopes with d :s 9. In tum, I thoroughly investigated the asymptotic sit uation: in a series of papers, I found new nontrivial upper estimates for the minimum number of parts of smaller di ameters needed to partition an arbitrary (0, 1 )-polytope, (0, 1,-1)-polytope, and polytope of a general "arithmetic" type. To conclude this section, I stress that the story is not over. First, I shall discuss some further details below, ex plaining some geometric and combinatorial methods. Sec ond, one can find many additional results and references in the survey paper [20] as well as in the books [8], [2 1], [22], and [23]. Finally, a wealth of interesting unsolved prob-
Cf2
You can g u ess what h ap
pened: the conject u re was
d ramatical ly d isproved .
6
THE MATHEMATICAL INTELLIGENCER
lems can be proposed, and I shall speak about some of them at the end of the paper. How to Cover Sets by Sets?
In this section, I want to exhibit a purely geometric ap proach to Borsuk's conjecture. This approach being intu itively clear and graphic, it will be unnecessary to go into technical detail: the numerous pictures will speak for them selves. In fact, many relatively interesting results had been obtained even before the Borsuk problem was proposed. But as I have already explained, the question served as a great stimulus to further rapid development. The approach I shall exhibit helps, in particular, to study quite thoroughly the partitions of small-dimensional sets. I first give the history of this specific aspect of the problem, and then proceed to illustrate the techniques.
In the dimension d = 1, the problem is solved obviously. In the case of d = 2, Borsuk himself showed that any set of diameter 1 can be divided into three parts of diameters smaller than or equal to 0.866 . . (see [ 1]). :he 3dimensional situation appears to be much more complicated. The first proof of Borsuk's conjecture for d = 3 was due to H. G. Eggleston [24], but it was impossible to derive from it any explicit bound like Borsuk's: Eggleston only proved that every set of diameter 1 could be partitioned into some four smaller subsets. Let us consider the following value:
v;
0'3
/
A'
-�' \'
,' ,' , f
.
B'�ii'riii;IIM
; ,�'
A'
-�'
, I J
I I
\' \ ... \
ncR3
inf
n�> ... ,n4
.
max diam ni .
i =l,
...
,4
Here the supremum is taken over all !1's of diameter 1, whereas the infimum is taken over all partitions of such a fixed !1. In other words, a3 is responsible for the biggest
, I
,
= sup
=
'
... \ ...... ...
A
)
\
C'
I
' ----18
' '
1:3
B
A
VOLUME 26, NUMBER 3, 2004
7
ii'ijii;IIW
I I I I I I I I I
:
I I I I I
8
1HE MA1HEMATICAL INTELLIGENCER
1
T3
value of the diameter of a part in an optimal partition of a "worst " set in R3. In these terms, the result of Eggleston is very disappointing: it only means that a3 < 1. The first im provement was provided by A. Heppes [25],who succeeded in estimating a3 from above by
±v9
+
4vs = o.9 9775 ...
B. Griinbaum [26] made amazingly precise computations and replaced the last bound by the inequality Y6129o3o- 93741 9V3 0.9887 ... 0'3 :s: 1518\12 Recently V. V. Makeev and L. Evdokimov [27] discovered the estimate a3 :s: 0.98.As far as I know, this result remains unsurpassed. At the same time, some theorems were proved by this author in a joint paper [28] with Yu. Kalnishkan. I shall formulate them a little later. Coming back to the special case of polytopes, it is worth noting that this case was discussed for d 2 in [29) by P. Erdos and for d = 3 in [30] by A. Heppes and P. Revesz. Although the methods used by Erdos, Heppes, and Revesz have no connection with the main idea of this section, they are so neat that I couldn't avoid mentioning them. After this digression, I now return to the promised general approach. Let us introduce the notion of a universal cover. =
=
Definition
A set U c is said to be a universal cover, if, for any body 0 c of diameter there exists an appropriate rigid motion 'P such that C U.
Rd Rd
1, q;(O) The term "universal cover " is entirely natural, for such a U does really cover any other point set in the Euclidean space: one must only move it properly. Note that we did not lose generality by assuming that all our O's were of di ameter 1. Now I can easily describe a procedure for at tacking the Borsuk problem. First of all, one should find a universal cover U in Rd. Then one should try to divide this U into the least possible number of parts of diameters smaller than 1. What is very important, U itself can, gener ally, have an arbitrary diameter greater than or equal to 1 . On the one hand, dividing a fixed universal cover is much more convenient than partitioning any arbitrary set. How ever, on the other hand, the art of constructing a "good" universal cover U is extremely delicate. Here "good" means of course that, any n being inscribed into U, the boundary of U must "fit snugly" on that of fl. Doubt may be even jus tified concerning whether the approach above works ef fectively. We shall see that this approach is remarkably fruitful and still worthwhile. Omitting the proofs, I proceed to illustrations. The two dimensional result of Borsuk follows immediately from the consideration of a regular hexagon whose distance between parallel sides equals 1. This hexagonal universal cover was actually proposed by J. Pal. In Fig. 3, the corresponding par-
tition is shown. For d = 3, A. Heppes and B. Griinbaum in dependently used the same U. This U can be obtained from a regular octahedron lying in R3 and having distance 1 be tween its parallel faces. One cuts three rectangular pyramids from it using three pairwise orthogonal planes, as depicted in Fig. 4. The partition of U due to Griinbaum is extraordi narily ingenious. For details, I refer the reader to the origi nal article [26) and to the book [8]; but I give the portrait of this partition in Fig. 5.Note that the study of an octahedron as a possible universal cover had already been suggested by D. Gale. However, it is also quite interesting that the octa hedron itself cannot be divided into four parts of a needed diameter. 3 The result of Makeev and Evdokimov is even more refmed: instead of an octahedron they considered a rhombic dodecahedron with analogous properties. The dis tance between its parallel faces was equal to 1, and some parts were cut from it with the help of similar pairwise or thogonal planes. In truth, the arguments for the fact that a rhombic dodecahedron covers universally the sets of diam eter 1 in R3 are nontrivial and nonelementary: they are based on some advanced topological background. A partition of the rhombic dodecahedron was done by computer, and I think it unnecessary to draw it here. Another series of wonderful illustrations showing the importance of covering sets by sets is provided by some upper bounds for the valuef(d). The simplest one isf(d) CVd + l)d, and to obtain it one should take as U C Rd a d dimensional cube whose edges are of length 1. To prove the inequality f(d) :s: 2d it suffices to show that a ball of ra dius is a universal cover in Rd.This was done in 1 901 by H.W.E. Jung [23).Lassak's modification of the last spherical universal cover is represented in Fig. 6.It is constructed as the intersection of Jung's ball with an arbitrary ball of radius 1 whose center lies on the boundary of the first ball. Fig. 7 depicts possible divisions of two- and three-dimen sional Lassak covers into three and five parts, respectively. Recall that, in the general case, such a cover must entail the estimate f(d) :s: 2d-l + 1. The reader might feel unsatisfied if I failed to elucidate one more question. Indeed, we know that a3 < 0.98. But what can be said about lower bounds for this value? The same question can be asked for d 2 and even for d = 1, for the analogous quantities a2 and a1. It is clear that a1 = t. and we also know that a2 v:}. To estimate a2 or a3 from below means to give an example of a two- or a three-di mensional body such that, in any of its appropriate parti tions, there is a part whose diameter is large enough. Sur prisingly, one can take balls as such examples, and nothing better has been proposed. An optimal partition of a ball in Rd proceeds as follows: one inscribes into this ball a regu lar d-simplex :1, and one builds d + 1 parts by intersecting the ball with the multi-dimensional (closed) angles con taining the d-simplices whose bases are at the (d 1)-faces of :1 and whose additional vertices are at the center of the ball (i.e., the angles have one common vertex; see Fig. 8). It is not hard to check that the diameters of such parts are :s:
1· d
\ 2 ri + 2
=
:s:
-
3 Don 't forget that its diameter exceeds 1 !
VOLUME
26, NUMBER 3, 2004
9
.
smaller than the diameter of the ball. Thus, for d = 2, they are exactly equal to so that a2 Ford = they co . . . , and, consequently, incide, in turn, with 3 +6V3 a3 2: . . . . Gale conjectured that, in any dimension, a ball is an example of a "worst" set; i.e., any other body can be partitioned more economically. Of course, Borsuk's con jecture being false, this guess fails as well. But, in the case ofd = nobody knows how to improve the inequality
J
0.888 3,
j3
=
=
+6\13 =
0.888
0.888 . . .
v:}.
3,
:S C¥3 :S
0.98.
Theorem
Let r(O) denote the minimum radius of a ball contain ing a set n c R3 of diameter 1. Assume that r(O)E u Then n can be divided into four parts whose diameters are smaller than
0.96.
t,
In principle, r(O) is always bounded from below by which is obvious, and from above by (Jung's theorem). But it is really surprising that we cannot apply our method to bodies whose radii of circumscribing balls lie in a cen tral part of the interval Our method generalizes the one I have described in this section. It is based on the con sideration of universal covering systems:
fs
[t, fg].
Definition
A system of sets u = { U } , U C Rd, is said to be a univer sal covering if, for any body 0 c Rd of diameter 1 , there exists an appropriate rigid motion cp such that cp(O) c U, where U is a set in u. To complete this sec " tion, I would like to em phasize the disappointing fact that for d = almost nothing has been proved. Even in the case of poly topes, no one knows how to work on the conjecture. We strongly believe that it should be true: some grounds for such faith are given, say, by the paper [28] . Also, some surveys concerning universal covers can be found in [31] and [32].
"P.
Erdos liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems." Kahn and Kalai's reasoning cer tainly falls under Erdos's definition. One can find circum stantial expositions of this reasoning and of many related con siderations in, say, [20] and [21 ] . However, I can't deny myself the pleasure of presenting here some ideas of the method. I shall show how to obtain a counterexample for d = Let n = 44, and let
946.
L
The only partial result is due to this author and to Yu. Kalnishkan:
[t,0.5 3] [ 0.6, fs].
BOOK [33]. The point is that
=
{x
=
Cx1, ..., Xn): XiE { - 1 , 1 }, i = 1, . . . , n; X1 = 1; X1
X . . .
X
Xn = 1 }.
In other words, I is the family of all n-dimensional ( -1, 1)-vectors such that, in every vector xE I, the first coordinate equals 1 and the number of negative coordinates is even. Therefore, the cardinality of I is card I = 242 . The following lemma holds: Lemma. Assume that Q {a1, ... , a8} is an arbitrary subfamily of I such that, for any two vectors Ri, lljE Q, the relation (Ri, llj) -=!= holds, where by triangular brack ets we denote the standard inner product in Rn. Then =
0
card Q :S
10
�0
(43k ) .
The Lemma will play a crucial part in our construction. Its proof is very ingenious. It is based on the so-called lin ear algebra method in combinatorics, and here the catalyst nature of Borsuk's problem has manifested itself once again. The method was invented about twenty years ago [34] for this purpose, but it was immediately put to work getting numerous further results. To each vector x = (x1, ..., Xn) in I we assign the vee, Yn,n), , Y2,n, , Yl,n, Y2,1, tor X* X= CY1,1, Y1,2, where Yi,J = Xi X Xj Thus, we have the new family I* C Rn2 of cardinality 242. First, we note that actually I* lies in an affine sub space of Rn2 whose dimen " sion does not exceed the value d = n(n =-ll = = 1 . Con This is evident, for always Yi,J = Yj,i and Yi,i sequently, we may consider I* as a family of vectors in Rd. Second, one can easily realize that, for any two vectors x * x, z zE I*, their inner product coincides with the quantity (x,z)2. The last observation means, in particular, that the diameter of I* is attained on those and only those pairs of vectors x * x, z z, for which (x,z) = Suppose we are able to divide I* into ·
·
·
·
·
·
·
·
·
·
Erdos l i ked to talk about
4
The Book, in which God main tains the perfect proofs . . .
How t o Construct Counterexamples?
Having learned some elements of the technique of cover ing sets by sets, we switch abruptly to studying an almost opposite aspect of the problem: from a purely geometric approach, we move to a purely combinatorial one. Of course this change-over only confirms the nature of Bor suk's question. So, in this section, I shall exhibit a con struction that will enable us to disprove the conjecture. The original construction of Kahn and Kalai was so beau tiful and subtle that one of its modifications was included in the book of Aigner and G. Ziegler Proofs from the
M.
10
THE MATHEMATICAL INTELLIGENCER
M.
=
�
*
0.
*
242
parts of smaller diameters. Then
I*=
ntU ... un;,
diam D'i < diam I*.
946.
Let !1; = {xE :>:::X* XE mJ. Clearly card !1; =card m, and,by the well-known pigeon-hole principle,there is some i such that ( ) card > 43 . !1; k�o k It follows from the lemma that there exist two different vectors x and in !1; with (x, z) = 0. Therefore, diam D7 = diam :>::*-a contradiction. Thus we obtain, for d =946, the lower bound f(d) 1 242 = 1649.8 ... > 947. 10
z
2::
0 �J�)
2::
Acknowledgment
It is a great pleasure for me to say how grateful I am to my friend Yu. Kalnishkan for his invaluable help in drawing wonderful illustrations for this paper. REFERENCES
[1 ] K. Borsuk, Ore! Satze uber die n-dimensionale Euklidische Sphare, Fundamenta Math. , 20 (1 933), 1 77-1 90. [2] F. Reuleaux, Lehrbuch der Kinematik I. Vieweg, Braunschweig 1 875 (Engl. trans! . : 1 876; Reprint: Dover, New York 1 963). [3] H . G. Eggleston, Convexity, Cambridge Univ. Press, Cambridge,
The construction is complete. Note that the convex hull of the vectors in I* is a ( -1,1) polytope. Obviously it can be transformed into a (0,1)-poly tope with the same "Borsuk properties."
1 958. [4] K. Borsuk, Ober die Zerlegung einer Euklidischen n-dimensionalen Vol/kugel in n Mengen,
In the previous sections, I outlined the seventy-year-long history of one of the most exciting problems in combina torial geometry. It remains to speak about future perspec tives. Here are several still unsolved questions. Prob lem 1. Construct a universal covering system proving Borsuk's conjecture ford =4.Prove the 4-dimensional con jecture at least for polytopes. Note that,for those polytopes whose vertices are at some rational vectors in R4 and whose diameters are rational as well, even a stronger assertion holds: any such polytope can be divided into four parts of smaller diameters [28]. Prob lem 2. What can be said about (0,1,-1)-polytopes in small dimensions? For instance, what is the maximum di mension in which the conjecture for such polytopes is true? Prob lem 3. Prove or disprove Gale's conjecture in R3. In other words,does there exist a set !1 c R3 such that,in any of its decompositions ililto four parts, there is one part of diameter greater than p +6V3 0.888 ... ? What is the min =
imum dimension where Gale's conjecture is false?
Let n = 2k. Consider the family of vectors i 1, .. . , n; � = {x (Xt, ... , Xn) :x; E (0,1,-1}, card (i = 1, . .. , n:x; = :±:1} = %}· Suppose we know this lemma: Lemma. Assume that Q = (a1, ... , a8) is an arbitrary
Problem 4.
=
subfamily of L such that, for any two vectors a;, lljE Q, the relation (a;,llj) =I= holds. Then
0
card Q s C. The linear algebra method gives here a very good bound on this C (see [18], [20]), but nobody knows whether this bound is tight. So the problem is to determine the precise value of C in the lemma. This problem being solved, new lower estimates for f(d) would be obtained, and the di mension of a counterexample to the conjecture could be
Verh. Internal. Math. Kongr. , Zurich, 2
(1 932), 1 92. [5] L. A. Lyusternik and L. G. Shnirel'man. Topological methods for variational problems,
Some Unsolved Problems
=
reduced. My guess is that, by using this approach, one should be able to find counterexamples for d 135.
ONTI, Moscow, 1 930.
[6] H. Lenz, Zur Zerlegung von Punktmengen in solche kleineren Durchmessers,
Archiv Math., 6 (1 955), N 5, 41 3-41 6.
[7] H . Hadwiger, Oberdeckung einer Menge durch Mengen kleineren Durchmessers,
Comm. Math. Helv., 1 8 (1 945/46), 73-75; Mit
teilung betreffend meine Note: Oberdeckung einer Menge durch Mengen
kleineren
Durchmessers,
Comm.
Math.
Helv.,
19
(1 946/47), 72-73. [8] V. G. BoltyanskiT, I. Ts. Gokhberg, Results and problems in com binatorial geometry,
Nauka, Moscow, 1 965; English trans!. , Cam
bridge Univ. Press, Cambridge, 1 985. [9] M. Lassak, An estimate concerning Borsuk's partition problem, Bull. Acad. Polon. Sci. Ser. Math., 30 (1 982), 449-451 . [1 0] C. A. Rogers, Covering a sphere with spheres, Mathematika, 1 0 (1 963), 1 57-1 64.
[1 1 ] 0. Schramm, Illuminating sets of constant width, Mathematika, 35 (1 988), 1 80-1 89.
[1 2] J. Bourgain and J. Lindenstrauss, On covering a set in Rd by balls of the same diameter,
Geometric Aspects of Functional Analysis
(J. Lindenstrauss and V. Milman, eds.), Lecture Notes in Math. ,
vol. 1 469, Springer-Verlag, Berlin, 1 991 , 1 38-1 44.
[13] P. Erdos, My Scottish book "problems", The Scottish Book, Math
ematics from the Scottish Cafe (R.D. Mauldin ed .), Birkhauser, 1 981 ' 35-43.
[1 4] C. A. Rogers, Symmetrical sets of constant width and their parti tions,
Mathematika, 1 8 (1 97 1 ) , 1 05-1 1 1 .
[1 5] D. Larman, Open problem 6, Convexity and Graph theory (M.
Rozenfeld and J. Zaks eds.), Ann. Discrete Math. , vol. 20, North Holland, Amsterdam and New York, 1 984, p. 336.
[1 6] J. Kahn and G. Kalai , A counterexample to Borsuk's conjecture, Bulletin (new series) of the AMS, 29 (1 993), N 1 , 60-62.
[1 7] A. Hinrichs and Ch. Richter, New sets with large Borsuk numbers, 2002, http://www.minet.uni-jena de/hinrichs/paper/1 8/borsuk.pdf. [1 8] A.M . RaTgorodskiT, On a bound in Borsuk's problem, Uspekhi Mat. Nauk, 54 (1 999), N 2, 1 85-1 86; English trans!. , Russian Math. Sur veys, 54 (1 999), 453-454. [1 9] G. M. Ziegler, Coloring Hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions,
Lect. Notes Com
put. Sci. , 2 1 22 (200 1 ) , 1 59-1 7 1 .
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26, NUMBER 3, 2004
11
[22] H. Hadwiger, H. Debrunner, V. Klee, Combinatorial geometry in
AUTHOR
the plane,
Holt, Rinehart and Winston, New York, 1 964.
[23] L. Danzer, B. Grunbaum, V. Klee, Helly's theorem and its relatives, Convexity 0/. Klee, ed.), Providence, R . I . , 1 963, 99-1 80. (24] H. G. Eggleston, Covering a three-dimensional set with sets of smaller diameter,
J. London Math. Soc., 30 (1 955), 1 1 -24.
[25] A. Heppes, Terbeli ponthalmazok felosztasa kisebb atmer6ju reszhalmazok 6sszegere,
A magyar tudomanyos akademia, 7
(1 957), 4 1 3-41 6. (26] B. Grunbaum, A simple proof of Borsuk's conjecture in three di mensions,
Proc. Cambridge Philos. Soc . , 53 (1 957), 776-778.
(27] V. V. Makeev, On affine images of a rhombo-dodecahedron cir ANDREi M. RAiGORODSKii
cumscribed about a three-dimensional convex body in R3,
Department of Mechanics and Mathematics
Moscow State University Moscow
Math. Sci. (New York), 1 00 (2000), 2307-2309.
11 9992
(28] A. M. Ra1gorodski1, Yu. A. Kalnishkan, On the Borsuk problem in
Russia
R3,
e-mail: arai9or®avangard.ru Andrei Raigorodskii was bom "kandidat" degree O i ke
sian
Mat. Zametki, 74 (2003), N 1 , 1 49-1 5 1 ; English transl., Math.
Notes, 2003. (29] P. Erdos, On sets of distances of n points, Amer. Math. Monthly,
in Moscow in 1976. He got his
53 (1 946), 248-250.
the American PhD) in 2001 and his Rus
doctor's degree as we go to press , and is
the Lomonosov State University.
Zapiski
Nauchn. Semin. POMI, 246 (1 997), 1 91 - 1 95; English transl., J.
[30] A. Heppes and P. Revesz, Zum Borsukschen Zerteilungsproblem,
on the faculty o f
Acta Math. Acad. Sci. Hung . , 7 (1 956), 1 59-1 62.
He h as pu b lished about 25 pa
[31 ] V. V. Makeev, Universal coverings and projections of bodies
pers on combinatorial geometry. His favorite animal is the toad.
of constant width,
Ukrain. Geom. Sb. , 32 (1 989), 84-88 (in
Russian). [20] A. M. Ra1gorodski1, The Borsuk problem and the chromatic numbers of some metric spaces,
[32] B. Weissbach, Polyhedral covers, Coli. Math. Soc. J. Bolyai 48 (In tuitive geometry), North-Holland, Amsterdam 1 987, 639-646.
Uspekhi Mat. Nauk, 56 (2001), N 1 , 1 07-146;
(33] M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer
English transl., Russian Math. Surveys, 56 (2001 ), 1 03-1 39. (21 ] V. G. Boltyanski, H. Martini, and P. S. Soltan, Excursions into com binatorial geometry,
Universitext, Springer-Verlag, Berlin Heidel
Verlag, Berlin, 1 998. (34] P. Frankl and R. Wilson, Intersection theorems with geometric con sequences,
berg, 1 997.
Combinatorica, 1 ( 1 98 1 ), 357-368.
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12
THE MATHEMATICAL INTELLIGENCER
Mathematically Bent
Colin Adams, Editor
A Proof of God Colin Adams
t was one of those Wednesday after
Inoons when Monday seems a distant
The proof is in the pudding.
Opening a copy of
The Mathematical
Intelligencer you may ask yourself uneasily, "What is this anyway-a mathematical journal, or what?" Or
you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
memory and Saturday has shrunk to an unreachable pinprick in the future. I had just finished teaching my seminar on Boolean Algebras and was settling down in my office with a much-needed cup of coffee. Having just placed my feet up on the desk, I was startled by a knock at the door. Oh no, I thought to myself, it's probably Bunsen. Bunsen was one of the weakest students ever to have graced a campus, and there was nothing more disheartening than the appearance of his hangdog face in my office door. I wearily stood up and swung open the door. To my relief, there stood an older gentleman in a stiff woolen suit. He had to be at least 80. "Hello," I said,putting on a pleasant smile. "Hello." He held out his hand. "I am Arthur Gottlieb." "Nice to meet you, Mr. Gottlieb.I am Professor Rasmussen." I shook his hand. "What can I do for you?" "It is what I can do for you," he said with a slight accent. "May I come in?"
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01 267 USA e-mail:
[email protected]
He motioned with the cane that he gripped in his left hand. I hesitated for just an instant, con sidering the work I needed to do. But curiosity won out. "Please." I waved him to the wooden chair I reserved for students. The less comfortable the chair, the less time they spent in my office. I seated myself back behind my desk. "So what can you do for me?" I asked, as I reached for my coffee and took a sip. He leaned forward conspiratorially. "I have discovered a mathematical proof of God." I coughed up the coffee. ©
"Excuse me?" "I have a mathematical proof of God." He nodded knowingly. I laughed nervously. Bunsen was starting to look a whole lot better. "Urn, I'm sorry," I said, "but you can't have a mathematical proof of God. God is not defined as a mathe matical object about which you can prove theorems. Just as you can't have a mathematical proof that there are atoms. Atoms and God, if they exist, are attributes of the real world. Math is just about math." "Well I have a proof," he said with finality, sitting back in his chair. I sighed and settled back in my own chair. It didn't look as though he was planning on leaving soon. "Okay," I said. "I'll bite. How does it go?" He smiled slightly and lifted his eye brows. "It is a proof by contradiction. " "Yeah?" I said, flicking a glance at my computer, wondering if I had any new e-mail. "Yes, I assume first of all that there is no God, and then ultimately I derive a contradiction. Therefore there must be a God." "And what is the contradiction?" "That my first wife's name was Gladys." "That's ridiculous." "Yes, it is. My first wife's name was Elba. It was my second wife who was named Gladys." "No, I mean that's not the kind of contradiction you get out of a mathe matical proof. You have to get a math ematical contradiction,like A is strictly greater than B and B is strictly greater than A. Or like 1 = 2." "But I have 1 = 2. I show that my first wife,wife number one,is named Gladys, but that is actually the name of my sec ond wife, wife number 2. So 1 = 2." "Okay, fine. But my point is that things like your wives' names shouldn't come up in a mathematical proof. Math ematical proofs should be about math ematical objects,like numbers,groups,
2004 SPRINGER-VERLAG NEW YORK, LLC, VOLUME 26, NUMBER 3, 2004
13
algebras, topologies. G ladys and Elba are not mathematical objects." " They are elements of the set of all wives that I have had." " Su re." " Then they are mathematical ob jects." I sighed. " Okay, sure. So they are mathematical objects. And so you use them to show that G od exists." He glanced over at my open door and then leaned forw ard again. " Actu ally, that is only one of the things I prove." "R eally?" I sai d. " Pray tell, what else do you prove?" " I prove that the parallel postulate follows from Euclid' s other axioms. I provide an algorithm for trisecting any angle. I give a formula for finding the roots of a fifth- degree polynomial." I threw up my hands. " What? Y ou don' t square the circle?'' " What does that mean?" " The one you left out, squaring the circle. It means constructing with straightedge and compass a square that has ar ea exactly equal to the area of a unit circle, which is 1r." " Oh, I can do that, too." I whistled. "Y ou ar e quite a piece of work." " Thank you," he sai d, smiling. " Okay," I said. " Now tell me. Does it bother you that all of these results, tri secting the angle, showing the parallel postulate follows from the other ax i om s, gi vi ng a form ula for the roots of a fifth-degree polynomial, they are all known to be false?" "If youm eant hat others have cla im ed to prove them false, I am awar e of that." " And it doesn' t bother you that mathematicians all over the world have accepted those proofs as correct?" "M athematics is not a democracy." Now I was exasperated. " No, it' s not. But the whole idea of a proof in mathematics is that it can be checked by any reasonable human being, and the conclusion will be that it is correct. A whole heck of a lot of human beings have looked at the proofs of these re sults, and they have all come to the sam e conclusion; they are correct." "Y es, but these same human beings don' t understandal gebraic lingerie." " Excuse me. Did you say lingerie?" 14
THE MATHEMATICAL INTELLIGENCER
"Y es, mathematical objects related to the set of all wives that I have ever had. I created them." I looked out my door to see if there was anyone in the hall in case I needed help. "Y ou see," he continued, " algebraic lingerie helps to contai n sets such as the set of all wives I have ever had. It is a meta set, used to restrain unruly subsets of the original set." Suddenly I was paying attention. Dickson had suggested the possibility of such sets in the 1930s, but nothing had ever come of the idea.
I sat open m outhed . Every t h i ng h e h ad said made sense . " How would these meta sets help you in your argument?" I asked, tryi ng not to show my interest. " The algebraic lingerie allows the creation of a homomorphism from the set of all names to the set of all wives that I have ever had. Assuming the nonexistence of G od, one can infer that na mes are not determ ined by a higher design. Therefore, some probability distribution govern s their choice. Ap ply ing a P oisson process, one gener ates a homogeneousM arkov chain." " Wai t a m inute," I sai d. " Why ho mogeneous? Isn't non-homogeneity im plied by the transitivity of the Lapla cia n?" " Only when the process is semi simple. When the process is simple, as in our case, the Laplacian is only de fined over a compact subdomain, and its transitivity there doesn' t even imply superellipticity, let alone homogene ity." " Oh." I didn' t kn ow where this was going, but it was clear the guy kn ew some math. " Please continue." " Well, then we factor by the kern el of the homom orp hism, yi elding an ab stract subvariety detem1 ined by the maxi mal idea l. The definition of this subvariety can be analytically contin ued and then completed to yield a sim-
plicial complex in a fundamental do main for the action of the cusp sub group of a hyperbolic orbifold com mensurable with a Bianchi group of arbitrarily large discriminant. The trace field generates a dilogarithmic map that lifts to the universal cover. Q uotienting out by the orientation reversing isometries yields a manifold of Hausdorff dimension 3/2. The coho mological sheaf of this manifold allows us to prove the existence of a bilocal diffeomorp hism onto the generators for the fundamental group of a CR manifold of dimension 12. The primary obstruction to a lifting of associated Steenrod algebra affords a means to define a weakly contractible map to the commutator. Suspending this map yields a cofibration of the associated Eilenberg-M acLane space. Projecting to the generic fiber yields a Lipschitz map from the set of names to the set of all wives. W hen the range is re stricted to just my wives, the commu tativity of the map forces my first wife to have the name G ladys. And that is a contradiction." He thumped his cane on the floor for emphasis. I sat open-mouthed. Everything he had said made sense. I had just heard the single most am azing mathematical exposition of my life, bar none. This guy made G auss look like a ditchdigger. I needed to stall while I fig u red out what to do. " The, ummm . . . the Steenroda lge bra. How do you get its primary ob struction?" " The algebra is defi nedi n term s of a generati ng set that depends only on the t angential bifur cat ion present in the ex treme values corr espondi ng to the com pact core. If there were not a primary obstruction, then transitivity would not hold in its cofm ite extension. But t he cofm ite extension is Hopfian by defi ni tion, and hence transitivity must hold." He thu mped his cane again. " Oh, yes, I see," I said, although in tru th, I would need a month to figure out his answer. " Now tell me. Have you told your proof to anyone else?" " No," he said. " I am no longer mar ried. M y fifth wife, Henri etta, she passed away three years ago. So I de cided, five wives, it is enough. Now I tum to mathematics. I work for three
"How dare you!" he cried. "How years. But you ar e the first person to dare you speak to me in this manner!" whom I tell my proof." "I see." I stood up and swung the His face was bright red now, and he door to my office closed. "Wouldn' t was glaring at me. Suddenly, he clawed at his chest. "I want the wrong person listening in," I said with a smile. He brightened con . . . I . . . " siderably, assuming I now believed him. For an instant, he looked surprised. I sat back down and leaned forward, Then he crumpled to the floor, his cane bouncing off the file cabinet and clat placing my elbows on the desk. "M r. G ottlieb, I appreciate your tering down beside him. I rushed coming by. It is inspiring to see such a around the desk and bent down, lifting vigorous interest in mathematics. And his head onto my knee. certainly you have absorbed a lot of, He looked up at me, his face ashen. shall we say, background." "Tell me the truth," he said. "Y ou know He tipped his head in acknowledg I proved it." ment. "But nowadays, M r. G ottlieb, origi nal mathematics cannot be produced by amateurs. It takes years of study to reach the level of understanding nec essary.M ath is just too abstract." "But I have done it," he said, his eye brows furrowing. "I just explained it to you." "M r. G ottlieb," I said firmly, "al though much of what you said would I hesitated for a second. Then I said, sound reasonable to a layperson, to a "Y es, you proved it." mathematician such as myself it is "G od exists.Y ou know it.Y ou must clearly nonsense. Y ou haven' t proved live with it." anything here. Y ou' ve merely tied to Then his head fell back, and I knew gether a string of words that sound he was dead. plausible, but in fact mean no more I slumped down to the floor, his than a nursery rhyme. I suggest you go head still resting in my lap. Here was home, and find a more productive pas perhaps the greatest mathematician of time. Perhaps you might enjoy watch all time, and I was the only person who knew it. Because of me and my actions, ing TV." G ottlieb turn ed red in the face. he was dead. I carefully lifted his head "What are you saying?" he said in a from my lap and placed it gently on the trembling voice. "Y ou know I am right. floor. Then I called security. Why do you deny it?" I won the Fields medal two years "In fact,M r. Gottlieb, I know you are later. It was for work on algebraic lin wrong. Y ou claim to have disproved gerie. I could have announced any one some of the most well-known and fun of the results, and it would have been damental results in mathematics, let enough to guarantee me the medal, but alone your claim of a proof of G od. If I was careful not to undermine too you were right, well, all of mathemat much of mathematics at once. I didn' t ics as we know it would come tumbling want to put myself and every other down, and the public would lose con mathematician out of work. Better to fidence in the mathematical commu leak the results one at a time, and hope nity. I am sorry. I am a busy man, and the underlying mathematical frame I don' t have time to listen to a de work would recuperate between mented old man who makes up math blows. ematics to distract himself from his M y renown spread quickly. I flew grief over the loss of a long chain of from one invited address to another. wives." M athematicians flocked to my talks. G ottlieb stood up, waving his cane. About once a year, I announced an Veins stood out prominently on his other result. Each time, it was as if someone had kicked the mathematical forehead.
