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JOHN F. RIGBY

Precise Colourings of Regular Triangular Tilings Dedicated to Professor H. S. M. Coxeter on the occasion of his 90th birthday, February 9, 1997 The regular tiling {3, n} is a tiling of equilateral triangles, with n triangles m e e t i n g at each vertex; it o c c u r s on the sphere, in the E u c l i d e a n plane, or in the h y p e r b o l i c plane, according as n < 6, n = 6, or n > 6. In a p r o b l e m in the American Mathematical Monthly in 1993 [8], the late Raphael Robinson a s k e d for a colouring of the faces of {3,7} using seven colours, such that each colour occurs just once at each vertex. Such a colouring had already been exhibited a n d discussed b y me ([3], Fig. 16; [4], Fig. 7; [6], p. 10, colour plate 2a), and it was no d o u b t k n o w n long before then, as it is closely associated with Klein's regular

m a p {3,7}8; it has also been discussed in detail in [2], together with the associated simple group of order 168. Robinson told me in a letter in October 1993 that he had, in fact, solved the p r o b l e m in 1984 after it was p o s e d by David Gale, who was interested in the following m o r e general problem: find a colouring of the regular triangular tiling {3, n} using n colours, such that each colour occurs j u s t once at e a c h vertex. I call such a colouring a precise colouring. In a l a t e r letter to me, R o b i n s o n outlined his m e t h o d of s o l u t i o n w h e n n is even. My o w n method, involving sets of f a c e s called trees, w o r k s for all n and was originally de-

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THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER-VERLAG NEW YORK

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signed to produce colourings of a type very different from Robinson's, namely chiraUy perfect or fuUy perfect colourings. But by sewing together suitably coloured trees, we can sometimes produce a precise colouring (not perfect) of {3, n} with only three or six different types of vertex. Robinson produced the same sort of colouring by constructing a map of type {3, n} with n faces and three vertices, coloured with n colours, and then unwrapping it to give a colouring of the universal covering {3, n}. I shall discuss Robinson's method in the fmal section.

This article is based mainly on [7] and on ideas explained in detail in the M. Phil. thesis of my research student Nergiz Yaz [9]. I shall not give constant specific references to [7] when the more basic ideas are explained below. Most of the content of the last two sections is n e w and is still under investigation. The tilings {3,10} and the face-centred tilings {3,9} and {3,8} were drawn by computer by Douglas Dunham, to w h o m I am m o s t grateful; the remaining tilings, and all the shading and colouring, were done by hand.

Introduction

A symmetry (direct or opposite) of {3, n} is an isometry (direct or opposite) of the plane, or of the sphere when n < 6, that maps the tiling to itself; it is an automorphism of the tiling. A colouring of the faces of {3, n} is chiraUy perfect if all direct symmetries of the tiling permute the colours but the opposite symmetries jumble them up, and fully perfect if all symmetries permute the colours. Fully perfect precise colourings exist only when n is even and n r 8 ([7], Sec. 5). In a chirally perfect colouring, the symmetries of the tiling that permute the colours form a subgroup of index 2 of the full symmetry group. There is another type of colouring, which I call semiperfect (see the final section), where the subgroup of colour-permuting symmetries is also of index 2. As far as I know, this type of colouring exists only when n is even. For this reason, we shall concentrate mainly on precise colourings that may be chirally perfect or fully perfect, as these exist for all n. I use the word "perfect" to mean "chirally perfect or fully perfect." This convention makes good sense when we consider the connection between perfect colourings and regular maps below.

VOLUME 20, NUMBER 1, 1998

5

Trees

A symmetry (direct o r opposite) of a set S o f tiles of {3, n} is a s y m m e t r y o f {3, n} that m a p s S to itself. Our main tool will be sets of tiles called trees. A tree is a set S of tiles satisfying the following conditions [9]: (i) S is edge-connected (i.e., the tiles o f S are c o n n e c t e d by edges a n d not j u s t by vertices; (ii) no v e r t e x of S lies in the interior of S; (iii) S is vertex-transitive (i.e., given any two v e r t i c e s A and B of S, there exists a direct s y m m e t r y of S m a p p i n g A to B); (iv) the tiles of S at a v e r t e x A of S form a c o n n e c t e d set when A is removed, so t h a t the tiles of S s u r r o u n d i n g A form a single block of tiles. It is easily p r o v e d that S is transitive on b o u n d a r y e d g e s (edges separating tiles o f S from tiles not in S). It follows from transitivity that the n u m b e r k of tiles of S at a v e r t e x o f S is the same for all vertices; this n u m b e r k is the valency of S, and we call S a k-tree. As an example, b e f o r e w e discuss h o w to c o n s t r u c t trees, Figure l a s h o w s the only type o f 5-tree, w h e n n = 8 (we shall a l w a y s use n to denote the n u m b e r of tiles at each v e r t e x o f the underlying tiling). The s a m e 5-tree (combinatorially s p e a k i n g ) exists for all n --- 8; b u t if w e try to c o n s t r u c t the s a m e tree w h e n n = 7, its b r a n c h e s join up in a consistent m a n n e r to form loops as in Figure lb, and something similar h a p p e n s w h e n n = 6 (Fig. lc). Even w h e n n = 5, w e c a n still exhibit this 5-tree as a tree with loops (Fig. l d ) , b u t n o w the b o u n d a r y e d g e s j o i n up with each other a n d t h e y no longer s e p a r a t e tiles in the tree from tiles not in the t r e e (only t h r e e of the six d o u b l e - b o u n d a r y edges are visible in the figure). We shall r e g a r d the t r e e s in Figure lb, lc, a n d l d as being the s a m e (at least in a local sense) as t h a t in Figure la. Once w e have c o n s t r u c t e d any k-tree, it will a l w a y s exist without l o o p s w h e n e v e r n -> k + 3. It s e e m s c l e a r b e c a u s e of transitivity that the s a m e tree will also exist, with loops, w h e n n = k + 2, k + 1, or k. This is p r o v e d in [7] for trees of a s p e c i a l type called s t a n d a r d trees, a n d it will be d i s c u s s e d again at the end of the section " N o n s t a n d a r d Trees and Pseudotrees." The black tiles in Figure l c can be used as the tiles of one colour in a perfect precise colouring o f {3,6}. The layout of the black tiles forms a clear pattern, and the s a m e pattern is followed by the tiles of the remaining five colours. This idea is important, so here is a s e c o n d example. The nonblack tiles in Figure 2 form a 7-tree w h o s e construction will be analysed later; it is a tree with loops b e c a u s e n = 8. When one is actually involved in painting the black tiles in this figure, it is useful to devise simple rules for proceeding from an edge or a vertex of one black tile to the neighbouring black tiles. Because of the transitivity o f the tree, these rules are the s a m e whichever edge or v e r t e x o f whichever black tile we start from; and because of the transitivity, any symmetry of {3,8} that m a p s a black tile to another black tile m a p s the set of b l a c k tiles to itself. Thus, the black tiles can be used as the tiles of one colour in a perfect precise colouring of {3,8}. If w e colour one of the remaining tiles green, say (ignoring the existing red and yellow colouring), we can use the same rules to obtain all the green tiles, and proceeding in this way, w e obtain the entire colouring. In general, s u p p o s e we have a perfect precise colouring of

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{3, n}, one colour being black. Then the nonblack tiles fgrm an (n - 1)-tree with loops, and, conversely, any (n - 1)-tree leads to a perfect precise colouring of {3, n}; so to obtain all perfect precise colourings, we need only obtain all trees.

Constructing Standard Trees E v e n though it is not t r e e - s h a p e d , a single tile f o r m s a 1t r e e w h i c h I shall denote b y 1. There are no 2-trees. T h e r e is a unique type of 3-tree forming a single band o f tiles, a n d I shall d e n o t e this t e m p o r a r i l y b y 3; 3-trees o c c u r as conn e c t e d sets of white tiles in Figure 2. It is c o n v e n i e n t to reg a r d a single edge as a 0-tree, d e n o t e d b y 0. A tile T of a tree S having a n edge on the b o u n d a r y of S is a boundary tile of S; its b o u n d a r y edge is t h e base o f T, a n d the opposite v e r t e x o f T is its apex. The o t h e r t w o sides o f T are its left and right sides. These definitions app l y only to trees of v a l e n c y at least 3. It can be s h o w n t h a t e v e r y v e r t e x of S is the a p e x o f j u s t one b o u n d a r y tile [9]. Figure 2 s h o w s a 7-tree (with n = 8) in which t h e b o u n d a r y tiles are coloured red. The white tiles form 3-trees, a n d the y e l l o w tiles are 1-trees. E v e r y edge of each c o p y o f P is j o i n e d to the left side o f a b o u n d a r y tile, and e v e r y edge o f e a c h c o p y of Q is j o i n e d to the right side of a b o u n d a r y tile. We therefore d e n o t e this 7-tree b y the s y m b o l (31). In a similar manner, Figure 3 s h o w s the 4-tree (10), built up, with the aid of r e d b o u n d a r y tiles, from white 1-trees a n d heavily d r a w n 0-trees; t h e 5-tree in Figure 1 is s e e n to b e (11). The 3-tree 3 is easily s e e n to be (00), so the 7-tree (31) of F i g u r e 2 is ((00)1). Let P and Q be any t r e e s o f valencies h and k. As long as n -> h § k § 3, we can p u t t o g e t h e r copies o f P a n d Q, w i t h the aid of b o u n d a r y tiles, in a m a n n e r similar to t h a t in F i g u r e 2, to form a t r e e o f v a l e n c y h § k + 3: e v e r y edge o f e a c h c o p y of P is j o i n e d to the left side of a b o u n d a r y tile, a n d every edge of e a c h c o p y of Q is j o i n e d to t h e right side o f a b o u n d a r y tile. We d e n o t e this tree b y (PQ). Thus, w e have an inductive m e t h o d o f constructing trees, starting with the t r e e s 0 and 1, a n d a n y tree c o n s t r u c t e d in this w a y is called a standard tree. A s t a n d a r d tree gives rise to a s t a n d a r d p e r f e c t p r e c i s e colouring. N o n s t a n d a r d t r e e s will b e c o n s i d e r e d in the n e x t section. A n y s t a n d a r d tree has a (0,1)-symbol consisting o f O's a n d l ' s suitably bracketed, a n d different (0,1)-symbols give different trees. It is e a s y to calculate the n u m b e r o f stand a r d k-trees b y m e a n s of a r e c u r r e n c e formula ([7], Sec. 3). F o r instance, the n u m b e r o f s t a n d a r d 19-trees is 1554; six of t h e s e have mirror s y m m e t r y (their (0,1)-symbols are p a l i n d r o m i c ) and the r e m a i n i n g ones o c c u r in left- a n d r i g h t - h a n d e d pairs. Thus, t h e r e are 1554 s t a n d a r d p e r f e c t p r e c i s e colourings of {3,20}, a n d six of these are fully perfect. Nonstandard Trees and Pseudotrees Figure 4 shows a nonstandard 9-tree with its boundary tiles co]outed red. It is not a typical nonstandard tree, but unfortunately it is the only n o n s t a n d a r d tree that can conveniently be visually illustrated: the n e x t simplest example is a 12-tree. The white tiles in Figure 4 form 3-trees, each of which has

two boundary arcs (maximal connected sets of b o u n d a r y edges). We shall designate the edges of one b o u n d a r y arc as positive edges (shaded yellow), and the remaining b o u n d a r y edges as negative edges. With the correct designation, every white 3-tree in Figure 4 has its positive edges j o i n e d to the left sides of red boundary tiles, and its negative edges joined to the right sides of r e d b o u n d a r y tiles. (Compare this with the w a y in which the s t a n d a r d 9-tree (33) is constructed according to the method in the previous section.) To c o n s t r u c t general n o n s t a n d a r d trees, w e n e e d to use pseudotrees [9]. A p s e u d o t r e e r e s e m b l e s a tree, b u t it has b o u n d a r y arcs of t w o different types, each of w h i c h is like the b o u n d a r y arc of a tree. F o r example, Figure 5 s h o w s a p s e u d o t r e e of type (4,5), w h i c h has b o u n d a r y a r c s of valencies 4 and 5. A p s e u d o t r e e of type (h,k), with v e r t i c e s o f v a l e n c i e s h and k, is required to be transitive on h-vertices and on k-vertices. We shall denote a p s e u d o t r e e o f t y p e (h,k) by the s y m b o l p(h,k), an imprecise notation, as t h e r e m a y be m o r e t h a n one p s e u d o t r e e of t y p e (h,k). Let P be a p s e u d o t r e e of type (h,k). Boundary edges of P joining h-vertices will be designated as positive edges, and the remaining b o u n d a r y edges as negative edges. If n --- h + k + 6, we can put t o g e t h e r copies of P, with the aid o f boundary tiles, in the following way, to obtain a n o n s t a n d a r d (h + k + 3)-tree without l o o p s [9]: every positive edge o f each c o p y of P is joined to the left side of a b o u n d a r y tile, and every negative edge of e a c h c o p y of P is j o i n e d to the right side of a b o u n d a r y tile. If n = h + k + 5, h + k + 4, o r h + k + 3, the s a m e construction yields a tree with loops. In the c o n s t r u c t i o n o f Figure 4, w e are e s s e n t i a l l y regarding the 3-tree as a p s e u d o t r e e p(3,3). Any s t a n d a r d tree can b e r e g a r d e d as a p s e u d o t r e e if its (0,1)-symbol contains at least two O's [9]. It s e e m s p r o b a b l e t h a t t h e r e are only two w a y s o f building up a tree, but I have n o t y e t set d o w n in detail the rea-

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sons for this assertion: if X is a tree, t h e n either X = (PQ) in the m a n n e r d e s c r i b e d in the p r e v i o u s section, w h e r e P and Q are t r e e s of smaller valencies, o r X is a n o n s t a n d a r d tree c o n s t r u c t e d from a p s e u d o t r e e in the m a n n e r j u s t described. If this is so, the a r g u m e n t u s e d in [7], Sec. 4, can be a d a p t e d to s h o w that if a k-tree w i t h o u t l o o p s has b e e n c o n s t r u c t e d w h e n n -> k + 3, t h e n "the same" tree with loops exists w h e n n = k + 2, k + 1, o r k.

Alternating Trees To c o n s t r u c t p s e u d o t r e e s , w e n e e d alternating trees; t h e s e are similar to t r e e s b u t they have v e r t i c e s of two different types t h a t o c c u r alternately along e a c h b o u n d a r y arc [9]; we require an alternating tree to b e transitive on each type of vertex. If the two t y p e s of v e r t e x have valencies h a n d

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VOLUME20, NUMBER1,1998

p r o c e s s is the consideration of pseudoalternating t~ees and quadruple trees [9].

Colour Permutations and Regular Maps

k, the alternating tree is of type (h,k), denoted imprecisely by a(h,k). Figure 6 shows an alternating tree of type (4,5). Alternating trees and pseudotrees can be built up in various inductive ways, starting with basic alternating trees which are constructed as follows. Let X be a k-tree, and assume n -> k + 2. If we adjoin a single tile (a spine) to each boundary edge of X we obtain an alternating tree of type (1, k + 2) [9]. Figure 7 shows a 4-tree with spines added to form an alternating tree a(1,6), and Figure 8 shows the alternating trees a(1,2) and a(1,3). There is space here to show examples of only two of the ways of constructing pseudotrees and alternating trees. Figure 9 shows a pseudotree p(6,7) constructed from copies of alternating trees a(1,3) (white) and a(2,1) (yellow) with the aid of red boundary tiles, in a manner similar to that described for standard trees above [9]. Figure 10 shows an alternating tree a(6,7) constructed by piecing together copies of a(2,4) (white) and a(4,3) (green) directly, without the aid of extra b o u n d a r y tiles [9]. These are examples of what are termed "the old and new methods of construction" in [9], but the two methods do not exhaust all possibilities: there is an example in [9] of an alternating tree a(3,9) that cannot be constructed by either method; the next stage in what is probably an unending

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The direct symmetry group of {3, n} induces a group G of permutations of the colours in any perfect colouring. Two faces P and Q in a perfect precise colouring are equivalent if there is a direct symmetry mapping P to Q and mapping each colour to itself. If we identify equivalent faces, we obtain a regular map of type {3, n} whose universal covering is {3, n}. If the colouring is chirally perfect, then the map m a y be chirally regular or reflexive (fully regular), but if the colouring is fully perfect, then the corresponding regular map is reflexive. (There are different terminologies here; I am using the definitions in [1], Chapter 8.) If this map has f faces, its direct s y m m e t r y group is isomorphic to G and has order 3fl See also [5]. The direct symmetry group of {3, n} is generated by rotations c~ and fl through an angle 2qr/n about two adjacent vertices A and B [9]. The cyclic permutations of colours ind u c e d by a and fl generate G. If a perfect precise colouring is associated with a tree constructed by one of the various methods described in [9], there are algorithms for finding these cyclic colour permutations [9].

Imperfect Colourings I conjectured in [7], Sec. 10, that any k-tree without loops has a perfect precise colouring in k colours; this means that each colour occurs once at every vertex, and every direct symmetry of the tree permutes the colours. This is easy to prove, as follows. When n = k + 1, the tree (with loops) can be used to determine the nonblack tries in a perfect precise colouring of {3, n}. The remaining colours in this colouring provide a perfect precise colouring of the tree in k colours.

This colouring can b e "unwrapped" to provide a p e r f e c t precise colouring of the tree w h e n n -> k + 3. The colouring of the 5-tree obtained in this w a y is s h o w n in Figure 11. Alternatively, given a k-tree without loops, w e can consider the same k-tree with loops w h e n n = k (as w e did in Fig. ld); the tree then covers the entire tiling. Colour the tiling using any perfect precise colouring with n -- k colours, then unwrap this colouring to give a colouring of the original k-tree. It is an interesting exercise to colour the 5-tree by this method. The converse p r o c e d u r e - - t a k i n g a colouring o f a tree w i t h o u t l o o p s and w r a p p i n g it a r o u n d the c o r r e s p o n d i n g t r e e with l o o p s - - c a n n o t always b e performed. S u p p o s e n -> 6 a n d 3 -< k -< n - 3. Let P b e a k-tree and Q an (n - k)-tree. We c a n s e w t o g e t h e r c o p i e s o f P and Q alternately to cover the entire tiling {3, n}. If t h e c o p i e s of P a r e precisely c o l o u r e d with k colours a n d t h e c o p i e s of Q with n - k different colours, the resulting colouring of {3, n} will b e precise, b u t n o t perfect. There is another important w a y of colouring certain trees. We start by looking at the colouring in Figure 11 from a different viewpoint. The stippled boundary tiles form a vertexconnected but not edge-connected set. These tiles can be coloured in an essentially unique w a y with three colours (A, B, and C in the figure) so that each colour occurs once at each vertex. The remaining tiles of the tree have been coloured 1 or 2, and this p r o d u c e s a perfect p r e c i s e colouring of the 5-tree with only three types of vertex (we say that t w o vertices are of the s a m e type if the direct s y m m e t r y mapping one to the other induces the identity permutation on the colours). It follows that the group of colour p e r m u t a t i o n s ind u c e d by the symmetries of the tree has o r d e r 3. We can n o w p r o v e b y induction that every standard tree whose (O,1)-symbol contains no O's has a perfect precise colouring with only three types of vertex. As an example, the n o n b l a c k tiles in Figure 12 s h o w the 9-tree ((11)1) ob-

tained b y putting t o g e t h e r 5-trees a n d 1-trees with the aid of b o u n d a r y tiles. One v e r t e x - c o n n e c t e d set of b o u n d a r y tiles h a s b e e n c o l o u r e d with c o l o u r s P, Q, and R. The adj a c e n t 5-trees can n o w be c o l o u r e d with five colours A, B, C, 1, a n d 2 as d e s c r i b e d in the p r e v i o u s paragraph, in s u c h a w a y t h a t colours A, B, and C are always a d j a c e n t to P, Q, a n d R. We continue in this w a y a n d finally colour t h e isolated w h i t e 1-trees with c o l o u r 3. F o r k = 4 r + 1, t h e r e is a l w a y s a k-tree w h o s e (0,1)-symbol c o n t a i n s no O's. It follows t h a t if n = 4t § 2, we c a n sew t o g e t h e r (4r + 1)-trees a n d (4s + 1)-trees c o l o u r e d in the w a y j u s t described, w h e r e r + s = t, to obtain a precise c o l o u r i n g of {3, n} with only three types of vertex. F i g u r e 12 i l l u s t r a t e s a m e t h o d o f colouring a 9-tree, b u t it c a n also b e l o o k e d at in a d i f f e r e n t w a y as b l a c k 1-trees and a c o l o u r e d 9-tree s e w n t o g e t h e r to give a p r e c i s e c o l o u r i n g o f {3,10}. Alternatively, t h e figure s h o w s 5-trees c o l o u r e d A, B, C, 1, a n d 2, a n d 5-trees c o l o u r e d P, Q, R, 3, a n d b l a c k , s e w n t o g e t h e r to give t h e s a m e colouring. This is j u s t a p a r t i c u l a r c a s e o f a g e n e r a l p h e n o m e n o n w h e r e t h e s a m e colouring c a n b e s e w n t o g e t h e r in various ways. All this is gradually leading to a linkup with R o b i n s o n ' s method, b u t let us first c o n s i d e r w h a t w e can do w h e n n = 4r. A single e x a m p l e m u s t suffice. W h e n n = 8, w e c a n cover the tiling {3, n} with 3-trees a n d 5-trees s e w n t o g e t h e r alternately, a s in Figure 13 ( w h e r e the 3-trees are stippled). Colour one o f the 3-trees with c o l o u r s X, Y, a n d Z. Colour an a d j a c e n t 5-tree as before, so that A, B, and C are adjacent to X, Y, and Z, as shown. If w e wish to c o l o u r the 5tree on the o t h e r side of the 3-tree, with A, B, a n d C adjacent to X, Y, a n d Z, we must use the mirror image of the previous 5-tree colouring. We can n o w c o m p l e t e the entire colouring in an obvious way, but we now have six types of vertex: three types and their mirror images.

VOLUME 20, NUMBER 1, 1998

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a

b

1l

Robinson's Method Here are p o r t i o n s o f a letter to m e from R a p h a e l Robinson in N o v e m b e r 1993; in the first paragraph, he is quoting from a letter written to David Gale in O c t o b e r 1992, and in the s e c o n d p a r a g r a p h he is revising one o f t h e i d e a s in that earlier letter. It is easy to find a coloring w h e n n is even. The n triangles at a v e r t e x f o r m a polygon with n sides. Identify pairs of sides in s u c h a w a y as to identify a l t e r n a t e vertices. This gives a surface tried by n triangles. There are three vertices, a n d all the triangles m e e t at e a c h vertex. This leads to t h e required coloring. F o r n = 8, 10, 12, suitable pairings o f the sides are given b y the f o r m u l a s

apbpaqbq,

abcdea-lb- tc- ld- le-1,

apbpcqaqbrcr.

The generalizations are clear. F o r n t w i c e an o d d number, we o b t a i n a coloring with j u s t t h r e e t y p e s o f vertices, but w h e n n is a multiple of 4, the c o n s t r u c t i o n given leads to six t y p e s o f vertices, since a n o n o r i e n t a b l e surface was used. I see n o w t h a t the extension to n -- 16 is n o t as direct as I s u p p o s e d then. But the f o r m u l a apbpcqdqarbscsdr m a y b e used. A similar formula m a y b e u s e d whenever n is a multiple o f 8, with the s e c o n d letter of the s e c o n d half r e p e a t e d at the end. We c o l o u r t h e m a p d e s c r i b e d a b o v e w i t h n colours, then w e unfold the colouring to give a colouring o f the universal covering {3,n}. In a r e v e r s e manner, w e c a n identify vertices of the s a m e t y p e (or o f mirror-image type) in the colourings d e s c r i b e d at the end of the p r e v i o u s section to obtain a m a p with n triangular faces a n d t h r e e vertices. When n = 4r, the c o r r e s p o n d i n g surface h a s Euler characteristic 4 r - 6 r + 3, w h i c h is odd, so it is a n o n o r i e n t a b l e surface. No s u c h m a p with n faces exists w h e n n is odd, since the n u m b e r o f edges is 3n/2.

10

THE MATHEMATICAL INTELLIGENCER

The first t w o of R o b i n s o n ' s formulae for identifying e d g e s o f the n-gon can b e illustrated as in Figures 14a a n d 15a: e d g e s to be identified have b e e n linked, by a d o t t e d line if the identification involves turning the n-gon over. Such a diagram of linked e d g e s p r o d u c e s a valid w a y of c o n s t r u c t i n g a m a p with n triangles at each of t h r e e vertices if, w h e n w e d r a w l o o p s to join a d j a c e n t e d g e s as in Figure 14b, in b o t h p o s s i b l e ways, w e obtain a single p a t h p a s s i n g t h r o u g h all n edges. We can unfold such a m a p in three ways, with any of its three vertices at the centre of the n-gon; so a map has three "link-diagrams." The other two diagrams associated with the m a p given by Figure 14a are s h o w n in Figure 14b, and the o t h e r t w o diagrams associated with Figure 15a are s h o w n in Figure 15b. The map associated with the colouring in Figure 13 has all three diagrams as s h o w n in H g u r e 16; this is w h a t w e should expect, as there is a s y m m e t r y (direct o r opposite) mapping any vertex to any other vertex that p e r m u t e s the colours. The map associated with the colouring in Figure 11 has all three diagrams as s h o w n in Figure 17. Given a colouring with only t h r e e t y p e s of vertex, cons t r u c t e d from s t a n d a r d t r e e s in the m a n n e r d e s c r i b e d in the p r e c e d i n g section, t h e r e m u s t be a simple algorithm for o b t a i n i n g its link-diagram from the (0,1)-symbols of the trees, b u t I have not investigated it. T h e r e is an obvious g e n e r a l i s a t i o n of Robinson's Figure 15a w h e n n = 4 r § 2. A g e n e r a l i s a t i o n of Figure 14a, that s e e m s s i m p l e r than those s u g g e s t e d b y Robinson w h e n n -4r, is s h o w n in Figure 18. G e n e r a l i s a t i o n s of Figures 16 a n d 17 a r e s h o w n in Figures 19 a n d 20. W h e n n = 4r, w e m u s t u s e a n o n o r i e n t a b l e s u r f a c e in R o b i n s o n ' s m e t h o d b e c a u s e o f the Euler c h a r a c t e r i s t i c m e n t i o n e d above. When n = 4 r + 2, w e can use a nonorie n t a b l e surface, but t h e r e is no need. W h e n n = 14, the link-diagram in Figure 21 is e s p e c i a l l y interesting; its s y m m e t r y group is the dihedral group o f or-

ti

e~q-a.Yb-'

@

IGURE 1

IGURE 1~

d e r 14, and all t h r e e d i a g r a m s a s s o c i a t e d with the corres p o n d i n g m a p are identical, so the s y m m e t r y group of the m a p has o r d e r 42 with 21 direct a n d 21 o p p o s i t e s y m m e tries. The c o r r e s p o n d i n g p r e c i s e colouring of {3,14} is w h a t I have e l s e w h e r e called "semiperfect" ([3], Fig. 18; [5], Fig. 16; [7], Section 9), b e c a u s e half the direct a n d half t h e opp o s i t e s y m m e t r i e s o f {3,14} p e r m u t e the colours. We n e e d a n a m e for the c o r r e s p o n d i n g t y p e of map; "semiregular" a l r e a d y h a s a different, well-established meaning. Does this situation o c c u r for other values of n? In Figure 21, with a suitable labelling of the edges, 0 is j o i n e d to 5, 2 to 7, and so forth. Consider a (4r + 2)-gon with edge 2i j o i n e d to edge 2i + 2k + 1 (mod 4 r + 2) (i = 0, 1 , . . . , 2r). This produces a valid link-diagram w h e n k and k + 1 are b o t h prime to 2 r + 1 (when 2r + 1 is prime, this condition is simply 1 -< k - 2 r - 1) and the s y m m e t r y group of the diagram is the dihedral group of order 4 r + 2. Denote this link-diagram by LD(k). A calculation s h o w s that the other two diagrams associated with the corresponding m a p are LD(g) a n d LD(h), where g -= - 1 - k -1, h -= - 1 - g - l , a n d k ~- - 1 - h -1 (mod 2r+l).Nowk=g=hiffk 2+k+l---0. (When2r+lis prime, this is equivalent to k 3 - 1, k r 1.) Note that if k 2 + k + 1-= 0, then ( - k - l ) 2 + ( - k - l ) + 1 ~- 0 also; but the link-diagrams L D ( - k - 1 ) and LD(k) are the same. It can b e s h o w n t h a t k exists such that k 2 § k + 1 = 0 ( m o d 2 r + 1) iff 2 r + 1 = 3 i p a q b . . . , where i = 0 o r 1, the p r i m e s p, q , . . . are c o n g r u e n t to 1 modulo 6, a n d a, b , . . . 0. F o r instance, 2r + 1 = 13, k = 3; or 2 r + 1 = 21, k = 4. F o r s u c h values o f r a n d k ( e x c e p t for r = k = 1), w e o b t a i n a m a p of n = 4 r + 2 triangles and 3 vertices, w h o s e full s y m m e t r y group h a s o r d e r 3n, and a c o r r e s p o n d i n g precise colouring of {3,n} t h a t has only three t y p e s o f v e r t e x a n d is semiperfect. These p a r t i c u l a r "Robinson colourings" c o m e as close as p o s s i b l e to being perfect, e x c e p t o f c o u r s e for the case r = k = 1, n = 6 w h e n w e have the only R o b i n s o n colouring t h a t is fully perfect. In one sense, chirally perfect and s e m i p e r f e c t colourings are equally p l a c e d on t h e l a d d e r of perfection: in each case, t h e s y m m e t r i e s o f the tiling that p e r m u t e the colours form a s u b g r o u p of i n d e x 2 o f the group of all symmetries.

:IGURE 2~

IGURE 1

IGURE 1~.

REFERENCES

1. H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, New York: Springer-Verlag (1980). 2. D. Mackenzie, A hyperbolic plane colouring and the simple group of order 168, Am. Math. Monthly 102 (1995), 706-71& 3. J. F. Rigby, Symmetry in geometry: a personal view, Symmetry: Culture and Science 1 (1990), 63-76. 4. J. F. Rigby, Compound tilings and perfect colourings, Leonardo 24 (1991 ), 53-62; also reproduced in M. Emmer (ed.), The VisualMind, Cambridge, MA: MIT Press (1993), 177-186. 5. J. F. Rigby, Perfect colourings and regular maps, in Proceedings of the International Conference on Geometry, Chulalongkorn University Research Report 17, Bangkok, (1991), pp. 297-313. 6. J. F. Rigby, Fun with tessellations, in R. K. Guy and R. E. Woodrow (eds.), The Lighter Side of Mathematics, Providence, RI: Mathematical Association of America (1994), 74-90. 7, J. F. Rigby, Perfect precise colourings of triangular tilings, and hyperbolic patchwork, Symmetry: Culture and Science (to appear). 8. R. M. Robinson, Problem 10349, Am. Math. Monthly 100 (1993), 952. 9. N. Yaz, Perfect precise colourings of regular triangular tilings, M. Phil. thesis, University of Wales (1997).

1GURE 2

VOLUME 20, NUMBER 1, 1998

11

Letters

to

The Mathematical encourages material

the

Editor

Intelligencer

comments

about the

in this issue. Letters

to t h e e d i t o r s h o u l d editor-in-chief,

be s e n t to t h e

Chandler

Davis.

Identifying the S.F.

and, thus,

This is in comment on the "Remembrance of Paul Erdds" in your spring issue, 38-48. I knew Paul since my days as a graduate student of Witold Hurewicz at M.I.T. in 1946. As I recall, Erd6"s explained that the "Supreme Fascist" was none other than God, the ruler of the Universe---even though Erd6"s was a confirmed atheist. MIRIAM LIPSCHOTZ-YEVICK 22 Pelham Street Princeton, NJ 08540

1+

2

=

i=1

Perron, although a specialist on continued fractions, does not mention here that Engel's infinite product is related to the well-known periodic continued fraction, convergent for positive real z Vz=

USA

1

1+ (z-1)/2 I

I 1 + I(z-11)/41 +l(z-11)/41 + . . .

An Infinite Product f o r S q u a r e Rooting w i t h Cubic C o n v e r g e n c e

In 1913, Friedrich Engel (1861-1941) gave [1] an infinite product for the positive square root ~nn,

(see his book [3], p. 320), whose approximants k~ a r e k i = a / / b / , where ai =

(1 + z l / 2 ) / + (1 - z l / 2 ) / 2 9 ( z - 1)i/2

b / = (1

+

zl/2) i - (1 - z l / 2 ) /

2 9 z 1/2 9 ( z -

1) i/2

There are the recursion equations . . . .

2 ai+2 = ( z - -

1 ) 1/2

"

ai+l + ai,

1 ) 1/2

"

bi+l + bi;

(n > 0, n # 1), where 2

bi+2 = ( z -

+____~1 n ql -

n-

l '

q i + l = 2 " q~ -

1.

the formulas for index-doubling Oskar Perron (1880-1975), in his book I r r a t i o n a l z a h l e n , p. 127 [2], refers to this result and gives Engel's elementary derivation: With the help of the identity

(1+1)(1+

1

a2i = a~ + z . b 2, b 2 i = 2 9 a i 9 bi;

and the formulas for index-trebling a3i = a i 9 ( a 2 + 3 9 z . b2i), b3i = b/ " ( 3 " a 2 + z . b 2 ) .

qi+l ) = qi+l -- 1 ( qi -- l l - 1 ' qi+l

\

qi

From index-doubling immediately follows

] k 2 . i = -~ k i +

he obtains co

1Fi(1 + i=1

Thus, Newton iteration for the polyno-

H(I + qi+l1 )=

mial P(x)

.=

.

\

12

THE MATHEMATICALINTELLIGENCER9 1998SPRINGER-VERLAGNEWYORK

ql

.

]

-~ x 2 - z ,

~'i+1 = 2

~,i

starting with ~0 = 1, c o i n c i d e s with index-doubling of the c o n t i n u e d fraction approximants, starting with kx = 1. The c o r r e s p o n d e n c e is k 2 i = ~'i

[note that a l = bl = 1 / k / z - 1 and a2 = (z + 1)/(z - 1), b2 = 2 / ( z - 1); thus, ~1 = k2 = (z + 1)/2]. Moreover, the following identity holds: a2-z"

=

Vzz = 3z + 1 z + 3

9 (3z 9 4- 297z s + 6156z 7 + 37044z 6 4- 87642z 5 4- 87390z 4 + 37212z 3 + 6084z 2 + 315z + 1 ) + (z 9 + 315z s + 6084z 7 + 37212z 6 + 87390z 5 4- 87642z 4 § 37044z 3 + 6156z 2 + 297z + 3) . . . . . This c o r r e s p o n d s to the cubic N e w t o n - S c h r S d e r iteration for ~i =

2=1

and a2.i=2"a

a 3 i / b 3 i,

2-1.

~72+3 " z

T h e r e results z

a2.i

ai

2 9a2

a2.i+l a2.i

w h e r e ~?i =

a2.i

=Z.(l+ ki

1 a2.i

k3 i,

2z z + 1

2.(z+ 1) 2 z 2 + 6z 4- 1

~/+1=~7i'(1+

k3.i

--

a3.i b3.i

_

z -

en mangeant."

2 . ( z . b 2 - a 2) 3 " a--2i-+ z - - b 2

a3 i+1

=

bt

(1 +

2

4.z.b2_3

3

4 9z

9 b2i

9 hi

9

h2 9 h3. "'"

h~1)'(1+

)

Of course, there is a n o t h e r infinite p r o d u c t for the analytic function ~zz;

3

.(1+

192900153617

) .... '

9 hi

9 hi+l.

4-z

h2 --

.h12.h~ . . . . .

h~-3..(1+

91267-0089).(1+ 2 76022378683214-7978143718729 ) . . . . give 8 c o r r e c t figures, the first three 27 c o r r e c t figures, the first four 80 c o r r e c t

z+3 z--l' Z3

the first four t e r m s give 33 correct figures. There is no n e e d to restrict the iteration to integers. F o r r o o t s of positive rational numbers, rational arithmetic can b e used. If n is close to 1, convergence is dramatic: F o r example, the first t w o t e r m s in

9(1+ h~-)'(l+~-4) .... , ~-=(1+ 2)'(1+ 9-~9) 2

hi+l = - -

hi-

.

-

h~ )

z - 1 Explicitly, w e have

)=

=

This i t e r a t i o n finally leads to the infinite p r o d u c t

where

2

9 b2i -

b3i+1 -- b3 i 9 hi+ 1 =

a 2 - z - b2 = - 1

k3.i = k i 9 (1 § 3 " a 2 + z " b 2 \

14045--~-

V~

a3{ 9 (hi+l + 2),

)"

(and a2.i = 2 9 a 2 + 1).

( 9(1+ 2770663499604051 ) .... ' 9 1 +

and

~ZZ=(i+

F o r o d d i,

Thus,

2 ) .... 9 1 + 467613464999866416197

4.z z -

4"z

9 ( a 2 + 3 9 z . b 2) bi'(3"a 2 + z - b 2)

ki"

_

v~

z "b2i - a2i

ai

= k i - (1 +

.(1+1+7)-(1+--22--7761797 )

the first four t e r m s give 46 correct figures; 3. V2+z

hi + 1

Is t h e r e an even faster w a y o f calculating square roots? Index-trebling should give cubic convergence, c o r r e s p o n d ing to s o m e cubically converging higher-order N e w t o n - S c h r S d e r iteration. In fact, index-trebling a m o u n t s to vient

h/-~l ) ,

where

2 9 (z 2 + 6z + 1) 2 z 4 § 28z 3 + 70z 2 4- 28z + 1 "L'appdtit

Calculating the n e x t h i + l n o w n e e d s only one a d d i t i o n a l multiplication if the previous results are stored. The cubic c o n v e r g e n c e is rapid. Three examples follow:

starting with

In o r d e r to bring the infinite p r o d u c t to Engel's form, we m a y w r i t e

Since the z / k 2 i = z/~i c o n v e r g e (from b e l o w ) t o w a r d V z , the infinite produ c t of Engel with ql = a2, q2 = a4, q3 = a s , . . , is obtained. Explicitly, ~zz=

-

_

~?0 = kl = 1. z. ki

+ 37212z 6 + 87390z 5 + 87642z 4 + 37044z 3 + 6156z 2 + 297z + 3) (z 1) ~

the first f o u r t e r m s give 62 correct figures;

7~i+ 1 = 1 ~ i " 3 " ~72 4- Z '

z 9 b2.i _ z 9 bi

k2.i

3z 3 + 27z 2 + 33z + 1 z 3 4- 33z 2 + 27z 4- 3

b2 = ( - 1 ) i.

Thus, for even i, a 2-z'b

it r e a d s [note that a3 (3z + 1)/ (%/z - 1) 3 and b3 = (z + 3 ) / ~ / ~ - 1)3; thus, ks = (3z + 1)/(z + 3)]

4- 33Z 2 + 27Z + 3 ( z - 1) 3

h 3 = (2' 9 4-

315z s + 6084z 7

(Letters: continued

on page 38)

VOLUME 20, NUMBER 1, 1998

13

JONATHAN

M. BORWEIN

Brouwer-I leyting Sequenccs Convergc The Occurrence of the Sequence 0 1 2 3 4 5 6 7 8 9 within ;T. In his p r i m e r on Intuitionism, Heyting [6] frequently relies on the o c c u r r e n c e or n o n - o c c u r r e n c e o f the sequence 0123456789 in t h e d e c i m a l e x p a n s i o n o f ~r to highlight issues of classical v e r s u s intuitionistic (or constructivist) mathematics. 1 At the time t h a t B r o u w e r d e v e l o p e d his t h e o r y (1908) and even at t h e t i m e that [6] w a s written, it s e e m e d wellnigh impossible that the first o c c u r r e n c e of any 10-digit sequence in ~- c o u l d e v e r be determined. The confluence o f faster c o m p u t e r s a n d b e t t e r algorithms, both for ~r a n d m o r e i m p o r t a n t l y for arithmetic (fast-Fourier-transform-based, c o m b i n e d with Karatsuba, multiplication [3]), have r e n d e r e d their intuition false. Thus, in June a n d July 1997, Yasumasa K a n a d a a n d Dalsuke Takahashi at the University of Tokyo c o m p l e t e d t w o computations of ~- on a massively parallel Hitachi m a c h i n e with 21~ p r o c e s s o r s . Details of Kanada and T a k a h a s h i ' s computation are l o d g e d at www.cecm.sfu.ca/personal/jborwein/Kanada_50b.html. Included are s o m e details of times, m a c h i n e r y used, digit distribution, etc. The k e y algorithms u s e d are as in the r e c e n t survey in this j o u r n a l [1], with the addition of significant numerical/arithmetical enhancem e n t s and subtle flow management. During their c o m p u t a t i o n a l tour-de-force, Kanada and Takahashi d i s c o v e r e d the first o c c u r r e n c e of 0123456789

in ~- beginning at the 17,387,594,880th digit (the '0') after t h e d e c i m a l point. It is w o r t h noting that y e a r s b e c o m e h o u r s on such a parallel machine, and also that even t h e l a b o r o f multiplying two seventeen-billion-digit integers tog e t h e r w i t h o u t F F T - b a s e d m e t h o d s is to be m e a s u r e d in years. The underlying m e t h o d r e d u c e s to roughly 300 s u c h multiplications ([3],[2]). A d d t h e likelihood of m a c h i n e s c r a s h i n g during a m a n y - y e a r sequential c o m p u t a t i o n . H e n c e without fast a r i t h m e t i c and parallel c o m p u t e r s , B r o u w e r and Heyting might indefinitely have r e m a i n e d safe in using this particular a n d s o m e w h a t natural e x a m p l e . I m a y emphasize h o w out o f r e a c h the question a p p e a r e d even 35 y e a r s ago with t h e following anecdote. S o m e t i m e after S h a n k s and Wrench c o m p u t e d 100,650 p l a c e s o f ~r (in 1962 in 9 hours on an IBM 7090), Philip Davis a s k e d Dan S h a n k s to fill in the b l a n k in the s e n t e n c e " m a n k i n d will n e v e r d e t e r m i n e the ... digit o f ~-." Shanks, a p p a r e n t l y alm o s t immediately, replied "the billionth."

