Letters
to
the
Editor
The Mathematical InteUigencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
T h e B e n e f i t s of N o n - B l i n d n e s s
The U s e s of R e f e r e e s
The opinion piece by "Lemme B. Bourbaki" on blind refereeing (Winter 1999) imputes only evil to giving a referee the identity of the author. But a bias can be used in a positive way. A good illustration was given by Saunders Mac Lane at the time of the AMS experiment with blind refereeing: he pointed out that when he knows that the author is a beginner, he goes out of his way to be helpful. I was happy to read this, especially as I was already doing the same thing. For instance, in my referee's report, under comments for the author, I would tend to write: (1) I feel sure that in the main theorem, you can dispense with the hypothesis that A is normal, and I suggest you spend a month or two looking into this question. (2) The symbol t' that appears in Theorem 3 has a different meaning from the same symbol in Theorem 1. Suggestion: call the new one t*. About the same paper some smart-alec referees might write, "The results tend to be weak and the notation is sometimes inconsistent," or worse, "Weak results, inconsistent notation." In the other direction, if I recognize the author as an established expert in the field, or deduce the fact from the bibliography, then I expect his proofs to be correct, and do not take the time to check every detail. (This shortens the prepublication process, for which both author and editor are grateful-though I get my comeuppance when the editor loses little time sending me another manuscript to referee.)
In her/his opinion piece "On Blindness" published in The Mathematical InteUigencer, vol. 21, no. 1, Lemme B. Bourbaki argues that the referees should not know who the authors of a paper are before making a judgment about its publication. These arguments look irrefutable if we consider the referee as a judge who decides (or helps to decide) whether the paper should be made accessible to mankind or discarded.' However, nowadays e-mall, ftp, and www servers are (in most parts of the world) easier to access than printed journals, so in reality all papers are made accessible. Journals in coming years may have more the role of recommended reading lists. Maybe one should consider a different question: not whether to make the authors invisible, but whether to make the referees visible. If what is really happening is that somebody is recommending papers from the electronic archive that she/he finds interesting, then it seems natural that such a recommendation be signed.
Leonard Gillman Department of Mathematics University of Texas at Austin Austin, TX 78712 USA email:
[email protected] Alexander Shen Institute for Problems of Information Transmission Ermolovoi 19 K-51 Moscow GSP-4, 101447 Russia e-mail:
[email protected] L e m m e B. Bou~baki responds:
I wonder how Professor Gillman et al. recognize which manuscripts are by beginners? Just by unfamih'arity of the authors' names? It seems more likely that they try to recognize the author as a beginner from the content of his or her paper. Blind refereeing would not impair this ability. Perhaps he would appreciate it if referees received manuscripts which
O 1999 SPRrNGER-VERLAG NEW YORK, VOLUME 21, NUMBER 3, 1999
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REFERENCES [1] L~vy-Leblond, J.-M., 1997, Math. Intelligencer 19(4), 63. [2] Bracewell, R.N., 1986, The Fourier Transform and its Applications, 2nd ed. (New York: McGraw-Hill). [3] Glasser, L., 1987, J. Chem. Educ., 64, A228. L. Glasser Centre for Molecular Design Department of Chemistry University of the Witwatersrand Johannesburg South Africa e-mail:
[email protected] -0,5-
-
10
time
20
Figure 1. The Fourier kernel multiplied by an exponential decay, f(~
(with or without authors' names) indicated authors' level of seniority? But whatever the procedure, Professor Gillman's letter is not such as to make a beginner hope to get him as referee. By his own account, he doesn't check all the details of an experienced mathematician's proof, but reserves his mordant scrutiny for the beginner. Exactly. If he didn't understand from my original article why some of us favor blind refereeing, perhaps his own example will make it clearer. Dr Shen's interesting proposal seems independent of the position in my original Opinion piece. Fourier and Argand in 3D J.-M. L~vy-Leblond has recently [1] depicted Fourier series, sums, and integrals in two-dimensional "portraits" and "movies" as Argand diagrams in the complex plane. The "portrait" describes the series as a sequence of vectors at equal angles to one another, so that the Fourier sum is the vector connecting start to end of this chain; the "movie" is obtained by altering each of the equal angles between segments in concert with the time variable, so that the sum vector rotates correspondingly. These "portraits" and "movies" provide such an instructive geometrical view of the processes of Fourier analysis that L~vy-Leblond wonders "why they have not been put forward earlier." In point of fact, diagrams with much
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THE MATHEMATICALINTELLIGENCER
=
exp(-x),
the same motivation have been presented, sometimes in three dimensions [2] with the time axis (the argument) providing the third dimension. I have used such three-dimensional diagrams [3] in attempting to elucidate the relationship between the physical action of spectral analysis and the mathematics of Fourier transformation. This approach shows the Fourier kernel exp(-i2~rxs) as a vector of unit magnitude which, with the passage of time x, coils with a frequency s; plotting time normal to the Argand plane, this appears as a helix. Then if the functionf(x) to be transformed is real, multiplying the kernel by it generates a space curve which is altered in the course of rotating around the time axis (Fig. 1). With a little more effort, the 3-dimensional figures can be displayed stereoscopically (Fig. 2).
Still More Simple and Straightforward? In "The simple and straightforward construction of the regular 257-gon," vol. 21 (1999), no. 1, 31-37, Prof. C. Go~lieb states that "the construction of the regular 257-gon does not seem to have been performed very often" and goes on to cite Richelot's 1832 series of 4 papers dealing with that construction. A very simple construction, without any use of Galois theory, and in some points quite similar to the author's, was described by K. Haage, "Einfache Behandlung der 257-teilung des Kreises," Z. Math. Naturw. Untercicht 41 (1910), 448-458. The method Gottlieb describes for the construction of two numbers given their sum and their product is attributed to the famous 19th-century historian Carlyle by D. W. DeTemple, "Carlyle circles and the Lemoine simplicity of polygon constructions," Amer. Math. Monthly 98 (1991), 97-108, where a different construction of the regular 257-gon is outlined. Victor Pambuccian Department of Mathematics Arizona State University West P.O. Box 37100 Phoenix, AZ 85069-7100 USA e-mail:
[email protected] Figure 2. Stereo pair representation of a periodic signal, showing both the p l o t of the
kernel and the plot o f the kernel multiplied by the signal. From [3], by permission.
Christian Gottlieb replies: I did not mean to claim that nothing had happened since Richelot! I expect that still more references will be communicated to me. Avoiding Galois theory does make the proof more accessible. My inten-
tion w a s that m y p a p e r s h o u l d b e readable b y those unfamiliar with Galois theory. On the o t h e r hand, I b r o u g h t s o m e Galois t h e o r y in b e c a u s e it helps u n d e r s t a n d w h a t is going on. The c o n s t r u c t i o n of t w o n u m b e r s given their s u m and p r o d u c t w a s found first b y Descartes (as I m e n t i o n e d in passing, on p. 33). This w a s an import a n t a c h i e v e m e n t in D e s c a r t e s ' s time, t h o u g h by Carlyle's time it w a s a m e r e exercise. A C o r r e c t i o n T e r m for t h e Biblical ~ ? In the Fall issue of the Intelligencer, G e o r g e C. Bush tries to e x o n e r a t e the a n c i e n t H e b r e w s from having the very c r u d e estimate ~r - 3.
This r e m i n d s m e o f w h a t I l e a r n e d from m y late colleague, P r o f e s s o r Shlomo Breuer, concerning the circ u m f e r e n c e o f the t e m p l e ' s m o l t e n sea, as it is written in the s a m e Biblical passage (1 Kings 7:23) " . . . a n d a line o f thirty cubits did c o m p a s s i t . . . " It is the c u s t o m among the J e w s to use the 22 alphabet letters for numbers: aleph for 1, beth for 2, and so on for 3,4, . . . , 1 0 , 2 0 , . . . , 9 0 , 1 0 0 , . . . , till the last letter, tav, for 400. The w o r d for line, kav, which is 106 in this method, app e a r s in the traditional text with an ext r a letter. This s e e m s to be one of the few h u n d r e d spelling mistakes which were the result of copying the Bible through the ages until getting the canonical version (about the 6th century),
where the traditional reading (qere) differs from the written text (ktiv). However, it m a y be that this particular "spelling mistake" was done on purpose b y a Hebrew scholar who, like G.C. Bush, w a s troubled b y the mathematical mistake. The e x t r a letter has the numerical value 5. Thus, the "correction" is: "relJlace 106 b y 106 + 5." Applying this correction factor to 30/10, one gets 333/106 = 3 . 1 4 1 5 . . . , an excellent rational approximation in its time. Dan Amir School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel e-mail:
[email protected] VOLUME 21, NUMBER 3, 1999
5
EDWARD BERTRAM AND PETER HORAK
Somc Applications of Graph Theocy to Other Parts of Mathematics
any mathematicians are now generally aware of the significance of graph theory as it is applied to other areas of science and even to societal problems. These areas include organic chemistry, solid state physics and statistical mechanics, electrical engineering (communications networks and coding theory), computer science (algorithms and computation), optimization theory, and operations research. The wide scope of these and other applications has been well documented (e.g., [4, 11]). However, not everyone realizes that the powerful combinatorial methods found in graph theory have also been used to prove significant and well-known results in a variety of areas of pure mathematics. Perhaps the best known of these methods are related to a part of graph theory called matching theory. For example, results from this area can be used to prove Dilworth's chain decomposition theorem for fmite partially ordered sets. A well-known application of matching in group theory shows that there is a common set of left and right coset representatives of a subgroup in
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THE MATHEMATICALINTELLIGENCER9 1999SPRINGER-VERLAGNEWYORK
a finite group. Also, the existence of matchings in certain infinite bipartite graphs played an important role in Laczkovich's affirmative answer to Tarski's 1925 problem of whether a circle is piecewise congruent to a square. Other applications of graph theory to pure mathematics may be found scattered throughout the llteraturel Recently, a collection of examples [10] showing the application of a variety of combinatorial ideas to other areas has appeared. There, for example, matching theory is applied to give a very simple constructive proof of the existence of Haar measure on compact topological groups, but the other combinatorial applications do not focus on graph theory. The graph-theoretic applications presented here do not overlap with those in [10], and no attempt has been made at a survey. Rather, we present five examples, from
set theory, n u m b e r theory, algebra, a n d analysis, w h o s e s t a t e m e n t s are well k n o w n o r are easily u n d e r s t o o d b y m a t h e m a t i c i a n s w h o are n o t e x p e r t s in the area. Additional criteria for c h o o s i n g t h e s e five e x a m p l e s w e r e t h a t the s t a t e m e n t can b e f o r m u l a t e d using few definitions and that the p r o o f c a n b e e x p l a i n e d in a relatively s h o r t space, w i t h o u t t o o m u c h technical detail. The p r o o f s h o u l d exhibit the strength a n d elegance o f graph-theoretic m e t h o d s , although, in s o m e cases, one m u s t consult t h e lite r a t u r e in o r d e r to c o m p l e t e t h e proof.
Preliminaries F o r t h e c o n v e n i e n c e of o u r readers, w e recall the n e c e s s a r y definitions from graph theory. An (undirected) graph G = (V, E ) is a p a i r in w h i c h V is a set, the v e r t i c e s o f G, a n d E is a set o f 2-element subsets o f V, the edges of G. An edge e ~ E is d e n o t e d b y e = xy, x a n d y being the e n d v e r t i c e s of e. Here, e is i n c i d e n t with x (and e is incident with y). T h e degree of a v e r t e x v, deg(v), is the n u m b e r of e d g e s i n c i d e n t with v. In a d i r e c t e d graph, o r simply digraph, G = (V, E ) , the (directed) e d g e s a r e o r d e r e d p a i r s of v e r t i c e s o f V and a r e d e n o t e d b y e = (x, y). A t r a i l of length n in a g r a p h G (digraph G) is a s e q u e n c e o f v e r t i c e s Xo, x l , x2, 9 9 9 Xn (xi E V), such that for i = 0, 1, . . . , n - 1, x i x i + l is an e d g e o f G ((xi,xi+l) is an orie n t e d edge o f G). If x0 = x,~, t h e n the trail is said to b e closed. When all the vertices in the sequence are distinct, the trail is called a path. A c l o s e d trail, all of w h o s e vertices a r e distinct e x c e p t for Xo a n d Xn, is called a cycle. A g r a p h G is c o n n e c t e d if a n y t w o vertices of G are j o i n e d by" a p a t h in G. Otherwise, G is said to b e d i s c o n n e c t e d . T h e c o m p o n e n t s of G a r e t h e m a x i m a l c o n n e c t e d subg r a p h s o f G. A tree is a c o n n e c t e d graph w i t h o u t cycles. A g r a p h G = (V, E ) is said to b e b i p a r t i t e if V can b e partit i o n e d into t w o n o n e m p t y s u b s e t s A a n d B such t h a t e a c h edge o f G has one end v e r t e x in A and one end v e r t e x in B. Then, G is also d e n o t e d b y G = (A, B; E). If ( H , . ) is a group and S a s e t of g e n e r a t o r s o f H, n o t n e c e s s a r i l y minimal, the C a y l e y g r a p h G(H, S), o f ( H , . )
with r e s p e c t to S, h a s vertices x,y,... E H, and x y is an edge if a n d only if e i t h e r x = y . a o r y = x . a for s o m e a E S. If G is any g r a p h and e = x y an edge o f G, t h e n b y a cont r a c t i o n along e, w e m e a n t h e graph G ' which arises f r o m G b y identifying t h e vertices x and y ( s e e Fig. 1). We say that a graph G1 is c o n t r a c t i b l e onto a graph G2 if t h e r e is a s e q u e n c e of c o n t r a c t i o n s along e d g e s w h i c h t r a n s f o r m s G1 to G2. T h e a u t o m o r p h i s m g r o u p of a g r a p h G is t h e group o f all p e r m u t a t i o n s p o f the vertices o f G with t h e p r o p e r t y that p ( x ) p ( y ) is a n edge of G iff x y is a n edge o f G. A g r o u p H of p e r m u t a t i o n s acting o n a set V is called s e m i r e g u l a r ff for e a c h x ~ V, t h e stabilizer H x : = {h ~ H I xh = x} consists o f the identity only, w h e r e x h d e n o t e s the image o f x u n d e r h. If H is transitive a n d semiregular, t h e n it is regular.
Cantor-Schr6der-Bemstein Theorem Our first e x a m p l e is a graph-theoretical p r o o f o f the classical result of SchrOder a n d Bernstein. Actually, the theor e m w a s s t a t e d b y Cantor, w h o did n o t give a proof. The t h e o r e m w a s p r o v e d i n d e p e n d e n t l y b y S c h r ~ d e r [1896) a n d Bernstein (1905). The i d e a b e h i n d the p r o o f p r e s e n t e d h e r e can b e f o u n d in [8]. Theorem (Cantor-SchrUder-Bernstein):
L e t A a n d B be sets. I f there i s a n i n j e c t i v e m a p p i n g f: A --->B a n d a n i n j e c t i v e m a p p i n g g: B ~ A, t h e n there i s a b i j e c t i o n f r o m A onto B, t h a t is, A a n d B h a v e the s a m e c a r d i n a l i t Y .