"Tell m e the
truth , " he said . "You know I p roved it ."
anthill, with all the mathematicians scurrying to repair the damage. Each time, my stock rose accordingly. I considered announcing the proof of G od as the coup de grace, but I couldn' t bring myself to do so. Initially, I reasoned that it wouldn' t make the world a better place. There are more than enough people already convinced there is a G od and happy to kill one an other because of it. Perhaps it is just as well to let people make up their own minds. But ultimately, I realized this was a rationalization on my part. M y reluc tance cam e more from the sense that claiming a proof of G od as my own would be a sacrilege an order of mag nitude greater than I had already com mitted. Somebody was keeping ac counts, and my tally wasn' t looking so good. It didn' t take long for me to stop en joying the attention. The question of my ultimate accountability was always there, nagging me in the background. I ceased accepting invitations to speak, unable to withstand the looks of admiration from students and faculty. I cringed inside each time my genius was prai sed, knowing I hadn' t earned it. I began to avoid my colleagues, skipping department meetings and seminars, claiming I was too busy with my work. No one dared complain. And now, I find myself spending most of the day sitting in my office with the door closed. Once in a while I work through the implications of G ottlieb' s work, and I realize some other basic tenet of mathematics is incorrect, and I see the further disintegration of the mathematical machinery that took thousands of years to build. I no longer work on my own math ematics, which I once enjoyed so much. What is the point? None of the results would compare to the results for which I am already famous. Nobody would care. And anyway, the mathe matical scaffolding upon which they would depend is itself corrupt. No, most of the time, I just sit, star ing at a blank pad on my desk. I sit and I stare, trying not to think about what I have done, trying not to think about anything, always under the watchful eye of G od.
VOLUME
26, NUMBER 3, 2004
15
lt�ftii ji§
..
ftih$1MQ.'I.Ji.W.Jhi4J
Coxeter and Friends Marjorie Senechal
This column is afornmfor discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We
Marj o ri e Senechal , Editor
O
n his 95th birthday, Asia Weiss presented Donald Coxeter, her friend and former thesis director, with a sheaf of congratulatory e-mail mes sages from geometer friends around the world. No mathematician has meant so much to so many for so long. Ninety-five years and counting: in Sep tember 2000, he had lectured at a con ference in Stockholm; he would give another in Budapest in July. Donald Coxeter's mathematical community is many-ringed, like ever widening ripples on a pond. Yet Regu lar Polytopes is no falling stone. It tugs
mathematical societies, student organizations, and informal communities of cardinality greater
thickets and wickets of h i g her d i mensions so d eftly and del ig htfu l ly that we (al most) feel at home there .
than one. Jthat we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
us upward through the thickets and wickets of higher dimensions so deftly and delightfully that we (almost) feel at home there. The Coxeter community includes the readers of this and his many other books and articles. It in cludes all the mathematicians for whom "Coxeter" is an adjective and Coxeter graphs and Coxeter groups are working tools. It includes the hardy band of professional and amateur geometers Donald sustained, by cheer and example, through the long mid twentieth-century years when fashion proclaimed abstraction in, and "geo" and "metry" out. ("The Moor has done his work, the Moor may leave," I. M. Yaglom noted wryly. And added sadly: "This revolution has driven out . . . sev eral investigators of a more geometric
Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal,
Department
of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail:
[email protected]
16
tum of mind.") The Coxeter commu nity includes Donald's vast network of correspondents: in long handwritten letters (never e-mail), he queried, crit icized, and suggested. No person, no point, was too large or too small. "SHEPHARD should have initials G . C. (instead of G . S.)," he corrected me in 1983. "I noticed this particularly be cause he once 'bawled me out' for mak ing the same mistake." And of course the Coxeter community includes all who worked, studied, or collaborated with Donald, including his sixteen doc toral students and the organizers and
It tugs us u pward th ro u g h the
include "schools" of mathematics, circles of correspondence,
I
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
participants in "The Coxeter Legacy: Reflections and Projections," a sympo sium in his memory held in Toronto in May 2004. The Coxeterbehind the adjective was a man of warmth and wit and boundless energy, who loved discussing geometry with anyone and everyone but rarely talked about himself. Not about the opera he composed at thirteen; nor about his teacher, the enigmatic Wittgenstein; nor when and where he met his beloved wife Rien; nor about his migration from Cambridge, England, to Toronto, Canada, where he taught for over sixty years. This issue's column, a biographical sketch by Donald's biogra pher Siobhan Roberts and Asia Weiss, is not about a mathematical community, but for it. For all of us.
1$ft'l.t§rrfhi¥119·1rrl,lii.lhi¥J
Donald in Wonderland: The Many Faceted Life of H. S. M. Coxeter Siobhan Roberts and Asia lvic Weiss
Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal,
Department
of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail: senechal@minkowski .smith.edu
Marj o r i e Senec hal , Editor
There is something pleasing to a mystic in such a land of mirrors. For a mystic is one who holds that two worlds are better than one. In the highest sense, indeed, all thought is reflection.
W
ith that quotation from G. K. Chesterton's Man Alive, Donald Coxeter invited many into his realm of geometry, and, once there, he happily rambled around in reminiscences of his past. A Precocious Childhood
One of the earliest photographs of Don ald Coxeter is as a boy of about three years old. He was all dolled up in a frilly collared shirt and knickerbockers, a swag of blonde curls to his shoulders, and seated at a grand piano with his feet dangling from the bench. By Cox eter's analysis, his hands were more smacking the keys than playing-he was posing for his mother, whose por trait of her son in this exact pose now resides at his alma mater, Trinity Col lege, Cambridge (the piano is at the Fields Institute). By the time Donald was 13, he had not only become a de cent pianist for his age, learning from one of his father's musician friends who frequented their cavernous studio living room that was furnished with not one but two grand pianos, he was also composing. One arrangement he titled "Autumn" and another "Devil," which was part of an opera called "Magic." Later, he composed a String Quartet in F Minor, as well as a few songs. Cox eter often recalled that his mother took him to see Gustav Holst for an evalua tion of his work Holst was then a com poser-in-residence at a girls' school just outside of London. "I don't know how she got to him," Coxeter said, "but she took me along and I showed him some of the music I had written, and played a little bit on the piano. On the whole he thought it was rather poor." They received much the same response from a visit to the Irish composer C. V.
I
Standford, who advised, "Educate him first." From there, as the story goes, Don ald's parents tried to shelter him from the unpleasantness of their divorce by sending him to boarding school, St. George's in Harpenden, just outside London. Twelve-year-old Donald, how ever, found a better escape in his head. He created his own language and called it Amellaibian-a cross between Latin and French. He filled a 126-page book, detailing the imaginary world in which Amellaibian was spoken, a mythic place for which he even included maps (this pre-dated Tolkien by decades). Written in impeccable upper-case let ters worthy of a draftsman, the book also contains vocabulary lists, histo ries, genealogies, short stories, and a section called "Fairies Birthdays and Other Events." Gradually, the text turns very numerical, with pages and pages of computations devoted to weights and measures, formulas, equa tions, and Amellaibian magic num bers-these were the numbers that fac tored into Donald's favourite number at the time, 250. Of his time at boarding school, Cox eter recalled, "I was incarcerated." He was miserable, but he admitted that his formative encounter with geometry oc curred at St. George's. Convalescing in the school infirmary from the chicken pox, Donald found himself lying next to John Flinders Petrie, son of the Egyptologist and adventurer Sir William Matthew Flinders Petrie. It was the beginning of a long friendship and collaboration. The two began chat ting about why there were only five Platonic solids, and they passed the time contemplating the possibilities of others in higher dimensions. A year or two later, Donald won a school prize for an essay on how to create shapes in higher dimensions. He called it "Di mensional Analogy." Donald's father subsequently decided his son deserved a more challenging ed ucational environment. He took Donald
© 2004 SPRINGER-VERLAG NEW YORK, LLC, VOLUME 26, NUMBER 3, 2004
17
Dear Professor Neville, Professor Edith Morley says I am to write to you and say she suggested it. I am going to buy your book on the Fourth Dimension, as I am awfully keen on that sort of thing. I am writ ing a book myself on Dimensional Analogy, of which I enclose an out line . . . Yours Hopefully, Donald Coxeter
Portrait of child Coxeter painted by his mother. (All illustrations are used by kind permission of Susan Coxeter Thomas.)
and his essay to see Bertrand Russell Russell and Donald's father were fellow pacifists and had met in London at a con scientious objectors' meeting during the First World War. Russell agreed that Donald held great potential in mathe matics and suggested he get in touch with E. H. Neville, the mathematician who helped bring Ramanujan from In dia to Cambridge. Among Coxeter's papers is a letter sent to Neville by a family friend, Pro fessor Edith Morley, dated September 1 1 , 1923. She wrote: Dear E.
H. ,
I have taken a liberty which I hope you will forgive! A certain Donald Coxeter, aged 1 5, who is supposed to be a rather unusual mathematician and musician for his years has spent his summer holidays in writing what I am told is an entirely original trea tise on the 4th dimension. The boy is a friend of my friend Mrs. McKillop: 18
THE MATHEMATICAL INTELLIGENCER
I don't known him personally, but I have heard a great deal about him and know that he does not get any real sympathy or understanding at school in his mathematical pursuits. I think you will forgive me for sending him word he may write to you and ask you to help him. Apparently he has read your little book (I think I'm right in saying this): at any rate, he has heard of it and feels you are the one person who can help him. If there is no promise in his work, you can easily choke him off if there is, your advice may be invaluable to him. He is to go to Cambridge later on. He will write to you direct when he plucks up courage to do so, and I hope you will not think either of us very presumptuous. Yours v. sincerely, Edith Morley On exactly the same day, September 1 1, 1923, the prodigy in question, age 16, also put pen to paper:
Donald had almost given up on any lifeline from Neville when, a month later, he received his reply. A meeting was arranged at St. George's. Neville asked Donald one question: "What has a limit?" As Coxeter recalled, when he did not reply "A sequence," Neville ad vised him to leave his current school ing (it is uncertain whether Neville's motivation at removing him was that he was impressed not to have a nai:ve answer to his question, or dismayed at a deficient one; Donald, from what he remembered of the scenario, modestly guessed the latter). Neville suggested Coxeter drop all subjects save mathe matics and German and be fast-tracked by private tutelage for Cambridge. A suitable tutor was found in Alan Robson at Marlborough College. Don ald rented a room with a family in town and rode his bicycle to the College where Robson coached him daily dur ing his spare period (the College would not admit a new student as old as 16). As far as marks went, he initially was ranked among the bottom of all Rob son's students-he was obsessed with the fourth dimension but he was dis mally behind on some of the funda mentals. Gradually he rose from the bottom to the top of the class, which was possible not only because he ne glected all other subjects but also be cause he had been expressly forbidden from daydreaming about polytopes un til all his homework was completed. Of course, Coxeter had been unable to ab stain entirely, as he noted a few years later: "I still spent a good deal of my spare time writing further volumes of Dimensional Analogy. I shall always re member the thrill, the excitement I felt, while sitting under a tree in the nearby Savemake Forest, when I rediscovered
Donald with his father.
the pure Archimedean polytopes in six, seven and eight dimensions." More than forty years later, this prize-win ning essay reached its telos when Cox eter published his book Regular Poly topes. His long-lost tutor Alan Robson sent him a postcard of congratulations: "I am glad to see your Polytopes actu ally printed; and I like it very much. The pictures and tables are very pleasing. What a long time it is since you made that resolution (do you remember it?), when you were working for the Trinity exam, not to work in 4 dimensions ex cept on Sundays. " From Marlborough, Donald was sent on his way with one final gift from his tutor. Robson suggested that Cox eter submit his work to the Mathe matical Gazette. His attempts to evaluate the volume of a spherical tetrahedron led to some definite inte grals, about which he confessed baf flement. In volume 13 of the Gazette, published in 1926, Coxeter proposed: "Can any reader give an elementary verification of the results which have been suggested by a geometrical con sideration and verified graphically?" Cambridge and Princeton and Beyond
For the Michaelmas term of 1926, Cox eter was offto Cambridge, bolstered by an entrance scholarship and a sub-
stantial supply of homemade marzipan from his mother. He settled into room G9 of Whewell's Court. What could be better, in the wildest dreams of a Trin ity fresher studying mathematics, than to receive, in November, a response to his Mathematical Gazette query. A reg istered letter arrived from none other than the great G. H. Hardy, then a pro fessor of geometry at Oxford. "I tried very hard not to spend time on your in tegrals," Hardy noted alongside his pages of calculations, "but to me the challenge of a definite integral is irre sistible." This was a rite of passage; Coxeter had entered the realm of the mathematical dialectic. At Cambridge, Coxeter kept to him self and the rigours of study. The first and only mention of his name in the Trinity yearbook was in 1928 when the Magpie and Stump debating club re ported, "We have two veteran new members, Mr. J. A. Todd, who is too funny for words, and Mr. H. S. M. Cox eter, who is always very good and un intelligible, but terribly brief." With J. E. Littlewood as his director of under graduate studies, Coxeter completed the Tripos examinations with "B Wran gler" standing. His PhD advisor was H. F. Baker, who had graduated from Cambridge himself, in 1888. He had re mained as a scholar and teacher, and was elected the Lowdean Chair of As-
tronomy and Geometry in 1914. Every Saturday morning Coxeter made the 10-minute bicycle ride from his resi dence (by now in Great Court), over the River Cam, to Baker's house on Storey's Way, where he reported his progress. Saturday afternoons were re served for Baker's famous geometrical "tea parties." Coxeter attended along with P. Du Val, G. de B. Robinson, J. A. Todd, D. W. Babbage, J. G. Semple, T. G. Room, W. J. Welchman, and William Hodge. As noted in one of Baker's obituaries in 1956, " [He) gath ered around him a group of young men, infected by his enthusiasm and his for ward vision . . . here gathered the in spiration which has made geometry the great subject which it is in many uni versities here and beyond the seas." Baker's followers were all very keen, though some found these in escapable meetings-on a Saturday, no less-rather tiring. Baker himself never tired, apparently, and kept the proceedings running at a clip. Each stu dent took one afternoon to present his most recent research, followed by dis cussion. During one of Coxeter's ses sions, in 1929, as he noted in his Per sonal Record Book of Fellows of The Royal Society, I described the sequence of 'pure Archimedean' polytopes in 3, 4, 5, 6, 7, 8 dimensions (later to be named ( - 1) 21. 021. 121. 221. 321. 421 ) with their number of vertices: 6, 10, 16, 27, 56, 240. Coxeter went on to detail that One of the algebraic geometers imme diately expressed interest because 6, 10, 16, 27, are the numbers of lines on the Del Pezza surfaces in 6, 5, 4, 3 di mensions. Du Val went one step far ther by declaring 2 X 28 to be the num ber of lines on the 'Del Pezza surface' in 2 dimensions, which is a repeated plane joined to itself along a qua dratic curve of genus 3; the lines are the repeated bitangents. This led to my paper on Pure Archimedean Poly topes. One day, during a solitary bi cycle ride on the "Gags, " I saw how these and other polytopes could be ex hibited as members of one family by
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means of the symbol npq (for a figure in n + p + q + 1 dimensions). This led to my long paper in the Philo sophical Transactions of this society. On another occasion when Cox eter's tum came at the tea party, he in vited his "Aunt Alice," as he called her, to deliver a joint lecture, using her fa mous models as props. She was more widely known as Alicia Boole Stott, a housewife-geometer whom Coxeter credited with introducing the word "polytope" to the English language circa 1902. Stott was the middle child of George Boole's five daughters. Her father, who became famous for his al gebra of logic as published in his book The Laws of Thought 150 years ago, died when she was four years old, thus indicating, by Coxeter's estimation, that her mathematical ability was purely hereditary. Forever giving credit and praise where it was due, Coxeter wrote an extensive biographical profile of Stott (as he did for many other pre decessors in the field) and included it in his book Regular Polytopes. Stott's influence on Coxeter's work is evident from the preface of his PhD disserta tion. He wrote: Chapters 7, 9, and 13 comprise an attempt to express in a more gen eral form some of the discoveries of Mrs. A. BOOLE STOTT and Prof. P. H. SCHOUTE. In Chapter 10, I have carried out to its logical conclusion a hint dropped by Mrs. STOTT. SCHOUTE apparently failed to see the importance of her "partial oper ations", and consequently missed an infinite family of uniform poly topes. . . . Ludwig Wittgenstein also took a lik ing to Coxeter, selecting him as one of six pupils for his seminar on the phi losophy of mathematics. "I had tea with Wittgenstein yesterday," he said in a letter home in his later years at Trin ity. "He talked very interestingly about blindness and deafness, and why you get seasick on a camel but not on a horse." Adding in closing, "He doesn't seem any more abnormal than before." Wittgenstein made on Coxeter a similar impression as Coxeter on the 20
THE MATHEMATICAL INTELLIGENCER
but he loved his mirrors. In the 1960 Geometry Project documentary, pro duced at the University of Minnesota, Coxeter and his colleagues constructed a number of gigantic kaleidoscopes. Into one he placed a triangle on which he had printed in bold letters the word NONSENSE. In another he positioned his wife's beloved dachshund, Nico, who, invariably growled (and under standably so) at Donald. Coexeter carried his mirrors around in felt pouches sewn for him by his mother. Every so often in his diaries he would note: "Repaired mirrors." The hinges that sealed one mirror to the next, were, perhaps, shaken out of alignment from his fondness for dri ving too fast. "Was copped for reckless driving (65 m.p.h.)," he noted one day, and on another, "Drove Pat [Du Val] to have tooth extracted (Skidded and dented fender of another car on the way)." From the vantage point of old age, Coxeter described his years at Prince ton, where he studied with Oswald Veblen, Hermann Weyl, George P6lya, J. W. Alexander, L. P. Eisenhart, J. H. M. W edderbum, Eugene Wigner, and Solomon Lefschetz, as the happiest times of his life. He was constantly back and forth to New York, pursuing women almost, but not quite, as much as mathematics. His courtships, how couldn't understand that kind of phi ever, were doomed due to their pre losophy," recalled Coxeter. "I thought dominantly metaphysical nature. After it was nonsense. It didn't appeal to me one disappointment, he wrote a long at all. The only thing I remember of his letter confiding in his father, relaying work was that his book Tractatus the romantic disaster detail by detail Logico-Philosophicus began with 'The and then signing off by saying, "I am world is everything that is the case,' writing this in bed in the middle of the and ended with the famous sentence, night. Being too bowled over to sleep. 'Whereof one cannot speak one must Now I will try to fmd solace in Klein's 'Lectures on the Icosahedron.' " remain silent.' " Not long after his second visit to After earning his PhD at Cambridge, Coxeter did fellowship stints back and Princeton, upon his return to Cambridge forth between Princeton and Cam in August of 1935, Coxeter met the "at bridge (1932-33 Princeton, 1933-34 tractive Dutch girl" who did become his Cambridge, 1934-35 Princeton, 1935- wife: Rien Brouwer. They met in March 36 Cambridge). On each campus he 1936, and with a mere two months kept with him a collection of mirrors courtship, he proposed, in a cemetery. he'd had cut specially for his purposes They were married under tragic circum (they are now at York University). The stances, in August at the Round Church mirrors had hinges fastened to them, at Cambridge, shortly after the sudden and without much effort they could be death of Donald's father, who drowned assembled into a crude version of a after suffering a heart attack while kaleidoscope. He was hardly a vain man swimming in the English Channel.
debaters: he was unintelligible. Witt genstein refused to lecture for 50 min utes, as was the custom, but required 150 minutes, partly because it took him an hour to warm up and partly due to his habit of stopping mid-sentence and holding his audience in wait as he worked through the next point or searched for the next word internally. Coxeter once timed a pause to reach more than twenty minutes, after which Wittgenstein carried on where he left off, as if all was normal, and with no apology or explanation. On another oc casion, Wittgenstein complained the lecture room was too formal, he said he preferred a private sitting room. Coxeter offered his in staircase I of Great Court. Wittgenstein made use of it several times, even after Coxeter dropped out of the class to spend more time on his mathematical research. "I
" I had tea with Wittgenstein
yesterday . H e
d oesn 't seem any more abnormal than before . "
Coxeter at Cambridge.
Early in 1936, Coxeter had refused the offer of an assistant professorship at the University of Toronto. Baker was retiring, making Coxeter a candidate for his Lowdean chair of geometry. It was an elected position, and that sum mer he learned he had lost the chair to William Hodge, who had won the Adams Prize for geometry in 1934. Upon consultation with Baker, Coxeter realized he had few options. He was persuaded to reconsider the Toronto offer. "Many good men have begun away from England," advised Baker, adding, "Europe seems now to be mad; and anyway, Toronto is an inspiring
place." On June 6, Coxeter wired Samuel Beatty, then the chair of the mathematics department at Toronto, and asked if he may accept the offer af ter all. A "Yes" cable arrived two days later. On September 3, the newlywed couple sailed for Canada. Coxeter spent almost all his r tthematical life at the University of Toronto, save for numerous positions as a visiting pro fessor around the world. Just before leaving Cambridge, Coxeter was asked by Littlewood to write the eleventh edition of W. W. Rouse Ball's Mathematical Recre ations & Essays. The notes left to Lit tlewood by Ball (who had been Little-
wood's tutor at Cambridge from 1903 to 1906) were passed on to Coxeter. In 1938 he completed the revision, which included a new chapter on polyhedra. This chapter was written very much in the same style as Coxeter's later Reg ular Polytopes. He also added a chap ter on cryptography, written mostly by A. Sinkov, with whom Coxeter had a life-long correspondence (they first be came acquainted, most likely, during Coxeter's second visit to Princeton). It was through Mathematical Recre ations that Coxeter first encountered John Horton Conway. Although Con way never studied with Coxeter, he al ways considered himself an honorary student, because of the very Coxeter ian nature of some of his work The seminal meeting of minds occurred in March 1957, when, as a student at Caius College, Cambridge, a teenaged Conway penned a letter to Coxeter, which began: Dear Sir, Over the past year or so my copy of your edition of Ball's "Mathematical Recrea tions " has accumulated an as tonishing number of notes and some corrections. Most of these can hardly be said to be suitable for publication in later reprints, but one or two may seem important. . . . The letter went on for five pages; tiny scrawls interrupted by only a select few diagrams, including a very sure handed rendering of a magic square. Conway finally signed off by saying,
My absolutely last remark is a ques tion. Where can Ifind the requisite in formation required to draw { 5,3,3), or do I have to work out the details for myself? I should be very thankful if you could supply me with some ac cessible information. Yours hopefully, J. H. Conway Mathematical Contributions
In the diaries that Coxeter kept for nearly three-quarters of his life, now part of his archives that are kept at the University of Toronto, he mostly
recorded social engagements, occa sionally seminars, books, and con certs. Very infrequently he made note of manuscripts underway and theo rems proved or in process. On Febru ary 22, 1933, for example, he wrote, "Proved (while getting up) that all continued products of generators (of finite g.g.r.) are conjugate." This product of generators has been called a Coxeter element and its order a Cox eter number. It will be for his work on regular poly topes, reflection groups, and related ar eas that Coxeter will be best remem bered. A group generated by involutions, and defined by relations specifying the period of the products of all pairs of gen erators, is known as a Coxeter group. In spired by a fellow student, J. A. Todd, with whom he later had a joint paper on the subject, Coxeter investigated the groups of symmetries of regular poly topes. Eventually, this led to a system atic study of the reflection groups. In a sequence of papers finished in 1933 [1,2,3], Coxeter gave the complete clas sification of discrete groups generated by reflections (finite and infinite) in spherical and Euclidean spaces. Coxeter's name is also associated with a graph, corresponding to a Cox eter group, where vertices of the graph represent involutory generators. When the generators commute, the corre sponding vertices are not connected. Otherwise, the vertices are connected and the edges of the graph are labeled with integers ;:::: 3 or with a symbol x. The label under the edge connecting two vertices of the graph indicates the order of the product of the corre sponding generators. Although Cox eter wrote that he started using graphs to represent reflections during his visit to Princeton in 1932, the first reference to the use of a graph can be found in a paper he published in the Journal of the London Mathematical Society [ 1 ] , which h e submitted o n his birthday, February 9, 193 1. The first published appearance of the graphs was in the Annals of Mathematics in 1934 [2]. He completed this paper during his first visit to Princeton in February 1933. E. B. Dynkin rediscovered essentially the same notation independently some years later. Coxeter greeted this news
VOLUME 26, NUMBER 3. 2004
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cordially rather than competitively, worked on constructing uniform poly duction to Geometry, published in and he was particularly pleased with topes. After several unsuccessful at 1961. It was translated into numerous the communication with Dynkin that tempts at proving the completeness of languages, is still in print, and is cur came about as a result. C oxeter liked their enumeration of uniform poly rently on the syllabus of a math course to recount details of a letter from hedra, C oxeter, Longuet-Higgins, and atM cG ill University. Towit: a librarian Dynkin, dated April 3, 1984, in which M iller wrote a paper [8] containing the at the University of Toronto once at Dynkin remarked, " It is striking that complete classification (as was proved tested that Introduction to Geometry my notation turn ed out to be so simi later). For a nice account of the im was the book most frequently stolen lar to yours. This probably shows how portance of this work, refer to G iiin from the mathematics library. C oxeter himself, however, consid natural these notations are." baum's contribution in [7]. C oxeter's masterpiece was the fa ered his Regular Complex Polytopes While at Princeton in 1933, C oxeter began to enumerate the stellations of mous Regular Polytopes. With its pub (RCP) to be his masterpiece. It had the icosahedron (he was certainly the lication in 1947, he made his mark as a been inspired, to some extent, by his first to complete the enumeration). great expositor, unifying with great el long-standing collaboration with G . C . Upon return to England, he collabo egance and clarity his own research on Shephard, which began in 1951 when rated with Petrie and Du Val, who ex polytopes as well as the findings of his C oxeter served as the extern al exam ecuted corresponding fine-line draw predecessors (whom he included in his iner of Shephard's dissertation Com ings, as well as Flather, who made treatise with wonderful historical plex Polytopes. Twenty years later, this models of these polyhedra (C oxeter re sketches). Regular Polytopes pro seed reached fruition in C oxeter' s called that Flather's small size, as a foundly influenced an extraordinary mind with the publication of RCP. In dwarf, facilitated his production of such number of mathematicians. In the comparing this book to its predeces intricate models). Flather completed memorial article in the Notices of the sor, Regular Polytopes, C oxeter re twenty-four of the models and shipped AMS, G riinbaum states, it is " possibly marked, " The sequel is more profound. them to C oxeter prior to the Second one of the most quoted geometry texts The exposition is beautiful, including World War for safe-keeping, fearing of the century," and Peter M cM ullen several outstanding drawings by M c they might have been destroyed if kept acknowledges the " deep influence" of M ullen." It is also with RCP that C ox in England (one was damaged in trans the book on his career. Others have re eter provided the most explicit and port, but the remaining twenty-three are ferred to it as their Bible, or as the vivid explanation of how he worked as now kept atY ork University). After the modem-day addendum to Euclid's El a mathematician. Upon the publication War, Flather made another set, this time ements. Of course, some would argue of the long-awaited revised second edi complete, of the fifty-nine stellations that C oxeter's masterp iece was Intra- tion, he commented to the press, " I (kept at Trinity C ollege, C ambridge). The manuscript of the " Fifty-nine Icosa hedra" was completed once C oxeter was in Toronto and it was submitted in 1938. G . de B.R obinson was instrumental in bringing C oxeter to Toronto (the two had met at Baker's tea parties in 1928). C oxeter,R obinson, andR ichard Brauer founded the Canadian Journal of Mathematics, with C oxeter serving as the first editor-in-chief. It was also through R obinson that C oxeter came upon Wythoffs construction, a topic of many subsequent lectures. According to C oxeter, Wythoff, in 1918, derived polytopes from the group (3,3,5], and remarked that " a similar investigation . . . may be undertaken . . . with regard to other polytope families. . . . " In 1930, R obinson provided the proof of the re sult. The construction comprised chap ter fourteen of C oxeter's PhD disserta tion and was also used in chapter fifteen in an attempt to classify convex uniform polytopes. Independently, Coxeter with his mirrors. M . S. Longuet-Higgins and C . P.M iller
J.
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THE MATHEMATICAL INTELLIGENCER
have made an attempt to construct the book like a Bruckner symphony, with crescendos and climaxes, little fore tastes of pleasure to come, and abun dant cross-references. The geometric, algebraic and group-theoretic aspects of the subject are interwoven like dif ferent sections of an orchestra." The aforementioned are but a few of the contributions Coxeter made to mathematics and geometry. Other im portant aspects of his work should be given at least a passing mention. He made many contributions to inversive geometry, exploring its connection to hyperbolic geometry. He was among the first to move from "real" to combi natorial geometry. His 1937 paper on regular skew polyhedra [ 4], following discussions with Petrie, extended the notion of a regular polyhedron to in clude infinite polyhedra with vertices adjacent to any given vertex belonging to a skew polygon (such as the poly gon formed by the lateral edges of an antiprism). Coxeter worked on sphere packing and extreme quadratic forms. His interest in projective geometry gave direction to several PhD disserta tions and resulted in two of his books: The Real Projective Plane in 1949, fol lowed by Projective Geometry in 1957. Coxeter's interest in discrete groups generated by involution naturally led him to investigate non-Euclidean geom etry. In 1950, he wrote a paper with Witrow [6], listing all fifteen honey combs of the hyperbolic 3-space. As a sequel to this, at the International Con gress of Mathematics held in Amster dam in 1954, Coxeter gave an invited lecture on the complete classification of tessellations in n-dimensional hy perbolic space [5] . A distinguished mention must go to William Moser, with whom Coxeter au thored Generators and Relations for Discrete Groups, published in 1957. Moser, on many occasions, such as at the conference held honouring Coxeter on his 70th birthday-which drew hun dreds of mathematicians from around the world and resulted in the produc tion of the book The Geometric Vein, The Coxeter Festschrift-has shared stories that nicely exhibit the apprecia tion and respect held for Coxeter, es pecially the admiration for him as a
teacher, disseminating his love of geometry and the joy he celebrated in its beauty, both in the classroom and at the weekly geometry seminars. On one occasion Moser bestowed this praise:
called me Donald. " I replied, "Yes, Pro fessor Coxeter. " And so it remained .for a few more years.