The Use of 0 1 2 3 4 5 6 7 8 9

by Brouwer and Heyting

F o r c o m p l e t e n e s s I list t h r e e o f the "classical" e x a m p l e s : 1. The sequence a -= {2 -n} is a Cauchy sequence. Let the s e q u e n c e b -~ {bn} be d e f i n e d as follows: If the n t h digit a f t e r the decimal p o i n t in t h e decimal e x p a n s i o n of ~- is t h e 9 o f the first s e q u e n c e 0123456789 in this expansion, bn = 1, in every other c a s e bn = 2 -n. b differs from a in at m o s t one term, so b is classically a Cauchy sequence,

~To quote Carl Boyer, A History of Mathematics, John Wiley (1968) pp. 661-662: "According to Brouwer, language and logic are not presuppositions for mathematics, a subject that has its source in intuition that makes its concepts and inferences immediately clear to us; a statement that an object exists having a given property means that there is a known method that enables the object to be found or constructed in a finite number of steps. In particular, he argued the method of indirect proof, to which transfinite arithmetic had frequent recourse, is invalid. Ever since the time of Aristotle the three basic laws of logic had been held sacrosanct: (1) the law of identity, A is A; (2) the law of contradiction, A cannot simultaneously be B and not B; and (3) the law of the excluded middle (or tertium non datur). Brouwer denied the last of these laws of logic and refused to accept results based on it. For example, he asked the formalists whether it is true or false that the sequence of digits 123456789 (sic) occurs somewhere in the decimal expansion of ~-. Since no known method exists for making a decision, one cannot apply here the law of the excluded middle and assert that the proposition is either true or false."

14

THE MATHEMATICALINTELLIGENCER9 1998 SPRINGER-VERLAGNEW YORK

b u t as long as we do n o t k n o w whether such a sequence 0123456789 occurs in 7r, we are not able to find n such t h a t Ibn+p - bnl < 1/2 for every p; we have n o right to assert that b is a Cauchy sequence in our sense. ([6], page 16.) 2. A proof of the impossibility of the impossibility of a property is not in every case a proof of the property itself. It will be instructive to illustrate this by a n example. I write the decimal e x p a n s i o n of ~- a n d u n d e r it the decimal fraction p = 0 . 3 3 3 . . . , which I b r e a k off as soon as a sequence of digits 0123456789 has a p p e a r e d in or. If the 9 of the first s e q u e n c e 0123456789 in ~r is the kth digit after the decimal point, p = ( 1 0 k - 1)/3.10 k. Now suppose that p could n o t be rational; then p -- (10 k - 1)/ 3.10 k would be impossible and no sequence could appear in ~r; but t h e n p = 1/3, which is also impossible. The a s s u m p t i o n that p c a n n o t be rational has led to a contradiction; yet we have n o right to assert that p is rational, for this w o u l d m e a n that we could calculate integers p and q so that p = p/q; this evidently requires that we can either indicate a sequence 0123456789 or demonstrate that n o such sequence can occur. ([6], page 17, referring to [4].) 3. However, m a n y other classical theorems are n o longer valid. I state an e x a m p l e that a b o u n d e d m o n o t o n e sequence need n o t be convergent. A simple counterexample is the sequence {an} which is defmed as follows: an = 1 - 2 - n if a m o n g the first n digits in the decimal expansion of ~ no sequence 0123456789 occurs; an = 2 - 2 - n if among these n digits such a s e q u e n c e does occur. Nobody k n o w s if the limit of this sequence, if it exists, will be 1 or 2; so we are not allowed to say that this limit exists as a well defined real n u m b e r generator. ([6], page 31.)

by Lakatos is more evident t h a n ever. And as Hersh has suggested the philosophy of m a t h e m a t i c s matters as n e v e r before.

Acknowledgments Research supported by NSERC a n d the Shrum e n d o w m e n t of Simon Fraser University. REFERENCES

[1] David H. Bailey, Jonathan M. Borwein, Peter B. Borwein, and Simon Plouffe, "The Quest for Pi," Mathematical Intelligencer 19, no. 1 (Winter 1997), 50-57. [2] L. Berggren, J. Borwein, and P. Borwein, PI: A Source Book, Springer-Verlag, New York, 1997. [3] J.M. Borwein, P.B. Borwein, and D.H. Bailey, "Ramanujan, modular equations and pi or how to compute a billion digits of pi," American Mathematical Monthly 96 (1989), 201-219. Reprinted in Organic Mathematics Proceedings, http://www.cecm.sfu.ca/ organics, April 12, 1996, with print version: CMS/AMS Conference Proceedings, 20 (1997), ISSN: 0731-1036. [4] L.E.J. Brouwer, "lntuitionische Zerlegung mathematischer Grundbegriffe," Jahresbericht deutsch. Math. Ver. 33 (1925), 251-256. [5] R. Hersh, "Fresh breezes in the philosophy of mathematics," American Mathematical Monthly 102 (1995), 589-594. [6] A. Heyting, Intuitionism: an Introduction, North-Holland Publishing Co., Amsterdam, 1956.

The sequence c o n t i n u e s to occur, n e x t at the 26,852,899,245th place. The first six o c c u r r e n c e s begin at the 17,387,594,880; 26,852,899,245; 30,243,957,439; 34,549,153,953; 41,952,536,161; 43,289,964,000 place respectively. A simple calculation suggests that the probability of the first i n s t a n c e being within the first 25 billion places was over 90%, with the probability of it b e i n g in the first 7 billion u n d e r 50%. Side notes: 09876543210 begins at the 42,321,758,803th digit of ~, and the digits ending at 50 billion are 85,133,987,127,510,930,042. At any rate, s o m e t i m e in July 1997, the three examples above began to behave well. Needless-to-say, the strength of the intuitionist a r g u m e n t is in no way diluted by the destruction of these specific examples. I am m a k i n g n o substantive criticism of the intuitionist/constructivist position or of the h u m a n i s t philosophy espoused by Hersh [5] and others. As effective c o m p u t a t i o n and experimental mathem a t i c s - o f which I a m a passionate e x p o n e n t - - p l a y s a larger and larger role in mathematics, it b e c o m e s less and less satisfactory to rely o n the law of the excluded middle. The "quasi-empirical" n a t u r e of mathematics as argued for

VOLUME 20, NUMBER 1, 1998

15

J.M. WILLS

Spheres and Sausages, Crystals and Catastrophes-and a Joint Packing Theory

a

ll packings in the real world are finite. However, in contrast to the usual study of infinite packings, for finite packings there is no unique theory, but various approaches for special problems.

For a large family of finite packings ("free packings," i.e., without a given container), a general theory has been developed since 1993. This theory is compatible with the classical theory of infinite lattice and nonlattice packings, and permits a joint theory of packings. The crucial point was the introduction of a parameter in [Wl], which led to the widely applicable "parametric density." This density gives a unifying approach to various strange packing phenomena. Further, it allows a description of crystal growth and shape, which completes the structural description of crystals by classical lattice packings. This article gives a brief introduction.

Seven Questions Which packing of four tennis balls is denser: the tetrahedral (Fig. 1) or the linear one? And the same problem for 56 balls (Fig. 2)? Is the linear packing (Fig. 3) the densest packing of five balls? What is the densest packing of n billard balls on a table? How can one compare the density of two crystals of the same structure but different shape, say of two diamonds? And what is the densest packing of identical coins on a table--and in space?

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THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER-VERLAG NEW YORK

These seven questions are special cases of the old problem: How can one measure the density of finite packings if no container or bin is given? To each of these questions, except the last one, one may get various answers, because there are various density definitions for finite packings of this type [GS, SCDH, GW]. They are constructed for special applications and are not based on volume, and hence not compatible with classical infmite packing theory. The joint theory of finite and infinite packings (and coverings), sketched below, will give clear-cut answers to the seven questions.

Finite Packings As usual, let us restrict ourselves to convex bodies, i.e., compact convex sets with interior points in Euclidean dspace E d for d -> 2. A family of convex bodies Ki C E d, i E I, is a packing if the Ki do not overlap, i.e., if int(Ki n Kj)

=

0

for

i #:j.

(1)

If I is finite, the packing is called finite, otherwise infinite. Here, we are especially concerned with finite packings. If a convex body C C E d (a bin or container) is given and

U i@ I

Ki C C

(2)

Figure 1. Tetrahedral packing of four balls.

is required, the finite packing is called a "bin packing" (Fig. 4); otherwise, a "free packing" or simply a packing. For any packing there is a minimal or natural bin

Cnat= conv ( U

Ki),

(3)

tice) packings (e.g., [R0, GL]) in various ways, as is well known. With ~nat, this only holds in E 2. The Norwegian mathematician A. Thue was the first (1892) to use ~nat. He s h o w e d

i.e., the convex hull of UieI hi, which is the intersection of all convex bodies C satisfying (2). Figure 5 s h o w s the convex hull of 11 circles, and in Figure 3, the convex hull and its shadow are s h o w n for a linear packing of 5 balls. Densities For any convex body K C its volume or Lebesgue measure V(K) > 0 is defined, and the density of a bin packing [formulas (1), (2)] is given by

Ea,

Z v(gi) ~bin -- iEl

V(C)

(4)

For any finite packing, one has from formulas (1) and (3) its natural density

~. Y(gi) iel

~nat-

(5)

V(Cnat) "

Obviously, for any bin packing, 0 < ~bin <---~nat --< 1 holds. From 6bin, one obtains the densities of infmite (nonlat-

Figure 2. A d e n s e packing of 56 balls, but not the d e n s e s t one.

VOLUME 20, NUMBERI, 1998

17

Figure 3. Sausage packing of five balls.

The obvious a n s w e r to our seventh question is: The densest packing o f n identical coins in E 3 is a b a n k roll and its density is ~nat = 1 for all n E ~. But t h e d e n s e s t space-

filling p a c k i n g o f identical coins in K 3 has density 0.90 . . . . Similarly, i n E d, d -> 4, for cylinders. Thus, the classical densities c a n n o t b e o b t a i n e d from ~nat. F o r p a c k i n g s of bails in E d and their various p h e n o m ena, s o m e intuitive n o t i o n s have b e e n i n t r o d u c e d by m a t h e m a t i c i a n s and crystallographers: If Cn = { c l , . . . . , Cn} is the set of centers o f the n balls, then the p a c k i n g is called a "sausage" for dim(conv(Cn)) = 1; a "pizza" for 2 -< dim(conv(Cn)) -< d - 1, a n d a "cluster" for dim (conv(Cn)) = d. L. F e j e s TSth's famous s a u s a g e conjecture of 1975 says t h a t for d -> 5 and all n E ~ , the d e n s e s t packing o f n balls is a s a u s a g e (Fig. 3). In K 3, t h e d e n s e s t packing o f n balls is a cluster if n = 56 a n d 59 -< n -< 62, o r n -> 65. It is conj e c t u r e d that for all o t h e r n, t h e sausage is optimal. So und e r this assumption, the s h a p e of d e n s e s t ball p a c k i n g c h a n g e s drastically w h e n n c h a n g e s from n -< 55 to n --- 56. Further, the d e n s e s t p a c k i n g o f 56 balls is n e i t h e r the s a u s a g e n o r the t e t r a h e d r a l p a c k i n g in Figure 2, h u t ( m o s t p r o b a b l y ) it has an o c t a h e d r a l shape. It is c o n j e c t u r e d that in E 4, this a b r u p t change f r o m sausage to cluster h a p p e n s a r o u n d 377,000 [GZ], w h i c h justifies the name "sausage ca-

Figure 4. Two packings of 10 circles in a square with same bin pack-

Figure 5. Dense packing of 11 circles and a p a r t of the hexagonal

ing density,

lattice packing.

t h a t d e n s e p a c k i n g s o f n c i r c l e s t e n d w i t h n ---) ~ to the d e n s e s t l a t t i c e p a c k i n g o f circles, t h e h e x a g o n a l p a c k i n g lattice. So he a n s w e r e d the sixth o f t h e s e v e n questions. In F i g u r e 5, a d e n s e p a c k i n g o f 11 c i r c l e s is shown, a n d Thue's r e s u l t is p l a u s i b l e , a l t h o u g h a n e x a c t p r o o f is difficult. Thue's p r o o f w a s c o m p l e t e d b y Segre, Mahler, and L. F e j e s TSth in 1940, a n d b e t w e e n 1950 a n d 1972, T h u e ' s a p p r o a c h w a s g e n e r a l i z e d b y Rogers, B a m b a h , Woods, Zassenhaus, Oler, G r o e m e r , Graham, a n d others, a n d a j o i n t t h e o r y o f finite a n d infinite p a c k i n g s ( a n d coverings) in E 2 w a s d e v e l o p e d (for details, s e e t h e s u r v e y in [GW]). And ~nat in E3? Our first questions c o n c e r n i n g the 4, 5, and 56 bails all have the s a m e answer: Linear packings of bails ("sausages," s e e Fig. 3) are denser. This p r e p a r e s us to find for d -> 3, that the density ~nat l e a d s to a j u m b l e of strange p a c k i n g p h e n o m e n a .

Sausages, Pizzas, Clusters, and Catastrophes

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THE MATHEMATICALINTELLIGENCER

Figure 6. Dissection of the convex hull of a 4-ball packing.

tastrophe." F o r ball p a c k i n g s in E d, d <- 10, it is k n o w n that either sausages or c l u s t e r s are densest, b u t n e v e r pizzas. In particular, the d e n s e s t p a c k i n g of billiard balls on a table is linear for all n E ~ . In sharp contrast to this result, there are, for all d -> 3, convex bodies (with center) such that their densest packings are always pizzas, as Gritzmann and Arhelger first observed. This "zoo" of strange p h e n o m e n a can be arbitrarily enlarged [GW, Zo], a n d it s h o w s that i n E d, d - 3, a j o i n t theo r y of fmite and infmite p a c k i n g s cannot b e e x p e c t e d . Parametric

Density

The crucial idea for o v e r c o m i n g these difficulties w a s the i n t r o d u c t i o n of a p a r a m e t e r in [Wl]. I explain the s i m p l e s t case: Let Ki = K + ci, i = 1 , . . . , n, be t r a n s l a t e s o f a conv e x b o d y K C E a, let Cn = {cl, 99 9 Cn}, and p > 0. Then

8(K, Cn,p) =

nV(K) V(conv(Cn) + pK)

is the p a r a m e t r i c density, w h e r e "+" is the v e c t o r s u m or M i n k o w s k i sum. F o r p = 1, one gets 8nat, and Figures 3 and 5 are exampies for p = 1. Figure 1 s h o w s a shell of thickness 0.3 around the tetrahedral packing o f four unit balls, h e n c e with p = 1.3. F r o m classical c o n v e x geometry, one k n o w s that

V(c~

+ pK))= Z

(1)

is a polynomial in p of degree d, and the coefficients are Minkowski's m i x e d volumes. Figure 6 shows the dissection o f the convex hull of the tetrahedral packing o f four balls. The teWahedron measures V(conv(C4)) and the four spherical s e g m e n t s in the figure s u m up to the ball, i.e., to V(B3), and these a r e the first and the last m i x e d volumes in t h e s u m in this special case. The other s u m m a n d s can be "seen" if one fills the gaps in this figure appropriately, namely b y four triangular p r i s m s o v e r the facets of the t e t r a h e d r o n and the remaining six cylindrical pieces. One also gets an i m p r e s s i o n of how t h e p a r a m e t e r p controls t h e influence of the b o u n d ary region of the packing. The t h e o r y of m i x e d v o l u m e s is a highly d e v e l o p e d a n d p o w e r f u l tool. After [Wl], various a p p l i c a t i o n s o f p a r a m e t r i c density were p r o v e d in [BHWl] and a series of subsequent articles, and a j o i n t t h e o r y of finite a n d infinite packings (and coverings) w a s d e v e l o p e d (cf. e.g., [BttW2, BS, BW, Sc]). The next s e c t i o n s give a brief survey.

Basic Properties

Of c o u r s e , one is m a i n l y i n t e r e s t e d in d e n s e s t p a c k i n g s . So let 8(K,n,p) d e n o t e the m a x i m a l d e n s i t y w h i c h is att a i n e d b y a p a c k i n g o f t r a n s l a t e s o f a given c o n v e x b o d y K a n d t h e p a r a m e t e r p > 0. In this case, t h e v o l u m e o f conv(Cn) + pK is minimal. A n y C,, with 8(K, Cr~,p)= 8(K,n,p) is d e n o t e d b y Cn,p o r Cn,p(K); a n d C**,p + K is c a l l e d a d e n s e s t p a c k i n g (of n t r a n s l a t e s of K with res p e c t to p). Clearly, Pn,p = conv(C**,p) is a c o n v e x polytope, p o s s i b l y degenerate; for example, in Figure 3 P,,,p is a segment of length 8. The crucial property, first o b s e r v e d in [W1] and later generalized in [BHW1], is n o w that the shape and, in particular, the d i m e n s i o n of the d e n s e s t packing P,,,p d e p e n d s on p roughly as follows: F o r any c o n v e x b o d y K in Euclidean E d and for any n u m b e r n of its translates, there is a "sausage p a r a m e t e r " p~,d(K) > 0 and a "critical parameter" p~,d(K) < ~ such that

VOLUME 20, NUMBER 1, !998

I g

(a) dim Pn,p(K) = 1 for 0 < p < ~,d(K), i.e., the best packings are linear (sausages), Co) 2 <- dim Pn,p(K) --< d - 1 for [2~t,d(g ) < p < [ ~ , d ( g ) , i.e., the best packings are flat (pizzas), (c) dimPn,p(K) = d for p~,d(K) < p < 0%i.e., the best packings are clusters. F o r B 2, i.e., for circle packings in the plane, t~,2 = / 9 ~ , 2 = V~/2, so for p < ~ / 2 , linear packings are densest, and for p > ~ / 2 , clusters are densest, as in Figure 5. For four balls in E 3 and p = 1.3, the tetrahedral packing in Figure 1 is densest [BW], and for five balls and p = 1, it is the sausage in Figure 3. Thus, the various packing phenomena of the "zoo" are unified by the parametric density, and to all questions for densest packings at the beginning, one can give the same philosophical answer: It depends--namely, on the parameter. Even for the packing of coins, the best packing is nonlinear if p is sufficiently large. Although the sphere packings are the most attractive ones, the bank roll (with p -- 1) is the most convincing example for sausage packings: In my earliest talks on fmite packings (1980/81), I started with some identical coins on the overhead screen, to show circle packings in the plane, and then I used the same coins to demonstrate sausage packings in 3-space, and the audience got the point. Next, we come to the asymptotic properties and results, which show the close relation to classical theory.

The Joint Packing Theory For asymptotic investigations, let -1

#(K,p) = lim sup #(K,n,p), Q(k, p) = lim n ~ Pn p(K). n---> ~

n.--)~

'

The central results on ~(K,p) are now that #(K,p) joins finite and classical infmite packings: If 3(K) denotes the classical packing density, then

zalike densest sphere packings are known. A third unsolved problem is the asymptotic shape of densest sphere packings if no restrictions to lattice packings are made [BS1]. I end this section with some remarks on generalizations and applications: 1. With slight modifications, parametric density also applies to coverings and general arrangements, and, in particular, to packings of balls of different size, which is relevant for application to crystal growth [Sc]. 2. Another generalization is 6(K, Cn,pC)= nV(K)/V(conv (Cn) + pC), where the convex body C is an isotropy parameter. This permits connections to bin packings, to anisotropic packings, and to lattice-point inequalities with mixed volumes [BoHW, H].

Application to Crystal Growth The fifth of our seven questions deals with crystals. This is perhaps the most interesting application Of parametric density. If one restricts to lattice packings of balls in a given lattice L, then Q(K,p) is just the Wulff-shape [La, W2, W3]. This is a convex polytope defined by the polar lattice L* of L, which models the shape of ideal crystals, and its shape is controlled by p. The lattice and, hence, the structure are given by the chemical properties of the crystal. Although no energy arguments are used, it is the parameter which simulates the energy flow and models the crystal shape surprisingly well. So the lattice describes the structure of the crystal, and the parameter its shape. Now only metals and rare gases are given by lattice packings of balls of one size. Even in the much more general case of periodic packings of balls (atoms) of different sizes (Fig. 7), one obtains corresponding Wulff-shapes [Sc]. One gets the regular octahedron and its truncations for the diamond (Fig. 8), the cube and its truncations for salt

(~(K,p) = (~(K) for p -> 2 and centrally symmetric K, and for p -> d 4- 1 and arbitrary K [BHW 1]. Similar results hold for restrictions to lattice packings or periodic packings. Consequently, one also obtains fmite analogs of classical packing theorems by Gauss, Blichfeldt, Rogers, and Minkowski-Hlawka--in some cases with weaker constants, which is not surprising, because these new results hold for fmite packings as well as for infinite packings. In one case, Gritzmann improved a classical result with the new methods [GW]. One also obtains relations to classical packings of caps on the sphere. Further, in [BHWl], L. Fejes T6th's famous sausage conjecture was proved for all d -> 13,387, which M. Henk later improved to all d -> 45. The best bound now is d-> 42. The problem for the remaining dimensions d = 5 , . . . , 41 is open. Another open problem is whether (except for n -< d) the densest packing of n balls in E d and any p > 0 can be different from either a sausage or a cluster. Presently, no piz-

20

THE MATHEMATICALINTELLIGENCER

Figure 7. Periodic packing with balls of different sizes: a typical model for crystal structures.

Figure 8. Wulff-shapes of crystals. From the left: table salt, pitchblende, diamond.

(NaC1), and the tetrahedron and its truncations for pitchblende (ZnS), all according to reality. One can model even extreme crystals like whiskers, needles, and prisms. Partial results indicate that also the shapes of microclusters can be modeled as densest packings with parametric density, coinciding with classical energy densities such as the Lennard-Jones potential. In these cases one has to consider packings of ellipsoids rather than balls; the few results obtained until now are encouraging. In the case of nonconvex crystals like snowflakes or fullerenes, the theory cannot be applied and one needs further ideas. One of the most difficult and most interesting questions is whether parametric density can give any information about the shape of quasicrystals. The structural aspects of quasicrystals have been discussed extensively (e.g., [Se]). Very recent investigations [BS2] show that one also gets Wulff-shapes, 10-gons, 20-gons, etc., in the planar case; and in 3-space, in particular, the dodecahedron and its truncations, again in agreement with reality. ACKNOWLEDGMENT

The computer graphics were done by Manuel Huber, a 16year-old student from Pfaffenhofen. REFERENCES

[BHWl] U. Betke, M. Henk, and J.M. Wills, Finite and infinite packings, J. Reine Angew. Math. (Crelle) 453 (1994), 165-191. [BHW2] U. Betke, M. Henk, and J.M. Wills, Sausages are good packings, Discrete Comp. Geom. 13 (1995), 297-311. [BoHW] J. Bokowski, H. Hadwiger, and J.M. Wills, Eine Ungleichung zwischen Volumen, Oberfl&che und Gitterpunktanzahl konvexer Mengen im n-dimensionalen euklidischen Raum, Math. Z 127 (1972), 363-364. [BS1] K. B6r6czky, Jr. and U. Schnell, Asymptotic shape of finite packings, Can. J. Math., in press. [BS2] K. B6r6czky, Jr. and U. Schnell, Quasicrystals and parametric density, in press. [BW] K. B6r6czky, Jr., and J.M. Wills, Finite sphere packings and critical radii, Belt. Geom. AIg. 38 (1997). [FK] G. Fejes Toth and W. Kuperberg, Packing and covering with convex sets, in Handbook of Convex Geometry, Amsterdam: NorthHolland (1993), Chap. 3.3. [GL] P.M. Gruber and C.G. Lekkerkerker, Geometry of Numbers, Amsterdam: North-Holland (1987). [GS] R.L. Graham and N.J.A. Sloane, Penny-packing and two-dimensional codes, Discrete Comp. Geom. 5 (1990), 1-11. [GW] P. Gritzmann and J.M. Wills, Finite packing and covering, in

Handbook of Convex Geometry, Amsterdam: North-Holland (1993), Chap. 3.4. [GZ] P.M. Gandini and A. Zucco, On the sausage catastrophe in 4-space, Mathematika 39 (1992), 274-278. [H] H. Hadwiger, Das Wills'sche Funktional, Monatsh. Math. 79 (1975), 213-221. [La] M. v. Laue, Der Wulffsche Satz for die Gleichgewichtsform von Kristallen, Z Kristallogr. 105 (1943), 124-133. [Re] C.A. Rogers, Packing and Covering, Cambridge: Cambridge University Press (1994). [Sc] U. Schnell, Periodic sphere packings and Wulff-shape, Mathematika, in press. [Se] M. Senechal Quasicrystals and Geometry, Cambridge Univ. Press, Cambridge (1995). [SCDH] N.J.A. Sloane, J.H. Conway, T.D.S. Duff, and R.H. Hardin, Minimal energy clusters of hard sphere, Discrete Comp. Geom. 14 (1995), 237-260. [Wl] J.M. Wills, Finite sphere packings and sphere coverings, Rend. Semin. Mat. Messina, Ser. II 2 (1993), 91-97. [W2] J.M. Wills, On large lattice packings of spheres, Geom. Dedicata 65 (1997), 117-126. [W3] J.M. Wills, Parametric density and Wulff-shape, Mathematika 43 (1996), 229-236. [Zo] Ch. Zong, Strange Phenomena in Convex and Discrete Geometry, New York: Springer-Verlag (1996).

VOLUME 20, NUMBER 1, 1998

21

II~'~lvi|,[~]i,~.~|[.-~-nl[,~.],,n,,,,.,nit[:;-1

The Continuing Silence of BourbakiAn Interview with Pierre Cartier, June 18, 1997 Marjorie Senechal

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and alsofrom scientists, historians, anthropologists, and others.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA; e-maih [email protected]

22

Marjorie

Senechal,

Editor

Nicolas Bourbaki, 1935-???? If you are a mathematician working today, y o u have almost certainly b e e n inf l u e n c e d b y Bourbaki, at least in style a n d spirit, and p e r h a p s to a g r e a t e r extent t h a n you realize. But if y o u are a student, y o u m a y never have h e a r d of it, him, them. What or w h o is, o r was, Bourbaki? Check as many as apply. B o u r b a k l is, o r was, as the case m a y be: 9 t h e d i s c o v e r e r (or inventor, if you p r e f e r ) of the notion of a m a t h e m a t ical structure; 9 o n e of t h e great a b s t r a c t i o n i s t movem e n t s of the twentieth century; 9 a small b u t e n o r m o u s l y influential c o m m u n i t y of mathematicians; 9 a collective that hasn't p u b l i s h e d for fifteen years. The a n s w e r is: all of the above, and t h e y are four closely woven s t r a n d s of an i m p o r t a n t c h a p t e r in intellectual history. Is it time to write that c h a p t e r ? Has the s t o r y of Bourbaki c o m e to an end? B o u r b a k i was born in Paris in 1935 w h e n a small group of m a t h e m a t i c i a n s at the l~cole Normale Sup~rieure, dissatisfied with the courses t h e y w e r e teaching, d e c i d e d to r e f o r m u l a t e them. Most m a t h e m a t i c i a n s have h a d that exp e r i e n c e at one time or another, b u t the s c o p e o f Bourbaki's d i s s a t i s f a c t i o n g r e w quickly and without b o u n d . By 1939, writing as an a n o n y m o u s collective u n d e r the p s e u d o n y m Nicolas Bourbaki, it b e g a n to publish a series of b o o k s i n t e n d e d to t r a n s f o r m t h e theo r y a n d p r a c t i c e of m a t h e m a t i c s itself. F r o m its beginning, B o u r b a k i w a s a fervent believer in the unity a n d univ e r s a l i t y of mathematics, a n d dedic a t e d itself to d e m o n s t r a t i n g b o t h b y r e c a s t i n g all of m a t h e m a t i c s into a unified whole. Its goals w e r e total formalization and perfect rigor. In the p o s t - w a r years, Bourbaki m e t a m o r p h o s e d from rebel to establishment. B o u r b a k i ' s own rules explicitly pro-

THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER-VERLAG NEW YORK

]

vided for self-renewal: from time to time, y o u n g e r m a t h e m a t i c i a n s w e r e invited to join and older members resigned, in a c c o r d a n c e with m a n d a t o r y "retirement" at age fifty. N o w B o u r b a k i itself is n e a r l y t w e n t y y e a r s older t h a n any of its m e m b e r s . The long-running Bourbaki s e m i n a r is still alive and well and living in Paris, but the voice o f Bourbaki i t s e l f - - a s e x p r e s s e d t h r o u g h its b o o k s - - h a s b e e n silent for fifteen years. Will it s p e a k again? Can it s p e a k again? Pierre Cartier w a s a m e m b e r of Bourbaki from 1955 to 1983. Born in Sedan, F r a n c e in 1932, he g r a d u a t e d from the l~cole N o r m a l e Supdrieure in Paris, w h e r e he s t u d i e d under Henri Caftan. His thesis, d e f e n d e d in 1958, w a s on a l g e b r a i c geometry; since t h e n he has contributed to many areas of mathematics, including number theory, group theory, probability, and mathematical physics. P r o f e s s o r Cartier taught at Strasbourg for a decade beginning in 1961, after which he j o i n e d CNRS, the Centre National de la Recherche Scientifique. Since 1971 he has been a p r o f e s s o r at IHES (Institut des Hautes l~tudes Scientifiques) at Bures-sur Yvette, and has taught at the

Pierre Cartier (photo by Marjorie Senechal).

l~cole Polytechnique and at the Ecole Normale, where among other activities he runs a seminar on epistemology. In 1979 he was awarded the A m p e r e Prize of the French Academy of Sciences. Professor Cartier has b e e n involved in various programs to help developing countries, including Chile, Vietnam, and India, build science at home; he is also an editor of a b o o k about art and mathematics. F e w people are better qualified to discuss the silence of Bourbaki. We are grateful to him for agreeing to do so with the r e a d e r s of The M a t h e m a t i c a l Intelligence~: The | n t e ~ i e w S e n e c h a h Please tell u s f i r s t about y o u r o w n connection to Bourbaki. C a r t i e r : As far as I r e m e m b e r , m y first a c q u a i n t a n c e with B o u r b a k i w a s in June 1951. I was a first-year s t u d e n t at the ]~cole Normale, Henri Cartan w a s m y p r o f e s s o r of m a t h e m a t i c s there, a n d at his request Bourbaki invited m e to j o i n their meeting at Pelvoux, in the Alps. I r e m e m b e r that w e d i s c u s s e d m a n y things, especially a t e x t written b y Laurent Schwartz on the foundations o f Lie groups; it w a s one o f the first drafts in the well-known series of B o u r b a k i on Lie groups. It w a s not m a n y years after Schwartz's invention of distributions, which m a d e him famous. You have to understand that the mathematics students at l~cole Normale w e r e all students of both Henri Cag2~ and Laurent Schwartz (who left Nancy for Paris in 1952). We a t t e n d e d their s e m i n a r s and c o u r s e s and t r i e d to use their n e w tools in all directions. F r a n c o i s Bruhat and I w e r e a m o n g the first to u n d e r s t a n d the i m p o r t a n c e of distributions in the theory of Lie groups and their representations. Bruhat devoted his thesis to these topics and I published my own contributions only m u c h later. F o r me, it w a s very i m p o r t a n t to be e x p o s e d from the inside. I w a s surp r i s e d to see all these great p e o p l e I h a d k n o w n from a distance. I w a s acc e p t e d v e r y freely. It t o o k t h r e e o r four m o r e y e a r s before I was formally acc e p t e d as a member. In the fifties and sixties, t h e r e w a s a c o n t i n u o u s spect r u m from the inside core B o u r b a k i to t h e outside. The w o r k that w a s p r i n t e d

in the b o o k s , w h a t was r e p o r t e d in the seminar, a n d t h e w o r k of the s t u d e n t s were closely linked, a n d I think that is one of the r e a s o n s for the g r e a t success o f F r e n c h m a t h e m a t i c s at that time. Of course, those times w e r e v e r y different. The scale w a s much smaller. Then t h e r e w e r e a b o u t ten d o c t o r a t e s a y e a r in m a t h e m a t i c s in France (comp a r e d to t h r e e h u n d r e d today). At t h a t first meeting I was w h a t t h e y call a cobaye, a guinea pig. I w a s v e r y enthusiastic a b o u t it. First of all, it w a s the first thing in m o d e r n m a t h e m a t i c s that I saw. I c a m e from a small city, from a difficult situation b e c a u s e o f the war. I h a d b e e n a student in a very provincial, v e r y o u t d a t e d high school. Some of m y t e a c h e r s were very g o o d but of c o u r s e t h e y w e r e very far a w a y from m o d e r n science. The m a t h e m a t ics I w a s t a u g h t was classical geometry, in t h e uncultivated, synthetic way. I did have the luck to have an imaginative t e a c h e r in physics, and so at first I w a n t e d to b y a physicist. Then I w a s a s t u d e n t at the Lyc~e Saint-Louis in Paris b e f o r e being a c c e p t e d at the Ecole Normale, and I t o o k private lessons in p h y s i c s from a very p e c u l i a r teacher, Pierre Aigrain. (A g r a d u a t e o f the Naval A c a d e m y , he was in 1950 an assistant p r o f e s s o r o f physics; eventually he b e c a m e S e c r e t a r y of State for science u n d e r President Giscard.) Usually a b r i g h t s t u d e n t c o m p l e t e s the p r o g r a m in t w o years, but I m a n a g e d to get t h r o u g h it in one. But b o t h the m a t h e m a t i c s a n d the physics I w a s taught w e r e totally o u t m o d e d at t h a t time, totally. I r e m e m b e r that, in a course called General Physics at t h e Sorbonne, the p r o f e s s o r m a d e a solemn declaration: " G e n t l e m e n " - - h e did not m e n t i o n ladies but there w e r e very few girl s t u d e n t s - - " i n m y class w h a t s o m e p e o p l e call the 'atomic hypothesis' h a s no place." That w a s 1950, five y e a r s after Hiroshima! So I w e n t to Aigrain a n d said, "What do I do?" and he said, "Well, of course, you have to get y o u r degree, b u t I will t e a c h y o u physics properly." This shows w h a t the F r e n c h university w a s at the time. In o r d e r to u n d e r s t a n d the influence o f Bourbaki, y o u have to u n d e r s t a n d that. Bourbaki c a m e into a vacuum. Many p e o p l e have d i s c u s s e d the r e a s o n s w h y

this w a s so; I d o n ' t think this is the place to discuss it again. But obviously in the fifties, t h e early fifties, the teaching of science w a s v e r y poor. It t o o k Bourbaki a b o u t five o r six years to subvert the w h o l e system. By 1957 or '58 the subversion h a d b e e n almost complete, in Paris. S e n e c h a l : B u t B o u r b a k i began i n the thirties... C a r t i e r : The first b o o k was published in 1939, but t h e r e w a s the war, which delayed things, a n d also Andr5 Weil was in the States, Claude Chevalley was in the States, and Lanrent Schwartz had to hide during the war b e c a u s e he is a Jew. B o u r b a k i survived during the w a r with only Henri Caftan and J e a n Dieudonn~. But all the w o r k that had been d o n e in the thirties bloss o m e d in the fifties. I r e m e m b e r how w e - - t h e young m a t h e m a t i c i a n s - - w e r e really eager to go to the b o o k s t o r e to buy the new books. A n d at that time Bourbaki p u b l i s h e d at least one or two v o l u m e s every year. When I formally b e c a m e a m e m b e r of Bourbaki in 1955, I had to abide by the rule that e v e r y o n e should leave at 50, and so I left in 1983, when I was alm o s t 51. I d e v o t e d a l m o s t 30 y e a r s of m y life, and at least one third of my work, to Bourbaki. The working habits of Bourbaki involved very m a n y preliminary drafts of a b o o k before it was published. At the time, w e had three meetings a year, o n e w e e k in the fall, one w e e k in the spring, and two weeks in the summer, which is already one month of hard work, ten o r twelve hours a day. The published b o o k s comprised about 10,000 pages, which means approximately 1000 to 2000 pages of preliminary reports and drafts written every year. I estimate that I contributed about 200 pages a year during all this time with Bourbaki. S e n e c h a l : H o w m a n y people belonged, at that t i m e ? C a r t i e r : A b o u t 12. It was always a small, well-delimited group. The semin a r w a s different, m u c h m o r e open. But still, in the 1950s, if y o u look at the table of c o n t e n t s o f the s e m i n a r volumes, about half the p a p e r s w e r e writt e n by m e m b e r s o f Bourbaki; in those days the interaction b e t w e e n the seminar and the group w a s very strong.

VOLUME 20, NUMBER 1, 1998

23

less a group of students of Grothendieck. Now that's no longer true: it's still a dis- They were the ones to reshuffle matheBut at t h a t time G r o t h e n d i e c k h a d altinguished series but it's usually written matics. The second generation had alr e a d y left Bourbaki. He b e l o n g e d to by people w h o have no direct connec- r e a d y been exposed to the n e w teachB o u r b a k i for a b o u t ten y e a r s b u t he left tion with the institution Bourbaki. But ing. My generation, the third generation, in anger. The personalities w e r e v e r y did not have to prove that the new at that time p e o p l e published in the strong at the time. I r e m e m b e r t h e r e seminar series p a r t o f their discoveries, m e t h o d was better than the old one bew e r e c l a s h e s very often. There w a s or preliminary a c c o u n t s of Bourbaki's cause we were taught with the n e w also, as usual, a fight of generations, m e t h o d basically. I think I w a s just on ideas that later a p p e a r e d in the books. like in a n y family. I think a small group the borderline, because in high school I I was typically a m e m b e r of t h e like t h a t r e p e a t e d m o r e or less t h e psyw a s still taught in the old method, b u t third generation. You can say that chological features of a family. So w e there have b e e n four. The first genera- w h e n I w e n t to Paris I w a s e x p o s e d to had c l a s h e s b e t w e e n generations, tion were the fathers: Andr6 Weil, the n e w thinldng. A n d so w e w e r e less c l a s h e s b e t w e e n brothers, a n d so on. a n d less dogmatic, b e c a u s e w e didn't Henri Cartan, Claude Chevalley, J e a n But t h e y did not distract B o u r b a k i Delsarte, and J e a n Dieudonn6, p e o p l e have to prove anything. The core of from his m a i n goal, even t h o u g h t h e y w h o f o u n d e d the group in the thirties. F r e n c h m a t h e m a t i c s h a d s u r r e n d e r e d w e r e quite brutal occasionally. At least (Others j o i n e d in t h e beginning, but left to Bourbaki. Bourbaki h a d a l r e a d y the goal w a s clear. There w e r e a few soon.) Then t h e r e w a s a s e c o n d gen- s e i z e d power, not only in intellectual p e o p l e w h o could n o t t a k e the eration, p e o p l e invited to j o i n b u r d e n of this p s y c h o l o g i c a l during or j u s t after the war: You can think of the first books of style, for instance Grothendieck Laurent Schwartz, Jean-Pierre Serre, Pierre Samuel, Jean-Louis Bourbaki as an encyclopedia. If you left and also Lang d r o p p e d out. S e n e c h a l : D i d the goals s t a y Koszul, J a c q u e s Dixmier, Roger consider it as a textbook, it's a clear i n people's m i n d s all the Godement, and Sammy disaster. time, or w e r e they c h a n g i n g ? Eilenberg. The third generation C a r t i e r : They changed. The was A r m a n d Borel, A l e x a n d r e first generation had first to creGrothendieck, F r a n q o i s Bruhat, ate a p r o j e c t from nothing. They h a d t e r m s but also in a c a d e m i c terms. It myself, Serge Lang, a n d J o h n Tate. to invent a method. Then in the forties Senechal: D i d these generations dif- w a s clear that from an institutional you c a n s a y that the m e t h o d h a d p o i n t of view, B o u r b a k i h a d won. f e r i n their a t t i t u d e s or outlook? e m e r g e d a n d Bourbaki k n e w w h e r e to If you l o o k at the v o l u m e s on Lie C a r t i e r : They w e r e very different. I groups, you will see t h a t t h e later ones go: his goal w a s to provide the founthink they b e c a m e m o r e and m o r e d a t i o n for mathematics. They h a d to pragmatic, a n d less a n d less dogmatic. have c h a p t e r s that y o u d o n ' t e x p e c t in s u b m i t all m a t h e m a t i c s to the s c h e m e Bourbaki. It b e c a m e m o r e a n d m o r e Senechal: A n d h o w did that s h o w u p of Hilbert; w h a t van der W a e r d e n h a d explicit; t h e r e are t a b l e s a n d drawings. in Bourbaki's work? I t h i n k this was basically t h e influence done for a l g e b r a w o u l d have to b e C a r t i e r : F r o m the beginning, the done for t h e rest of mathematics. What Bourbaki t r e a t i s e w a s conceived as o f one person, A r m a n d Borel. He w a s s h o u l d b e i n c l u d e d was m o r e o r less comprising t w o parts. The first part is f o n d of quoting Shaw, "It's the Swiss clear. The first six b o o k s o f B o u r b a k i on f o u n d a t i o n s a n d consists of six n a t i o n a l character, m y d e a r lady," and c o m p r i s e the basic b a c k g r o u n d knowlbooks, on set theory, algebra, general v e r y often during a d i s c u s s i o n he edge o f a m o d e r n graduate student. topology, e l e m e n t a r y calculus, topo- w o u l d say, "I'm the Swiss peasant." Of course at that time differential The m i s u n d e r s t a n d i n g w a s that logical v e c t o r s p a c e s , a n d (Lebesgue's) m a n y p e o p l e thought that it s h o u l d b e integration theory. The last four of geometry was blossoming, and it had altaught t h e w a y it w a s w r i t t e n in the these b o o k s give the foundations of w a y s been a great challenge to Bourbald. books. You can think of the first b o o k s analysis, as p e r c e i v e d by Bourbaki, You have to r e m e m b e r that t h e father of of B o u r b a k i as an e n c y c l o p e d i a of with a strong bias t o w a r d functional Henri Caftan was Elie Caftan, the m a t h e m a t i c s , containing all the n e c e s geometer, and the Bourbaki recognized analysis. The s e c o n d part, falling s h o r t s a r y information. That is a g o o d deCaftan, and had of more ambitious projects, consists of only one godfather, r scription. If y o u c o n s i d e r it as a texttwo very s u c c e s s f u l series, on Lie m u c h dislike for all the other French book, it's a disaster. groups and on commutative algebra. mathematicians of the thirties. Bourbaki S e n e c h a l : Were you aware o f that Looking b a c k at the list of the Bourbaki c a m e to terms with Poincar6 only after w h e n y o u were a m e m b e r o f Bourbaki? m e m b e r s of the s e c o n d and third gen- a long struggle. When I j o i n e d the group Did people i n B o u r b a k i r e a l i z e that erations, you realize that some of the in the fifties it was not the fashion to this w a s not a textbook? world's leading e x p e r t s of the time were value Poincar6 at all. He w a s old-fashC a r t i e r : More o r less, b u t n o t so ioned. Of course, the o p i n i o n a b o u t there, and that accounts for the breadth clearly as now. There was s o m e misand depth of the second part of Poincar6 has completely changed. But u n d e r s t a n d i n g a b o u t that, I s u p p o s e it's clear that his style and Bourbaki's Bourbaki's work. b e c a u s e w e didn't have t e x t b o o k s . I reThe older generation had learned style were totally different. m e m b e r v e r y well h o w I l e a r n e d algeThe fourth generation was more or mathematics in the old-fashioned way.