Proof. Without loss of generality, we m a y a s s u m e that A a n d B are disjoint. Define a bipartite graph G = (A, B; E ) , w h e r e x y E E if and only if e i t h e r f ( x ) = y o r g ( y ) = x, x E A, y E B. By o u r hypothesis, 1 -< deg v -< 2 for each v e r t e x v of G. Therefore, each c o m p o n e n t o f G is either a one-way infinite p a t h (i.e., a path of the form x0, Xl, 9 9 9 Xn, 9 9 9 ), o r a twow a y infinite p a t h (of the f o r m . . . X - n , X-n+1, 9 9 9 x - l , Xo, xl, 9 9 9 Xn, 9 9 .), o r a cycle o f even length with m o r e than two vertices, o r a n edge. Note that a finite path of length -->2 cannot be a c o m p o n e n t of G. Hence, t h e r e is in each com-
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ponent a set of edges s u c h that each vertex in the component is incident with precisely one o f these edges. Hence, in each component, the s u b s e t of vertices f r o m A is o f the same cardinality as the s u b s e t of vertices from B. []
C o r o l l a r y . I f J is a subgroup o f a group H, then a n y G(H, S) is contractible onto G(J, T) f o r some set T o f generators o f J. P r o o f . H R , the regular r e p r e s e n t a t i o n of H, acts naturally
Fermat's (Little) T h e o r e m There are m a n y p r o o f s o f F e r m a t ' s Little Theorem, even s h o r t algebraic o r n u m b e r - t h e o r e t i c proofs 9 The first k n o w n p r o o f of the t h e o r e m was given by Euler, in his lett e r of 6 March 1742 to Goldbach. The i d e a o f t h e graphtheoretic one p r e s e n t e d b e l o w can be f o u n d in [5] where this method, t o g e t h e r w i t h s o m e n u m b e r - t h e o r e t i c results, w a s used to p r o v e E u l e r ' s generalization to n o n p r i m e modulus. T h e o r e m (FermaO: L e t p be a p r i m e such that a is not divisible by p. Then, a p a is divisible by p. -
-
Proof. Consider the g r a p h G = (V, E), w h e r e V is the set o f all sequences ( a l , a 2 , 9 9 9 , ap) of natural n u m b e r s bet w e e n 1 a n d a (inclusive), with a i r aj for s o m e i C j . Clearly, V h a s aP - a elements. F o r any u E V, u = (Ul, 9 9 9 Up-l, Up), let us s a y t h a t u v E E j u s t in c a s e v = (Up, Ul, 9 9 9 Up-i). Clearly, e a c h v e r t e x o f G is of d e g r e e 2, so each c o m p o n e n t o f G is a cycle, of length p. But then, the numb e r of c o m p o n e n t s m u s t b e (a p - a)/p, so pla p - a. []
Nielson-Schreier T h e o r e m Let H be a group and S b e a set o f g e n e r a t o r s o f H. Then, a p r o d u c t o f g e n e r a t o r s and their inverses w h i c h equals (the identity) 1 is c a l l e d a trivial relation a m o n g the gene r a t o r s in S if 1 can b e o b t a i n e d from that p r o d u c t by rep e a t e d l y replacing x x -1 o r x - i x b y 1, o t h e r w i s e such a p r o d u c t is called a nontrivial relation 9 A group H is f r e e if H has a set o f g e n e r a t o r s such t h a t all relations among the g e n e r a t o r s a r e trivial. In [1] Babai p r o v e d the N i e l s o n - S c h r e i e r T h e o r e m on s u b g r o u p s of free groups, as well as o t h e r results in diverse areas, from his "Contraction Leinma." The p a r t i c u l a r case of this l e m m a w h e n G is a tree, and its use in p r o v i n g the N i e l s o n - S c h r e i e r Theorem, w a s also o b s e r v e d b y Serre [12, Chap 9 1, Sec. 3]. The p r o o f o f the Contraction L e m m a b e l o w is s o m e w h a t technical, although it uses only t h e i d e a s from group t h e o r y a n d graph t h e o r y w e have a l r e a d y recalled, and is o m i t t e d here 9 C o n t r a c t i o n L e m m a . Let H be a s e m i r e g u l a r subgroup o f the a u t o m o r p h i s m group o f a connected g r a p h G. Then, G is contractible onto s o m e Cayley graph o f H. If H is a group a n d h E H, c o n s i d e r the p e r m u t a t i o n hR o f H o b t a i n e d b y multiplying all the e l e m e n t s o f H on the right b y h. The collection HR = {hR: h ~ H} is a regular group o f p e r m u t a t i o n s ( u n d e r c o m p o s i t i o n ) a n d is called t h e (right) regular p e r m u t a t i o n r e p r e s e n t a t i o n o f H. It is k n o w n [1] t h a t G is a Cayley graph o f t h e group H if and only if G is c o n n e c t e d and H R is a s u b g r o u p of the a u t o m o r p h i s m group o f G.
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as a r e g u l a r p e r m u t a t i o n group o n G(H, S), w h i c h is connected. Thus, the s u b g r o u p of H R c o r r e s p o n d i n g to the ele m e n t s o f J i s a semiregular s u b g r o u p of the a u t o m o r p h i s m group o f G(H, S). N o w apply t h e Contraction Lemma. [] T h e o r e m (Nielson-Schreier): A n y subgroup o f a f r e e group is free. Proof. We first s h o w that in a n y group H a n d for any set S of g e n e r a t o r s of H, the Cayley g r a p h G(H, S) contains a cycle of length > 2 if a n d only if t h e r e is a nontrivial relation among t h e generators in S. To s h o w this, s u p p o s e x0, Xl, 9 9 9 Xn = Xo is a cycle of G(H, S). Then, there are ai E S, 1-i-n, such that X i - l a ~ = x i , w h e r e e i E { 1 , -1}. Hence, X n = X n - l a n ~n ---- X n - 2 a nOn-1 _ 1 a ~ n = "'" _-- xoa e1l a e22 . . . an• n , i.e., the identity 1 = a~l a~2 ... agn. If this w e r e a trivial relation, t h e n there w o u l d exist an integer i, 1 -< i -< n, such that ai = a i + l and ei = - e / + t . However, this implies that xi-1 = xi+t, a contradiction. Similarly, if a~la~2 ... a~n = 1 is a nontrivial relation, then Xo, Xl, 9 9 9 Xn-1, Xn, w h e r e xi = x i - l a ~ , 1 0 such that f o r every x E R, IxI < & M U (M + x) r 0 . Proof. Find a c l o s e d set F a n d an open set G with F C_ M a n d F C G such that 3A(G) < 4A(F) ( w h e r e h is Lebesgue m e a s u r e ) . Since G is a c o u n t a b l e union o f disjoint o p e n intervals, t h e r e is one a m o n g them, say I, such that 3A(/) < 4A(F n I ) . Let 8 = 89 a n d s u p p o s e that Ixl < & Then, I U (x + I) is an interval o f length less than ~h(I) w h i c h contains b o t h F n I a n d x + ( F n I ) . The last two sets c a n n o t b e disjoint, since o t h e r w i s e
~h(I) = ~h(I) + ~A(I) < h((F n I ) u (x + (F n I))) -< ~(z u (x + I ) ) --- ~ ( 1 ) ,
. . . . . .
, 2 n, 2 n - l , "'', 22, 2, 1.
IGURE
f3(x) I
which is a contradiction. Hence, O r (F fq I ) f-) (x + (F fq I ) ) C_ M n (x + M), c o m p l e t i n g the proof.
[]
R e m a r k . It is well k n o w n t h a t a n o u m e a s u r a b l e set cannot b e c o n s t r u c t e d w i t h o u t using the a x i o m o f choice. Our graph T is not connected, and, in fact, e a c h c o m p o n e n t o f T h a s only a c o u n t a b l e n u m b e r o f vertices. Thus, to define A a n d B, w e n e e d to m a k e use o f this axiom.
Sharkovsky's Theorem Let f: R --> R b e a c o n t i n u o u s function. A p o i n t x E R is called a k-periodic p o i n t o f f i f f k ( x ) = X a n d f i ( x ) r x for i = 1, 2 . . . . , k - 1. Here, f n is t h e n t h iterate off, i . e . , f n =
fofn-1. fffhas a k-periodicpoint,is itnecessary thatfhave an m-periodic point for some m r k?
f (X)~
I, f2(x )'
II
x=O
ii
f2(x)
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f (x) 13
f3(x)
VOLUME 21, NUMBER 3, 1999
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FIGURE ;
11
13 of the existence of a periodic point into a problem about the corresponding digraph. Theorem, [14]. I f the digraph associated w i t h a k-periodic point of a f u n c t i o n f has a nonrepetitive dosed trail of length m, then f has an m-periodic point. Figure 4 shows the digraph associated with any 3-periodic point of a function. Clearly, this digraph contains a nonrepetitive closed trail of arbitrary length, showing that the existence of a 3-periodic point o f f implies t h a t f has periodic points of all orders. This special case, and other results on systems with 3-periodic points, were proved in 1975 by Li and Yorke [9], when Sharkovsky's theorem was still little noticed. The reader is referred to Straffin's one-page proof of his theorem above, which is modeled after Li and Yorke's. Straffm's proof makes essential use of two lemmas which are standard in analysis courses: L e m m a 1. Suppose I and J are dosed intervals, f continuous, and J C f ( I ) . Then there is a closed interval Q c I such that f(Q) = J. L e m m a 2. Suppose I is a closed interval, f continuous, and I C f(1). Then f has a f i x e d point in I. Using his theorem above, Straffm proved some parts of Sharkovsky's Theorem, and his approach subsequently allowed several authors to complete the proof (see [3, 6]). In the proof of Sharkovsky's Theorem presented in [2] graphs were used without applying Straffm's result. To give some of the flavor of the proofs in [3, 6] we sketch the proof of a partial result, showing that in the ordering S, all even integers lie after all the odd integers (see [6]). Theorem. I f a continuous function f: R---> R has a point of odd period 2n + 1 (n >- 1), then it has periodic points of all even periods. FIGURE 4
I1
10
THE MATHEMATICAL INTELLIGENCER
Proof (sketch). For n = 1, the proof was given above. Now, suppose n > 1 and assume by way of induction that the theorem is true w h e n e v e r f h a s a point of odd period 2m + 1, where 3 -< 2m + 1 < 2n + 1. Straffm proved generally that the digraph corresponding to a periodic point of period k contains a closed trail of length k in which some vertex is repeated exactly twice. In our case, k = 2n + 1, and this closed trail can, therefore, be decomposed into two closed nonrepetitive trails, one of which has odd length, say 2m + 1 < 2n + 1. If this closed trail is of length greater than one, the assertion follows by our induction assumption and the previous theorem. If not, then Straffm proved that our digraph must contain the directed subgraph given in Fig. 5. This subgraph has a cycle of length 2, and one of length 4. For any even number t > 4, we may begin a nonrepetitive closed trail of length t at the bottom right-hand vertex, traverse the 4-cycle once, and follow this by traversing the 2-cycle exactly (t - 4)/2 times. By the previous theorem, the existence of all even periods follows. [] REFERENCES
1. L. Babai, Some applications of graph contractions, J. Graph Theory 1 (1977), 125-130. 2. L. Block, T. Guckenheimer, M. Misiurewicz, and L.-S. Young, Periodic points and topological entropy of one dimensional maps, in Global Theory of Dynamical Systems, Lecture Notes in Mathematics vol. 819, Springer-Verlag, Berlin (1980), pp. 18-34. 3. U. Burkart, Interval mapping graphs and periodic points of continuous functions, J. Combin. Theory (B) 32 (1982), 57-68. 4. L. Caccetta and K. Vijayan, Applications of graph theory, in Fourteenth Australasian Conference on Combinatorial Mathematics and Computing (Dunedin, 1986); Ars. Combin. 23 (1987), 21-77. 5. K. Heinrich and P. Horn.k, Euler's theorem, Am. Math. Monthly 101 (1994), 260. 6. C.-W. Ho and C. Morris, A graph theoretical proof of Sharkovsky's theorem on the periodic points of continuous functions, Pacific J. Math. 96 (1981), 361-370. 7. W. Imrich, Subgroup theorems and graphs, in Combinatorial Mathematics V, Proceedings of the Fifth Australian Conference, Lecture Notes in Mathematics Vol. 622, Springer-Verlag, Berlin (1977), pp. 1-27. 8. D. K6nig, Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, Leipzig (1936); reprinted by Chelsea, New York (1950).
9. T.-Y. Li and J. A. Yorke, Period three implies chaos, Am. Math. Monthly 82 (1975), 985-992. 10. L. L6vasz, L. Pyber, D. J. A. Welsh, and G. M. Ziegler, Combinatorics in pure mathematics, in Handbook of Combinatorics (R. L. Graham, M. Gr6tschel, and L. Lovasz, eds.), ElsevierScience B.V., Amsterdam, (1996). 11. F. S. Roberts, Graph Theory and Its Applications to the Problems of Society, CBMS-NSF Monograph 29, SIAM Publications, Philadelphia, 1978. 12. J.-P. Serre, Groupes Discretes, Extrait de I'Annuaire du College de France, Paris (1970). 13. A. N. Sharkovsky, Co-existence of the cycles of a continuous mapping of the line into itself, Ukr. Math. Zh. 16 (1964), 60-71 (in Russian). 14. P. D. Straffin, Periodic points of continuous functions, Math. Mag. 51 (1978), 99-105. 15. R. Thomas, A combinatorial construction of a non-measurable set, Am. Math. Monthly 92 (1985), 421-422.
IL'~:--~.~L.-~_'.:..--
Jeremy
Gray,
Sale of the Century?
Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England
12
Editor
I
n October 29, 1998 Christie's in New York auctioned the Archimedes Palimpsest for $2 million, which, with their commission, means that an as-yet-unknown buyer paid $2.2 million for it. What is this text--surely one of the most expensive mathematical manuscripts in existence---and what were the circumstances surrounding its sale? Archimedes (287?-212 BC) was one of the great Greek mathematicians; many would say the greatest. His high status in medieval times helped ensure the transmission of his works down the centuries, and today they fill a book of respectable size. He wrote on many things, but the volumes of pieces of the solids of revolution associated with conic sections was a major theme of his work, as was the study of their centres of gravity and of buoyancy. At the end of the nineteenth century the energetic and meticulous Danish scholar Johan Ludvig Heiberg embarked on a programme of producing the most accurate possible editions of surviving works of Greek mathematics. This involves tracking down all extant manuscripts and early printed books and sorting out the differences in the texts (and there are always some differences) by examining their known histories--not at all an easy task in the days before photocopiers, scanners, and the like. The aim is to determine which manuscripts were copied from others that have survived, and then what lost manuscripts they were ultimately copied from. The next job is to reconstruct the lost manuscripts, to the extent possible. In the case of Archimedes, however, the thread was rather tenuous (the textual basis for Heiberg's edition of Euclid's Elements is richer, for example). The existing manuscripts were all copies or translations of two manuscripts, and both were lost. The first of these was one of two sources used by William of Moerbeke in the early fourteenth century when
O
THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK
he made his medieval Latin translation of a number of the Archimedean treatises. No trace of it is known after 1311. The second of Moerbeke's sources was copied several times during the Italian Renaissance, but it too disappeared, sometime in the 16th century; some of the copies survive. So the discovery of a new text, the Archimedes palimpsest, in 1899 caused real excitement when it was listed in a catalogue of the library of the Metochion of the Holy Sepulchre in Istanbul. Heiberg was able to examine the manuscript in 1906 and 1908, and he published the results of his study in the second edition of his critical text of Archimedes's works (in series of texts published by Teubner, 1910-15). Most excitingly, the new text is independent of the two lost manuscripts. The palimpsest not only gave alternative readings of four mathematical treatises, it included the original Greek text of On Floating Bodies, until then known only from the medieval Latin translation. Better yet, it contained the text of a treatise called the Method of Mechanical Theorems, in which Archimedes explained how he used mechanical means to discover the theorems for which he subsequently provided logical mathematical proofs. This provides exceptional insight into how Archimedes worked, and on the careful distinction he observed between discovery and subsequent proof in mathematics. It is unique among ancient scientific writings for its treatment of methodology. It also contains a fragment of the Stomachion, otherwise only known in Arabic, a treatise on transposing figures. The palimpsest itself carries the Archimedean text, as it was copied in Constantinople in the mid-10th century, on vellum leaves, originally 300 • 200 mm. These leaves had been washed clean in the twelfth century, folded in haft to make a smaller book, and covered with Greek religious texts; the lines of the second script run
p e r p e n d i c u l a r l y a c r o s s t h o s e of the first. Cicero's Republic survives in the s a m e w a y in the Vatican Library. The A r c h i m e d e s p a l i m p s e s t f o r m s a "book" 71/2 b y 6 inches, with 174 p a g e s of text, a n d to r e a d the A r c h i m e d e a n material one m u s t r o t a t e the pages, p~eer bet w e e n the lines, and grapple w i t h the fact t h a t the "Archimedean" o r d e r was s c r a m b l e d in t h e process. T h e r e the m a t t e r might h a v e rested, with a m o d e r n scholarly edition in place, and the p r e c i o u s m a n u s c r i p t h o u s e d in Istanbul. But the t e r r i b l e hist o r y o f t h e 20th century intervened. After the First World War, a brutal struggle e r u p t e d b e t w e e n G r e e c e and Turkey, innocent civilians w e r e mass a c r e d in great numbers, a n d a rich c o s m o p o l i t a n mix of c u l t u r e s a r o u n d the A e g e a n was, if the h i d e o u s m o d e m e u p h e m i s m m a y m a k e t h e s e events m o r e poignant, "ethnically cleansed." The survival o f the library of the M e t o c h i o n could not be assured, n o r i n d e e d could t h a t of the m o n k s . The catalogue of 1890 listed 890 Greek manuscripts, and in 1922 the Greek Orthodox Patriarch of Jerusalem sent a letter to the Exarchus of the Metochion, asking him to send them all secretly to the National Library of Greece for safekeeping. A total of 823 manuscripts w e r e sent, and they are n o w in the Library, the p r o p e r t y of the Patriarchate. The Archimedes palimpsest w a s not one of them. Officially it is one of the 67 that were lost, b u t the m y s t e r y s u r r o u n d i n g its j o u r n e y to New York is p a r t o f the drama. At all events, the m a n u s c r i p t w a s i n a c c e s s i b l e to s c h o l a r s from s o m e time in the 1920s until quite recently. R u m o u r s circulated to the effect t h a t it w a s being privately offered for sale, b u t t h e doubtful legal situation m a d e it i m p o s s i b l e for universities to p r o c e e d . Other r u m o u r s s u g g e s t e d t h a t t h e r e w e r e no legal o b s t a c l e s but t h a t v a r i o u s a t t e m p t s to b u y the manu s c r i p t c a m e to nothing. Meanwhile it s e e m e d m o r e d e e p l y lost t h a n it had b e e n for m a n y years. (In this connection it is interesting that four p a g e s of the m a n u s c r i p t n o w c a r r y icons. These w e r e n o t m e n t i o n e d b y Heiberg, and one o f t h e m is on t o p of a p a g e of the Method. One can only s p e c u l a t e on w h e t h e r these w e r e a d d e d recently,
Page from the Archimedes Palimpsest. 9 Christie's Images, Ltd. 1999.
p e r h a p s w i t h a view to increasing the value of t h e m a n u s c r i p t in a sale.) Christie's, w h o i n f o r m e d the G r e e k G o v e r n m e n t of the f o r t h c o m i n g sale on August 13, argued that the manuscript w a s b o u g h t in the 1920s b y an u n n a m e d F r e n c h m a n . The w a y in which he c a m e b y the m a n u s c r i p t is entirely unclear. The G r e e k Minister for Culture, Evangelos Venizelos, w a s quoted b y t h e Athens News A g e n c y on O c t o b e r 24 as saying that P a t r i a r c h Diodoros of Jerusalem had informed the Minister that t h e r e w a s no r e c o r d o f it ever having b e e n sold. If so, the possibilities w o u l d s e e m to be that the F r e n c h m a n , o r a n o t h e r collector, stole
it, o r t h a t it was illegally s o l d by one o f the monks, p e r h a p s to b u y his w a y out of the raging war. However, the m a t t e r is actually m o r e c o m p l i c a t e d than that. Christie's agreed t h a t p r o o f of theft w o u l d m a k e the sale i m p o s s i b l e a n d ff e s t a b l i s h e d they w o u l d w i t h d r a w it. But, t h e y said in a s t a t e m e n t on Octob e r 26, it was well k n o w n that manuscripts f r o m the M e t o c h i o n h a d left t h e library a p p a r e n t l y quite legally at various times. A p a g e o f the p a l i m p s e s t was acquired b y t h e G e r m a n c o l l e c t o r Tischendorf and on his death sold by his executors to Cambridge University Library in 1876. A r o u n d the turn of the century the Biblioth~que Nationale de
VOLUME 21, NUMBER 3, 1999
13
Page from On Floating Bodies, the Archimedes Palimpsest. 9 Christie's Images, Ltd. 1999.