Professor Coxeter is a teacher in the broadest sense of the word. . . . Sixteen students have completed Ph.D. theses
On Donald Coxeter's second-last trip out into the world he traveled to Banff, Alberta, with his daughter Susan, his devoted caretaker since the death of Rien in 1999, for a conference on "As pects of Symmetry." Introducing his paper, he flicked on the overhead pro jector and slid on his first trans parency. At that moment, Coxeter was bathed in a gigantic colour projection ofM. C. Escher's "Circle Limit III." "The topic of my paper," began Coxeter, "is one that has intrigued me and preoc cupied me for nearly five decades. It's about what I call the 'intuitive geome try' of my friend M. C. Escher." After meeting at the International Congress in 1954, Coxeter and Escher struck up a collaboration of sorts, mostly through correspondence. Cox eter asked Escher if he could use one of his tessellations in a paper he was publishing. Escher agreed, and when he received his complimentary copy, other diagrams therein done by Coxeter served to break Escher's long-standing creative block. Escher did not compre hend what he called Coxeter's "hocus pocus text." But in a letter of thanks to Coxeter he exclaimed,
Like Any Other Artist
Introduction to Geometry was the book most frequently stolen from the mathematics l i brary . under his direction. He has taught se lected groups of gifted high-school stu dents, breathed mathematical life into high-school teachers, and inspired gen erations of students during his years at the University of Toronto. At Coxeter's 80th birthday party Moser told this story about his mentor:
In 1955, Donald Coxeter and I danced together as partners, not in each other's arms, but holding hands at a respectable distance. This was not a Though the text of your article on University of Toronto Ph.D. require "Crystal Symmetry and its General ment. That summer we were in Still izations" is much too learnedfor a sim water, Oklahoma, where Donald ple, self-made plane pattern-man like taught an N.S.F. summer course and me, some of the illustrations, and es I was his assistant. As a divers1:on pecially.figure 7, gave me quite a shock. .from the intense work-he was Since a long time I am interested in preparing the lectures, I was prepar patterns with "motives" getting smaller ing notes of his lectures, and we were and smaller till they reach the limit of also completing our book-Donald de irifinite smallness. The question is rel cided to learn ballroom dancing, and atively simple if the limit is a point in the course required participation in the centre of a pattern. Also a line-limit pairs. You have noticed that I call him is not new to me, but I was never able Donald, as all of you probably do. But to make a pattern in which each "blot" I do so by invitation! At the end of is getting smaller graduallyfrom a cen that summer, he said to me: "William, tre towards the outside circle-limit, as we have known each other .for six shows your .figure 7. years, you have been my student .for .four years, we have worked closely to Subsequently, Escher credited Coxeter gether, we have written a book to with inspiring his Circle Limit draw gether. I think it is time tha t you ings. While working on his Circle Lim-
VOLUME 26, NUMBER 3, 2004
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its Escher would say, "I'm Coxetering today." And in a letter to his son, George, Escher effused,
My woodcut inspired by the Coxeter system is finished, and to me it is the most beautiful one that I have made of the "smaller and smaller" type. I can't stop looking at that circular all encompassing limit of infinitely small shapes, all so logical and or dered. The work approaches absolute beauty and simplicity. I am anxious to hear the reaction of Mr. "Cokes eater" himself, to whom I sent a copy. Coxeter, for his own part, considered himself at once a mathematician and an artist. "I am like any other artist," he once told the Globe a,nd Mail. "It just so happens that what fills my mind is shapes and numbers." Though Coxeter did geometry for the sheer beauty and art of it, not for any practical purpose, his work often did find application in varied fields. Architect and polymath Buckminster Fuller, or Bucky, as Coxeter called him, encountered Coxeter's work in the con-
struction of his geodesic domes. Fuller later bestowed grandiose praise on Coxeter in the dedication of his book Synergetics, on the geometry of thought:
By virtue of his extraordinary life's work in mathematics, Dr. Coxeter is the geometer of our bestirring twenti eth centu'ry. [He is] the spontaneously acclaimed terrestrial curator of the historical inventory of the science of pattern analysis. Though flattered, Coxeter viewed this dedication as Bucky doing some name dropping in order to attract a mathe matical audience to his book. Donald read snippets of it and deemed Fuller would have done better to consult a mathematician in the writing. Coxeter should also be remembered for his keen sense of social justice. A pacifist like his father, he declined an opportunity to work as a code-breaker during the Second World War. On sev eral occasions he spoke of the unfair treatment theoretical physicist Leopold Infeld, a colleague at the University of Toronto, received from Canada. Infeld
Coxeter in Toronto with one of his great-grandchildren.
24
THE MATHEMATICAL INTELLIGENCER
was forbidden from leaving the coun try to spend his sabbatical leave in Poland, his birth country, because Poland was at the time "behind the iron curtain." Infeld was denounced by con servative critics in parliament as a po tential traitor to the Canadian people who would provide the communists with atomic secrets. After resigning from his position at the university and remaining in Poland, the Infelds and their Canadian-born children were stripped of their citizenship. Coxeter was acknowledged for his support in the publication of Infeld's book Why I left Canada. Helen Infeld, after Leopold's death, kept in touch with Coxeter and gave him these kind words in a letter dated January 6, 1976:
Do you know, my life has been such that I have come to highly evaluate some human qualities and feel that it is good to tell people who have them so. I'd like to tell you that I do admire you as a person of principle, not swayed by general prejudice, emo tional blindness or temporary hyste ria of others in important matters.
Would that people as a whole had such rational understanding, everywhere!
A U T H O it l
And finally we return to Coxeter's piercingly dry sense of humour. He was fond of nonsense and particularly Lewis Carroll's The Adventures of Al ice in Wonderland. His most common request was for the "Jabberwocky" passage. He said that word-"Jabberr wOckAy"-with such relish. He would recite from memory with the same dra matic intonation, cranking the volume on his otherwise sedate self:
One, two! One, two! And through and through The vorpal blade went snicker-snack! He left it dead, and with its head He went galumphing back. ''And, hast thou slain the Jabberwock? Come to my arms, my beamish boy! 0 frabjous day! Callooh! Callay! " He chortled in his joy. Once, when asked why he never tired of Alice in Wonderland, he replied, "It's like reading about a part of mathemat ics that you know is beautiful, but that you don't quite understand. Like string theory. That's as much a mystery to me as it is to anyone else who can't make head nor tails of the 16th dimension." During the final weeks before his death, Donald Coxeter persevered in putting the final touches on a paper he had delivered in Budapest the summer before. He could not quite believe it when no further errors or typos could be found-he always took great pleasure in seeking out the mistakes in his papers
and books, to be corrected in subsequent printings, of which there were always many. With the paper finally finished, Coxeter died two days later. REFERENCES
mental regions are simplexes, J. London 6 (1 93 1 ) , 1 32-1 36.
[2] H. S. M. Coxeter, Discrete groups gener ated by reflections, Ann. of Math. 35
ation of finite groups of the form R? =
Proc.
London
=
1 , J. London Math. Soc. 1 0
(1 935), 2 1 -25.
[4] H . S. M . Coxeter, Regular skew polyhedra
[1 0] J. A. Todd , The groups of symmetries of regular polytopes, Proc. Camb. Phil. Soc. 27 (1 93 1 ) , 2 1 2-231 .
43 (1 937), 33-62.
[5] H. S. M. Coxeter, Regular honeycombs in hyperbolic space; pp. 1 55-1 69 in Proc. Internal. (1 954),
Congress
Math.
Amsterdam
North-Holland, Amsterdam, 1 956.
[6] H. S. M . Coxeter and G. J. Whitrow, World structure
and
non-Euclidean
honey
Books Mentioned
(with P. Du Val, H. T. Flather, and J. F. Petrie) The
Fifty-nine
and Essays
don, 1 939.
2003), Notices of the AMS 50 (2003), 1 234-1 240.
[8] M . S. Longuet-Higgins, J. C. P. Miller, and H. S. M . Coxeter, Uniform Polyhedra, Phi
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[9] J. A. Todd and H. S. M. Coxeter, A prac
University of
W. W. Rouse Ball's Mathematical Recreations
201 (1 950), 41 7-437. and A. I. Weiss, H. S. M . Coxeter (1 907-
Icosahedra,
Toronto Studies, Toronto, 1 938.
combs, Proc. Roy. Soc. London Ser. A
los. Trans. Roy. Soc. London Ser. A
(1 934), 588-621 ' [3] H. S. M. Coxeter, The complete enumer (R,R1Jk•1
Math. Soc.
analogues,
[7] E. W. Ellers, B. Grunbaum, P. McMullen,
[1 ] H. S. M. Coxeter, Groups whose funda Math. Soc.
in three and four dimensions and their topological
(1 1 th edition), Macmillan, Lon
Regular Polytopes,
Methuen, London, 1 948.
The Real Projective Plane,
McGraw-Hill, New
York, 1 949.
(with W. 0. J. Moser) Generators and Relations for Discrete Groups,
Springer-Verlag, Berlin,
1 957. Introduction to Geometry, Wiley, Projective Geometry,
New York, 1 961 .
Blaisdell, New York, 1 964.
Regular Complex Polytopes,
Cambridge Uni-
tical method for enumerating cosets of a
versity Press, London, New York, 1 97 4. (2nd
finite abstract group, Proc. Edinburgh
edition reprinted with corrections and a new
Math. Soc.
1 4th chapter).
5(2) (1 936), 26-34.
VOLUME 26, NUMBER 3, 2004
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lj¥1(¥·\·i·i David
E . Rowe , E d itor
Coxeter on People and Polytopes David E . Rowe
H
I
.
S. M. Coxeter, known to his friends as Donald, was not only a remarkable mathematician. He also en riched our historical understanding of how classical geometry helped inspire what has sometimes been called the nineteenth century's non-Euclidean revolution. Coxeter was no revolution ary, and the non-Euclidean revolution was already part of history by the time he arrived on the scene. What he did experience was the dramatic after shock in physics. Countless popular and semi-popular books were written during the early 1920s expounding the new theory of space and time in Ein stein's general theory of relativity. Gen eral relativity and subsequent efforts to unite gravitation with electromagne-
Send submissions to David E. Rowe, Fachbereich 1 7 - Mathematik, Johannes Gutenberg University, 055099 Mainz, Germany.
26
H. S. M. Coxeter
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
tism in a global field theory gave re search in differential geometry a tremendous new impetus. Geometry became entwined with physics as never before, and higher-dimensional geometric spaces soon abounded as mathematicians grew accustomed not just to four-dimensional space-times but to Hilbert space and its infinite dimensional progeny. Seen from this perspective, Cox eter's work on polytopes surely must have looked quaint to many contem porary observers. Still, his research was by no means a lonely adventure; he found plenty of others who shared his fascination with the symmetry properties of geometrical configura tions. Aided by the wonders of com-
puter graphics, this classical geomet ric style of mathematics has recently enjoyed a tremendous resurgence. Donald Coxeter surely saw this not as a personal triumph but as a triumph for geometry and its practitioners. One has only to read Coxeter with an his torian's eye to appreciate this part of his legacy. The most noteworthy example is his Regular Polytopes, which first ap peared in 1948, marking the culmina tion of 24 years of loving labors in a field that was considered exotic and well outside the mainstreams of math ematical research. Coxeter seems to have relished his role as the consum mate outsider. He was an old-fashioned geometer who thoroughly grasped the significance of modem methods, some of which he invented (Coxeter groups, Coxeter diagrams). Yet he identified with gifted amateurs and artists all his life. Indeed, he spent as much effort resurrecting the work of forgotten he roes as he did rehabilitating the theory of polytopes, his main field of research. In Regular Polytopes Coxeter aimed to reach a broad audience, and by all accounts he succeeded. His text offers a reasonably self-contained introduc tion to the classical background, much of which would have been familiar to Johannes Kepler. It begins with a tour of polyhedra and their symmetry groups, tessellations and honeycombs, kaleidoscopes, and star-polyhedra. Gently guiding the reader through the first six chapters, Coxeter hoped to provide the novice with adequate preparation for the adventures that fol low. Clearly he knew that the land of polytopes was fraught with difficulties, and so he suggested for all those who might feel "at all distressed by the multi-dimensional character of the rest of the book" that they consult the text books [Manning 19 14] and [Som merville 1929]. Now, both of these were seriously dated in 1948; this ad vice betrays how unfashionable higher geometry had become since the heyday of Italian projective geometry, when hyperspace constructions abounded. At some early point in his career, Coxeter clearly was drawn to the lore surrounding this branch of geometry. In Regular Polytopes he reaches across
the boundaries of time, embracing kin dred spirits, both living and dead, who shared his geometrical and aesthetic vision. However skeptical one might be of this bow toward a fictive community of polytope afficianados, one cannot help but admire Coxeter's enthusiasm and the generosity he showed toward those who shared in the enterprise. This liberal attitude was a natural one for Coxeter, who came from a Quaker family. He saw himself as an interna tionalist and felt that mathematical knowledge enriches humankind as a
H e saw h i mself as an i nternational ist and felt that mathematical knowledge en riches h u manki nd as a who l e . whole. A particularly striking comment reflecting his views can be found in the preface of Regular Polytopes:
The history of polytope theory pro vides an instance of the essential unity of our western civilization, and the consequent absurdity of interna tional strife. The Bibliography lists the names of thirty German mathe maticians, twenty-seven British, twelve American, eleven French, seven Dutch, eight Swiss, four Italian, two Austrian, two Hungarian, two Polish, two Russian, one Norwegian, one Danish, and one Belgian [Coxeter 1973, p. vii}. The book is strewn with historical summaries that give us a glimpse of the human side of the polytope industry. His remarks are always informative, often warm, witty, and erudite, but never stuffy. In several cases they pro vide important autobiographical clues as well, a few of which will be men tioned below.
Coxeter's Heroes
The Preface of Regular Polytopes opens in typical Coxeter style with some historical remarks. He notes that the term "polytope" was coined by Reinhold Hoppe in 1882 and intro duced into the English-language litera ture around the tum of the century by Alicia Boole Stott (1860-1940). Cox eter reminds us that the subject had its roots in Greek mathematics. Before Euclid's time the regular polygons and polyhedra were shrouded in Pythagorean lore, as demonstrated by the prominent role played by the Pla tonic solids in Plato's Timaeus. Euclid took up the construction of these per fect bodies, first studying the con structible polygons in Book IV and then describing the five regular polyhedra in Book XIII. By making this the culmi nating topic of the Elements-which ends with the observation that the tetrahedron, cube, octahedron, dodec ahedron, and icosahedron are the only such polyhedra-Euclid helped pre serve their exalted status within clas sical geometry. Coxeter clearly identified with this tradition; he flatly stated that the main motivation for studying the Platonic solids was aesthetic. We are drawn to these figures, just as the ancient Pythagoreans were, because "their symmetrical shapes appeal to our artis tic sense." True, Felix Klein's Lectures on the Icosahedron cast the theory of the quintic equation in a fresh new light, but Coxeter saw this as wholly unnecessary: "if Klein had not been an artist he might have expressed his re sults in purely algebraic terms" [Cox eter 1973, p. vi]. Klein surely would have disagreed: Leopold Kronecker had done just that, but Klein insisted that one needed geometry in order to do Galois theory properly! Coxeter's much-sung hero in Regu lar Polytopes is the Swiss mathemati cian Ludwig Schlafli (1814-1895), whose work on this topic appeared posthumously in [Schliilli 190 1 ] . Dur ing his lifetime, Schliifli was best known for his investigation of the "dou ble-six" configuration of 12 lines which bears an intimate connection with the 27 lines on a cubic surface [Hilbert and Cohn-Vossen, 1932, 146-151], whereas
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his monograph on polytopes remained virtually unnoticed for some 50 years. As Coxeter tells it,
The French and English abstracts of this work, which were published in 1855 and 1858, attracted no atten tion. This may have been because their dry-sounding titles tended to hide the geometrical treasures that they contain, or perhaps it was just because they were ahead of their time, like the art ofvan Gogh [Coxeter 1973, p. 143}. Schliifli studied the general poly tope (which he called a "Polyschem") and developed a criterion for determin ing regular polytopes in dimensions four and higher. For this purpose he intro duced the now-standard Schliifli sym bol, which contains all the information needed to characterize a polytope. Let's consider the case of a regular polyhe dron. Ifp denotes the number of vertices or edges on a face and q the number of edges or faces that pass through a point, then the Schliifli symbol {p, q} can take on just five values satisfying the condi tion (p - 2)(q - 2) < 4. The general Schliifli symbol is defined analogously. Proceeding to dimension four, we de fine r as the number of faces or solids that meet at an edge. Schlafli could then show that the only admissible values of {p,q,r} are precisely {3,3,3 ), {4,3,3), {3,3,4), {3,4,3), {5,3,3), and {3,3,5). In a laborious fashion, he also showed that it was possible to construct these six four-dimensional figures. Finally, Schliifli proved that only three types of regular polytope exist in dimensions five and higher. These are the n-sim plex, the general measure polytope, and its dual figure, all of which are eas ily constructed as the analogues of the tetrahedron, cube, and octahedron in ordinary 3-space. Thus, Schliifli was the first to recognize that dimensions three and four are unique in that they contain "exotic polytopes." Coxeter's book gives a lucid ac count of these matters and much more. He provides a brief synopsis of Schliifli's lengthy monograph, along with a few biographical remarks on his career. He also notes that ignorance of Schliifli's work meant that it had virtu-
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THE MATHEMATICAL INTELLIGENCER
ally no impact on other researchers, in cluding an American named Washing ton Irving Stringham. A student of J. J. Sylvester's at Johns Hopkins Univer sity during the late 1870s, Stringham wrote his dissertation on four-dimen sional regular polytopes and published his main results in the American Jour nal of Mathematics. After graduation he went to Leipzig where he presented these latest findings in Felix Klein's seminar. Stringham's approach was based on an analysis of the number of regular polyhedra that can meet at a vertex point without filling up the spatial re gion surrounding it. This gave his work a visual appeal, and numerous others began to experiment with solids that bound a 4-D cell. Stringham was able to show that there are three "exotic polytopes" in dimension four-the self dual 24-cell and the dual 120- and 600cells. Although partial to Schliifli's pio neering achievement, Coxeter readily admitted that
[Stringham 's} treatment was far more elementary and perspicacious, being enlivened by photographs of models and by drawings similar to our Figs. 7.2a, b, c. The result was that many people imagined Stringham to be the discoverer of the regular poly topes [Coxeter 1973, p. 143]. Coxeter further noted that another seven authors independently rediscov ered the six four-dimensional regular polytopes between 1881 and 1900, that is before the publication of [Schliifli, 190 1 ] . Only two of them, however, came up with a notation as elegant as the Schliifli symbol. This research activity on polytopes went hand-in-hand with a surge of in terest in non-Euclidean and higher dimensional geometries during the last two decades of the nineteenth century. Stringham's drawings captured the imagination of numerous amateurs, some of whom were artists. As Linda Dalrymple Henderson has shown, these new geometrical ideas had a pro found impact on modern art that was independent of the discussions on rel ativity theory and space-time geome tries [Henderson 1983]. In 1909 the
Scientific American opened a prize competition for the best popular essay explaining the fourth dimension. The hefty prize of $500 helped attract 245 entries, four of which were published in the magazine. In the wake of the rel ativity revolution, the editors opted to produce a full book of essays from the contest. Henry Parker Manning from Brown University, author of [Manning 1914], was asked to choose a suitable set of essays from the original entries and to write an introduction to the vol ume [Manning 192 1 ] . In the publisher's promotional preface, the Scientific American noted that the subject of the fourth dimension "has unfortunately been classed with such geometrical ab surdities as the squaring of a circle and the trisection of an angle" [Manning 1914, p. 4]. Yet in one important respect those who sought to plumb the mys teries of the fourth dimension were quite unlike ordinary circle squarers and angle trisectors: the latter activi ties have always been attractive to am ateur puzzle-solvers, whereas the for mer theme has a mysterious quality about it that appeals to a different kind of mindset. Coxeter on the Intuitive Approach to the Fourth Dimension
Coxeter had a very matter-of-fact atti tude about the fourth dimension. He asserted that we can approach higher dimensional Euclidean spaces in three different ways: axiomatically, alge braically, or intuitively. The first two methods pose no real difficulties, whereas the third relies on dimen sional analogy, which can easily lead one astray. He nevertheless took the in tuitive approach seriously because it can be "very fruitful in suggesting what results should be expected." He even cited Edwin Abbott's Flatland approv ingly in this connection, but he also is sued these words of warning to the mystically minded:
Many advocates ofthe intuitive method fall into an . . . insidious error. They assume that, because the fourth di mension is perpendicular to every other direction known through our
senses, there must be something mys tical about it. Unless we accept Hou dini's exploits at theirface value, there is no evidence that a fourth dimension ofspace exists in any physical or meta physical sense [Coxeter 1973, p. 1 19]. Coxeter surely had many encoun ters with those who believed other wise. His friendship with the afore mentioned Alicia Boole Stott, which began in 1930, no doubt gave him ad ditional insights into the role of the fourth dimension in Victorian culture. Alice was only four years old when her father died, and she grew up with her four sisters in poverty. Coxeter gave this vivid description of her early life:
She spent her early years, repressed and unhappy, with her maternal grandmother and great uncle in Cork. When Alice was about thirteen the five girls were reunited with her mother (whose books reveal her as one of the pioneers of modern pedagogy) in a poor, dark, dirty, and uncomfortable lodging in London. There was no pos sibility of education in the ordinary sense, but Mrs. BoJle's friendship with James Hinton attracted to the house a continual stream of social crusaders and cranks. It was during those years that Hinton's son Howard brought a lot of small wooden cubes and set the youngest three girls the task of memorizing the arbitrary list of Latin words by which he named them, and piling them into shapes. To Ethel, and possibly Lucy too, this was a meaningless bore; but it inspired Al ice (at the age of about eighteen) to an extraordinarily intimate grasp of four-dimensional geometry. Howard Hinton wrote several books on higher space, including a considerable amount of mystical interpretation. His disciple did not care to follow him along these other line r!f thought, but soon surpassed him in geometrical knowledge. Her methods remained purely synthetic, for the simple rea son that she had never learnt analyt ical geometry [Coxeter 1973, p. 258]. Coxeter presumably obtained much of this information firsthand from Ali cia Boole Stott. Alicia's mother was
Mary Everest Boole ( 1832-1916), whose father was a minister (see [Michalowicz 1996]). The Everest fam ily is mainly remembered, however, for the exploits of Mary's uncle, Colonel Sir George Everest, who worked for many years as the Surveyor General of India. In 1841 he charted the Himalayas and determined the position and height of its highest peak, known today as Mount Everest. Mary lived in France until she was eleven, taking mathe matics lessons from a private tutor. Af ter returning to England she worked as her father's assistant, teaching Sunday school classes and helping him prepare his sermons. She kept her interest in mathematics alive by reading books from her father's library. Another uncle, John Ryall, was pro fessor of Greek at the newly estab lished Queens College in Cork, West Ireland. When Mary Everest went to visit him, he introduced her to his colleague, the mathematician George Boole, who found she had a burning de sire to learn higher mathematics. Boole went to England two years later to give Mary Everest private lessons, and she eventually assisted him in writing his Investigation of the Laws of Thought (1854). Mary's father died around the time it appeared, and in 1855 she and Boole married. Their marriage was a happy one, but it lasted only nine years: Boole died of pneumonia in 1864 leav ing her with the five children to care for. Soon afterward, Mary was offered a job as a librarian at Queens College. She also began writing books on vari ous topics. One of her interests was psychic phenomena and the spirit world, which led to local controversy at the college when she tried to pub lish a book called The Message of Psy chic Science for Mothers and Nurses. As a consequence, she was forced to give up her position at Queens College, but her father's friend, James Hinton, hired her as his secretary. As Coxeter duly noted, the Hintons, father and son, made life in the Boole household, well, different than it had been. James Hinton drew Mary's at tention to evolution and the art of thinking, ideas she began to develop in a series of articles and books. Charles Howard Hinton (1853-1907) was at
least as flamboyant as his father. His interests in four-dimensional geometry no doubt deeply influenced Alicia Boole, but he married Mary's eldest daughter, Mary Ellen, instead. Perhaps his preoccupation with relationships in four dimensions caused him to be somewhat disoriented when it came to sorting things out in ordinary three space. At any rate, when it was discov ered that he was also married to a woman named Maud Wheldon, Howard was put on trial for bigamy. He and Mary Ellen thereafter fled to Japan, but even tually he found his way to Princeton. A prolific writer about the fourth dimen sion, C. H. Hinton influenced contem poraries as diverse as Edwin Abbott and the Theosophist Rudolf Steiner. Some of his writings are still available in [Hinton 1980]. Among other works, Mary Everest Boole wrote a book titled Philosophy and Fun of Algebra, described by Ivars Peterson in [Peterson 2000]. She also in vented what she called "curve stitching" to help children learn basic geometry. Her daughter Alicia shared a similar in terest in geometrical visualization, cre ating various models for projections of four-dimensional constructs to satisfy her own curiosity. She had little time for this after 1890, however, when she mar ried Walter Stott, an actuary. As Coxeter described it, "for some years she had a life of drudgery, rearing two children on a very small income" [Coxeter 1973, p. 258]. Her return to mathematics came about through her friendship with Pieter Hendrik Schoute (1846-1913), the lead ing Dutch expert on polytopes. Citing Coxeter again:
Mr. Stott drew his wife's attention to Schoute's published work; so she wrote to say that she had already deter mined the whole sequence of [middle] sections . . . for each polytope agree ing with Schoute's result. In an en thusiastic reply, he asked when he might come over to England and work with her. He arranged for the publi cation of her discoveries in 1900, and a friendly collaboration continued for the rest of his life. Her cousin, Ethel Everest, used to invite them to her house in Hever, Kent, where they spent many happy summer holidays.
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Mrs. Stott's power of geometrical vi sualization supplemented Schoute's more orthodox methods, so they were an ideal team. After his death in 1 913 she attended the tercentenary celebra tion of his university of Groningen, which conferred upon her an hon orary degree and exhibited her mod els [Coxeter 1973, pp. 258-259]. Alicia Boole Stott was mathemati cally inactive after Schoute's death, but that changed when she met Donald Coxeter in 1930. He was then working on his doctorate in Cambridge under the supervision of H. F. Baker. Their acquaintance was facilitated by Alicia's nephew, G. I. Taylor, whose mother Margaret was the second of George and Mary Boole's five daughters. Cox eter and Stott collaborated on an in vestigation of Thorold Gossett's semi regular polytope s{3, 4, 3}, which he had recently rediscovered. She showed that its vertices lie on the edges of the regular polytope (3, 4, 3 }, dividing them in the ratio of the golden section. Coxeter as Promoter of Geometrical Art
Geometry and art were intimately con nected in Coxeter's mind, and so it was natural that he should befriend the Dutch artist M. C. Escher. Their "col laboration" was wonderfully recounted in [Coxeter 1979], which explains how Escher came to produce his woodcut "Circle Limit III" after Coxeter explained the general procedure for constructions in Poincare's model for the hyperbolic plane. In turn, Coxeter appreciatively analyzed "Circle Limit III" (which went beyond the framework he had presented to the artist) in several articles [Coxeter 1979], [Coxeter 1985] , [Coxeter 1996]. See, too, the article by Siobhan Roberts and Asia Weiss in the present issue. For a more recent account of the mathe matics involved, illustrated with beauti ful color pictures, see [Dunham 2003]. Shortly before his death, Coxeter was honored by the Fields Institute at Toronto University, where a complex geometric sculpture by Marc Pelletier was unveiled. For the trained eye its aesthetic appeal was enhanced by the virtuosity of the achievement: for this sculpture shows an orthogonal projec-
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tion of the 120-cell into three-space, a model much appreciated by Coxeter, John Conway, and the other geometers present. Fittingly enough, Pelletier was on hand, too, and spoke about the work of another artist, whose work both he and Coxeter greatly admired, Paul Donchian (see the tribute to Donchian in [Coxeter 1973, p. 260]). Born in Hartford, Connecticut, in 1895 into a family of Armenian descent, Donchian took over the rug business established by his father. Many of his ancestors had been jewelers and craftsmen, a tradition that came to fruition in his wire models showing three-dimensional projections of the four-dimensional regular polytopes. Coxeter had this to say about the artist's techniques.
Donald Coxeter's Regular Polytopes, one of the great books of twentieth century mathematics. One must hope it will continue to find an appreciative audience for many years to come. REFERENCES
[Coxeter 1 973] H. S. M. Coxeter, Regular Poly topes,
3rd ed. , New York: Dover, 1 973.
[Coxeter 1 979] H. S. M. Coxeter, "The non Euclidean symmetry of Escher's Picture 'Cir cle Limit Ill ' , " Leonardo, 1 2 (1 979), 1 9-25, 32. [Coxeter 1 985] H. S. M. Coxeter, "Review of M. C. Escher: His Life and Complete Graphic Work," Mathematical lntelligencer,
7(1) (1 985)
59-69. [Coxeter 1 996] H. S. M. Coxeter, "The Trigono metry of Escher's Woodcut 'Circle Limit I l l ' , " Mathematical
lntelligencer,
1 8(4)
(1 996)
42-46; erratum ibid. 1 9(1 ) (1 997), 79. [Dunham 2003] Douglas Dunham, "Hyperbolic
[Donchian 's models use} straight pieces of wire for the edges and glob ules of solderfor the vertices. The ver tices are distributed on a set of con centric spheres (not appearing in the model), one for each pair of opposite sections. Donchian did not attempt to ·indicate the faces, because any kind of substantial faces would hide other parts (so that the model could only be apprehended by a four-dimensional being). The cells appear as "skeletons, " usually somewhat flattened by fore shortening but still recognizable. Parts that would fall into coincidence have been artificially separated by slightly altering the direction of proj ection, or introducing a trace of per spective [Coxeter 1973, p. 242].
Art and the Poster Pattern," Math Awareness Month-April 2003, http://mathfoum.org/mam/ 03/essay1 . html. [Henderson 1 983] Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art,
Princeton: Prince
ton University Press, 1 983. [Hilbert and Cohn-Vossen 1 932] David Hilbert and Stephen Cohn-Vossen, Anschauliche Geometrie,
Berlin: Springer, 1 932.
[Hinton 1 980] Charles Howard Hinton, Specu lations on the Fourth Dimension: Selected Writings of C. H. Hinton,
ed. Rudolph Rucker,
New York: Dover, 1 980. [Manning 1 9 1 4] Henry Parker Manning, Geom etry of Four Dimensions,
New York: Dover,
1 91 4. [Manning 1 92 1 ] Henry Parker Manning, ed. , The Fourth Dimension Simply Explained,
New
York: Scientific American Publishing, 1 92 1 .