24

THE MATHEMATICALINTELLIGENCER

b r a and topology. W h e n I was a student, every time t h a t B o u r b a k i published a n e w book, I w o u l d j u s t b u y it o r b o r r o w it from t h e library, and learn it. F o r me, for p e o p l e in m y generation, it was a textbook. But t h e misunderstanding was that it s h o u l d be a textb o o k for everybody. That w a s the big disaster. Anyway, by t h e n the s c o p e of the p r o j e c t w a s m o r e o r less clear. But w h a t should B o u r b a k i d o after that? The s e c o n d generation h a d an existing m e t h o d , a n d had j u s t to develop a proj e c t with clearly d e l i n e a t e d boundaries. The third g e n e r a t i o n had to go b e y o n d that, to go into the o p e n world, w h i c h meant, at that time, g e o m e t r y in a general way: algebraic geometry, differential geometry, s e v e r a l c o m p l e x variables, Lie groups, m o d u l i spaces, a n d so on. I think I'm responsible for the idea that Bourbald should devote a special chapter to the geometry of crystallographic groups. The reasons for that are clearly stated in the introduction to the series on Lie groups. C o x e t e r was the first to understand the relation of Lie groups to the crystallographic groups a n d their classification. Certainly the p e o p l e w h o were w o r k i n g on Lie g r o u p s were, by spirit, m o r e geometrical a n d m o r e p r a g m a t i c t h a n the others. But I r e m e m b e r that I h a d to fight quite h a r d to convince m y colleagues within Bourbaki that crystallographic groups should be given p r e e m i n e n c e . S e n e c h a l : What w a s B o u r b a k i ' s opini o n o f Coxeter? C a r t i e r : I think that b y the sixties peop l e realized the i m p o r t a n c e of his work. Borel had m a n y o f the s a m e i d e a s a n d Jacques Tits also p l a y e d a role. Tits w a s m u c h c l o s e r in spirit, in his w a y of doing m a t h e m a t i c s , to C o x e t e r than to Bourbaki. He w a s n ' t formally a m e m b e r o f B o u r b a k i b u t he h a d a long c o l l a b o r a t i o n with us. So w e c o u l d t h a n k him, in t h e books, for his c o l l a b o r a t i o n without b r e a k i n g the rule o f anonymity. Tits w a s very generous: he supplied us with m a n y o f the exercises, and m a n y o f his results w e r e p u b l i s h e d for the first time in Bourbaki volumes. But of c o u r s e he h a d a very different w a y of thinking a b o u t mathematics.

In t h e s e c o n d generation a n d third generation, the two main s e r i e s w e r e c o m m u t a t i v e algebra (with a l g e b r a i c g e o m e t r y in the b a c k g r o u n d ) on the one hand, and Lie groups on t h e o t h e r hand. A n d there is an obvious difference o f style and of emphasis, d e s p i t e t h e fact t h a t at that time B o u r b a k i w a s really a collective and e v e r y o n e cont r i b u t e d to every book, m o r e o r less. Serre w a s a m a s t e r of b o t h sides; he w a s n o t an e x p e r t in Lie g r o u p s at first b u t he b e c a m e one. Serre w a s t h e natural l e a d e r in the s e c o n d g e n e r a t i o n b e c a u s e , like Weil in the first generation, he w a s the only one with a really universal a p p r o a c h to m a t h e m a t i c s . But n e i t h e r of t h e m w a s an analyst. Certainly the contents o f B o u r b a k i w e r e m u c h m o r e about algebra, algebraic geometry, than a b o u t analysis. By t h e fourth generation t h e goal w a s less visible. G r o t h e n d i e c k h a d dev e l o p e d his own program, o u t s i d e of Bourbaki, so the n e e d for a B o u r b a k i w a s less obvious. And t h e r e w a s also s o m e l a c k of a global u n d e r s t a n d i n g of m a t h e m a t i c s . The m e m b e r s h a d bec o m e m o r e specialized in their interests. T h e r e w e r e various a t t e m p t s within t h e g r o u p to focus on n e w projects. F o r instance, for awhile the i d e a w a s that y o u s h o u l d develop the t h e o r y o f several c o m p l e x variables, a n d m a n y drafts w e r e written. But it n e v e r matured, I think partly b e c a u s e it w a s t o o late. T h e r e were already m a n y g o o d t e x t b o o k s on several c o m p l e x varia b l e s in the seventies, b y G r a u e r t and o t h e r people. By the end o f the seventies, the m e t h o d of Bourbaki h a d b e e n so well u n d e r s t o o d that e v e r y o n e k n e w h o w to write in this spirit. There w a s a w h o l e generation o f t e x t b o o k s , a n d b o o k s , w h i c h were u n d e r his influence. B o u r b a k i was left w i t h o u t a task, a n d so he d e c i d e d to d e v o t e p a r t of his e n e r g y to revising his o w n books, the so-called "New Edition." The revision w a s m o s t l y completed; t h e s e w e r e really t h o r o u g h revisions. S e n e c h a l : Do the r e v i s i o n s i n c l u d e a change o f style? C a r t i e r : No, no. But for instance, the s e c t i o n on the topology o f m e t r i c spaces was much more developed and d e e p e n e d , the proofs w e r e improved,

a n d there is a small v o l u m e that t r i e d to bridge the gap b e t w e e n p r o b a b i l i t y t h e o r y a n d the w a y that Bourbaki presented Lebesgue integration theory. That was a n a t t e m p t to c o r r e c t one obviously m i s t a k e n point o f view o f Bourbaki. S e n e c h a l : What other areas o f m a t h e m a t i c s do y o u see n o w as h a v i n g been left outside? C a r t i e r : First o f all analysis, although there is an elementary calculus text, a very good book, that was the influence of Jean Delsarte. There is essentially no analysis b e y o n d the foundations: nothing about partial differential equations, nothing a b o u t probability. There is also nothing a b o u t combinatorics, nothing about algebraic topology, nothing a b o u t concrete geometry. And Bourbaki never seriously c o n s i d e r e d logic. Dieudonn~ himself w a s very vocal against logic. Anything c o n n e c t e d with m a t h e matical p h y s i c s is totally a b s e n t f r o m B o u r b a k i ' s text. In the Bourbaki seminar, I c o n t r i b u t e d a long series of pap e r s with e m p h a s i s on questions o f m a t h e m a t i c a l physics. But I was t h e only one, a n d m y contributions w e r e not always a c c e p t e d without a fight. But even in t h e a r e a s o f m a t h e m a t ics that w e r e n o t c o n s i d e r e d b y Bourbaki, looking b a c k w a r d s over t h e last thirty years, it is obvious that t h e i r d e v e l o p m e n t h a s b e e n v e r y m u c h influenced b y the Bourbaki spirit. S e n e c h a l : Was there a bias a g a i n s t p h y s i c s , or did B o u r b a k i j u s t n o t t h i n k about it? C a r t i e r : Well, o f course there w a s a strong bias against it, for m o s t people. At the beginning I s u p p o s e I w a s slightly h e t e r o d o x within the B o u r b a k i group. I h a d a longstanding interest in m a t h e m a t i c a l physics. A few y e a r s ago, in a d i s c u s s i o n with Andr~ Weil, j u s t after he p u b l i s h e d his o w n memoirs, I said, "You m e n t i o n e d that in 1926 y o u w e r e at G(ittingen . . . in 1926 s o m e thing h a p p e n e d in GSttingen." A n d Weil asked, "What did h a p p e n in GSttingen?" a n d I said "Oh! Q u a n t u m mechanics!" A n d Weil said, "I d o n ' t k n o w w h a t t h a t is." He was a s t u d e n t of Hilbert in 1926 and Hilbert h i m s e l f was i n t e r e s t e d in quantum mechanics, Max Born w a s there, Heisenberg w a s there, a n d others, but a p p a r e n t l y

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Andr~ Weil didn't p a y any attention to it. I recently h a d a n o c c a s i o n to give a public lecture a b o u t the p h i l o s o p h y o f s p a c e of H e r m a n n Weyl, so I r e a d the literature a b o u t him carefully. There is an obituary of H e r m a n n Weyl written by Chevalley a n d Well. They praise him, for g o o d r e a s o n s , b u t there is no mention of his w o r k in physics, not even his w o r k in general relativity. By all accounts, the t w o b e s t b o o k s o f Weyl are his b o o k on general relativity and his b o o k on q u a n t u m mechanics! S e n e c h a l : Bourbaki's last publication w a s i n 1983. Why doesn't it p u b l i s h anything now? C a r t i e r : There are several r e a s o n s for that. First, t h e r e w a s a clash b e t w e e n Bourbaki and his publisher, a b o u t royalties and t r a n s l a t i o n rights, ending in a long and u n p l e a s a n t legal process. When the m a t t e r w a s settled in 1980, Bourbaki w a s a l l o w e d to m a k e a deal with a n e w publisher. Using the extensive w o r k d o n e in the seventies tow a r d s the revision o f the old books, w e were able to r e p u b l i s h t h e m in a n e w edition. We c o m p l e t e d the existing series by two or t h r e e m o r e volumes, but t h e n . . , silence. Beyond the easily s t a t e d goal of a

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"final edition," Bourbaki struggled in the seventies and the eighties to form u l a t e n e w directions. I m e n t i o n e d alr e a d y a failed p r o j e c t a b o u t several c o m p l e x variables. T h e r e w e r e att e m p t s at h o m o t o p y theory, at s p e c t r a l t h e o r y of operators, at t h e i n d e x theorem, at symplectic geometry. But n o n e o f t h e s e projects w e n t b e y o n d a preliminary stage. B o u r b a k i could n o t find a n e w outlet, b e c a u s e they had a d o g m a t i c view o f mathematics: everything s h o u l d b e s e t inside a secure f r a m e w o r k . That w a s quite r e a s o n a b l e for general topology and general algebra, w h i c h w e r e a l r e a d y solidified a r o u n d 1950. Most p e o p l e agree n o w that y o u do n e e d general foundations for m a t h e m a t i c s , at least if you believe in the unity of m a t h e m a t i c s . But I believe n o w that this unity should be organic, while Bourbaki advocated a structural point of view. In accordance with Hilbert's views, set theory was thought b y Bourbaki to provide that badly n e e d e d general framework. If you n e e d s o m e logical foundations, categories are a m o r e flexible tool than set theory. The point is that categories offer both a general

philosophical f o u n d a t i o n - - t h a t is ,the encyclopedic, or taxonomic p a r t - - a n d a very efficient mathematical tool, to be used in mathematical situations. That set theory and structures are, by contrast, m o r e rigid can be seen b y reading the fmal c h a p t e r in Bourbaki set theory, with a m o n s t r o u s endeavor to formulate categories without categories. It is amazing that c a t e g o r y t h e o r y was m o r e o r less the b r a i n c h i l d o f Bourbaki. The two f o u n d e r s w e r e Eilenberg a n d MacLane. MacLane w a s n e v e r a m e m b e r of Bourbaki, b u t Eilenberg was, and MacLane w a s c l o s e in spirit. The first t e x t b o o k on h o m o logical a l g e b r a was Cartan-Eilenberg, which w a s p u b l i s h e d when b o t h w e r e very active in Bourbaki. Let us also m e n t i o n Grothendieck, who d e v e l o p e d c a t e g o r i e s to a very large extent. I have b e e n using categories in a c o n s c i o u s o r u n c o n s c i o u s w a y in much o f m y work, and s o h a d m o s t of the B o u r b a k i m e m bers. But b e c a u s e the w a y o f thinking w a s t o o dogmatic, or at least t h e pres e n t a t i o n in the b o o k s was t o o dogmatic, B o u r b a k i could not a c c o m m o date a c h a n g e of emphasis, o n c e the p u b l i c a t i o n p r o c e s s was started. I t h i n k the eighties were a n a t u r a l limit. U n d e r the p r e s s u r e o f Andr~ Weil, B o u r b a k i insisted t h a t every m e m b e r s h o u l d retire at fifty, a n d I rem e m b e r that, in m y eighties, I said, as a joke, t h a t Bourbaki s h o u l d retire w h e n he r e a c h e s fifty. S e n e c h a l : It seems that this m o r e or less happened. C a r t i e r : Yes, I think one of t h e m a i n r e a s o n s is that its stated goal, to provide f o u n d a t i o n s for all existing mathematics, w a s achieved. But also, if y o u have s u c h a rigid format it is v e r y difficult to i n c o r p o r a t e n e w developments. If the e m p h a s i s d o e s n ' t change, it's still possible. But of course, after fifty years, the e m p h a s i s had changed. S e n e c h a l : Would y o u say a little m o r e about that? C a r t i e r : Andr~ Weil w a s f o n d of s p e a k i n g o f the Zeitgeist, the spirit of the times. It is no a c c i d e n t that B o u r b a k i lasted from the beginning o f the thirties to the eighties, while the Soviet s y s t e m lasted from 1917 to 1989. Andr~ Weil does not like this c o m p a r ison. He says repeatedly, "I've n e v e r

b e e n a communist!" T h e r e is a j o k e t h a t the 20th c e n t u r y lasted from Sarajevo 1914 to Sarajevo 1989. The 20th century, from 1917 to 1989, has b e e n a century of ideology, the ideological age. S e n e c h a l : By ideology, do you m e a n the idea of a blueprint that can serve f o r all purposes and f o r all time? C a r t i e r : A f i n a l solution. There are g o o d r e a s o n s to h a t e t h a t expression, b u t it w a s in the p e o p l e ' s m i n d s that w e could reach a f'mai solution. There is a b o o k by H.G. Wells called A Modern Utopia, w h i c h ought to be reprinted. By c h a n c e I w a s reading it j u s t at the time of the c o l l a p s e of the Soviet system. As y o u know, H.G. Wells w a s certainly v e r y friendly tow a r d s the O c t o b e r 1917 revolution, he w a s a friend of the Soviets, admittedly. But he h a d a very s h a r p m i n d and he h a d such a sharp historical view that he could envision d e v e l o p m e n t s . Even t h o u g h he w a s e x c i t e d b y this n e w era, he u n d e r s t o o d that the final solution d o e s n ' t exist and t h a t it w a s a mistake to c o n s i d e r that you c a n r e a c h such a s t a t e of social historical equilibrium that from then on s o c i e t y will stay as it is forever. Wells a r g u e d very well against this idea. If y o u r e a d his books, y o u will see that as one of his obsessions. Hilbert, I think, r e f l e c t e d this Zeitgeist. There is one r e c o r d i n g of his voice; in Constance Reid's b o o k a b o u t Hilbert t h e r e is a f l o p p y disk of it, a r e c o r d o f s o m e s p e e c h t h a t Hilbert gave in G e r m a n y in the thirties. It's v e r y ideological. At the time Hilbert w a s aging and so his views w e r e solidifying. If you p u t the m a n i f e s t o o f the surrealists and the i n t r o d u c t i o n of B o u r b a k i side by side, as well as o t h e r m a n i f e s t o s of the time, t h e y l o o k very similar. My daughter is c u r r e n t l y translating a b o o k a b o u t t h e birth of cinematography, and in a c h a p t e r a b o u t the Italian futurists there is a v e r y similar statement. In science, in art, in literature, in politics, e c o n o m i c s , social affairs, there was the s a m e spirit. The s t a t e d goal of B o u r b a k i w a s to create a new mathematics. He d i d n ' t cite any o t h e r m a t h e m a t i c a l texts. B o u r b a k i is self-sufficient. Of c o u r s e at the time

the c o m m u n i s t s in the Soviet Union w e r e claiming the same. We k n o w n o w it w a s a lie, and that the l e a d e r s k n e w at the t i m e they were lying. Certainly B o u r b a k i was n o t lying, b u t still, the spirit w a s the same. It was the t i m e of ideology: Bourbaki was to b e the New Euclid, h e w o u l d write a t e x t b o o k for the n e x t 2000 years. S e n e c h a l : Why is there a lack o f any kind o f visual illustration in m o s t of Bourbaki? C a r t i e r : I think the best a n s w e r w o u l d be the d e s c r i p t i o n of Chevalley given by his d a u g h t e r [see insert]. The B o u r b a k i w e r e Puritans, a n d Puritans are s t r o n g l y o p p o s e d to p i c t o r i a l repr e s e n t a t i o n s of truths o f their faith. The n u m b e r of P r o t e s t a n t s a n d J e w s in the B o u r b a k i group was o v e r w h e l m ing. A n d you k n o w that the F r e n c h P r o t e s t a n t s especially are v e r y c l o s e to J e w s in spirit. I have s o m e J e w i s h b a c k g r o u n d and I was r a i s e d as a Huguenot. We are p e o p l e of the Bible, of the Old Testament, a n d m a n y H u g u e n o t s in F r a n c e are m o r e enamo r e d o f t h e Old T e s t a m e n t t h a n o f the New Testament. We w o r s h i p J a w e h m o r e t h a n J e s u s sometimes. So, w h a t were the reasons? The general philosophy as developed b y Kant, certainly. Bourbaki is the brainchild of G e r m a n philosophy. Bourbald was founded to develop and propagate German philosophical views in science. Andrd Weft has always b e e n fond of German science and he w a s always quoting Gauss. All these people, with their o w n tastes and their o w n personal views, w e r e proponents of G e r m a n philosophy. A n d t h e n t h e r e was t h e i d e a that t h e r e is an o p p o s i t i o n b e t w e e n art a n d science. Art is fragile and mortal, bec a u s e it a p p e a l s to emotions, to visual meaning, a n d to u n s t a t e d analogies. But I think it's also p a r t o f the E u c l i d e a n tradition. In Euclid, y o u find s o m e d r a w i n g s but it is k n o w n that m o s t of t h e m w e r e a d d e d after Euclid, in l a t e r editions. Most of the d r a w i n g s in the original are a b s t r a c t drawings. You m a k e s o m e reasoning a b o u t s o m e p r o p o r t i o n s and you d r a w s o m e segments, b u t they are not i n t e n d e d to b e g e o m e t r i c a l segments, j u s t r e p r e s e n t a tions of s o m e a b s t r a c t notions. Also

Lagrange p r o u d l y stated, in his textb o o k on m e c h a n i c s , "You will not find any drawing in m y book!" The analytical spirit w a s p a r t o f the F r e n c h tradition and p a r t of the G e r m a n tradition. And I s u p p o s e it w a s also due to t h e influence of p e o p l e like Russell, w h o claimed t h a t t h e y could prove everything f o r m a l l y - - t h a t so-called geometrical intuition w a s not reliable in proof. Again B o u r b a k i ' s a b s t r a c t i o n s a n d disdain for visualization w e r e part o f a global fashion, as illustrated by the abstract t e n d e n c i e s in the music and the paintings o f that period. S e n e c h a l : Did the members o f Bourbaki appreciate abstract m u s i c and abstract art? C a r t i e r : I d o n ' t think there was m u c h taste for a b s t r a c t music o r art. You could s a y that on the w h o l e they h a d s t a n d a r d b o u r g e o i s tastes. E d u c a t e d b o u r g e o i s - - n o t philistine. F o r instance, b o t h Cartan and Dieudonn~ w e r e lovers a n d p r a c t i t i o n e r s of music, b u t they w e r e v e r y classical. Caftan certainly, in his P r o t e s t a n t education, w a s very fond o f Bach, and Dieudonn~ w a s quite a g o o d p i a n o player, at an a m a t e u r level, b u t quite good, and he h a d a fantastic memory. He k n e w hundreds and h u n d r e d s o f p a g e s of s c o r e b y heart a n d could follow every single note. I r e m e m b e r I h a d a few o c c a s i o n s to go to the c o n c e r t hall with him. It w a s fascinating, he w o u l d look at t h e score in his h a n d a n d exclaim "OH!" if a note w a s missing from the orchestra! He d e v o t e d the last six m o n t h s of his l i f e - - w h e n he d e c i d e d that his mathematicai life w a s finished, he had written his last book, and he r e t r e a t e d to his h o m e - - t o listening to recordings and following t h e s c o r e s and the notes. It's interesting to k n o w that revolutionaries in m a t h e m a t i c s w e r e not revolutionaries in o t h e r things. I s u p p o s e that the only p e r s o n in the B o u r b a k i group w h o w a s really a w a r e of the connections o f the B o u r b a k i ideology with other ideologies w a s Chevalley. He was a m e m b e r o f various avant-garde groups, b o t h in politics a n d in the arts. As the e d i t o r o f Chevalley's work, I have decided, at the urging of his daughter, to include a special v o l u m e a b o u t his w o r k outside mathematics. He had w r i t t e n various pamphlets, a n d

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various notes; Catherine Chevalley will have to w o r k h a r d to collect t h e s e things and w e will p u b l i s h t h e m as p a r t of his c o l l e c t e d works. Chevalley w a s t h e only one w h o p e r c e i v e d the c o n n e c t i o n b e t w e e n Bourbaki a n d the rest, and that m a y b e why, in the seventies, he was m o r e critical than o t h e r p e o p l e . In the seventies a sensible p e r s o n c o u l d already see the end of a long historical trend, and I think he w a s v e r y sensitive to this. Mathematics w a s t h e m o s t i m p o r t a n t part of his life, b u t he did n o t d r a w any b o u n d a r y b e t w e e n his m a t h e m a t i c s and the rest o f his life. Maybe this w a s b e c a u s e his f a t h e r w a s an a m b a s s a d o r , so he h a d m o r e c o n t a c t with various people. S e n e e h a l : Could you state the m a i n reasons for the decline of Bourbaki? C a r t i e r : As I said, in the eighties there was no longer a s t a t e d goal, e x c e p t for the long legal battle. I think it was one of the c a s e s o f the century! We hired a famous l a w y e r w h o h a d fought for the heirs of P i c a s s o a n d Fujita. We survived artificially: w e h a d to win this battle. But it w a s a pyrrhic victory. As usual in legal battles, b o t h parties lost and the l a w y e r got rich. In fame and in pocket. In a s e n s e B o u r b a k i is like a dinosaur, the h e a d t o o far a w a y from t h e tail. When Dieudonn~ was the scribe of Bourbaki, for m a n y m a n y years, every p r i n t e d w o r d c a m e from his pen. Of c o u r s e t h e r e h a d b e e n m a n y drafts and p r e l i m i n a r y versions, b u t the p r i n t e d v e r s i o n w a s a l w a y s from the p e n o f Dieudonn& A n d with his fantastic m e m o r y , he k n e w every single word. I r e m e m b e r , it w a s a joke, y o u could say, "Dieudonn~, w h a t is this result a b o u t so a n d so?" a n d he w o u l d go to the shelf a n d t a k e d o w n the b o o k a n d open it to t h e right page. After Dieudonn~ ( a n d an interlude b y Samuel and D i x m i e r ) I w a s the secret a r y of Bourbaki, a n d it w a s m y d u t y to do m o s t of t h e proofreading, I think I p r o o f r e a d five to t e n t h o u s a n d pages. I have a g o o d visual m e m o r y . I wouldn't c o m p a r e m y s e l f with Dieudonn~, b u t t h e r e w a s a time w h e n I k n e w m o s t of the p r i n t e d m a t e r i a l in Bourbaki. But no o n e after me w a s able to do this. So B o u r b a k i lost the

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THE MATHEMATICALINTELLIGENCER

a w a r e n e s s of his o w n body, the 40 p u b l i s h e d volumes. A n d as I said before, B o u r b a k i w a s m o r e or less like a family. The s e c o n d o r third o r fourth g e n e r a t i o n in any family or any social group follows definite sociological patterns. My o w n family was typical. My g r a n d f a t h e r w a s a self-made man, a v e r y successful b u s i n e s s m a n . My f a t h e r a n d m y uncle w e n t into the business, b u t t h e y w e r e n o t s o d e v o t e d to t h e fight. A n d peop l e in m y g e n e r a t i o n - - w e l l , I s u p p o s e I m a d e the right d e c i s i o n n o t to engage in it. Indeed, p e o p l e in m y generation w h o did go into our family b u s i n e s s did n o t do so well, b e c a u s e t h e y didn't have anything to fight for. But t h e s e are the i n n e r workings. Of c o u r s e the outside w o r l d also has an influence. That the o u t s i d e m a t h e m a t ical w o r l d has c h a n g e d is obvious. We all k n o w that what Stalin c o u l d never achieve with his army, to c o n q u e r the world, the collapse o f the Soviet Union h a s achieved for m a t h e m a t i c s . The Russian m a t h e m a t i c i a n s have brought a different style to the West, a different w a y of looking at t h e p r o b l e m s , a n e w blood. It's a different time, with different values. I have no regrets: I think it was w o r t h w h i l e to live in t h e t w e n t i e t h century. But n o w it's finished. S e n e e h a l : How would you describe your j o u r n e y with Bourbaki ? C a r t i e r : I have been personally very happy, because when I r e a c h e d the time of normal retirement from Bourbaki, I h a d the very fortunate opportunity to be a s k e d to deliver the lecture on behalf of Vladimir Drinfel'd at the International Congress of Mathematicians at Berkeley in 1986 (Drinfel'd was p r e v e n t e d from coming for political reasons). It was a great challenge and a great h o n o r for me; his p a p e r is one o f t h e m o s t imp o r t a n t p a p e r s in the proceedings. Overnight that changed m y m a t h e m a t ical life. I said, "This is w h a t I have to do now." Of course I k n e w the basic m a t e r i a l b u t the p e r s p e c t i v e w a s new. I c a n n o t claim that within the few h o u r s I h a d to p r e p a r e t h e lecture I c o u l d really m a s t e r it, b u t I u n d e r s t o o d e n o u g h to explain to the people, "This is new, it is important." When I began in mathematics the

main t a s k of a mathematician was to bring o r d e r and m a k e a synthesis of existing material, to create w h a t T h o m a s Kuhn called n o r m a / science. Mathematics, in the forties and fifties, was undergoing w h a t Kuhn calls a solidification period. In a given science there are times w h e n you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The p u r p o s e of mathematics, in the fifties and sixties, was that, to create a new e r a of n o r m a l science. N o w we are again at the beginning of a n e w revolution. Mathematics is undergoing major changes. We don't k n o w exactly where it will go. It is not yet time to m a k e a synthesis of all these things---maybe in twenty or thirty y e a r s it will b e time for a new Bourbald. I consider myself very fortunate to have had two lives, a life of normal science and a life of scientific revolution. BIBLIOGRAPHY

Nicolas Bourbaki, Faits et I~gendes, by Michele Chouchan, I~ditions du Choix, Argenteuil, 1995. "NicholasBourbaki,CollectiveMathematician:an interviewwith ClaudeChevalley,"by DenisGuedj, translated by Jeremy Gray, The Mathematical Intelligencer, vol. 7, no. 2, 18-22, 1985. Les Math~matiques et I'Art, by Pierre Cartier, Institut des Hautes #tudes Scientifiques preprint IHES/M/93/33. "The Continuing Silence of a Poet," by A. B. Yehoshua, in The Continuing Silence of a Poet." collected stories, Penguin Books, 1991 ..

various notes; Catherine Chevalley will have to w o r k h a r d to collect t h e s e things and w e will p u b l i s h t h e m as p a r t of his c o l l e c t e d works. Chevalley w a s t h e only one w h o p e r c e i v e d the c o n n e c t i o n b e t w e e n Bourbaki a n d the rest, and that m a y b e why, in the seventies, he was m o r e critical than o t h e r p e o p l e . In the seventies a sensible p e r s o n c o u l d already see the end of a long historical trend, and I think he w a s v e r y sensitive to this. Mathematics w a s t h e m o s t i m p o r t a n t part of his life, b u t he did n o t d r a w any b o u n d a r y b e t w e e n his m a t h e m a t i c s and the rest o f his life. Maybe this w a s b e c a u s e his f a t h e r w a s an a m b a s s a d o r , so he h a d m o r e c o n t a c t with various people. S e n e e h a l : Could you state the m a i n reasons for the decline of Bourbaki? C a r t i e r : As I said, in the eighties there was no longer a s t a t e d goal, e x c e p t for the long legal battle. I think it was one of the c a s e s o f the century! We hired a famous l a w y e r w h o h a d fought for the heirs of P i c a s s o a n d Fujita. We survived artificially: w e h a d to win this battle. But it w a s a pyrrhic victory. As usual in legal battles, b o t h parties lost and the l a w y e r got rich. In fame and in pocket. In a s e n s e B o u r b a k i is like a dinosaur, the h e a d t o o far a w a y from t h e tail. When Dieudonn~ was the scribe of Bourbaki, for m a n y m a n y years, every p r i n t e d w o r d c a m e from his pen. Of c o u r s e t h e r e h a d b e e n m a n y drafts and p r e l i m i n a r y versions, b u t the p r i n t e d v e r s i o n w a s a l w a y s from the p e n o f Dieudonn& A n d with his fantastic m e m o r y , he k n e w every single word. I r e m e m b e r , it w a s a joke, y o u could say, "Dieudonn~, w h a t is this result a b o u t so a n d so?" a n d he w o u l d go to the shelf a n d t a k e d o w n the b o o k a n d open it to t h e right page. After Dieudonn~ ( a n d an interlude b y Samuel and D i x m i e r ) I w a s the secret a r y of Bourbaki, a n d it w a s m y d u t y to do m o s t of t h e proofreading, I think I p r o o f r e a d five to t e n t h o u s a n d pages. I have a g o o d visual m e m o r y . I wouldn't c o m p a r e m y s e l f with Dieudonn~, b u t t h e r e w a s a time w h e n I k n e w m o s t of the p r i n t e d m a t e r i a l in Bourbaki. But no o n e after me w a s able to do this. So B o u r b a k i lost the

28

THE MATHEMATICALINTELLIGENCER

a w a r e n e s s of his o w n body, the 40 p u b l i s h e d volumes. A n d as I said before, B o u r b a k i w a s m o r e or less like a family. The s e c o n d o r third o r fourth g e n e r a t i o n in any family or any social group follows definite sociological patterns. My o w n family was typical. My g r a n d f a t h e r w a s a self-made man, a v e r y successful b u s i n e s s m a n . My f a t h e r a n d m y uncle w e n t into the business, b u t t h e y w e r e n o t s o d e v o t e d to t h e fight. A n d peop l e in m y g e n e r a t i o n - - w e l l , I s u p p o s e I m a d e the right d e c i s i o n n o t to engage in it. Indeed, p e o p l e in m y generation w h o did go into our family b u s i n e s s did n o t do so well, b e c a u s e t h e y didn't have anything to fight for. But t h e s e are the i n n e r workings. Of c o u r s e the outside w o r l d also has an influence. That the o u t s i d e m a t h e m a t ical w o r l d has c h a n g e d is obvious. We all k n o w that what Stalin c o u l d never achieve with his army, to c o n q u e r the world, the collapse o f the Soviet Union h a s achieved for m a t h e m a t i c s . The Russian m a t h e m a t i c i a n s have brought a different style to the West, a different w a y of looking at t h e p r o b l e m s , a n e w blood. It's a different time, with different values. I have no regrets: I think it was w o r t h w h i l e to live in t h e t w e n t i e t h century. But n o w it's finished. S e n e e h a l : How would you describe your j o u r n e y with Bourbaki ? C a r t i e r : I have been personally very happy, because when I r e a c h e d the time of normal retirement from Bourbaki, I h a d the very fortunate opportunity to be a s k e d to deliver the lecture on behalf of Vladimir Drinfel'd at the International Congress of Mathematicians at Berkeley in 1986 (Drinfel'd was p r e v e n t e d from coming for political reasons). It was a great challenge and a great h o n o r for me; his p a p e r is one o f t h e m o s t imp o r t a n t p a p e r s in the proceedings. Overnight that changed m y m a t h e m a t ical life. I said, "This is w h a t I have to do now." Of course I k n e w the basic m a t e r i a l b u t the p e r s p e c t i v e w a s new. I c a n n o t claim that within the few h o u r s I h a d to p r e p a r e t h e lecture I c o u l d really m a s t e r it, b u t I u n d e r s t o o d e n o u g h to explain to the people, "This is new, it is important." When I began in mathematics the

main t a s k of a mathematician was to bring o r d e r and m a k e a synthesis of existing material, to create w h a t T h o m a s Kuhn called n o r m a / science. Mathematics, in the forties and fifties, was undergoing w h a t Kuhn calls a solidification period. In a given science there are times w h e n you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The p u r p o s e of mathematics, in the fifties and sixties, was that, to create a new e r a of n o r m a l science. N o w we are again at the beginning of a n e w revolution. Mathematics is undergoing major changes. We don't k n o w exactly where it will go. It is not yet time to m a k e a synthesis of all these things---maybe in twenty or thirty y e a r s it will b e time for a new Bourbald. I consider myself very fortunate to have had two lives, a life of normal science and a life of scientific revolution. BIBLIOGRAPHY

Nicolas Bourbaki, Faits et I~gendes, by Michele Chouchan, I~ditions du Choix, Argenteuil, 1995. "NicholasBourbaki,CollectiveMathematician:an interviewwith ClaudeChevalley,"by DenisGuedj, translated by Jeremy Gray, The Mathematical Intelligencer, vol. 7, no. 2, 18-22, 1985. Les Math~matiques et I'Art, by Pierre Cartier, Institut des Hautes #tudes Scientifiques preprint IHES/M/93/33. "The Continuing Silence of a Poet," by A. B. Yehoshua, in The Continuing Silence of a Poet." collected stories, Penguin Books, 1991 ..

GARY LAWLOR

A New AreaMaximization Proof for the Circle

0 ~

n this article, I give what I believe to be a new proof for the planar isoperimetric problem. The basic idea is to find a geometric strategy for dividing the circle and any competitor of equal perimeter into tiny corresponding pieces, in such a way that each piece of the competitor will contain the same length from the boundary and will

have less area than the corresponding piece of the circle. Since it directly compares an arbitrary curve with the circle, this proof does not require a separate proof of the existence of an optimum solution. This geometric decoupling of a global problem into related local problems, in such a way that local minimization implies global minimization, is the basic philosophy of "directed slicing," introduced as a general method of proving optimization by the author [L1] and used for other optimization proofs; see, for example, [L2, KL]. I recommend V.M. Tikhomirov's book, Stories about M a x i m a and M i n i m a IT], for a nice discussion of optimization problems, including the isoperimetric problem. Tikhomirov notes that Blaschke [B] gives a good list of historical references, and he also recommends the books [CR] and [RT]. Measuring

the Area of a Unit Circle

Figure 1 shows the geometric basis for perhaps the simplest demonstration that the area of a circle of radius 1 is equal to half the perimeter; here, the circle is approximated by a large number of narrow isosceles triangles radiating out from the center.

Figure 1. Finding the area of a circle.

9 1998 SPRINGER-VERLAG NEWYORK, VOLUME 20, NUMBER 1, 1998

29

the reflection argument leads quickly to a complete proof. Instead, let us study the first-quadrant picture by a slicing and covering argument.

Figure 2, Among triangles with fixed base and fixed opposite angle, the one with greatest area is isosceles,

My proof that the circle has most area among all regions with fixed perimeter will build on this same picture. The Two Main Ideas The maximization proof at the local level will come from the fact that among all triangles with fLxed base and fLxed opposite angle, the one with greatest area is isosceles (see Fig. 2). The idea we need for linking the local facts to global facts is a method for covering a region with triangles, each having its short base along the boundary of the region and each having a given fLxed angle opposite this base. The Proof Let S be any simple closed curve in the plane, having length 2~-. We need to show that the area of the region R which it encloses is at most zr. We will assume that R is convex and symmetric across both the x and y axes. These properties can be obtained by a simple reflection and replacement argument; indeed, provided we know that there exists a maximizing solution,

Constructing the Covering Triangles Let $1 be the curve of length r which is the portion of S lying in the first quadrant (see Fig. 3). Let p be its endpoint on the x axis, and q be the endpoint on the y axis. Let R1 be the region bounded by $1 and by the two axes. Choose a large whole number N. Find points pi along $1 which divide it into segments of length r with p -Po and q = PN. For each i, let Pi denote the segment of $1 having endpoints Pi and Pi+l. We will consider each segment Pi to be a straight-line segment, forgoing the accompanying limit argument. Now, from each point pi, draw a ray Li down toward the x axis, each making an angle Oi = (i/2N)~r with the horizontal. For 0 --< i < N, let Ti be the triangle bounded by Pi, Li, and Li+v Figure 3a shows what the result looks like for an example curve which is shorter and wider than the quarter-circle. Figure 3b shows what happens for a taller and narrower curve. W h y t h e T r i a n g l e s C o v e r All of t h e Region

We need to know that for an arbitrary point x in R1, there is one of the triangles Ti which contains it. To see this, note that as you face the origin from x, L0 is to your left and LN to your right. This means that somewhere in between i -- 0 and i = N, there is a switch; that is, for some j, the point x is on or above Lj but is below Lj+ 1. Thus, x is in the triangle Tj, as long as it is on the correct side of the short segment Pj. To see that it is, first, note that by the convexity properties, the entire region R1 (and, in particular, the point x) is to the left of any tangent line of R1. Second, since the tangent vector to $1 points up and to the left, the two rays Lj and Lj+ i will cross to the left of Pj rather than above and to the right. Thus, x is in the triangle Tj.

PN = q ~ PN = q

Q u a r t e r circle for comparison L~

LN uarter circle P~

~

IIII/// o

/

\

n

.