France acquired another, and two more migrated to America in the 1920s and 1930s. At no time did the Jerusalem Patriarchate protest at their disappearance or allege theft, and it never claimed any of them back. Christie's concluded that they were convinced that "the consignors have proper title and every right to sell," adding, a little obscurely, that they hoped that "Greece, the fountain-head of democracy, Western art and science, will be able to purchase the codex." In fact, Greece itself has no title to the manuscript. At least in 1890 it be-
14
THE MATHEMATICALINTELLIGENCER
longed to the Greek Orthodox Patriarchate of Jerusalem. The Archimedean text was written in a monastery of that order in the tenth century--not, that is to say, byArchimedes. But it could only be the Greek government, and not the Patriarchate, that could afford to buy it if it came up for sale. Felix de Marez Oyens, the head of Christie's books and manuscripts department, held a news conference in Athens on October 26, and the Athens News (27 October 1998) reported him saying, "We hope it will end up in Greece or a great national or university institution else-
where where it can be properly studied and preserved. It would be a great pity for it to go back to private ownership." Sums of over $1 million were by now being talked about, and the Greek government began to look into its pockets. Meanwhile the legal matter went before the courts, and the Ministry of Culture appointed a lawyer, Stavros Dimas, to coordinate legal action. He had successfully opposed the sale in 1993 of 308 artefacts stolen from the Aidonia region near Corinth by antiquity smugglers in the 1970s. An out-ofcourt settlement with the Greek state in that case led to the Michael Ward gallery in New York donating the collection to the "Greek Heritage Protection Association" in Washington. The palimpsest was by now on display at Christie's in New York. It had gone on display on Monday 26 October, and interested parties, not all of them with the money to buy it, could take it out of its case and examine it. Because the binding had been broken open it was now possible to read Archimedean text that had been trapped in the spine when Heiberg saw it, although it seems very likely that his reconstructions of such passages will be found to be valid. There was not much literary licence in a Greek mathematical work. The diagrams are more interesting. Even in the catalogue two diagrams from On Floating Bodies can be seen that differ significantly from Moerbeke's, so they are presumably derived from a common predecessor. Some scholars have begun to wonder ff there are not two versions of On ~ o a t i n g Bodies. Heiberg's diagrams are not from the manuscripts. Late on Wednesday 28 October Judge Kimba Wood, the federal judge hearing the appeal, ruled against the Jerusalem Patriarchate, on the grounds that in French law someone who has had an object for 30 years acquires a permanent right of possession and is therefore entitled to sell it. Indeed, according to some accounts they may already have done so, for rumours circulate that the present owner, who wants to be anonymous, purchased the manuscript from the family of the collector not very long ago. The auction began the next day at
2 p.m. Bidding w e n t quickly from $400,000 to $1 million, with t h e c u r r e n t b i d as usual d i s p l a y e d (in lights) in dollars a n d several o t h e r currencies, including F r e n c h francs, Swiss francs, D e u t s c h m a r k s , pounds, a n d yen. Then it s e e m e d as if the smaller bidde~rs (pres u m e d to be A m e r i c a n universities) w e r e burned off, and the battle was bet w e e n a man at the front with a cordless p h o n e and s o m e o n e at the back. When the bidding reached $2 million, the man at the front asked the auctioneer to walt j u s t one minute while he consulted his client. The auctioneer agreed and the tussle for extra time was r e p e a t e d a few times until he closed the bidding, and the Archimedes palimpsest w e n t to the m a n at the back. This w a s Simon Finch, an u p m a r k e t L o n d o n b o o k d e a l e r . He w a s a s k e d the n a m e o f the n e w o w n e r a n d r e f u s e d to say, b u t he did s a y that the b u y e r und e r s t o o d h o w i m p o r t a n t it w a s for s c h o l a r s to r e a d and i n t e r p r e t the p a l i m p s e s t and that the d o c u m e n t w o u l d b e m a d e available to scholars. There, for the moment, the m a t t e r rests. How, and where, the n e w o w n e r
will m a k e the manuscript available for scholars to consult is n o t known. The palimpsest is our only source for the Method, a n d the only G r e e k source of On l~oating Bodies. Heiberg's scholarship does n o t leave much r o o m for n e w textual discoveries, b u t the whole nature of G r e e k mathematical diagrams is not well understood. It is currently being studied b y Reviel Netz, and the n e w diagrams in the palimpsest will call for a thorough examination. The text is also interesting to philologists: A r c h i m e d e s w r o t e in Doric, not Attic, Greek. Perh a p s m o s t important, it is easily o u r oldest a n d m o s t substantial c o n n e c tion to the writings of one of the greatest of all m a t h e m a t i c i a n s - - t r u l y a maj o r achievement of h u m a n history and one that belongs, somehow, to us all.
Postscript The Baltimore Sun, 11 March 1999, reported that Will Noel, curator of manuscripts and rare b o o k s at the Waiters Art Gallery, and Abigail B. Quandt, senior conservator of mannscripts and rare b o o k s at the Waiters, ~11 be taking care of the Palimpsest. It is on display there
from 20 June to 5 September 1999. Quandt will then conserve it before returning it to its presently still unidentified owner. She will spend a year and a half trying to arrest the deterioration o f the pages and stabilizing them, which involves taking the palimpsest apart, separating the leaves, and dealing with the fungus that affects many pages. Then she will tend the edges, which have b e e n damaged by fire and b e c o m e brittle. When the repairs are done, a decision will be taken about whether to rebind the book. Advanced methods of digital image enhancement will be used to produce the Archimedean text, which m a y well be presented as photographs.
Acknowledgments I w o u l d like to t h a n k David Fowler, Henry Mendell, and Reviel Netz for their helpful-conunents on an earlier draft, and for bringing m e up to date on the Archimedean sources, and Rosalind Mendell for providing an account of the sale. Of course all mistakes that remain are mine. I would also like to thank Christie's for supplying the illustrations of the Palimpsest.
OSMO PEKONEN
he
irst
Bourbak' ? Autograph he story of Andrd Weil's arrest in Finland in 1939 has been often told. The version he gave himself [2] has a certain acceptance, and is repeated f o r example in the obituary in The Times (12 August 1998). For m a n y years none of us, however curious, could do m u c h to fill in the details. No government easily releases records on suspected espionage, and Finland has been especially cautious. In 1991 it became possible---though still difficult--for me to see some of the dossier. The full story, a good deal more dramatic and even more ironic than Weil's own reconstruction, has been published [1] with an afterword by Andr~ Weft himself, confirming that my account is consistent with his recollection. Andr~ and Eveline Weft arrived in Finland 15 June 1939. They made long visits to Lars Ahlfors and Roll Nevanlinna, then a touristic jaunt to the North. Eveline Weil left for
16
THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAG NE'W YORK
France 20 October. Andr6 was arrested 30 November on suspicion of spying for the Soviet Union (which had opened hostilities that very day), and remained in detention until he was expelled 12 December. (The grounds for suspicion were incomprehensible references in his notes to the Poldevian Academy and such, and compromising correspondence such as the letter dated 31 August 1939 from P.S. Alexandrov ending, "I hope your illustrious colleague M. Bourbaki will continue sending me the proofs o f his magisterial work.") Pursuing the records of the Weils' visit, I consulted
SA:
Figure 1. The entry in Rolf Nevanlinna's guest book at Korkee. Reproduced by permission of Arne Nevanlinna.
Arne Nevanlinna, an architect, son of the mathematician Rolf Nevanlinna. He showed me the entry in the guest book for 21 July, which is reproduced here. May this be the earliest extant autograph of the great N. Bourbaki? Notes on documentation. Many records of the S6minaire Bourbaki were burned by Maoists who celebrated the centenary of the Paris Commune by occupying the Ecole Normale Sup6rieure 20-21 March 1971; possibly some early Bourbaki autographs would have been there. The Finnish dossier on Weil has now been opened and is at the National Archives. REFERENCES
1. O. Pekonen, "L'affaire Weil & Helsinki en 1939," Gazette des Math~maticiens 52 (April 1992), 13-20. 2. A. Weil, Souvenirs d'apprentissage, Birkh&user, 1991. Department of Mathematics University of Jyv&skyl& 40351 Jw&skylA Finland Figure 2. Andr6 and Eveline Weil in 1939. Photo courtesy of the National Archives of Finland.
VOLUME 21, NUMBER 3, 1999
17
J. W. NEUBERGER
C'ontinuous Newton's Method for Po ynomia s In the early 1980s, m a n y o f us w e r e s u r p r i s e d to hear of the fractal nature of d o m a i n s of a t t r a c t i o n for N e w t o n ' s m e t h o d a p p l i e d to polynomials. To recall this shock, t a k e p ( z ) = z3 - 1
zEC.
(1)
F o r 1, w, and oJ2, the t h r e e r o o t s o f p, c o l o r a p o i n t z0 E C r e d if {zn}~ = 0 c o n v e r g e s to 1, Zn+l = zn - p(zn)/p'(Zn),
n = 0, 1 , . . . .
(2)
Color z0 blue if {z,~}n=0 converges to ~o, a n d c o l o r it green if {Zn}n=O c o n v e r g e s to ~o2. Any remaining p o i n t gets colo r e d b l a c k [e.g., if p ' ( Z n ) = 0 for s o m e non-negative integer n]. Figures 1 a n d 2 were g e n e r a t e d b y the c o d e "croots.for" b y R o b e r t Renka. Figure 1 s h o w s (with the lightest shading s t a n d i n g for red, the d a r k e s t for green, and t h e i n t e r m e d i a t e shading for blue) d o m a i n s o f attraction for Eq. (2). Figure 2 indicates c o r r e s p o n d i n g d o m a i n s o f a t t r a c t i o n for the modified N e w t o n ' s m e t h o d (for ~ = 89 Zn+l=zn-~*p(Zn)/p'(zn),
n=0,1,....
(3)
As ~ is c h o s e n smaller, one gets c o r r e s p o n d i n g pictures with even s m a l l e r jewels. E n c o u r a g e d b y this, m a t h e m a t i cians naturally divided b y t~ and let ~ --> 0. This is the "continuous N e w t o n ' s method." In its b a s i c form, it consists of finding a function z on a s u b s e t o f R so that z'(t) = -p(z(t))/p'(z(t)),
t E D(z).
(4)
Then, t ---) ~ should give p ( z ( t ) ) --> O. F o r this note, I u s e an i m p r o v e d version o f c o n t i n u o u s N e w t o n ' s method: F o r z0 ~ C, a continuous function z from a / / o f R to C is s o u g h t so that
18
THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK
p ( z ( O ) ) = Zo,
p(z)'(t) = -p(z(t)),
t E R.
(5)
The i m p r o v e m e n t is in t h e handling o f singularities. Solutions o f (5) m a y not b e s o l u t i o n s o f p ' ( z ) z ' = - p ( z ) , for t h e y m a y be such that p ' ( z ( t ) ) = 0 and z ' ( t ) d o e s n o t exist. The analysis o f (5) allows us to sail right t h r o u g h t h e s e singularities. N o w l o o k h o w m u c h b e t t e r c o n t i n u o u s N e w t o n (4) o r (5) d o e s w i t h the e x a m p l e p ( z ) = z 3 - 1. We w o u l d n ' t exp e c t c o n v e r g e n c e starting from the rays 0 = ~r/3, 0 = - ~r/3, 0 = 7r. However, let us start in M, the c o m p l e m e n t o f t h e union o f t h e s e three rays. F o r z0 in one of the t h r e e c o m p o n e n t s o f M and z satisfying (5), u = ~-mt--.~ z ( t ) exists a n d is t h e r o o t o f p in that c o m p o n e n t . These d o m a i n s o f a t t r a c t i o n are j u s t w h a t any right-thinking p e r s o n w o u l d (wrongly) s u s p e c t for Eq. (2). This n o t e gives a r a t h e r c o m p l e t e d e s c r i p t i o n of dom a i n s o f attraction for c o n t i n u o u s N e w t o n ' s m e t h o d for polynomials. Results are e x p r e s s e d in three theorems. We a r e seeking r o o t s o f p, a n o n c o n s t a n t c o m p l e x polynomial. The m e t h o d is to s e e k c o n t i n u o u s functions z f r o m R to C w h i c h solve the differential equation p(z)'(t) = -p(z(t)),
t E R.
(6)
Let Q b e t h e set of such functions. T h e o r e m 1. I f z ~ Q, t h e n u = lim z ( t ) e x i s t s a n d p ( u ) = O.
In short, continuous N e w t o n ' s m e t h o d d o e s find roots! The r a n g e of a m e m b e r of Q will b e called a trajectory.
f
Figure 1. Newton's method for roots of p:
p(z) = z 3 -
1, z E C .
T h e o r e m 2. E v e r y m e m b e r o f C is contained i n s o m e trajectory.
Figure 2. Damped Newton's method for p:
p(z) = z 3 -
1, z E C .
gleaned from these cited papers, in the presellt note I try to give a rather serf-contained treatment of the polynomial case,
A subset G of C is called an incoming trajectory if there are x E C, d ~ R, and z ~ Q so that p ( x ) r O,
p'(x) = O,
z(d) = x,
and
G = z ( ( - % d]).
Such an incoming trajectory is said to end at x. It connects to an outgoing trajectory starting at x, namely z([d, ~)). Denote by M the set of all members of C which belong to no incoming trajectory. T h e o r e m 3. E v e r y component ( m a x i m a l connected subset) o f M contains j u s t one root of p. I f z ~ Q and R ( z ) intersects S, a component o f M, then
u = t---> limoo z(t) is the root o f p w h i c h is i n S. The set of all continuous solutions z to Eq. (6) form a generalized flow in the sense of [1]. This concept helps in organizing situations like the present one, in which uniqueness under initial conditions doesn't hold. I am grateful to John Mayer and Hartje Kriete for leading me to a n u m b e r of recent references to Newton's methods. It is safe to say that any of the authors of [2, 3, 5, 6] could have written the present note had they decided to do so. All of these papers deal with discrete Newton's m e t h o d (3), and most c o n c e r n various aspects of continuous Newton's method as well. In [2], there is a treatment of h o w Julia sets connected with (3) converge to portions of the set M above. In [3, 5, 6] interesting connections between the two Newton's m e t h o d s are given, for polynomials as well as for analytic functions more general than polynomials. Even though some of the present results can be
R e m a r k . I have not treated the root structure of polynomial maps from C k to C k f o r K > 1. Continuous Newton's method might apply, but I expect both formulations and proofs would be different.