Donchian's models were displayed at the 1934 Century of Progress Expo sition in Chicago. At the Coxeter cele bration, Marc Pelletier pointed out that many of Donchian's works are held in storage at the Franklin Institute in Philadelphia, where they were last dis played in 1967, the year of his death. Perhaps they should be dusted off and put back on display. For those willing to settle for two-dimensional images, it should not be forgotten that Coxeter adorned his classic book with pho tographs of the Donchian models, an other form of tribute to the artist who brought them to life. These are only a few of the lovely things to be found in
[Michalowicz 1 996] Karen Dee Michalowicz, "Mary Everest Boole (1 832-1 9 1 6): An Erst while Pedagogist for Contemporary Times," Vita Mathematica: Historical Research and Integration with Teaching,
ed. Ronald Calinger,
Washington, D.C.: Mathematical Association of America, 1 996. [Peterson 2000] lvars Peterson, "Algebra Phi losophy, and Fun, " www. ma.org/mathland/ mathtrek_1 _1 7_OO.html. [Schlafli 1 90 1 ] Ludwig Schlafli, "Theorie der vielfachen Kontinuitat , " Denkschriften der Schweizerischen naturforschenden Gesell schaff,
38 (1 901 ): 1 -237.
[Sommerville 1 929] Duncan M. Y. Sommerville, An Introduction to Geometry of N Dimen sions,
New York: Dover, 1 929.
CHRIS RORRES
Comp eting Book I I of Archimedes' s On Floating Bodies
rchimedes (c. 287-212121 1 B. C.) lived in the Greek city-state of Syracuse, Sicily, up until the time it was conquered by the Romans, a conquest that led to his death. Of his works that survive, the second of his two books of On
Floating Bodies1 is considered his most mature work, commonly described as a tour de force [6,15]. This book contains a detailed in vestigation of the stable equilibrium positions of floating right paraboloids2 of various shapes and relative densities, but restricted to the case when the base of the paraboloid lies either entirely above or entirely below the fluid sur face. This paper summarizes the results of research in which I completed Archimedes's investigation to include also the more complex cases when the base of the floating para boloid is partially submerged. Modem scientific computing and computer graphics enabled me to construct a three-di mensional surface that summarizes all possible equilibrium positions (both stable and unstable) for all possible shapes and relative densities. This equilibrium surface contains folds and cusps that explain certain catastrophic phenom ena-for example, the sudden tumbling of a melting ice berg or the toppling of a tall structure due to liquefaction of the ground beneath it-that have long been observed but not previously explained fully. Books I and II
Book I of On Floating Bodies begins with a derivation of Archimedes's Law of Buoyancy from more fundamental principles, and finishes with a simple, elegant geometric
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proof that a floating segment of a homogeneous solid sphere is always in stable equilibrium when its base is par allel to the surface of the fluid, either above the fluid sur face or below it. Book I introduced the concept of fluid pressure and initiated the science of hydrostatics. It took almost eighteen centuries before this work on the nature of fluids was continued by such scientists as Simon Stevin (Dutch, 1548-1620), Galileo Galilei (Italian, 1564-1642), Evangelista Torricelli (Italian, 1608-1647), Blaise Pascal (French 1623-1662), and Isaac Newton (English 16421 727). In the interim, Book I served mainly as the basis for determining the density of objects, such as gemstones and precious-metal artifacts, by comparing their weights in air and in water. In Book II Archimedes extended his stability analysis of floating bodies from a segment of a sphere to a right pa raboloid. However, Book II contained many sophisticated ideas and complex geometric constructions and did not have the appeal of Book I. Only after Greek geometry was augmented with algebra, trigonometry, and analytical geometry and the field of mechanics reached the maturity to handle the concepts of equilibrium and stability that Archimedes introduced was Book II seriously studied. It then became the standard starting point for scientists and
naval architects examining the stability of ships and other floating bodies. 3 To describe the results Archimedes obtained in Book II, let me first precisely define his object of study: Definition
A paraboloid is a homogeneous solid convex object bounded by a surface obtained by rotating a parabola about its axis of symmetry and by a plane that is not parallel to the parabola's axis of symmetry. If the plane is perpendic ular to the axis of symmetry it is called a right paraboloid, otherwise it is called an oblique paraboloid. The planar por tion of the surface, which is either circular or elliptical, is called the base of the paraboloid. Let R be the radius of the base of a right paraboloid and let be its height (Fig. 1A). Define its base angle ¢ as the angle between and for which tan ¢ = I R. In a pro file view of the paraboloid it is the angle between its base and the tangent line to the parabolic cross section at the base (Fig. lB). This base angle determines the shape of the parabola. Next, let Pbody be the mass-density of the parab oloid, and let PJZuid be the mass-density of the fluid in which it is floating within a uniform gravitational field. Following Archimedes, let us neglect the density of the air above the fluid4 and define the relative density (or specific gravity) of the paraboloid as s = Pbody l PJZuid• which is a number in the interval for a floating paraboloid. Finally, let e be the tilt angle (or heel angle), by which is meant the angle of inclination in the interval of the axis of the pa raboloid from the vertical with corresponding to the base above the fluid level (Fig. lB). As with Archimedes, let us confine the rotation of the paraboloid so that its axis al ways lies in a fixed vertical plane. Below is an example of one of the ten propositions in Book II, in which I first give a very literal translation of the Greek text and then a very liberal modem translation. In the literal translation the "axis" is a line segment whose length is the height of the paraboloid and the "line-up to-the-axis" is the semilatus rectum of the paraboloid, which is a line segment of length R2 I The last sentence in my translation actually consists of seven excerpts from the beginning of the proof of Proposition 8 where a geo metric construction is described.
H
0°
90°
2H
[0,1]
[0°, 180°) 0°
[22]
H
2H.
Figure
Archimedes's Proposition 8. Literal Translation:
A right segment of an orthoconoid, when its axis is greater than one-and-a-half times the line-up-to-the-axis, but small enough so that its ratio to the line-up-to-the axis is less than fifteen to four, and when further its weight has to that of the fluid [of equal volume} a ratio less than that which the square of the amount by which the azis exceeds one-and-a-half times the line-up-to-the axis bears to the square of the axis, will, when so placed in the fluid that the base does not touch the surface of the fluid, not return to the vertical position and not remain in the inclined position except when its axis makes with the surface of the fluid a certain angle to be described. [This angle is EJNr in the diagram (Fig. in which} ED. is equal to the axis; EK is twice Kb.; (3) KP is equal to the line-up-to-the-axis; (4) the weight of the body is to that of the fluid [of equal volume} as the square of side
Z is to that of side ED.; (5) X is twice XZ; (6) X is equal to P'JI; and (7) the square of side 'I'E is half of the rectangle of sides KP and E'Jf.
(2)
2)
(1)
Archimedes's Proposition 8. Modem Translation:
A right paraboloid whose base angle ¢ satisfies 3 < tan2 ¢ < and whose relative density s satisfies s < (1 - 3cot 2 ¢)2 has precisely one stable equilibrium position with its base completely above the uid sur ace. The corresponding tilt Vs) tan2 ¢ - 2. angle is e = tan� 1
15/2
� (1 -
8
Archimedes's objective in Proposition was to describe a geometric construction using compass and straightedge that begins with three line segments describing the shape and relative density of the paraboloid (the axis, the line-up to-the-axis, and the line segment Z whose length is VsH) and ends with a diagram in which the tilt angle is revealed. My objective in the modem translation, however, was to summarize the geometric construction in a single analyti cal expression in which the equilibrium tilt angle e is ex pressed as an explicit function of s and ¢. My modem trans lation incorporates centuries of algebraic, trigonometric, and analytical developments and considerably alters how
the Greeks would have grasped Archimedes's results. It also shows the limitations of Greek geometry in formulat ing and describing complicated physical phenomena.
1 . A and B: Two views of a floating right paraboloid with
Proposition 8. C: The paraboloid has been rotated clockwise from this equilibrium position while the weight of the displaced fluid was kept equal to the weight of the paraboloid.
VOLUME 26, NUMBER 3, 2004
33
P roposition 8 of Book I I of Arch i medes's
On Floating Bodies
The following statements appear in the proof of Proposition
8:
1. 86 is equal to the axis.
2. 8K is twice K6.
3. KP is equal to the line-up-to-the-axis.
4. The weight of the body is 5.
to that of the fluid [of equal volume] as the square of side
half of the rectangle of sides KP and 8'¥.
E
d
B
X
•
•
•
'II
p
z
:
•
K
These statements describe a compass-and-straight-edge construction b ginning with: 1. The "axis" of th
paraboloid. This i
a li n
egment of length H, the height of the paraboloid.
2 . The "line-up-to-the-axis. " This is the emilatus rectum of the paraboloid, which is a line segment of 1 ngth R212H,
3.
wher R is the radius of the base of the R and construct the lin -up-to-the-axis. The magnitude
s.
paraboloid. Alternatively, we could begin with a line
segment of
I ngth
This is the ratio of the weight of the paraboloid to that of an equal volume of fluid. Floating
the p araboloid in the fluid vertically with the base up, we have that
merged portion of the paraboloid. The line segment
s = (h!H?, where h is the h ight of th
ub
Archimedes shows that the paraboloid's angle of inclination (the angle its axis makes with the surface of the fluid) is angle E8'1'. The complementary angle BE'¥ is my tilt angle e. Using algebraic notation, where AB appearing in an
ment, the seven statements above become 1. B6
2. BK
3. 4. 5. 6. 7.
KP
=
=
=
H
2(K6)
R2/2H
(
s =
=
=
=
From Eqs. 1-7 and the fact that the base angle tan2 e
THE MATHEMATICAL INTELLIGENCER
(J of the paraboloid satisfies tan
_ _ 2 _ __8 _' _ l'_ � � 8"1J1 .!. (KP)(8"1J!) 2
- ( 'l'E )
_
(2
2 3B6 -
34
quation repre ents the length of the conesponding line seg
_
2
3
KP
- KP)
28'1' KP
(2(BK - P'l' - KP) KP
(2
2. r 2 -86 - - v s8 6 3 3
KP
-
KP)
4> = 2HIR, we obtain 2(BK -
X
KP
- KP)
2('!:..H - '!:.. VsH - .!£._) 3 3 2H
R2
2H
•
•
;J
B
X
E •
W
z
:
K
p
•
Figure 2. Diagram for the statement of Proposition 8, scaled for the paraboloid in Figure 1 .
Archimedes's other propositions in Book I I complete his study of the stable equilibrium tilt angles when the base is either completely above or completely below the fluid sur face for appropriate values of the base angle and the rela tive density. The main geometric tools he used were the formulas for the volumes and centroids of right and oblique paraboloids, formulas that he himself derived in other works. 5 The mechanical tools he used-again, tools that he himself first formulated-were his Law of Buoyancy for a floating body, his Law of the Lever, and the equilibrium condition that the center of gravity of the floating body must lie on the same vertical line as its center of buoyancy. (Because a paraboloid is a homogeneous convex body, its center of buoyancy coincides with the center of gravity of its submerged portion.) Righting and Energy Arms
The numerical techniques I used required the evaluation of the moment acting on an unbalanced floating paraboloid. In Figure lC a right paraboloid is floating in a fluid with the weight of the displaced fluid equal to the weight of the right paraboloid. However, it is not in equilibrium because the center of gravity G of the body is not on the same ver tical line as its center of buoyancy B. Rather, the weight of the paraboloid and the buoyancy force form a couple that will cause the paraboloid to rotate in a counterclockwise direction toward the equilibrium position shown in Figure l B. The value of the couple, called the righting moment, is the weight of the paraboloid times the horizontal dis placement GZ between G and 8, taken as positive if 8 is to the right of G. This horizontal displacement is called the righting arm and its use is preferred by naval architects to the righting moment. If a wave causes a ship to heel, the righting arm expressed as a function of the heel angle af fects the dynamics of how the ship will return to its verti cal equilibrium orientation. One of the standard specifica tions of a ship is a graph of its righting arm for a wide range of heel angles. If the base is completely above or below the fluid sur face, it is possible to determine an exact expression for the righting arm of a floating right paraboloid using the exact formulas for the volume and centroid of an oblique para boloid. For example, if the base is above the fluid surface then (1)
Righting Arm
H
=
sin 8 tan2 ¢
r2 - 2 L 3 c1 -
v� s)tan2 ¢ + tan2
'
8Jl
Setting this equal to zero determines all equilibrium tilt an gles with the base above the fluid surface and, in particu lar, returns the expression for the tilt angle determined by Archimedes's Proposition 8 above. When the base is com pletely submerged, symmetry principles can be used to ob tain an analogous expression.6 While the righting arm provides the necessary information for the stability analysis of a floating body, its potential en ergy also provides some insight. Taking the fluid surface as the level of zero potential energy, the potential energy of the paraboloid/fluid system is the sum of the potential energy of the paraboloid and the potential energy of the displaced fluid. The potential energy of the paraboloid is its weight multiplied by the height of its center of gravity G above the fluid sur face. Likewise, the potential energy of the displaced fluid is its weight (the same as the weight of the paraboloid) multi plied by the distance of its center of gravity 8 below the fluid surface. The total potential energy is then the weight of the paraboloid multiplied by the vertical distance between 8 and G. For a homogeneous convex paraboloid, G will always lie above 8 if the relative density is less than one, and so the po tential energy will always be positive. By analogy with the term "righting arm," I shall call the vertical displacement from 8 to G (BZ in Figure lC) the energy arm of the floating paraboloid. The fundamental re lationship between force and energy shows that when the righting arm and energy arm are expressed as functions of the tilt angle 8, then (2)
d(energy arm)
d8
. . = nghtmg arm .
In order to work with dimensionless units, let us divide both the righting arm and the energy arm by the height H of the paraboloid. Thus one unit of the normalized energy arm is the energy needed to raise the paraboloid in air a distance equal to its height. Figure 3 is an example of the normalized righting arm and the normalized energy arm as a function of the tilt angle for a right paraboloid with base angle 74.330° and relative den sity 0.510. When its base is above the fluid surface (0° ::; 8 ::; 28.2°) I used Eq. (1), and when the base is below the fluid sur face (151.0° ::; 8 ::; 180°) a similar exact expression was used. When the base is cut by the fluid surface, I used nu merical integration to determine the volume and first mo ments of the unsubmerged portion of the paraboloid, from which the center of buoyancy and resulting righting arm and righting-arm curves were determined. 7 The six roots of this righting-arm curve, or, equivalently, the six stationary points of the energy-arm curve, determine the six equilib rium positions of the corresponding paraboloid. Because a positive righting arm produces a counter clockwise rotation and a negative righting arm produces a clockwise rotation, the way in which the algebraic sign changes through a root determines the stability classifica tion of the corresponding equilibrium configuration. In par ticular, a root is asymptotically stable (AS), neutrally sta ble to first order (NS), or unstable (US) if the slope of the righting curve at the root is positive, zero, or negative, reVOLUME
26, NUMBER 3, 2004
35
E <( .....
C> c: :;:: ..c: C>
0::
E
<(
'-
>. C>
(]) c: w '-
Tilt Angle (9)
0.01 0 -0 . 0 1
0 . 1 92
0 . 1 90
0 . 1 88
0 . 1 86
00
35.0°
56.8°
94.5°
1 3 1 .5°
1 80°
Figure 3. Top: The normalized righting-arm curve of a paraboloid with base angle 74.330° and relative density 0.510. Middle: The normalized energy-arm curve of the paraboloid. Bottom: The six equilibrium configurations of the paraboloid determined by the roots of the righting-arm curve (or the stationary values of the energy-arm curve) together with their stability classifications (AS, US, or NS). The base is partially sub merged if 28.2° < 0 < 1 51 .0°.
spectively.8 None of the six equilibrium positions for the particular paraboloid described in Figure 3 were present in Archimedes's studies, because they are either unstable or correspond to the base being cut by the fluid surface. Archimedes's Results
Let me next summarize Archimedes's results in Book II in graphical form. In Figure 4, I have plotted a surface in (cp,s, O)-space in the region [0°,90°] X [0, 1 ] X [0°,180°] in
which each point identifies a combination of base angle, relative density, and tilt angle for an AS equilibrium con figuration of a paraboloid whose base is not cut by the fluid surface. The bottom portion of this equilibrium surface is associated with the base lying above the fluid surface, and the top portion is associated with the base lying below the fluid surface. Because of certain symmetry considerations6 the top portion of the equilibrium surface is a rotation of its bottom portion about the line s = 112 and (J = 90°.
Relative Density
Base Angle
Figure 4. Archimedes's results in graphical form. Each point on the surface identifies an AS equilibrium configuration of the paraboloid in which the base is not cut by the fluid surface. Typical configurations are shown for different parts of the surface. The red curves identify those limiting configurations in which the base touches the fluid at one point.
36
THE MATHEMATICAL INTELLIGENCER
The curved piece of the bottom portion of the equilib rium surface, as partially determined by Archimedes's Proposition 8 above, has the explicit equation 6 = tan- 1
(3)
J �(1 - Vs)tan2
2
restricted to the appropriate domain in
(4)
s
=
(
6 + tan2 6 6 + 5 tan2 6
- - 1(
)4
'
6 + 5 tan2 6
4 tan 6
)
'
where at 6 = goo the limiting values s = 1/625 and
-
-
Complete Equilibrium Surface
My own research involved completing the equilibrium sur face in Figure 4 by appending those points corresponding to AS configurations in which the base is cut by the fluid surface and also all points corresponding to US and NS con figurations. The result is shown in Figure 5. The construction of Figure 5 required determining all of the roots of the righting-arm curves for a large number of base angles and relative densities using numerical tech niques. The base angle
1 80° 1 so·
e
1 20°
Angle Tilt
go• 60° 30° o·
Figure 5. Top: The complete equilibrium surface of a floating paraboloid with the equilibrium tilt angles () plot
Relative Density
go•
Base Angle
ted against the base angle <J> and the relative density s. The AS points are in blue and the US points are in gray. The NS points lie on the winding curve separating the two regions. The yellow vertical line cuts through the six equilibrium points in Figure
3. The red vertical line to the
right is the jump (b to c) in Figure 8 associated with a tumbling iceberg, and the red vertical line to the left is the jump (c to d) in Figure 9 associated with a toppling structure. Bottom: Stereoview of the equilibrium surface.
VOLUME 26. NUMBER 3. 2004
37
other than 0°, goo, and 180°. For those three exceptional values of 8, I used the facts that (1) the entire planes (} = oo and (} = 180° are part of the equilibrium surface, indi cating that the right paraboloid is always in equilibrium when its axis of symmetry is vertical, and (2) the cross sec tion of the equilibrium surface at (} = goo consists of these three line segments: {s = 112, (} = goo }, where the parabo loid is on its side, half in and half out of the fluid; {
38
THE MATHEMATICAL INTELLIGENCER
where a US point and an AS point meet and annihilate each other, forming a fold catastrophe. If the parameters of a floating paraboloid change in such a way as to pass over a fold, the equilibrium configuration will jump catastrophi cally from the NS point on the fold to an AS point lying on the vertical line through the NS point. The NS points on the curved portion of the equilibrium surface and the corre sponding fold catastrophes arise only when the base of the paraboloid is partially submerged, and so did not enter into Archimedes's consideration. Cusp Catastrophes, Bifurcations, and Hysteresis Loops
Figure 6 (left) is a projection of a portion of the curve of fold catastrophes onto the
0.55 ,..----,..---,
:§:
� ·u; c Q)
0 0 . 50 Q) >
�
Qi 0::
0.45 '-------'---'-' 75° 74° 73°
Base Angle (4J)
Figure 6. Left: The projection of a portion of the curve of fold catastrophes onto the %-plane. Its three cusps identify three cusp catastro
phes at tilt angles of so.o•, go•, and 1 20.0'. Right: An oblique view of the topmost cusp catastrophe at fJ
The vertical lines in Figure 7 H and L are at s = 0.510 and 1> = 74.330°, respectively, and pass through the six tilt angles shown in Figure 3. A slight increase in either the rel ative density or the base angle from these values causes the structurally unstable NS point at (} = 131.5° to be anni hilated, while a slight decrease causes it to split into an AS-US pair.
1 20.0•.
=
Tumbling of Icebergs due to Melting
Icebergs are notoriously unstable and may tumble over for no apparent reason [2,3,20]. Jules Verne gave an explana tion of this phenomenon in his 1870 novel 20, 000 Leagues Under the Sea. After a tumbling iceberg strikes the Nau tilus, Captain Nemo explains, "An enormous block of ice; a mountain turned over. When icebergs are undermined by
F ixed Tilt Angle (e) 1
�
�
a = 55•
0 75 U> . <:
Q) D Q) >
e
e = so·
=
0
s1·
=
1 60°
0.5
� 0. 2 5 Q) 0::
0
D
B
A o•
60°
30°
60°
30°
so• o•
'
so• o•
60°
30
30°
so· o·
so•
60°
Base Angle (�)
Fixed Base Angle ($) 1 80°
§: 1 3 5°
�
Q)
c;, go• <:
<( i=
o·
0
E
0.5
1
0.5
0
1
0.5
0
0
Fixed Relative Density (s) 1 ao·
§: 1 35° Q)
c;, <:
<( i=
go•
o·
72°
74°
76°
J
b n·
74 °
330
7
0.5
.
I
---1
4
.
Relative Density (s)
l
s = 0.460
45°
H
=
76 °
K
n•
76°
L
°
72
74°
76°
Base Angle (¢)
Figure 7. Slices of the equilibrium surface. The black curves are AS points, the gray curves are US points, and the black dots are NS points. A hysteresis loop (a through d) is shown in (J), and the common vertical line in (H) and (L) cuts through the six equilibrium tilt angles de scribed in Figure
3.
VOLUME 26, NUMBER 3 , 2004
39
�
�
Ql
c;, c:
1 80° ,....'9----.., s = 0.9 1 35°
i=
45°
. 00 ____...__....... 60° 70° 80° a Base Angle (�)
90°
a
b
c
Figure 8. A paraboloidal iceberg of relative density 0.9 tumbles over as it melts and its base angle passes through 82.65° (b to c).
warmer waters or by repeated collisions, their center of gravity rises, with the result that they overturn completely" [2 1]. Figure 8 quantifies this phenomenon for a paraboloidal iceberg with uniform relative density of 0.9, melting in such a way that its base angle slowly increases (i.e., it gets nar rowerfl). The cross section of the equilibrium surface at this relative density shows that for base angles less than 82.54° the iceberg can float stably in a vertical orientation with its base above water (a). As its base angle slowly melts from 82.54° to 82.65°, its tilt angle slowly increases from oo to 12.3° (a to b), and then suffers a catastrophic jump to 98. 1 o when the base angle increases past 82.65° (b to c). The paraboloidal iceberg will tumble, rather than grad ually roll over, only if its relative density is greater than 0.467 (Figure 7 I and J). The tumbling then takes place al most immediately after the base cuts the fluid surface. Toppling of Structures due to Soil Liquefaction
During an earthquake, loose, water-saturated soil can be have like a viscous fluid, a phenomenon known as soil liq uefaction. Structures originally supported by the soil begin to float on it when it liquefies and can then sink and top ple as the density of the liquefied soil decreases.
Figure 9 illustrates this phenomenon for a paraboloidal structure with a base angle of 80° initially standing verti cally on solid ground at a tilt angle of 180° (a) . Let us con sider solid ground as a liquid with infinite density, so that the relative density of the structure is zero. As the ground liquefies its density slowly decreases from infinity through large finite values and the relative density of the structure increases from zero through small finite values. The cross section of the equilibrium surface at a base angle of 80° shows that as the relative density of the structure increases from 0 to 0. 177 the structure slowly sinks into the ground in a vertical position (a to b), then starts to tilt slowly un til it reaches a tilt angle of 162.6° at a relative density of 0. 187 (b to c), at which point its base is barely above ground. If the relative density increases further, the structure top ples catastrophically to a tilt angle of 79.9° (c to d). This toppling is irreversible. If the soil returns to its solid state, the structure, if still in one piece, ends up at a tilt angle of 77.2° (d to e). As with the iceberg, the paraboloidal structure cannot topple until its base is partially exposed above the soil level. Additionally, this toppling can only occur if the base angle of the structure is greater than 74. 194° (cf. Fig. 7G). For smaller base angles the paraboloidal structure gradually
e
�
1 80°
b
a
-a; 1 35° c;, c:
90° 0
e
c
d
0.1 0.2 Relative Density (s)
0.3
Figure 9. A paraboloidal structure with base angle 80° topples as the soil under it liquefies and the relative density of the structure passes through 0.1 86 (c to d).
40
THE MATHEMATICAL INTELLIGENCER
sinks and tilts into the soil without toppling as the soil's density decreases.
nus
1 850) [5,8]. Moerbeke's Latin translation was the source of all ver
sions of On Floating Bodies from his time until the twentieth century. Moerbeke's translation of both books of On Floating Bodies was first
Conclusion
One need only glance at Archimedes's Proposition 8 above to see that On Floating Bodies is several orders of magni tude more sophisticated than anything else found in ancient mathematics. It ranks with Newton's Principia Mathe matica as a work in which basic physical laws are both for mulated and accompanied by superb applications. However, Archimedes's investigation of floating para boloids had to await the computer age for its continuation, just as did his famous Cattle Problem [24]. This latter prob lem has an integer solution with more than 200,000 digits that needed modem computers to determine. Likewise, I needed advanced computing and graphics systems to deter mine all possible equilibrium positions of Archimedes's float ing paraboloids and to represent them in a single diagram. No doubt Archimedes would have been interested in see ing the results in this paper, but one could ask how much of the mathematics developed in the last two millennia he would need to learn to understand them. At the very least he would have to learn about three-dimensional Cartesian coordinate systems, although he should have no trouble with this concept considering how close he came to defin ing a polar-coordinate system in his description of the spiral that bears his name. Unhooking him from the strait jacket of compass-and-straightedge construction to explain how the relationship among three variables can be repre sented by the points on a surface might take a little longer. He could then see how the equilibrium surface in Figure 5 presents a global picture of the behavior of his floating parabolas and how the twists and turns of that surface lead 1 to catastrophic 0 transitions. He could also then appreciate some of the advances made in mathematics in the last 23 centuries, although my guess is that he would have ex pected more, considering the enormous advances that he alone made in his lifetime. Acknowledgments
My sincere thanks to Professor C. Henry Edwards of the University of Georgia and Prof. Dr. Horst Nowacki of the Technical University of Berlin for their advice and support.
printed in 1 565, independently by Curtius Troianus in Venice and by Federigo Commandino in Bologna [4]. A palimpsest from the tenth cen tury, discovered and edited by J. L. Heiberg in 1 906, contains the only extant Greek text [1 6,25]. The texts by Dijksterhuis [8] and Heath [1 4] are the only translations/paraphrases presently available in English. 2. Also called parabolic conoids or orthocono1ds. 3. Some classic works concerned with how things float are: Chris tiaan Huygens
(Dutch, 1 629-1 695), De iis quae liquido supernatant;
Pierre Bouguer
(French, 1 698-1 758), Traite du Navire, de sa Con
struction, et de ses Mouvements; Leonard Euler Scientia navalis; Jean Le Rond d'Aiembert
(Swiss, 1 707-1 783),
(French, 1 71 7-1 783),
Traite de l'equilibre et du Mouvernent des Fluides; Fredrik Henrik af Chapman
(Swedish, 1 72 1 -1 808), Architectura Navalis Mercatoria;
George Atwood
(English, 1 7 45-1 807); The Construction and Analy
sis of Geometrical Propositions Determining the Positions Assumed by Homogeneal Bodies Which Float Freely. and at Rest, on the Fluid's Sur . face; a/so Determining the Stability of Ships and of Other Floating Bod ies; Pierre Dupin
(French, 1 784-1 873), Applications de geometrie et
de rnecanique; August Yulevich Davidov
(Russian, 1 823-1 885); The
Theory of Equilibrium of Bodies Immersed in a Liquid
[in Russian] . More
recent works include [7,9-1 2 , 1 8 , 1 9].
4. If Pa�r is the mass-density of the air, then, because the paraboloid
is a uniform convex body, the buoyancy effect of the air can be ac counted for by defining the relative density as s
Pa�rl· Actually, Archimedes's description of
s
=
(Pbody - Pair)I(PIIuid -
as the ratio of the weight
of the body to the weight of an equal volume of fluid results in this ex pression if the weighing is done in air, but it is doubtful that he was aware of the buoyancy effects of air. 5. Archimedes's proof for the volume of a right or oblique paraboloid is contained in Propositions 2 1 -22 of On Conoids and Spheroids. He gave a "mechanical" proof of the location of the centroid of a right pa raboloid in Proposition 5 of The Method. He used the correct expres sion for the centroid of an oblique paraboloid in On Floating Bodies II, but no proof survives [8, 1 4] .
6 . Symmetry considerations show that if e i s an equilibrium tilt angle
when the relative density of a floating body of revolution is s, then 1 80° IJ is an equilibrium tilt angle for the body when its relative density is 1
s.
-
Thus only tilt angles in the range [0°,90°] need be explicitly com
puted. Although Archimedes does not mention this fact, it is clear that he was aware of it for his paraboloids, for his proofs when the base is below the fluid surface are the same, mutatis mutandis, as his proofs
Web site
A Web site maintained by the author containing QuickTime movies animating the figures is available at http://mail.vet. upenn.edu/-rorres/. The author also maintains an exten sive website on Archimedes at http://www.math.nyu.edu/ -crorres/Archimedes/contents.html.
when the base is above the fluid surface. 7. The integrals determining the volume and centroids of the unsub merged portion can be found in closed form using symbolic algebra programs, but they are page-long monstrosities, and numerical inte gration yields results much more quickly and with greater accuracy. Ad ditionally, numerical techniques were used to determine when the weight of the displaced fluid is equal to the weight of the paraboloid
NOTES
1 . A Greek manuscript dating from about the ninth century and con taining both books of On Floating Bodies was translated into Latin by the Flemish Dominican William of Moerbeke in 1 269, along with other
and to find the roots of the righting-arm curve. The symbolic calcula tions were performed with MapleTM and Mathematica™, and the nu merical calculations and graphs were performed with MatLab™ .
8. Points N S t o first order may b e AS o r U S when higher-order terms
manuscript were lost in the fourteenth century, but Moerbeke's holo
are considered. In particular, the NS points when e
graph remains intact in the Vatican library (Codex Ottobonianus Lati-
nonhyperbolic, degenerate,
works of Archimedes from other manuscripts. The tracks of the Greek
=
oo and 1 80a are
actually AS and the rest are US. These NS points are also classified as and structurally unstable [1 , 1 3].
VOLUME 26, NUMBER 3, 2004
41
Latin Tradition, The University of Wisconsin Press, Madison, 1 964, pp. 3-8.
A U T H O R
6. M. Clagett, Biographical Dictionary of MathematiCians. (article on Archimedes) Charles Scribner's Sons, New York, 1 991 , Vol. 1 , p. 95. 7 . R. Delbourgo, "The floating plank," A merican Journal of Physics 55 (1 987), 799-802. 8. E. J. Dijksterhuis, Archimedes, (In Dutch: Chapters 1-V, I. P.