Po - " P b Figure 3, (a) Slicing a wider region; (b) slicing a taller region

30

THE MATHEMATICALINTELLIGENCER

I

-

Po

Conclusion

Finally, we compare the triangles Ti with the isosceles triangles which would be obtained by slicing the circle by the same procedure. Since they all have the same short base length and the same angle opposite the base, the isosceles triangles composing the circle have more area than the triangles Ti. Because the latter triangles cover all of R1 (perhaps even with overlap and/or extension beyond R1), we see that the area of the quarter-circle is greater than that of R1, and thus the circle's entire area is greater than that of our original region R. REFERENCES

[B] W. Blaschke, Kreis und Kugel, Leipzig: Veit and Co. (1916); 2 nd ed., Berlin: Walter de Gruyter & Co. (1956). [CR] R. Courant and H. Robbins, What is mathematics? Oxford: Oxford University Press (1978). [KL] Michael Kerckhove and Gary Lawlor. A family of stratified areaminimizing cones, Duke Math. Jour. (to appear). ILl] Gary Lawlor, Proving area-minimization by directed slicing, preprint. [L2] Gary Lawlor, A new minimization proof for the brachistochrone, Am. Math. Monthly 103(3) (1996), 242-249. [RT] H. Rademacher and 9 Toeplitz, The Enjoyment of Mathematics, Princeton, NJ: Princeton University Press (1966). IT] V.M. Tikhomirov, Stories about Maxima and Minima (transl. A. Shenitzer), Providence, RI: American Mathematical Society (1990).

VOLUME 20, NUMBER 1, 1998

31

mmJ~l~|,[~ll~U, ln[.-~-umm~u-~,[.~i~-un,ni,u[~,l,,[.-:'-m A l e x a n d e r

The Generalized Towers of Hanoi for Space-Deficient Computersand Forgetful Humans Timothy R. Walsh

This column is devoted to mathematics for fun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions from readers--either new notes, or replies to past columns.

Please send all submissions to the Mathematical Entertainments Editor, A l e x a n d e r Shen, Institute for Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:[email protected]

32

Shen,

Editor

The T o w e r s of Hanoi puzzle is frequently given in the l i t e r a t u r e as an exa m p l e of a p r o b l e m w h o s e solution is b e s t d e s c r i b e d recursively. The argum e n t a d v a n c e d by the r e c u r s i o n i s t s is t h a t t h e well-known iterative solution is m o r e difficult to p r o g r a m and to p r o v e c o r r e c t than the r e c u r s i v e o n e - in fact, the first c o r r e c t n e s s p r o o f was given b y P. B u n e m a n a n d L. Levy in 1980 [1], roughly 100 y e a r s after the solution was discovered. As an unrepent a n t iterationist, I offer t h e following counter-argument: e x e c u t i n g the recursive solution on n rings b y comp u t e r requires storing at l e a s t n items in a s t a c k (at the v e r y m i n i m u m the c u r r e n t destination peg for e a c h o f the n rings), and executing it b y h a n d requires r e m e m b e r i n g all t h e s e items simultaneously, w h e r e a s e x e c u t i n g the iterative solution requires r e m e m b e r ing only one piece o f information: w h e t h e r or not you j u s t m o v e d the s m a l l e s t ring. And in [6] I offered a w a y of avoiding having to r e m e m b e r even t h a t one piece of information. Surely it is a w o r t h w h i l e p r o j e c t to s p a r e the u s e r s o m e work, even if w e mathem a t i c i a n s have to w o r k h a r d to do it! The puzzle was generalized by F. Scarioni and M. G. S p e r a n z a [5], and i n d e p e n d e n t l y b y M. C. Er [2], to allow the initial position of t h e rings to b e any legal position. Recursive solutions w e r e p r e s e n t e d in t h o s e t w o articles, a n d an iterative one w a s f o u n d b y Er [3] a n d i n d e p e n d e n t l y b y y o u r s truly [7]. Neither of us tried to p r o v e that o u r s o l u t i o n s used the m i n i m u m n u m b e r of moves; this omission w a s c o r r e c t e d b y A. M. Hinz [4], who further generalized t h e puzzle to allow b o t h the initial and final p o s i t i o n of the rings to be any legal position. Since Hinz w a s kind e n o u g h to improve upon my work I hereby return the favour. His s o l u t i o n (as well as

THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER VERLAG NEW YORK

I

Er's) r e q u i r e s the storing of at l e a s t one size-n a r r a y of d a t a in a d d i t i o n to w h a t e v e r d a t a is u s e d to r e p r e s e n t t h e p r o b l e m (the c u r r e n t p o s i t i o n o f t h e rings and, in t h e m o s t general case, the d e s t i n a t i o n p e g for each ring). I pre-

sent a solution in which both the amount of extra data that has to be stored and the time needed to decide on each move except the f i r s t one are independent of the n u m b e r of rings (well, almost independent: t h e r e are a total o f O ( n ) o p e r a t i o n s w h i c h w o u l d have to b e amortized). F o r the s a k e o f forgetful h u m a n s I modify the s o l u t i o n so t h a t t h e e x t r a d a t a has to b e upd a t e d o n l y O(n) times, m a k i n g it e a s y to w r i t e d o w n a n d c o n s u l t w h e n needed. I'll b e g i n with the classical T o w e r s of Hanoi puzzle ( w h e r e the rings begin and e n d on a single peg), a n d in subsequent s e c t i o n s I'll discuss generalized versions. In all of these v e r s i o n s there are t h r e e pegs, labelled A, B, a n d C, a n d n rings of different sizes, labelled 1 , 2 , . . . ,n in increasing o r d e r o f size. In a legal position of the rings, t h e rings a r e s t a c k e d on the t h r e e p e g s so that no ring sits on t o p of a ring s m a l l e r than itself. In the notation I will b e using here, t h e legal p o s i t i o n for 6 rings in w h i c h the o d d - n u m b e r e d rings a r e on p e g A a n d the e v e n - n u m b e r e d rings are on p e g B will be d e n o t e d b y t h e following line: 1

3

5

AI

2

4

6

BI

C]

A legal move t a k e s the t o p m o s t ring of s o m e n o n - e m p t y peg a n d p u t s it on a p e g w h i c h d o e s n o t c o n t a i n a n y ring s m a l l e r t h a n t h e ring w h i c h is b e i n g m o v e d . In t h e n o t a t i o n I will b e u s i n g here, t h e m o v e w h i c h t a k e s t h e topm o s t ring o f p e g A a n d p u t s it o n p e g B will b e d e n o t e d b y AB. S t a r t i n g from t h e p o s i t i o n d e n o t e d b y t h e a b o v e line, AB is a legal move, as a r e

AC and BC, but BA is not, because this move would put ring 2 on top of ring 1, nor is CA or CB, because peg C is empty. From a Single Peg to a Single Peg In the original Towers of Hanoi problem, all the rings are initially stacked on a single peg, called the source peg, and have to be stacked on another peg, called the destination peg; the remaining peg is called the spare peg. The classical iterative solution to this problem, which was first proved to be correct by P. Buneman and L. Levy [1], can be represented by the following set of rifles for moving the rings. FIRST-MOVE RULE {source-to-destination version). I f n is odd, then move ring 1 onto the destination peg; otherwise move ring 1 onto the spare peg. NEXT-RING-TO-MOVE RULE. Never move the same ring twice in a row. Thus, i f you have j u s t moved ring 1, then either all the rings are now stacked on the destination peg and you are done, or else there is one legal move to make: move the secondsmallest topmost ring onto the peg not containing ring 1. A n d i f you have not j u s t moved ring 1, move it. CYCLIC-PEG.ORDER RULE. Ring 1 always moves f r o m peg to peg in the s a m e cyclic order (established by the f i r s t - m o v e rule). To solve the puzzle by hand, follow the first-move rule and write down the cyclic order of the pegs, and then apply the next-ring-to-move rule and the cyclic-peg-order rule (when it is applicable) until all the rings have been stacked on the destination peg. The second-smallest topmost ring is easily identified in O(1) time by examining the topmost ring on each peg. The only piece of information which must be updated and is not obtainable by examining the topmost ring on each peg is whether you have j u s t moved ring 1. It is infuriatingly easy to forget this bit of information when you're tired or interrupted, and frustratingly wasteful to update it on paper after every move. Fortunately there is another rule

which can either replace the cyclicpeg-order rule or be used in conjunction with it to make it possible to decide in O(1) time whether ring 1 should next be moved [6]. EVEN-TARGET RULE. Ring 1 should be moved on top of the second-smallest topmost ring i f and only i f this ring has an even label. For example, suppose the source peg is A, the destination peg is B, and the current position of the rings is 1

2

AI

3 6 BI

4 5 Cl.

Since n = 6, which is even, the fwst move was AC, establishing the cyclic peg order ACBA. Moving ring 1 according to the cyclic-peg-order rule would put it on peg C, in accordance with the even-target rule; so you know that you should make this move. The rings are now in the position

2 AI

3 6 BI

1

4

5

cI.

Moving ring 1 according to the cyclicpeg-order rule would put it on peg B, violating the even-target rule; so by the next-ring-to-move rule the next move is AB. The rings are now in the position

AI

2 3 6 BI

1

4

5

C[.

Once again, the cyclic-peg-order rule and the even-target rule both say to move ring 1 onto peg B. Continuing in this way, you can complete the solution without ever having to write down anything more than once (the cyclic peg order) or remember anything but the rules. Of course, a computer has no problem storing in a single variable the information as to whether it is ring 1 which is to be moved; so either the cyclic-peg-order rule or the even-target rule can be used. The former has the advantage that the second-smallest topmost ring need be located only every second move; the latter has the advantage that the cyclic peg order need not be stored. In order to be able to locate the peg containing the second-smallest topmost ring in O(1) time, the topmost ring of each peg is stored in a size-3 array Top, and the ring beneath each ring is stored in a

size-n array Beneath [3]; in either case, an empty peg is represented by n + 1. For the last position above, Top would be (7,2,1) and Beneath would be (4,3,6,5,7,7). Examining the array Top one can determine in O(1) time that ring 1 is on peg C and that the secondsmallest topmost ring is ring 2, on peg B, and then the correct next move--CB--can be made in O(1) time, changing Top to (7,1,4) and Beneath to (2,3,6,5,7,7). If the second-smallest topmost ring is greater than n - - a n d this too can be determined in O(1) t i m e - then all the rings are stacked on a single peg. From An Arbitrary Legal Position To a Single Peg In this version the rings begin in an arbitrary legal position and must be stacked on a designated destination peg. All the iterative solutions [3,4,7] depend upon the calculation of target pegs for each of the rings, using the target-calculation algorithm below. TARGET-CALCULATION ALGORITHM. The target f o r ring n is the destination peg. For each ring i -- n - 1 , . . . , 2,1: i f ring i § 1 is on its target peg, then this peg is also the target peg f o r ring i, and otherwise the target peg f o r ring i is the peg other than the peg containing ring i + 1 and the target peg f o r ring i + 1. For example, if the destination peg is B and the current position of the rings is

AI

2 3 6 BI

1

4

5

cI,

then the target pegs for all the rings are calculated as follows: ring 6 5 4 3 2 1

target B B A B B B

location B C C B B C

The iterative algorithm in [4] for stacking the rings on the destination peg is essentially the one shown in Figure 1. It uses the array Target to store the target pegs and Peg to store the current peg containing each ring.

VOLUME 20, NUMBER 1, 1998

all that is required to apply the firstmove nile. Since the rings are scanned from largest to smallest by the targetcalculation algorithm, their position cannot be initially represented by the arrays Top and Beneath; however, the array Beneath can do double duty, playing the role of the array Peg until the target for ring 1 has been determined. For example, for the initial position Figure 1. Er-Hinz iterative algorithm for stacking n rings onto the destination peg from any legal position.

The iterative algorithm in [3] is essentially the same, except that an explicit implementation is given, using the arrays Top and Beneath as well as Peg and Target so that each move can be made in O(1) time. The array Target is constantly updated to keep ring 1 moving in the correct cyclic direction; it would have sufficed to update the cyclic direction for ring 1 for each new value of i. Examining this algorithm you can see that it obeys the next-ring-to-move rule: it is obeyed during the execution of each instruction 'Stack all the smaller rings on Peg[i],' and the moves of ring 1 that end one execution of this instruction and begin the next one are separated by a move of a larger ring made by the instruction 'Peg[i]: =Target[i].' All the moves of ring 1 except 'Peg[i]= Target[I],' which would necessarily be the first move made, are made during the execution of the instruction 'Stack all the small rings on Peg[i]' for some i, and so they obey the even-target rule as well. It follows that these two rules determine every move except the first one! The first move is 'Peg[i]: =Target[i]' for the smallest ring i which is not on its target peg, if there is one, leading to a new first-move rule: FIRST-MOVE

RULE

(general-to-desti-

nation version). I f ring 1 is not on its

target peg, then move it there; otherwise foUow the next-ring-to-move rule f o r moving a larger ring. As an example of this rule, recall that from the position

AI

2 3 6 BI

1

THE MATHEMATICAL INTELLIGENCER

4

5

CI

the next move required to stack the rings on peg B (assuming they began on peg A) was CB. As I have shown above, the target-calculation algorithm puts the target for ring 1 on peg B, so that the first-move nile would lead to the same move CB. If the initial position happens to be the one with all the rings stacked on the source peg, then the target-calculation algorithm puts the target for ring 1 on the destination peg if n is odd and on the spare peg otherwise, so that the source-to-destination version of the first-move rule falls out as a special case of the general-to-destination version. To stack the rings on the destination peg from any legal position, find the target for ring 1 using the targetcalculation algorithm, then determine the first move using the general-to-destination version of the first-move-rule, and then apply the next-ring-to-move rule and the even-target rule (when applicable) until all the rings are stacked on the destination peg. This is the algorithm which appears in [7], where it was derived from the recursive algorithm of [5]; a correctness p r o o f that is independent of any other algorithm appears in [8]. This algorithm can be implemented using only the arrays Top and Beneath. It does not require storing the array Target: during the application of the target-calculation algorithm, a single variable (which contains the target peg of the current ring i) is initialized to the destination peg and updated for each new value of i, and ends up being the target peg for ring 1, which is

AI

2 3 6 BI

1 4 5 cl,

the array Beneath would initially be (C,B,B,C,C,B), with the pegs A, B, and C c o d e d by convenient n u m b e r s like 0, 1 and 2. Then, once the target peg for ring 1 has been established, another pass through the rings f r o m largest to smallest suffices to convert the representation of the position of the rings to agree with the meaning of the w o r d s top and beneath: Top bec o m e s (7,2,1) and Beneath b e c o m e s (4,3,6,5,7,7). In this way, only O(1) extra space is needed and every m o v e except the first one can be e x e c u t e d in O(1) time. Of course, this is all well and g o o d for a c o m p u t e r which can r e m e m b e r whether it has just moved ring 1, but not for a forgetful human. Examining the algorithm of Figure 1, one can see that before the instruction 'Peg[i]:= Target[i]' has been executed, only rings smaller than ring i have ever been moved. Keep track of one number: the label on the largest ring which has so far been m o v e d (the variable i in the algorithm of Figure 1). This n u m b e r increases whenever the instruction 'Peg[i]:=Target[i]' gets executed. If i > 1, then the smaller rings which were stacked on one peg get stacked on another, so that you can use the memoryavoidance tricks outlined for that case. The first move following 'Peg[i]:= Target[i]' is a move of ring 1 which obeys the even-target rule; this move establishes a cyclic order on the pegs which will be valid during the execution of 'Stack all the smaller rings on Peg[i ]', after which either all the pegs are stacked on the destination peg or else a ring larger than ring i b e c o m e s free to move. Write down this cyclic order; then the cyclic-order rule can be c o m p a r e d with the even-target rule to

d e t e r m i n e w h e t h e r ring 1 is to be m o v e d n e x t - - u n t i l a r i n g larger than r i n g i is free to move. At this point, this larger ring gets moved, i increases, a n d a n e w cyclic o r d e r i s e s t a b l i s h e d b y the following move. The a m o u n t of information that n e e d s to b e written d o w n is greater than that w h i c h w o u l d b e required to r e c o r d w h e t h e r ring 1 h a s just b e e n moved, but it need be done only n times. F o r example, s u p p o s e y o u are moving the rings onto peg B from the position AI

2

3

6

BI

1

4

5

C I.

The first move has a l r e a d y b e e n calculated: since the target p e g for ring 1 is B, the first m o v e is CB. Now i, the largest ring that has b e e n moved, is 1; w r i t e it down, but no cyclic o r d e r n e e d b e w r i t t e n down. After y o u m a k e this move, the position b e c o m e s AI

1

2

3

6

B[

4

5

C[.

A l r e a d y a larger ring is free to m o v e - ring 4. Do so and r e p l a c e 1 by 4. The p o s i t i o n is n o w 4

AI

1

2

3

6

BI

5

C I.

By the next-ring-to-move rule, you m u s t n o w move ring 1, a n d b y the event a r g e t rule, it m o v e s to p e g A:

1 4 AI

2 3

6

BI

5

C I.

This e s t a b l i s h e s the cyclic o r d e r BACB on the pegs; w r i t e it down. Now, if the t e l e p h o n e rings a n d y o u forget w h e t h e r you've j u s t m o v e d ring 1, obs e r v e t h a t the c y c l i c - o r d e r rule is n o w i n c o m p a t i b l e with the even-target rule, so t h a t ring 1 m u s t n o t b e m o v e d yet. The advantage of this algorithm over the one in Figure 1 is that you don't have to store an array of target pegs and you don't have to both u p d a t e the cyclic peg o r d e r a n d r e m e m b e r w h e t h e r you have j u s t m o v e d ring 1: you can opt to do only one or the other, depending upon w h e t h e r you are a forgetful h u m a n o r a space-deficient computer. From

a Single

Arbitrary

Legal

Figure 2. An algorithm for taking n rings which are stacked on a source peg and putting each ring i on a specified peg Dest[l'J,

Peg To An Position

N o w s u p p o s e that t h e rings begin on a single p e g - - t h e source p e g - - a n d are to b e m o v e d to s o m e a r b i t r a r y legal posi-

tion. Hinz's solution to this p r o b l e m [4] is: find the sequence of m o v e s that w o u l d s t a c k the rings on the s o u r c e peg if t h e y s t a r t e d in their d e s i g n a t e d final position, and then r e v e r s e t h a t seq u e n c e o f moves. The a l g o r i t h m of Figure 2 solves the given p r o b l e m directly, a n d it is easier to p r o g r a m and to e x e c u t e by hand, for n o t a r g e t p e g s have to b e calculated and no s e q u e n c e of m o v e s written d o w n o r s t o r e d and then e x e c u t e d in reverse order. To p r o v e t h a t this algorithm works, I u s e t h e loop invariant: all the r i n g s f r o m n d o w n to i + 1 are on t h e i r dest i n a t i o n pegs, and all the r i n g s f r o m i d o w n to 1 are stacked on a single peg. At t h e beginning, w h e n i = n, this i n v a r i a n t is true b e c a u s e all the rings begin s t a c k e d on the s o u r c e peg. S u p p o s e it to b e true for a given i. If ring i is on its destination peg, t h e n all the rings from n d o w n to i a r e o n their d e s t i n a t i o n pegs, and all t h e rings f r o m i - 1 d o w n to 1 are s t a c k e d on a single peg; the l o o p invariant is r e t r i e v e d w h e n i d e c r e a s e s to i - 1. If ring i is n o t on its d e s t i n a t i o n peg, s t a c k i n g all the s m a l l e r rings on the "other" peg a r r a n g e s t h a t all the rings f r o m i - 1 d o w n to 1 are s t a c k e d on a single peg a n d m a k e s it p o s s i b l e to p u t ring i onto its d e s t i n a t i o n peg, and again t h e l o o p invariant is retrieved w h e n i d e c r e a s e s to i - 1. The algorithm t e r m i n a t e s w h e n i d r o p s to 0, at w h i c h p o i n t the l o o p invariant says that all t h e rings are o n t h e i r d e s t i n a t i o n pegs, a s required. The a r r a y Dest is p a r t o f the problem; s o it c a n n o t be c o n s i d e r e d e x t r a storage. The arrays Top a n d B e n e a t h

are sufficient to r e p r e s e n t the p o s i t i o n o f the rings: to d e t e r m i n e w h e t h e r ring i is on the p e g Dest[i], use the fact t h a t all the rings f r o m i d o w n to 1 are o n the s a m e peg, s o that all you have to do is e x a m i n e the t o p m o s t rings o n each peg to d e t e r m i n e the location o f ring 1 a n d t h e n c o m p a r e it with Dest[i]. The e x t r a s t o r a g e is thus O(1). It m a y h a p p e n t h a t s e v e r a l rings in a row will turn out to b e on their destination pegs, so that s e v e r a l d e c r e m e n t s of i will b e m a d e b e f o r e the n e x t move; however, the failure of this algorithm to t a k e only O(1) t i m e p e r m o v e is limited to O(n) o p e r a t i o n s p e r f o r m e d t h r o u g h o u t the entire e x e c u t i o n of the a l g o r i t h m - those w h i c h t e s t w h e t h e r each ring i is on Dest[i]. To e x e c u t e this algorithm by hand, write d o w n the fmal position of t h e rings, a n d t h e n c r o s s e a c h ring off w h e n it is e i t h e r m o v e d to its destination peg o r t u r n s out to b e there already. W h e n e v e r y o u get to a ring i which has n o t y e t b e e n c r o s s e d off, if i = 1 t h e n m o v e ring 1 to its destination peg a n d y o u are done, and otherwise the n e x t m o v e is to move r i n g 1 to the d e s t i n a t i o n peg o f ring i i f i is odd and to the other peg i f i is even (this is the first m o v e in the instruction 'Stack all the s m a l l e r rings on t h e "other" p e g b e s i d e s Dest[i]'). This move e s t a b l i s h e s a cyclic o r d e r on t h e pegs; write it d o w n a n d c o m p a r e it with the even-target rule to d e t e r m i n e the n e x t m o v e until ring i gets m o v e d to its d e s t i n a t i o n peg. F o r example, s u p p o s e y o u w a n t to move the rings from peg B to the position

VOLUME 20, NUMBER 1, 1998

35

1 4 AI

2 3 6 BI 5 C[.

The initial position is AI

1 2 3 4 5 6 B]

CI

Ring 6 is already on its destination peg; cross it off the final position. Ring 5 is not on its destination peg, and since 5 is odd, ring 1 must n o w be moved to peg C. This establishes the cyclic order BCAB on the pegs, so that by comparing this cyclic order with the even-target rule you can move the pegs with nothing to r e m e m b e r until the rings reach the position

1 2 3 4 AJ

6 B I 5 CI.

Now ring 5 has been moved to its destination peg and ring 4 turns out to be already there; so these rings are crossed off and you get to ring 3, which needs to be moved from peg A to peg B. From

One Arbitrary

Position

Legal

To Another

The final problem I consider is moving the rings from one legal position to another. Hinz solves this problem by presenting two sequences of moves [4]. Let ring i be the largest ring that is not on its destination peg Dest[i]. Ring i is moved onto Dest[i] in either one move or two. Before ring i can be moved, all the smaller rings m u s t be stacked on the appropriate "other" peg. Before the first move of ring i, the smaller rings are in an arbitrary legal position; they must be stacked on a single peg. Before the second move of ring i, if there is

one, the smaller rings are stacked on one peg and must be stacked on another one. Once ring i is on Dest[i], the smaller rings must be taken from the single peg on which they are n o w stacked and moved to their destination pegs. One of these two sequences of moves (moving ring i once or twice) is minimal; so just calculate in advance the n u m b e r of moves each sequence will take and then execute the shorter sequence. If each of the c o m p o n e n t parts of the Hinz algorithm is implemented using the algorithms p r o p o s e d here, the entire algorithm uses O(1) extra storage and each move takes O(1) time, apart from the O(n) operations required to decrease i from n to 0 during the execution of the algorithm of Figure 2, and can be executed by hand with O(n) updating of extra information. So how do you go about calculating in advance the number of moves that each of the component algorithms will make? Moving n rings from one peg to another takes 2 n - 1 moves ([1] and elsewhere). The following formula for the number of moves required to m o v e n rings from an arbitrary legal position to a destination peg was presented in [4] (but I presented it in [7], which was published six years earlier): it is the binary n u m b e r b n . . 9 b 2 b b where b i = 0 if and only if Peg[i]=Target[i]. It is possible to compute this binary number

Figure 3. An algorithm for finding m, the number of moves needed to stack n rings from an arbitrary legal position onto a destination peg, and for finding the target peg for ring 1.

while computing the target for ring 1 without storing the array Target:' as soon as the single variable that contains -Iarget[i] for the current value of i is updated it is compared with Peg[i] to determine if the next bit bi is equal to 0 or 1. Working by hand, the entire binary n u m b e r is written d o w n and then evaluated. Working by computer, the bits are not stored; instead, the n u m b e r is evaluated using H o m e r ' s method for evaluating a polynomial in x for x = 2 (see Figure 3). Incidentally, not every textbook which teaches both H o m e r ' s method and the evaluation of numbers written in base b points out that the same algorithm is being used! To c o m p u t e the number of moves required to move n rings from a source peg to an arbitrary legal position, Hifiz [4] p r o p o s e s computing the array Target which would result from moving the rings from their fmal position to the source peg, and then computing the binary n u m b e r from the formula (you can avoid storing the array Target by using the algorithm in Figure 3). The algorithm of Figure 4 evaluates directly the n u m b e r of moves made by the algorithm of Figure 2, by keeping track of the peg on which rings i - 1 . . . . . . will be stacked before moving ring i to Dest[i]. Examining Figure 2, you can see that if ring i is not on its destination peg, then 2 i-1 - 1 moves will be made to stack the smaller rings, and one m o r e to move ring i, for a total of 2 ~-1, and otherwise no moves will be made. The n u m b e r of moves thus has the binary expansion b n . 9 9 b 2 b l , where b i = 0 if and only if ring i is already on its destination peg when this condition is tested. Comparing Figures 3 and 4, you can see that these two algorithms differ only in the names of the variables; the advantage of executing the algorithm of Figure 4 is that it acts directly on the problem at hand. For example, suppose that the rings begin in the position

1 4 5 AI

2 6 BI

3 Cl

and are to end in the position

3 6 AI

1 4

5

BI

2 CI.

Suppose y o u decide to m o v e ring 6 from peg B to peg A in one step. You must first stack all the smaller rings

36

THE MATHEMATICAL INTELLIGENCER

tination), the n u m b e r of rings (up to 10), whether you or the computer sets up t h e initial a n d final position, and w h e t h e r you or the c o m p u t e r moves the rings. If you let the c o m p u t e r set up the initial and final position and you move the rings yourself, then you can score points depending upon the puzzle version, the number o f rings, the n u m b e r of m o v e s made in excess of the minimum, and the number of illegal moves attempted. Send m e a diskette and enough m o n e y Figure 4. An algorithm for finding m, the number of moves needed to execute Figure 2.

on p e g C. Here is a t r a c e o f the comp u t a t i o n o f the t a r g e t for ring 1 tog e t h e r w i t h the n u m b e r o f m o v e s required. ring 5 4 3 2 1

target C B C C A

location A A C B A

bit 1 1 0 1 0

moves 1 3 6 13 26

One m o r e move p u t s ring 6 on peg A. After 27 m o v e s the p o s i t i o n will be

6 AI

BI

t

1

2

3

4

5

AI

BI

6

C I.

It will t a k e 31 m o r e m o v e s to s t a c k rings 1 t h r o u g h 5 on peg B a n d one m o r e m o v e to p u t ring 6 on p e g A. After 38 m o v e s the position will b e

6 AI

1 2 3 4 5 BI

Cl.

Finally t h e rings must b e b r o u g h t to t h e i r final position, which is

3 6 AI

1

4

5

BI

2 CI.

2 3 4 5 C[.

N o w y o u m u s t m o v e rings 1 through 5 to their final positions. A t r a c e of the c o m p u t a t i o n of the n u m b e r of m o v e s follows. ring 5 4 3 2 1

One m o r e move p u t s ring 6 on p e g C. After 6 moves, the position will b e

stackpeg d e s t i n a t i o n bit m o v e s C B 1 1 A B 1 3 C A 1 7 B C 1 15 A B 1 31

It will t a k e 58 m o v e s to p u t the rings on their destination pegs. N o w s u p p o s e y o u d e c i d e to move ring 6 from peg B to p e g A in 2 steps. Before moving ring 6 f r o m p e g B to peg C you m u s t s t a c k all the s m a l l e r rings on peg A. The trace follows. ring 5 4 3 2

target A A A B

location A A C B

bit 0 0 1 0

moves 0 0 1 2

1

B

A

1

5

ring stackpeg destination bit m o v e s 5 B B 0 0 4 B B 0 0 3 B A 1 1 2 C C 0 2 1 C B 1 5 It will t a k e only 43 m o v e s to p u t the

rings in t h e i r f'mal position, c o m p a r e d with 58 m o v e s if you m o v e ring 6 only once. The target for ring 1 h a s a l r e a d y b e e n c a l c u l a t e d - - i t ' s B; so the first m o v e to m a k e is AB. I have written a computer game for the Macintosh 68000 series which implements the algorithms p r e s e n t e d here. The only arrays used are Top, Beneath, and Dest (Dest is used only w h e n the rings are to be moved to an arbitrary legal position), and except for the display and the O(n) extra operations performed dining the execution of the algorithm of Figure 2, each move is m a d e in O(1) time. You get to choose b e t w e e n the four versions of the puzzle (single or multiple source, single or multiple des-

VOLUME 20, NUMBER 1, 1998

37

t~ c~ p~ and Y~ receive the executable file, the source code (about 900 lines, in C), and a copy of this paper. If you score the maximum 100 points, save the output file and send it to me and you'll be duly inducted into the Timewasters' Hall of Fame!

~

(__~) = 1§

( ___~_~71 ) 9 1+

.(1+

2 ) .(1+ -912670091 2 -760223786832147978143718731

REFERENCES

1. P. Buneman and L. Levy, The towers of Hanoi problem, Information Processing Letters 10 (1980), 243-244. 2. M.C. Er, The generalized towers of Hanoi problem, J. Inform. Optim. Sci. 5 (1984), 89-94. 3. M.C. Er, An iterative solution to the generalized towers of Hanoi problem, BIT 23 (1983), 295-302. 4. A.M. Hinz, The tower of Hanoi, L'Enseignement Math6matique 35 (1989), 289-321. 5. F. Scarioni and M.G. Speranza, A probabilistic analysis of an error-correcting algorithm for the towers of Hanoi puzzle, Information Processing Letters 18 (1984), 99-103. 6. T.R. Walsh, The towers of Hanoi revisited: moving the rings by counting the moves, Information Processing Letters 15 (1982), 64-67. 7. T.R. Walsh, A case for iteration, Proceedings of the 14th Southeastern Conference on Computing, Graph Theory and Combinatorics, 1983, Congressus Numerantium 40 (1983), pp. 38-43. 8. T.R. Walsh, A simple sequencingand rant~hg method that works on almost all Gray codes, Research Report No. 243, Department of Mathematics and Computer Science, University of Quebec in Montreal, April 25, 1995.

What one observes here holds quite generally: The cubic approximants of X / ] ~ are the reciprocals of the approximants of ~zz. The cubic approximation could be used with advantage for numerical calculation of square roots ~zz with highprecision arithmetic of floating-point binary numbers, where the mantissa is restricted to the interval [1,4]. It is, therefore, sufficient to consider the case z E [1,2] and to reduce the case z E [2,4] to the case 4/z E [1,2]. Computationally, in high-precision arithmetic with floating-point numbers, division is the most demanding operation. Now, two Newton steps give quartic convergence, but they need two divisions; the iteration above has the slight advantage that it reaches a cubic order of convergence with only one division per step. Approximations of still higher order for square-rooting are possible, like one with quintic convergence using the formulas 9 a i2b i2 + 5z 2" b~), + 10z 9 a i2b i2 + z2 9 b~).

a5i = ai 9 ( a ~ + l O z b5i = bi 9 (5a~

Practically, they do not seem to be of importance; the gain in convergence can be made up by repeated use of the simpler cubic iteration.

(Letters: continued

from

page 13)

figures. Furthermore, for 0 < n < 1, the a~ are negative, for example in

F.L. BAUER N6rdliche Villenstrasse 19 Munich, Germany REFERENCES

9

-7761799 )

.(1+

2 -467613464999866416199

38

THE MATHEMATICALINTELLIGENCER

1. Friedrich Engel, in: Verhandlungen der 52. Versammlung deutscher Philologen und Schulm~nner in Marburg 1913. 2. Oskar Perron, Irrationalzahlen, Berlin: de Gruyter (1921 ). 3. Oskar Perron, Die Lehre von den Kettenbr[Jchen, Leipzig: Teubner (1913).

STEVE SMALE

Finding a I lorseshoe on the Beaches of Rio What is Chaos? A mathematician discussing chaos is featured in the movie Jurassic Park. James Gleick's book Chaos remains on the best-seller list for many months. The characters of Tom Stoppard's celebrated p l a y A r c a d i a discourse on the meaning of chaos. What is the fuss about? Chaos is a new science that establishes the omnipresence of unpredictability as a fundamental feature of common experience. A belief in determinism, that the present state of the world determines the future precisely, dominated scientific thinking for two centuries. This credo was based on mechanics, where Newton's equations of motion describe the trajectories in time of states of nature. These equations have the mathematical property that the initial condition determines the solution for all time. This was taken as proving the validity of deterministic philosophy. Some went so far as to see in determinism a refutation of free will and hence even of human responsibility. At the beginning of this century, with the advent of quantum mechanics, the untenability of determinism was exposed. At least on the level of electrons, protons, and atoms, it was discovered that uncertainty prevailed. The equations of motion of quantum mechanics produce solutions that are probabilities evolving in time. In spite of quantum mechanics, Newton's equations govern the motion of a pendulum, the behavior of the solar system, the evolution of the weather, many macroscopic situations. Therefore the quantum revolution left intact many deterministic habits of thought. For example, well after the Second World War, scientists held the belief that long-range weather prediction would be successful when computer resources grew large enough.

In the 1970s the scientific commtmity recognized another revolution, the theory of chaos, which seems to me to deal a death blow to the Newtonian picture of determinism. The world now knows that one must deal with unpredictability in understanding common experience. The coin-flipping syndrome is pervasive. "Sensitive dependence on initial conditions" has become a catchword of modern science. Chaos contributes much more than extending the domain of indeterminacy, just as quantum mechanics did more than half a century earlier. The deeper understanding of dynamics underlying the theory of chaos has shed light on every branch of science. Its accomplishments range from analysis of electrocardiograms to aiding the construction of computational devices. Chaos developed not from newly discovered physical laws, but by a deeper analysis of the equations underlying Newtonian physics. Chaos is a scientific revolution based on mathematics----deduction rather than induction. Chaos takes the equations of Newton, and uses mathematical analysis to establish the widespread unpredictability in the phenomena described by those equations. Via mathematics, one establishes the failure of Newtonian determinism by using Newton's own laws!

Taxpayers' Money In 1960 in Rio de Janeiro I was receiving support from the National Science Foundation (NSF) of the United States as a postdoctoral fellow, while doing research in an area of mathematics which was to become the theory of chaos. Subsequently questions were raised about my having used U.S. taxpayers' money for this research done on the beaches of Rio. In fact none other than President Johnson's science adviser, Donald Hornig, wrote in 1968 in Science:

1This is an expanded version of a paper to appear in the proceedings of the Intemational Congress of Science and Technology--45 years of the National Research Council of Brazil.

9 1998 SPRINGER-VERLAGNEWYORK, VOLUME20, NUMBER1, 1998 3 9

This blithe s p i r i t leads m a t h e m a t i c i a n s to s e r i o u s l y propose that the c o m m o n m a n who p a y s the taxes ought to feel that m a t h e m a t i c a l creation should be supported w i t h public f u n d s on the beaches o f Rio . . . What happened during the eight years between the work on the beaches and this national condemnation? The 1960s were turbulent in Berkeley where I was a professor; my students were arrested, tear gas frequently filled the campus air; dynamics conferences opened under curfew; Theodore Kaczynski, the suspected Unabomber, was a colleague of mine in the Math Department. The Vietnam War was escalated by President Johnson in 1965, and I was moved to establish with Jerry Rubin a confrontational antiwar force. Our organization, the Vietnam Day Committee (VDC), with its teach-in, its troop train demonstrations and big marches, put me onto the front pages of the newspapers. These events led to a subpoena by the House Unamerican Activities Committee (HUAC), which was issued while I was en route to Moscow to receive the Fields Medal in 1966. The subsequent press conference I held in Moscow attacking U.S. policies in the Vietnam War (as well as Russian intervention in Hungary) created a long-lasting furor in Washington, D.C. Let me hark back to what actually happened in that Spring of 1960 on those beaches of Rio de Janeiro. Flying Down to Rio In the 1950s there was an explosion of ideas in topology, which caught the imagination of many young research students such as myself. I finished a Ph.D. thesis in that domain at the University of Michigan in 1956. During that summer I, with my wife, Clara, attended in Mexico City a conference reflecting this great movement in mathematics, with the world's notables in topology present and giving lectures. There I met a Brazilian, Elon Lima, who was writing a thesis in topology at the University of C h i c a g o - where I was about to take up the position of i n s t r u c t o r - and we became good friends. A couple of years later, Elon introduced me to Mauricio Peixoto, a young Visiting professor from Brazil. Mauricio was from Rio, although he had come from a northern state of Brazil where his father had been governor. A goodhumored pleasant fellow, Mauricio, in spite of his occasional bursts of excitement, was conservative in his manner and in his politics. As was typical for the rare mathematician working in Brazil at that time, he was employed as teacher in an engineering college. Mauricio also helped found a new institute of mathematics (IMPA), and his aspirations brought him to America to pursue research in 1957. Subsequently he was to become the President of the Brazilian Academy of Sciences. Mauricio was working in the subject of differential equations or dynamics and showed me some beautiful results. Before long I myself had proved some theorems in dynamics. In the summer of 1958, Clara and I with our newborn son, Nat, moved to the Institute for Advanced Study (IAS)

40

THE MATHEMATICALINTELLIGENCER

in Princeton, New Jersey. I was supposed to spend two yeats there with an NSF postdoctoral fellowship. However, due to our common mathematical interests, Mauricio and Elon invited me to fmish the second year in Rio de Janeiro. So Clara and I and our children, Nat and newly arrived Laura, left Princeton in December, 1959, to fly down to Rio. The children were so young that most of our luggage consisted of diapers, but nevertheless we were able to realize an old ambition of seeing Latin America. After visiting the Panamanian jungle, the four of us left Quito, Ecuador, Christmas of 1959, on the famous Andean railroad down to the port of Guayaquil. Soon we were flying into Rio de Janeiro, recovering from sicknesses we had acquired in Lima. I still remember vividly, arriving at night, going out several times trying to get milk for our crying children, and returning with a substitute, cream or yogurt. We later learned that, in Rio, milk was sold only in the morning, on the street. At that time Brazil was truly part of the "third world." However, our friends soon helped us settle into Brazilian life. We arrived just after a coup had been attempted by an air force colonel. He fled the country to take refuge in Argentina, and we were able to rent, from his wife, his luxurious 11-room apartment in the district of Rio called Leme. The U.S. dollar went a long way in those days, and we were even able to hire the colonel's two maids, all with our fellowship funds. Sitting in our upper-story garden veranda, we could look across to the hill of the favela (called Babylonia) where Black Orpheus was filmed. In the hot humid evenings preceeding Carnaval, we would watch hundreds of the favela dwellers descend to samba in the streets. Sometimes I would join their wild dancing, which paraded for many miles. In front of our apartment, away from the hill, lay the famous beach of Copacabana. I would spend my mornings on that wide, beautiful, sandy beach, swimming and body surfmg. Also, I took a pen and paper and would work on mathematics. M a t h e m a t i c s on the Beach Very quickly after our arrival in Rio, I found myself working on mathematical research. My host institution, Instituto da Matematica, Pura e Aplicada (IMPA), funded by the Brazilian government, provided a pleasant office and working environment. Just two years earlier IMPA had set up its own quarters, a small colonial building in the old section of Rio called Botafogo. There were no undergraduates and only a handful of graduate mathematics students. There were also a very few research mathematicians, notably Peixoto, Lima, and an analyst named Leopoldo Nachbin. There was also a good math library. But no one could have guessed that in less than three decades IMPA would become a world center of dynamical systems, housed in a palatial building, as well as a focus for all Brazilian science. In a typical afternoon I would take a bus to IMPA and soon be discussing topology with Elon or dynamics with Mauricio, or be browsing in the library. Mathematics re-

s e a r c h typically d o e s n ' t require m u c h - - a p a d of p a p e r and a ballpoint pen, library r e s o u r c e s , and colleagues to query. I w a s satisfied. Especially enjoyable w e r e the times spent on t h e beach. My w o r k w a s mostly scribbling d o w n ideas a n d trying to s e e h o w arguments could b e p u t together. I w o u l d sketch crude diagrams of g e o m e t r i c objects flowing t h r o u g h space, a n d try to link the p i c t u r e s with formal deductions. Deep in this kind of thinking a n d writing on a p a d o f paper, I w a s n o t b o t h e r e d by the distractions of the beach. It w a s g o o d to b e able to take time off from the r e s e a r c h to swim. The surf was an exciting challenge and even s o m e t i m e s quite frightening. One time w h e n Lima visited m y "beach ofrice," w e entered the surf a n d were both caught in a current which t o o k us out to sea- While Elon felt his life fading, b a t h e r s shouted the advice to swim parallel to the shore to a s p o t where we w e r e able to return. (It was 34 y e a r s later, j u s t before Carnaval, that once again those s a m e b e a c h e s alm o s t did m e in. This time an oversized wave b o u n c e d me s o hard on the sand it injured m y wrist and tore m y should e r tendon; and then that s a m e big wave carried m e out to sea. I was lucky to get b a c k using my good arm.) L e t t e r from A m e r i c a At that time, as a topologist, I p r i d e d myself o n a p a p e r that I h a d j u s t p u b l i s h e d in dynamics. I was d e l i g h t e d with a c o n j e c t u r e in that p a p e r w h i c h had as a c o n s e q u e n c e that (in m o d e m terminology) "chaos doesn't exist"! This euphoria w a s s o o n s h a t t e r e d by a letter I r e c e i v e d from N o r m a n Levinson. I k n e w him as c o a u t h o r o f the main g r a d u a t e t e x t in o r d i n a r y differential equations a n d as a scientist to be taken seriously. Levinson w r o t e m e o f a n earlier result o f his w h i c h effectively c o n t a i n e d a c o u n t e r e x a m p l e to m y conjecture. His p a p e r in turn w a s a clarification of extensive w o r k of the British m a t h e m a t i c i a n s Mary Cartwright a n d J. L. Littlewood done during World War II. Cartwright and Littlewood had b e e n analysing s o m e equations t h a t a r o s e in war-related studies involving radio waves. T h e y h a d found unexpected and unusual behaviour of solutions of t h e s e equations. In fact Cartwright and L i t t l e w o o d had f o u n d signs of chaos, even in equations that a r o s e naturally in engineering. But t h e w o r l d w a s n ' t r e a d y to listen. I never m e t Littlewood, b u t in the mid-sixties, D a m e Mary Cartwright, then h e a d o f a w o m e n ' s college (Girton) at Cambridge, invited m e to high table. I w o r k e d day and night to try to resolve the challenge to m y beliefs t h a t t h e l e t t e r posed. It w a s n e c e s s a r y to t r a n s l a t e Levinson's analytic arguments into m y o w n geom e t r i c w a y of thinking. At least in m y o w n case, understanding m a t h e m a t i c s d o e s n ' t c o m e from r e a d i n g o r even listening. It c o m e s f r o m rethinking w h a t I s e e o r hear. I m u s t r e d o the m a t h e m a t i c s in the c o n t e x t o f m y particular background. And t h a t b a c k g r o u n d c o n s i s t s of m a n y threads, s o m e strong, s o m e weak, s o m e algebraic, s o m e visual. My b a c k g r o u n d is s t r o n g e r in g e o m e t r i c analysis, but following a sequence of f o r m u l a e gives m e trouble. I t e n d to b e s l o w e r than m o s t m a t h e m a t i c i a n s to u n d e r s t a n d an

argument. The m a t h e m a t i c a l literature is useful in that it p r o v i d e s clues, and one can often u s e these clues to p u t t o g e t h e r a cogent picture. When I have reorganized t h e m a t h e m a t i c s in m y own terms, then I feel an u n d e r s t a n d ing, n o t before. I eventually convinced m y s e l f that i n d e e d Levinson w a s correct, a n d that m y conjecture w a s wrong. Chaos was alr e a d y implicit in the analyses o f Cartwright a n d Littlewood. The p a r a d o x was resolved, I h a d g u e s s e d wrongly. But while learning that, I d i s c o v e r e d t h e horseshoe! The H o r s e s h o e The h o r s e s h o e is a natural c o n s e q u e n c e of a g e o m e t r i c a l w a y of looking at the equations o f Cartwright-Littlewood and Levinson. It helps u n d e r s t a n d the m e c h a n i s m of chaos, and e x p l a i n the w i d e s p r e a d u n p r e d i c t a b i l i t y in dynamics. Chaos is a characteristic o f dynamics, that is, of t i m e evolution o f a set o f states o f nature. Let m e t a k e time t o be m e a s u r e d in discrete units. A state of nature will be ideaiized as a p o i n t in the t w o - d i m e n s i o n a l plane. I will s t a r t by describing a non-chaotic linear example. The i d e a is to t a k e a square, Figure 1, and to study w h a t h a p p e n s to a point on this square in one unit of time, und e r a t r a n s f o r m a t i o n to be d e s c r i b e d . The v e r t i c a l dimension is s h r u n k uniformly t o w a r d s t h e c e n t e r o f t h e square and the h o r i z o n t a l is e x p a n d e d uniformly at t h e s a m e time. Figure 2 s h o w s the d o m a i n obt a i n e d b y this process, A*B*D*C*, s u p e r i m p o s e d over t h e original square ABDC. I have also s h a d e d in the set o f points w h i c h don't move out of the square in this process. The s e c o n d of o u r three s t a g e s in u n d e r s t a n d i n g is t h e p e r t u r b e d linear example. N o w the square is m o v e d into a b e n t v e r s i o n of the elongated r e c t a n g l e of Figure 2. Thus Figure 3 d e s c r i b e s the m o t i o n of o u r square o b t a i n e d b y a small m o d i f i c a t i o n of Figure 2. The h o r s e s h o e is the fully non-linear version of w h a t h a p p e n s to p o i n t s on the square, b y an extension of t h e p r o c e s s e x p r e s s e d in Figures 2 a n d 3. This is the situation w h e n m o t i o n m a k e s a qualitative d e p a r t u r e from the linear model. See Figure 4. IGURE

A

C

B

D

VOLUME 20, NUMBER 1, 1998

41

"visual motion." The results in the next section c o n c e r n visual motions. In summary, the horseshoe is a fully non-linear motion. In the next section, I will s h o w how chaos c o m e s out of this picture.