Next are some vector field pictures for five examples. These are done with Mathematic& The Mathematica commands for Fig. 3 are p[z_] := z 3 - 1, n[z_] := -p[z]/p'[z], n l [ x _ , y_] := Re[n[x + Iy]] n2[x_, y_,] := Im[n[x + Iy]] PlotVectorField [{nl[x, y], n2[x, y]}, {x, -1.2, 2}, {y, -1.5,1.5}1. Change the first line to get a vector field for another polynomial. Change limits on the last line to examine other regions in C. The reader is invited to sketch in the union of any incoming trajectories in each case. Perhaps the main points of interest in these pictures are the points of attraction, i.e., the roots, and the hyperbolic points. Hyperbolic points are precisely the ends of incoming trajectories. The example in Fig. 3 has three points of attraction and one hyperbolic point. Two of the roots are imaginary. The hyperbolic point is zero, the sole root of p'. It turns out that any incoming trajectory can be continued as a trajectory to converge to any of the three roots. This illustrates some typical behavior. The example in Fig. 4 has three roots (two complex) and two hyperbolic points. Note that either of the incoming trajectories converging to the rightmost root o f p ' can be con-
VOLUME 21, NUMBER 3, 1999
19
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Next is the technique I p r o m i s e d for patching solutions t o g e t h e r (nonuniquely) at a singularity. L e m m a 3. S u p p o s e x E C, p ' ( x ) = 0 a n d p ( x ) r O. There i s ~ > 0 so that i f s E R a n d 0 < Isl < ~, then
~),z~ C.
is a r o o t of p. This says t h e t r a j e c t o r y m u s t shuttle infinitely often b e t w e e n distinct r o o t s of p. Let U b e a union of s e p a r a t e d o p e n disks e a c h containing e x a c t l y one root. Continuity of z obliges it to have infmitely m a n y v a l u e s in C\U. Then it m u s t have ~ l i m i t points there---a contradiction. [] Following a r e s o m e l e m m a s on w h i c h a r g u m e n t s for T h e o r e m s 2 a n d 3 depend. L e m m a 1. S u p p o s e x E C, q E R, and p "(x) r O. There is a m a x i m a l o p e n i n t e r v a l (a, b) c o n t a i n i n g q o n w h i c h there is a f u n c t i o n w so that w(q) = x
and
p ' ( w ( t ) ) r O, p(w)'(t) = -p(w(t)),
t E (a, b).
(10)
Moreover, s u c h a f u n c t i o n w i s u n i q u e o n this m a x i m a l interval.
P r o o f . Note that if (10) holds, then w is differentiable on (a, b) and, consequently, w(q) = x
and p ' ( w ( t ) ) r O, w'(t) = -p(w(t))/p'(w(t)),
(11)
The result follows from classical e x i s t e n c e and uniqueness t h e o r y for o r d i n a r y differential equations. [] L e m m a 2. S u p p o s e x E C, q E R, p ' ( x ) r 0 w i t h w a n d (a, b) as i n (10). I f b < ~, t h e n lim w ( t ) e x i s t s a n d lim p ' ( w ( t ) ) = 0. t---->bt-->bIf a ~ -% then
(12)
lim w ( t ) e x i s t s a n d lim p ' ( w ( t ) ) = 0. t-*a+ t-)a+
(13)
Proof. First, a s s u m e that b < ~. Then, {w(t): t E [q, b)} is b o u n d e d as n o t e d in the a r g u m e n t for T h e o r e m 1. S u p p o s e that
lim w ( t )
t-->b-
i f s > O, there i s f: [0, s] --->C so that f(O) = x a n d p ( f ) ' ( t ) = - p ( f ( t ) ) , t E [0, s],
(14)
and i f s < 0 a n d v = - s , t h e ~ is g: [0, v] --> C so that g(v) = x, p ( g ) ' ( t ) = - p ( g ( t ) ) , t ~ [0, v].
(15)
Proof. Select ro > 0 so that if y E C, 0 < lY - xl < to, t h e n p ( y ) r p ( x ) a n d p ' ( y ) r 0. F o r s n e a r 0, w e will s t u d y Ps, x as in (9). Denote b y x8 a r o o t of Ps,x c h o s e n to minimize IXs - x I. To see w h y
x = lim xs,
(16)
8---->0
factor Ps, x c o m p l e t e l y a n d n o t e that if (16) failed, it c o u l d not be that lim Ps,x(X) = 0;
8--->0
but that is true b e c a u s e Ps, x(X) = p ( x ) - e x p ( - s ) p ( x ) . F o r s r 0 s u c h that Ix8 - x I < ro, t a k e as, bs, and Ws so that ws(O) = xs,
t E (a, b).
t E (a, b);
p(Ws)'(t) = - P ( W s ( t ) ) ,
t E (as, bs),
with as < 0 < bs and (as, bs) m a x i m a l in the s e n s e of (10). Choose r > 0 s o that r < r0. A s s e r t i o n . There is ~ > 0 so that if 0 < s < & t h e n IWs(t) - x I < r
if as < t < O.
To p r o v e this assertion, a s s u m e it is n o t true. Denote b y {Sk}~=O a d e c r e a s i n g sequence of positive n u m b e r s converging to 0 s u c h that if k is a positive integer, there is t ~ (ask, 0) such that IWsk(t) - x I >- r.
F o r e a c h positive integer k, d e n o t e b y tk the largest numb e r t E (ask, 0) so that Iwsk(t) - x I = r,
a n d note t h a t if tk - t -< 0, t h e n Iwsk(t) - x I -< r a n d so p'(Wsk(t)) r 0,
p(Wsk(t)) r p ( x ) .
VOLUME21, NUMBER3, 1999 21
Then for all k, p(Wsk(t)) = e x p ( - t ) p ( x s k ) = e x p ( - ( t + Sk))P(X), ask - % repeat the extension process only going to the left. In any case the e n d result is a function z in Q. C a s e 3. Finally, s u p p o s e that p ( x ) r 0 and p ' ( x ) = O. P i c k f a n d g satisfying (14) and (15), respectively, and defree w o n I - v , s] so that w(0) = x, w ( t ) = g ( t + v) (t ~ [ - v , 0)), w ( t ) = f ( t ) (t ~ (0, s]). Note that p ( w ) ' ( t ) = - p ( w ( t ) ) , t ~ [ - v , s] (there is something to reflect u p o n c o n c e r n i n g the differentiability of
22
THE MATHEMATICALINTELLIGENCER
p ( w ) at 0). Extend w to the left a n d fight as n e e d e d using Lemmas 1, 2, and 3 to arrive at a n extension z ~ Q. [] P r o o f [Theorem 3]. By definitions, M contains every root ofp. Also, note that i f z E Q, S a c o m p o n e n t of M, and R ( z ) intersects S, then R(z) C S, so the root u o f p such that u = l i m t _ ~ z ( t ) must also be in S. It follows that every comp o n e n t S of M contains at least one root of p. Suppose a c o m p o n e n t S of M contains more t h a n one root of p, say ul, 9 9 9 Ub for s o m e integer b > 1. Partition S into S1 . . . . , Sb with S j = { x E S : z E Q, x E R ( z ) ,
u j = l~=z(t)},
j=
l, . . . , b,
and note that no two m e m b e r s o f $ 1 , 9 9 9 S b intersect. Since S is connected, there are integers m, n E { 1 , . . . b} s u c h that v = limk_~= Vk for some v E S m and vl, v2, 9 9 9 ~ Sn. Choose z, zl, z2, 9 9 9 E Q such that z(0) = v a n d Zk(O) = Vk, k = 1, 2 , . . . . Then, lira z ( t ) =Um,
b--~ oo
lim Zk(t) = Un,
t - - ) cr
k = 1, 2 . . . . .
There are tl, t2, . 9 9 E R such that lim Zk(tk) = Um and, consequently, ~
k--~
p(zk(tk)) = O.
However, for fixed positive integer k, lira zk(t) = Un,
t-..)~
~'m~=p ( z k ( t ) ) = O.
Because of the continuity of each of {Z k}k=l and the local c o m p a c t n e s s of C, we arrive at infmitely m a n y roots of p, a contradiction. Thus, S c o n t a i n s only one root of p. [] Using similar arguments b u t without using Lemma 3, one c a n prove the weaker result that through every x with p ( x ) #= 0 and p ' ( x ) r 0 goes a solution of z ' ( t ) -- p ( z ( t ) ) / p ' ( z ( t ) ) converging either to a root of p or to a root of p ' , and that for some x E C, the flint alternative holds. N o w Lemma 3 is the only place in the d e v e l o p m e n t which uses the F u n d a m e n t a l T h e o r e m of Algebra, so one can d e d u c e that result from the o n e j u s t stated.
Ulterior Motive The above development gives a w a y to tag a solution u to p ( u ) = 0 with a region in C, roughly its d o m a i n of attraction in the d e s c e n t process p(z)' = -p(z). This p r o b l e m is analogous to the following. Suppose each of H a n d K is a Hilbert space, F is a C (2) function from H to K, a n d
r
= I~(u)l~2,
u E H.
I have in mind, for example, cases in which H is a Sobolev space of functions on s o m e region in Euclidean space, K is a n L2 space on that region, a n d the p r o b l e m of finding u ~ H so that F(u) = 0
(17)
characterized by a constant struggle for a few crumbs of compactness. In contrast, in arguments for continuous Newton's m e t h o d for polynomials I could relax: there w a s plenty of compactness. I felt like a kid again. REFERENCES
t. J.M. Ball, Continuity properties and global attractors of generalized semifiows and the Navier-Stokes equations, Nonlinear Sci. 7 (1997), 475-502. 2. F. von Haeseler and H. Kriete, The relaxed Newton's method for rational functions. Random Computat. Dynam. 3 (1995), 71-92. 3. H. Jongen, P. Jonker, and F. Twilt, The continuous, desingularized Newton method for meromorphic functions, Acta AppL Math. 13 (1988), 81-121. 4. J.W. Neuberger, Sobolev Gradients and Differential Equations, Springer Lecture Notes in Mathematics Vol. 1670, Springer-Verlag, New York, 1997. 5. H. Peitgen, M. Prufer, and K. Schmitt, Global aspects of the continuous and discrete Newton method: A case study, Acta Appl. Math. 13 (1988), 123-202. 6. D. Saupe, Discrete versus continuous Newton's method: A case study, Acta Appl. Math. 13"(1988), 59-80.
represents a system of partial differential equations with some but perhaps not enough boundary conditions to imply existence of one and only one solution. It is this kind of root-fmding which has been a major focus of attention for me recently [4]. t. 4Y(u)h = (F'(u)h, F(u))K = (h, F'(u)*F(U))H,
u, h ~ H,
where F'(u)* E L(K, H ) is the Hilbert space adjoint of F ' ( u ) , u E H. This leads to a Sobolev gradient V~b for ~b satisfying the identity r
=(h, (Vr
U, h E H,
= F'(u)*F(u), u E H. by taking ( V r Seek u E H such that F ( u ) = 0 by means of continuous steepest descent, i.e., consider z: [0, oo) __) H so that z(O) = x E H,
z ' ( t ) = -(V~b)(z(t)),
t -- 0, (18)
in the hope that u = lim z(t) exists and F ( u ) = 0.
(19)
t---*~
In analogy with continuous Newton's method for polynomials, one says that x, y E H are equivalent relative to (18) provided that they lead, through (18) and (19), to the same element u E H. Granted that the limit exists for each x E H and z in (18), one has H p a r t i t i o n e d in such a way that each leaf in the partition contains precisely one solution. These leaves are analogous to the components of M in the first section above. Numerical, geometric, and algebraic studies of these leaves m a y provide an a p p r o a c h to the general boundary value problem for the system (17). There are some results in this direction in [4]. The study of partial differential equations, however, is
VOLUME 21, NUMBER 3, 1999
23
IIl.[.an~',eli,[~.,r-~l[.-~411-.],~-~--~l
The Hidden Pavements of Michelangelo's Laurentian Library .?_
Jay Kappraff
Does your hometown have any mathematical tourist attractions such
Dirk Huylebrouck,
Editor
since publishing m y b o o k
~nections: the Geometric Bridge between Art and Science [1], I have b e c o m e u s e d to playing t h e role o f a "mathematical tourist." I a m frequently c o n t a c t e d b y r e s e a r c h e r s keen on discussing s o m e d i s c o v e r y t h a t they have m a d e and seeking m y a d v i c e as to t h e m a t h e m a t i c a l a s p e c t o f their work. S o m e t i m e s the w o r k a p p e a r s to s h e d n e w light on ancient o r m o d e r n geometry; other times it s e e m s to lead nowhere. It is in this context that Ben Nicholson telephoned me three years ago. He had b e c o m e privy to a set of facsimiles of fifteen 8'6" • 8'6" pavement d e s i g n s - possibly created by Michelangelo-t h a t lay hidden beneath the floorboards of the Laurentian Library in Florence [2,3]. He was trying to decipher their geometries in order to enable him and an artist, Blake Summers, to reconstruct t h e m at full scale. This project definitely piqued m y interest. One thing led to another, and I soon found myseff a part of Nicholson's team of r e s e a r c h e r s devoted to the study of the p a v e m e n t s - -
I
Blake Summers; Rolf Bagemihl, a n archivist living in Florence; David Krell, a philosopher, ArieUe Salber, a graduate student at Yale specializing in Renaissance history; and Saori Hisano, a graduate student at I]linois Institute o f Technology, the college w h e r e Nicholson teaches. F r o m time to time w e have also s o u g h t the help of o t h e r students o f Nichoison's, Salvatore Camporeale, a F l o r e n t i n e theologian, a n d E r n e s t McClain, a musicologist. I s a w on m y first visit to the studio o f Blake S u m m e r s that he w a s recreating the p a v e m e n t designs with t h e aid of a giant aluminum b a r w h i c h s e r v e d as a compass. S u m m e r s a n d Nicholson w e r e using intuitive geometry very m u c h in the spirit of Boethius, w h o t r a n s l a t e d t h e Elements of Euclid into a language u n d e r s t a n d a b l e to t h e guilds of m a s o n s during Middle Ages. In the p r o c e s s o f analyzing the pavem e n t s w e feel t h a t we have d i s c o v e r e d a t a x o n o m y o f ancient g e o m e t r y t h a t c o m m i n g l e s all of the g e o m e t r i c syst e m s h a n d e d d o w n from antiquity into an i n t e g r a t e d whole. We have identi-
as statues, plaques, graves, the cqfd where the famous conjecture was made,
the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? I f so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42,
Figure 1. Entrance stairway at the Laurentian Library. (Figs. 1-7 are from Firenze Biblioteca
8400 Oostende, Belgium e-mail:
[email protected] Medicea Laurenziana, salone de Michelangelo. By permission of the Minister of Culture. Further reproductions are strictly forbidden.)
24
THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEW YORK
fled six principai geometries upon which the pavements appear to be based: (1) the Vesica Pisces [1,2]; (2) the law of repetition of ratios popularized by the 20th-century designer Jay Hambridge under the name dynamic symmetry [4,5]; (3) the eight-pointed Bnmes star discovered by Tons Brunes, the late Danish engineer [6,7,8];'(4) a set of constructions based on ~ and referred to by Brunes as the sacred cut [2,5,9]; (5) the ad-quadratum squarewithin-a square; and (6) the golden mean [1,5]. Let me summarize what I have learned about this remarkable set of designs and briefly describe the structure of two of them. The Laurentian Library, which was designed by Michelangelo, is situated on the second floor of the San Lorenzo church complex in the heart of Florence. Work on the library was begun in 1523 by Pope Clement VII, alias Guilio Medici, the nephew of Lorenzo di Medici, as a monument to his uncle; it was opened to the public 48 years later by his distant cousin, Grand Duke Cosimo I. The Library was meant to be a home for the books from antiquity that survived to the Renaissance. The modest seeing of the Library leaves one utterly unprepared for what one encounters upon entering. First one is confronted with a massive staircase (Fig. 1) calculated to provoke a numerological trance: There are two steps to get into the building, then series of 3 steps, 7 steps, and 5 steps, with 9 steps to the left and right. After mounting the staircase one enters the Reading Room (Fig. 2). Here the seeming regularity and normalcy hides a frenzy of paradox and ambiguity. Just look at the walls. There is no predominant structure. The wall consists of seven planes, completely disorienting the viewer. In 1774 a portentous accident occurred in the Reading Room of the Laurentian Library. The shelf of desk 74, overladen with books, gave way and broke. In the course of its "repair, workmen found a red and white terracotta pavement which had lain hidden for nearly 200 years beneath the floorboards. The librarian had trapdoors, still operable today, built into the floor, so future generations could view these un-
usual pavements. Further details of the history and significance of the pavements can be found in Nicholson's CDROM, Thinking the Unthinkable House [101. Overall, the pavement consists of two side aisles and a figurative center aisle (Fig. 2). Desks situated on a raised wooden dais have been placed over the pavements. On the side of each desk are listed the books that were to be stored in it. Beneath the desks are a se-
ties of fifteen panels, of different designs, each about 8'6" x 8'6". The fifteen panels along one aisle mirror the ones on the other aisle, but differ in subtle ways. When juxtaposed, the 15 pairs of panels appear to tell a story about the essentials of geometry and number. In 1928 the pavements were photographed for the fLrSt time when the desks were removed temporarily whilst structural repairs were made to the subflooring (Fig. 3).
Figures 2 and 3. The Laurentian Library Reading Room--with and without desks.
VOLUME 21, NUMBER 3, 1999
25
Figure 4. a) The Index Panel 1; and b) the Cross Panel 15. (Figs. 4-7 are details of the pavements.)