Noordhoff, Groningen, 1 938: Chapters Vl->0/ , Euclides, VX->0/1 1 ,
1 938-44). English translation b y C. Dikshoorn, Princeton University
Press, NJ, 1 987. 9. B. R. Duffy, "A bifurcation problem in hydrostatics," American Jour nal of Physics
CHRIS RORRES University of Pennsylvania
symmetrical objects and the breaking of symmetry. Part 1 : Prisms,"
New Bolton Center
American Journal of Physics
382 West Street Road
symmetrical objects and the breaking of symmetry. Part 2: The
USA
cube, the octahedron, and the tetrahedron , " American Journal of
wwwsite: w3.vet.upenn.edu/faculty/rorres
Physics
e-mail: [email protected]
Joseph B. at
with
Keller on scattering theory, then for 33 years taught
University of Pennsylvania's School of Vet
erinary Medicine, modeling spread of diseases in food animals (such as avian flu in poultry). All the while, he has been fasci nated with the life and to appearances on
work of Archimedes. These studies led
two BBC documentaries, and his Web site
has received more than 250,000 hits just in the last 1 2 months.
Here
he is seen with the Archimedes-inspired finials he de
signed for his fence at hom e. on the area and volume
60 (1 992), 345-356.
1 2. E. N. Gilbert, "How things float," The American Mathematical
Drexel University in Philadelphia. Now emeritus at Drexel ,
he is working at the
60 (1 992), 335-345.
1 1 . P. Erdos, G. Schibler, and R. C. Herndon, "Floating equilibrium of
Kennett Square, PA 1 9348
Chris Rorres did his doctorate at the Courant Institute
61 (1 993), 264-269.
1 0. P. Erdos, G. Schibler, and R. C. Herndon, "Floating equilibrium of
The theorems they evoke -that
of a sphere, and that on the volume
of a right paraboloid-were among Archimedes's favorites.
Monthly
98 (1 991), 201 -2 1 6.
1 3. J . Hale and H. Ko<;:ak, Dynamics and Bifurcations, Springer-Ver lag, New York, 1 991 , p. 1 9. 1 4. T. L. Heath, The Works of Archimedes, Dover Publications, New York, 2002. Unabridged republication of The Works of Archimedes (1 897) and The Method of Archimedes ( 1 9 1 2), both published by The Cambridge University Press. 1 5. T. L. Heath, A History of Greek Mathematics, Dover Publications, New York, 1 98 1 , Vol. I I , p. 95; Unabridged republication from Clarendon Press, Oxford, 1 92 1 . 1 6. J. L. Heiberg, Archimedis Opera Omnia, with corrections by E. S. Stamatis, B. G. Teubner, Stuttgart, 1 972. 1 7. C. W. Keyes, Cicero: De Re Publica, Book I, Section 22, Loeb Clas sical Library, Harvard University Press, Cambridge, 1 928. 1 8. H. Nowacki, Archimedes and Ship Stability, Max Planck Institute for the History of Science, Berlin, 2002. 1 9. H. Nowacki and L. D. Ferreiro, Historical Roots of the Theory of Hydrostatic Stability of Ships,
9.
Unlike Verne's iceberg, the center of gravity of the paraboloidal
iceberg remains fixed relative to its size at a distance of one-third of its
Max Planck Institute for the History
of Science, Berlin. 2003. 20. J. F. Nye and J. R. Potter, "The use of catastrophe theory to an
height along its axis from its base.
alyze the stability and toppling of icebergs," Annals of Glaciology 1
1 0. In Greek:
(1 980), 49-54.
CATASTROPHE =
KATAkTIIOct>H = a downward turn
REFERENCES
1 . D. M. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems,
Cambridge University Press, Cambridge, 1 990, p. 79.
2. R. C. Bailey, "Implications of iceberg dynamics for iceberg stability
Brunetti), Penguin Putnam Inc., New York, 1 969, p. 368.
22. E. S. Stamatis, The Complete Archimedes, (In Greek: E. k kTA
MATHk, AIIXIMHAOEk AIIANTA) Volume B ' , Athens, 1 973.
23. S. Stein, "Archimedes and his Floating Paraboloids," in Mathe
estimation , " Cold Regions Science and Technology 22 (1 994),
matical Adventures for Students and A mateurs,
1 97-203.
sociation of America, Washington, D.C. , 2004.
3. D. W. Bass, "Stability of icebergs," Annals of Glaciology 1 (1 980). 43-47. 4. M. Clagett, "The impact of Archimedes on medieval science," Isis 50 (1 959), 4 1 9-429. 5. M. Clagett, Archimedes in the Middle Ages. Volume I, The Arabo-
42
2 1 . J . Verne, 20,000 Leagues Under the Sea. (translated by M. T.
THE MATHEMATICAL INTELLIGENCER
Mathematical As
24. H. C. Williams, R. A. German, and C. R. Zarnke, "Solution of the Cattle Problem of Archimedes, " Mathematics of Computation 1 9 (1 965), 6 7 1 -674. 25. N. Wilson, The Archimedes Palimpsest, Christie's Auction Catalog, New York, 1 998.
JEAN-MARC LEVY-LEBLOND
Dimensiona Variations on Themes of Pythagoras , Euc id , and Archimedes N
odern culture is characterized by a lively interest in its past. This continual going back to sources is made possible by a radically new situation: for the first time in its history, humanity can have access to most of what it has produced. Current technologies permit
the literary, musical, and artistic creations of all times to be reproduced and distributed at low cost, so that every one-at least in the prosperous parts of the world-has available the whole of the human cultural heritage. There is surely another reason for our archeophilia, namely, the weakening of common cul tural values under the pro found historic changes we are undergoing planet wide. Feeling great uncer tainty about the future, one naturally turns to the past to find inspiration and direction, or just comfort. The Renaissance is the archtype of such a backward look yielding progress. Without a past we lose the future. Hence the importance of regular revisiting of masterpieces: Euripides and Shakespeare, Cervantes and Hugo, Mon teverdi and Schubert, Giotto and Delacroix will always help us to live, love-and die. Provided, that is, that these great works are properly re-created (reinterpreted), not in the vain attempt of finding their original meaning, but rather seeking to get new meanings from them. We must listen, read, and look at what comes from the past with the pres ent's ears, eyes, and minds; Bach cannot be the same after Stravinsky, or Titian after Picasso.
These ideas may be evident as regards art, but they are not so clear when applied to science, looked at in its rela tion (or lack thereof) with culture. Indeed science, at least since the beginning of the twenty-first century, often vaunts its absolute modernity and demands a radical contempo raneousness, even an es sential amnesia, relegating all interest in the past to the status of an optional em bellishment. Scientists to day show a lack of histori cal culture unmatched in any other intellectual pro fession. To be sure, the most creative minds may feel an active interaction with their forerunners, and many of the great advances of the last century show the mark of an explicit dialogue with the past. Einstein was perfectly aware of confronting Galileo and Newton; and more particularly Abraham Robinson, in developing non-standard analysis, explicitly related it to Leibniz. But at the ordinary level of teaching, popular writ ing, and even research, such ties to the past are unusual. We may teach and publish on history of science, but ordi narily without connection to actual scientific practice. This is too bad. Modem advances can give us unexpected in sight into old results, casting light on recent developments,
Modern advances can g ive
us u nexpected i nsig ht i nto
old resu lts , casting l i g ht on recent developments .
© 2004 SPRINGER-VERLAG NEW YORK. LLC. VOLUME 26. NUMBER 3. 2004
43
A
0
Figure 1 . Pythagoras in the plane.
c
B
just as a modem production of Antigone or King Lear can reveal new meanings and have current impact. This may be too pompous an introduction for a few ex amples I want to offer. They are drawn from one of the old est fields in the world, geometry-understood in the physi cist's sense, as "measure of space." Some classical (even ancient) results have interesting generalizations to N di mensions. I think they show the permanence of these tra ditional theorems and also may help in forming a better in tuition for higher space dimensions. The collective amnesia I have been complaining about makes it hard to know how much originality to claim for the present results. That most mathematician and physicist colleagues had no references to give me indicates mostly that they shared the amnesia. I found by good luck that one of the results anyway (the first), was published in a long-ago article by the great geometer H. S. M. Coxeter. I will be grateful for any fur ther information on the questions taken up here. Pythagoras and the Orthosimplex in N Dimensions
Let us begin by revisiting the very ancient theorem of Pythagoras, and generalizing it to arbitrary dimension. Of course, I am not talking about the metric version, which is now taken as an axiom in defining euclidean spaces, but of a result which may be pleasant if not profound. 1 The classical result will first be presented in the form which is to be generalized. Let two lines in the plane meet orthogonally at 0. For any segment AB joining these lines (Fig. 1), the square of its length is the sum of the squares of the lengths of the two segments OA and OB, which it cuts off on the lines-its two orthogonal projections:
Figure 2. Pythagoras in space.
(Area ABC)2 = (Area OAB)2 + (Area OBC? + (Area OCA)2
A fairly easy proof could be given starting from Heron's formula for the area of an arbitrary triangle in terms of the lengths a, b, c of its sides: 1 A2 = 16 (a
+
b + c)(b =
+
c - a)(c + a - b)(a + b - c)
1 (2b2 c2 - a4 + circular perm.). 16
Theorem of Pythagoras 3D. The square of the area
of triangle ABC equals the sum of the squares of the areas of its three projections:
(3)
Getting the sides of the triangle ABC in terms of those of its projections by the (usual) Pythagorean theorem, a2 = b '2 + c ' z ,
b2 = c '2 + a '2 ,
c2 = a '2 + b '2
(4)
with the notations of Figure 2. Putting this back into eq. (3),
which is just the result announced in eq. (2). Actually, as often happens, giving a more appropriate proof enables us to see the generality of the result. We can express the area of triangle ABC as a vector orthogonal to its plane whose length is the vector product of any two of its sides: (5)
(1)
Now in Euclidean 3-space consider three lines meeting orthogonally at 0. Let ABC be a triangle with a vertex on each of these lines, whose projections on the planes de termined by pairs of these lines are then OAB, OBC, and OCA (Fig. 2). Then
(2)
Now we just notice that ' ' a = c - b ' , b = a' - c , c = b ' - a' ,
(6)
and we see that
A = - (c ' - b ') 1\ (a ' - c ' )
1 2
= - b ' 1\ c ' + - c ' 1\ a ' + - a ' 1\ b '
1 2
1 2
1 2
(7)
1The result was, I thought, new. I then happened to find it in the article by H. S. M. Coxeter and P. S. Donchian, "An n-dimensional extension of Pythagoras's theo rem," Math. Gazette 1 9 (1 935), 206. Their result is just the same, but their proof is more straightforward.
44
THE MATHEMATICAL INTELLIGENCER
The three terms are the vector areas of triangles OAB, OBC, OCA, respectively. These vectors being mutually perpen dicular, the usual Pythagorean theorem immediately gives the result in eq. (2). This demonstration fits into a more general interpretation. Let us define the "vectorial area" of a surface 2. as the vector Al : =
Jl n d2s
(8)
Theorem of Pythagoras ND. The square of the
(N - I)-dimensional volume of the hypotenusal face of an orthosimplex equals the sum of the squares of the volumes of its N right faces: N
,L CVolN- 1 Fk)2 • 1
=
(rz 1\ r3 1\ . . . 1\ rN) - (r1 1\ r3 1\ . . . 1\ rN) . . . ::'::: (r1 1\ rz 1\ . . . 1\ rN- 1) ( 1 1)
The first term here gives the volume of face F b the N-sim plex (0, P2, P3, . . . , PN), as vector A1 orthogonal to that face. The same applies to each of the other terms, one cor responding to each of the right faces. This gives (with ap propriate choice of orientations) (12)
where n is the normal to the surface. Then for every closed surface, the vectorial area is zero. The physical interpreta 2 tion is simple: for any vector u, the quantity u · Al is the flux of the constant field u across the closed surface 2., which is zero (if it is necessary to convince oneself of this, one may transform the flux to the volume integral of the field's divergence, which is zero). Applied to any tetrahe dron, this says that the sum of the vectorial areas of the four faces is zero. If three of these faces are mutually or thogonal, as they are for the tetrahedron OABC under con sideration here, this means that the vectorial area of the fourth face is the vector sum of those of the three mutu ally orthogonal faces; now eq. (2) follows by applying the usual 3-dimensional Pythagorean theorem. Now take N orthogonal lines meeting at 0 in N-dimen sional Euclidean space, and anN-simplex formed by N points Pt, P2, . . . , PN, one on each line. Together with 0, these make an (N + 1)-simplex, all of whose face angles at 0 are right angles, which we accordingly call an "orthosimplex." Call its (N - 1)-face H : = (Pt, P2, . . . , PN), its "hypotenusal" face, and call its N faces Fk : = (0, Pt, Pz, . . . , Pk, . . . , P,rv),3 k = 1, 2, . . . , N, the "right" faces; these are the projections of the hypotenusal face parallel to the axes; they too are or thosimplexes, in (N - I) dimensions. Then
(VolN- 1 H)2 =
A
(9)
The proof is almost trivial if one invokes the exterior calculus. Setting rk = OPk for the vector to the point Pk (k = 1, 2, . . . , N), one may express the (N - I)-dimensional volume of the N-simplex (Pt, P2 , . . . , PN) as a vector or thogonal to the (N - I )-dimensional subspace containing it, by taking the exterior product of (N - I) of its edges, for example those away from the point P1:
A simple proof by recursion shows that, by antisymmetry of the exterior product, the right-hand member is equal to the sum of the terms given by the exterior product (up to sign) of (N - 1) of the N vectors rk (k = I , 2, . . . , N):
Exactly as in the 3-dimensional case, this is valid for every simplex. But here, starting from an orthosimplex, we have the added feature that the projections (the right faces) are or thogonal: � is parallel to the kth axis; the usual Pythagorean theorem now gives eq. (2). Note, however, that the generality of the result is lim ited: the N-simplexes, which can be defined by N points on N orthogonal lines, are, for N > 3, very special. Indeed, they are determined by specifying N parameters rt, r2, . . . , rN, while the general N-hedron depends on N(N - I )/2 param eters (for example, the lengths of its edges). Euclid and the N-Dimensional Pyramid
Conceptual mastery of space goes by way of geometry, re garded in the first place as science of the measure of forms-thus fundamentally a physics of space. Estimation of lengths, areas, and volumes of the simplest objects in 3space is its core. 4 The importance of the formulas giving the area of a triangle and the volume of a pyramid cannot be overstated; these are, after all, the first examples of in tegration in 2 and 3 dimensions, as was realized later. Now the geometric reasoning which establishes these two re sults generalizes easily to N dimensions, furnishing both a new proof of elementary results and an approach which ought to instill some N-dimensional intuition. The case of the triangle is quickly disposed of, for every triangle has evidently area half that of the rectangle with the same base and altitude (Fig. 3). But I prefer here to start from a special case which will be easily generalizable: an isosceles right triangle, half of the square having the same side. The formula giving its area, Area of triangle =
t
Altitude
X
Length of base
(13)
then extends to an arbitrary triangle by suitable affine transformations, which make the triangle skew and dilate its dimensions, retaining validity of eq. (13) and the factor t (Fig. 4). The extension to 3 dimensions is still easy. Consider a cube, and make use of the 3-fold symmetry about an axis joining two opposite vertices. A clever dissection into three equal pyramids joined along that main diagonal (Fig. 5) shows that the volume of a right pyramid with square base
2Thanks to Jean-Paul Marmora! for this remark. 3The notation means that the face Fk contains all the points OP 1 P2 . . . PN with the exception of Pk. 4Euclid,
Elements,
Book XI I .
VOLUME 26, NUMBER
3. 2004
45
(x11 x2 , . . . , xN)E f(N) (that 2, . . . , N) belongs to one and only one of the hype:rpyramids, namely, that Il/N) such that Xt = sup (xb Xz , . . . , XN). An evident consequence is that each of the hype:rpyramids has volume equal to liN. The re equal pieces. Indeed, any point is, one such that 0
� xk � 1 V k
= 1,
sult generalizes to an arbitrary hype:rpyramid in the same way
as above, showing therefore that Volume of N-hype:rpyramid =
Figure 3. The area of a triangle.
�
Altitude X Volume of (N - 1 )-hyperbase.
( 15)
In algebraic terms, this is evidently equivalent to inte grating a monomial of exponent (N - 1). Indeed, consider
and with altitude equal to the side of the square is indeed a third of the volume of the cube: Volume of pyramid
=t
the section of the hype:rpyramid
ITNCN) (for example) by the (0 < t < 1): it is an (N - !)-hypercube whose vertices are the points (ta 1 1 taz , . . . , taN- b t) with each ak = 0 or 1 (k = 1 , 2, . . . , N - 1); its � N ··;'$.' volume is t - 1 • Then the total volume of the plane
Altitude X Area of base
(14)
Again, suitable affine transformations carry the pyramid into any other pyramid with parallelogram base. Finally, a
XN = t
pyramid is obtained by integrating with respect to the height
t of the plane of
Cavalieri-style comparison of such a pyramid with another having the same altitude, and any base provided it has the same area as the parallelogram (Fig.
6),
pletes the proof that eq. (14) holds for any pyramid.5 Seeing the factor
t in 3 dimensions,
t in 2 dimensions and the factor
�
..�. ,..,..... � .,...!II· ...."... � ,........ .. ·�·�·�·..···· ,....... ' ·
,.,
com
....,..
t
�
:/·"
...
·
!
one is tempted to generalize. In 4
dimensions, the picture can still be visualized. The 4dimensional hypercube (often called a "tesseract"
when one wants to sound mysterious) is easily pro jected down to
2 dimensions (Fig. 7).
One can see that
it can, in fact, be dissected into four equal hype:rpyra mids with (3-dimensional) cubic base, joined along a main diagonal of the hypercube. Thus the volume of each is equal to a quarter of the volume of the hypercube, and this relation generalizes to other hyper-
Figure
pyramids as above.
4. The area of a triangle, again.
Now there is a fairly simple formal proof in arbitrary dimension space
�
N
N. In the euclidean
with an orthonormal basis at
0, consider the hypercube fCN) of unit edge whose 2N vertices are given by their coordinates a 1 1 a2 , . . . , aN, with each ak being 0 or 1 (k = 1, 2, N . . . ' N). Denote by rkC - 1 ) the hyper cubic (N - 1 )-face of fCN) defined by setting ak = 1; these N hypercubes are the N faces of f(N) which meet at the vertex 0' = (1, 1, . . . , 1) opposite the origin 0 (0, 0, . . . , 0). Further, consider the N hype:rpyramids IlkCN) (k 1, 2, . . . , N) having 0 as vertex and as N bases the N hypercubes fkC - 1 ) respectively. These equal hype:rpyramids, with common edge 00', partition fCN) into N the origin
=
=
5By shameless use of "physicist"s reasoning'' implicitly relying on the infinitesimal, this argument totally bypasses the negative solution of H ilbert's Third Problem:
the impossibility (unlike the case of 2 dimensions) of dissecting two tetrahedra of
the same volume into congruent elements.
46
THE MATHEMATICAL INTELLIGENCER
Figure 5. The dissection of the cube and the volume of a pyramid.
Figure 6. The volume of an arbitrary pyramid.
Figure 7. The dissection of the hypercube and the volume of a hyperpyramid.
VOLUME 26, NUMBER 3, 2004
47
the section. In conclusion, this shows by purely geometric methods that
f
dt tN- 1 =
�-
(16)
The calculation of the volume of the pyramid and ob taining the factor 1/3 by dissection of the cube into pyramids is not exactly the method Euclid gives. According to an ear lier proof of Eudoxus, whose idea is generally attributed to Democritus, 6 the result is obtained by trisecting a prism into three pyramids (not necessarily equal) of the same volume. I leave to the reader the pleasure ofverifying that this method also generalizes to N dimensions. Archimedes and the N-Dimensional Sphere
The double jump of
7r
Among the many mysteries afforded by the number 1r,7 not the least is the following: not content to relate the circum ference and the area of the circle, 8 the very same number appears in the formulas for the area of the sphere and its volume. 9 But better yet, the expressions for area and vol ume of the N-dimensional sphere get along without requir ing any other transcendental number-as they perfectly well might, after all, at least before one has done any cal culations. These formulas all appeal only to 1r, raised to powers which jump in every other dimensionality, thus gen eralizing the behavior observed in dimensions 2 and 3. Table 1 displays this strange phenomenon:10 N
AN VN
.. .
0
0
1
1
2
2
2R
.
2nfl nfi2
3
4 71fi2
i 7TR3
3
4
.·� ·
2-i'R3 2
_1_ -i'R"
5
�-i'R4
6
_§_-i'Rs
_1_7i3R"
3
15
7i3Rs 6
7
�7i3R" 15
�1f3R7 1 05
The most-used explicit calculation throws little light on these results. The classical "dodge" for calculating the area of the sphere in N dimensions consists of integrating over Eu clidean space the Gaussian function, which combines the properties of factoring into functions (Gaussian) of a single variable and of having spherical symmetry. The first of these properties allows one to write
( dNr exp( - r2) )fRN =
=
L L [L . . .
The second gives the expression
( dNr exp( -r2) ) fRN =
lx 0
2 a]'/Y'N- 1 dr exp( -r )
=
1 - aNf(N/2)RN. 2
(18)
This yields the final result (for the volume, one integrates the surface of the sphere of radius r, from 0 to R):
f(l/2) IN aN = 2 f(N/2) [f( l/2))N [f( l/2))N N = 2 Nr(N/2) = f(1 + N/2)
V
(19)
Now let us make this more explicit, distinguishing accord ing to whether the dimensionality is even or odd. In the for mer case, N = 2p, the denominator is a factorial of an in teger, and in the numerator there are N factors v;., so p factors 1r. In the latter case, a factor y;. comes into the denominator which takes care of one of the (2p + 1) fac tors y;. in the numerator, finally leaving only p factors 1r. This compensation, which in this account seems perfectly accidental, gives rise to the "double jump" of 1r, without giving any understanding of its geometric necessity. I would like to present now two other methods of cal culation which clear up the mystery of the double jump of powers of 1r; one of them is based on a simple and elegant recursion which has the virtue of generalizing some results more than two millennia old. In 3 dimensions (Archimedes)
To begin with, recall the classical calculation of the area of the ordinary sphere (the sphere in our space). Archimedes was the first to show1 1 that the area of a sphere is exactly equal to the lateral area of the circumscribed cylinder of the same radius R and with altitude equal to the sphere's diameter (Fig. 8). The simplest proof consists in considering on the sphere of radius R the parallel of latitude determined by the polar angle e. The infinitesimal zone between this parallel and another distant dl from it (on the sphere) has area dA = 21rR sin e dl, because the small circle at that latitude has radius r = R sin e. But the parallel planes containing the two circles cut off on the cylinder circumscribed at the equator a band of altitude dh = sin e dl. The area of the band is therefore dA' = 2 1TR dh dA, equal to that of the infinitesimal spherical zone. The sphere and the cylinder thus have the same total area: =
dx1 dxz . . . dxN exp( -x r . . . - x�)
dx exp(-xz)
r
= [f( 112) JN = 1TN/2RN. (17)
(20)
6See the long note of Thomas L. Heath in The Thirteen Books of Euclid's Elements, Dover, New York, vol. Ill, pp. 365-368. 7See Jean-Paul Delahaye, Le fascinant nombre Pi, Belin, 1 999. 81 know. I know: I said it that way when I was a child, but nowadays we are supposed to say "the area of the disk." Fashions of the times. 9Beg pardon. The volume of the ball. 10" . 7T . . seeing that the circumference of a circle is 271fi while the surface of a sphere is 4 R2, we might be tempted to expect the hypersurface of a hypersphere [in 4 dimensions] to be 671fi3 or 8�. It is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression, 2-i'R3 " H . S . M. Cox . eter, Regular Polytopes, Macmillan, 1 963, p. 1 1 9. In his course on Mathematical Methods of Physics in the 1 960s, Laurent Schwartz scored a success with his ex pressions of regret at not living in a space of 6 dimensions, where the formula for surface area is "so beautiful." 1 1 Archimedes, Oeuvres completes, val. 1 : La sphere et le cylindre, ed. Ch. Mugler, Les Belles Lettres, Paris,
48
THE MATHEMATICAL INTELLIGENCER
1 970.
R
(23) whence
Now we can get to the heart of the argument, estab lishing a generalized Archimedean cylindrical projection.
r = (x1 , Xz, . . . , 'fP, and introduce polar coordinates for its first two
To this end, consider the variable point
xN)
E:
coordinates:
Figure 8. The area of the sphere.
As to the volume
(25)
of the interior of the sphere, it suffices to
consider it as cut up into infinitesimal pyramids having ver
In the new coordinates, the volume element becomes
tex at the center and bases on the surface: for each of these
dV = ± R dA, hence for the whole V = ± RA, so
pyramids, of course, sphere one has also
(2 1)
Now we set up a bijection between the sphere tion at
Archimedes expressed this result by declaring equal the vol
is, the circle parametrized by angle
ume of the sphere and that of the cylinder after removing
torus:
from it the pyramids with vertex at the center and bases the
of these results-rightly-that he wanted the key figure of sphere with circumscribed cylinder carved on his tombstone.
Three centuries after his assassination at the siege of Syra cuse, Cicero recognized his grave by this insignia; it has not been rediscovered in recent centuries, alas. The correspon dence which comes in here between the sphere and the cir cumscribed cylinder is just the cartographer's "cylindrical par allel'' (or "normal") projection, which, we have seen as an incidental extra, preserves area locally, not only globally; but, as we knew, it distorts angles, worse and worse as one ap proaches the poles, where it is singular.13
In N dimensions
�N let us consider the sphere :JN of radius R, denoting its area by AN(R), and its interior ball VAN of volume VN(R). Again denoting by aN and VN the area
In euclidean N-space
and volume of the sphere of unit radius, we have for evi dent reasons of homogeneity
{
AN(R) = aNR N - l N VN(R) = VNR .
{ r = (p, cp,r' ) ; lri = R ]
E:
:JN �
{'P
) , in other words, a
E:
g2 r' E: VAN- 2 ; ,r' l < R P = YR2 - lr' 2
(27)
This really does generalize the usual cylindrical projection: in the case of 3 dimensions, the sphere
cg3) is projected to
the cylinder, which is the Cartesian product of the circle
(g2) and the segment (VA 1), which we can just as well re
gard as a torus. In the general case too, the projection pre serves measure; for the integral over the sphere
gN (with
its uniform measure) of an arbitrary function F, can be writ ten in terms of the Dirac 8-function as
NN ( ( ) N d l u F(r) = ) �N d r 8(lrl - R)F(r) � N = pdpdcpd - 2r' 8(Yir'l2
J
I I
= R d cp
r-'
+
p2 - R)F(cp,r' )
N d - 2 r' F( cp,r' ),
(28)
where the last equality requires the standard result that
(22)
Finally, simple integration just as in the 3-dimensional case, allows us to write
'P
p = 0) and the Cartesian product of the ball VAN- 2 1r' ', < R) with the sphere g2 (that
(parametrized by r' , with
bases of the cylinder. 1 2 It is said that Archimedes was so proud
gN (omit
ting its poles to avoid the singularity of the parametriza
8[u(x) - u(a)] = [u' (a)] - 1 8(x - a). Integrating here the (F = 1 ) , one obtains the area of the sphere as the product of the area of the sphere g2 with the vol ume of the ball VAN - Z', that is, the charming formula identity function
AN(R) = 2 TTRVN - 2 (R), so that aN= 27TVN- 2·
(29)
1 2The equality in this form can be demonstrated directly by slicing the two volumes by planes perpendicular to the axis of the cylinder and observing that the two sections (a disk and an annulus respectively) have equal area. The Cavalieri theory of indivisibles (already known before the 1 7th century), a heuristic form of a rigorous theory of in tegration, allows one then to assert that the two solids -the sphere and the hollowed-out cylinder-made up of sections of equal areas, have the same volume. 1 31t is commonly but mistakenly said that this projection is Mercator's. That great cartographer invented in
1 569
a cylindrical projection which is conformal (preserves
angles), much less trivial. It is curious, in view of its appearance 1n Archimedes's result, that the cylindrical normal projection was apparently not used in cartography until the work of J. H. Lambert. See John P. Snyder,
Flattening the Earth (Two Thousand Years of Map Projections).
University of Chicago Press,
1 993.
VOLUME 26. NUMBER 3 , 2004
49
This formula is the essential result; despite its simplicity, I have not seen it in any book. l4 In view of one can now give the following recur rence relations:
(24),
(34), 27T VN N VN-2, (30) 2 7Tr fU1dtk) dx2p+1 o(R2 - � tk - �p+l) = ( V p +1 2 2 15 ;� fR (R2 - �p+1tdxzp+1 2 ;� R2P+1f( 1 - u2rdu, 7T (36) 0 1
and
=
which lead easily to the table above, and also lead back to the formulas ( 1 9). This shows us at once why the power of jumps at every other step. Namely, the dimensionalities and are trivial and do not involve at all; after that, one gets from di mension 0 to all the even dimensions, introducing an addi tional power of each time (note that eq. (29) does hold for N and similarly one gets from dimension 1 to the odd dimensions. =
1r
One
7T
2),
7T
per plane!
Here is another viewpoint on this curious difference be tween spaces of even dimensionality and of odd dimen sionality. The volume of the sphere :J can be written
N
VN l�N dNr8(R2 - r2) = L dx1dx2 . . . dxN 8(R2 - xr - X� �N 8 =
�
. . . -x ),
(31)
where is the Heaviside function. First look at even dimensionality N = 2p. Change vari ables to polar coordinates in each of the p planes defined by pairs of coordinate axes:
Xzk -1 Pk 'Pk, X2k Pk 'Pk =
=
cos
Further, let
sin
(k = 1 ,
(k =
1, 2, . . . , p). (32)
2, . . . ' p)
.
(33)
(31) Vzp Jr�N fi Pk dpk dipk o(R2 - IP�) ( 227T r tudtk o(R2 - � tk} (34)
It follows at once by =
k�l
that
1
=
and in the last integral the integrand is 1 on the p-dimen sional volume interior to the pyramid described by the pos itive half-axes (0 < k = 1, . . . , p) and the hyperplane = cutting them at That volume is + + . . . + giving us finally
(p!) -1 R2,
tk> tk tz
2,
odd, 2
N= p + If on the other hand the dimensionality is one makes the same change of variables as before in p planes corresponding to p pairs of coordinates, but now one coordinate is left an old maid. Thus we get, instead of eq.
1,
tp R2•
(35)
=
=
or, finally,
p+1 p 2p+1 - 22(2p +p.'1)!7T R2p+1'
v:
_
(37)
in agreement with the general expressions. The preceding changes of variable can be regarded as projecting, with preservation of Euclidean measure, •
•
2
!Jl32p
in even dimensions, the ball onto the Cartesian prod uct of p spheres :J (circles) with the interior of a p dimensional polyhedron; in odd dimensions, the ball onto the Cartesian product of p spheres :J (circles) as before with the in terior of a segment of a + 1 )-dimensional paraboloid
(
given by
!Jl32p+ 1 2(p ktl tk + �p+1 :'5, R2, 0 tk (k = 1 , 2, . . . , p)} <
In both cases, the volume of the polyhedron or paraboloid is rational, so there are as many factors as there are circles. The situation is simple in essence: there are as many fac tors in the expression for volume (or for area) of the sphere in N dimensions as there are "independent circu larities" in the space, meaning by that, the number of in dependent ways to tum around in space, that is, simply the number of independent planes. That is evidently the inte gral part of half the dimensionality.