The Horseshoe and Chaos: Coin Flipping The laws of chance, with good reason, have traditionally b e e n expressed in t e r m s of flipping a coin. Guessing w h e t h e r heads or tails is the o u t c o m e of a coin toss is the paradigm of pure chance. On the other hand it is a deterministic process that governs the whole m o t i o n of a real coin, a n d hence the result, heads or tails, depends only o n very subtle factors of the initiation of the toss9 This is "sensitive d e p e n d e n c e on initial conditions 9 A coin-flipping e x p e r i m e n t is a sequence of coin tosses each of which has as o u t c o m e either heads (H) or tails (T). Thus it can be represented in the from HTTHHTTTTH. . . . 2 A general coin-flipping e x p e r i m e n t is thus a sequence So sl s2 999 where each of So, sl, s2 999 is either H or T. Here is the result of the h o r s e s h o e analysis that I f o u n d o n that C o p a c a b a n a beach. Consider all the p o i n t s which, u n d e r the horseshoe mapping, stay in the square, i.e., d o n ' t drift out of our field of vision. These "visual motions" corr e s p o n d precisely to the set of a l l coin-flipping experiments! This discovery d e m o n s t r a t e s the o c c u r r e n c e of unpredictability in fully n o n - l i n e a r m o t i o n a n d gives a m e c h a n i s m of how d e t e r m i n i s m p r o d u c e s uncertainty. The correspondence is the following. To each visual motion there is a n associated coin-flipping experiment. If :Co Xl x2 999 is the visual motion, at time i = 0, 1, 2, 3 , . . . associate H or heads if xi lies in the top half of the square a n d T or tails if it lies in the b o t t o m half. Moreover, and this is the crux of the matter, every possible sequence of coin flips is represented by a horseshoe motion. Therefore the d y n a m i c s is as unpredictable as coin-flipping 9 In the n a t u r a l o n e - o n e c o r r e s p o n d e n c e

FIGURE:

The horseshoe is the domain s u r r o u n d e d by the dotted line. Instead of a state of nature evolving according to a mathematical formula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dynamics was with the use of algebraic expressions. But a description given by formulae would be c u m b e r s o m e . That form of description w o u l d n ' t have led m e to insights or to perceptive analysis. My b a c k g r o u n d as a topologist, trained to b e n d objects like squares, helped to m a k e it possible to see the horseshoe. The dynamics of the horseshoe is described by moving a point in the square to a point in the horseshoe according to Figure 4. Thus the c o m e r marked A m o v e s to the point marked A* in one unit of time. The m o t i o n of a general point x in the square is a sequence of points x0 Xl x2 . . . Here x0 = x is the present state, Xl is that state a unit of time later, x2 that state two units of time later, etc. Now imagine o u r visual field to be j u s t the square itself. When a point is m o v e d out of the square we will discard that motion. Figure 5 shades in the points which don't leave the square in o n e unit of time. I will call a s e q u e n c e which n e v e r leaves the square a

X0ZlX2

IGURE:

A

m$

B

...... ""--..... "-., ............................................................

. . . . . . . . --.

...--->S

0sl

82 ...,

Xo x l x2 999 is a m o t i o n lying in the square a n d So Sl s2 9 is a sequence of H's a n d T's. On the left is a deterministically generated motion a n d o n the right a coin-flipping experiment. We have seen complete unpredictability pop up within deterministic motion, the horse, "''''"-.. ,.,, shoe 9 This is chaos.

C*

""'"-,,,.,.

B*

D* D

The Hidden Origins of Chaos As chaos is a mathematically b a s e d revolution, it is n o t surprising that a m a t h e m a t i c i a n first saw evidence of chaos in dynamics.

2To give a complete picture in this section, one needs to reverse time and consider sequences of heads and tails which go back in time as well,

42

THE MATHEMATICAL INTELUGENCER

A

A*

B

"'"'"'"'",

..................................................................

,.....L

P

...........................................................

,

"'"

C*

"' ,

D • ................................................................

i..,'"

S*'- .................................................................

"" FIGURE 6

c

D

,=ULmaL~ Henri Poincar~ w a s (with David Hilbert) one of the two f o r e m o s t mathematicians in the world active at the end of the last century. I h e a r d of him first as an originator of topology, who had written an article claiming that a manifold with the s a m e algebraic invariants as the n-dimensional sphere w a s topologically identical to the n-dimensional sphere. Then he found a m i s t a k e in his proof. Restricting himself n o w to 3 dimensions, he formulated the assertion as a problem, n o w called Poincar~'s Conjecture, still one o f the three o r four great unsolved p r o b l e m s in mathematics today. More germane to m y p r e s e n t story is his c o n t r i b u t i o n to the s t u d y of dynamics. Poincar~ m a d e e x t e n s i v e studies in celestial m e c h a n i c s , t h a t is to say, the m o t i o n s o f the planets. At that time it w a s a c e l e b r a t e d p r o b l e m to p r o v e the solvability o f t h o s e underlying equations, a n d in fact Poincar~ at o n e time thought that he had p r o v e d it. Shortly thereafter, however, he b e c a m e t r a u m a t i z e d b y a discovery w h i c h n o t only s h o w e d him wrong b u t s h o w e d the impossibility of ever solving the equations for even three bodies. This d i s c o v e r y he c h r i s t e n e d "homoclinic point."

A h o m o c l i n i c p o i n t is a m o t i o n t e n d i n g to an equilibrium as time i n c r e a s e s and also to t h a t sarae equilibrium as time r e c e d e s into the past. See Figure 6. Here p is an equilibrium a n d h m a r k s the homoclinic point. The a r r o w s represent the direction o f time. This definition sounds harmless enough but carries amazing consequences. Poincar~ wrote concerning his discovery: 3

One will be struck by the complexity of this figure, which I won't even try to draw. Nothing can more dearly give an idea of the complexity of the three-body problem and in general of all the problems of d y n a m i c s . . . In a d d i t i o n to showing the impossibility of solving the equations o f p l a n e t a r y motion, the homoclinic point has turned out to b e the t r a d e m a r k o f chaos; it is found in essentially e v e r y chaotic d y n a m i c a l system. It w a s in t h e first half of this c e n t u r y that A m e r i c a n m a t h e m a t i c s c a m e into its own, a n d traditions s t e m m i n g from Poincar~ in t o p o l o g y a n d d y n a m i c s were central in this d e v e l o p m e n t . G.D. Birkhoff w a s the m o s t well-known FIGURE 7

:IGURE ,=

3My own translation from the French.

VOLUME 20, NUMBER 1, 1998

American mathematician before World War II. He came from Michigan and did his graduate work at the University of Chicago, before settling down at Harvard. Birkhoff was heavily influenced by Poincar~'s work in dynamics, and he developed these ideas and especially the properties of homoclinic points in his papers in the 20s and 30s. Unfortunately, the scientific community soon lost track of the important ideas surrounding the homoclinic points of Poincar& In the conferences in differential equations and dynamics that I attended in the late 50s, there was no awareness of this work. Even Levinson never showed in his book, papers, or correspondence with me that he was aware of homoclinic points. It is astounding how important scientific ideas can get lost, even when they are aired by leading mathematicians of the preceding decades. I learned about homoclinic points and Poincar~'s work from browsing in Birkhofs collected works, which I found in IMPA's library. It was because of the recently discovered horseshoe that the homoclinic landscape was to sink into my consciousness. In fact there was an important relation between horseshoes and homoclinic points. If a dynamics possesses a homoclinic point then ! proved that it also contains a horseshoe. This can be seen in Figure 7. Thus the coin-flipping syndrome underlies the homoclinic phenomenon, and helps to comprehend it. The Third Force I was lucky to fmd myself in Rio at the confluence of three different historical traditions in dynamics. These three strains, while dealing with the same subject, were isolated from each other, and this isolation obstructed their development. I have already discussed two of these forces, Cartwright-Littlewood-Levinson and Poincar~-Birkhoff. The third had its roots in Russia with the school of differential equations of A. Andronov in Gorki in the 1930s. Andronov had died before the first time I went to the Soviet Union, but in Kiev, in 1961, I did meet his wife, AndronovaLeontovich, who was still working in Gorki in differential equations. In 1937 Andronov teamed up with the Soviet mathematician L. Pontryagin. Pontryagin had been blinded at the age of 14, yet went on to become a pioneering topologist. The pair described a geometric perspective of differential equations they called "rough," subsequently called structural stability. Chaos, in contrast to the two previously mentioned traditions, was absent in this development because of the restricted class of dynamics. Fifteen years later the great American topologist Solomon Lefschetz became enthusiastic about Andronov and Pontryagin's work. Lefschetz had also suffered an accident, that of losing his arms, before turning to mathematics, and this perhaps generated some kind of bond between him and the blind Pontryagin. They first met at a topology conference in Moscow in 1938, and again after the war. It was through Lefschetz's influence, in particular from an article of his student De Baggis, that Mauricio Peixoto in Brazil learned of structural stability. 4~

THE MATHEMATICALINTELLIGENCER

Peixoto came to Princeton to work with Lefschet~ in 1957, and this is the route which led to our meeting each other through Elon. After this meeting, I studied Lefschetz's book on a geometric approach to differential equations, and eventually came to know Lefschetz in Princeton. Thanks to Pontryagin and Lefschetz, there was the specter of topology in the concept of structural stability of ordinary differential equations. I believe that was why I listened to Mauricio. Good Luck Sometimes a horseshoe is considered an omen of good luck. The horseshoe I found on the beach of Rio certainly seemed to have such a property. In that spring of 19601 was primarily a topologist, mainly motivated by the problems of that subject, and most of all driven by the great unsolved problem posed by Poincar& Since I had started doing research in mathematics, I had produced false proofs of the 3-dimensional Poincar6 Conjecture, returning again and again to that problem. Now on those beaches, within two months of fmding the horseshoe, I found to my amazement an idea which seemed to succeed provided I returned to Poincar6's original assertion and then restricted the dimension to 5 or more. In fact the idea not only led to a solution of Poincar6's Conjecture in dimensions greater than 4, but it gave rise to a large number of other nice results in topology. It was for this work that I received the Fields Medal in 1966. Thus "... the mathematics created on the beaches of Rio ..." (Hornig) was the horseshoe and the higherdimensional Poincar6 Conjecture.

I'~.lt--~_.,:,.--

Jeremy

Gray I

Orbits of Asteroids, a Braid, and the First Link Invariant Moritz Epple

n 22 January 1833, Carl Friedrich G a u s s w r o t e a short p a s s a g e in one o f his m a t h e m a t i c a l n o t e b o o k s w h i c h w a s to b e c o m e widely k n o w n among mathematicians and physicists s o o n a f t e r it w a s fn'st p u b l i s h e d in 1867:

O

O f the g e o m e t r i a situs, w h i c h L e i b n i z f o r e s a w a n d into which only a p a i r o f geometers (Euler and Vandermonde) were granted the privilege o f taking a f a i n t glance, w e k n o w and have, after a century and a half, little m o r e than nothing. A centralproblem in the overlapping area o f geometria situs a n d g e o m e t r i a magnitudinis will be to count the intertwinings [Umschlingungen] o f two closed or i n f i n i t e curves. Let the coordinates o f an undeterm i n e d p o i n t o f the f i r s t curve be x, y, z; o f the second x', y ', z ;" a n d let H I ( x , _ x)2 + @, _ y)2 + (z' - z)2] [(x' - x ) ( d y d z ' - dzdy') + (y' - y) dzdx' - dxdz') +

(z' - z) (dxdy' - dydx')] = V; then this integral taken along both curves is = 4m~r,

m being the n u m b e r o f i n t e r t w i n i n g s . The value is reciprocal, i.e., it rem a i n s the s a m e i f the curves are interchanged. 1 The elusive science o f g e o m e t r i a situs w h i c h G a u s s w a s referring to w a s s o o n a f t e r w a r d given the m o d e m n a m e of Topologie--topology--by one o f G a u s s ' s students, J o h a n n B e n e d i k t Listing. 2 Geometria m a g n i t u d i n i s , on

Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

the other hand, d e n o t e d the kind of analytical g e o m e t r y w h i c h the 18th cent u r y had e l a b o r a t e d so impressively, b a s e d on the 17th-century ideas o f Descartes, Newton, and others. The beautiful f o r m u l a Gauss w r o t e d o w n c o n n e c t e d the g e o m e t r y of magnitude with that o f position: A linking number, d e p e n d e n t only on the relative positions of two curves in the topological sense, was calculated by an integral involving the coordinates o f points on these curves; topological information was extracted from analytical information. The t e x t o f G a u s s ' s fragment p o s e s several historical riddles. As in m a n y o t h e r p a s s a g e s o f his n o t e b o o k s , Gauss gave n o i n d i c a t i o n o f any p r o o f or a r g u m e n t for his claim, n o r did he give any r e a s o n s w h i c h h a d led him to c o n s i d e r t h e linking o f s p a c e curves at all. Without f u r t h e r information, w e cannot even b e s u r e h o w his claim should b e i n t e r p r e t e d mathematically: Is it a definition; i.e., did Gauss w a n t to say that the p o s s i b l e values of t h e double integral on t h e left side of his f o r m u l a are i n t e g e r multiples of 47r, and that, therefore, the integer appearing on the right side could be def m e d as the linking n u m b e r of the t w o curves involved? Or is Gauss's f o r m u l a a theorem, c o m p u t i n g an independently defined n u m e r i c a l invariant o f intertwined curves b y analytical means? We thus have t h e following four questions: 1. When, a n d how, did Gauss find the integral? 2. How did he k n o w that the values o f this integral w e r e integer multiples of 4~r?.

lWerke, Vol. V, p. 605. All emphasis in this and the following quotations is in the originals. Square brackets are used to indicate my omissions or additions. 2First in a letter of 1836, then in Listing's essay Vorstudien zur Topologie, published in 1847. The name geometrfa situs, or analysis situs, however, was retained by Riemann and later Poincar& Only in the first decades of the 20th centuw, topology gradually replaced analysis situs. Gauss's reference to Euler is to the latter's Solutio problematis ad geometriam situs pertinentis of 1736, dealing with the K6nigsberg bridges; the reference to Vandermonde is to a paper entitled Remarques sur les problemes de situation of 1771, in which Vandermonde studied various weaving patterns and their symmetries, along with the problem of circuits of knight's moves. Both papers are reprinted in English translation in (Biggs, et aL 1976).

9 1998SPRINGER-VERLAGNEWYORK,VOLUME20, NUMBER1, 1998 45

3. Did Gauss think of an independent notion of a linking number? 4. Why did he write down the fragment in early 1833? The historical literature has not addressed these questions in any detail. 3 Fortunately, it is possible to give substantial answers to all four questions. We will find that the line of thought which eventually induced Gauss to document his insight originated almost 30 years earlier, and in the context of astronomy. Let us start with the fourth question, and the circumstances of the publication of the passage in 1867, in the fifth volume of Gauss's Werke. This volume was edited by Ernst Schering, the GOttingen physicist who was in charge of Gauss's papers until his death in 1897. It was devoted to published and unpubfished work on electricity and magnetism. The topological fragment was placed in a section containing other unpublished notes, mainly on electromagnetic induction, and Schering obviously believed that the fragment belonged in this context. His editorial choice was certainly reasonable, though not beyond criticism, as we shall see below. In fact, in the years following Faraday's discovery of induced currents in 1831, Gauss was working intensely on electromagnetic induction, together with his young friend and colleague, the physicist Wilhelm Weber, who had arrived in G6ttingen in September 1831. After more than a year of theoretical and experimental work, the two set up the first telegraph in Germany in Spring 1833, leading from the G6ttingen Observatory (in which Gauss both lived and experimented) to Weber's physics laboratory. One of the crucial laws governing the physics of electromagnetic induction was Biot and Savart's law, describing the force which an infinitesimal current element exerted on an "element of positive northern magnetic fluid," as Gauss wrote. His posthumous

fragments gave several forms of this interaction. For instance, a current element situated at a point R and directed to an infinitesimally near point R', of strength /~ = RR', acts on a magnetic element at P with a force of strength /z sin R ~ P (RP) 2 and direction orthogonal to the plane determined by P, R, and R'. Rewritten in modern symbolism, this infinitesimal force is f_

~xf

llfIF

if we denote the "vectors" R R ' and RP by dg and f, respectively. This fragment was written in the same notebook as the one on the linking integral and published by Schering immediately following it. Indeed, the integrand of the linking integral is proportional tofdg' if we choose d~ as the line element of one of the curves while d~' denotes the line element of the other. Thus, there was a compelling electrodynamical interpretation of the linking number: It was proportional to the work V done when a fictitious magnetic test particle was carried along a closed curve in the magnetic field induced by a constant current running through another closed curve. From Amp~re's researches, it was known that this work could also be determined by adding the oriented intensities of the currents intersecting a surface bounded by the path of the magnetic test particle. 4 Therefore, it followed that the double integral expressing V was an integer multiple of some constant, independent of the metric details of the situation. Against this background, it is quite understandable that Schering chose to place the fragment on the Linking integral among Gauss's electrodynamical writings. There is even further evidence showing that Gauss was thinking of topological matters in the years

of his cooperation with Weber. In 1847, another astronomer and mathematician interested in topology, August Ferdinand M6bius, wrote to Gauss: As I have heard f r o m W. Weber, already some years ago you intended to write a treatise on all possible interlacings [Verschlingungen] of a thread, as an introduction to, or preparation for, the theory of electrical and magnetic currents. May we hope that this treatise will soon appear? The fulfillment of this hope would be most desirable f o r myself and certainly f o r m a n y others, too.5 Apparently, Gauss was less cautious in his remarks to colleagues than in his own notebooks. No texts have survived which could be regarded as parts of a treatise on topology. Nevertheless, Weber's information was probably right. Well before his electrodynamical work, Gauss had had similar plans to write on geometria situs (see below). The resigned tone of the first lines of his fragment on the linking integral may well represent an admission that he was not yet in a position to realize his intentions, according to his high standard of pauca sed matura. All this seemed to confirm the electrodynamic interpretation of Gauss's note. Accordingly, the first readers of Gauss's fragment were physicists. The most important one was in Scotland. Within one year of the publication in 1867, James Clerk Maxwell read Gauss's fragment, and communicated the idea to his scientific friends, including Peter Guthrie Tait in Edinburgh, who was to embark on the classification of knots about 10 years later. Maxwell also reported on it to the London Mathematical Society in early 1869, and worked out the idea of the passage in great detail in his major work, the Treatise on Electricity and Magnetism (Maxwell 1873, w167 409422). Among other things, he pointed out that there could be nontrivial links

3The best treatments of Gauss's topological fragments are still (St&ckel 1918) and (Pont 1974); both give little more than a listing of some relevant sources. 4This was an early, magnetostatic particular case of what came to be known as "Stokes's theorem." Gauss discussed this situation in his AIIgemeine Theorie des Erdmagnetismus of 1838. 5MObius to Gauss, 2 February 1847. The letter is in Gauss's papers and was probably available to Schering. No answer to M6bius's letter seems to be extant, The contact between Weber and M6bius was the consequence of Weber's exile from the K6nigreich Hannover. With six of his colleagues, Weber had lost his chair at the university in 1837 for his refusal to accept the abolition of Hannover's liberal constitution. In 1843, he accepted a call to Leipzig where M6bius was working.

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Figure 1. Maxwell's link.

o f t w o c o m p o n e n t s with vanishing linking number, by an e x a m p l e which tod a y is often called the "Whitehead link" (see Fig. 1). Maxwell's discussion c l o s e d with a variation on Gauss's t h e m e of the e m e r g e n c e o f a n e w mathematical field: I t w a s the discovery by G a u s s o f this v e r y integral, e x p r e s s i n g the w o r k done on a m a g n e t i c pole w h i l e des c r i b i n g a closed c u r v e i n p r e s e n c e o f a d o s e d electric current, a n d indicati n g the geometrical c o n n e x i o n bet w e e n the two closed curves, that led h i m to l a m e n t the s m a l l progress m a d e i n the G e o m e t r y o f P o s i t i o n s i n c e the t i m e o f L e i b n i t z , E u l e r a n d Vandermonde. We have now, however, s o m e progress to report, chiefly due to R i e m a n n , Helmholtz, a n d Listing. 6 Through Schering's edition and Maxwell's reception, the electrodyn a m i c a l i n t e r p r e t a t i o n o f the linking integral was well established. If this was t h e full story, the linking n u m b e r w o u l d have to be defined b y the double integral, and w e s h o u l d n o t e x p e c t to find r e f e r e n c e s to the situation in question in Gauss's e a r l i e r writings. But w e do find such references, and t h e y allow us to develop a s e c o n d w a y of looking at the m a t h e m a t i c s of the linking integral, in a p u r e l y g e o m e t r i c fashion. In o r d e r to present the first such reference, I will go b a c k to one of the maj o r breakthroughs in Gauss's scientific

career, his calculation of the orbit o f the fLrst o b s e r v e d asteroid, Ceres, in 1801. A s t r o n o m y held a leading p o s i t i o n in the public appreciation of science at the time, and Gauss's success did m o r e to p r o m o t e his c a r e e r than his earlier recognition by mathematicians as a leading number-theorist. In the y e a r s to follow, a s t r o n o m e r s found a large numb e r of similar celestial bodies, and Gauss---still in B r a u n s c h w e i g - - c o n t i n ued to think of asteroids. In August 1804, he published a small treatise entitled g?ber die Grenzen der geocentrischen O t t e r der Planeten, w h i c h t o o k up a rather practical question, namely t h e determination of the celestial region in which a given n e w asteroid, or planet, might possibly appear. 7 This s h o r t article is a striking e x a m p l e of the diversity and density of a r g u m e n t which Gauss was able to achieve in a single text. Published in an astronomical journal, the treatise addressed, at the s a m e time, issues of practical astronomy, such as recent observational d a t a a n d the making of star maps, a n d m a t h e m a t i c a l topics in geometry, differential equations, and g e o m e t r i a situs. In fact, Gauss had already b e e n dir e c t e d to this latter field in the c o n t e x t of his first p r o o f of the f u n d a m e n t a l t h e o r e m o f algebra, in 1799. On 12 O c t o b e r 1802, he a d d r e s s e d the s u b j e c t in a l e t t e r to the a s t r o n o m e r Heinrich Olbers, with w h o m he had s t a r t e d a c o r r e s p o n d e n c e on the o c c a s i o n o f Olbers's observations of Ceres. G a u s s m e n t i o n e d that he e x p e c t e d Carnot's Gdomdtrie de p o s i t i o n to a p p e a r soon, wrongly taking this title in the s e n s e of g e o m e t r i a situs. He added: This still almost unexploited subject, i n w h i c h w e only have a f e w f r a g m e n t s f r o m E u l e r and a geometer w h o m I highly appreciate, Vandermonde, m u s t open a completely n e w field a n d f o r m a separate and highly interesting branch o f the sublime science o f quantity. 8

Let m e t a k e the liberty of presenting the p r o b l e m o f Gauss's p a p e r in m o d e r n m a t h e m a t i c a l language. Let the orbit o f the e a r t h ' s m o t i o n a r o u n d the sun b e given b y X C ~3, and let X ' C ~3 b e the o r b i t of a n o t h e r celestial body, planet, comet, o r asteroid (the sun being at the c e n t e r of a suitable s y s t e m o f Cartesian coordinates). Determine the region on the s p h e r e given by

This region w a s called z o d i a c u s b y Gauss. Its d e t e r m i n a t i o n helped to limit the effort n e e d e d b o t h in the observation o f the given celestial b o d y and in the p r o d u c t i o n of an atlas of t h e smallest p a r t o f the celestial s p h e r e into which the orbit of the b o d y could b e drawn. In o r d e r to solve his p r o b l e m (topologists will already have recognized h o w it is c o n n e c t e d to the linking integral), Gauss derived a differential equation for the boundary curve or curves of the zodiacus, implicitly assuming the orbits to be s m o o t h curves. If ~ = ( x , y , z ) and s = ( x ' , y ' , z ' ) den o t e the c o o r d i n a t e s of orbit points, a n e c e s s a r y c o n d i t i o n that a pair o f p o i n t s (~,~') c o r r e s p o n d to a b o u n d a r y p o i n t of t h e z o d i a c u s is that the triple consisting of t h e t w o tangent v e c t o r s to the orbits at :~ a n d :~' and the disp l a c e m e n t v e c t o r P := ~' - 9 be linearly dependent. Gauss e x p r e s s e d this condition b y saying that the two tangents at ~ a n d ~' h a d to b e coplanar. Translating this condition into a form u l a led to the differential equation (x' - x ) ( d y ' d z - d y dz') + (y' - y ) ( d z ' d x - dz d x ' ) + (z' - z ) ( d x ' dy - d x dy') = O. F o r later use, let us abbreviate the differential form on the left-hand side b y w. Obviously, this form is, up to a change o f sign, nothing but the num e r a t o r of the integrand in the linking integral! At this point, Gauss inserted a very typical remark: He had under-

6From (Maxwell 1873, w 421). That Helmholtz's name appears in this extension of the list of topologists points to another development which had made British physicists aware of topology, namely research on vortex motion in perfect fluids. See my Topology, Matter, and Space, to appear in Archive for History of Exact Sciences, for a detailed study of this development. 7The article is reprinted in Werke, Vol. Vl, pp. 106-118. 8Schilling and Kramer (1900/1909, vol. 1, p. 103).

VOLUME 20, NUMBER1, 1998 4 7

t a k e n a m a t h e m a t i c a l study of this equation in its o w n fight, but for the s a k e o f brevity, he did n o t wish to go into that now. It turned out t h a t n o t all solutions of the above differential equation repr e s e n t e d actual b o u n d a r y lines of the zodiacus. Gauss distinguished three p o s s i b l e cases: (1) The minimal dist a n c e of the celestial b o d y to the sun was greater t h a n the m a x i m a l distance b e t w e e n earth a n d sun, (2) vice versa, (3) the two orbits w e r e linked. He s h o w e d that in t h e first t w o cases, the solutions r e p r e s e n t e d two disjoint closed curves on the sphere, w h e r e a s in the third, he f o u n d a single closed curve. But, G a u s s r e m a r k e d , n o n e of the regions b o u n d e d b y this curve could b e the zodiacus o f a case (3) celestial body: In this case, one could s h o w "for r e a s o n s of the g e o m e t r y o f position" that t h e zodiacus w a s the whole celestial sphere. Consequently, the solution of the differential equation oJ = 0 n o w h a d a different meaning. This, too, Gauss left as a p r o b l e m to the reader. He r e m a r k e d that none of the orbits of t h e k n o w n p l a n e t s was linked with t h a t o f the earth, but "comets of the s o r t exist in abundance. "9 The article c l o s e d with a calculation of the zodiacus of the recently discovered Pallas a n d Ceres. What w e r e t h e t o p o l o g i c a l r e a s o n s alluded to above, a n d w h a t w a s the c o n t e n t of G a u s s ' s further study of w? Gauss k e p t quiet on this point, but we can m a k e a p r o b a b l e guess on the basis of his later r e m a r k s concerning this differential form. 1~ G a u s s s e e m s to have thought o f s o m e t h i n g like the following g e o m e t r i c a l situation: If one looks at the image o f a c l o s e d orbit X' of a celestial b o d y on the celestial s p h e r e of the earth, w i t h the earth's position fixed at a p o i n t ~, one gets an oriented, c l o s e d curve that m a y be den o t e d by 7~. This curve encloses an

oriented area, A(~), w h i c h m e a s u r e s w h a t Gauss in later writings called the "solid angle" enclosed b y 75. As with p l a n e angles, this a r e a is well defined only up to a multiple of the total surface of the sphere, 47r. [In m o d e r n terms, 7~ is a 1-cycle o n the sphere; hence, it b o u n d s a 2-chain ~ , 7~ = Oa~. The a r e a A(~) in question is the integral of the canonical a r e a form o v e r this chain. Since a~ is well d e f m e d only up to a 2-cycle, i.e., up to a multiple of the s p h e r e itself, A(~) is determ i n e d only up to a multiple of 47r.] W h e n the earth moves a small distance along its orbit, say f r o m ~ to ?~, the curve 7~ is continuously d e f o r m e d into a n e a r b y curve 7#, p r o d u c i n g a continu o u s change hA of the a r e a enclosed. This a r e a change can b e c a l c u l a t e d analytically. Up to a sign, it is given b y the integral hA

=

eo

w h e r e r d e n o t e s the d i s t a n c e b e t w e e n the t w o integration a r g u m e n t s of w, a n d the first integration follows the e a r t h ' s orbit b e t w e e n 9 a n d ?~. When t h e e a r t h has c o m p l e t e d one revolution, t h e planet's orbit a p p e a r s again at its original position. Therefore, the a r e a change a s s o c i a t e d with a comp l e t e revolution m u s t b e an integer multiple of the total s u r f a c e of the sphere. If this multiple is different from zero, the zodiacus of t h e p l a n e t is the w h o l e sphere. In the c a s e o f linked ellipses, it is easy to s e e t h a t the w h o l e s p h e r e is covered w h e n t h e e a r t h m o v e s a r o u n d its orbit once. But since this p r o p e r t y does n o t d e p e n d on the "measure," i.e., metric s t r u c t u r e o f the w h o l e setting, it pertains to geometria situs. Here, w e have a second, g e o m e t r i c w a y o f deriving the linking integral and its behavior, n In m o d e r n terms, w e m a y d e s c r i b e it as the calculation of

the m a p p i n g degree of the m a p p i n g defining t h e zodiacus,

Of course, w e should be c a u t i o u s with s u c h reconstructions. But it is evident that G a u s s k n e w a g o o d d e a l m o r e t h a n h e w r o t e d o w n in his article of 1804. Even if he was n o t y e t thinking in t e r m s of linking n u m b e r s , etc., w e m a y c o n c l u d e that the phen o m e n o n of linking and the r e l e v a n t differential f o r m were k n o w n to G a u s s at this time. Before leaving the article on t h e zodiacus, let m e a d d a r e m a r k on t h e notion o f o r i e n t e d a r e a u p o n w h i c h this r e c o n s t r u c t i o n hinges. In a l e t t e r to" Olbers o f 1825, Gauss m e n t i o n e d t h a t he h a d k n o w n the notion of an o r i e n t e d a r e a o f p l a n e figures for about 30 years. The crucial p r o b l e m w a s to d e a l w i t h self-intersections of their "circumference," i.e., to d e t e r m i n e the a r e a of a figure o u t l i n e d b y an arbitrary c l o s e d curve w i t h a finite n u m b e r o f transverse d o u b l e points. In this case, t h e different regions of the p l a n e separated b y t h e a r c s of t h e given curve h a d to be w e i g h t e d with a p p r o p r i a t e coefficients. G a u s s e x p l a i n e d s o m e e x a m ples o f this p h e n o m e n o n - - t o d a y captured in t h e homological t e r m i n o l o g y of chains, cycles, and b o u n d a r i e s - - i n his l e t t e r to Olbers. i2 This l e t t e r belongs to t h e p e r i o d in w h i c h G a u s s w o r k e d on his Disquisitiones generales circa superficies curvas (1827). There, the n o t i o n of oriented a r e a w a s n e e d e d for t h e calculation of t h e total curvature o f a curved surface in 3space, d e f i n e d as the oriented a r e a of its image o n the sphere u n d e r w h a t is usually called the "Gauss map" today. In this way, Gauss w a s led to reconsider t h e g e o m e t r y of c l o s e d p l a n e curves (or o f c l o s e d curves on t h e

9Werke, Vol. Vl, p. 11 lf. In 1847, Listing counted 25 pairs of asteroids, whose orbits were known to be linked by then (Listing 1847, p. 64f.). lOSee in particular AIIgemeine Theorie des Erdmagnetismus of 1838, in Vol. V of the Werke, w 38. 11This geometric interpretation was first defended against Schering's electrodynamical interpretation by Schering's student, Otto B6ddicker. In his lectures on potential theory in 1874/75, Schering had communicated Gauss's result to his students, among them BOddicker, who decided to choose the linking integral as a dissertation topic. To a large extent, his dissertation (G6ttingen, 1876) is a lengthy elaboration of the geometric interpretation of the linking integral, based on the discussion of solid angles in Gauss's AIIgemeine Theorie des Erdmagnetismus. Note that Maxwell also included the geometrical interpretation in his treatment of the linking integral (Maxwell 1873, w167 417-421). ~2Gauss to OIbers, 30 October 1825, in Werke, Vol. VIII, p. 398.

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THE MATHEMATICALINTELLIGENCER

s p h e r e ) in s o m e detail, a n d it was in this p e r i o d that he a p p a r e n t l y first t h o u g h t of preparing a t r e a t i s e on t o p o l o g i c a l themes. This is again d o c u m e n t e d in his letters to Olbers, and, m o s t explicitly, in a l e t t e r to Schumacher: Some time ago I started to take up again a part of m y general investigations on curved surfaces, which shall become the f o u n d a t i o n of m y projected work on higher geodesy. [...] Unfortunately, I f i n d that I will have to go very far afield since k n o w n things m u s t also be developed i n a diff e r e n t form, adapted to m y investigations. One has to follow the tree down to all its root threads, and some of this costs me week-long intense thought. Much of it even belongs to g e o m e t r i a situs, an almost unexploited field. The w i s h I have always had i n all m y works, to give them such perfection ut nihil amplius desiderari p o s s i t [that nothing more can be desired], makes it even more difficult, as well as the necessity to leave m y work f o r other matters. 13 The p u b l i s h e d treatise on c u r v e d surf a c e s did not contain the m a t e r i a l ref e r r e d to, b u t Gauss s i g n a l e d his int e n t i o n to p u r s u e these m a t t e r s further at the end of the sixth p a r a g r a p h of the Disquisitiones, dealing with the calc u l a t i o n of total curvature. It should be emphasized that these considerations, rather than the p r o b l e m of the classification of knots, formed the b a c k g r o u n d to Gauss's a t t e m p t s to study the topology of c l o s e d plane curves, d o c u m e n t e d in s o m e fragments on geometria situs which w e r e published in Vol. VIII of the Werke in 1900.14 Late in his career, G a u s s r e t u r n e d to the a s t r o n o m i c a l p r o b l e m of the limits o f the zodiacus. In J a n u a r y 1848, at the age o f 70, he p u b l i s h e d a s h o r t note in the Astronomische Nachrichten. 15

Again, a n e w asteroid, Iris, h a d app e a r e d in t h e firmament, and G a u s s ' s assistant at t h e G6ttingen observatory, Goldschmidt, h a d calculated its zodiacus. G a u s s h i m s e l f did not m i s s t h e o c c a s i o n to c o m m u n i c a t e s o m e n e w m a t h e m a t i c s . In his note, Gauss tells us in his 44-year-old article on the limits of the zodiacus that there h a d b e e n a p r o b l e m w h i c h he h a d not dealt w i t h on the e a r l i e r occasion, "because a further d i s c u s s i o n w o u l d only have b e e n a hors d'oeuvre there, and b e c a u s e I really w a n t e d to let o t h e r p e o p l e have the pleasure of occupying themselves with a m a t h e m a t i c a l p r o b l e m which in m y opinion w a s not uninteresting." The p r o b l e m he w a s speaking of w a s t h a t of the e x c e p t i o n a l solutions o f the differential equation oJ = 0, i.e., t h o s e which do not r e p r e s e n t b o u n d a r y comp o n e n t s o f t h e zodiacus. What w a s their g e o m e t r i c a l meaning? N o w Gauss gave the answer: A given geocentric p o s i t i o n p of a celestial b o d y could arise in different ways; in o t h e r words, t h e r e might be m o r e t h a n one pair of p o s i t i o n s (2,2') which w a s m a p p e d to p = r Implicitly supposing everything to be s m o o t h a n d generic in an a p p r o p r i a t e sense, G a u s s p o i n t e d out that the celestial sphere w a s divided into several regions, e a c h o f which w a s c o v e r e d a different n u m b e r of times b y the mapping (P. The solutions of the differential equation ~o = 0 r e p r e s e n t e d the set of singular p o i n t s of this b r a n c h e d covering, i.e., the lines w h e r e the n u m b e r of p r e i m a g e s changed b y 2. Along the s o l u t i o n curves, an i n t e r m e d i a t e n u m b e r o f p r e i m a g e s existed. (Of course, G a u s s did n o t s p e a k o f "branched coverings" and their singular points, b u t he des c r i b e d t h e g e o m e t r i c a l situation v e r y clearly.) G a u s s e n d e d with a qualitative d i s c u s s i o n o f the case of two c o n i c sections as orbits; here, the m a x i m u m n u m b e r o f p r e i m a g e s of ap is 4; the dif-

ferent configurations o b t a i n e d d e p e n d on the relative p o s i t i o n of t h e s e orbits. F o r instance, in t h e case of linked ellipses, o~ = 0 is satisfied along a single c l o s e d curve. Therefore, Gauss conc l u d e d his note: in s u c h a case, there e x i s t e d two regions in the celestial sphere, one o f w h i c h was covered once, the o t h e r thrice. We can t a k e this s e c o n d p a p e r on the zodiacus as further evidence for the view that already in 1804 Gauss had a rather detailed picture of the geometrical meaning o f his differential form, oJ. Up to this point, w e have a n s w e r e d questions 4, 1, a n d 2 relating to the fragment on the linking integral. We have seen t w o interpretations, one physical and one geometrical, of the linking integral, w h i c h explain h o w Gauss k n e w that its values w e r e integer multiples o f 4~-. The crucial third question remains: Did Gauss think of an i n d e p e n d e n t n o t i o n of the linking n u m b e r of t w o s p a c e curves so that his formula represents a computation r a t h e r than a definition? A s e a r c h in Gauss's m a t h e m a t i c a l n o t e b o o k s in s u m m e r 1994 b r o u g h t to light an unpublished p a g e on a topological topic which reveals the answer.t6 The editor of Gauss's f r a g m e n t s on geometria situs, Paul St~ckel, d i d not see its imp o r t a n c e in 1900, for the simple r e a s o n that the m a t h e m a t i c a l world h a d not yet directed its a t t e n t i o n to the type of m a t h e m a t i c a l o b j e c t s d i s c u s s e d by Gauss on this page: braids. Even today, it s e e m s to b e a w i d e s p r e a d opinion that braids w e r e i n t r o d u c e d into mathematics by Emil Artin in 1926, despite Wilhelm Magnus's r e p e a t e d indications that Adolf Hurwitz had studied the braid group (without its name) in a seminal p a p e r on Riemann surfaces in 1891.17 A close reading of Tait's papers of 1877-1885 and Listing's Vorstudien reveals, however, that braidlike obj e c t s had b e e n o f interest to b o t h of

~3Gauss to Schumacher, 21 November 1825, in Werke, vol. VIII, p. 400. 14Well (1979) has called attention to an interesting letter of 1863 in which Betti reported that Riemann had told him about some attempts by Gauss to study knots. Riemann's communication, however, refers only to the last years of Gauss's life, i.e., the period after Listing had discussed the knot problem in his Vorstudien zur Topologie. No documents seem to survive in Gauss's papers which definitely relate to studies of knots rather than closed plane curves. 15Reprinted in Werke, Vol. Vii, pp. 313-316. 16NSUB GSttingen, Cod. Ms. Gauss, Handbuch 7, p. 283. 17For a reference to Artin (1926), see (Burde and Zieschang 1985, p. 161). Magnus's allusions to (Hurwitz 1891) can be found in his survey on braid groups (Magnus 1974) and in his book with Chandler (Chandler and Magnus 1982).