The spatial conundrums, paradoxes, and errancies of the building fabric reappear in the geometry of the pavements. We think that the apparent raggedness of the panels can be explicated as accurate and premeditated interpretation of antique geometry in terms of the philosophical concerns of the 1500s. Michelangelo was working with themes well understood at the time, a "secret art of geometry" which could be read in the pavement by the knowledgeable, but which is much more inscrutable today. The books in the Library were organized with the sacred books to the East and the profane to the West, a throwback to the ancient "tree of knowledge". On the East side, tucked behind the projecting entrance door, is a single desk whose books include the Koran, Kaballah, Machiavelli, and books of magic. These subjects escape the tidy categories used to order the Laurentian collection, and they were placed out of sight. The sequence hits its stride with 13 desks containing the works of Italian, Latin, and Greek poets, and continues on through books devoted to the quadrivium (music, astronomy, geometry, and arithmetic), leading to 27 desks loaded with Latin and Greek books of theology, ending with books devoted to the Pentateuch. On the West side of the Reading Room,
~)6
THE MATHEMATICALINTELLIGENCER
the books make a counterpart to those to the East and follow the epistemological form devised by Aristotle. Across from the poets are the texts related to the trivium (granunar, rhetoric, and oratory), then on to logic, medicine, history, ethics, and metaphysics. Aristotle's books were not bound in a single compendium as we might fmd them in a bookstore today, but were found in six different locations in the library, according to the part of his epistemology that they addressed. To traverse the length of the Reading Room could be thought of as a journey through the hill extent of the world's wisdom and knowledge. Monsignior Comporeale feels that movement from Panel 1 (see Fig. 4a) near the door to Panel 15 (see Fig. 4b) on the other side of the Room may have represented the Christian's journey from baptism to enlightenment. Panel 1 consists of octagons and crosses symbolic of baptism, while Panel 15, adjacent to the Pentateuch, also contains crosses and 10 concentric sets of squares surrounding a central square. That the Hebrew Pentateuch is placed here may be deliberate metaphor for the 10-ness that pervades it: the 10 Commandments, 10 generations to Abraham, and 10 more to the Flood. Also Ernest McClain has found that much of the numerology of the Hebrew Bible can be re-
lated to an ancient musical scale based on the first 10 numbers [11]. Let's analyze two of the pavements. Panel 14 is referred to by Nicholson as the Timaens panel. It is composed of four diamonds set within circles that are cut with segments of circles, and the whole design is framed with a white border. In Fig. 5, this panel is shown juxtaposed with a reproduction by Fabbrini, from the circle of Michelangelo, of Michelangelo's system of proportions. Michelangelo felt that the system of proportions developed at the time by Dtirer was inadequate to describe the supple human body, and that his own system better allowed for flexible joints within the body. You will notice that Michelangelo provides a scale on the right subdivided first into 2 parts, then 4 and 8 parts, with each unit further subdivided into 3 parts for a total of 24 equal parts. This is reminiscent of the lambda figure 1
2 4 8
3 6
12
9 18
27
found in Plato's Timaeus and referred to there as the World Soul. This was one of the neo-Platonic ideas brought to the Renaissance by Ficino's academy. It forms the basis of the musical system studied by Pythagoras and writ-
Figure 5. The geometric construction of the Timaeus Panel 14.
t~n about by Nichomachus [12,13,14], and it was used by Alberti as the basis of his system of architectural proportions [5,9]. So we expect the number 27 in Michelangelo's system rather than 24. Sure enough, the head of the model projects higher, the foot projects beneath the floor plane, and an extra unit is intercalated at the hip. Now we have the 27 units of the World Soul.
Figure 6. The Medici Impressa.
The 27 units of Michelangelo's system of proportions can be found in Panel 14. The space from bottom to top
is subdivided into 27 equal parts. However, it is surely deliberate that the seemingly similar space from left to
Figure 7. The Cosimo Panel 2.
VOLUME21, NUMBER3, 1999
27
right has an extra, incommensurate interval. The four circles of the figure echo the coat of arms of Cosimo di Medici, which is also emblazoned on the pavements of the central aisle (Fig. 6). Each bf the 27 traits is exactly 3 soldis in width resulting in an 81-square grid for the entire design. The panel across from this on the other side of the library appears to be based on an 80square grid. Is it coincidental that the ratio 80:81, known in musical parlance as the syntonic comma, is exactly the ratio by v(hich the tones of the ancient scale, attributed to the followers of Pythagoras based o n t h e primes 2 and 3, differs from the Just scale, based on primes 2, 3, and 5? Such conundrums are found over and over in the structure of the pavements. Panel 2, the Medici panel (Fig. 7), is a rosette form typical of many such antique rosette forms that appeared at the time in Florence. The pavement is a rectangle of dimension 12 x 13. Nicholson feels that these numbers are significant as the number of months in the solar and lunar calendars. The small difference between a square and a rectangle is crucial to its construction. 1. In the first step in the construction, the rectangle is extended to a 13 x 13 square concentric with a 12 x 12 square, and the horizontal and vertical axes are placed in the squares. An equilateral triangle is drawn to a side of the 12 x 12 square. The distance from the center of the square to the vertex of the triangle is the radius of a standard circle of the construction called the pitch circle (Fig. 8a). Beginning where the pitch circle cuts the horizontal axis, six circles o f radius equal to the pitch circle are drawn (Fig. 8b). 2. Next six additional circles are drawn beginning where the pitch circle cuts the vertical axis. 3. Twelve additional circles are drawn by repeating steps 2 and 3 for the pair of perpendicular diagonals of the squares resulting in a 24-rosette pattern (Fig. 8c). 4. Twenty-four additional circles are drawn haft-way between the circles of the rosette. These will be widened into the white bands appearing in the
28
THE MATHEMATICAL INTELLIGENCER
panel. For that step, the small difference between the diagonal of the t2 x 13 rectangle and the 13 x 13 square is exploited. Circles with this small difference as radii are drawn where the 24 circles intersect the pitch circle to yield 48 n e w points on the pitch circle (Fig. 8d). 5. Forty-eight additional circles with radius equal to the pitch circle are n o w drawn. These demarcate the white bands of the Medici panel (Fig. 8e).
6. In the final step eight varieties of ellipses are created to fill the diamond shapes. Nicholson's and Summer's reconstruction of panel 2 is s h o w n in Fig. 9. What appeared as a kind of errancy in the deviation of the rectangle from a square exploded into the entire design. Furthermore, in the steps leading to its creation, a series of 3, 6, 12, 24, 48, and 96 circles are created. This is the series that led to the Titius-Bode
Figure 8. Construction of Panel 2. a) A triangle in a square establishes the pitch circle; b) a rosette of six circles; c) a rosette of 24 circles; d) the mismatch of the diagonals of the 12 x 12 square and the 12 x 13 rectangle generates 48 additional circles; e) ninety-six circles create a set of spiral bands in which eight classes of ellipses are placed.
Acknowledgments
As you c a n see, this p r o j e c t has materialized for m e into the ultimate o f m a t h e m a t i c a l tours. I wish to ack n o w l e d g e the fruitful c o l l a b o r a t i o n that I have u n d e r t a k e n with m y colleagues, Ben Nicholson a n d Saori Hisano, that has m a d e this w o r k a g r e a t pleasure. REFERENCES
Figure 9. Nicholson's reconstruction of the Cosimo Panel.
law t h a t p r e d i c t e d the p o s i t i o n s of the p l a n e t s up until Saturn. Again, is this c o i n c i d e n c e o r p r e s c i e n c e ? The pavem e n t mirroring Panel 2 is p l a c e d in an 11 • 12 square p e r h a p s symbolizing the 12 disciples and the 11 disciples o n c e J u d a s w a s excluded. The question begging to b e a s k e d is w h y 30 magnificent p a v e m e n t s w o u l d b e c o n s t r u c t e d and t h e n c o v e r e d up. P e r h a p s s o m e cryptic s y m b o l s w e r e c o n c e a l e d within the p a v e m e n t s in the m a n n e r of U m b e r t o E c o ' s Name of the Rose and then h i d d e n to p r e v e n t their revelation. Nicholson has a m o r e mund a n e hypothesis. When the library was conceived there were approximately 1000 b o o k s in existence. W h e n it w a s c o m p l e t e d , the Library c o n t a i n e d over 3000. The Library had to b e reconfigu r e d to a c c o m m o d a t e the n e w books. In the original plan for the Library, a triangular r o o m w a s s u p p o s e d to have b e e n c o n s t r u c t e d to h o u s e the r a r e s t o f t h e books. However, this r o o m w a s n e v e r built. Unfortunately we do not k n o w who created the pavements. The b o o k s listing the financial transactions in connection with them would m o s t likely have listed the designer, but they have been lost. However, surely Michelangelo w o u l d at least have had a major say concerning such a crucial c o m p o n e n t of his library.
1. Kappraff, J. Connections/ The Geometric Bridge between Art and Science. New York: McGraw-Hill Books. (1991). 2. Nicholson, B., Kappraff, J., and Hisano, S. "A Taxonomy of Ancient Geometry Based on the Hidden Pavements of Michelangelo's Laurentian Library." In Art and Science: The Proceedings of the Second Conference on Art and Science edited by J. Barrallo, San Sebastian, Spain: Univ. of the Basque Country Press (1998), and in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings edited by R. Sarhangi, Arkansas City, KS: Gilliland Publ. (1998). 3. Nicholson, B., Kappraff, J., and Hisano, S. "The Hidden Pavements of the Laurentian Library." In Nexus I1: Architecture and Mathematics edited by K. Williams. Fuccechio, Italy: Edizioni dell'Erba (1998). 4. Hambridge, J. The Elements of Dynamic Symmetry. Originally published by Brentano's (1929) (rpt. By New York: Dover). 5. Edwards, E.B. Pattern and Design with
Dynamic Symmetry. Originally published by Dynamarhythmic Design (1932) (rpt. New York: Dover) 6. Brunes, T. The Secrets of Ancient Geometry and its Use. Copenhagen: Rhodos (1967). 7. Kappraff, J. "A Secret of Ancient Geometry." In Geometry at Work: A Collection of Papers in Applied Geometry edited by K. Gorini. Math. Assoc. of Amer. Notes (In press). 8. Kappraff, J. Mathematics Beyond Measure: A Guided Tour through Nature, Myth, and Number. New York: Plenum Press (In press). 9. Kappraff, J. "Musical Proportions at the Basis of Systems of Architectural Proportion both Ancient and Modern." In Nexus: Architecture and Mathematics edited by K. Williams. Fuccechio: Edizioni dell'Erba (1996). 10. Nicholson, B. "Architecture, Books + G-'~bmetry." In CD-Rom: 'Thinking the Unthinkable House. Renaissance Society at the University of Chicago (1997). 11. McClain, E. "The Star of David as Jewish Harmonical Metaphor." Intemational Journ. of Musicology. Vol. 6, pp. 24-49. (1997). 12. McClain, E. Private communication. 13. McClain, E. The Pythagorean Plato. York Beach, ME: Nicolas-Hays (1978). 14. McClain, E. "Temple Tuning Systems." InternationalJoum. of Musicology, 3 (1994). New Jersey Institute of Technology Newark, NJ 07102 USA
VOLUME 21, NUMBER 3, 1999
29
More or Less Mathematics Wanted in Engineering?
Two images, one of the medieval ceUarium at Fountains Abbey, England, and another of a tiled forecourt in Vernon, France, inspired two engineers to different conclusions. Thefirst author considers that the importance of abstract mathematics is somewhat overrated, and he even makes an appeal to the scientific community to support his view. The second author calls for another kind of intervention: he asks for some mathematical assistance by proposing an unsolved problem to readers. Perhaps engineers and mathematicians will have reactions, outraged or otherwise, to contribute to The Intelligencer's letters.--D.H.
A. Less Mathematics and More Numeracy Wanted in Engineering
Edward Reed To give academic respectability to a trivial piece of engineering research, it is standard practice to add some mathematics to make it appear more significant than it is. Mathematics is the silicon implants of academia. This is not to be confused with numeracy. Numeracy is highly relevant for engineers and unforttmately is often lacking (see Figure 1). Most readers will have looked at the drawing, read the caption and answered the question before even reading the first paragraph. I understand this is a question in Trivial Pursuit, although I cannot confirm it. The answer universally accepted is the Great Wall of China. The real question is, is this true? I ask students, colleagues, and friends this question. Some look it up in books, like Hutchinson's New Century Encyclopaedia (Helicon Publishing Ltd., 1995), or the Readers' Digest Book of Facts. They choose between the supporters of the c o m m o n positive answer, as proposed in the first reference, or the opponents, as in the latter. Few give a reasoned answer. What I am really after is some c o m m o n sense, ap-
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THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK
plied numeracy. We will return to this question later. At school I liked and was good at mathematics. Once the prettiest girl in the class allowed me to ldss her in exchange for letting her copy my maths homework. That is the only time in m y life, and I am a sexagenarian, that I've ever found a use for mathematics. More seriously, it has been my experience that very few engineers use mathematics in practice. They have of course to be numerate. The insistence on mathematics for all engineers has been a damaging deterrent to some highly capable, imaginative, and creative people. Engineering is after all about making things work. The empirical approach, trial and error, the ability to experiment quickly is of much more importance. Most problems cannot be solved by mathematics anyway, or they have to be reduced to too simple a model to be useful. I expressed this view in a rather light-hearted manner in an article published internally within my university. To my surprise, many colleagues said that what I had said needed saying again and again. The professor of mathematics got the vapours, refused to speak to me, and shortly afterwards took early retirement. I also likened mathematics in engineering to the wowing of ladies: as there is supposed to be a lot of it about and since m o s t men find little of it coming their w a y
Figure 1. What is the only man-made object that can be seen on Earth from the Moon?
Figure 2. The medieval cellarium at Fountains Abbey, England (photograph reproduced by kind permission of Yorkshire Post Newspapers).
t h e y believe s o m e b o d y else m u s t b e doing awfully well. After learning m y views on mathematics, m a n y o f m y critics tell m e that t h e y w o u l d n o t like to p a s s over a b r i d g e that I h a d designed. Neither w o u l d I: My a r e a is m e c h a t r o n i c s . However, bridge-building will serve as ~m example. No bridge w a s ever built b y mathematics. They are designed and built b y t e a m s of engineers requiring a v a s t range of skills; the m o r e grandiose the bridge, the bigger the range of
Figure 3. Map of France, iocating Vernon.
course numerate: they n e e d e d to calculate such m a t t e r s as t h e n u m b e r o f eggs required for the m o r t a r (see Figure 2). B a c k to the Great Wall problem. Here is one solution. Put y o u r thumb in front o f you and hold it up to the Moon. The Moon a p p e a r s as a small disc a b o u t one-sixth the size of your thumbnail. By estimation it can b e said that an o b j e c t as far a w a y and as big as the M o o n - which is about 3000 k m across---appears about 3 m m across to the eye. If we estimate that the wall is 5m across, we c a n calculate b y p r o p o r t i o n w h a t thickness it w o u l d a p p e a r to s o m e b o d y on the Moon. The a n s w e r is that it is much t o o small for even the most p o w erful o f telescopes, let alone the n a k e d eye. An often e x p r e s s e d view that the wall c a n be seen b e c a u s e it is over 2000 km long is patently daft. It is h o p e d that the wall e x a m p l e has d~monstrated what is meant b y numeracy.
skills. Bridges are built as something fit for the p u r p o s e at the m o s t economicai price, a n d they have b e e n since time immemorial. The medieval builder h a d only w o o d and stone, b u t he could construct s t o n e arches and k n e w that if a shape, k n o w n to us as a catenary, could b e d r a w n so as to go through every stone, t h e n his arch w o u l d s t a n d up. The great medieval bridges and cathedrals of E u r o p e were built without mathematicians, but the builders had to be imaginative, creative, sldlled, and of
Figure 4. The tiled forecourt in Vernon--is there a pattern?