7T
7T
16
The surface effect
It may be interesting to show how the expression for the volume of the sphere in N dimensions illuminates the phys ical nature of high-dimensional spaces. One might call this the "surface effect": the higher the dimensionality, the more points there are close to the surface-in a sense to be clar ified by the following examples. Naturally this is true for most sufficiently regular N-dimensional bodies; the only ad vantage of the sphere is in permitting explicit calculations leading to simple results. First let us ask this question: what is the radius of the concentric ball inside a ball of radius which contains half
R
14Thus Coxeter (op. cit., p. 1 26) gives the result, in the form of the first of equations (30), but derived from the expressions obtained by the "gaussian" method, not from the geometric meaning. ' 5Note in particular the case of the sphere in 4 dimensions: its area is equal to the volume of a torus in 3-space, and I wonder if the corresponding cartography would be a useful geometric tool.
1 6We may be permitted to wonder whether the difference in behavior brought out here between even- and odd-dimensional spaces is related to the Euler-Poincare characteristic, which also distinguishes them.
50
THE MATHEMATICAL INTELLIGENCER
2
It is a monotone increasing function, showing that the area of a sphere is larger (relative to the volume of the section) the larger the dimensionality. Here is another way to look at this. Consider a uniform probability density on the unit ball in N dimensions. Let us ask what is the mean distance rN between a random point in the ball and the center. This is given, of course, by
.
;-���+----+--�---r---4--1--t--+ ··.
.4
2
0
• ·•• •
...
6
4
...
. • · ·•
8
-
·
12
10
.•...
14
N
Figure 9. The "effective size" of the unit ball in N dimensions.
its volume? This requires that tvNRN, or
R'
=
V(R')
=
tV(R),
2 - IIN R.
or
I 16
vNR'N
=
(38)
Now plainly when N grows indefinitely, the radius R' ap proaches R. It is more striking to think instead of the outer shell between the inner and outer sphere, and containing half the volume of the latter. The higher the dimensional ity, the more the thickness DR = R R' shrinks relative to the radius R, as shown in Table 2.
-
We see that the outer half of the volume is crowded into a thinner and thinner layer. The reason is evidently the avail ability of a high number-namely 1)-of directions or thogonal to the radial direction, allowing a transversal "spreading," enabling volume to be large with small thickness. In the same order of ideas, it is worthwhile to compare the smface of the sphere '.:fN with the volume of its "prin cipal section," the ball ?13N- b by calculating the ratio AN/VN- I = fNIVN- I (it doesn't depend on the radius be cause of homogeneity, otherwise the quantity would have no significance). By Eq. (19), it turns out that
(N-
r(l\r(N- ) ( , 1
eN"'N- ·
�
_
�m2
�B
1 N- 1) , -2
(39)
an Eulerian function, whose first few values are given in Table 3.
.., . ,.:�
N
/N /VN- 1
1
2
2
.L
11'
(3 , 1 4 .
.)I
3
4
4 3 71'/2 (4.71 . . . )
.
,
5 1 6/3 (5 . 33 . . . )
. . . I N>>1 !
2vr;tV
r dr r AN(r)
_
rN
I ( > > 1�
=
1
J0 dr AN(r)
which becomes, by eq. (22),
rN
=
fN
�f� dr
__ _
eN
r r
= __ �o d r _ _ _ l
J0
(40)
,
1 -
N,
1
N
'
(41)
In other words, the mean distance to the center, for large values of the dimensionality is very close to the radius of the ball, which means that most of the points are very close to the boundary. That means that the radius of the ball gives a mislead ing idea of its effective size, making it seem smaller than it is, because the abundance of available dimensions some how compensates for the concentration near the surface. Consider this rather surprising illustration of this inter pretation. If we think that a good idea of the "effective size" of an N-ball is given by the length eN of the side of a hyper cube of volume equal to that of the ball, which we may take to have unit radius, then by eq. (19) we get this length as
("! : =
(VN) IIN
=
[f(l
f ( l/2) +
N/2)]11N '
(42)
which turns out to be a decreasing function of N (Fig. 9). Notice the obvious special cases €1 2 (in 1 dimension, the ball of unit radius is the same as the cube of side 2) and e2 = v;., =
Equation (42) is of course not defined in dimension zero. However, its limit as N --;; 0 is well defined: €0
=
y;. e"Y = 2.3656 . . . ,
(43)
where y is Euler's constant. This apparently obliges us to regard the curious constant (eq. 43) as the effective size of the 0-dimensional unit ball (?). Dimensionality and Orthogonality
I close with some elementary geometric considerations on N-dimensional euclidean space
26, NUMBER 3. 2004
51
Figure 10. The equiaxial angle. Figure 1 1 . The isogonal angle.
thogonal axes), having to add to unity, must generically all be small, making the angle between them close to a right angle. Let us check this in some special cases. Equiaxiality
Consider a system of N orthogonal axes in �N, and call N "equiaxes" referred to them the 2 - 1 lines making equal an gles with all of them; in 2 dimensions, the equiaxes are the two bisectors of the angle between the axes. Denote by aN the (acute) angle between an equiaxis and a coordinate axis; call it the "equiaxial angle" (Fig. 10). Thus the unit vector along an equiaxis projects onto each coordinate axis to a segment of ± cos aN so that the angle aN is given by N cos 2 aN = 1, or cos aN =
1
VN'
(44)
The numerical values of the equiaxial angle for some low di mensionalities are given in Table 4. Indeed, the equiaxes are closer and closer. to orthogonality to the coordinate axes. N
aN
.
.
1
2
0
45"
3
4
5
.
.
.
54.7"
60"
63.4"
.
.
.
N> > 1 1r
=2-
1
v7V
lsogonality
Just as natural is the figure which we may call "isogonal" N formed in � by N + 1 half-lines making equal angles with each other. In 2 dimensions this gives the Mercedes trigon; in 3 dimensions, the directions to the vertices of a regular tetrahedron from its center-in general, the directions of the vertices of a regular simplex seen from its center. Let us denote by f3N the 'isogonal angle" between any two of these lines (Fig. 1 1). By symmetry, the N + 1 unit vectors uk (k = 1 , 2, . . . , N + 1) on isogonal rays are linearly de pendent by
N+ l
L 1
uk
=
0.
(45)
Projecting this relation onto any one of the vectors, one gets the relation defming the isogonal angle, namely 1 COS f3N = - . N
(46)
These values may be tabulated as shown in Table 5 N
f3N
.
.
1
1 80°
2
. . .
1 20°
3
4
5
.
.
.
1 09.4"
1 04 S
101S
.
.
.
N> > 1
= .E: + .l 2
N
Again, the increase of dimensionality brings with it a ten dency toward orthogonality. Uniformity
Given the sphere :JN in N dimensions, what is the mean an gle between two vectors taken randomly on the sphere, given its uniform measure? A simple symmetry argument shows that it is a right angle. But the probability distribu tion of this angle 8, call it pN( 8), rewards attention. Let us parametrize the sphere in generalized spherical coordi nates:
x1 = cos81, x2 = sin81 cos82 , X3 = sin81 sin82 cos8:3, . . . . . . XN- 1 = sin81 sin82 . . . cos8N - 1 , (47) XN = sin81sin82 . . . sin8N- 1 ·
In these coordinates the uniform measure on the sphere appears as
N d - 1u N N = d81 dfh . . . d8N- 1 sin -2 81 Sin - 3 fh . . . Sin8N-2>
(48)
Of course, integration on all of the angles would give back the area of :JN calculated above. Here we are look-
17Note that the familiar relation [33 = 2a3, which determines several features of spatial symmetries in our world (for example in crystallography), is altogether confined to the 3· dimensional case. The equation Arccos (N- 1) = 2 Arccos (N- 112) , whose only solution is N = 3, must be added to the other particularities of our space, along with the fact that N = N(N 1 )/2 (whence the exterior product of two vectors can be a "vector product") and the fact that N + 1 = 2N- l (whence two equiaxial lines can also be isogonal).
-
52
THE MATHEMATICAL INTELLIGENCER
A U T H O R
1-+---- N = =
2
8
0,5Jr
0
7r
Figure 1 2. The probability density of the angular distance of two ran
JEAN-MARC LEVY-LEBLOND
dom points on an N-dimensional sphere.
ing at the angle between a randomly chosen variable di rection and a fixed direction. Let us take the zenith axis Ox1 as reference direction; so it will be the random vari able 8�, which is at issue. Now integrating over all the other angular variables, we see that the desired proba bility density, which determines the random distribution of the angular distance between two arbitrary directions, is given by
pN(8)
=
K sinN-2 8.
(49)
This is a distribution with mean value 8 = TTI2, more and more "sharply peaked" as the dimensionality N grows (Fig. 12); for large values of N its width is 88 O(N- 112), and it approaches the Dirac distribution 8( 8 TTI2) in the limit N ----> XJ. That is to say, the higher the dimensionality of a Eu clidean space, the more likely arbitrary directions are to be close to orthogonal. This idea may have some effect on the intuition we can have (or can't have!) about an infinite-dimensional space like the Hilbert space of quantum mechanics. 18 =
Physique Theorique
Universite de Nice Sophie-Antipolis Pare Valrose
061 08 Nice Ceclex France
e-mail: [email protected]
Jean-Marc Levy-Leblond describes himself as a theoretical physicist and an experimental epistemologist, but also an es sayist and editor on the cultural aspects of science-in short,
a "science critic." The review he co-founded, Alliage, has been
endeavoring to blend sciences and humanities for more than a dozen years. He has recently published a collection of off
trail chronicles on science, lmpasciences,
Seuil 2003. ,
-
1 81t is my pleasure to extend my deep thanks to Chandler Davis for translating this article into English.
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VOLUME 26, NUMBER 3, 2004
53
RABE-RUDIGER VON RANDOW
P aited Po yh ed ra ay back in 1 959, when I visited Burma and India as a student on my way from New Zealand to Germany, I picked up a rattan ball as de picted in Figure 1 . It features in a game called Sepak Raga ("to kick a rattan ball," in Malay) , a game known to have originated in Malaysia
500 years ago. The game involved players standing in a cir
cle, and the aim was to keep the ball aloft with any part of the body except the hands. The ball and the game were, of course, also well-known in all the other South East Asian countries, but it was not until the twentieth century, and especially after the Second World War, that the game be came popular in the West. It is now played with a plastic ball and a net, and is known as Sepak Raga Jaring (Jaring means "net" in Malay) or Sepak Takraw (Takraw means "ball" in Thai) or "kick volleyball." A cursory look at this ball shows that it is made by plait ing bands, each band consisting of rattan strands. How many bands are there? One easily sees that there are six bands, all exactly the same. The rattan ball also has twelve pentagonal holes, and we will see later that this number has to be twelve for topological reasons. There are also twenty "point-holes" on the rattan ball where three bands meet. Because of its twelve pentagonal holes, the rattan ball looks like a dodecahedron (see Fig. 2): to get exact agreement, imagine each pentagonal hole enlarged up to the five point-holes surrounding it. If, instead, we poke our ' finger through each of the twenty point-holes to create lit tle triangular holes, the rattan ball looks like an icosido decahedron (see Fig. 2), one of the archimedean polyhe dra, with twelve pentagonal and twenty triangular faces.
Are there any other plaited balls which we could regard as generalizations of the rattan ball? Before we can answer this question, we need to analyze the basic nature of the plaiting method used for the rattan ball. If we take strips of strong paper or stiff plastic material, all of equal width, we can plait a tight mat with a hexagonal pattern, as shown in Figure 3. By "tight" I mean that the hexagonal holes can not be made any smaller-unless, of course, we reduce the width of the strips. If we wish to introduce curvature into
Figure 1 . Rattan ball.
The author wishes to thank the referee for many constructive comments and suggestions for improvement.
54
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK. LLC
Dode ca h e d ro n Figure
3. Plane mat.
ball, I will prove later that such a plaited ball will always have exactly twelve pentagonal holes for topological rea sons. •
I c osidode ca h e d ro n
Figure
2.
The method for making plaited balls described in the previous paragraph will, with the sole exception of three Remarks, be the only method considered in this article.
Just as the rattan ball has dodecahedral symmetry, we would also like any other plaited ball to have symmetry properties akin to those of the platonic or archimedean polyhedra. (All sorts of sausages and cages can be plaited if we give this symmetry up. Four years ago an article ti tled "Molecular Modeling of Fullerenes with Hexastrips" [2) by Paulus Gerdes appeared in The Mathematical Intelli gencer. It deals with such plaited objects in connection with
Trun c ated i c osa h e d ro n
the mat, we can achieve this most easily by introducing a pentagonal hole (see Fig. 4), which is done by omitting a strip from a hexagonal hole . We can go on introducing pen tagonal holes, thus obtaining more and more curvature in the form of local "bumps," and if we are lucky we can form a closed surface, a topological sphere. As with the rattan
Figure
4. Mat with pentagonal hole. VOLUME 26, NUMBER 3, 2004
55
chemical structures. In that article there is a picture of a sausage-shaped structure obtained by halving the rattan ball into two equal caps of six pentagonal holes and join ing them again by inserting two plaited rings of hexagonal holes in between.) •
In passing I should note that there are obviously two ba sic ways of plaiting, relating to which strips go over and which go under. In the case of the rattan ball these two ways yield balls that are mirror images of each other, and clearly there is no point in taking this choice into con sideration. In the rest of our discussion I will ignore this basic choice in plaiting.
The idea above, of a hexagonal mat with curvature in troduced by means of pentagonal holes, leads to a plaited ball with hexagonal and pentagonal holes. So it would seem to be a good idea to look for a symmetry candidate among the archimedean polyhedra. The only suitable polyhedron is the truncated icosahedron (see Fig. 2) with twelve pen tagonal and twenty hexagonal faces, the common soccer ball. The plaited ball with this symmetry is also well-known (Fig. 5) and has ten bands, all exactly the same. To get full agreement between it and the truncated icosahedron (Fig. 2), again imagine all its holes (pentagonal and hexagonal) enlarged up to the point-holes surrounding them. A number of years ago I saw a very unusual picture (Fig. 6) in the Frankfurter Allgemeine Zeitung of 20 March 1996. It depicts holiday bathers on the beach of Deauville, France, in July 1936, within a sculpture constructed of metal bands, which bears an obvious affinity to the rattan ball and the plaited soccer ball. The sculpture is in fact plaited as described above, but with one essential differ ence: there are also triangular holes (as if someone has been along and poked his fingers into the point-holes) be cause the strips have not been plaited as close together as possible and are held in place with rivets. It is essential to understand that this feature applies only to the triangular holes. Neither the pentagonal nor the hexagonal holes can be shrunk to point-holes by plaiting more tightly. If, then, we plait this sculpture as tightly as possible, what does it become? One soon realizes that it becomes the plaited soc cer ball (Fig. 5). But the sculpture is not in fact plaited
Figure 5. Plaited soccer ball.
56
THE MATHEMATICAL INTELLIGENCER
Figure 6. Bathers with plaited ball sculpture on Deauville beach, 1 936.
tightly. And just as we considered the rattan ball with tri angular holes, which yielded icosidodecahedral symmetry, we would naturally like to know what sort of symmetry we have here. Letting point-holes become triangular holes in troduces a new type of face. In the case of the rattan ball, this forces us from the platonic to the archimedean poly hedra. In the case of the plaited soccer ball we are looking for a polyhedron with twelve pentagons, twenty hexagons, and sixty triangles, because each pentagon is surrounded by five triangles, none of which is shared. The sculpture is indeed the symmetric joining of twelve pentagrams by their tips, as was pointed out to me by the referee. There is cer tainly no archimedean polyhedron to match, and a look at the list of Johnson polyhedra (all the convex polyhedra with regular faces) on the internet also draws a blank So this polyhedron has nonregular faces, and a brief study of the symmetry of the sculpture shows that it must be the triangles. •
In the rest of this discussion I will consider all plaited balls to be tightly plaited.
Before I introduce the next step in our hunt for more plaited balls, a practical word in connection with the ac tual plaiting of such balls: one needs a lot of strips and a lot of paper clips. The strips should be of strong paper, 1.3 em wide and at least 60 em long. I cut up the pages of a company calendar by hand, which took much longer than the actual plaiting! Then one begins to plait, preferably with hexagonal holes, using paper clips to prevent the plaiting from undoing spontaneously. Pentagonal holes are easy to introduce by bending the strips into place and using paper clips to keep them there. As one advances, one can remove earlier clips as the structure becomes stable. One also finds increasingly that certain strips must go where there already is one from earlier plaiting, which can then be removed. Also, when one strip is used up, another is simply plaited in as a continuation. Of course, one needs a pattern to fol low, and this will be a graphical representation introduced
in the next section. The plaiting is really simple and quick; the difficult part is keeping track of where one is in the pat tern and it helps if there are no interruptions. But even with a pattern to follow, one still does not know how many bands there will be and how they will join up. Nor would it really help: if one follows the pattern faithfully, then as the structure begins to close up, everything falls into place. Halfway through, however (see Fig. 24), the whole thing looks like a giant octopus! When one has finished with the actual plaiting, what remains is to tighten the bands so as to give a tightly plaited model with a nice symmetrical shape. With some models made of stiff plastic strips, it is difficult to adjust the band lengths so that no little trian gular holes appear, but this does not happen when using paper strips. When one is satisfied with the look of the model, one should staple the band ends together so that the plaiting cannot loosen again. The referee has drawn my attention to the work of Pro fessor Jean Pedersen on braided and woven polyhedra. These are conceptually but not directly related to the plaited balls considered here. Those depicted in Figures 1 and 5 in fact appear as braided polyhedra in one of Peder sen's papers [5], in an entirely different context, where the braided rattan ball is called a golden dodecahedr-on. The next question is how to proceed at all, as there are no more Johnson polyhedra to inspire us. The method of making plaited balls introduced above results in balls with twelve pentagonal holes and some number of hexagonal ones. If we are to have any symmetry at all, the twelve pen tagonal holes should be distributed symmetrically over the ball. Because there are twelve of them, we thus can either distribute them singly, resulting in twelve "evenly spaced" bumps or "comers," making the ball look icosahedral, or distribute them in pairs, resulting in six "evenly spaced" comers, each comer consisting of two bumps, making the ball look octahedral, or distribute them in triples, resulting in four "evenly spaced" comers, each comer consisting of three bumps, making the ball look tetrahedral (see Fig. 7). Examples of these are depicted in Figures 5 and 8. The two models in Figure 8 both have 58 hexagonal holes. The tetra hedral-looking one is very clearly tetrahedral, and this is typical of a distribution in triples, as each of these makes a good comer and, moreover, fits the comer well geomet-
Icosa hedron
Figure 8. Oct-R58 and Tet-A58.
rically, as there are three faces and three edges of the tetra hedron meeting in each vertex. The other model in Figure 8 is not very obviously octahedral, because a pair of pen tagonal bumps does not make that good a comer and does not fit the comer of an octahedron except by straddling it. Satisfactory symmetry will result only if the other pairs are arranged so that the result is a plaited ball which looks the same whichever comer of the octahedron is turned towards us. This is typical of a distribution in pairs. A very clearly
Octa h ed ron
Tetra hedron
Figure 7.
VOLUME 26. NUMBER 3 , 2004
57
octahedral plaited ball is shown later in Figure 24. It has 88 hexagonal holes. Finally, the reason that the rattan ball in Figure 1 does not look icosahedral but dodecahedral is that it has no hexagonal holes at all, and thus the pentago nal ones look like the faces of a dodecahedron rather than the vertices of an icosahedron. •
•
•
These three types of distribution of the pentagonal holes over the ball will form our basic symmetry requirement, to which we add two further requirements: that the plaited ball is to look the same to us, whichever comer we have facing us and whichever rotational posi tion it is in (five different positions if it is a singleton, two if a pair, and three if a triple of pentagonal holes); i.e., the plaited ball is to be invariant under the rotation group of the corresponding regular polyhedron; and that, whichever pentagonal hole we have facing us, we can find a rotational position in which the plaited ball looks the same to us.
each pentagonal hole is like any other in the plaited baU. One could, for example,
The third condition ensures that
suggest the distribution consisting of two sets of six pen tagonal holes, like two polar caps with a belt of hexagons in between. An example of such a plaited ball was men tioned earlier: it is the sausage-shaped structure shown in [2], which can obviously be as long as we like. It clearly has two big comers and looks the same viewed from either one, but it has two different types of pentagonal holes, the two poles and those that border the belt of hexagons. The third condition excludes these structures. The third condi tion also has the important consequence that it enables us to represent the plaited ball graphically in a far more eco nomical form by considering only a basic cell of the graph. This will play an important role in the combinatorial treat ment given below. Thus our symmetry requirements are closely related to the three platonic polyhedra whose number of vertices di vides twelve, which are also those with triangular faces. •
Let us call plaited balls satisfYing the above three sym metry requirements plaited polyhedra.
While the tetrahedral model in Figure 8 is as symmetri cal as an abstract tetrahedron, right down to the way the bands run, one cannot say the same for the octahedral model. It isn't just the pair-comers themselves, but also the way they are aligned to one another. The model has a twist to it, so that there is a right-handed version and a left handed version. And yet it obviously fulfills the above three symmetry requirements. More and more plaited polyhedra have this twist or skewness as the number of hexagonal holes increases. It also reveals itself in their faces: the bands running from right to left are at a different angle from the bands running from left to right. The smallest plaited polyhedron with this property looks tetrahedral and has only twelve hexagonal holes (Fig. 9). As with the octahedral model in Figure 8, all these plaited polyhedra have a right- and a left-handed ver sion, which reduces the polyhedral symmetry.
58
THE MATHEMAnCAL INTELLIGENCER
.�
Figure
•
.
,,. • �. � .
.
___
.
.. ' �
' •.
r
4
.
'
•
9. Tet-A12.
In the rest of our discussion we will not distinguish be tween right- and left-handed versions of plaited polyhe dra with skewness.
Now that we know what we are looking for, namely plaited polyhedra, how does one go about finding them? After in troducing what seemed to me to be an efficient graphical representation (see below), I set an arbitrary bound at sixty hexagons and looked for them empirically with ad hoc methods of trial and error, becoming more methodical with time but having no system. Much later, I suddenly had the inspiration which led to a combinatorial method for ob taining all cases systematically (see Combinatorial Treat ment, below). It was an uplifting experience after all the drudgery, and although it comprises only a tiny drop in the ocean of knowledge, it was as described by Gerty Theresa Cori, 1947 Nobel laureate in medicine [ 1 ] : . . .
For a research worker the unforgotten moments of his life are those rare ones, which come after years ofplod ding work, when the veil over nature's secret seems suddenly to lift and when what was dark and chaotic appears in a clear and beautiful light and pattern. The next step, therefore, will be to develop a graphical representation for a plaited polyhedron which will enable one to go ahead and plait it. But first this remark. Remark
The plaited polyhedra which we are considering are all tightly plaited, with hexagonal holes which are plane regions, and twelve pentagonal holes which introduce curvature. However, instead of omitting one strip from the basic hexag onal plaiting so as to introduce curvature with a pentagonal hole (see Fig. 4), one can also omit two strips to produce a square hole. This introduces a bump with stronger curva ture. One can even go one step further, omitting yet another strip and obtaining a triangular hole, which produces an even shruper bump. Such a triangular hole has nothing to do with those obtained from point-holes as discussed earlier. Here I have only considered plaited balls with hexagonal and square holes, or with hexagonal and triangular holes. The re sulting shapes are rather pretty (see Fig. 10). Just as plaited
Figure 1 0. Figure 1 2a. Graph of rattan ball (Fig. 1).
balls with hexagonal and pentagonal holes have exactly twelve of the latter, we have the following result, whose proof is analogous to that for the pentagonal case: plaited balls with hexagonal and square holes (resp. triangular holes) have exactly six (resp. four) of the latter. 0 Graphical Representation
If we consider the problem of representing plaited polyhedra like those depicted in Figures 1, 5, 8, and 9 in as condensed a form as possible in order to obtain a pattern to work from, it seems natural to represent the hexagonal holes by black dots, say, and the pentagonal ones by "white" dots (circles), and to join each black dot to the six dots surrounding it and each white dot to the five dots surrounding it. The result is a graph incorporating all the information we need for plait ing the model. Of course, the same holds for any plaited ball, not just for plaited polyhedra. The graph for the plane mat is depicted in Figure 1 1 ; six edges emanate from each of its ver tices, i.e., all its vertices are of degree six. A white dot has five edges emanating from it, so it is a vertex of degree five.
*
*
*
*
*
*
is evident from the plaiting pattern. The graphs of the rattan
Figure 12b. Graph of plaited soccer ball (Fig. 5).
Figure 1 1 . Graph of plane mat.
Figure 12c. Graph of plaited polyhedron in Figure
Furthermore, all such graphs have triangular faces only,
as
9.
VOLUME 26, NUMBER 3, 2004
59
5), and the plaited polyhe 9 are depicted in Figure 12a-c, where the six
D. The given plaited polyhedron thus has four evenly spaced
starred vertices of graph 12b represent a single vertex. Clearly,
to cover the rest of its surface. For example, the tetrahe
ball, the plaited soccer ball (Fig. dron of Figure
triples of type A (or of type B, etc.) with hexagonal holes
9 has
the first two graphs are beautifully symmetrical but the third
dral-looking plaited polyhedron depicted in Figure
one is not, a result of the skewness discussed earlier.
four triples of type A at its vertices; its graph is given in Fig
•
Any plaited ball with pentagonal and hexagonal holes can thus be represented by a graph with triangular faces and vertices of degree
•
5 or 6.
The converse is readily seen to be true as well: any finite graph, all of whose faces are triangular and whose vertices are of degree
5 or 6 only, represents a plaited ball. All one
needs to do is to start plaiting, and sooner or later the bands must close up; otherwise the graph would not be finite.
Theorem. Any plaited ball as defined above, having only pentagonal and hexagonal holes, and possibly none of the latter, has exactly twelve pentagonal holes.
G be its graphical rep G drawn on a sphere. Then G rep
resents a spherical polyhedron to which Euler's polyhedral
V be
number of its edges, and
the number of its vertices,
F the
E the
number of its faces. Fur
thermore, let x : = the number of pentagonal holes of the
plaited ball, and
y
:
= the number of its hexagonal holes.
Then x = the number of vertices of
G of degree 5, y = the number of vertices of G of degree 6, and G has the follow
ing three simple properties: =
bounds exactly two faces, we get
3F = 2E.
2E
2
=
gives
6 (x
+ y)
12 =
in increasing size, the first four have the graphical repre sentations P-S (see facing page), where the pairs are high
6(V - E + F) - (5x + 6y) x.
=
6V - 6E
Q and S.
etc.) with hexagonal holes to cover the rest of its surface.
For example, the octahedral-looking plaited polyhedron de picted in Figure 13 has six pairs of type P at its vertices, and its graph, given in Figure 14, has sixteen black dots. In the
Oct-P16.
Finally, consider an icosahedral-looking plaited polyhe dron. Its twelve pentagonal holes are distributed singly and evenly. The plaited polyhedron thus has twenty congruent triangular faces which clearly can only be of type
A,
or of
type B, etc., and the face-type fully characterizes it. The smallest of these, which, of course, has A-faces, is the rat tan ball of Figure 1, which in fact looks dodecahedral be cause it has no hexagonal holes, so the pentagonal holes look like faces instead of vertices. Its graph (Fig. 12a) has no black dots. The next one, with B-faces, is the plaited
5. It has twenty hexagonal holes, and
its graph (Fig. 12b) has twenty black dots (as remarked ear lier, the six starred vertices represent a single vertex). These two plaited polyhedra are thus listed as
Icos-B20, respectively, in the table below.
Icos-AO and
The following table lists all plaited polyhedra with up to 60
V-
E
+
+ 4E = 6V -
fore discovering the systematic method, which was gratifying. Tet-A
4
10
12
18
24
28
36
Tet-B
28
32
34
40
50
52
56
Remark
Tet-e
24
36
46
48
52
60
I remarked above that the analogous result holds with anal
Tet-0
60
36
40
52
=
The given plaited poly
hedron thus has six evenly spaced pairs of type P (or of type
hexagons. There are 36 of them, and I had found them all be
Then substitution in Euler's polyhedral formula
F=
then its twelve pentagonal holes are distributed evenly in six pairs. Again, the possible pairs are restricted and, arranged
soccer ball of Figure
(a) G has no other vertices, therefore x + y V. (b) The number of edges of G emanating from the vertices of G is 5x + 6y with each edge counted twice, so 5x + 6y = 2E. (c) All the faces of G are triangles, and as each edge
Tet-A12.
If we consider an octahedral-looking plaited polyhedron,
table below it will thus be listed as
Given such a plaited ball, let
formula applies. Let
the table given below it will thus be listed as
Q,
easily obtain the following theorem:
resentation, and imagine
holes, and we see that its graph has twelve black dots. In
lighted with dashed lines in
Using this graphical representation, we can now very
Proof
ure 12c. I mentioned earlier that it has twelve hexagonal
ogous proof: any plaited ball with hexagonal and square holes (triangular holes) has exactly six (four) of the latter.
Oct-P
16
If these plaited balls are symmetric, the ones with hexagonal
Oct-O
48
and square holes (resp. triangular holes) have octahedral
Oct-R
58
(resp. tetrahedral) shape with identical triangular faces. D
By means of the graphical representation of a plaited
Ieos
AO
820
40
42
48
58
60
'
C30 060
polyhedron, we also obtain a systematic approach to the
Some of the plaited polyhedra in the above table can be
distribution of its pentagonal holes.
taken to be either tetrahedral or octahedral, they are bor
Consider a tetrahedral-looking plaited polyhedron. Then
derline cases. From a systematic point of view, we can and
its twelve pentagonal holes are distributed evenly in four
should consider only tetrahedral plaited polyhedra-apart
triples. Because of our symmetry rules, the possible triples
from the much simpler icosahedral series, of course-as I
are restricted and, arranged in increasing size, the first five
shall make plain in a moment. We will also see that, in gen
have the graphical representations A-E (see facing page),
eral, octahedral plaited polyhedra are even less symmetri
where the triples are highlighted with dashed lines in B and
cal than we had found earlier.
60
THE MATHEMATICAL INTELLIGENCER
I
I
I
I
\
\
\
A
\
B
""
c
""
I
�
..---;
�
�
�
""
""
E
D
�
Graphical representations of the distribution of the pentagonal holes of a tetra hedral-looking plaited polyhedron (see page 60).
o�---o
p
Q
o-------•�--�o
R
s
Graphical representations of the distribution of the pentagonal holes of an octa hedral-looking plaited polyhedron (see page 60). VOLUME 26, NUMBER 3 , 2004
61
Figure 13. Oct-P16.