VOLUME20, NUMBER1, 1998 4~

them. ~s Given that Listing had been in direct contact with Gauss, we cannot exclude the possibility that these ideas were connected with the even earlier ones of Gauss, which I will now present. On the page in question is nothing less than a nicely drawn picture of the four-strand braid which we would write as 0"30-10-220-3, using Artin's presentation of the braid group, together with some mathematical comments. It probably belongs roughly to the period of the Disquisitiones circa superficies curvas. In the notebook, it immediately precedes some pages with geodetical calculations, dated 1830. On the preceding pages, we find further, undated, geodetical calculations. The last date appearing on previous pages is 1815; apparently, the n o t e b o o k in question had been out of use for long periods of time. On the back of the page, there is a reading of a weaving pattern of bands. The signs "St" above and below the passage are St/~ckel's, showing that he had seen it while editing volume VIII of the Werke. Since this fragment has never been discussed in the literature, I will give a detailed interpretation with a complete translation as I go along.

The drawing shows that Gauss thought of the braid as being divided into six segments, extending from one crossing to the next. Gauss numbered these segments and labeled the four strands a, b, c, and d. Then he wrote d o w n a table with the title "change of coordination." Its rows correspond to the labeled strands, its columns to the n u m b e r e d segments.

not yet distinguish between the two possible orientations of a half-twist of two strands. That Gauss used c o m p l e x integers to encode the composition of the braid might be motivated by his well-known fascination with these numbers; it might, however, also have a more serious reason (see below). Gauss himself seems to have been dissatisfied with his table. Immediately below it, he wrote

ba i i + i + i i +i

What matters is to represent the whole [Inbegriff] of the entanglement [Verwicklung] in such a way as the aggregate of its parts that one sees which parts destroy each other.

c d

i

+i

+2i

!+2i +2i

2+2il 3 4+3i

Obviously, Gauss attempted to develop a notation for braids keeping track of (1) the permutations of the strands as one follows the braid and (2) the twists of the strands around each other. The real parts of the numbers appearing in the table specify the positions of the strands, and the imaginary parts were intended to record the twists. The assignment of an i to one of the strands at a crossing of the diagram is ambiguous, however; there seems to be no definite convention a d o p t e d in the table (this can be seen already at the first two crossings). The ambiguity is probably due to the fact that Gauss did

~SSee Listing's discussion of "vielfache Helikoiden" and "Helikoiden h6herer Ordnungen" (Listing 1847, pp. 43-51), or Tait on "clear coils" (Tait 1877, w167 25, 26).

*

/,|~[~'

cl~l~l-

II

I'

lj~ I ~

/ r

I,~

1"..

,I~'~.

In "parts" of a braid which m a y "destroy" each other, we can perceive the idea of a composition of braids and of braid equivalence. The meaning of the sentence is thus: Starting from its elementary parts (involving only one crossing), fmd a representation of a braid as a whole which allows one to decide w h e t h e r it (or any part of it) is trivial or not. In other words, this passage contains a first formulation of the classification problem for braids (or, more anachronistically, of the w o r d problem in the braid group). The next remark formulated a conjecture:

Probably it will suffice to list the half twists of one line around the other according to a certain sense of rotation. Here, Gauss came back to the idea underlying his table, n o w explicitly addressing orientation. The conjecture can be read in two ways. A minimal reading is to understand it as a new proposal for a concise notation of braids; for a stronger reading, see below. In fact, Gauss returned to his example and wrote down

cd, ab, da, ad - - b u t then he stopped and struck out the whole line. Also, the second try ended without success; we read

cd, ab.

J

c ~

Figure 2. Page 283 of Gauss's Handbueh 7.

50

THE MATHEMATICALINTELLIGENQER

It seems clear what happened. Gauss stumbled at the third crossing, where the orientation of the half-twists of the strands changes for the first time (in m o d e r n terms: the inverse of a gen-

e r a t o r a p p e a r s in the b r a i d word). At this point, Gauss s e e m s to have realized the necessity o f finding an adequate w a y of keeping t r a c k o f the orie n t a t i o n s of the half-twists c o m p o s i n g a braid. On the margin of t h e page, Gauss d r e w another s k e t c h showing a curve winding a r o u n d t w o points, p r o b a b l y illustrating t h e w i n d i n g of a b r a i d s t r a n d around two o t h e r s as s e e n from above. Then follows the last sent e n c e o f the fragment: One only has to count i n e v e r y line h o w often + changes w i t h - . Given that Gauss did n o t s p e c i f y w h a t his signs 4- and - mean, the interpretation o f this r e m a r k can only be tentative. The earlier m e n t i o n i n g o f a "sense of rotation," the d r a w i n g on the right margin, and the n o t e s "south before/north behind" on the left margin m a k e it probable, however, t h a t the signs do, indeed, refer to the t w o possible orientations of a half-twist. On this reading, Gauss p r o p o s e d to count the difference b e t w e e n the n u m b e r s o f positive and negative haft-twists a given s t r a n d e x p e r i e n c e s in a p o s s i b l y c o m p l e x braid. F o r a t w o - s t r a n d braid, a n d on division by 2, this a m o u n t s to the m o d e r n c o m b i n a t o r i a l definition of the linking number. The drawing on the m a r g i n and G a u s s ' s k n o w l e d g e o f the i n d e x o f p l a n e curves m a k e it p r o b a b l e that alr e a d y at the time of writing, he w a s a w a r e o f an analytical m e t h o d for the c o m p u t a t i o n of the linking n u m b e r , using t h e p r o j e c t i o n o f a b r a i d to a transv e r s a l plane: If the c o o r d i n a t e s of two b r a i d s t r a n d s in ~3 ~ C~)~ a r e given b y (zt(t),t) and (z2(t),t), respectively, t h e n t h e i r linking n u m b e r is, o f course, t h e (half-integer-valued) i n d e x o f the curve t ~-> z l ( t ) - z2(t) with r e s p e c t to zero. In this light, the f r a g m e n t of 1833 would, indeed, be j u s t a n o t h e r comput a t i o n o f the linking number, a n d n o t a definition. The second reading o f G a u s s ' s conj e c t u r e (and the w h o l e fragment) w o u l d b e m u c h stronger. On this reading, w e w o u l d ascribe to G a u s s the belief t h a t in o r d e r to solve t h e classific a t i o n p r o b l e m for braids, it w o u l d p o s s i b l y suffice to determine, in addition to the p e r m u t a t i o n a s s o c i a t e d

with a braid, j u s t the n u m b e r s o f (signed) half-twists b e t w e e n all p a i r s of strands. The r e m a r k s following t h e conjecture d o not c o n t r a d i c t this s t r o n g e r interpretation. Even t h o u g h the c o n j e c t u r e is false, it w o u l d s h o w a r e m a r k a b l e insight (and note t h a t Gauss qualified his conjecture with t h e cautious " P r o b a b l y . . . " ) . There is a sort of a middle course between the t w o readings discussed: While Gauss w a s looking for a n o t a t i o n for braids w h i c h enabled one to decide w h e t h e r o r n o t two braids were equivalent, he c a m e close to defining a nontrivial invariant for braids, namely the last r o w o f the table he set up (with a defmite convention on crossings adopted). Allow m e to sketch a corresponding elaboration of Gauss's ideas, freely using m o d e r n mathematical language. These variations on Gauss's theme are n o t intended to r e p r e s e n t an historical reconstruction of his line o f thought. At best, they might indicate the space o f m a t h e m a t i c a l speculation in which Gauss w a s m o ~ n g when he gave an afternoon's thought to the first braid. A Gaussian Invariant of Braids Let the n - s t r a n d b r a i d group Bn b e gene r a t e d b y ( r l , . . . , Crn-1, with defining relations

O-kO-k4_lO-k

=

O'k+lO-kO-k+ 1

1, 1, 2, 3 - i ,

z, a(o-k)(Z)

:=

1, z-l+i, Z4-

aii b c d

2 4 3+i

(k = 1, 2 , . . . , n -

1).

The a c t i o n o f the inverse is given b y z,

RezCk,k+ 1 = k Rez k4-1.

a(O-k t)(Z ) := z + l - i , R e z z-l,

+i

One easily verifies the defining relations o f Bn for this action. Given a b r a i d w o r d w in the generators (rk, w e can c o n s i d e r the corresponding p a t h of a point z E 77[i]. In Gauss's e x a m p l e , we have w = q3q1~22q3, so for z = 1, w e get t h e sequence

3 1+ 4

+i

i'

1+i

2+

2-i 1+i 3+i 4

The columns o f this table can easily be r e a d off a d i a g r a m o f the given braid: J u s t t a k e the p o s i t i o n s of the strands as real parts a n d at e a c h crossing, add _+i to the lower s t r a n d according to the orientation o f t h e crossing. By construction, the last c o h m m of the table is an easily c a l c u l a t e d invariant of the given braid. It is d e t e r m i n e d by the perm u t a t i o n a s s o c i a t e d with w and, for each strand, the s u m o f the signs of its undercrossings. Of course, this invariant is not complete, as it is d e t e r m i n e d by the linking n u m b e r s b e t w e e n the s t r a n d s of the braid. The above c o n s t r u c t i o n m a y evidently be m o d i f i e d in various ways. F o r instance, w e c o u l d c o n s i d e r a s y m m e t r i c version of t h e action: i'

Re z r k,k + 1 Re z = k Rez=k+l

2-i.

The paths o f the n p o i n t s 1, 2 , . . . , n E 77[i] d e t e r m i n e the b r a i d w o r d completely, since in e a c h step w e can see which g e n e r a t o r h a s been acting. In fact, w e can t a k e t h e s e p a t h s as the entries of an i m p r o v e d version of Gauss's table, making e a c h p a t h one r o w of the table. In the e x a m p l e , the modified table reads

fork=l,...,n-2. An a c t i o n c~ o f Bn on the lattice o f Gaussian integers Z[i] m a y be defined by

2-i,

fi((&)(z) :=

Re z r k,k + 1 + 1 + i, R e z = k -1+i, Rez=k + l (k = 1, 2 , . . . , n -

1).

F o r this action, the last column of the table for a b r a i d w o r d is of the form 9rw(1) + j ( 1 ) i ,

ww(2) + j ( 2 ) i , . . . , ~r~(n) + j ( n ) i ,

w h e r e ~rw is the p e r m u t a t i o n associa t e d with a braid, a n d j ( k ) is the sum of the signs o f all crossings in which the kth s t r a n d is involved; in o t h e r w o r d s , j ( k ) is t w i c e t h e s u m of the linking n u m b e r s b e t w e e n the kth strand and all other s t r a n d s o f the braid. Very probably, t h e few lines in Gauss's n o t e b o o k a r e the only trace of a m a t h e m a t i c a l activity in which one of the genuine o b j e c t s o f t o p o l o g y was first conceived. It w o u l d be difficult and u n n e c e s s a r y to d e c i d e which of

VOLUME 20, NUMBER 1, 1998

51

the above readings c o m e s closest to "what Gauss really did." Nevertheless, this fragment d o c u m e n t s the invention of a n e w type of m a t h e m a t i c a l object, which would b e called b r a i d s a century later. To be sure, t h e r e is no detailed discussion of the i d e a of c o m p o s i n g braids, and, in any case, no explicit group-theoretical thinking. F r o m the letter to Olbers w r i t t e n in 1802, w e also k n o w that Gauss h a d previously read V a n d e r m o n d e ' s p a p e r on weaving patterns. But still t h e r e is a certain acuity in Gauss's f r a g m e n t w h i c h m a r k s the borderline b e t w e e n v a g u e i d e a s and a fairly definite o b j e c t o f thought. The least w e can s a y is that the fragm e n t reveals that G a u s s h a d an i d e a of h o w to describe the p h e n o m e n o n of linking in braids b y a n u m b e r counting the signs of d i a g r a m crossings. This closes the gap in o u r interpretation of the fragment on the linking integral. We have thus o b t a i n e d a r a t h e r clear picture of h o w this f r a g m e n t was situated in Gauss's m a t h e m a t i c a l practice. Having first e n c o u n t e r e d the differential form dominating the mathematics of linking in the a s t r o n o m i c a l context of the zodiacus o f asteroids, Gauss was led to recognize its topological content w h e n he c o n s i d e r e d linked orbits of celestial bodies. Then, during his studies of the differential g e o m e t r y of curved surfaces, m o t i v a t e d b y geodetical work, he felt b o u n d to r e c o n s i d e r topological issues. A m o n g o t h e r things, he discussed the n o t i o n o f an oriented a r e a of closed curves in the plane or on the sphere, a topic w h i c h he m o s t probably had c o n s i d e r e d a l r e a d y in the context of his investigation of the zodiacus. Around this time, he also gave s o m e hours of thought to his braid and discovered the c o m b i n a t o r i a l w a y of determining the linking number. When he finally was led b a c k to geometria situs in the c o n t e x t o f his j o i n t w o r k with Weber on e l e c t r o m a g n e t i c induction, it w a s easy for him to recall his earlier results. If we are to believe the indirect r e p o r t by MSbius, G a u s s r e n e w e d his earlier wish to write a treatise on linked (and p e r h a p s k n o t t e d ) s p a c e curves, b u t (as before) he w a s unable to produce something w h i c h satisfied his extraordinary publication standards. In his notebook, he m a d e only this single

52

THE MATHEMATICALINTELLIGENCER

mention, in a tone of resignation, and it gave the core result o f Gauss's own contributions to the n e w discipline. What is striking a b o u t this s t o r y are the r o l e s of the various disciplines in the c o n t e x t of which the linking phen o m e n o n w a s discussed. Although G a u s s was, throughout his career, very clear t h a t in the c o n c e p t u a l h i e r a r c h y of m a t h e m a t i c s , geometria situs was fundamental, the p r a c t i c a l m o t i v a t i o n of dealing with linking a n d r e l a t e d topics c a m e from the e x a c t s c i e n c e s of ast r o n o m y , geodesy, and e l e c t r o m a g n e t ism. The m a t h e m a t i c a l activity of w h i c h the fragment o f 1833 is an ext r e m e l y c o n d e n s e d t r a c e w a s one of mathematization, of a d o m a i n of very intuitive p r o b l e m s w h i c h h a d n o t yet b e e n t r e a t e d within the "sublime scie n c e o f quantity," with t h e p a r t i a l exc e p t i o n of t h o s e i m m e d i a t e l y r e l a t e d to c o m p l e x integration. That t h e s e problems c o n t i n u e d to r e a p p e a r in several o f t h e leading sciences of G a u s s ' s day m u s t b e one of the r e a s o n s for t h e very high value he p u t on the s l o w l y emerging s c i e n c e of topology. REFERENCES

1. E. Artin, Theorie der Z6pfe, Abhandlungen Math. Sem. Hamburg. Univ. 4 (1926), 47-72. 2. N. L. Biggs, E. K. Lloyd, and R. J. Wilson, Graph Theory, 1736-1936, Oxford: Clarendon Press (1976). 3. G. Burde and H. Zieschang, Knots, Berlin: de Gruyter (1985). 4. B. Chandler and W. Magnus, The History of Combinatorial Group Theory, New York: Springer-Verlag (1982). 5. C. F. Gauss, Werke, 12 Vols., Leipzig: B. G. Teubner, (1863-1933). 6. A. Hurwitz, 0ber Riemannsche Fl~chen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1-61. 7. J. B. Listing, Vorstudien zur Topologie, Gdttinger Studien 2 (part 1) (1847), 811-875; reprinted as a book by Vandenhoeck & Ruprecht, G6ttingen (1848). 8. W. Magnus, Braid groups: A survey, in Proceedings of the Second International Conference on the Theory of Groups, Lecture Notes in Mathematics No. 372, Berlin: Springer-Verlag (1974). 9. J. C. Maxwell, Treatise on Electricity and Magnetism, 2 vols., Oxford: Clarendon Press (1873). 10. J. C. Pont, La topologle algebrique des

11.

12.

13.

14.

origines a Poincare, Paris: Presses Un!: versitaires de France (1974). C. Schilling and I. Kramer, Briefwechsel zwischen Gauss und Olbers, 2 vols., Berlin: Springer-Verlag (1900/1909). P. St~ckel, Gauss als Geometer, in Materialien for eine wissenschaftliche Biographie von Gauss, Eds. F. Klein, M. Brendel and L. Schlesinger, Leipzig: B. G. Teubner (1918), Vol. 5, pp. 26-142. P. G. Tait, On knots, Trans. Roy. Soc. Edinburgh 28, (1877), 145-190; reprinted in Tait's Scientific Papers, Vol. I, Cambridge: Cambridge University Press (1898), pp. 273-317. A. Well, Riemann, Betti and the birth of topology, Arch. History Exact ScL 20 (1979), 91-96.

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The Borromean Rincjs Peter Cromwell Elisabetta Beltrami Marta Rampichini

Editor J

h e r e is an o b j e c t that o c c u r s in l o w - d i m e n s i o n a l topology, o f p a r t i c u l a r i n t e r e s t in knot theory, k n o w n as the B o r r o m e a n R i n g s - - s e e Figure l ( a ) . The t h r e e rings t a k e n together are inseparable, but r e m o v e a n y one ring a n d the o t h e r two fall apart. Because o f this property, t h e y have been u s e d as a s y m b o l in m a n y fields, including heraldry, physics, theology, psychology, a n d art. Although one occasionally s e e s a p h r a s e s u c h a s "The rings are the emblem o f an Italian family o f t h e Renaissance," the t e r m "Borromean" is often i n t r o d u c e d in t e x t b o o k s w i t h o u t any e x p l a n a t i o n at all. In this article, we p r o v i d e a m o r e detailed a c c o u n t o f their history, and r e p o r t on s o m e surprising discoveries.

Welcome!

The Borromean Rings in Mathematics The link f'wst a p p e a r e d in a mathematical c o n t e x t in the earliest w o r k on knots: P e t e r Tait's enumeration o f 1876 [28]. Talt u s e d the Borromean rings a n d another link o f similar construction to show that w h a t he called "belinkedness" ( n o w called linking number) is not sufficient to distinguish links. His two figures are r e p r o d u c e d in Figures l(a) and (b). Each one shows an alternating, 3-component link such that e a c h c o m p o n e n t is unknotted and no t w o c o m p o n e n t s are linked. Yet, he concludes that the two links are non-trivial and different. Here he implicitly uses the idea that an irreducible, alternating diagram h a s minimal crossing num-

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Figure 1. Some 3-component links from Tait

Readers of The Mathematical Intelligencer are accustomed to the

name of lan Stewart--not only as prolific author, but as editor of the Mathematical Tourist for many years. A tip of the hat to him in appreciation for his major role in the magazine's life, and in hope of more contributions from him in the future. He is giving up his job as Mathematical Tourist editor to Dirk Huylebrouck, who has become familiar to our readers too, as author of varied and lively Intelligencer pieces.

ber---one of the recently p r o v e d Tait conjectures [13, 19, 29]. To i n c r e a s e the b e l i n k e d n e s s he changes s o m e of t h e crossings in the B o r r o m e a n rings a n d p r o d u c e s the (3,3)-torus link (Figure l(c)). True B o r r o m e a n circles a r e impossible [15], so it is not surprising that he o b s e r v e s [28, p.183] t h a t

T

(a)

Whereas [the B o r r o m e a n link] if m a d e of wire, is particularly stiff, the n e w figure is eminently flexible. This s e e m s to have b e e n practically k n o w n to t h e m a k e r s of chain armour. Links which b e c o m e trivial after the removal of any c o m p o n e n t were studied b y Hermann Brunn in another early w o r k [4], and such links are n o w called Brunnian links. He referenced Tait's examples, but neither he nor Tait used the term "Borromean". The earliest use of the name w e have found in the mathematical literature is in a 1962 overview of knot theory [10]: on pages 131-132, Ralph F o x uses the A l e x a n d e r polynomial to show that the B o r r o m e a n rings are truly linked. The inseparability of the rings can also b e s h o w n using simple colouring arguments [20], or deduced from general results on alternating links [18]. The other mathematical claim to fame for the B o r r o m e a n rings concerns their universality. A link L is called univ e r s a / i f every closed, orientable 3-manifold can be obtained as a branched covering over the 3-sphere with branch set

(b)

(c)

[28].

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 1, 1998

53

I

I

Borromeo BORROMEO (-1442)

Giovanni BORROMEO

I

Margherita BORROMEO (-1447)

Giacomino VITALIANI

Vitaliano

BORROMEO(-1449) [

Ambrosina FAGNANI

Fi|ippo BORROMEO ( - 1464) I Vitaliano BORROMEO (1451-1495)

Bianca di SALUZZO

Gian Maria VISCONTI

I Giustina BORROMEO

Francesca VISCONTI

I Giovanni BORROMEO

Ludovico VISCONTI-BORROMEO Figure 2. Partial family-tree of the Borromeo family showing the relationships of people mentioned in the text.

L. The B o r r o m e a n rings and the Whitehead link were the fLrst links shown to p o s s e s s this property [11]. The Borromeos and their Crest The B o r r o m e o family's a s s o c i a t i o n with the rings can b e t r a c e d b a c k to the fifteenth century. The story begins with Vitaliano, s o n of Margherita, one of the B o r r o m e o s in San Miniato in Tuscany, and G i a c o m o Vitaliani, a m a n from an a n c i e n t family with roots in P a d u a (see Figure 2 for a basic familytree.) In 1396, Vitaliano w e n t to live in Milan with his m o t h e r ' s brother, Giovanni B o r r o m e o . Vitaliano t o o k the s u r n a m e B o r r o m e o w h e n he w a s given the citizenship o f the city in 1406 b y Filippo Maria Visconti, Duke of Milan. Fifteen y e a r s l a t e r his uncle n a m e d him as heir of his estate. Vitaliano held office at the Duke's court, first as t r e a s u r e r (1418) and then as counsellor (1441). Through this position he was given land around Arona, and, in 1445, the Duke invested him as Count of Arona. When the Duke died in 1447, the senate of the n e w Ambrosian Republic a p p o i n t e d Vitaliano as one of the twelve b a i l i a (magistrates). He died in 1449. His son Filippo c o n t i n u e d the successful b a n k i n g b u s i n e s s he had started, with b r a n c h e s in London and Barcelona, a n d the family b e c a m e increasingly wealthy. Filippo was m a d e a C a v a l i e r e A u r a t o (Golden knight) by

54

THE MATHEMATICALINTELLIGENCER

F r a n c e s c o Sforza in 1450, w h o succ e e d e d Visconti as Duke o f Milan. This is p o s s i b l y the o c c a s i o n w h e n the rings w e r e a d d e d to the B o r r o m e o crest. Several of the a c c o m p a n y i n g figures s h o w the family crest in various forms. Like the crests of m a n y aristocratic families, it developed over time, recording events in the family's history. The rings are sometimes shown inset with three diamonds, although these are sometimes r e d u c e d to crude spikes. Besides the three rings, the p h o t o g r a p h s in Figures 9-11 also s h o w o t h e r heraldic s y m b o l s (unicorn, crown, horse's bridie, camel), and many of t h e s e elements can be traced b a c k to this period. In 1442 on the d e a t h o f B o r r o m e o Borromeo, b r o t h e r o f Margherita and Giovanni, the diagonal b a n d s o f w a v e s from the Vitaliani crest, a n d the horizontal b a n d s from the c r e s t of the B o r r o m e o s in Svevia, w e r e j o i n e d to f o r m a s y m b o l c o m m o n to b o t h families. This is the earliest p a r t of the crest. The unicorn, with the gold c r o w n a r o u n d its neck, looking at the Visconti s y m b o l of the s e r p e n t eating a boy, w e r e a d d e d at Duke Visconti's r e q u e s t w h e n Vitaliano w a s c r e a t e d a Count. The p o i n t e d c o u n t ' s c r o w n w a s also a d d e d at this time. The three rings w e r e a gift from F r a n c e s c o Sforza as r e c o g n i t i o n of the s u p p o r t that the B o r r o m e o family had given in defence of the city. The tourguides at the Palazzo B o r r o m e o on

Isola Bella say that the rings r e p r e s e n t the t h r e e families Visconti, Sforza, a n d B o r r o m e o who, after m u c h fighting, f o r m e d an "inseparable union" t h r o u g h intermarriage. I n d e e d Filippo m a r r i e d F r a n c e s c a Visconti. Their d a u g h t e r Giustina also m a r r i e d into the Visconti family, a n d h e r b r o t h e r Vitaliano having no children, n o m i n a t e d h e r family as heir a n d s t a r t e d the ViscontiB o r r o m e o family line. The silver bridle was a n o t h e r gift from t h e Sforzas. It was a w a r d e d to Giovanni, a n o t h e r son of Filippo, w h o d e f e a t e d t h e Swiss at the River T o c e in 1487. There are several stories to explain the camel a n d the feathers, one of which relates to the adoption of Vitaliano by Giovanni. After the young Vitaliano had run out o f money, he went to Milan to fend his rich uncle, even though his earlier requests for help had b e e n denied. He sold w h a t possessions he had, and bought s o m e donkeys and s o m e elaborately d e c o r a t e d covers. When he arrived, his uncle asked him to explain the symbols on the covers. Vitaliano h a d his a n s w e r ready: the camel sitting in the b a s k e t signified himself and his poverty. His uncle found this amusing and t o o k him in. The Borromean Islands A r o n a is a small t o w n seated at the foot of the Alps, on the southwest shore of Lake Maggiore in northern Italy. Nearby, in the lake, the B o r r o m e o family o w n s three islands: Isola Bella, Isola Madre, a n d Isola Superiore, also k n o w n as Isola dei Pescatori as it was a small fishing community. You can c a t c h a b o a t to visit t h e m from Stresa, a few miles up the shore. Isola Bella contains an impressive Baroque palazzo built in the seventeenth century by Vitaliano B o r r o m e o (1620-1690). As y o u app r o a c h the island from Stresa, look out for one o f the statues on the shore (Figure 3). Isola Madre, by far the m o s t relaxing of the islands, is set out like a p a r k with gardens, exotic birds, a n d a simpler palazzo that contains s o m e family portraits. Two of us (PC + EB) visited the islands in S e p t e m b e r 1996. It did n o t t a k e us long to fred our fwst example o f the famous emblem: it even a p p e a r s on the

Figure 4. Entrance ticket showing the non-alternating 'Borromean' rings.

Figure 3. One of the statues greeting visitors to Isola Bella.

entrance ticket (Figure 4). However, a closer inspection revealed that the depicted link is not the one known to mathematicians! The crossing on the lower right is switched. This produces a split link: one ring can be separated from the others. The remaining two linked nngs are known as the Hopflink. Having made this discovery, we studied all the examples of the emblem that we came across. To our surprise, we found several variations. The three circles were always present and, ignoring the crossings, they were usually arranged with 3-fold symmetry. This

pattern of three circles can be interlaced in ten different ways (up to various symmetries) producing five topologically distinct links. We found examples of all five on Isola Bella! We also saw the underlying 4-valent graph with no crossings indicated. Even this underlying design is not used consistently: w h e n the space available is too small, the craftsmen just made a chain of three rings. We saw both of the patterns s h o w n in Figure 5. Figure 6 shows the ten geometrically distinct patterns that can be derived by choosing the crossings in different ways. In this case, we are considering patterns up to symmetry: 3-fold rotation, reflection, and reflection in the plane of the pattern. This last s y m m e t r y operation means that the sense of all the crossings is switched. Considered from the topological viewpoint, these ten patterns have only five distinct link types: the mathematicians' Borromean rings (a), the (3,3)-torus link (e and f), a 3-component chain (g, h and j), a Hopf link with a split component (b, c and d), and the 3-component trivial link (i). All the patterns in the left column can be found on Isola Bella. Unfortunately, photography was not allowed inside the palazzo. A large example of the alternating pattern (a) is inlaid in a floor; it can also be found in a floral form in the garden. Pattern (c) appears very frequently. It is used on the tickets, and also decorates the flowerpots in the garden. Just after you enter the palazzo, y o u climb a staircase embellished with large coats-of-arms. The

Borromeo crest, at the top of the stairs, contains pattern (e). Beneath the palazzo is a grotto decorated with shells and other objects from the sea. The floor of one cave contains pattern (g) made of black and white shells. To fend an example of the trivial link (i), we had to cheat a little: a modern gate in the garden is topped with the three rings which have been laid flat one on another and welded together. The

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 6. The ten interlaced patterns that can be derived from three circles. Those in the left column appear in the Palazzo Borromeo Figure 5. Two chains of three circles.

on Isola Bella,

VOLUME 20, NUMBER 1, 1998 5 5

method of construction is still clearly visible. This variety of designs is unusual. Almost all interlaced ornament is alternating, but here this simple principle for choosing the crossings seems to be violated deliberately. It is said that the three rings symbolise an inseparable union of three families. However, as we have seen, the inseparability of the link is not always preserved in the designs. Mistakes are easily made, 1 and, if there were only one "incorrect" example, it could be overlooked. However, so many instances need more explanation. When the rings are carved in relief they have a strong three-dimensional quality, and to give the impression of genuinely circular, linked rings it is much more satisfactory to mimic a geometrically realisable object. Compare the rings in Figures 8 and 10 to see the different effect. The craftsmen seem to have known that the alternating motif cannot be constructed from circular rings; physical exmnples of all the other links can be made from planar circles. When the design is inlaid into a fiat surface, the alternating pattern can be used without its impossibility coming to mind. The Rings in Cremona and Milan The o d ~ of the emblem can be traced a little further. In the fifteenth centre% northern Italy was composed of several autonomous city-states, the Venetian Republic extended down part of the eastern coast of the Adriatic, the southeast corner of France up as far as Lake Geneva formed the ldngdom of the Savoy, and to the south were the papal states and the kingdom of the two Sicilies. Cabrino Fondulo (1370-1425) was a mercenary. He conquered the city-state of Cremona in 1406, taking advantage of feud in the ruling family. In 1413, he was made Count of Soncino, Marquis of Castiglione, and Vicario (representative) for the Emperor Sigismund (1368--1437). The three interlaced rings seem to have been an emblem of the city, and are said to symbolise the close friendship between Fondulo, Sigismund, and "antipope" John XXIII (1370-1419).

According to [27] the rings appear on a coin (Figure 7) in the collection of the Gherardesca in Pisa, but we were unable to locate the collection. When the city came under attack from the Milanese, Fondulo surrendered the city to Filippo Maria Visconti (1392-1497), Duke of Milan, for 35,000 scudos, retaining for himself the title of Marquis and the town of Castiglione. With Pandolfo Malatesta, he plotted against the Duke, but in 1424, he was betrayed by his two "close friends," taken to Pavia, and beheaded. In this way, Cremona became incorporated into the duchy of Milan in 1420. u sought a husband for his illegitimate daughter Bianca Maria (1423-1468). At her engagement to Francesco Sforza (1401-1466) in 1432, she was formally given Cremona to be her dowry. Francesco fought his prospective father-in-law on many occasions, and in 1440 he conquered Cremona for himself, except for the small town of Casaimaggiore. The following year, he reached agreement with the Duke to resolve the dispute. They ratified the accord in Cavriana near Mantua and decided to conclude the marriage. The ceremony took place in the Church of San Sigismondo in Cremona the same year. In 1446, the fighting began again. Cremona was retaken for the Duke by Francesco Piccinino, and this started a protracted war between the duchy of Milan and the Venetian Republic. The Duke, in desperate trouble, asked the King of Naples, the ruler of Rimini, the Pope, and even his son-in-law for help. Having no legitimate male heir, the Duke promised Sforza he would inherit everything. After the Duke's death the following year, the duchy of Milan was replaced with a popular government--the Ambrosian Republic. However, this was short-lived. In 1450, Francesco was staying at the Borromean castle at La Peschiera, south-east of Milan. The Milanese surrendered and he was welcomed in by the burghers as absolute ruler, and offered the title Duke of Milan. Francesco rewarded his friends and allies in the noble families, granting them

Figure 7. The C r e m o n e s e coin (reproduced f r o m [27]).

his own symbol: the three rings. They were the emblem of the only land to which he had legal title, Cremon& Besides the Borromeo family, he gave the honour to the families Cavazzi della Somaglia, Sanseverino, and Birago. Consequently, the rings can be found on many monuments in and around Milan. The three of us spent some pleasant hours in April 1997 visiting some of them. 9 In one of the courtyards of the Sforza Castle, there is a "fountain" in the wail. The front of the basin is decorated with five squares containing symbols of the Sforza (Figure 8). The left-hand square contains three interlaced rings. They are the (3,3)torus link of Figure 6(f). This is the only other design which, like the real Borromean rings, has 3-fold rotational symmetry when the crossings are observed. 9 The most famous member of the Borromeo family was Carlo (1538-1584), who became cardinal and archbishop of Milan in 1566, and was canonised in 1610. His statue stands by the Church of Santa Maria Podone on Piazza Borromeo. Symbols of the family are engraved in the four sides of the plinth: three uninterlaced rings are on the back. On the opposite side of the piazza, on Via Borromei, stands the Palazzo Borromeo where Vitaliano brought his donkeys and was taken in by his uncle. Much of it was damaged by fire after being bombed in World War II. The remaining wall of the inner courtyard is decorated with rings grouped in chains of three, and crosses of five. The capital of one of the original columns is preserved on the left wall (Figure 9). It shows a winged version of the pattern in Figure 6(c).

1In the book Gauge Fields, Knots and Gravity [2], the picture of the "13orromean" rings actually shows the (3,3)-torus link in the form shown in Figure 6(0.

THE MATHEMATICAL/NTELUGENCER

Figure 8 (left). The fountain at the Castello Sforzesco, Milan. Figure 9 (top). The capital of a column atthe Palazzo on Via Borromei, Milan.

On the north side o f the piazza is a m o r e m o d e r n palazzo, which is decorated with the B o r r o m e a n c r e s t - Figure 10. This contains the alternating rings. 9 A few s t r e e t s away, on Piazza San Sepolcro, is the B i b l i o t e c a Ambrosiana, f o u n d e d b y c a r d i n a l F e d e r i g o B o r r o m e o in 1607. The B o r r o m e o c r e s t a d o r n s the e n t r a n c e (Figure 11), this time containing the (3,3)t o r u s link in the form o f Figure 6(e). 9 At 39-41 Via Manzoni, y o u will find the palazzo o f the B o r r o m e o d'Adda. This is an eighteenth century building that was restored in the 1820's. The crest on top of the building contains the three rings overlayed on a bird of s o m e kind. The rings a r e non-alternating but the details a r e t o o b a d l y e r o d e d to identify the p a t t e r n completely. 9 In the Church of Santa Maria della P a s s i o n e on Via C o n s e r v a t o r i o there is the t o m b o f Daniele Birago, archb i s h o p of Mitilene in Egeo. The monument, m a d e by A n d r e a Fusini in 1495, is d e c o r a t e d with the t h r e e alternating rings.

9 A c c o r d i n g to [27], the rings a p p e a r on the b r e a s t of one of the s t a t u e s on the Duomo: one of the large statues at t h e top of a column s u p p o r t ing a spire, on the side facing t h e

Palazzo Reale. However, the cathedral is d e c o r a t e d with several thousand statues, so good luck in finding it. Inside, n e a r the south door, you will find an altar to the Saints Auxanus

Figure 10. The Borromeo crest on a palazzo in Piazza Borromeo, Milan.

VOLUME 20, NUMBER 1, 1998 5 7

Figure 11. T h e B o r r o m e o c r e s t o n t h e Biblioteca Ambrosiana, Milan.

and Mansuetus, which contains the three rings. 9 The Palazzo degli Atellani at 65, Corso Magenta has some interesting windows (Figure 12). The central design has the non-interlaced rings, although these date from a 1920 restoration. The house has a long history and has been largely rebuilt. For a time, it was owned by Duke Lodovico Sforza "I1 Moro" (1452-1508), second son of Francesco. He was a patron of the arts, and from 1482 to 1499 Leonardo da Vinci (1452-1519) worked at his court. Leonardo stayed at the house while working on a commission to paint the refectory at the little monastery of Santa Maria delle Grazie just across the street: you can join the queue to see his famous masterpiece The Last Supper. The Duke later gave the house to his equerry, Giacomo Atellano. 9 Cremona lies about 90 km southeast of Milan. Bianca Maria Visconti and Francesco Sforza married in the Church of San Sigismondo on Via Marmolada, in 1441. The new Duchess wanted to c o m m e m o r a t e the event by extending the church to create a monastery. Started in 1463, the work was completed in 1492. The monks left long ago and the church is n o w a parish church. In the right side of the church is a doorway into the cloisters. Made from walnut, the doors are elabo-

THE MATHEMATICALINTELLIGENCER

rately carved with Sforza emblems, including three alternating rings (Figure 13). The doors were made by local Cremonese craftsmen Paolo and Giuseppe Sacha in the period 1536-42. The intarsia in the choir are also worth a look. Also in Cremona, the rings can be found in the decorations under the colonnades in the courtyard of the Palazzo del Comune (townhall), opposite the cathedral.

Circles in Trinitarian Iconography The mystery of the Christian Tr~Jlity is

expressed in the Athanasian Creed: we

worship one God in Trim'ty, and Trinity in unity; neither confounding the Persons, nor dividing the substance. Trying to depict this triune nature without leaving oneself open to accusations of polytheism was problematic, and geometrical symbols became popular. The equilateral triangle, consisting of three equal parts, equally joined, was used as an early symbol of the Trinity. It was often inscribed in a circle, a symbol for God for many centuries. To the Greeks, the circle symbolised perfection; its never-ending form also encapsulates the idea of eternity. The association of rings with the

Figure 12. Window of the Palazzo degli Atellani on Corso Magenta, Milan.

are omitted. Common expansions are Yahweh and Jehova. Alfonsi, writing the tetragrammaton as IEUE, split it to produce the names of the three persons: IE, EU and UE. These are written into his diagram. J o a c h i m of Fiore (1132-1202) t o o k the splitting of the sacred name from Alfonsi, and arranged the labels on a design of three interlaced circles. His basic idea is sketched in Figure 15; for manuscripts showing the diagram see [25]. The c o m p o n e n t rings are actually topologically equivalent to each other, although this is not apparent in Joachim's figure. It is more obvious when the link is redrawn as a symmetric diagram--Figure 15(b). It is suggested in [25] that this image of God as three interlaced rings inspired Dante Alighieri (1265-1321). At the climax of his Divina Commedia he reveals a vision of God: [Paradiso, w 115-120] Figure 13. Door of the Church of San Sigismondo, Cremona.