VOLUME 21, NUMBER 3, 1999
31
As a teacher of engineering, I have problems with students and their numerical skills even if their mathematics is good. I recently did an overseas exchange with another teacher, which i n v o l v e d ' u s taking one another's classes. I found his students similarly poor at numerical work, so maybe this is a universal problem. Numeracy might be boring and mathematics exciting: I confess to having found this so at school. However it is numerical skills that are most lacking in engineering students. Perhaps mathematicians should address themselves to this problem. Leeds MetropolitanUniversity Leeds LS1 3HE United Kingdom
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THE MATHEMATICAL INTELLIGENCER
B. More Mathematics Required to Explain an Ingenious Tiling Pattern in Vernon, France
R.J. Holroyd While on holiday in France I visited Vernon, a small town that stands on the Seine about midway between Paris and the sea. (See figure 3.) About a mile upstream is the small village of Giverny, with the house and gardens where Monet spent the last half of his life. While walking around the town, I came across a recently built arts centre (the Espace Culturel Philippe Auguste) with an intriguingly tiled forecourt. I was told that the workmen had a plan for laying the blocks, but no one could tell me how the plan was produced. The pattern, shown in figure 4, is made
up of five different blocks: 3 X 2, yellow; and 2 x 2 and 2 x 1 in both black and white, but their arrangement appears to be neither regular nor irregular. A mathematician suggested that the pattern might be an example of asymptotic periodicity and referred to me to some algebra textbooks, but so far as I could see these did not shed any light on the problem. Out of my depth about these mathematician's mathematics, apparently so different from the usual engineering computations, I would welcome any enlightenment about this fascinating pattern-or lack of pattern. 104 Arbury Road Cambridge CB4 2JF United Kingdom
AVRAHAM FEINTUCH AND ALEXANDER MARKUS
Tho Toeplitz-I lausdorff Theorem and Robust Stability Theory Introduction In his Hilbert Space Problem Book, Paul Halmos wrote [9],
a convex set? He then states that the result is false for the case n -> 3. Halmos reports these facts as follows [9, p. 110]:
p. 108]: In early studies of Hilbert space (by Hilbert, Hellinger, Toeplitz and others) the objects of chief interest were quadratic forms. Nowadays they play a secondary role. First comes an operator A on a Hilbert space Y~and then, apparently as an afterthought, comes the numerical-valued function f--* (Af, f ) on Y~. This is not to say that the quadratic point of view is dead; it still suggests questions that are interesting with answers that can be useful. As Halmos points out, the object of importance is the numerical range of an operator ("operator" throughout this article means a b o u n d e d linear operator on a c o m p l e x Hilbert space), the set
W(A) = {(Af, f ) : [[f[[ = 1} C C, where ( , ) denotes the inner p r o d u c t on Y~and II [Ithe n o r m induced by it. The fundamental property of this set, k n o w n as the Toeplitz-Hausdorff Theorem, is that it is convex. Toeplitz [13] first proved that the boundary of W(A) is a c o n v e x curve. Hausdorff [10], using a different approach, s h o w e d that, in fact, W(A) is a convex set. He also suggested looking at the theorem in the following way. Write A as A = B + i C , where B = (A+~*) and C = (A-A*) are 2 2i Hermitian operators. Then, the Hermitian forms (Bf, f ) and (Cf, f ) take on real values only~ and the point sets
W(A) C C
and
{((Bf, f ) , (Cf, f ) ) : Ilfll = 1} C R 2
are identical. This suggested to Hausdorff the following possible generalization: If A1..., An are Hermitian operatots, is the set {((Atf, f ) , (A2f, f ) , ..., (Anf, f)): IJfH = 1} C R n
(8)
It is a pity that it is so very false. It is false for n = 3 in dimension 2; counterexamples are easy to come by. The irony is that although false for dimension 2, it turns out to be true for all other dimensions, for n = 3 (for n > 3 Halmos's statement is precise). This was s h o w n by various authors from different points of view ([1], [3], [11]). In fact, the difference between the two-dimensional case and that of higher dimensions was already pointed out in [4]. Also, [5] proves the following result on "quasi-convexity" for the set (*): it contains, with any two points, an ellipsoid (perhaps degenerate) joining them. There are also variations of Hausdorff's suggested generalization which hold for all n (see the section Further Connections). Recently, connections 'have been made between these convexity results and the ideas arising in the theory of robust stability of systems. The purpose of this article is to highlight these relationships. In the next section, we present an elementary p r o o f of the generalized Toeplitz-Hausdorff T h e o r e m for three selfadjoint operators and a connterexample for the case n = 4, which is appropriate for any dimension. The main idea of the p r o o f given here is from [11] and is in the spirit of the original p r o o f of Hausdorff [10]. In order to shorten the proof, we assume a dimension of at least 4. The three-dimensional case needs additional efforts. In the following section, we give a brief discussion of robust stability of linear feedback systems and formulate a central problem. We go on to s h o w h o w the generalized Toeplitz-Hausdorff Theorem is used to solve this problem. The final section gives a brief survey of further generalizations and connections.
9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 3, 1999
We complete this introduction with an exercise, which supplies the counterexample alluded to by Halmos. Let A]:[~
10],
A2=[~-~],
A3=[10
_~].
Show tha~ {((A1 f, f ) , (A2f, f ) , (A3f, f ) ) : Ilfl] = 1} is not convex (it is in fact the unit sphere in R3).
The Joint Numerical Range is a complex Hilbert space and ~ ( ~ ) denotes the bounded linear operators on Y~. The j o i n t n u m e r i c a l range W(A1, A2, A3) of three Hermitian operators A1, A2, A3 ~(Y~) is the set [((Alf, f ) , (A2f~f), (A3f, f)): ]]fil = 1}.
then h ._L A f and h ._L A * f imply (by the above) that h and f are path connected in Z(A). Also, (h, Ag - A ' g ) = 0 implies that I(Ag, h)l = [(Ah, g)[, so h and g are path connected in Z(A). Thus, so are f and g. Assume h ~_ Z(A) or (Ah, h) r 0. We show the existence of a path in Z(A) of the form s(t) = tf + (1 - t)g + u(t)h, where u(t): [0, 1]---) R satisfies u ( 0 ) = u ( l ) = 0. The assumptions on h gave I(Ag, h)l = I(Ah, g)l. Replacing h by ei*h, for s o m e #s E R, if necessary, allows us to assume that (Ag, h) + (Ah, g) = O. This, together with the assumption that h _L A f and h _L_A'f, gives that (As(t), s(t)) = t(1 - t){(Af, g) + (Ag, f ) } + u2(t)(Ah, h). Thus, (As(t), s(t)) = 0 if and only if
We assume that Y~ h a s dimension at least 3.
u2(t ) =
t(1 - t)[(Af, g) + (Ag, J)] (Ah, h)
T h e o r e m 1: W(A1, A2, As) is a convex set i n It 3. For Zl We m a k e the observation that if x, y E W(A1, A2, A3) and t E (0, 1), we can construct an aff'me transformation T on R 3 such that Tx = (0, O, 1 - t), Ty = (0, 0, - t ) , and T(tx + (1 - t)y) = (0, 0, 0). T m a p s the triple (A1, A2, As) to a new triple (B1, B2, B3) of Hermitian operators. Thus we can reduce the p r o o f of the t h e o r e m to showing that if p, /z > 0 with (0, 0, p), (0, O, - t z ) ~ W(A1, A2, A3), then (0, O, O) E W(A1, A2, A3). We give an alternate formulation. Write A E L(YEJ as A = A1 + iA2 and define Z(A) = { f E ~: Hf[I = 1, (Af, f ) = 0}. It is to be shown that if (A3f, f ) takes on positive and negative values on Z,(A), then it takes on the value zero as well. This will follow from showing Z(A) is connected or, equivalently, that Z(A) = {f r 0: (Af, f ) = 0} is. This was proved by Hausdorff [10] forA Hermitian and in [9] for any operator A with Y~ of dimension at least 3 (in case of dimension 2, A = [~
T h e o r e m 2. For A E D~(Y~) w i t h d i m e n s i o n ~. >--4, Z(A) is a connected set.
(Af~ and z ~ = (Ag.f) c h o o s e a real n u m b e r 0 such
that fl -- e i~ Zl + e -i~ z2 4). If h E Z(A),
THE MATHEMATICALINTELLIGENCER
Thus, for [if i[ = 1, ~. (Bkf~f) 2 > 0; SO 0 {~ W(B1, B2, B3, B4). k=l
On the other hand, f = (2 -v2, 2 - l e , 0 , . . . , 0) gives ((Bkf, f))~ = (1, 0, 0, 0), and f = (2 -1/2, - 2 -1/2, 0 , . . . , 0) gives ((Bkf, f))~ = ( - 1 , 0, 0, 0), so that W(B1, B2, B3, B4) is not convex. Robust Stability Theory A m a j o r p u r p o s e of control engineering is that of regulation: external disturbances act on a physical system, and one m u s t design m e c h a n i s m s that keep certain to-be-controlled variables within certain bounds. The temperature in houses is regulated by a t h e r m o s t a t so that the inside t e m p e r a t u r e remains within certain bounds. This thermostat illustrates a central concept of control, feedback. The value of one variable in the system is m e a s u r e d and is "fed back" in order to take appropriate action through a con-
trol variable at a n o t h e r p o i n t in the system. The t h e r m o s t a t s e n s e s the r o o m t e m p e r a t u r e , c o m p a r e s it with t h e des i r e d temperature, and f e e d s b a c k the result to the furnace, w h i c h then starts or shuts off, in o r d e r to raise o r l o w e r t h e t e m p e r a t u r e so that it will c o m p l y with w h a t is desired. We p r e s e n t a s t a n d a r d m a t h e m a t i c a l m o d e l for a linear f e e d b a c k system. The signals are m o d e l e d by v e c t o r s in a Hilbert s p a c e ~ and the s y s t e m s are linear o p e r a t o r s on Y~. Stable linear s y s t e m s c o r r e s p o n d (at this stage) to b o u n d e d linear o p e r a t o r s on Y~, and instability m e a n s that the relevant o p e r a t o r is n o t bounded. A f e e d b a c k s y s t e m is d e s c r i b e d b y the equations u
e
~
y
l y = Me, e = u + Ay.
Now, in o r d e r for the d e s c r i p t i o n to m a k e sense, the outp u t signal m u s t d e p e n d on the external input signal u, so it m u s t b e p o s s i b l e to solve t h e s e equations to o b t a i n y in t e r m s o f u. In o r d e r to do this, one m u s t b e able to invert I - AM, and if that is possible, w e obtain y = M ( I A M ) - l u . If M and A are s t a b l e and ( I - ~ ) - 1 is stable, w e s a y that the s y s t e m is internally stable. In the last t w o decades, c o n t r o l engineers have b e e n c o n c e r n e d with the fact t h a t modeling e r r o r s c a n highly d i s r u p t conclusions a b o u t stability of f e e d b a c k systems. T h e y have d e v e l o p e d a t h e o r y o f "robust stability": constructing f e e d b a c k s y s t e m s t h a t will r e m a i n stable in the p r e s e n c e of a p r e s c r i b e d class o f m o d e l i n g errors. One w a y of doing this is b y taking M to r e p r e s e n t a k n o w n stable s y s t e m and letting h r e p r e s e n t an uncertainty system which is k n o w n only to belong to a certain o p e r a t o r ball, s a y IIAII < 1. In this case, it is i m m e d i a t e that if NMII -< 1, then the given f e e d b a c k system is stable, for then IIAMII < 1 a n d ( I AM) -1 is given by the norm-convergent geometric series cc
~ . (AM) n. In fact, this c o n d i t i o n on M is also necessary. n=0
If IIMII > 1, t h e r e exists A with IIA]] < 1 such t h a t / - A M i s n o t invertible. This is s e e n as follows: let p = IIMII > 1 a n d T = (1/p2)M *. Then, IITII < 1 and I - T M = ( 1 / p 2 ) ( p 2 I M ' M ) . Because M * M is a non-negative o p e r a t o r a n d p2 -IIMH2 = HM*MII 9 o-(M*M) (the s p e c t r u m o f M ' M ) , I - T M is n o t invertible. However, w e naturally w a n t the t h e o r y to treat the situ a t i o n w h e n t h e r e are multiple inputs and outputs e a c h giving rise to its o w n uncertainties. Then M is an n • n opera t o r matrix, a n d the u n c e r t a i n t i e s m o d e l e d by h arise as an n X n diagonal o p e r a t o r matrix. Even for n -- 2, this s e t u p is challenging. In o r d e r to d i s c u s s this situation, we i n t r o d u c e s o m e term i n o l o g y and notation. The s t r u c t u r e d n o r m ( i n t r o d u c e d b y Doyle [6] and Safonov [12]) o f A 9 ~(Y~), relative to a given s u b a l g e b r a ~ C ~s with identity, is the n u m b e r
p~(A)
1
inf{llTIl: T 9 ~ , I - TA n o t invertible}"
This n u m b e r h a s p r o v e d t o b e a p o w e r f u l tool in the s t u d y o f r o b u s t stability. ~ gives a m e a s u r e of stability with res p e c t to the t y p e s of uncertainties d e s c r i b e d above. Using this term, w e c a n reformulate our p r e v i o u s d i s c u s s i o n a s the formula/z:~(vO (M) = NMII. It is easily s e e n that if ~ 1 C_ ~2, then t z ~ ( A ) - / ~ 2 ( A ) , and that for 50 = {M: A 9 C}, Ix~r(A) = p(A), the s p e c t r a l radius o f A. Thus, in general, p(A) Of o r k = 1, 2, 3, then there e x i s t n o n - n e g a t i v e n u m b e r s ~1, a2, a n d a3 s u c h that at + ol2 Jr ol3 > 0 a n d atA~ + c~2A2 + a3A 3 --< 0. Proof. Let S+ = {<xl, x2, x3> E R3: Xk > 0, k = 1, 2, 3}. By hypothesis, S+ A W(A1, A2, A3) -- ~ . Since W(A1, A2, A3) is convex, t h e r e exists a p l a n e alXl + a2x2 + aax3 = 0 in R 3 such t h a t W(AI, A2, A3) C {<Xl, x2, x3>: a l x t + a2x2 + c~3x3 --< 0}
and S + C [(Xl, x2, x3>: o/ix 1 -F o/2x 2 + o~3x3 ~ 0}.
VOLUME 21, NUMBER 3, 1999
The first of these conditions gives alA1 + a~A2 + a~A~ 0, k = 1, 2, 3. []
should be mentioned that in [7] computational algorithms for finding the distance from the origin to the convex set W(A1, A2, A3) are used to compute/~3(A).
T h e o r e m [8]. For Further Connections
I
-All A12 A131 A = A21 A22 A23/ LA31 A32 A33J
with Aij E ~(Y0,/~3(A) =/23(A). Proof. The diagonal algebra ~3 has a commutator (~ which is all scalar diagonal 3 x 3 matrices. For At, h2, h3 :# 0, define
.o11, 0 ,] i ,0 ]
AA =
h2 ]I
L0
00
A
h21
0
9
0
Since /~3(A) - 0, as in Figure 6. Using the f o r m u l a 0r2/2 for the area o f a s e c t o r o f angle 0 a n d radius r, each shell h a s a r e a (O - O)rh + R(r,h), w h e r e the e r r o r R(r,h) is less than the area of the small shell e l e m e n t o f a r e a (r + h)h~q(r + h) - g(r)), s e e Figure 6, a n d this is less than Crh 2, for s o m e c o n s t a n t C, a s s u m ing t h a t g(r) is well behaved. It follows that, ignoring t e r m s of o r d e r h 2, the a r e a o f e a c h shell is (O - O)rh, w h i c h is the length of the bottom a r c o f the shell multiplied b y h. This s h o w s w h y the first p a r t o f A s s u m p t i o n 2 holds. All these shells have a r e a a linear function of h up to an e r r o r t e r m o f l o w e r order, a n d form a disjoint union of ~ , w h i c h s h o w s w h y the s e c o n d p a r t o f A s s u m p t i o n 2 holds. Letting h --> 0, it follows that the area o f ~ is
j
:
rR (O - O)rdr = Jo [g(R) - g(r)]rdr.
The s t a n d a r d derivation of this f o r m u l a uses the f o r m u l a rdrdO for the area e l e m e n t in polar c o o r d i n a t e s A r e a o f ~ = f f d x d y = f f rdrdO =
s;s; =0
rdOdr =
=g(r)
s;
[g(R) - g(r)lrdr.
A circle is simply g(r) = 0, w h i c h yields 21r f R r d r = 7rR29 A spiral, in p o l a r c o o r d i n a t e s , is given b y the equation r = aO, for s o m e constant a, which can b e written as O = kr, w h e r e k = 1/a. By the above, the area o f the spiral is (kR - k r ) r d r = k R
rdr - k kR 3 2
r2dr = k
R3 3
kR 3 1 OR 2 - -- 6 3 2 '
w h e r e the t e r m on the right is seen to be 1/3 the a r e a of the s e c t o r of the circle o f r a d i u s R and angle O, yielding Proposition 2. A n y p r o o f of this f o r m u l a is equivalent to evaluating s u c h integrals. A r c h i m e d e s e v a l u a t e d fR r2dr b y d e c o m p o s i n g it into Riemann s u m s a n d obtaining a c l o s e d form for the s u m 12 + ... + n 2. In P r o p o s i t i o n 2 this integral is c o m p u t e d b y realizing it as t h e m o m e n t of a triangle and evaluating this as its weight multiplied b y the d i s t a n c e o f its c e n t e r of gravity from t h e fulcrum.
The Way of Archimedes The Calculus Reform m o v e m e n t has e m p h a s i z e d experim e n t a t i o n over rigor in calculus e d u c a t i o n and h a s b e e n criticized as a result [53]9 To d e f e n d its p o s i t i o n t h a t physical p r o b l e m s should b e u s e d t o d i s c o v e r m a t h e m a t i c a l results, Harvard Calculus a p p e a l s to A r c h i m e d e s a n d The Method [35, p. vii]: T h e W a y o f A r c h i m e d e s : F o r m a l d e f i n i t i o n s a n d procedures evolve f r o m the i n v e s t i g a t i o n o f p r a c t i c a l problems.