All the plaited polyhedra in the first four rows of the table on p. 60 have the following property in common: if we regard their pentagonal holes only, they exhibit the structure of a truncated tetrahedron '?! (see Fig. 15). In the plaited polyhe dra the triangular faces of '?! are either all A or all B or all C etc., and the hexagonal faces of '?! are filled with hexagonal holes in some way. Here we do not want the hexagonal faces of '?! to be regular hexagons, so that we can imagine the trun cation of '?! to be deeper or less deep. The less deep the trun cation is, the more tetrahedral the plaited polyhedron looks. If we now let the truncation of '?! become steadily deeper, then the triangular faces of '?! will look steadily less like comers and more like faces in the plaited polyhedron. At the same time the common edges of the hexagonal faces of '?! will be come shorter, and in the plaited polyhedron these are either all P or all Q or all R etc., so they will look increasingly like the comers of an octahedral plaited polyhedron. In brief, as we truncate '?! more deeply, its triangular faces grow in size and thus lose their character as tetrahedral vertices, and the six edges joining them shrink in length and thus gain their character as octahedral vertices, and '?! looks increasingly oc tahedral (see Fig. 16 which was taken from the book Polyhe dra by P. R. Cromwell, Cambridge University Press, 1997). •
Thus, every octahedral plaited polyhedron is implicitly also a tetrahedral one, and conversely.
But a closer look at the eight faces of an octahedral looking deeply truncated '?! reveals that they are not iden tical: four of them are the same and come from the four tri angular faces of '?!; the other four are also the same and are given by the four hexagonal faces of '?!-see Figure 14, which depicts the two sets of four faces in the graph of Oct p1 6 (Fig. 13). If one type of face is rather smaller than the other, the octahedral plaited polyhedron acquires a rather lopsided look! This problem is, of course, restricted to oc tahedral plaited polyhedra; the four faces of a tetrahedral plaited polyhedron are identical, and so are the twenty faces of an icosahedral plaited polyhedron. One could ask how the octahedral plaited polyhedra in the above table would be listed if one regarded them as tetrahedral ones. The answer is:
Oct-P16 = Tet-B16 Oct-P36 = Tet-D36 62
THE MATHEMATICAL INTELLIGENCER
Figure 1 4. Graph of Oct-P1 6 (Fig 1 3).
Oct-P40 = Tet-D40 Oct-P52 = Tet-E52 Oct-Q48 = Tet-D48 Oct-R58 Tet-D58 =
One finds these by studying the corresponding graphs. Fig ure 14, for example, depicts the graph of Oct-P1 6 (Fig. 13), and the cross-hatched B-triangles are the triangular faces of '?!, so we get Tet-B1 6. The hexagonal faces stand out very clearly, and I note in passing that each one contains a C-triangle involving every second vertex of its boundary. This will play an important role in my systematization. First, however, we must return to one important ques tion pertaining to plaited balls in general, namely the num ber of bands required. Even in the very special case of plaited polyhedra, I know of no simple formula for an swering this question, or the allied question of whether all the bands are the same. The data in the table below were determined from the finished plaited models. We see that
all plaited polyhedra with up to sixty hexagonal holes re quire three or more bands, so it seems very unlikely that there will be any with fewer than three.
Figure 1 5. Truncated tetrahedron.
�(!)-�- 4!7\ - - -
.
_ _ _
/
,__
,
Figure 1 6. Sequential truncation from tetrahedron to octahedron.
'
Clearly, each band flanks holes, alternately on the right and on the left. Define the length of a band as the number of holes that it flanks. If a plaited polyhedron has N hexag onal holes, then its total number of "hole-flankings" is 6N + 5 · 12 = 6(N + 10), which must also be the sum of the lengths of its bands. We can distinguish different types of bands by the sequence of pentagonal and hexagonal holes which they flank, even if they have the same length. Let
xl
X2 x3
=
=
=
number of different types of bands, number of bands of each type, lengths of bands.
Then we have the following results: x1
x2
3,4
12,12
Xa
Tet-856
Tet-A 10
1
4
30
Tet-A 12
1
3
44
Tet-A 18
2
4,6
Tet-A24
1
Tet-A28
1
Tet-A36
x1
x2 3
1 32
Tet-C24
1
3
3,4,4
20, 1 8, 1 8
Tet-C36
1
6
46
18,16
Tet-C46
3
4,4,6
24, 24, 24
3
68
Tet-C48
1
3
116
6
38
Tet-C52
1
3
1 24
3
3, 4, 6
20, 24, 20
Tet-C60
1
3
Tet-A40
2
3,4
36,48
Tet-060
1
3
Tet-A42
1
4
78
Oct-P16
1
3
Tet-A48
1
3
116
Oct-P36
2
6,6
Tet-A58
3
4, 6, 6
30, 24, 24
Oct-P40
1
3
1 00
Tet-A60
1
6
70
Oct-P52
4
3,4,4,4
28. 24, 24, 2 4
Tet-828
2
3,4
20,42
Oct-Q48
2
6,8
26,24
Tet-832
1
6
42
Oct-R58
1
4
1 02
Tet-834
1
12
22
lcos-AO
1
6
10
Tet-840
2
3,4
68,24
/cos-820
1
10
18
Tet-850
2
4,6
54,24
lcos-C30
1
12
20
Tet-852
2
4,6
54,26
lcos-060
1
15
28
Tet-A4
2
I
I I
I
Xa
1 40 ' i
I
1 40 52 22,24
Although the data for Tet-C60 and Tet-D60 in the above table are identical, the two plaited polyhedra are not! Finally, before turning to the combinatorial treatment of plaited polyhedra, I should briefly mention fullerenes Cn in this context. These are usually represented as a spherical graph-structure (a cage) with pentagonal and hexagonal holes and n vertices, all of degree 3. If we take the dual graph of the cage, obtained by representing each hole by a vertex and joining two vertices if the corresponding holes have a common edge, then we have a finite graph with tri angular faces and vertices of degree 5 or 6, as considered earlier. On the one hand, let V be the number of vertices of this graph, E the number of its edges, and F the number of its faces. On the other hand, let Cn have H hexagonal holes (clearly it has twelve pentagonal holes). Then we have F = n, V = H + 12, 2E = 3F, and substitution in Euler's polyhe dral formula V - E + F = 2 gives 2(V - E + F) = 2 V - 3F + 2F = 2H + 24 - n = 4 or H = n - 10. Most fullerenes are, however, nonsymmetrical in our sense. Two exceptions are C60, the famous first fullerene, which corresponds to our Icos-B20 (Fig. 5), and the fullerene C540 depicted in the VRML Gallery of Fullerenes [6], which clearly corresponds to Icos-L260 (see the table below for the definition of L).
A beautiful example of a nonsymmetrical bucky ball was featured in an article [4] by Istvan Hargittai and was de picted on the cover of that issue of The Mathematical In telligencer. Further examples can be found in the VRML Gallery of Fullerenes [6] mentioned above. Combinatorial Treatment
I will now develop the methodology necessary for system atically enumerating all possible plaited polyhedra. We saw earlier, with the help of the truncated tetrahedron CZf (Fig. 15), that the octahedral plaited polyhedra need not be treated separately; they are, in fact, best regarded as tetra hedral plaited polyhedra. The same holds for the icosahe dral plaited polyhedra, as I will now show. All icosahedral plaited polyhedra clearly have the fol lowing property in common: if we regard their pentagonal holes only, they exhibit the structure of an icosahedron 'f6 (see Fig. 7). In the plaited polyhedra the triangular faces of 'f6 are either all A or all B or all C etc., and there is exactly one way to choose four of these triangular faces so that they are symmetrically distributed over 'f6; see Fig. 17a, in which the four chosen triangles of 'f6 are hatched (the triangle at the back is hatched more lightly, and the shaded triangles (including the cross-hatched one!) should be ignored for the moment). If we regard these as the triangular faces of the truncated tetrahedron CS, we have matched 'f6 combinatori ally with CZf and have thus included the icosahedral plaited polyhedra as a special case of the tetrahedral ones. In Figure 1 7a the four triangles on 'f6 making up one of the hexagonal faces of CZf are shaded. Furthermore, within this hexagonal area there is a triangle (cross-hatched in Fig. 1 7a) involving every second vertex of its boundary. There are four such triangles, one in each of the four hexagonal areas, and Figure 1 7b depicts two of them on CS. Thus we have now matched 'f6 and CZf completely with respect to edges as well. We therefore need only consider the tetrahedral case in detail. The key to the further de velopment is the truncated tetrahedon CZf depicted in Fig ure 17b. By the symmetry of our plaited polyhedra intro duced earlier, the triangles in the four hexagonal faces of ':J are all equilateral and congruent. In a given plaited poly hedron they are therefore either all A or all B or all C etc., but they are in general not the same as the set of four "cor-
t
Figure 1 7a.
Figure 1 7b.
VOLUME 26, NUMBER 3, 2004
63
ner" triangles. We saw an example of this earlier when dis cussing Oct-P16 (Fig. 13) whose graph is depicted in Fig ure 14. These four triangles can in fact also act as the "cor ners" of the plaited polyhedron, so there is in general more than one way of choosing four congruent triangles to act as the triangular faces of '5". How do we deal with this choice? We simply choose the set of smallest triangles for the triangular faces of '5", smallest meaning earliest in the series A, B, C, . . . . This has brought us a decisive step further: we have sub divided each of the four hexagonal faces of '5" into an equi lateral "central" triangle, which we will call {3, and three congruent "flanking" triangles, each of which we will call y. Finally, let us denote by a each of the four triangular faces of '5". We have thus combinatorially dissected the plaited polyhedron via the surface of '5" into four times a + {3+ 3y. Note that y is not in general equilateral, except of course in the case of an icosahedral plaited polyhedron. As a consequence we can represent a plaited polyhedron graphically by a much smaller graph, given by a + {3 + 3y and having a quarter of the area of the full graph (see Fig. 18, which depicts a choice of a+ {3+3y for Tet-A12 (Fig. 9), whose graph was given in Fig. 12c). This reduced graph is all that is required in order to plait a model: by symmetry it is simply repeated four times, fitting together as in Fig ure 17b. We could in fact also leave off two of the y's, and for large a or {3, these could be reduced as well! Moreover, we now have the basis for a systematic enu meration of all plaited polyhedra. The following steps re main to be dealt with, namely to 1. Consider all cases of a, {3 E {A, B, C, D, . . . } with {3 :::::: a, starting with the smallest a, namely A. 2. List systematically all ways of positioning {3 with respect to a, which will then also determine 'Y· 3. Derive a formula for the number of hexagons in the re sulting plaited polyhedron. The graphical representation of plaited polyhedra in volves graphs with triangular faces only; hence it will be natural to work with a hexagonal coordinate system as in Figure 19, where D is depicted: one vertex of D is at (0,0), another at (2, 1), and the third one then has to be at ( - 1,3). Thus, D is fully characterized by the coordinates (2, 1), or more briefly by the number pair 21. We can also very easily calculate the area :il(D) of D in unit triangles. I will do this for a general equilateral trian gle � given by pq, whose vertices thus are the points with the coordinates (0,0), (p,q), ( -q,p+q). Then, by vector al gebra, :il(�) is given by the vector product of the two 3vectors (p,q,O) and ( -q,p+q,O), i.e., by det
(!
q
p !q
Figure 1 8. Graph of Tet-A12 (Fig. 9) with choice of
10, s!l(A) 1, D 21, s!l(D) 7, G 31, s!l(G) 13, J 41, s!l(J) 21, M 42, s!l(M) 28,
A
=
=
=
=
=
=
=
=
=
=
B E H K N
=
=
=
=
=
1 1, s!l(B) 30, s!l(E) 40, s!l(H) 50, s!l(K) 51, s!l(N)
=
=
=
=
=
'!fi
=
4(:il(a) + :i1({3)
THE MATHEMATICAL INTELLIGENCER
I
L
=
=
=
=
20, s!l(C) 4, 22, s!l(F) 12, 32, s!l(l) 19, 33, s!l(L) 27, =
=
=
=
+ 3:il( y)).
Moreover, from Euler's polyhedral formula 'V - � + '!fi 2 and the relation 2� 3'!fi, we have 4 = 2('V - � + '!fi) = 2'V - 3;Ji + 2;Ji = 2'V - ;Je or 'V = + 2. Let 'iJe denote the number of hexagonal holes of this plaited polyhedron. Then 'iJe 'V - 12, and with the above formula for '!fi we obtain =
=
tg;
=
'iJe
=
2(:il(a)
+ :il(f3) + 3:il(y) - 5).
or
64
C
F
Let us leave the question of how to position {3 with re spect to a until later, and assume that we have a fixed choice of a, {3E {A, B, C, D, . . . } and a fixed position of {3 with respect to a. Then the corresponding 'Y is also de termined, and I will derive a formula for :il( y) later. Then, clearly, the number of triangular faces '!fi of the graph of the corresponding plaited polyhedron is given by its total area in unit triangles, namely
)
The first fourteen members of the series A, B, C, D, E, . . . together with their areas, listed in increasing size, are:
3, 9, 16, 25, 31.
a + {3 + 3-y.
( 0 , 0)
Figure 19. Hexagonal coordinate system.
This formula gives the number of hexagonal holes of the
{3 is chosen. So it can appear once, twice, or even four times
plaited polyhedron determined by a, {3, and
in the general list, but the latter case only happens once for
are known, and the formula for
y. .'il(a) and .'il(/3)
.'il(y) will be given later.
In special cases this formula takes very simple forms,
!cos series, where {3 y let X ab, then 'Jf 1 0(.'il(X) - 1) a =
e.g., for the
=
. . . }: b2 - 1 ), and as special cases we get b 0 � 'Jf 10(a2 - 1) b 1 'Jf 10a(a + 1) b = a - 1 'Jf 30a(a b a � 'Jf = 10(3a2 - 1). =
=
=
=
=
XE {A,B, C,D,
=
10(a2 + ab +
=
�
=
=
=
(Fig. 9). The graphs are identical, and in both of them
the A's are hatched diagonally. In the first graph the {3's are Cs and are hatched horizontally, and in the second they are
portant here: the associated truncated tetrahedron 1)
hexagonal faces. Thus the two {3's can be drawn into one and
the same reduced graph, as shown in Figure 22.
This is typ
ical of all tetrahedral plaited polyhedra with two possible {3's.
2 + ab + b2 -
10(a
?I is the
same in both cases, except for the way {3 is chosen in the
Remark
Note that the formula 'Jf
Tet-A60. Figure 2 1 shows how two different {3's arise for Tet
A12
D's and are hatched nearly vertically. One point is very im
=
�
plaited polyhedra with up to 60 hexagonal holes, namely for
To derive a formula for .'il( y), we must represent the pos
1) for the
!cos series can be written in the form 'Jf t Y(a2 + ab + b2 - 1 ) , where Y is the number of faces of an icosahedron. =
sible positions of
a
with respect to {3 systematically in our
be given by ab and {3 cd, and let {3 be fixed in position in such a way that the
hexagonal coordinate system. Let
a
The same formula is easily seen to be valid if we allow only
by
square or only triangular holes instead of pentagonal ones
x-axis goes through {3, as shown in Figure 23 for a
=
10 and
{3
=
earlier in Figure 20, along with the corresponding trian
balls with hexagonal and square holes, or with hexagonal
a
and triangular holes.
need not be considered, as it appears naturally as the
(see earlier
Remarks).
hedral series, so so
Y = 4.
Y=
In the first case we have an octa
8, and in the second a tetrahedral one,
This completely settles the symmetric plaited
D
mentioned to me that the formula
'Jf
10(a2 + ab + b2 -
=
1)
!cos series is contained in a paper [3] by Michael Gold
berg from 1937, where he studied the corresponding graphs in connection with the isoperimetric problem for polyhedra.
Another special case of the formula for 'JC is discussed
under
Further Deductions at the end of the article.
It remains to discuss the ways of positioning {3 with re spect to
a
ing areas
gles
y. But we should note that the a
third possible position first
in the case given by
a =
10 and {3
=
21 (see fourth reduced graph in Fig. 20), so only two posi tions are relevant here. This is so in general: there are three possible positions for
a,
but only the first two or the last
two are relevant in any particular case. To make precise the positions of a in the coordinate sys
tem which I have just introduced, let us pick the
y that is
spanned by an edge of a and an edge of {3; then its vertices
and also to obtain formulae for the correspond
.'il( y).
Let us assume that we have a certain choice of
{A, B, C,
(c,d), (c + d, -c). I depicted the three possible positions for
possible position of
In a recent conversation in Bonn, Professor Hirzebruch for the
12. More precisely, let the vertices of {3 be fixed at (0,0),
D, . . . } with {3 2:
a.
Then
a
a,
{3E
has a vertex in com
mon with {3, and if we look at the three reduced graphs in Figure 20 (ignoring the fourth one for the moment), then we see that there is a
pentagon
of triangles around this
common vertex, not a hexagon, because it represents a pen tagonal hole in the plaited polyhedron. Moreover, the three examples in Figure 20 show clearly that
a
can occupy sev
eral different positions within this pentagon with respect to {3, and the three congruent triangles
y are immediately
obvious in each case. The two different reduced graphs for
Tet-Al2
Tet -A24
Tet-AJ8 in Figure 20 will be elucidated below. •
Thus one gets all possible plaited polyhedra by consid ering all choices of
a,
{3E
{A, B, C,
and all possible positions of tating
a
a
D, . . . } with {3
2: a
with respect to {3 by ro
around its common vertex with {3.
As y is
then
fixed in each case, the reduced graph of the corre sponding plaited polyhedron is immediate.
An icosahedral plaited polyhedron is, of course, uniquely a alone. A given tetrahedral plaited polyhe
characterized by
dron has a unique
a,
but in general there can be one, two or,
rarely, even. four different {3's which characterize it, depend ing on how the hexagonal face of
?I is chosen, and on how
Figure 20. Reduced graphs.
Tet-Al8
VOLUME 26. NUMBER 3. 2004
65
Figure 22. Reduced graph for Tet-A12 with two alternative {3's.
Figure 21 . Two choices of {3 for Tet-A12 (Fig. 9).
a
Tet-A12
or
AQ ( 2 )
are the origin, the vertex ( c,d) of {3, and the vertex P of the first edge of a met by rotating anticlockwise from {3. We define a to be in Position (1) if P = (a, b), in Position (2) ifP = ( - b,a +b), and in Position (3) ifP = ( - a -b,a). Thus the third vertex of y has the following coordinates for po sitions (I)-(3) of a: (1) : (a,b),
(2) : ( - b,a + b),
(3) : ( - a -b,a).
As we saw earlier, of the three possible positions for a, only positions (1) and (2), or positions (2) and (3), are rel evant in any particular case. The area 31(y) is now immediately available by means of the determinant method used earlier. The three cases for the three positions are as follows: (1) : 31( y) = [ad - be[ , (2): 31(y) = ac + be + bd, (3) : 31( y) = ae + ad + bd.
From the geometry we also obtain the fact that position (1) is possible if !!_ > !!, i.e., be > ad; otherwise position (2) a
c
is the first. Any plaited polyhedron can now be given in the form af3(j), where a, {3E (A, B, C, D, . . } with f3 � a and (J) is the position of a with respect to {3, and the number 'de of its hexagonal holes can be calculated by means of the for mula given earlier. Just one small point needs to be men.
66
THE MATHEMATICAL INTELLIGENCER
b
Tet-A24
or
AQ ( 3 )
Figure 23. Reduced graphs in hexagonal coordinate system.
tioned: iffor example a = D = 21, then a = 12 is also a pos sible choice, for which we may write !1., and similarly for {3. We in fact used a = A = 10 and {3 !1 = 12 in Figures 20 and 23, as this is the simplest case having the proper ties we wanted to exhibit. As I mentioned earlier, different {3's can give rise to the same plaited polyhedron. The following table lists all plaited polyhedra for 'de ::::; 60, paying regard to this property: =
One {3:
Icos-AO : AA(2), Tet-A36 : AG(2), Tet-B32 : BE(l), Tet-C46 : CF(2),
Tet-A4 : AB(2), Tet-A58 : AJ(2), Tet-B50 : BE(2), Tet-C48 : C[2(3),
Tet-AlO : AB(3), Icos-B20 : BB(2), Tet-C24 : C[2(2), Icos-D60 : DD(2),
Tet-A18 : AD(2), Tet-B28 : BC(2), Icos-C30 : CC(2), Oct-P52 : EQ.(2),
Two [3's: Tet-A12 : AC(2), A[2(2), Tet-A40 : AF(3), AJ_(2), Oct-P1 6 : BC(l), BD(l), Tet-B52 : BQ_(2), BH(l), Tet-C52 : CE(2), CJ_(2), Oct-P40 : DE(l), [lF(l), Tet-D60 : [lG(l), DH(l).
Tet-A24 : A[2(3), A Q.(2), Tet-A42 : AQ.(3), Al(2), Tet-B34 : B[2(2), BG(l), Tet-B56 : BF(2), BJ(l), Tet-C60 : CG(2), C[(2), Oct-Q48 : D[2(2), D/(1),
Tet-A28 : AE(2), AF(2), Tet-A48 : AH(2), A/(2), Tet-B40 : BD(2), B/(1), Tet-C36 : CD(2), CQ_(2), Oct-P36 : [lD(l), DG(l), Oct-R58 : [lE(l), DJ(l),
Four {3's:
Tet-A60 : A/(3), Ati(2), Al(3), AM(2).
I close with an example illustrating the procedure for find ing all plaited polyhedra of a certain size. Example
When we fill all the gaps in the systematic list with 'iff ::::: 60, many members with 'iff > 60 appear; e.g., there are five with 88 hexagonal holes, namely CJ(2) , [2F(2), DJ_(2) and its mir ror image [lJ(I ), and EG(2). How many of these give dif ferent plaited polyhedra? For a start, there are at least three different ones, as a = C, D, or E, and one can verify that there are in fact four different ones among them, which I list with their alternative classification: CJ(2) and CL(2), [2F(2) and D(52)(1), DJ_(2) and [lJ(I), EG(2) and E!Y..( 2). Three of these are not in our list, namely CL(2), D(52)(1) and E!Y..( 2). How many more exist? A brief algebraic search using 'iff soon reveals just one more, namely HJ_(2), with just one {3. Thus there are five different plaited polyhedras with 88 hexagonal holes, and from their reduced graphs we can im mediately begin to plait them. Figure 24 shows the strongly octahedral HJ_(2) at the halfway point and completed. Further Deductions
Note that the second and the third plaited polyhedras with 'iff = 88 both have a = D, i.e., they have the same a and the same 'iff , a phenomenon we have met only once before, namely with Tet-D60 and Icos-D60. We also see, by checking its reduced graph, that the third plaited polyhedron with 'iff 88 has a regular 2!, i.e. , the hexagonal faces of 2J are regular hexagons. This is the fourth plaited polyhedron with this property: the other three have a = A, B, and C and are Tet-A4, Tet-B32, and Tet-C46. The general formula for such plaited polyhedra is very easily derived. Let a = ab, then from the regularity of the hexagon we see that {3 is given by (a,b) + ( - b,a + b) = (a - b, a + 2b). We immediately derive .'11 ({3) = 3.
Figure 24. The strongly octahedral ball at the halfway point (top) and
completed.
VOLUME 26, NUMBER 3, 2004
67
b(a - b) + b(a +2b) sl(a). This yields 'iff = 2{ 7(a2 + ab + b2) - 5 l . Alternatively, the area of the regular hexagon is 6sl(a), so that sl(a) + .9'1({3) + 3sl( y) sl(a) + sl(hexagon) 7sl(a) . =
A U T H O R
=
=
REFERENCES
[ 1 ] Carl F. Cori, "The Call of Science,"
Annual Review of Biochemistry
38 (1 969), 1 -2 1 . [2] Paulus Gerdes, "Molecular Modeling of Fullerenes with Hexastrips," The Mathematical lntelligencer
21 (1 999), no. 1 , 6-1 2 .
[3] Michael Goldberg, " A class o f multi-symmetric polyhedra," Math. J.
T6hoku
43 (1 937), 1 04-1 08.
[4] Istvan Hargittai, "Fullerene Geometry under the Lion's Paw," Mathematical lntelligencer
[5] Jean J. Pedersen, "Visualising Parallel Divisions of Space, " matical Gazette
The
RABE-RUDIGER VON RANDOW
1 7 (1 995), no. 3, 34-36.
Forschungsinstitut fOr Diskrete Mathematik Mathe
Universitat Bonn, LennestraBe 2
62 (1 978), 250-262.
D-531 1 3 Bonn, Germany
[6] http://jcrystal.com/steffenweber/gallery/Fullerenes/Fullerenes. html
e-mail: [email protected]
NOTE ADDED IN PROOF
The data of the systematic classification table can be obtained purely algebraically, along with the determination of the common edge of any
two of the hexagonal faces of '3 and of the alternative {3. I have pro grammed the formulae and obtained all the plaited polyhedra ior �
=
0, . . . . . , 200; there are 279. Many of these are members of a par
ticular series, like the icosahedral series or those with regular '3 men
Rabe von Randow was born in Shanghai and lived there until the age of fifteen when his family emigrated to New Zealand . He stud
ied mathematics, physics, and chemistry at Auckland University
and graduated wijh an M.Sc. in mathematics. He then went to
Germany and had the gOOd fortune to beoome
one
of the first
doctoral students of Professor F. Hirzebruch in Bonn. After com pleting his Ph.D. in algebraic topology in 1 962, he held teaching
tioned above. The formula for the latter series is a special case of a much
posts at Otago University,
more general formula for many plaited polyhedra with one {3. Also the
in Tucson, the Universijy of Durham, England, and the University
the University of Arizona
New Zealand,
interesting new cases. I will be happy to provide further information and
of Cologne . Since 1 972 he has been in Bonn. In recent years he
to supply reduced graphs for any of the above plaited polyhedra.
again. He is married and has two school-age children.
fact that .stl(53)
=
.st1(70)
=
49 and .si1(65)
.stl(9 1 )
=
=
91 gives rise to very
has widened his interests to include physics and chemistry once
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'{ I4.1) -
2
Yr !
\
, )l + I
�
� J
n \ -\\') druaya-W�t> ndtUOtlffi to apply the ' f the co � • e'_ 'l' o estun�t
l�[email protected]§r.Sih¥11@1§4fh,j,i§.id
M i chael Kleber and Ravi Vaki l , Ed itors
a
Solution for Impl icit Proof (published in vol . 2 6 , n o . 2 , p. 68) Kevin Wald
a
This column is a place for those bits of contagious mathematics that travel from person to person in the
b
community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
b-a a(a/b) =
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil,
Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 1 25, USA e-mail: [email protected]
56A Cedar Street, Somerville, MA 02143 USA e-mail: [email protected]
© 2004 SPRINGER-VERLAG NEW YORK, LLC, VOLUME 24. NUMBER 3, 2004
69
FRANK MORGAN
Streams of Cy 1 nd nca Water soap bubble likes to be round because a round sphere minimizes area for a given volume (Fig. 1). Salt crystals are cubes for a similar reason, but the energy they want to minimize depends on orientation with respect to the underlying crystal lattice, so that the three axis plane directions are much cheaper than other directions, and these are the ones that occur in the ideal cubical shape, called the Wulff shape (see [T] and [M3, Chapt. 10]). Since a round sphere minimizes surface area A for given volume V, for every smooth variation (perturbation param etrized by time), dA/dV must be the same, or you could pick up volume where dA/dV were cheap and return it where dA!dV were more expensive, reducing area. Geometrically,
A theorem of A. D. Alexandrov [A] says that the unit sphere is the only compact (connected, embedded) surface of unit mean curvature. There are however other noncom pact surfaces of unit mean curvature, even among surfaces of revolution, such as the celebrated unduloids of Delau nay ([D], see [E]), as in Figure 2. Where they are widest, the longitudinal and cross-sectional curvatures might be roughly the same. Where they are thinnest, the cross
this constant is the sum of the principal cUIVatures, twice
sectional curvature is very positive, whereas the longitudi
the mean curvature. A unit sphere has principal curvatures both 1 and hence mean curvature 1 at all points.
nal curvature is negative. The mean remains 1. A special case is the cylinder.
Figure 1 . A round soap bubble minimizes area (John M. Sullivan). A cubical salt crystal minimizes a different en ergy, which favors the axis directions. Both are exam ples of Wulff shapes.
70
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
Figure 2. Unduloids have constant curvature. They are unstable; a stream of water breaks up into water droplets
(Moments of Vision
by Harold E. Edgerton and James R. Killian, Jr., MIT Press, 1 979).
Unlike the sphere, these unduloids are unstable. A stream of water breaks up into water droplets. There is a similar story for the cylindrical energy E, which has unit cost in the horizontal and all vertical direc tions, the directions which occur on a cylindrical can, and is much more expensive in other directions. As a result, the cheapest way to enclose a given volume, the Wulff shape, is a cylindrical can. For the unit cylinder of height 2, the "mean curvature" 1 dE 2 dV is constantly 1 . For example, raising the top at unit speed, dE!dt = 27T, dV/dt = 7T, and dE!dV = 2. Pushing out the sides increases the area of top, bottom, and sides: dE/dt = 2 7T + 2 7T + 271{2) = 81r, dV/dt = 47T, and dE/dV = 2. Actually for this cylindrical energy it is not quite true that all smooth variations of the Wulff shape have dE/dV = 2; introducing expensive directions sometimes yields dE!dV > 2. In any case, the derivative of E - 2V is always nonnegative, and this is the official definition of unit mean curvature. Another difference from ordinary area is that unit mean curvature is no longer a local, infinitesimal con dition: one has to consider variations of large support which move whole faces. The recent news [Ml] is that there are cylindrical analogs of the unduloid (see Fig. 3), which consist of chains of similar cylinders of alternating radii R, r and heights H R
h
r
2 R-r
with R + r = 1 . For example, raise the lid of one of the large cylinders at unit speed. Then dE!dt = 2 7TR - 27rr' while dV!dt = 1rR2 - �. so that
Figure 4. Do streams of liquid crystals break up into cylindrical
droplets? (Cylinder from http://mathworid.wolfram.com/Cylinder.html.)
dE dV
2 7T(R 7T(R2
-
-
r) r2)
=
2 =2 R+r ·
Like the classical unduloids, these cylindrical unduloids are unstable. I wonder whether in Nature one can find streams of liquid crystals breaking up into cylindrical droplets? (See Fig. 4.) Perhaps not. In contrast to the classical Rayleigh insta bility of a long cylinder, the infinite cylinder is stable for cylindrical energy, as I learned from Jean Taylor and John Cahn (see [C]). Indeed, for nonzero normal variations, the first variation is strictly positive [Ml, Rmk. 3.3]. This cylin drical stability may explain the occurrence in crystals of long thin fibers. To prove that these cylindrical unduloids have unit mean curvature for all variations, one may restrict attention to surfaces of revolution, which reduces the proof to a ques tion about curves in the plane. Using the special nature of the energy, one can show that the curves must consist of horizontal and vertical pieces. Now the problem reduces to discrete geometry, the sort of calculations we've already made. There is also a family of nonembedded cylindrical "nodoids" of revolution [Ml]. Other energies admit gener alizations of the families of compact constant-mean-curva ture immersions of Kapouleas [M2].
Figure 3. These new cylindrical unduloids have constant mean curvature for cylindrical energy.
VOLUME 26, NUMBER 3, 2004
71
Polthier and Rossman [PRJ give other examples within certain classes of polyhedral surfaces for the isotropic Eu clidean norm. During the refereeing process, I learned from Jean Tay lor that she had worked out such results and mentioned them in talks. Also, I had not seen the stability of the cylinder for cylindrical energy, which she learned from John Cahn [C].