Trinity can be traced back to Saint Augustin of Hippo (354-430). In his w o r k De Trinitate [IX, 5, 7], he described h o w three gold rings could be three rings but of one substance. A diagram in the Dialogi Contra Iudaeos (Dialogues against the Jews) by Petrus Alfonsi (1062-1110) has three circles connected in a triangle as shown in Figure 14; a manuscript showing the diagram can be seen in [30], p. 38. Alfonsi was brought up as a Jew in the Muslim part of Spain, then converted to Christianity and emigrated to Aragon, England, and France. He was educated in Arabic and Hebrew and was interested in science, particularly astronomy. Originally called Moses, he took the name Peter at his baptism in 1106. Soon after this he wrote the dialogues, which take the form of a discussion between Moses and Peter, to show that his adopted religion was compatible with reason and natural philosophy. In the sixth dialogue he discussed the Trinity. The sacred name for God was written with consonants alone in the Hebrew alphabet: Yod, H~ Vav, H~. Since it was forbidden to p r o n o u n c e the name, it is u n k n o w n what vowels

Ne la p r o f o n d a e chiara sussistenza de l'alto lume parvermi tre giri di tre colori e d'una contenenza; e l'un da l'altro come iri da iri parea reflesso, e'l terzo parea foco che quinci e quindi igualmente si spiri.

Figure 14. Symbol of the Trinity by Petrus Alfonsi.

(Translation: Within the profound and shining subsistence of the lofty light appeared to me three circles of three colours and one magnitude; and one seemed reflected by the other, as rainb o w by rainbow and the third seemed

fire breathed forth equally from the one and the other. [6]) One medieval interpretation of the rainbow held that it was composed of three fundamental colours: red, green, and blue. Reeves and Hirsch-Reich suggest that Dante saw the red, green, and blue of Joachim's three circles as iridescent, each reflected in the others, together malting one rainbow appearing as three [25]. Dante needed to be careful here, as Joachim was condemned by the fourth Lateran Council (1215) for giving the circles different colours and hence making them unequal. With this progression in the right direction, it is not surprising to find that the Borromean rings finally appear as a symbol of the Trinity. A thirteenth-century manuscript in the Municipal Library at Chartres contained four such figures, one of which is reproduced in Figure 16. In the centre, inside all the circles, is the word "unitas'; the three syllables of"trini-tas" are distributed in the outer sectors. Unfortunately, the manuscript was destroyed in the fire of 1944. The copy shown here was reproduced in a manual of Christian iconography [8], along with descriptions of the other three. The labels on these other figures are shown in Figure 17. They are (a) "God is Life" surrounded by "Father," "Son" and "Holy Spirit"; (b) "God is" surrounded by "Word," "Light," and "Life' (c) the phrases "Trinitas Unitate" (three in one) and "Unitas Trinl'tate" (one in three) distributed over the diagram.

Other Symbolic Uses of the Rings The symbolism of the rings has been recognised and applied in a variety of areas.

Michelangelo Bounarroti (1475-1564) used a symbol of the three circles with an M in one of the outer sectors to mark his stones. According to Giorgio

(b) Figure 15. Symbol of the Trinity by Joachim of Fiore (left) and a diagram of a topologically equivalent link showing the symmetry of the three components.

VOLUME20, NUMBER1, 1998 59

Figure 16. Figure o f the Trinity from a t h i r t e e n t h - c e n t u r y manuscript ( r e p r o d u c e d f r o m [9]).

Vasari [31], the circles signified the three professions of sculpture, painting, and architecture, and that they were so close as to be inseparable. After his death, the Florentine academici, judging him to have attmned the highest rank in each, changed the circles into crowns o f laurel leaves. The crowns can be seen on his tomb, designed by Vasari, which is at the b a c k of the right aisle in the Church of Santa Croce in Florence. The pattern is not alternating. 9 The F r e n c h m a n J a c q u e s Lacan (1901-1981) w a s a F r e u d i a n psychoanalyst w h o gave a series of seminars over t h e p e r i o d 1973-80. T o w a r d s the e n d o f his w o r k he m a d e frequent use o f topological obj e c t s such as the MSbius b a n d and the Klein bottle, and, in particular, knots and links. He used the three c o m p o n e n t s o f the B o r r o m e a n rings to s y s t e m a t i s e the relationships bet w e e n the Real, t h e Symbolic, and the Imaginary [14]. 9 The m e t a p h o r is also applied in particle physics [1, 32]. F o r a nucleus to be stable, it n e e d s to contain roughly equal numbers o f p r o t o n s and neutrons. In a h e a v y isotope such as Lithium-11, which contains three protons and eight neutrons, the quantum

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THE MATHEMATICALINTELLIGENCER

tunnelling effect allows t w o neutrons to extend beyond the surface of the nuclear core and form a cloud or halo. The two halo n e u t r o n s are very w e a k l y bound, and the halo nucleus is a b o u t twice the a v e r a g e size for a nucleus of the s a m e mass. If one of t h e halo neutrons is r e m o v e d , the o t h e r c o m e s a w a y too. Thus the halo nucleus can be thought o f as a threeb o d y system (a Lithium-9 core plus t w o neutrons) w h i c h c o m e s a p a r t if any one particle is r e m o v e d . Mikhail Zhukov christened t h e s e o b j e c t s B o r r o m e a n nuclei.

Other Examples of the Rings The following e x a m p l e s o f the rings a r e k n o w n to us. We w o u l d like to h e a r o f any others. 9

The rings were u s e d in a triangular f o r m by the Vildngs (Figure 18).

(~)

Known a s the "Walknot" ( m e a n i n g k n o t o f the slain), the s y m b o l w a s u s e d in p i c t u r e s o f battles to identify w a r r i o r s passing to the n e x t w o r l d to find their p l a c e in Odin's castle Valhalla [5]. 9 In North America, the design is k n o w n as the Ballantine rings: the rings are u s e d by the Falstaff Brewing C o r p o r a t i o n on their Ballantine ale. This t e r m is used b y Louis Kauffman in his book Formal Knot Theory [12], p. 113. 9 The rings feature p r o m i n e n t l y in the video Not Knot p r o d u c e d b y the G e o m e t r y Center at the University o f Minnesota. Using c o m p u t e r animation, it t a k e s you on a tour o f hyperbolic space, and explains the construction of a hyperbolic tiling o f th& c o m p l e m e n t of the B o r r o m e a n rings. The fly-through sequences are filmed in true hyperbolic perspective. 9 The Australian artist John Robinson has m a d e v a r i o u s forms of "Borromean" s c u l p t u r e from interlinked squares (entitled Creation), triangles (Intuition), a n d r h o m b i (Genesis) [26]. An edition o f each one has b e e n don a t e d to the N e w t o n Institute in C a m b r i d g e by his p a t r o n s D a m o n de Laszlo a n d Robert Hefner III. More i n f o r m a t i o n a b o u t the artist a n d his w o r k c a n b e found in the Symbolic Sculpture and Mathematics exhibition on the m a t h e m a t i c s d e p a r t m e n t w e b p a g e s of the University o f Wales at B a n g o r [3]. 9 The alternating rings appear in the Cappella Rucellai in the Church of San Pancrazio in Florence. The 1467 tempietto was designed by Leon Battista Mberti (1404-1472); the rings, in b l a c k on white marble, are said to b e a symbol of the Medici family. 9 A similar design of three interlaced crescent m o o n s (Figure 19) can b e

(b)

Figure 17. Sketch reconstructions of the lost Chartres figures.

(c:)

/// Figure 18 (left). The 'Walknot' of the Vikings. Figure 19 (right). The crescent moons of Diane de Poitiers. Figure 20 (below). A motif of the Ricordi music company. Property of Casa Ricordi--reproduced by kind permission.

seen at the Palace of Fontainebleau. Sometimes the points of each moon meet, completing the rings. The motif was designed by the architect Philibert de l'Orme, and is based on the moon emblem used by Diane de Poitiers (1499-1566), mistress of King Henry II of France. There may be also an Italian influence since the Queen of France was Caterina II de Medici. The Milanese music printing company of G. Ricordi used the alternating rings as their logo (Figure 20). The motif was used with many variations: the first edition of Giacomo Puccini's opera Madame Butterfly (1904) was decorated with three interlinked dragons. The Prussian Alfred Krupp (18121887) inherited a small iron works from his father, and transformed it into a large empire of heavy industry. In 1875 he registered three overlapping circles, as in Figure 60), as the trademark of his firm: they represented seamless railway tyres, one of the company's masterworks in the 1850's [17]. As the production of weapons increased, the same symbol came to be interpreted, especially in this century, as the muzzles of three cannon. The symbol is still used by the new corporation Fried. Krupp AG Hoesch-Krupp. Acknowledgments We are very grateful to the Borromeo family, and the library staff in the Biblioteca Trivulziana. Flavia Rampichini and Nicoletta Onida also provided useful information. The first author would also like to thank the Beltrami family for their warm hospitality during his visits to Italy. REFERENCES 1. S. M. Austin and G. F. Bertsch, "Halo nuclei," Scientific American 272 June (1995), 62-67. 2. J. Baez and J. P. Muniain, Gauge Fields, Knots and Gravity, series on Knots and Everything vol. 4, World Scientific, 1994. 3. R. Brown, J. Robinson and C. Quinton,

Symbolic Sculpture and Mathematics, 1996. URL http://www.bangor.ac.uWma/. 4. H. Brunn, "0ber Verkettung," Sitzungberichte der Bayerischer Akad. Wiss. Math-Phys. Klasse 22 (1892), 77-99.

VOLUME20, NUMBER1, 1998 (}1

5. P. R. Cromwell, "Borromean triangles in Viking art," Math. Intelligencer 17 (1995), no. 1, 3-4. 6. Dante, The Divine Comedy: Paradise, vol. 1 (Italian text and translation by C. S. Singleton), Bollingen Series, Princeton Univ. Press, 1975. 7. Y. Delaporte, Les Manuscripts Enlumin~sde la Bibliotheque de Chartres, Chartes, 1929. 8. M. Didron, Iconographie Chr6tienne, Imprimerie Royale, Paris, 1843. 9. M. Didron and A. N. Didron, Christian Iconography, or the History of Christian Art in the Middle Ages, George Bell and Sons, London, 1886. 10. R. H. Fox, "A quick trip through knot theory," in Topology of 3-manifolds and Related Topics, ed: M. K. Fort, PrenticeHall, Inc., 1962, 120-167. 11. H. M. Hilden, M. T. Lozano, and J. M. Montesinos, "The Whitehead link, the Borromean rings and the knot 946 are universal," Collect. Math. 34, no. 1 (1983), 19-28. 12. L. H. Kauffman, Formal Knot Theory, Princeton Univ. Press, 1983. 13. L. H. Kauffman, "State models and the

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THE MATHEMATICALINTELLIGENCER

Jones polynomial," Topology 26 (1987),

23. P. Priest, Dante's Incarnation of the Trinity,

395-407. 14. J. Lacan, Le S6minaire de Jacques Lacan, vol XX "Encore" (1972-73), I~ditions du

Longo Editore, Ravenna, 1982. 24. M. Reeves, Joachim of Fiore and the Prophetic Future, SPOK, London, 1976. 25. M. Reeves and B. Hirsch-Reich, The Figurae of Joachim of Fiore, Clarendon

Seuil, Paris, 1975. 15. B. UndstrOm and H.-O. Zetterstr6m, "Borromean circles are impossible," Amer. Math. Monthly 98 (1991), 340-341. 16. G. Lopez and S. Severgnini, Milano in Mane, second edition, Mursia, 1990. 17. W. Manchester, The Arms of Krupp 1587-1968, Michael Joseph, London, 1969. 18. W. W. Menasco, "Closed incompressible surfaces in alternating knot and link complements," Topology 23 (1984), 37-44. 19. K. Murasugi, "Jones polynomials and classical conjectures in knot theory," Topology 26 (1987), 187-194. 20. O. Nanyes, "An elementary proof that the Borromean rings are nonsplittable," Amer. Math. Monthly 100 (1993), 786-789. 21. G. Petrocchi (editor), Enciclopedia Dantesca, Institute Treccani, Rome, 1970. 22. P. Portaluppi, La Casa de gli Atellani in Milano, Bestetti e Tumminelli editori, Milan, 1922.

Press, Oxford, 1972. 26. J. Robinson, Symbolic Sculpture, Edition Limitee, Carouge, Geneva, 1992. 27. D. Sant'Ambrogio, "Dell'impresa araldica dei tre anelli intrecciati," Archivio Storico Lombardo, 7 anne XVlII (1891), 392-398. 28. P. G. Tait, "On knots," Trans. Royal Soc. Edinburgh 28 (1876), 145-190. 29. M. B. Thislethwaite, "A spanning tree expansion of the Jones polynomial," Topology 26 (1987), 297-309. 30. J. Tolan, Petrus Alfensi and his Medieval Readers, University Press of Florida, 1993. 31. G. Vasari, The Lives of the Painters, Sculptures and Architects (translated by A. B. Hinds), revised edition, voL 4, Evewman's Library, Dent, London, 1963. 32. M. Zhukov, et al., "Bound-state properties of Borromean halo nuclei," Physics Reports, Review section of Physics Letters 231, no 4. (1993), 151-199.

KARL GUSTAFSON AND TAKEHISA ABE

Tho Third Boundary Oondition Was it Robin's? G u s t a v e Robin (1855-1897) died 100 years ago in obscurity. Why w a s the third b o u n d a r y condition o f p a r t i a l differential equations n a m e d after him? Who w a s Gustave Robin?

Boundary Conditions of Partial Differential Equations W h e n one s e e k s a h a r m o n i c function u, i.e., Au = 0 w h e r e 02

A d e n o t e s the Laplacian o p e r a t o r 0X~l + ... + ~

02

over

s o m e n-dimensional d o m a i n ~1, one c o m m o n l y e n c o u n t e r s t h r e e b o u n d a r y conditions. The first is the Dirichlet b o u n d a r y condition: u(x) = f ( x ) is given at all x in the b o u n d a r y 01~. The s e c o n d is the N e n m a n n b o u n d a r y condition: the o u t w a r d n o r m a l derivative Ou/On = f i x ) is given on 0gl. Then there is a third b o u n d a r y condition: Ou/On + ~u = f ( x ) is given on O~, w h e r e a is a given positive coefficient. This third boundary condition is variously designated, b u t frequently it is called Robin's b o u n d a r y condition. Dirichiet was a p r o m i n e n t m a t h e m a t i c i a n a n d his contrib u t i o n s to m a t h e m a t i c s and s c i e n c e are well known. Less p r o m i n e n t b u t well k n o w n for his contributions to partial differential equations w a s (Carl) Neumann, for w h o m the s e c o n d b o u n d a r y c o n d i t i o n is named. But w h o w a s Robin? Why is his n a m e a t t a c h e d to t h e third b o u n d a r y condition? In this two-part essay, w e will a n s w e r t h e s e questions. Early r e s e a r c h in 1976 b y the fkrst-named a u t h o r (K.G.) w h e n writing the b o o k [1] r e v e a l e d that (Victor) Gustave Robin w a s a p r o f e s s o r o f m a t h e m a t i c a l p h y s i c s at t h e S o r b o n n e in Paris. After his death in 1897, Robin's Collected Works w e r e p u b l i s h e d in the p e r i o d 1899-1903 b y his friend and colleague Louis Raffy. We quote from the A v e r t i s s e m e n t of I, P a r t 2, o f t h o s e Collected Works:

Le prdsent fascicule contient la partie maitresse de l'oeuvre de Robin, la Thermodynamique gdndrale, d laqueUe il ne cessa gu~re de penser, depuis l'dge de vingt ans jusqu'aux dernier mois de sa vie. Ndanmoins, dans cet intervalle de vingt-deux anndes, pendant lequel ses vues se modifi~rent profonddment en s'dlargissant de plus en plus, il n'en a rien publid, s a u f deux Notes de trois pages, imprimdes en 1879. Par contre, ~ diverses reprises, il br~tla presque tout ce qu'il avait dcrit, de sorte qu'une faible partie de ses papiers a dchappd ~ la destruction. There is p o i g n a n c y here. It begs a fuller explanation. Why didn't Robin p u b l i s h more? Why did Robin, n e a r the end, burn m o s t o f w h a t he h a d written? Was it a result of scientific disappointment; the illness of t h e last m o n t h s of his life; p e r s o n a l d e p r e s s i o n ? 1997 m a r k s the 100th anniversary o f Robin's death. In the first p a r t o f this e s s a y we will tell o f o u r s e a r c h for exp l a n a t i o n s o f the link b e t w e e n Robin a n d the b o u n d a r y condition t h a t n o w carries his name. We have u n c o v e r e d a strong "Russian Connection," although it d o e s not provide a c o m p l e t e answer. In the s e c o n d p a r t of this essay, in the n e x t issue, we will look at R o b i n ' s w o r k in totality, which i n c l u d e s significant c o n t r i b u t i o n s to potential theory and t h e r m o d y n a m i c s . Robin's i d i o s y n c r a c i e s and early death left him a l m o s t u n r e m e m b e r e d . We w e r e unable to find any a c c o u n t o f him in any o f the archives of Paris.

The Third Boundary Condition The first b o u n d a r y condition is e a s y to visualize: it is prescribing the b o u n d a r y values of the h a r m o n i c function u. Physically the solution u to the f a m o u s Dirichlet boundaryvalue p r o b l e m m a y be thought o f as the s o a p film stretch-

9 1998 SPRINGER-VERLAG NEWYORK, VOLUME 20, NUMBER 1, 1998

63

ing across a closed wire loop. The Laplacian h is a linear approximation to the minimal surface equation. See [1]. The second (Neumann) boundary condition is less easy to visualize, but in its homogeneous form (f = 0) it corresponds to a "free end" condition in vibrating systems or, for example, at the end of a whip. In its nonhomogeneous form (f r 0) it represents a specified flux through the boundary of ~. In acoustics, the pressure perturbation satisfies a Neumann condition Op/an = 0 at solid walls. The third boundary condition linearly combines the Dirichlet and Neumann boundary conditions. As such, it may be thought of as an elastic response, or a partially absorbing boundary. An especially important instance of the third boundary condition that recently came to our attention: the fundamental equation of physical geodesy has the OT form - - + a T = - A g on the earth's geoid. Here T is the On anomalous gravitational potential, i.e., the deviation from a spherical or ellipsoidal reference potential. At the end of this paper we present a worked-out example of how things change in vibrating systems when a Dirichlet or Neumann boundary condition is replaced by the third boundary condition. We suggest the following easier exercise which the reader may do right now! Consider a harmonic function u ( x ) in one dimension on the domain ~: 0 < x < 1. This means u ( x ) = mx § b, a line. Dirichlet boundary conditions, u(0) = f0, u(1) = fl, determine the solution u ( x ) = ( f l - f o ) x +fo. Neumann boundary conditions - u ' ( O ) = fo, u'(1) = f l , determine solutions u ( x ) = f i x + b, b an arbitrary constant. Note that necessarily f0 = - f l for a solution to exist. Robin boundary conditions, - u ' ( 0 ) + au(0) =f0, u'(1) + au(1) = f l , determine the solution

u(x)

-

(fl - fo)

2 + a

x +

(1 + ~)f0 + f l a(2 + a)

For a > 0, the Robin boundary condition, solutions to the third boundary value problem are uniquely determined. The case a = 0 reduces to the Neumann problem. For a < 0, which we will call the Steklov boundary condition, one must be more careful: one may encounter an eigenvalue. The eigenvalue occurs in this simple example at a = -2. Solvability then obtains only for certain dataj~ in this case, f0 = fl. Then solutions are determined only up to the eigenfunction U - 2 ( X ) = - 2 x + 1. The Search for Robin As mentioned above, our search began 20 years ago in the preparation of the book [1]. Immediately the Oeuvres S c i e n t i f i q u e s of G. Robin were found in the University of Colorado Library! These are three beautifully bound volumes published by Gauthier-Villars in Paris almost 1O0 years ago. See Figure 1. However, prior to our search in 1976, the library check-out record for these three volumes contains only 3 entries: October 24, 1910; August 1, 1911 and July 15, 1912. Apparently 34 library systems in the United States possess at least one of these volumes. The question of Robin and why he is linked to the third boundary condition was then placed into the book [1] as a historical teaser. A little information was provided there, and in the second edition [2]. During the period 1980-1990 we acquired some further information about Robin. Independently during the same period, while preparing the Japanese-language third edition [3], the second-named author (T.A.) and colleagues obtained further perspective on

Figure 1. The Collected Works of Gustave Robin. These were compiled shortly after Robin's death by his friend and colleague Louis Raffy.

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THEMATHEMATICALINTELLIGENCER

Robin's life and work. These were reported (in Japanese) to scientific historical societies in China and J a p a n in [4,5,6]. Then, while working together to complete [3], we jointly published a short four-part account (in Japanese) of our search for Robin and his work, in the popular science magazine GITOH in J a p a n [7]. The present two-part essay is the English-language presentation of Robin's story as w e know it to the present time.

A Continually Recurring Historical Question Just after the manuscript for [1] was finished, the query [8] appeared in the Notices of the American Mathematical Society, raised by s o m e o n e well versed in differential equations. I (K.G.) immediately answered [9], although it must n o w be confessed that some information was withheld in order to sustain the curiosity in the historical problem. Just recently, s u m m e r 1994, as we were beginning to prepare this account, the same question "Who was Robin" appeared again, this time [10] in the NA Digest, a widely used electronic mail n e t w o r k for numerical analysts. We of course quickly responded; in later correspondence related to [10], J. Crow reported to us that he received m a n y messages wanting to k n o w the answer to the question, and only a very few, mostly erroneous it turns out, supplied pieces of information or speculation about Robin. Essentially all of Robin's work, including his doctor's thesis, is reproduced in his Oeuvres Scientifiques [11]. These Collected Works of Gustave Robin were compiled, published, and in places substantially rewritten by Louis Raffy, a friend and colleague of Robin. Raffy w o r k e d from Robin's few short publications, class notes taken by students in Robin's lecture at the Sorbonne, some unpublished fragmentary notes and manuscripts of Robin, and Raffy's o w n editing and additions to his knowledge of Robin's lecture, writings, and thinking. The collected works are, as s h o w n in Figure 1: I. Physique (one volume, published as two separate books) Part I. Physique mathdmatique (Distribution de l'~lectricit~, Hydrodynamique, Fragments divers), 1899. Part 2. Thermodynamique gdndrale (]~quilibres et modification de la matiSre), 1901. II. Mathdmatiques (Th~orie nouvelle des fonctions, exchisivement fond~e sur l'id~e de nombre), 1903. III. Chimie (never published) (Lemons de Chimie physique, profess~es & la Facult~ des Sciences de Paris).

The Dieudonn6 Correspondence After fmding the collected w o r k s and papers of Robin, and placing that information into the b o o k [1], the first author (K.G.) found himself in Paris for the month of November, 1978. A n u m b e r of eminent French mathematicians were asked about Robin. Surprisingly, no one knew anything. However, H. Br~zis over lunch suggested a letter to Jean Dieudonn& Not only was Professor Dieudonnd one of the greatest French analysts and expositors, but he also was

much our senior, and knew m u c h Paris history, though that concerning Robin was before his time. Three questions were directed to Professor Dieudonn& Professor Dieudonnd took the time to look into them [12]. To the first question, did Robin ever actually use the third boundary condition, there was no reply. To the second question, w h o was Robin's thesis advisor, Dieudonnd confirmed that it was Picard, and we k n o w the thesis committee: Emile Picard, one year younger than Robin; Hermite was the President of Robin's thesis committee; and the third m e m b e r was Darboux. To the third question, what was the nature of Robin's death, there was no answer.

Potential Theory Robin's thesis for the doctorate of mathematical sciences, submitted at the Sorbonne, July 13, 1886, and published as Sur la distribution de I'dlectricitd ~ la surface des conducteurs fermds et des conducteurs ouverts in the Annales scientifiques de l'Ecole Normale supdrieure, supplement (1866), was some of his best work. We n o w believe that it was on the basis of this work that Robin's name was later placed on the third boundary condition. Therefore let us turn to Robin's place in potential theory. Robin's thesis was concerned with finding an integral equation for the static surface charge density over a conductor with convex surface. In other words, one seeks a single-layer charge distribution on such a closed bounded surface S --- 0 ~ which produces no force at any of the interior points of the b o d y ~. This can be posed mathematically as solving the following two boundary-value problems, one the interior Neumann problem in ~ and the other the exterior Neumann problem on ~e, the exterior of ~: find a potential v via a single density layer p on S which satisfies both

{

Av=Oin~ Ov

7n = f o n ogz

{ A_~_nV= 0 in ~e 0v = f on 0~ for any given function f which is continuous on 0~. Necessarily (by the divergence t h e o r e m ) f must satisfy the compatibility condition ? o a f ds = O. Here Ov/On denotes the outer normal derivative for each problem, i.e., Ov/On = (grad v).n, where the unit vector n points outward for the interior problem, inward for the exterior problem. The solution is sought in the form of a single-layer potential 1

v(qo) = - ~ ~ a

P(ql) do', r

r = d(qo, ql).

This potential formulation of the solution is transformed by Robin into an integral equation for the charge density,

P(qo) = ~

P(ql) ~-n

do" + f(q0),

z = - I.

VOLUME 20, NUMBER 1, 1998 6 5

Because of the normal derivative of the fundamental singularity, the integral term here is now a double-layer potential. This integral equation is solved by means of a series P(qo) = Po(qo) + zpl(qo) + "'" + zk Pk(qO) + "".

Substituting and equating like powers of the expansion parameter yields the relations P0(q0) = f(q0) Pl(qo)

=

-

2~

Po(qo) ~ n

Pk(qO) = -- 2-~

do-

Pk-1 (qo) ~ n

do-.

In the same way, the potential v can be recovered from the expansion v(qo) = vl(q0) + zv2(qo) + "'" + z k-1 Vk(qO) + "'"

by integrating the coefficients

1 ~ Pi-l(q0) do-.

vi(qo) = - ~

r

In his thesis Robin gives conditions for the convergence of the series and applies it to a conducting spheroid. Picard's Version Robin's considerations of the convergence of the series are involved. The version used nowadays is that of Picard, who we now know was Robin's thesis advisor. Picard [13, p. 204] calls Robin's method of solution "une des plus ~16gantes" for this problem and refers to Robin's "remarquable Th~se sur le probl~me de la distribution de l'61ectricit6." We refer the reader to [13, pp. 203-210] for Picard's exposition of this problem. Picard's solution is more general than Robin's. Picard also notes similar work by Steklov [14] and E. Neumann (1899). (By the way, there are several Neumanns, and this is not Carl Neumann.)

These were the days when Poincar6 and others (see the references in [14, (1900)]) were still trying to prove general rigorous statements about the solvability of the Dirichlet Problem. Robin's methods were constructive, assuming the existence of a solution. Thus there are some differences among the assumptions made about the region and its boundary in these papers at this time. Some of the researchers tacitly assumed ~ convex, others (e.g., Lyapunov of [15]) proudly claimed to have overcome such restrictions, although the exact generality of their results is still sometimes vague. Until Lebesgue gave his counterexample in 1913, the extent of the solvability of the Dirichlet Problem remained cloudy. We also note the competitiveness of some of these papers. For example, there seem to have been priority disputes between Lyapunov and Steklov (Lyapunov's student), not only about convexity or nonconvexity of s but about who was first. Lyapunov [15, p. 243] more or less claims that Steldov, by following Robin's approach, was restricted to convex surfaces. Meanwhile, Lyapunov [15] also relies on Robin's method. For his part, Steklov [14, (1900), p. 208] claims he knew, long before Lyapunov published his results, how to solve the problems! All were trying to outdo Poincar6 (1896). Steklov's Version A main finding: Steklov actually states [14, (1900), pp. 252-253] the third boundary condition: E n e m p l o y a n t les rdsultats o b t e n u s dares les C h a p i t r e s prdcddents, n o u s p o u v o n s rdsoudre c o m p l ~ t e m e n t des d i v e r s p r o b l # m e s i m p o r t a n t s de la thdorie a n a l y t i q u e de la chaleur. . . . I1 e x i s t e u n e f o n c t i o n u, c o n t i n u e avec ses ddrivdes des d e u x p r e m i e r s ordres d l ' i n t d r i e u r de la s u r f a c e d o n n d e (S), . . . , s a t i s f a i s a n t a u x c o n d i t i o n s Au + f = 0 Oui On

+

hui = 0

~ l ' i n t d r i e u r de (S) sur S

The Russian Connection The reference by Picard to Steklov's work was the first hint of a "Russian Connection" in the Robin story. Our hypothesis then became: it was some important Russian mathematician who assigned Robin's name to the third boundary condition. We have neither confirmed nor eliminated this conjecture. As will become clear below, we have also an alternative hypothesis. But there is no doubt that the strong intellectual ties between France and Russia at the turn of the century helped carry Robin's name forward in this connection.

Steklov points out that this result solves steady heat conduction problems, but says that he defers a full development to another M6moire. It is important to note that Steklov does not, however, explicitly connect this third boundary condition in any way to Robin. Further, in these Russian papers in this period, although the single- and double-layer potential method are attributed to Robin, there are no specific literature citations to him at all.

Lyapunov's Version Lyapunov [15] points out that for the electrostatic potential problems (Dirichlet and Neumann) he considers, the approach of C. Neumann would not suffice, but that the method of G. Robin would. Note that the date of [15] is 1898, the year after Robin died.

Giinter's Version The Russian connection is well illustrated by the book of N. M. Gttnter [16]. This book originated from a seminar on potential theory that the author (alias Gunther) held in Leningrad (alias St. Petersburg) in the early 1920's. Gfinter's book is not only an excellent book on classical

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THE MATHEMATICALINTELLIGENCER

oit f est u n e f o n c t i o n donnde, . . . , h est u n e c o n s t a n t e p o s i tive...

p o t e n t i a l theory, b u t it p r o v i d e s us with further links to Robin, who is m e n t i o n e d at several points. Let us discuss t h o s e points [pp. 128, 154, 156, 157, 171]. C h a p t e r III [pp. 122-177] is entitled The N e u m a n n Problem and the R o b i n Problem. The former refers to the classical N e u m a n n b o u n d a r y value p r o b l e m m e n t i o n e d in the introduction to this essay. But the latter d o e s n o t refer to any "Robin" b o u n d a r y condition. Rather, it refers to the p r o b l e m of the integral equation for potentials t h a t Robin t r e a t s in his thesis, for a p p l i c a t i o n to the N e u m a n n b o u n d ary value problem. In [16, Chapter III], Gtinter starts with the N e u m a n n b o u n d a r y value problem, treating the general c a s e o f multiply c o n n e c t e d domains. He t h e n converts it to an integral equation. It is a s s u m e d t h a t this integral equation for the d e n s i t y / z o f the u n k n o w n p o t e n t i a l v has a solution w h i c h is m o r e o v e r p r o v i d e d b y a series which c o n v e r g e s uniformly on S for small e n o u g h Izl. Then Section 3 is essentially the iterated algorithm w e gave above b y R o b i n for finding the potential. On p. 128 Gtinter calls t h e s e potentials the Steklov-Robin Potentials. When inspecting this theory, one sees that the k e y is the a s s u m e d series e x p a n s i o n for the density p =-/~ at q0 -= 0: ~(o)

= po(O) + z p l ( O )

+ ... + z n p n ( O ) + . . . ,

w h i c h is to converge tmiformly for all sufficiently small Izl. F r o m this one arrives at (p. 154) the Robin Principle: ]Pm - Phi < a~n for m > n / > N, 0 < r < 1. This criterion is to h a n d l e the pole at z = 1, for it can be s h o w n that when Robin's Principle is satisfied, the limit p o f t h e Pn is zero. On p. 156 w e again have the r e f e r e n c e to this m e t h o d of single- and double-layer p o t e n t i a l s as the Steklov-Robin Method. The equivalent p h y s i c a l R o b i n Problem (p. 156) is p r e c i s e l y stated: to f i n d the charge dist r i b u t i o n on a surface S such that there is no f o r c e exerted at a n y p o i n t i n the i n t e r i o r o f the region enclosed by S. On p. 171 we find r e f e r e n c e to the R o b i n F u n c t i o n p: t h e charge density a r r i v e d at as the limit o f the Pn as des c r i b e d in the Robin Principle. It p a y s to r e m e m b e r t h a t n o w a d a y s we d e s c r i b e this proc e d u r e generally as t h e m e t h o d of successive a p p r o x i m a tions, o r Picard's method, a p p l i e d to integral equations using i t e r a t e d kernels. As Robin p o i n t e d out in his thesis, e.g., [11, p. 61 of the first v o l u m e o f the Collected Works], Carl N e u m a n n was one of the early ones to use t h e s e techniques, in 1877. Thus w e see that Gtinter's r e f e r e n c e s to Robin a r e all in t h e c o n t e x t of Robin's thesis, and n o t to any Robin b o u n d a r y condition. By the way, it is gratifying to s e e Gtinter use the F r e d h o l m t h e o r e m (p. 249). The b i o g r a p h y o f Gttnter, w r i t t e n by Smirnov a n d S o b o l e v at the end o f [16], is also interesting. According to them, Gtinter w a s k n o w n to love teaching, and to a p p r e c i a t e m a t h e m a t i c a l rigor, m u c h as Robin did. Gtinter's b o o k [16], originally s e m i n a r n o t e s in Russian, was p u b l i s h e d in F r e n c h in 1934, in G e r m a n in 1957, and in English in 1967. Thus it h a d wide circulation o v e r half a century. Gtinter certainly gives Robin g o o d

name-identification; b u t never for any third b o u n d a r y condition. Gtinter d i d not ignore the third b o u n d a r y condition. In [16, pp. 231-238], he solves the third boundary-value problem {~n Au = O Ou + hu = f

in on0~

by use of the m e t h o d o f single-layer potentials. Gtinter d o e s this in the c o n t e x t of a stationary t e m p e r a t u r e p r o b l e m (we have t a k e n t h e a m b i e n t t e m p e r a t u r e to b e zero in the equation above, for simplicity). This w o u l d have b e e n a g o o d c o n t e x t ( t e m p e r a t u r e ) in which to m e n t i o n Robin's name, b e c a u s e o f Robin's interest in t h e r m o d y n a m i c s which w e will d e s c r i b e later, b u t Gtinter d o e s not. He t h e n goes on to find the Green's (kernel) function for the third boundaryvalue p r o b l e m , which n o w a d a y s w e w o u l d call the Robin's (kernel) function (note: this is n o t the s a m e Robin's F u n c t i o n p m e n t i o n e d above). A l t h o u g h Gtinter uses the terms G r e e n ' s function for the Dirichlet P r o b l e m a n d N e u m a n n ' s function for the N e u m a n n problem, he j u s t refers to the "Green's" function for the k e r n e l function corr e s p o n d i n g to the third b o u n d a r y condition. In c o n t r a d i s t i n c t i o n to its fine e x p o s i t o r y qualities, Gtinter's b o o k d o e s n o t contain a bibliography.

Hille's, Goluzin's, and Krzy~'s Versions As Robin's t h e s i s was c o n c e r n e d with p o t e n t i a l theory, it is not u n e x p e c t e d that his n a m e might also a p p e a r in comp l e x function theory. One of the m o s t interesting references in this c o n t e x t that w e f o u n d w a s that o f Hille. In [17, p. 280], Hille introduces the c a p a c i t y C(E) = e -V(E)

of a ( c o m p a c t ) set E, w h e r e V(E) is defined b y =

loglz,-

z21-'d

Czl)d

Cz2),

/z being a n o r m a l i z e d m e a s u r e on E. (That i s , / z is an arbitrary Borel m e a s u r e c o n c e n t r a t e d on E w i t h / z ( E ) = 1.) The e x p o n e n t V(E) is called Robin's Constant. One could easily s u s p e c t this to have c o m e f r o m t h e r m o d y n a m i c s r a t h e r than c o m p l e x function theory, b u t Hille [17, p. 318] refers explicitly to Robin's 1886 Paris thesis. Hille volunteers that "Robin's p a p e r is mainly of historical interest." The singularity loglzl - z21-1 generalizes to Ix - YI-(n-2) for n = 3, 4, 5 , . . . , and Ix - Yl is r e g a r d e d as the "energy" of the m e a s u r e /z. P r o v i d e d that V(E) < % one can then u n d e r general conditions establish the e x i s t e n c e of a unique minimizing measure. Generalizations can be m a d e to u n b o u n d e d sets E, for which the Robin Constant m a y remain finite if OE is compact. Generalizations to Riemann surfaces have also b e e n developed. Robin's Constant for E the disk o f r a d i u s a in 2 d i m e n s i o n s is - l n a, and for E the ball of r a d i u s a in n > 2 dimensions, Robin's c o n s t a n t is a-(n-2). Goluzin [18, p. 310] also gives us Robin's Constant in

VOLUME 20, NUMBER 1, 1998

67

this context of logarithmic capacity, and notes that the Green's function for the domain E can be written conveniently in terms of the fundamental logarithmic singularity and Robin's Constant. This underlies the generalizations just mentioned. Krzyi [19, p. 126] gives a nice specific example of Robin's Constant for the exterior domain E outside of a system of long conducting cylinders. The Robin Constant for the exterior of the ellipse x2/a 2 + y2/b2 = 1 is just - l o g ( l ( a + b)). Goursat's, Kellogg's, and MacMillan's Versions We have not turned up reference to a Robin boundary condition anywhere in the French school from 1900 to 1930. For example, going back to 1923, Goursat [20, p. 234] calls the third boundary condition. Hu +K

Ou = L On

the m i x e d b o u n d a r y condition, not the Robin condition. Nowadays "mixed boundary condition" normally means that on one portion of the boundary, you have one of the three usual boundary conditions, whereas on another part of the boundary, you have a different one. Goursat [20, p. 240] does mention the boundary condition a --u R

0u -U On

and, with credit to Tommaso Boggio [21], shows how to resolve it by coupling that boundary condition, which we now call the Steklov boundary condition, due to the minus sign, to a conjugate problem with what we now call a Robin boundary condition, with a plus sign. Goursat [20, p. 515] does mention Robin, but only in connection with his single-layer potential method. Kellogg [22], also treats the third boundary condition. There is, in the wealth of bibliographical citations, no mention of Robin connected to it, nor any mention of Robin at all. MacMillan [23, p. 27], mentions Robin's Integral Equation, in the same sense as treated in Gtinter [16]. He gives a nice explicit treatment of Picard's solution to Robin's equation (pp. 228-233). Boundary Conditions and Names Our conclusion is that Robin never used the third boundary condition 0u On

--

+ au

= f

on

Ot2,

a>0,

which is named after him. We do not find it in his work, and his peers and mentors from the period 1890-1940 do not (as far as we have found) connect it to his name. On the other hand, Robin's method of single- and double-layer potentials for boundary-value problems became an important one.