This principle a c c u r a t e l y r e p r e s e n t s the w o r k s o f Archimedes, b u t a disparity arises in t h a t H a r v a r d Calculus p o s t p o n e s m a t h e m a t i c a l rigor indefinitely; A r c h i m e d e s ' s name should not b e a s s o c i a t e d with s u c h an endeavor. F o r example, the m e t h o d of e x h a u s t i o n u s e d by A r c h i m e d e s is essentially t h e E-ti a r g u m e n t a b a n d o n e d b y H a r v a r d Calculus, as B.L. van der W a e r d e n w r i t e s [58, p. 220]: 9 . . the e s t i m a t i o n s , w h i c h occur i n the s u m m i n g o f i n f i n i t e series a n d i n l i m i t operations, the 'epsilontics', as the calculation w i t h a n a r b i t r a r y s m a l l E i s s o m e t i m e s called, w e r e f o r A r c h i m e d e s an open book. I n this respect, h i s t h i n k i n g i s entirely modern. Moreover, A r c h i m e d e s h e l d in c o n t e m p t t h o s e w h o did not furnish p r o o f s o f their results. In t h e i n t r o d u c t i o n to O n Spirals, A r c h i m e d e s r e v e a l s that he intentionally ann o u n c e d false t h e o r e m s in o r d e r to e x p o s e s o m e of his cont e m p o r a r i e s [6]: 9 I w i s h n o w to p u t t h e m i n r e v i e w one by one, p a r ticularly as i t h a p p e n s t h a t - t h e r e are two a?nong t h e m w h i c h [are w r o n g and w h i c h m a y serve as a w a r n i n g to] those w h o c l a i m to discover e v e r y t h i n g but produce no proofs o f the s a m e m a y be confuted as hading actually pretended to discover the impossible. Harvard Calculus fails m i s e r a b l y w h e n m e a s u r e d against this Way of Archimedes. A p a r t from the p a s s a g e quoted above, the w o r d "theorem" a p p e a r s in [35] only in the n a m e " F u n d a m e n t a l T h e o r e m o f Calculus." C o m p a r e this with a s t a n d a r d calculus t e x t [22], which lists 130 t h e o r e m s in its index. Even m o r e revealing, the only instance o f the w o r d "proof" I l o c a t e d in [35] w a s in A r c h i m e d e s ' s i n t r o d u c t i o n to the m e t h o d q u o t e d above and u s e d in [35] to justify "The Way of Archimedes." In fact, this quote e m p h a s i z e s that d i s c o v e r y of t h e a n s w e r to a p r o b l e m leads to a t h e o r e m w h o s e p r o o f is facilitated b y k n o w l e d g e of the answer. My i n t e r p r e t a t i o n is n o t Calculus Reform b u t P r o b l e m - S o l v i n g : W h e n faced w i t h a p r o b l e m , use a n y m e t h o d t h a t allows y o u to c o n j e c t u r e the answer, t h e n find a r i g o r o u s proof. A r e c e n t d e v e l o p m e n t : The s e c o n d edition o f [35] h a s t a k e n a m o r e m o d e r a t e a p p r o a c h to Calculus Reform and n o w includes s o m e c o m p l e t e p r o o f s [35, 2nd Edition, p. 78] and the e-8 definition of a limit [35, 2nd Edition, p. 128]. However, this n e w edition no longer includes "The Way o f Archimedes."
Popular Misconceptions It m u s t be noted that the p e n u l t i m a t e r e m a r k o f the previous s e c t i o n p a r a p h r a s e s E.T. Bell [11, p. 31]: "In s h o r t he u s e d m e c h a n i c s to a d v a n c e his m a t h e m a t i c s . This is one o f his titles to a m o d e m mind: he used a n y t h i n g a n d everything that suggested i t s e l f as a w e a p o n to attack h i s problems." However, strong opinions s u c h a s t h o s e e x p r e s s e d
VOLUME 21, NUMBER 3, 1999
43
in [11] are fraught with danger, and it is instructive to include the continuation of this passage: To a modern all is f a i r in war, love, and mathematics; to m a n y of the ancients, mathematics was a stultified game to be played according to the p r i m rules imposed by the philosophically-minded Plato. According to Plato only a straightedge and a p a i r of compasses were to be permitted as the implements of construction in geometry. No wonder the classical geometers hammered their heads f o r centuries against 'the three problems of antiquity': to trisect an angle; to construct a cube having double the volume of a given cube; to construct a square equal to a circle.
This has~ since been discredited, see [24] [41] (better yet, look at original sources, e.g., as collected in [54, Vol. 1, Chapter 9]); and van der Waerden writes [58, p. 263], The idea, sometimes expressed, that the Greeks only permitted constructions by means of compasses and straight edge, is inadmissible 9 It is contradicted by the numerous constructions, which have been handed down, f o r the duplication o f the cube and the trisection of the angle9
In particular, Archimedes trisected the angle with rifler and compass in Proposition 8 of The Book of L e m m a s [6, p. 309], see [20] [31, Section 31]. The history of this misconception might prove an interesting subject for further study. Unfortunately, it is only one of a number of popular misconceptions about the limitations of Greek science [56]. For example, Isaac Asimov (1920-1992) has written [5], To the Greeks, experimentation seemed irrelevant. It interfered with and detracted f r o m the beauty of pure ded u c t i o n . . . To test a perfect theory with imperfect instruments did not impress the Greek philosophers as a valid w a y to gain k n o w l e d g e . . . The Greek rationalization for the "cult of uselessness" m a y similarly have been based on a feeling that to allow m u n d a n e knowledge (such as the distance f r o m Athens to Corinth) to intrude on abstract thought was to allow imperfection to enter the Eden of true philosophy. Whatever the rationalization, the Greek thinkers were severely limited by their attitude9 Greece was not barren of practical contributions to civilization, but even its great engineer, Archimedes of Syracuse, refused to write about his inventions and discoveries . . . to m a i n t a i n his amateur status, he broadcast only his achievements in pure mathematics.
This passage is contradicted by numerous examples of Greek scientific experiments, for example, Eratosthenes's measurement of the earth [4]9 Asimov may be excused for paraphrasing Plutarch's account of Archimedes in his Life of Marcellus, written circa 75 AD [49] [54, Vol. 2, p. 31]: Yet Archimedes possessed so lofty a spirit, so profound a soul, and such a wealth of scientific inquiry, that al-
44
THE MATHEMATICALINTELLIGENCER
though he had acquired through his inventions a n a m e and reputation f o r divine rather than h u m a n intelligence, he would not deign to leave behind a single writing on such subjects. Regarding the business of mechanics and every utilitarian art as ignoble or vulgar, he gave his zealous devotion only to those subjects whose elegance and subtlety are untrammeled by the necessities of life
Despite Plutarch's ancient credentials, he had no better insight into Archimedes's scientific contribution, which contradict his story. The reader is already aware that The Method shows that physical considerations played an important role in Greek mathematics. But Asimov and Plutarch are completely refuted by Archimedes in The Sand Reckoner [6] [ 18]: While examining this question I have, f o r m y part tried in the following manner, to show with the aid of instruments, the angle subtended by the sun, having its vertex at the eye. Clearly, the exact evaluation of this angle is not easy since neither vision, hands, nor the instruments required to measure this angle are reliable enough to measure it precisely. But this does not seem to me to be the place to discuss this question at length, especially because observations of this type have often been reported. For the purposes of m y proposition, it suffices to f i n d an angle that is not greater than the angle subtended at the sun with vertex at the eye and to then f i n d another angle which is not less than the angle subtended by the s u n with vertex at the eye. A long ruler having been placed on a vertical stand placed in the direction the rising sun is seen, a little cylinder was put vertically on the ruler immediately after sunrise. The sun, being at the horizon, can be looked at directly, and the ruler is oriented towards the sun and the eye placed at the end of the ruler. The cylinder being placed between the sun and the eye, occludes the sun. The cylinder is then moved further away f r o m the eye and as soon as a small piece of the s u n begins to show itself f r o m each side of the cylinder, it is fixed. I f the eye were really to see f r o m one point, tangents to the cylinder produced f r o m the end of the ruler where the eye was placed would make an angle less than the angle subtended by the sun with vertex at the eye. But since the eyes do not see f r o m a unique point, but f r o m a certain size, one takes a certain size, of round shape, not smaller than the eye and one places it at the extremity of the ruler where the eye was p l a c e d . . , the width of cylinders producing this effect is not smaller than the dimensions of the eye. 9 It is therefore clear that the angle subtended by the sun w i t h vertex at the eye is also smaller than the one hundred and sixty fourth part of a right angle, and greater than the two hundredth part of a right angle.
The correct value of the angular diameter of the sun is now known to average about 34' [26, p. 95], i9 the 159th part of a right angle. It is important to note that this shows not
:IGURE
:
:IGURE ,
m
only that ancient Greeks frequently performed experiments, but that Archimedes dealt with experimental error and also compensated for the fact that the human eye is part of the observational instrument, thus anticipating scientists such as Hermann von Helmoltz (1821-1894) [34]. A translation and analysis of The Sand Reckoner is given in [56]. A n s w e r s to E x e r c i s e s E x e r c i s e 1. A naive approach leads to incorrect results, evidence of the dangers of using infinitesimals, and indicating why Archimedes did not consider his method to be rigorous. For example, taking the radii of a circle of radius R, with respect to the circumference, and reordering them to form a rectangle, yields area 2qrR2. For a general figure, it's not even clear how to pick the radii. To make sense of what is going on, one regards radii as limits of sectors, i.e., infinitesimal triangles. In the case of the circle, this means that the weight of a radius, with respect to the circumference, is equal to one half its length. This can be loosely interpreted as the argument Archimedes used to compute the area of the circle [1]. In the general case, the following is justified: A s s u m p t i o n 3. The weight of a radius is proportional to the square of its length. In modern notation, this is simply i f rdrdO = f :
=0
rf(~
rdrdO = l f : [f ( O)]2dO,
Jr=0
where the radii have been chosen with respect to the unit circle. Given Assumption 3, one can compute the area of the spiral by using Pappus's argument [48, Book 4, Proposition 21], see also [32, p. 377] [41, p. 162]. To compute the weight of a spiral region, take each radius of the spiral, starting from the fmal radius, and place a disk with diameter equal to this radius at height the current angle so the resulting figure is a cone. Similarly, for each radius of the sector place a disk with diameter equal to this radius at height the current angle, resulting in a cylinder with the same base and height as the cone. Since Euclid's Proposition 2 6f Book 12 proves that "circles are to one another as the squares on the diameter," Assumption 3 shows that the ratio of the weight of the spiral region to the weight of the sector is the same as the ratio of the volume of the cone to the volume of the cylinder. But Euclid's Proposition 10 of Book 12 proved that the volume of a cone is one third the cylinder with the same base
and height, so the spiral weighs one third of the sector, which is the statement of Proposition 2. (Note that equilateral triangles could have been used instead of circles resulting in a pyramid whose volume is easier to compute.) Knorr [40] comments that this appeal to three-dimensional figures might have been considered inelegant by Archimedes as it uses volumes to compute areas. On the other hand, reversing this argument and using the evaluation above shows that the volume of a cone can be computed by the mechanical method, a result which does not appear in The Method. E x e r c i s e 2. In modern notation, Archimedes'$formulation of Proposition 1 is Area of circle of radius R = f [ 2~'rdr, for the integral represents the area of a right triangle with base R and height 2~rR. E x e r c i s e 3. This is equivalent to the fact that the length of an arc of fixed angle is proportional to its radius. In particular, ~r exists, see [45] [56]. The proof is similar to [23, Book 12, Proposition 2] cited in Exercise 1, and is implicit in Archimedes's M e a s u r e m e n t o f the Circle. Similarly, the length of an arc of fixed radius is proportional to its angle. E x e r c i s e 4. By analogy with Assumption 2, consider a sphere as being composed of spherical shells centered at the center of the sphere, where each shell weighs the same as a circle of equal area. The justification follows exactly as in Proposition 2: Consider two pans suspended at equal distances from the fulcrum of a balance. On one pan, place a sphere of center A and radius AB and on the other a line CD of length equal to AB. For each E on AB there is a spherical shell passing through E, and consider a circle of area equal to this spherical shell with center at F lying on CD, where CF equals AE, and such that the circle is perpendicular to CD. The resulting figure is a cone with base the area of the sphere and height the radius of the sphere; since it balances the sphere, the claim is justified. The similarity of this argument to the one of Proposition 1 suggests that Archimedes may have been implicitly aware of the ideas of this paper. Moreover, the reader may verify that the heuristic of this exercise and its justification directly generalize to higher dimensions (a different generalization is given in [19]): P r o p o s i t i o n 3. The volume o f a n n - d i m e n s i o n a l ball is equal to the volume o f a cone whose base has n - 1-
VOLUME 21, NUMBER 3, 1999
REFERENCES
dimensional volume equal to the (n - 1)-dimensional volume of the boundary of the ball and height equal to the radius of the ball. E x e r c i s e 5. The procedure, when applied to the spiral, yields a section of a parabola. The general formula for such areas was computed by Archimedes in The Quadrature of the Parabola, and in this case it states that the resulting area is four-thirds the triangle with same base and height as the section of the parabola. Since the height and base are equal to the final radius and half the final radius, respectively, Proposition 2 follows. E x e r c i s e 6. Further extensions of Archimedes's method could be a subject for investigation. As Archimedes wrote in The Method [6, Supplement, p.13], I deem it necessary to expound the method partly because I have already spoken of it but equally because I a m persuaded that it will be of no little service to mathematics; f o r I apprehend that some, either of m y contemporaries or of m y successors, will, by m e a n s of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me. Acknowledgment I would like to thank Main Herreman, Reviel Netz, and David Wilkins for helpful comments.
THE MATHEMATICAL INTELLIGENCER
[1] A. Aaboe and J. L. Berggren, Didactical and other remarks on some theorems of Archimedes and infinitesimals, Centaurus 38 (1996), 295-316. [2] K. Andersen, Cavalieri's method of indivisibles, Arch. Hist. Exact. Sci. 31 (1985), 291-367. [3] T. Apostol, Calculus, voL I, 2nd Edition, John Wiley & Sons, New York, 1967. [4] I. Asimov, How did we Find out that the Earth is Round?, Walker & Co., New York 1972. [5] I. Asimov, Asimov's new Guide to Science, Basic Books, New York 1984. [6] Archimedes, The Works of Archimedes, edited in modern notation with introductory chapters by T.L. Heath. With a supplement, The method of Archimedes, recently discovered by Heiberg, Dover, New York, 1953. Reprinted (translation only) in [36]. [7] Archimedes, Opera Omnia, IV vol., cum commentariis Eutocii, iterum edidit I.L Heiberg, corrigenda adiecit E.S. Stamatis, B.G. Teubner, Stuttgart, 1972. [8] Archim~de, Oeuvres, 4 vol., texte 6tabli et traduit par C. Mugler, Les Belles Lettres, Paris, 1970-72. [9] M. Balme and G. Lawall, Athenaze, An Introduction to Ancient Greek, 2 vols., Oxford University Press, New York, 1990. [10] A. Avron, On strict constructibility with a compass alone, J. Geometry 38 (1990), 12-15. [11] E.T. Bell, Men of Mathematics, Simon and Schuster, New York, 1937. [12] L. Bers, Calculus, vol. I, Holt, Rinehart and Winston, New York, 1969. [13] L. Bieberbach (1886-1982), Theorie der geometrischen Konstruktionen, LehrbQcher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Band 13, Verlag BirkhAuser, Basel, 1952. [14] Cleomedes (circa 150 B.C.), De Motu Circulari Corporum Caelestium, 2 vols., H. Ziegler, editor, Teubner, Leipzig 1891. See [32, p. 106] [43] [54, Vol. 2, p. 267]. [15] H.S.M. Coxeter, Introduction to Geometry, John Wiley & Sons, New York, 1989. [16] H.T. Croft, K.J. Falconer, and R.K. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York 1991. [17] P.J. Davis, The rise, fall, and possible transfiguration of triangle geometry: A mini history, Amer. Math. Monthly 102 (1995), 204-214. [18] E.J. Eijksterhuis, Archimedes, Princeton University Press, Princeton 1987. [19] M. Djori~ and L. Vanhecke, A Theorem of Archimedes about spheres and cylinders and two-point homogeneous spaces, Bull. Austral. Math. Soc. 43 (1991), 283-294. [20] U. Dudley, A budget of trisections, Springer-Verlag, New York, 1987. [21] C.H. Edwards, The Historical Development of the Calculus, Springer-Verlag, New York, 1979. [22] C.H. Edwards and D.E. Penney, Calculus and Analytic Geometry, second edition, Prentice Hall, Englewood Cliffs, NJ, 1988. [23] Euclid, The Thirteen Books of Euclid's Elements, translated with introduction and commentary by T.L. Heath, Dover 1956. [24] D.H. Fowler, The Mathematics of Plato's Academy: a New Reconstruction, Clarendon Press, Oxford 1990.