A U T H O R
REFERENCES
[A] A D. Alexandrov, Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ. Mat. Mekh. A stronom.
1 3 (1 958), 5-8; AMS
Trans!. 21 (1 962), 4 1 2-4 1 6. [C] J. W. Cahn, Stability of rods with anisotropic surface free energy, Scripta Metallurgica
FRANK MORGAN Department of Mathematics
1 3 (1 979), 1 069-1 07 1 .
[D] C. Delaunay, Sur Ia surface de revolution dont Ia courvure moyenne
Williams College
est constante, J. Math. Pures Appl. 6 ( 1 941 ), 309-320.
Williamstown, MA
[E] James Eells, The surfaces of Delaunay, Math. lntelligencer 9, no. 1
USA
(1 987), 53-57.
01 267
e-mail : Frank. [email protected]
(M 1 ] Frank Morgan, Cylindrical surfaces of Delaunay, preprint (2003). [M2] Frank Morgan, Hexagonal surfaces of Kapouleas, Pacific J. Math., to appear; arXive.org.
(M3] Frank Morgan, Riemannian Geometry: a Beginner's Guide, A K. Peters, Ltd, 1 998.
[PR] Konrad Polthier and Wayne Rossman, Discrete constant mean curvature surfaces and their index, J. Reine Angew. Math. 549 (2002), 47-77. [T] Jean Taylor, Crystalline variational problems, Bull. AMS 84 (1 978),
Among Frank Morgan's books are Riemannian Geometry (cited in this article), Geometric Measure Theory, and
and a column he still does at the
.
.
The
M at hChat .org .
He is director of
NSF Undergraduate Research Project at Williams College.
After he advised a
torus" (Exp. Math.
paper on "Double bubbles in the three
1 2), the authors commissioned a stained
glass window of one of hangs in his office
568-588.
.
Math Chat Book, based on a live call-in TV show he once did
their beautiful illustrations, and it now
window
(see the photo).
Mathematics and Cu ltu re Mathematics and Culture Michele Emmer, Univer ity of Rome 'La
apienza', Italy (Ed.)
This book stresses the trong links between mathematics and culture, as mathematics l i n k theater, literature, architecture, art, cinema, medicine but also dance, cartoon and mu ic. The articles i ntroduced here are meant ro be interesting and amusing starting points ro research the trong connection be[\veen scientific and literary culture. Thi collection gathers contribution from cinema and theatre director , musicians, architect , historians, physicians, expert in computer graphics and writer . In doing so, it highlights the cultural and formative character of mathematics, its educational value. Bur also its imaginative aspect: it is mathematics that is rhe creative force behind the screenplay of film such as A Beautiful Mind, theater play like Proof, musicals like Fermat's Last Tango, succes ful
books such as Simon Singh's Fermat's Last Theorem or Magnus Enzensberger's The
umber Devil.
Mathematics, Art, Technology and Cinema
Michele Emmer, Univer ity of Rome 'La Sapienza', Italy; and
Mirella Manarcsi, Univer ity of Bologna, Italy (Eds.)
This book i about mathematics. But al o abour
art, technology and i mages. And above all, abour cinema, which i n the past year , rogether with
theater, has d iscovered mathematics and mathe maticians. The authors argue rhar rhe discussion abour the d i fference be[\vecn the so-called two culture of cience and humanism is a thing of the pa t.
hey hold rhat both cultures are truly
linked through ideas and creativity, nor only through technology. In doing so, they succeed in reaching out to non-mathematician , and those who are not particularly fond of mathematics. An insightful book for mathematicians, film lover , those who feel pa si nate about i mages, and those with a que rioning mind. 2003/242 PP./HARDCOVER/599.00/ISBN 3-540-00601 -X
2004/352 PP., 54 I LLUS./HARDCOVER/S59.95/ISBN 3-540-01 770-4
' Springer '
www.springer-ny.com
72
THE MATHEMATICAL INTELLIGENCER
EASY WAYS TO ORDER: CALL Toll-Free 1 -800-SPRI GER • WEB www.springcr-ny.com E-MAIL orders@'springer·ny.com • WRITE to Springer·Verlag New York, Inc., Order Dept. 780-, PO Box 2485, ecaucus, NJ 07096-2485 VISIT your local scientific bookstore or urge )'Our librarian ro order for your dep3rtment. Price ubjccr to change wirhom notice. Please mention 57805 when ordering to guarantee listed prices.
Promotion 57805
la§lh§l.lfj
Osmo Pekonen , Ed itor
I
M. C. Escher's Legacy: A Centennial Celebration by Doris Schattschneider and Michele Emmer (Editors) NEW YORK, SPRINGER-VERLAG, US
$99.00,
ISBN
3-540-42458-X
2003 458 pp.,
REVIEWED BY HELMER ASLAKSEN
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
D
o we need another Escher book? Ever since Escher was "discov ered" by the mathematical community in the early 1950s, he has been the ar chetypal mathematical artist. His work
quality books, and if you see the names of Doris Schattschneider and Michele Emmer on the cover, you can expect top quality! Schattschneider's article [5] on wallpaper groups is a classic, and is, together with the book by Griin baum and Shephard [3] , required read ing for anybody interested in tilings and patterns. Schattschneider has writ ten an excellent book on Escher [6] , and Emmer has edited another confer ence proceeding [ 1 ] and directed a video on Escher [2]. What makes an Escher book good or bad? Some 50 years ago the novelty of his work was enough to excite peo ple, but now I believe that the mathe-
H ow d i d Escher do it? Did he th i n k of h is art i n a mathematical way? H ow d i d he feel about the way mathemat icians i nterpreted h i s art?
Column Editor: Osmo Pekonen, Agora Center, University of Jyvaskyla, Jyvaskyla,
40351 Finland e-mail: [email protected]
can b e found on the walls of many mathematics departments, and when mathematicians need to demonstrate a link between mathematics and art, they usually use Escher pictures. Is this healthy? Some mathemati cians feel that the ubiquitousness of Escher prevents other aspects of math ematical art from getting the attention they deserve. It is also important to re alize that arts specialists do not share our fascination with Escher. Many of them simply don't consider him to be an artist! When I was mathematical con sultant for the exhibition "Art Figures. Mathematics in Art" at the Singapore Art Museum, I had to adhere to a strict no-Escher policy. Our preoccupation with Escher may hinder us when reach ing out to the arts community. If you say that Escher is your favorite artist, most people will probably think that you have not looked at a lot of art. So, do we need another Escher book? My answer is simply that no mat ter the topic, there is always room for
© 2004
matical community expects more in sight. Mathematicians look for clear, enlightening explanations, but much of what is written about Escher fails to clearly connect his work with mathe matics. If it is mathematical art, there has to be some mathematics in it! One crucial aspect of the study of mathematics and art is to compare the way mathematicians and artists think of art. How did Escher do it? Did he think of his art in a mathematical way? How did he feel about the way mathe maticians interpreted his art? Many mathematicians seem to be so caught up with the mathematical viewpoint that they have a hard time realizing that most people don't think the way they do! In my opinion, a good Escher book is a book that manages to link these two aspects both by interpreting his work mathematically and by explain ing his own approach. Schattschnei der's earlier book [6] is perhaps the best example of this.
SPRINGER-VERLAG NEW YORK, LLC, VOLUME 26, NUMBER 3, 2004
73
The current book is the proceedings of the Escher Centennial Congress held in Rome and Ravello in 1998. I must confess that I'm generally not a big fan of conference proceedings. How many proceedings from confer ences that you did not attend do you have on your bookshelf? There's a big difference between giving a great lec ture and writing a great article. Most of the articles are brief, and do not go into detail or explain things carefully. The wide range of speakers and topics en sures that there will be something for everybody, but that also makes the book a bit scattered. The volume is really three books in one. Part One, "Escher's World," con tains a wide range of articles which I believe will appeal primarily to hard core Escher fans. I personally loved Veldhuysen's retracing of Escher's path in Italy; the article by Schattschneider and Hoolist about how C.v.S. Roo sevelt, a grandson of Theodore Roo sevelt, helped Escher conquer America; and Hofstadter's description of his ex perience with Escher's work. But if you are a more casual Escher fan, you may want to skip to Part Two, "Escher's Artistic Legacy." Some peo ple may enjoy reading about artists who have been inspired by Escher, but there are also several articles with a clear mathematical angle. I believe that most mathematicians will find a lot of interesting material here. I was most attracted by the wonderful article by Rice about pentagonal tilings, the de scription of anamorphic art by Houle, and Orosz's discussion of mirrors and perspective. Part Three is called "Escher's Sci entific and Educational Legacy," and it's a real gold mine! Coxeter continues his study of hyperbolic trigonometry in Escher's work, Lee writes about the TesselMania software, and Eisenstein and Loeb discuss notations for sym metry groups. And these are but three of the 14 articles in Part Three. I en joyed almost every one of them, and I believe that even the most jaded math ematician will find something of inter est in this part. Unfortunately, the book is both thick and expensive. Would it have
74
THE MATHEMATICAL INTELLIGENCER
been possible to have moved more ma terial to the CD? How about selecting half of the articles for the book and putting the rest on the CD? It would not have made the editors popular among the contributors, but I think it would have made the book more exciting and manageable for the reader. How many readers look at the color plates in the book while reading it? Would it have been better to move the color plates to the CD? That would probably have reduced the price. Does it make sense to have both a CD and color plates? I'm also unclear about the selection of the color plates. Are they included because the authors wanted to refer to them? Or was the intention to select the most attractive plates? I personally felt that there were other pictures more deserving of color re productions. The CD is very nice and very well or ganized. The CD symbol in the text was a good idea, but it would have been even better if it had indicated exactly what was on the CD. Just the text of the arti cle, or something really exciting? When using the CD, I was looking for "value added" material. I would for instance have liked a listing of the movies and an imations. I didn't find it, but I advise peo ple who are adventurous to bypass the menu, and instead go to the "Daten" di rectory, and open all the subdirectories. Just do it; you will thank me! For some reason the table of con tents appears after the two prefaces, and I kept having to tum pages to find it. It's a minor issue, but Springer-Ver lag has traditionally set the standard for quality production of mathematics books, so I always expect the best when I open a Springer book. I enjoyed the book, and I'm confi dent that any mathematician will find something of interest in it. REFERENCES
[ 1 ] H .S.M. Coxeter, M. Emmer, R. Penrose and M.O. Teuber, M.C. Escher: Art and Science North-Holland, 1 986. [2] Michele Emmer, The Fantastic World of M.C.
Escher,
VHS tape, Acorn Media,
2000. [3] Branko Grunbaum and G.C. Shephard, Ti/ings and Patterns,
W. H. Freeman, 1 987.
[4] Caroline H. MacGillavry, Fantasy & Sym metry:
The Periodic Drawings of M. C.
Escher,
Harry N. Abrams, 1 976.
[5] Doris Schattschneider, The Plane Symme try Groups: Their Recognition and Notation, American Mathematical Monthly
439-450. [6]
--
85 (1 978),
, Visions of Symmetry: Notebooks,
Periodic Drawings, and Related Work of M.C. Escher,
1 990.
W. H. Freeman & Company,
Department of Mathematics National University of Singapore Singapore 1 1 7543 Singapore e-mail: [email protected]
Discrete Convex Analysis Kazuo Murata SIAM, 2003 US $1 1 1 .00. ISBN 0-89871 -540-7 389 pp.
xxii +
REVIEWED BY JENS VYGEN
T
his monograph brings together continuous optimization and dis crete optimization. More precisely, it establishes a new theory of nonlinear discrete optimization by importing techniques of continuous optimization, in particular convex analysis. On the other hand, viewed from the continu ous side, it proposes a theory of con vex functions with some combinator ial properties. The theory has been developed mainly by the author since 1995. It provides many new insights and has already led to interesting ap plications. While discrete optimization prob lems are generally NP-hard, several very important types of problems have been solved well, in the sense of opti mality conditions, duality relations, and-most important-efficient poly nomial-time algorithms. This area, known as combinatorial optimiza tion, has been very fruitful in the last fifty years. However, many results in combina torial optimization are problem-spe cific. A special combinatorial structure
is analyzed, which then leads to algo rithms for optimizing (usually linear) objective functions over this structure. Examples are network flow algo rithms, algorithms for matching short est paths or optimum spanning trees in graphs. In some cases, the reason why some structures are easy to deal with and others are not could be explained by the theory of matroids and sub modular functions: we know how to optimize linear objective functions over discrete sets defined by submod ular functions. Submodular functions, i.e., real functions! defined on the power set of a finite set U satisfying f(X U Y) + f(X n Y) s j(X) + f(Y) for all X, Y � U,
f(x V y) + f(x 1\ y) s f(x) + f(y) for all x, y E ( 0, 1 )" , where V and 1\ denote component-wise maximum and mini mum, respectively. They are now gen eralized as follows: A functionj : zn � Ill U (x) is called L-convex if there ex ists an r E Ill with .f(x + (1, . . . , 1)) = j(x) + r for all X E zn, and j(x V y) + f(x 1\ y) S f(x) + f(y) for all X, y E 2". The second class of functions consid ered are also natural discrete ana logues of convex functions: A function f : lZ" � Ill U ( x ) is called M-convex if for all :r, y E zn withj(x)J(y) < X and for each index i with X; > Y; there is an index j with :x:J < yj and f(.r - P; + eJ) + f(y + e; - ej) s f(x) + f(y ) , where e ; denotes the ith unit vector. L-
ADVE N T U R E S I N MAT H E MAT I C S
It general izes the theory of
(poly)matroids and submodu lar fu n ctions, but it can also be viewed as a theory of convex fu nctions with some com bi n atorial properties . have nice properties, similar to convex functions in the continuous world. In deed, Lovasz proved the important characterization that a discrete set function is submodular if and only if its so-called Lovasz extension (sometimes also called Choquet integral) is convex. Frank showed a sandwich theorem: just as for a concave function f and a convex function h with .f s h there is always a linear function g withf s g s h, the same holds if we replace convex by submodular and concave by super modular (fis called supermodular if -.f is submodular). Moreover the linear function g can be chosen integer-val ued iff and h are. This can be regarded as equivalent to an earlier theorem by Edmonds. A discrete analogue of Fenchel duality was proved by Fu jishige, who also wrote the standard reference for sub modular functions (at least up to the appearance of Murata's book). If we identify subsets of an n-ele ment ground set by their characteristic vectors, submodular functions can be viewed as functions .f : ( 0, 1 )" � Ill with
convex and M-convex functions are conjugate to each other by a discrete analogue of the Legendre-Fenchel transformation. Although there were a few isolated results on nonlinear discrete optimiza tion problems, there was no unified theory. Discrete convex analysis pro vides such a theory. L-convex and M convex functions are the main subject. One can view it from two sides: it gen eralizes the theory of (poly)matroids and submodular functions, but it can also be viewed as a theory of convex functions with some combinatorial properties .
This book provides a theory gener alizing many important results and yielding new insights on them. It will be interesting for researchers in con tinuous optimization, in particular con vex analysis, as well as for anybody in terested in discrete optimization. The book is exceptionally well con ceived. It is completely self-contained and written very carefully. There are only very few monographs of such outstanding quality. Although the ma-
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VOLUME 26. NUMBER 3, 2004
75
terial is quite difficult and the book contains many deep results, it can be read and understood by nonspecial ists and can be used for advanced graduate courses. The author invested a lot of effort in the excellent intro ductory chapter to acquaint the reader with the theory to come and to create interest in the book. Consequently, this book should be read from the be ginning! After the introductory part, the au thor reviews fundamental concepts. Then he introduces M-convex sets, L convex sets, M-convex functions, and L-convex functions in detail, generaliz ing known concepts. A central chapter deals with conjugacy and duality. The book then continues with applications to (nonlinear) network flows, minimiz ing M-convex and L-convex functions (related to minimizing submodular functions, where the author reviews re cent breakthroughs) and generaliza tions of submodular flows. The con cluding chapters contain applications to economics (equilibria theory) and engineering (systems analysis). Rela tions to matrix theory and electrical networks may also be interesting for many readers. The material has not appeared in any English book before, parts have been published in a Japanese book by the same author, and in journal papers, mainly by Murota himself, some with Japanese colleagues. This important theory is now accessible, which will lead to a better understanding of the subject. No mathematical library should be without this book. Everyone interested in optimization should consider it, but also researchers from related fields will benefit from reading (parts of) Murata's book. It needs some time to understand the beauty of the theory, but it is worth spending this time as it leads to a better understanding of clas sical and new concepts. I recommend the book warmly. Research Institute for Discrete Mathematics, University of Bonn Lennestr. 2 D-531 1 3 Bonn, Germany e-mail: [email protected]
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THE MATHEMATICAL INTELLIGENCER
KdV & KAM Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Thomas Kappeler and
Jurgen Poschel BERLIN, HEIDELBERG, SPRINGER-VERLAG 2003 279 pp.
€99.95, ISBN: 3-540-02234-1
REVIEWED BY BENOiT GREBERT
T
he title may appear intriguing for non-specialists: what does this jux taposition of two acronyms, "KdV" and "KAM," represent? The Korteweg-de Vries (KdV) equa tion is perhaps the most famous and most studied nonlinear partial differ ential equation (PDE); indeed, this equation is often used as a model to test a new theory: if it works for KdV, one can believe it will work for most PDEs in one space dimension. Kolmogorov-Amold-Moser The (KAM) theorem, on the other hand, is certainly one of the most celebrated mathematical results of the latter half of the twentieth century. Announced in 1954 by A. N. Kolmogorov at the ICM in Amsterdam (and then fleshed out, extended, and applied in numerous contexts by V. Arnold and J. Moser), this theorem put an end to the ergodic hypothesis by proving that most of the invariant tori of an integrable finite-di mensional Hamiltonian system persist under small Hamiltonian perturba tions, and thus the perturbed systems admit non-trivial invariant sets of pos itive measure. This book brings together these two aspects: after studying the complete in tegrability of KdV as an infinite-dimen sional Hamiltonian system, the authors apply a suitable generalization of KAM techniques in infinite dimensions to perturbations of the KdV equation. As a principal result, they prove that the majority of finite-dimensional invari ant tori of KdV persist under small reg ular Hamiltonian perturbations, and, in particular, that such perturbed equa tions admit large families of quasiperi odic solutions.
In fact, during the last fifteen years, perturbation theory of integrable par tial differential equations has been quite extensively studied, and remark able results have been obtained. The existence of quasiperiodic solutions has been proved by essentially two dif ferent approaches: the first one, pio neered by S. Kuksin, consists in a suit able extension of KAM techniques to the infinite-dimensional case; the sec ond one, pioneered by W. Craig and E. Wayne, uses a Lyapunov-Schmidt de composition scheme. Until recently, these results were accessible only by consulting the original papers, which are very arduous to read. However, a few years ago, both W. Craig [ 1 ] and S. Kuksin [2] almost simultaneously pub lished more pedagogical monographs on this difficult and technical subject. In my opinion, the longer gestation of this new monograph has brought it to maturity. This book has been awaited for many years by the Hamil tonian PDEs community, but the au thors preferred to take the time neces sary to attain complete mastery of the concepts and results, of their presen tation and of their proofs. The result was worth waiting for. Let me survey the contents of the book. That KdV admits a complete set of integrals in involution is a well known result of P. Lax. In the first part of this book, Kappeler and Poschel go much further, constructing global Birk hoff coordinates for KdV with periodic conditions. The (long) proof uses the angles in troduced by Dubrovin (whose con struction is very clearly explained in the introduction of chapter 3) and the actions introduced by Flaschka and McLaughlin. The cartesian coordinates associated with these action-angle vari ables define the Birkhoff coordinates. The difficulty consists, then, in proving that these coordinates are globally de fined, hi-analytic, and canonical. It is important to insist on the glob ality of the Birkhoff normal form they provide for KdV. This means that, in the new variables, KdV is globally re duced to a very simple model, namely an infinite system of coupled harmonic oscillators, in which only the frequen-
cies contain the information specific to the KdV dynamics. In particular, it is evident that every solution of the KdV equation with periodic boundary con ditions is almost periodic, and that the phase space is foliated by invariant tori generically of infinite dimensions. Let us notice that this also repre sents a new feature compared with the book of S. Kuksin mentioned above: there, the author uses a "local" normal form theory, in the sense that it re mains valid only in a neighborhood of a finite-dimensional torus. Actually the globality of the normal form simplifies the KAM procedure that is imple mented later on. With the help of this global Birkhoff normal form, the authors consider in chapter 4 small Hamiltonian perturba tions of the KdV equation and its hier archy. They prove that most of the fi nite-dimensional invariant tori survive such small and regular perturbations. Unfortunately, only finite-dimen sional tori can be considered by these techniques. As one can expect, the nu merous constants intervening in the classical KAM procedure hardly de pend on the dimension of the phase space. Roughly speaking, considering perturbations of a finite-dimensional torus, we can expect to recover a part of the finite-dimensional context (the real proof is, however, much more sub tle, the nonlinearity mixing all the modes). The KdV equation has other hidden subtleties: because the Poisson struc ture, which allows writing KdV as a Hamiltonian system, contains the de rivation operator, when you perturb the Hamiltonian function you generate an unbounded vector field. This difficulty was solved by S. Kuksin by the fabulous "Kuksin Lemma," which allows one to solve a homological equation where the normal forms depend on the angles. The proof of this lemma represents a tech nical tour de force, and an entire chap ter of the book is devoted to it. When this problem is settled, the proof of the abstract KAM theorem in infinite dimensions remains technical (as any proof of a KAM result must be), but the authors make the effort to sep arate the heart of the proof from all the
technicalities intervening in its imple mentation. Much of the interest of this book lies in its form of exposition. The first chap ter consists of an illuminating overview of approximately 20 pages whose read ing conveys the feeling that all these very new theories are already well-under stood and digested. A second chapter is devoted to the classical background in finite dimensions. The presentation is again very clear and one could recom mend it to anyone who would like to penetrate the world of integrable Hamil tonian systems and KAM theory in finite dimensions. Each chapter contains its own introduction in such a way that it
The fabu lous " Kuksin Lem ma" allows one to solve a homolog ical equation where the normal forms depend on the ang les. becomes self-contained and can be read separately. In the same spirit of peda gogy, the book contains numerous use ful appendices (in fact, the last 4 chap ters), from a short course on the concept of analyticity in infinite-dimensional space to a calculation of the Birkhoff normal form at order 6 for KdV, to a pre cis on symplectic formalism. In short, one feels that the authors wrote this book with the aim to make this difficult and technical subject accessible and in teresting for non-specialists. REFERENCES
[ 1 ] Walter Craig, Problemes de petits diviseurs dans les equations aux derivees partielles, Panoramas et syntheses 9, Paris: Societe Mathematique de France, 2000. [2] Serge B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathemat ics and its Applications 1 9. Oxford: Oxford
University Press, 2000.
Laboratoire de Mathematiques Jean Leray Universite de Nantes 44322 Nantes, France e-mail: [email protected]
Most Honourable Remembrance: the Life and Work of Thomas Bayes by Andrew I. Dale SPRINGER-VERLAG, NEW YORK, ISBN
2003. 667
0-387-00499-8
pp. US
$99
REVIEWED BY DAVID BELLHOUSE
F
or well over twenty years Professor Dale has been carrying out re search into the life and work of that frustratingly obscure mathematician and Nonconformist minister Thomas Bayes (ca. 1701-1761), famous today for the theorem in probability that bears his name. Information on Bayes is hard to find. There are no records of his birth; very few of his papers have survived; and he seems to have lived a fairly quiet life, mainly outside London, so that references to him in contem poraneous memoirs or diary entries are exceedingly rare. Dale's book is a compendium of source material on al most everything that has been col lected on Bayes; he also provides a de tailed and extensive commentary on much of the material, especially on Bayes's published work. In some of the places where the commentary is somewhat thin, Dale gives appropriate references to fuller discussions in the research literature. Here, finally, in one place, the Bayes enthusiast has ac cess not only to the famous essay on probability published in Philosophical Transactions of the Royal Society for 1 763, but also to his two anonymously written tracts, one defending Newton's fluxions against the attacks of George Berkeley by attempting to put the doc trine of fluxions on a firm philosophi cal foundation, and the other a theo logical tract exploring the motivating source of God's actions in the world, which Bayes argued was divine benev-
VOLUME 26, NUMBER 3, 2004
77
olence. Moreover, for the first time we have printed access to Bayes's note book, which is not in any public archive but rather in the vaults of a London insurance company. In addressing his own style in writ ing the book, Professor Dale takes a passage from Charles Dickens's Amer ican Notes: There are many passages in this book where I have been at some pains to resist the temptation of trou bling my readers with my own de ductions and conclusions; preferring that they should judge for them selves, from such premises as I have laid before them. Professor Dale has tried to minimize the historical interpretation of some of his source material. Despite the quotation, some interpretations have been given. Recently I wrote a biography of Thomas Bayes that will appear in the fulness of time in Statistical Science. (It was first written in 2001 to celebrate the ter centenary of Bayes's birth. History is ap parently not a fast-moving subject, so I guess it has been preempted some where in the publication queue by more "timely" material.) For the most part in my own biography of Bayes I agree with Professor Dale's interpretation of the source material on Bayes's life. Enu merating these areas does not, however, make for a lively review. It is where we differ that most readers will fmd inter esting. I will provide two instances. Our first point of difference is where Bayes was born. Professor Dale makes a strong argument for Bovingdon in Hertfordshire, about 25 miles from London. There are indications that this is so. For example, there is evidence of the involvement of Bayes's father Joshua in the Nonconformist chapel in Bovingdon as early as 1697. On the other hand, Joshua Bayes married a woman from London in 1700. Further, he obtained his marriage licence not from the local diocese but instead from the Archbishop of Canterbury's office. The licence was necessary to him as a Nonconformist, because neither would the banns of marriage be read nor would the ceremony take place in the
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THE MATHEMATICAL INTELLIGENCER
local church. My interpretation is that the Bayes family lived in London, where Thomas Bayes was likely born, and the father rode out weekly on horseback, or perhaps boated partway by river, to Bovingdon to take the ser vice. From the historical evidence cur rently available both interpretations are feasible. These two reasonable interpretations also underline the paucity of source material on Bayes. A second point of difference is per haps more interesting to those whose interests lean to the history of mathe matics. Most Bayes biographers at tribute his election to fellowship in the Royal Society to his anonymously writ ten tract defending Newton's fluxions.
Bayes ' s election cert ificate de scri bes h i m as "a Gentleman of known merit , wel l skil l ed i n Geometry . " Professor Dale seems to fall into agree ment with these biographers. The prob lem with the interpretation is that Bayes's election certificate describes him as "a Gentleman of known merit, well skilled in Geometry and all parts of Mathematical and Philosophical Learning," and the tract on fluxions has nothing to do with geometry. Until re cently there has been no other infor mation on which to base an interpre tation. A few years ago I found several manuscripts of Thomas Bayes in the Centre for Kentish Studies (described in Chapter 10 of Dale's book). One of the manuscripts was a transcription of a theorem that Philip Stanhope (2nd Earl of Stanhope and first sponsor for Bayes to fellowship in the Royal Soci ety) attributed to Bayes. The theorem was about factoring a more general form of a certain polynomial first con sidered by Roger Cotes. Bayes used geometrical arguments to obtain his
solution. I think it was the circulation of this manuscript among Stanhope's friends that led to Bayes's being dubbed "well skilled in Geometry," and that eased his subsequent election. I think we can agree to disagree. With respect to Dickens's dictum, in the places where Professor Dale has decided to follow Dickens and present the "facts" only, for my own taste I would prefer to have seen more inter pretation and synthesis. In view of the exhaustive treatment Professor Dale has given to source ma terial related to Thomas Bayes, what is left for this reviewer can only be to add new material found since the book's publication. I went back to the Centre for Kentish Studies in September 2003. I was trying to piece together Stan hope's circle of mathematical friends from his mathematical papers. Over a two-day visit I read through Stanhope's twenty-five or so folders of mathemati cal papers. I hesitated over calling up one folder well down the list labeled in the catalogue, "Annotations on sundry Works of Chaucer." It was the afternoon of the second day and I was getting tired. I decided, however, that I had bet ter be thorough and look at what Stan hope thought of Chaucer. It turned out that what was written on the folder was actually "Annotations on sundry Works of Chance." The very frrst manuscript was Stanhope's transcription of Bayes's solution to the problem of runs in the theory of probability; and the solution was not only incorrect, it was not even close! Consequently, what we now have for Bayes regarding his work in proba bility is a very insightful paper pub lished posthumously in 1 763 and a wildly incorrect jotting possibly written in the late 1740s. Professor Dale has brought together in one place a collection of diverse and dispersed archival material. He should be congratulated for his efforts. His book is a welcome addition to Bayesiana. Department of Statistical and Actuarial Sciences University of Western Ontario London, Ontario N6A 587 Canada e-mail: [email protected]
K1flrr1 .1$•h:i§l
Rob i n W i l s o n
The Philamath' s Alphabet-E E
cole polytechnique: An impor tant consequence of the French Revolution was the founding of the Ecole polytechnique in Paris. There the country's finest mathematicians, in cluding Monge, Laplace, Lagrange, and Cauchy, taught students destined to serve in both military and civilian ca pacities. The textbooks from the E cole polytechnique were later widely used in France and the United States.
I
Education: Many stamps feature the teaching of elementary mathematics arithmetic, algebra, geometry, weights and measures, and computing. This stamp, issued for the International Year of the Child (1979), illustrates the teaching of the geometry of a circle. Egyptian accotmtants: Our knowl edge of Egyptian mathematics is scanty, deriving mainly (apart from the Pyra mids) from the Rhind papyrus and the Moscow papyrus (c. 1850-1650 BC). These include tables of fractions and several dozen solved problems in arith metic and geometry, probably designed for teaching scribes and accountants. ENIAC: The modem computer age started in World War II, with COLOSSUS in England, used for deciphering German military codes, and ENIAC [Electronic Numerical Integrator And Computer] in the United States. These machines were enormous: ENIAC was eight feet high and contained 17,000 vacuum tubes, 70,000 resistors, 10,000 capacitors, 1,500 relays, and 6,000 switches. Euclid: The first important mathe matician associated with Alexandria
was Euclid (c. 300 BC), who wrote on optics and conics but is mainly re membered for his Elements. The most influential and widely read mathemat ics book of all time, the Elements con sists of thirteen books on plane and solid geometry, number theory, and the theory of proportion. A model of de ductive reasoning, it starts from initial axioms and postulates and uses rules of deduction to derive each new propo sition in a logical and systematic order. Euler: Leonhard Euler (1707-1783) taught in St. Petersburg and Berlin and was probably the most prolific mathe matician of all time. He reformulated the calculus using the idea of a function, and contributed to number theory, dif ferential equations, and the geometry of surfaces. In his analysis book of 1748 Euler related the exponential and trigonometric functions by means of the fundamental equation ei'l' = cos cp + i sin cp, shown on the stamp.
Ecole polytechnique
Euclid Egyptian accountants
Education
Please send all submissions to the Stamp Comer Editor, Robin Wilson,
Faculty of Mathematics,
The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]
80
EN lAC
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Euler