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What are we to make of this situation, where a named mathematical entity does not correspond to the actual contribution made? Not a whole lot, especially for the historical period in question. Was Dirichlet the first to consider the Dirichlet boundary condition? The answer is clearly no. Earlier, Poisson, Green, Gauss, and others had considered these problems. On the other hand, Dirichlet and Neumann made substantial contributions to the problems carrying their names in the boundary condition. Then why Robin, and not one of many others who worked in this theory? There could have been several factors which led to his name being placed on the third boundary condition by French mathematicians in the early part of this century. First, we found that Robin's family was well placed within the French intelligentsia of the period. Secondly, he did his thesis under Picard, Hermite, and Darboux, all well established in the field of mathematical physics. Third, many of the other possible scientific names had already been anchored (repeatedly) to mathematics: Dirichlet, Neumann, Poisson, Poincar~, etc. Fourth, his illness could have prompted a sympathetic placing of his name on the boundary condition. These are good rationalizations--or would be if French scholars of the first half of the twentieth century had attached Robin's name to the third boundary condition. They did not. The Russian mathematicians of the same period were more generous, with their designations of Steklov-Robin (note the order) Potentials, Steklov-Robin method, Robin Problem, Robin Function. However, there is no link to the third boundary condition. As concerns Robin's Constant in the case of two dimensions and from the point of view of analytic function theory, we do not press the issue of who first placed Robin's name on the logarithmic capacity V = - l o g C; see the references cited by Hille [17, p. 318], e.g., Tsuji (1959), Nevanlinna (1953), Frostman (1935), or conceivably Szeg6 (1924) as referenced by Goluzin [18]. Clearly Robin's (capacity) Constant and his original problem concerning fmding the constant static electricity equilibrium potential are implicitly related. But any connection to the third boundary condition is not evident. Who First Attached Robin's Name to the Third Boundary Condition? Who first named the third boundary condition for Robin? Find the needle in the haystack! We hope that the reader will agree that by asking, we have gained many related mathematical perspectives and facts. Let us now state what little we do know about this question. As a graduate student, the first author (K.G.) first saw the Robin boundary condition named as such in the book Bergman and Schiffer [24]. In his recent query [10] Crow conjectured the book of Duff [25]. But Duff [26] says that he first heard the term Robin's boundary condition from Schiffer. So does Garabedian [27]. Schiffer [28] says he heard it from Bergman. Bergman died some years ago. As to the Russian connection, Il'in [29] states he first saw Robin's boundary condition in the book by Bitsadze

[30]. The original Russian language version of that book was published in 1966. Numerous other contemporary Russian mathematicians were asked about Robin but, like the French mathematicians, little was known. The first encyclopedia reference to Robin's boundary condition that we have found is [31]. The encyclopedia reference [32] at about the same time gives excellent descriptions of the Robin Constant and Robin Problem (electrostatics) but no mention of a Robin boundary condition. The fact that there already existed a Robin's Problem (electrostatics), e.g., as in Gtmter's book [16] which was available in Russian, French, German, and finally English, from the 1920's through the 1960's, would have worked against anyone's also naming the third boundary-value problem, the Robin Problem. Where does that leave us, if we must make a best guess as to who placed Robin's name on the third boundary condition? Stefan Bergman. In 1948 we find the kernel function [33, equ (19), p. 541] satisfying the third boundary condition called Robin's function. We have checked some of Bergman's earlier work and have found no earlier references to Robin, but they could exist. Bergman worked with kernel functions (e.g., Green's fmmtions) from 1920 to 1950, traveled extensively in Germany and Russia before World War II, was very knowledgeable in boundary-value problem theory from several cultural and linguistic viewpoints, and was well versed in both the pure and applied literature. Once you assign Robin's name to the third boundary-value problem's kernel function, it is but a short step to call the third boundary condition, Robin's. Until someone comes with an earlier citation, Bergman must be our best guess. T h e C a s e for N e w t o n It must be admitted that it is convenient to have names such as Dirichlet, Neumann, Robin, Steklov (and Sommerfeld, for the radiation boundary conditions), to help remember and distinguish these boundary conditions. If, on the other hand, one is seeking at least historical appropriateness, one would have to consider renaming the third boundary condition as Newton's. Almost everyone sees "Newton's" cooling law in a calculus or first ordinary differential equations course: the rate at which a body loses heat through its surface by conduction and convection is proportional to the temperature difference between the body's surface and its surroundings. Newton was clearly thinking of higher dimensions and hence partial differential equations when he formulated this law. As in many things, he was probably the first to discuss it with any rigor. Newton's research on cooling laws was published in 1701 in [34]. See [35, p. 268] for the following translation from the Latin:

Having discovered these things; in order to investigate the rest, there was heated a pretty thick piece of iron redhot, which was taken out of the f i r e with a p a i r of pincers, which was also red-hot, and laid in a cool place, where the w i n d blew continually upon it, and putting on

it particles of several metals, and other fusible bodies, the time of its cooling was marked, till all the particles were hardened, and the heat of the iron was equal to the heat of the h u m a n body; then supposing that the excess of the degrees of the heat of the iron, and the particles above the heat of the atmosphere, f o u n d by the thermometer, were in geometrical progression when the times are in arithmetic progression, the several degrees of heat were discovered... Newton apparently never wrote this law in standard mathematical terminology, but he was the first to formulate it. A critical review of this subject is [36]. It is noteworthy that early work was calibrated against an instrument constructed by Hooke and known as "Royal Society's Standard Thermoscope." This is interesting because most of us have always looked at Newton's cooling law as just an example of the more general phenomenon of "Hooke's Law," under which at a crucial moment, one forces a linearization onto a physics! Perhaps Fourier [37] was the first to write and promulgate Newton's cooling law in mathematical terms. Most likely there were, between Newton's time and Fourier's, other mathematical formulations of the equations of heat propagation and their conditions at a boundary. Fourier discusses Newton's analysis down to molecular detail. Fourier also explicitly discusses the "linearization" we mentioned above. Compare the discussion in [1, Section 1.7.1]. Note that Newton's law of cooling is usually stated in terms of time rate of temperature change, whereas the third boundary condition states a spatial rate of temperature change. But the arguments for both are essentially the same, and always involve a "Hooke's" linearization move. Of Men and Mountains Would a better long-term approach in science be to emulate the evolving practice in mountain naming: remove all haman names from mountains and climbing routes and use instead more authentic or natural names? For example, Mount Everest is now Chomolungma (Goddess Mother of the World), Mount McKinley is Denali (the Great One), the first ascent of one of us on Long's Peak is now Window T r a v e r s e , . . . . In that more enlightened culture, the fLrst, second, and third boundary conditions then become: fixed boundary condition, free boundary condition, elastic boundary condition. Even though the physical names do not fit all physical circumstances, they are certainly more descriptive than Dirichlet, Neumann, and Robin. At the beginning, a new climbing route or a new mathematical result is naturally personalized. We all know who did it and we refer to it most easily that way. But, over time, we do not know those individuals. Still, how do we get rid of names like Pike's Peak or Long's Peak? Example of the Third Boundary Condition We close with a mathematical example of a boundary-value problem to show the features that change when a fixed

VOLUME 20, NUMBER 1, 1998

69

(Dirichlet) or free (Neumann) h o m o g e n e o u s boundary condition is replaced by an elastic (Robin) one. Consider the vibrating string problem

I

c~2U ~x2 02U= 0 ,

~-

J

O<xO

u(O,t) = O, t > 0

l (i,t)

+ a u ( i , t ) = 0,

t> 0

u(x,O) = f ( x ) ,

0 < x < 1

/ | I

|

au ( x , 0 ) = 0, 0t

0<x
Here u(x,t) is the vertical height of a taut (e.g., violin) string undergoing small displacements from the horizontal, the string is fixed (Dirichlet) at its left end, but it is constrained only elastically (Robin) on its right end. The initial displacement f(x) (the pluck) is given, and for simplicity we have assumed that it is instantaneously and vertically applied (the last condition above). The solution, as easily worked out by the standard method of separation of variables, is

u(x,t) = Z Cn s i n ( ~ - ~ x ) c o s ( ~ n n t ) . n--1

The Cn are then found by the standard Fourier method of setting u(x,O)=f(x) and equating coefficients. Had the usual (e.g., violin) Dirichlet fLxed right-end condition been assumed, the eigenvalues An would have been just ~.2, 4~.2, 9~r 2, . . . , n27r 2, . . . , and the Cn the usual sine series Fourier coefficients. The third (Robin) boundary condition on the right end leads instead (the details are easily verified) to the eigenvalue equation a tan \ / A = - ~/A. Sketching the graph of this (transcendental) equation quickly guarantees the existence of and qualitative locations of an infinite sequence of eigenvalues )tn resulting from the intersections of the curves in this equation. But we do not have simple expressions for the relative spacings of these An. Thus the violin with a "loose end" will produce a superposed solution of the usual sine modes, but at frequencies from a transcendental equation which can only be solved approximately! The music will be correspondingly discordant.

Acknowledgments We appreciate the patience of countless French, European, Russian, and American mathematicians to w h o m we directed our query. In the second part, we will look at the totality of (Victor) Gustave Robin's contributions to science, and what we k n o w of his life. REFERENCES

1. Karl Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods, Wiley, New York (1980). 2. Karl Gustafson, Introduction to Partial Differential Equations and Hi/bert Space Methods, 2nd Edition, Wiley, New York (1987).

70

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3. Karl Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods, 3rd Edition, Kaigai, Tokyo, published in two parts (in Japanese) as: Applied Partial Differential Equations, Introduction to Modern Methods, 1 (1991) 2 (1992), T. Abe and I. Onda, translators. Also published later in a limited English version by International Journal Services, Calcutta, India (1993). To appear (revised), Dover (1998). 4. T. Abe and I. Onda, Gustave Robin and his Contributions to Mathematics, International Symp. History of Math. Edu., Gunma University, Kiryu, Japan (1987), 5. T. Abe and I. Onda, Gustave Robin and his Contribution to Mathematical Sciences, International Conf. History of Science and Technology, Normal University, Huhhot, China (1992). 6. T. Abe, About the Work of G. Robin and its Evaluation, BUTSURI (J. Phys. Soc. Japan) 49 (1994), 934-936 (in Japanese). 7. T. Abe and K. Gustafson, Unknown Scientist Gustave Robin: I-1 (T.A.), GITOH 40, #9-10 (1994), 7-11 ; I-2 (T.A.) GITOH 40, #11-12 (1994), 12-17; II (T.A. and K.G.) GITOH 41, #1-2 (1995), 22-27; III (T.A. and K.G.) GITOH 41, #3-4 (1995), 20-27. 8. S. Hastings, Query 150, Notices American Mathematical Society 25 (1978), 506. 9. K. Gustafson, Responses 162, Notices American Mathematical Society 27 (1979), 103, 228. 10. John Crow, "Who was Robin?," NA Digest 94, No. 26, June 26 (1994). 11. Gustave Robin (1855-1897), Publications. See Part II. 12. J. Dieudonn6, Private correspondence (1978-1979). 13. E. Picard, Traite d'Analyse, 2rid Edition, Vol 1, Gauthier-Villars, Paris (1901). 14. W. Steklov, Comptes Rendus CXXV (1897). See also W. Steklov, Les Methodes Gen6rales les Problemes Fondamentaux de la Physique Mathematique, Ann. Fac. Sci. Univ. Toulouse, II (1900), 207-272. 15. A. Lyapunov, Joum. de Math. IV (1898), 241. 16. N. M. G0nter, Potential Theory, Ungar, New York (1967). Earlier published as La theorie du potentiel et ses applications aux problemes fendamentaux de la physique mathematique, GauthierVillars, Paris (1934). Originally published in Russian and later also translated into German (1957). 17. E. Hille, Analytic Function Theory, Vol. tl, Ginn, Boston (1962). 18. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, AMS Translations, Providence (1969). 19. J. Krzy~,, Problems in Complex Variable Theory, Elsevier, New York (1971). 20. E. Goursat, Cours d'Analyse Mathematique, 3rd Edition, Voh 3, Gauthier-Villars, Paris (1923). 21. T. Boggio, Accademia delle Scienze di Torino, (1911-1912). 22. O. D. Kellogg, Foundations of Potential Theory, (1929), Dover, New York (1953). 23. W. D. MacMillan, The Theory of the Potential, McGraw Hill, New York (1930). 24. S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic, New York (1953). 25. G. Duff, Partial Differential Equations, University of Toronto, Canada (1956). 26. G. Duff, Private communication, 21 November, 1994. 27. P. Garabedian, Private communication, 27 January 1995. 28. M. Schiffer, Private communication, 13 January 1995. 29. V. II'in, Private communication, 2 March 1996.

30. A. Bitsadze, Boundary-Value Problems for Second-Order Elliptic Equations, North-Holland, Amsterdam (1968). 31. The International Dictionary of Applied Mathematics, D. Van Nostrand, Princeton (1960). 32. Mathematisches Wdrterbuch, Band II L-Z, Teubner, Berlin (1961). 33. S. Bergman and M. Schiffer, Kernel Functions in the Theory of Partial Differential Equations of Elliptic Type, Duke Math. J. 15 (1948), 536-566.

34. I. Newton, Scale Graduum Caloris, Phil. Trans. Roy. Soc. 22 (1701 ), p. 838. 35. B. Cohen, Isaac Newton's Papers and Letters on Natural Philosophy, CUP, Cambridge (1958). 36. J. Ruffnor, Reinterpretation of the genesis of Newton's "Law of Cooling," Archive for History of Exact Sciences 2 (1964), p. 138. 37. J. Fourier, Theorie analytique de la chaleur, Didot, Paris (1822).

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II;,~a,t[:a,,,~.| J e t W i m p ,

Editor

I

NoncommutativeGeometry by Alain Connes SAN DIEGO: ACADEMIC PRESS, 1994. Xlll + 661 PP. US $59.95, ISBN 0-12-185860-X

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Yhat is geometric about noncommutative geometry? Gelfand [Gelfand] and Grothendieck [Gr] taught us the value of treating a commutative ring as though it were the ring of functions on an underlying geometric object. But why stop there? For many years, algebraists and algebraic geometers have tried to extend the tools of their trade--localization, sheaves, differentials, schemes, and so forth--to the context of noncommutative rings. The question then becomes whether or not this is valuable. The exciting development of the past two decades, initiated in the arena of functional analysis and differential geometry by the author of the book under review, is the appearance of a host of interesting examples of rings in which one feels there ought be geometry. These rings arose in applications to operator algebras, differential topology and geometry, and theoretical physics. As such, the "geometric" study of these rings is rewarded by the success of the applications. Another bonus of this example-based approach is that the examples suggest the tools that facilitate their study, much the same way that sheaves were invented to study topological spaces and algebraic varieties. Thus, many of the tools that come out of Connes's theory are quite unexpected. This beautiful, ambitious, and erudite book explains, through many examples, the phenomena, tools, and some of the applications of noncommutative geometry. Noncommutative Geometry is Counes's answer to the

W

question, "What is geometric about noncommutative geometry?" The mathematical framework from which Connes's theory arose and from which it is, for the most part, still cast is the theory of operator algebras or, more specifically, C*-algebras. A C*-algebra is simply a subalgebra A C L(H) of the bounded linear operators on a Hilbert space H which is norm-closed and closed under the adjoint operation on operators. There is an abstract characterization of a C*-algebra as a Banach *-algebra in which the algebra structure and Banach space structure are compatible. This compatibility is expressed by the condition that for all x 9 A one has Ilxx*ll = IIxll2; see [Gelfand]. However, this definition hides one of the features that makes C*-algebras so natural: although the norm may appear to be an additional structure, it is, in fact, uniquely determined by the algebra structure and can be derived from it by the formula Ilxll"= (spectral radius xx*) u2. Another significant feature of C*algebras, at least for most of the deep applications, is the spectral theorem for continuous functions acting on normal elements of A. (An element x of A is called normal if xx* = x*x.) Any commutative C*-algebra is isomorphic to the ring of continuous functions Co(X) which vanish at infinity on a locally compact set X; see [Gelfand]. Thus, a commutative C*-algebra contains exactly the information of a locally compact topological space. It is in this way that noncommutative C*-algebras are assumed to play the role of noncommutative topological spaces. A basic example of a noncommutative C*-algebra is Mn(C),the algebra of n • n complex matrices, or a direct sum of these. (This exhausts all finitedimensional examples, by Wedderburn's theorem.) A slightly more interesting example is Mn(Co(X)), the algebra of n • n matrices of functions

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 1, 1998

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on a locally compact space X which vanish at infinity. A typical way to form C*-algebras is to represent a *-algebra on a Hilbert space and take its norm closure. For example, let F be a discrete group. Then, the algebraic group algebra C[FI is represented naturally, via the left regular representation, on 12(F), and its norm closure C*(F) is a C*-algebra called the reduced group C*-algebra of F. In the case where F is abelian, Cr*(F) is isomorphic to C0(F), the continuous functions which vanish at infinity on the Pontrjagin dual of F. For general F, one likes to think of C*(F) as describing the noncommutative space F, which for many nonabelian F is a very badly behaved classical topological space. Connes's book starts out appropriately with one of the motivations of the theory of operator algebras, namely quantum physics. In the very first section, he shows how Heisenberg rediscovered matrix multiplication from empirical results [H]. Classically, observable physical quantities are represented by functions on phase space. Comms shows that if you assumed this classical picture, then the set of frequencies emitted by an atom would form a semigroup inside the real n ~ m bers ~; that is, the sum of two emitted frequencies would also be one. The al~ gebra of observables would be the corn volution algebra on the group generated by this semigroup, which (since it is abetian) is then C0(F). Experiments show, however, that this is not the case. They show that the frequencies can be represented by the differences of a snmll number of terms Pij = T i - - ~j (i, j in some index s e t / , which is discrete so as to conform to the observed discreteness of possible energies of an atom). They, therefore, obey the Rydberg-Ritz combination principle, uq = v~l + r~j. One then sees that the set of possible frequencies does not form a group, but rather a groupoid. (A groupoid is like a group in having an associative multiplication and inverses, but unlike a group, only certain pairs of elements can be multiplied.) Let 5 = {(i, J)li, j e I}, and assume the product (i, j).(k, l) exists if and only i f j = k and then the product is (i, l). The algebra of observables is

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then given by sums Za(ia) with product dictated by the combination principle,

(a'b)(i,j) = >~ a(i, k)b(k, Z), k

which one recognizes as the usual rule for matrix multiplication. This is actually a special case of a general construction of a groupoid algebra, similar to a group algebra. Thus, if G is a groupoid, C[G] consists of finite formal sums a = ~T~r a~, a T ~ C. The multiplication is given by (a 'b) T =

2 TI'T 2 = T

avlbT2"

defined as follows. The elements of are just the pairs (x, y) ~ 9t; the 'multiplication (x, y).(z, w) exists if y = z, and the product is then (x, w), which, again, is in ~ by transitivity. The inverse of (x, y) is (y, x); and we see another feature of the notion of groupoid: there is not just one unit, but many. In the case of ~, the units are the elements (x, x) and can therefore be identified with X. In general, we can think of a groupoid as a sort of equivalence relation on the set of units. If, for example, Xis a discrete set, then the set of equivalence classes of 91 can be naturally identified with the set of irreducible representations of C~(~). In general, we think of Cr*(G) as the quotient of the set X of units of G by the groupoid. However, the identification of the set of equivalences with the irreducible representations breaks down. Although the quotient space may be really badly behaved, the algebra is free. To bring things down to earth, consider the space [0, l] x / 0 , 1}, the disjoint union of two copies of the unit interval. Define an equivalence relation - by (t, j) - (s, i) if 0 < s = t < 1.

One can then form a C*-algebra C*(G) much the same way as we did for a group, by taking the norm closure in some representation. Also, let me remark that there is a version of the group and groupoid C*-algebra when G has a topology. The matrix algebras and group algebras are, thus, special cases of groupoid algebras. Let me try to explain some of the situations where noncommutative spaces arise and some of the examples that Cormes addresses in his book. One basic operation throughout mathematics which causes considerable difficulties is the process of forming 9 quotients. Quotients of topo0 n logical spaces by subspaces 9 or, more generally, by equiv- An equivatence relation with a non-Hausdorff quotient alence relations are often poorly behaved. They can fail to be X is the space on the left above, and Hausdorff or can easily have the triv- the interiors of the two intervals get ial topology (only two open sets). identified to a single open interval Algebraic geometers are familiar with yielding the figure on the right. The the process of forming moduli spaces. groupoid algebra of the corresponding This process itself is a quotient equivalence relation can be identified process. Again, many problems are with {f : [0, 1] ---)M2(C) If(0) and f(1) faced because the quotient of a variety are diagonal matrices}. Why is this is not a variety or the quotient of a groupoid algebra a better description of scheme is not a scheme. Notions such this quotient than the quotient itself?. For as stack or algebraic space are gener- example, the actual continuous funcalizations of schemes invented so as to tions on the quotient are exactly the include ce~"min quotients. One of same as on a closed interval, whereas Connes's basic ideas is that quotients the groupoid algebra retains a richer are often best expressed as noncom- structure which encodes interesting features of the space. mutative spaces. Another example is a favorite of An important framework for dealing with quotients is, again, the notion Connes's. Aperiodic titings of the plane of groupoids. They also arise in alge- arose in the 1950s in connection with braic geometry in the theory of stacks. the following problem. Suppose you So, for example, an equivalence rela- are given a finite number of isometry tion on a set X, ,~ C X x X, gives rise types of tiles (prototiles) and are asked to a groupoid (which I will also call ~ ) if you can tile the plane with this set of

tiles. (Of course, you are given an infinite n u m b e r of copies of each type of tile.) Is there an algorithm to decide if this set tiles the plane? Wang [GS] s h o w e d that there was no such decision procedure if and only if there was a set of tiles which tiled the plane, but could only tile the plane aperiodically. The search was on, and s o o n there was found an example of a set of prototiles which was aperiodic (that is, tiled only aperiodically). This first example had a ridiculously large n u m b e r of prototiles in it. This n u m b e r was gradually reduced until Penrose found his beautiful examples which had only two tiles. An example of a set of Penrose tiles is

a~a 1

Here, a -- (1 + ~X/5)/2. We will not go into w h y they tile the plane, but they do. In fact, they tile the plane in an infinite n u m b e r of ways, an infmite number of distinct ways. We say two filings are equivalent if there is a rigid motion of the plane which brings one set of tiles onto the other. Connes constructs the moduli space of tilings of the plane as a noncommutative space. Although Penrose tilings are aperiodic, they possess a property called quasiperiodicity, which we will express by the fact that any finite patch of tiles in one filing by these two prototiles occurs infinitely often in any other tiling by the same prototiles. Thus, if you know all the possible ways to tile the plane with these two tiles, and you are given a tiling, it would still be impossible to determine which tiling you have in hand by looking at a finite piece. This is an expression of the fact that every tiling is arbitrarily close to every other tiling, so that the classical quotient has the trivial topology, with only two open sets. According to Connes, the way to

make sense of this moduli space is as follows: First, one can identify the space of such tilings with K, the space of sequences Zn of l's and O's which don't have two l's in a row. (I will not tell you h o w this is done, but it is explained in Connes's book.) Then the tiling corresponding to z,~ will be equivalent to the tiling corresponding to z~ if and only if, eventually, these two sequences coincide; that is, there is an n > 0 such that zj = z~ for Penrose tiling. From Connes, Noncornmutative Geometry, p. 89, all j _> n. Hence, the 9 Academic Press. moduli space is the quotient KAJt, where ~ is the equivatechnicalities, and for those w h o lence relation of eventual equality o f want to b e c o m e experts, there are sequences. One then forms the C*-alm a n y references and an extensive gebra o f this equivalence relation bibliography pointing the reader to (qua groupoid). This is the c o r r e c t the right place. F o r all these people, "quotient." Unlike the n o n - H a u s d o r f f Connes has a c c o m p l i s h e d the wonexample above, where the algebra derful feat o f explaining in a simple was n o t v e r y noncommutative, this aland c o h e r e n t w a y 20 years (or so) of gebra is highly noncommutative. In his impressive work. I r e c o m m e n d fact, it is a simple C*-algebra. Out o f this b o o k m o s t highly. this algebra, one can read off s o m e beautiful properties of the space o f REFERENCES tiles. F o r example, for a finite p a t c h [Connesl] A. Connes, Sur la th~orie non comof tiles, one can interpret the density mutative de I'int6gration, Algebres d'operaof o c c u r r e n c e of this patch in a n y teurs, Lecture Notes in Math. 725, New York: tiling as a sort of dimension associSpringer-Verlag (1979), 19-143. ated to the patch, and b e c a u s e of the [Connes2] A. Connes, Noncommutative differproperties o f this algebra, the density ential geometry, Pub/. Math. I.H.ES. 62 (1985), must be an element of the s u b g r o u p 257-360, + a S o f ]r [Gelfand] I.M. Gelfand and M.A. Nafmark, On Connes describes m a n y m o r e exthe embedding of normed rings into the ring of amples in his book. He also describes operators in Hilbert space, MaL Sb. 12 (1948), some w a y s in which n o n c o m m u t a t i v e 445-480. g e o m e t r y enters into the index the[Gr] A. Grothendieck and J. Dieudonne, ory, the N o v i k o v conjecture, harElements de g6ometrie algebrique, PubL Math. monic analysis, geometry, and theoI.H.E.S. 8 (1961). retical physics. Along the way, he [HI W. Heisenberg, The Physical Principles of develops in a tourist-friendly w a y the the Quantum Theory, New York: Dover (1969). tools, like K-theory, asymptotic mor[GS] B. GrOnbaumand G.C. Shephard, Tilings phisms, and cyclic cohomology. One and Patterns, New York: Freeman, 1989. can read this material on m a n y levels. The basic b o o k is written in a w a y that a n y o n e can get s o m e of the feelDepartment of Mathematics ing and ideas o f the subject. For t h o s e University of Pennsylvania who w a n t m o r e details, there are Philadelphia, PA 19104 m a n y a p p e n d i c e s that cover m o r e USA

VOLUME 20, NUMBER 1, 1998

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Undergraduate Analysis: Second Edition by Serge A. Lang New York: Springer-Verlag, 1996. 656 pp. US $54.95, ISBN 0-38794-841-4

REVIEWED BY DAVID M. BRESSOUD

erge Lang's Undergraduate A n a originally published in 1968 /, has b e e n a r o u n d long enough to have acquired the status o f a classic. This is really its third edition. He has c o n t i n u e d to refine it. The text is s p r i n k l e d with wonderful tidbits: B o n n e t ' s mean-value theor e m (p. 107), an e l e m e n t a r y p r o o f o f Stirling's f o r m u l a (p. 120), the asymptotics of the log integral (p. 124), an ele m e n t a r y p r o o f o f the Bruhat-Tits fixed-point t h e o r e m (p. 150), P e a n o ' s curve (p. 225), the Tietze extension t h e o r e m (p. 231), Bessel functions (pp. 245 and 347), the Stieltjes integral (p. 259), the g a m m a function (p. 346), the h e a t kernel (p. 347), the Poisson inversion f o r m u l a (p. 352), the Poisson s u m m a t i o n f o r m u l a (p. 358), the functional equations for the t h e t a and z e t a functions (pp. 358-9). There is Khintchine's t h e o r e m that if Zqf(q) converges, then the set of x with rational a p p r o x i m a t i o n s , p/q, with arbitrarily large d e n o m i n a t o r s that satisfy Ix - P/ql
S lysis, as Analysis

(pp. 420f). The b o o k goes on and on, a feast of delights for a n y mathematician. This is precisely its p r o b l e m : this b o o k is written for m a t h e m a t i c i a n s , n o t for u n d e r g r a d u a t e s t u d e n t s of mathematics. What w a s a l r e a d y refined w h e n it first e m e r g e d in 1968 is n o w so highly p o l i s h e d t h a t it is inaccessible to students c o m i n g at the subj e c t for the first time. What are stud e n t s to m a k e of the s e c t i o n on integration that begins, "The p r o o f of the existence of the integral is b e s t p o s t p o n e d until w e have t h e language o f n o r m e d v e c t o r s p a c e s a n d uniform approximation."? Or o f the differential form defined as a m a p f r o m an o p e n set in R ~' into the v e c t o r s p a c e genera t e d b y images u n d e r a multi-linear alternating m a p of r-tuples o f basis elem e n t s from an n - d i m e n s i o n a l v e c t o r s p a c e ? H o w much m o r e enlightening to begin b y saying t h a t differential f o r m s exist to be integrated, and then to build their structure u p o n the stud e n t ' s intuition a n d t h e g e o m e t r i c a l constraints that arise. 1 T h r o u g h o u t this book, all of the m o t i v a t i o n s a n d the intuitions and the c o n n e c t i o n s have b e e n wrung out o f t h e mathematics. I w a s particularly d i s m a y e d to s e e the derivation of the functional equation for the zeta function set up as an illustration of the F o u r i e r inversion f o r m u l a without any i n d i c a t i o n of w h a t the zeta function m e a n s o r w h y anyone w o u l d c a r e that it h a s a functional equation. I have found that m y m o s t successful c o u r s e s are gritty. Students learn b e s t w h e n t h e y are d r a w n into the material b y an incongruity, b y an unexp e c t e d connection, b y a n intriguing p r o b l e m that eludes their grasp. I have j u s t c o m p l e t e d one o f the m o s t successful c o u r s e s I have e v e r taught. It w a s the s e c o n d s e m e s t e r of underg r a d u a t e real analysis. I h a d twelve students, m i x e d juniors a n d s e n i o r s and one high school student, a n d I gave t h e m as text T h o m a s H a w k i n s ' s

Lebesgue's Theory of Integration: its origins and development.2

F o r a n y o n e w h o is not familiar with this book, it is an e x a m p l e o f s o m e of the b e s t s c h o l a r s h i p available in t h e history o f mathematics. Its title des c r i b e s its content. It twists a n d t u r n s t h r o u g h struggles with the c o n c e p t of the integral in the s e c o n d haft of t h e n i n e t e e n t h century, and e m e r g e s into the t w e n t i e t h with the insights a n d solutions o f Lebesgue, Vitali, Baire, Fubini, a n d Stieltjes. It is a b o o k t h a t a s s u m e s t h a t the r e a d e r is familiar with m e a s u r e theory. Because of this, I s u p p l e m e n t e d it with B a t t l e ' s The

Elements of Integration and Lebesgue Measure, running quickly t h r o u g h The Elements of Lebesgue Measure b e f o r e we o p e n e d H a w k i n s ' s book. My s t u d e n t s f o u n d H a w k i n s ' s b o o k difficult a n d frustrating. They c o m p l a i n e d t h a t he often states a t h e o r e m and t h e n follows it with a m a t h e m a t i cal d i s c u s s i o n in which it is n o t clear, on first reading, w h e t h e r he is p r o v i n g the t h e o r e m , o r sketching a proof, o r d e s c r i b i n g a detail o f the proof, o r exploring s o m e c o n s e q u e n c e s o f the proof. Many times, H a w k i n s w o u l d des c r i b e a r e s u l t or an e x a m p l e , a n d it w o u l d n o t b e until several p a g e s l a t e r that he w o u l d m e n t i o n a n o t h e r mathe m a t i c i a n w h o h a d f o u n d a count e r e x a m p l e to the t h e o r e m o r a theor e m t h a t c o n t r a d i c t e d the e x a m p l e . My s t u d e n t s w e r e n e v e r quite s u r e w h e t h e r a r e s u l t d e s c r i b e d in H a w k i n s w a s true o r h a d only b e e n t h o u g h t to be true. E a c h p i e c e of the t e x t w a s assigned to a s t u d e n t w h o w a s r e s p o n sible for digesting it and p r e s e n t i n g it to the r e s t of the class. More t h a n once, t h e p r e s e n t a t i o n r e v e a l e d confusion a b o u t w h a t h a d b e e n s a i d a n d w h a t w a s really true, and it w a s the c l a s s ' s r e s p o n s i b i l i t y to argue this out as I p l a y e d the roles of devil's advocate a n d a r b i t e r of last resort. As a result, the c l a s s r o o m w a s vib r a n t with "what if?". Within t h e class, e x a m p l e s w e r e constructed, critiqued, dismantled, a n d r e a s s e m b l e d . P r o o f s were proposed, flaws revealed, p a t c h e s found. My students l e a r n e d only a few o f the t h e o r e m s c o n t a i n e d

1Whether or not I succeeded, this was my intention in Second Year Calculus. 2This provides one possible answer to the question of what to do with a two-semester course in real analysis that begins with A Radical Approach to Real Analysis.

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THE MATHEMATICALINTELLIGENCER9 1998 SPRINGERVERLAGNEW YORK

in Lang's book, but they developed the skills that will enable them to read and appreciate Lang's book. Hawkins's book or even an historical a p p r o a c h to real analysis is not necessarily the b e s t - - a n d certainly not the o n l y - - w a y to teach analysis effectively. An engaging and challenging course could be built a r o u n d Lang's b o o k if the professor is willing to w o r k with this text, to grapple and experim e n t with it, to choose one of the challenging sections--I would be tempted by the chapters on Dirac s e q u e n c e s - and then to provide students with b a c k g r o u n d and motivation, to dive into it with the students while remaining vigilant for the m o m e n t when it b e c o m e s necessary to pull them out and circle the text for a while before jumping back in, to t h r o w some grit into this book. What I fear is the course in undergraduate analysis that begins by opening this b o o k to Part 1, Chapter 0, Section 1: "A collection of objects is called a set," and then bulldozes through it, t h e o r e m by theorem, until the professor finally loses steam. Such a course would leave a trail s t r e w n with casualties and marked by m o r e relief than enlightenment at its end. REFERENCES

Bartle, Robert G. The Elements of Integration and Lebesgue Measure. John Wiley & Sons. New York. 1966; reprinted in Wiley Classics Library Edition. 1995. Bressoud, David M. Second Year Calculus: From Celestial Mechanics to Special Relativity. Springer-Verlag. New York. 1991. - - - - - . A Radical Approach to Real Analysis. The Mathematical Association of America. Washington, DC. 1994. Hawkins, Thomas. Lebesgue's Theory of Integration: Its Origins and Development, second edition. University of Wisconsin Press. Madison. 1975; reprinted by Chelsea. New York. Lang, Serge. Analysis I. Addison-Wesley. Reading, Mass. 1968. Department of Mathematics and Computer Science Macalester College St. Paul, MN 55105 USA e-mail: [email protected]

Introduction to the Modern Theory of Dynamical Systems by Anatole Katok and Boris Hasselblatt ENCYCLOPEDIAOF MATHEMATICSAND ITS APPLICATIONS, NO. 54 NEW YORK: CAMBRIDGEUNIVERSITYPRESS, 1996, 822 PP. US $39.95, ISBN 0-521-57557-5 pb US $85.00, ISBN 0-521-34187-6 hb REVIEWED B Y ROBERT L. DEVANEY

Ynamical systems mean different things to different people. Perhaps all would agree with the authors' description that a dYnamical system involves a "phase space," whose points represent particular states of the system; a "time variable," perhaps continuous, perhaps discrete; and an "evolution law" that prescribes the state of the system at subsequent times. Beyond this, there are as many definitions of dYnamical systems as there are researchers in the field. Smooth dYnamicists study discrete or continuous systems that are sufficiently differentiable. Topological dynamicists dispense with differentiability and assume only continuity. Ergodic theorists don't worry about continuity, but assume the existence of a measure preserved by the system. Smooth ergodic theorists live in the intersection of all three worlds. Hamiltonian mechanists live here too, but their evolution laws are vastly different. A large group of smooth dYnamicists, the ODE crowd, consider only conlJnuous-time systems. Their world is finite- (usually low-) dimensional, whereas their cousins, the PDE crowd, inhabit infinite-dimensional space. That's a far cry from the 1D-iterators, who study iterations on the real line or the circle. This group would seem to have a natural ally in the complex dynamicists, but the complex people usually look at holomorphic systems while the 1D crowd prefers their systems to be continuous, C 3, or C ~176 but usually not analytic. This is just the beginning of the classification of dYnamicists, and I haven't

D

even mentioned the dynamicists in such fields as physics or engineering (they're usually called nonlinear scientists) or in the behavioral or social sciences (the chaos theorists). As I said, there are thousands of types of dynamical systems, and the scientists and mathematicians who study one type often have little or nothing in comm o n with their next-door dynamicists. So how could one book claim, as the current b o o k does, to be "the first selfcontained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with m o s t of the main areas of mathematics"? Well, the b o o k would certainly have to be long, probably longer than a calculus book. The reader would need to master plenty of prerequisites in order to read it coverto-cover. The authors would, of necessity, have a definite idea of what dynamical systems theory is all about, and they would have to make some choices as to which material to include and/or emphasize. The reader probably could not make it through the b o o k linearly, but would need a map resembling the New York subway system to guide him or her through the various topics and subtopics. Finally, the b o o k would have to be carefully and masterfully written to avoid becoming just a series of specialized results without motivation, proof, or coherence. That, in a nutshell, is exactly what Katok and Hasselblatt have achieved in A n Introduction to the Modern Theory of Dynamical Systems. The book is long, over 800 pages, and the pathways through it are m a n y and varied as the book's interdependency graph shows (see figure 1). The b o o k is destined to b e c o m e a classic compendium of late twentieth-century research in dynamics. It is a must-have for any researcher in the field. I emphasize the word researcher here. The authors claim that some portions of the book can be used by undergraduates or grad students in other disciplines, but these readers would need to have a substantial mathematical background. To assist them, the authors provide eight appendices spread over 40 pages reviewing the necessary background material. These appendices ad-

VOLUME20, NUMBER1, 1998

t~

dress basic topology (including homo- whiz by like an army of topy theory), functional analysis (in- Valkyries (horseshoes, genercluding the Krein-Milman and Choquet icity, bifurcation theory, the theorems), differentiable manifolds stable and unstable manifold (including tensor bundles and transver- theorem, topological entropy, sality theory), differential geometry, linearization, degree theory, topology of surfaces, measure theory, homoclinic points, Nielsen homology theory, and Lie groups. Whew! theory, billiards, minimal geThe book reminds me of Wagner's odesics, to mention just a Ring cycle; not only because there are few). as many chapters as there are hours of Chapters 9-16 form a music in Wagner's operas, but also be- more relaxing collection of F IGURE cause of the rich interweaving of themes specific types of low-dimenF,GUR~ and leitmotifs throughout the book9 sional dynamical systems, inThe book begins tamely enough, in cluding circle maps, twist maps, interDas Rheingold fashion, with a series of val maps, and flows on surfaces. Like diverse, important, and illustrative ex- Siegfried's Rhine journey, the ride here amples (horseshoes, toral automor- is smoother, necessitating fewer anaphisms, shift maps, circle maps)9 We're lytic and topological prerequisites. meeting the Rhinemaidens and the Chapters 17-20 plus a final 50-page gods and other characters that will supplement combine the ergodic and reappear throughout. By the end of hyperbolic theories to study certain chapter two, however, the mathemat- specialized subareas of smooth erics has become significantly more ad- godic theory. I'll resist any analogies to vanced as we encounter Poincar6- G6tterdfimmernng here, but I cannot Siegel linearization, cocycles, and help but wish that the authors had recohomological equations. I had the turned at the end of the book to apply feeling at this juncture that I had made some of the vast machinery built up it to the Rainbow Bridge, but that over these 20 chapters to the beautiful things were not going to go as smoothly examples they began with. The Ring in Valhalla. Chapters 3-5 introduce er- would be back in the R h i n e . . . godic theory and chapters 6-9 deal with A few final comments. There are hyperbolic systems. These are the two some major omissions in this treatise. principal themes of the book9 Chapter Partial differential equations rightfully 5 contains a particularly welcome col- are eliminated, as their treatment would lection of examples of dynamical sys- double the size of the book. Complex dytems that preserve a smooth measure, namics receive little mention, and there including Hamiltonian systems, New- is no discussion of the pathbreaking tonian and Lagrangian dynamics, con- work of Douady, Hubbard, and Sullivan tact systems, and geodesic flows9 on the Mandelbrot set. More curious is Although the hyperbolic systems chap- the fact that the Lorenz system is not disters fill a mere 140 pages, the topics cussed at all. Lorenz's name first appears

on page 760, 4 pages from the end of the book, and we read nothing about Birman, Holmes, and Williams's description of the important Lorenz attractor, or other work on strange attractors. The bibliography is excellent, though perhaps a bit skewed toward the authors' school of dynamicists. For example, fifteen of one of the authors' papers are cited, while the pioneering dynamicist Poincar6 is cited only twice. Smale and Birkhoff fare slightly better (4 and 5 citations), but Yoccoz is mentioned only once, and Yorke, Holmes, and Lorenz's work is not cited at all. As I said, the authors have a definite point of view about what dynamical systems is all about, and without a doubt, their book is a reflection of this. But this does not diminish the authors' achievement: a first-rate text with more than enough dynamics to suit just about anyone's taste. Department of Mathematics Boston University Boston, MA 02215 USA e-mail: [email protected]

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78

THE MATHEMATICALINTELLIGENCER

l--"ie~,,~,~e,,z,]~,i[.~---

Robin

Chinese Mathematics I Raymond Flood and Robin Wilson

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

Wilson

I

hinese m a t h e m a t i c i a n s have app e a r e d on a n u m b e r o f stamps, a n d the ones s h o w n h e r e illustrate m a t h e m a t i c a l developments over many centuries.

C

Zhang

Heng

[Chang

Heng]

( 7 8 - 1 3 9 A D ) , inventor o f t h e seismograph, w a s one o f s e v e r a l Chinese m a t h e m a t i c i a n s w h o d e v o t e d attention to finding the value o f ~-. He p r o p o s e d the value ~ / ~ ( a p p r o x i m a t e l y 3.16), a v a l u e also found by Indian m a t h e m a t i c i a n s a couple of c e n t u r i e s earlier. The Chinese fascination with 7r r e a c h e d its c l i m a x with the w o r k of Z u Changzhi [Tsu Ch'ung-Chih] (429-500), who f o u n d that 7r lies b e t w e e n 3.1415926 a n d 3.1415927; the value given on the s t a m p is the average o f t h e s e bounds. He also d i s c o v e r e d the a p p r o x i m a t i o n 355/113, which also gives ~- c o r r e c t to six d e c i m a l places; this a p p r o x i m a t i o n w a s r e d i s c o v e r e d in E u r o p e over a t h o u s a n d y e a r s later. B o t h o f these s t a m p s a p p e a r e d in 1995. Two 20th-century m a t h e m a t i c i a n s a p p e a r in series of s t a m p s on Chinese scientists, issued in 1988 a n d 1992. T h e y are X i o n g Q i n g l a i , w h o m a d e i m p o r t a n t contributions to m e r o m o r p h i c and entire functions; a n d his student, Hua Loo-Keng [Hua Luogeng] an outstanding n u m b e r - t h e o r i s t - - l e a -

Zhang Heng

THE MATHEMATICAL INTELLIGENCER9 1998 SPRINGERWERLAG NEW YORK

t u r e d in o u r article in vol. 16 (1994), no. 3, 36-46. Raymond Flood Department of Continuing Education Kellogg College Oxford, OX1 2JA UK

Hua

Loo-Keng

Xiong Qinglai

Zu

Changzhi