[25] S.H. Gould, The Method of Archimedes, Amer. Math. Monthly 62 (1955), 473-476. [26] R.M Green, Spherical Astronomy, Cambridge University Press, Cambridge 1985. [27] J. Hadamard (1865-1963), Legons de g~om6trie el#mentaire, 2 vols., A. Colin, Paris, 1937. [28] D. Hammett, The Maltese Falcon, Penguin, Middlesex, 1930. [29] E. Hayashi, A reconstruction of the proof of Proposition 11 in Archimedes method, Historia Scl. 3 (1994), 215-230. [30] G.H. Hardy (1877-1947), A Mathematician's Apology, Cambridge University Press, New York, 1985. [31] R. Hartshorne, A Companion to Euclid, a course of geometry based on Euclid's Elements and its modem descendents, AMS, Berkeley Center for Pure and Applied Mathematics, 1997. [32] T. Heath, A History of Greek Mathematics, vol. II, Dover, New York, 1981. [33] I.L. Heiberg, Eine neue Archimedes-Handschrifl, Hermes 42 (1907), p. 235. [34] H.L.F. von Helmholtz, Helmholtz treatise on physiological optics, translated from the 3d German ed., edited by J.P.C. Southall, Dover, New York, 1962. [35] D. Hughes-Hallett, A.M. Gleason, et al., Calculus, John Wiley & Sons, New York, 1994. Second edition, 1998. [36] Great Books of the Western World, voL 11, R.M. Hutchins, editor, Encyclopaedia Britannica, Inc., Chicago, 1952. [37] G. Johnson, The Big Question: Does the Universe Follow Mathematical Law? The New York Times, February 10, 1998. [38] F. Klein (1849-1925), Elementary Mathematics from an Advanced Viewpoint, 2 vols, Macmillan, New York, 1939. [39] W.R. Knorr, The evolution of the Euclidean elements, Synthese Historical Library 15, D. Reidei, Dordrecht-Boston, MA, 1975. [40] W.R. Knorr, Archimedes and the Spirals: The Heuristic Background, Historia Math. 5 (1978), 43-75. [41] W.R. Knorr, The Ancient Tradition of Geometric Problems, Birkh&user, Boston, 1986. [42] W.R. Knorr, The method of indivisibles in ancient geometry, in "Vita mathematica" (Toronto, ON, 1992; Quebec City, PQ, 1992), 67-86, MAA Notes 40, Math. Assoc. America, Washington, DC, 1996. [43] K. Lasky, The Librarian who Measured the Earth, Little, Brown & Co., Boston 1994. [44] H, Lebesgue, LeGons surles constructions g~ometriques, GauthierVillars, Pads, 1950. Reissued, Jacques Gabay, Paris, 1987. [45] E. Moise, Elementary Geometry from an Advanced Viewpoint, Addison-Wesley, Reading, MA, 1974. [46] R. Netz, personal communication. [47] Nicomachus of Gerasa (circa 100 A.D.), Introduction to Arithmetic, translated by M.L. D'Ooge, in [36]. See [54, Vol. 1, p. 101]. [48] Pappus, La Collection Math~matique, traduit avec une introduction et notes par P. Ver Ecke, Descl6e de Brouwer, Paris, 1933. [49] Plutarch, Lives, vol. 5, translated by B. Perrin, Loeb Classical Library 87, Harvard University Press, Cambridge, MA, 1917. Also, translated by John Dryden (1631-1700), http://www.oed.com/ plutarch.html.
[50] A. Seiberg, Collected Works, 2 vols, Springer Verlag, New York, 1989, 1991. [51] G.F. Simmons, Calculus with Analytic Geometry, McGraw Hill, New York, 1985. [52] H.M. Stark, On the "gap" in a theorem of Heegner, J. Number Theory 1 (1969), 16-27. [53] H. Swann, Commentary on rethinking rigor in calculus: The role of the mean value theorem, Amer. Math. Monthly 104 (1997), 241-245. [54] I. Thomas, Greek Mathematical Works, 2 vol., Loeb Classical Library 335, 362, Harvard University Press, Cambridge, MA, 1980. [55] G. Toussaint, A new look at Euclid's second proposition, Math. Intelligencer 15 (1993), 12-23. [56] I. Vardi, A classical reeducation, in preparation. [57] I. Vardi, APXIMHz~OY:SrlEPI TQ,N EMKD,N E~O&O2:, in preparation. [58] B.L. van der Waerden, Science Awakening I, Scholar's Bookshelf Press, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. [59] A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995), 443-551
VOLUME 21, NUMBER 3, 1999
47'
Ii'iFIli~li[~,,e-:ml[.-~-~m=rpli(~-'~-~.~..,=a.~ .
-
~z':.|
This column is devoted to mathematics for fun. What better purpose is there for mathematics? To gppear here, a theorem or problem or remark does
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~e-lB
Alexander
Shen,
Editor
Unexpected Proofs
Now, a s s u m e w e have two b o x e s , one inside another. Then, the e-neighb o r h o o d o f t h e first b o x will be inside the e - n e i g h b o r h o o d of the second, so
n e o f the nice things a b o u t mathe m a t i c s is that s o m e t i m e s a question l o o k s very simple b u t t h e a n s w e r u s e s an u n e x p e c t e d and e l e g a n t argument. Let m e s h o w two e x a m p l e s .
This is true for any 6, even for a large one w h e n t h e e 2 t e r m is t h e m a i n t e r m (note that the e 3 t e r m s are the s a m e for b o t h n e i g h b o r h o o d s and cancel e a c h other). Therefore, 11 = w l + hi + d l d o e s n o t e x c e e d / 2 = w2 + h2 + d2. The s e c o n d p r o o f u s e s r a n d o m n e s s . Let X b e a c o n v e x set in R 3. C o n s i d e r a r a n d o m line m in R 3. The orthogonal p r o j e c t i o n o f X onto m is a segment. Let us d e n o t e by d(X) the e x p e c t e d length o f this segment. Let Xm b e a s e g m e n t of length m. Then, d(Xm) is p r o p o r t i o n a l to m, i.e., d(Xm) = e m for s o m e c. (In fact, c = 1/2, but the e x a c t value is n o t i m p o r t a n t
O
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.
Boxes in a Train Rules o f the M o s c o w u n d e r g r o u n d say t h a t y o u are a l l o w e d to bring o n a rect a n g u l a r b o x o f size w • h • d only if w + h + d d o e s not e x c e e d 150 cm. Question: Is it p o s s i b l e to c h e a t by p a c k i n g one b o x into a n o t h e r ? The ans w e r is no: If a rectangular b o x w I X hi X dt c a n b e p l a c e d inside a n o t h e r one o f size w 2 x h 2 x d2, then Wl + hi + d l --< w2 + h2 + d2. We p r e s e n t t w o c o m p l e t e l y different p r o o f s of this fact. The first cons i d e r s t h e ~ n e i g h b o r h o o d of a b o x (inchiding the interior part). Its volume V(e) is defined for non-negative ~. It is e a s y to s e e that V(e) is a p o l y n o m i a l in
V(~) = V + Se + r
2 + (4/3)~-~.
Here, V is the volume of the box, S is the a r e a o f its surface, a n d 1 is t h e sum o f t h e d i m e n s i o n s (w + h + d). Indeed, the n e i g h b o r h o o d c o n s i s t s o f
Please send all submissions to the Mathematical Entertainments Editor, Alexander Shen, Institutefor Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:
[email protected] 48
I
9 t h e b o x itself (V) * six r e c t a n g u l a r b o x e s (of t h i c k n e s s e) coveting the faces and having total v o l u m e S e * t w e l v e p i e c e s n e a r the e d g e s that c a n b e c o m b i n e d into t h r e e cylind e r s o f radius e a n d lengths w, h, and d; t o t a l volume zre2(w + h + d) 9 eight p i e c e s n e a r the v e r t i c e s that f o r m a ball of radius e having total v o l u m e (4/3)~re3.
THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEW YORK
Vl + SI~ § 7r/1~2 4- (4/3)zre3 -< V2 + S2e + zr/2*"2 + (4/3)~-e3.
nOW.) Now, let X b e a b o x o f size w x h x d. F o r e a c h line m, the p r o j e c t i o n o f X onto m h a s l e n g t h p w + Ph § Pal, w h e r e Pw, Ph, and Pd are p r o j e c t i o n s o f segm e n t s of length w, h, a n d d, the e d g e s o f the box. By averaging, w e get
d(X) = c(w + h + d). If a b o x 0(1) is p l a c e d inside a n o t h e r one (X2), t h e n the p r o j e c t i o n o f X1 o n t o a line m is i n c l u d e d in t h e p r o j e c t i o n o f X2 onto m, s o d(X1) ~ d(X2). Combining this o b s e r v a t i o n with the preceding one, w e see that w l + hi + dl -< w2 § h2 + d2. (End o f the s e c o n d proof.)
Square Split into Triangles It is e a s y to split a square into n equal triangles if n is even. However,
it is impossible to split a square into n triangles of equal area i f n is odd. However, t h e p r o o f o f this fact is n o t s t r a i g h t f o r w a r d and uses s o m e topoiogy and algebra.
We start with a special case where (a) the triangles form a triangulation and Co) all vertices have rational coordinates. (Later, we'll see h o w these assumptions can be removed.) For any rational number r, defme its 2-valuation 114]as follows: i f r = 2k(p/q), where p and q are odd, I1~]is 2 -k. By defmition, ]]0H= 0. In a sense, 1141measures "oddness" of ~. for example, 3/2 is "odder" than 1, and 2 is "odder" than 4. Now, divide all rational points (x, y) (both x and y are rational) into three classes. If both x and y (represented as irreducible fractions) have even numerators, the point (x, y) belongs to class A. If at least one of x and y has an odd numerator, compare the "oddness" o f x and y: when x is "more odd," we get a B point, otherwise a C point. Formally, A: Ilxl] < 1 and IlYll < 1 B: Ilxll > Ilyll and Ilxll-> 1
C: Ilxll---IlYll and IlYll-> 1 Let us return to our square 12 = [0, 1] • [0, 1] and its triangulation with rational vertices. L e m m a . There exists a triangle in the triangulation whose vertices are labeled w i t h all three labels A, B, and C. Proof. Our classification can be considered as a mapping a from the set of vertices into the set {A, B, C}. Imagine that A, B, and C are vertices of some triangle ABC. Then, a can be uniquely extended to a mapping of the whole square into the triangle ABC that is piecewise affme (affme on each triangle of the triangulation). N o w the statement of the lemma can be reformulated as follows: a covers the interior part of the triangle ABC. To prove this statement (a version of Sperner's lemma), let us consider the restriction of a to the boundary of the unit square. We k n o w its values on the square's vertices: (0, 0) has type A, while (1, 0) has type B, and both (1, 1) and (0, 1) have type C (see Fig. 1). Moreover, it is easy to see that any vertex on the lower side of the square has type A or B and any vertex on the left side has type A or'C, whereas all
(0, 1) C"
Bor
C
_ (1.1) 'C )
AorC
BorC
A (0,0) w
VA or B
B (1,0) v
Figure 1 vertices on the remaining two sides have type B or C. Therefore, the restriction a]0I 2 of a to the boundary of the s q u a r e / 2 maps it into the boundary of the triangle ABC and has degree 1. Therefore, alOI 2 is not homotopic to a constant mapping. On the other hand, if the image a ( I 2) were contained in the boundary of triangle ABC, a would provide a h o m o t o p y between c~]0I2 and a constant mapping. (End of the p r o o f of the lemma.) Now we k n o w that our triangulation contains a triangle whose vertices are labeled A, B, and C. Let their coordinates be (al, a2), (bl, b2), and (cl, c2), respectively. This triangle has area s=ldet
~:-al
al
b2-a2 C2 -- a2 '
and I]511> 1 (as we'll see). On the other hand, S = 1/n, because all n triangles of the triangulation have the same area. Therefore, n is even. It remains to prove that I~1 > 1. Recall two main properties of the 2valuation:
9 Ilabll
=
Ilall'llbll
9 Ila + bll--< max(llall, Ilbll); this inequality turns into equality if Nail r Ilbll Using these properties, it is easy to check that the point (b~, b~) = (bl - al, b 2 - a2) belongs to type B and the point (c~, c ~ ) = (C1 --al, C 2 - a2) belongs to type C (bl is "more odd" than al, so subtracting al we do not change "oddness" of bl, etc.) By definition of types B and C, we have
So the statement is proved for the case of triangulation with rational vertices. Let me say, briefly, what could be done for the general case. Assume that the triangles do n o t form a triangulation, e.g., vertex Q of one triangle lies on side P R of another one. (See Fig. 2.) What can we do? We can admit "degenerate" triangles like PQR, get a triangulation, apply our argument, and find a triangle that is ABClabeled. This triangle cannot be degenerate since for its area S, we have proved that I111 > 1. What should we do if the coordinates of the vertices are irrational? In this case, one can extend the 2-valuation to an extension of Q that contains all the coordinates, and use the same argument. (I omit the details.) I found.this problem (and its solution) iIi an article of B. Bel~ker and N. Netsvetaev; they attribute it to J o h n Thomas (A dissection problem, Math. Mag. 41 (1968), 187-190) for the case of rational coordinates and Paul Monsky (On dividing a square into triangles, Am. Math. Monthly 77 (1970), 161-164) for the general case. I.ette~
Concerning Poncelet's theorem and your article in the Intelligencer, I wonder if you know this. Poncelet's initial theorem concerned a pencil of circles, and he stated it like this: let I, II, and III be three circles in a pencil. Start from a point m in I, draw the tangent to II, get a second point n in I; from n draw a tangent to III, get another point p in I. Then, the line mp, when m runs through I, envelops a circle IV (from the initial pencil). All closure theorems follow from this one. Now, the p r o o f of the InteUigencer applies to this, one has only to remark that the lengths of tangents drawn from points of I to circles in the pencil are proportional with universal constants. Then, the associR
IIb;ll > IIb ll; IIb;ll >- 1; Ilchll-> IIc;ll; Ilchll -> 1. ' ' ' ' II and IIb'1c211-' Therefore, IIb,c=ll > flb2c, 1, so 112sll = IIb;c5 - b c;ll = IIb;c ll--- 1 and 1511 = 211211 > 1.
P
Figure 2
VOLUME21, NUMBER3, 1999 4 9
ated measures on the circle I, given by H, III, etc., are proportional. Otherwise stated: for such a measure the line joining two points differing by a translation of this measure always envelops some circle of the pencil. Elliptic functions are at the core of Poncelet's theorem; here the elliptic function is the new measure. Marcel Berger Institut des Hautes Etudes Scientifiques 91440 Bures-sur-Yvette France e-mail: ber'
[email protected] Preparing an article on the story of Poncelet's theorem, I read some of the original papers. My following notes sketch the historical background. 1. Poncelet's original theorem is not about a triangle inscribed in a circle and circumscribed around another circle, but about an n-gon inscribed in a conic section and circumscribed around another conic section. Moreover, the theorem in Poncelet's approach is a consequence of a more general theorem. This general theorem is about a pencil of conic sections. L e t C, cl, c2, 99 9 Cn-1 be the elements of this pencil. Poncelet states: there is a conic section Cn such that whenever points A1, A2, 9 9 A n
THE MATHEMATrCALINTELUGENCER
are on C, a n d line A1A2 touches Cl, l i n e A2A 3 touches c2, 99. , An-l, An touches Cn- 1, thenAnA1 will touch Cn. The first publication was in 1822. 2. Poncelet was an officer of Napoleon. He was imprisoned in Russia, in Saratov, for more than a year. At this time, without books and equipment, he created many notions of projective geometry: ideal and imaginary points, for example. One of his practical results: Circles are exactly the conic sections containing the points (1, i, 0) and (1, - i , 0). One can transform two conics into two circles simply by projecting two of their common points to (1, i, 0) and (1, - i , 0). 3. The proof you found in Prasolov and Tikhomirov's textbook goes back to Jacobi (CreUe J. M a t h . 3 (1828), 376). Here is the short history of his proof: In Bd. 2 of Crelle's J o u r n a l (1827), Steiner proposed the problem of finding the algebraic relation of the radii of circles Cl and c2, and the distance of their centers, if there is a 4-gon, 5-gon, 9 8-gon inscribed in cl and circumscribed around c2. (In fact, these problems had been partly solved previously by Fuss, the academic secretary of St. Petersburg. Euler solved the problem for n = 3, his student Fuss for n = 4, and Fuss was able to solve the prob-
lem for n = 5, 6, 7, 8 if the n-gon is symmetric about the center of the circles.) Steiner gave the appropriate equations without proof in Crelle's J o u r n a l of the same year. In this issue, Abel and Jacobi had many articles on elliptic integrals and on their inverse, the elliptic functions. In Bd. 3, Jacobi wrote three articles on this topic. Then, he wrote a fourth one: he proved Poncelet's theorem for two circles by integrals and he could even check the equations of Steiner (and Fuss). When the old Poncelet refers shortly to Jacobi's proof, he uses essentially your arguments. Jacobi's article is longer. As I can see, Jacobi tried to solve geometrically the problem proposed by Steiner. He could set up equations, and these equations reminded him of Legendre's addition formulas of elliptic integrals. If Jacobi could find the connection, he could set up an elliptic integral related to Poncelet's theorem. It was only after this that he perceived the geometric meaning of the integrand: the reciprocal of the length of the tangent. Andras Hrask6 Tornoc u. 17 1141 Budapest Hungary e-mail:
[email protected] RODRIGO DE CASTRO AND JERROLD W. GROSSMAN
rails to Paul Erdds ar-Rous
~
he notion of Erdds number has floated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientific research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals. In this
p a p e r w e e x p l o r e s o m e (fairly short) c o l l a b o r a t i o n p a t h s t h a t one can follow from Paul E r d 6 s to r e s e a r c h e r s inside a n d outside of m a t h e m a t i c s .
An Outstanding Component of the Collaboration Graph The collaboration graph C h a s as vertices all r e s e a r c h e r s ( d e a d o r alive) from all a c a d e m i c disciplines, with an edge j o i n i n g vertices u and v if u a n d v have j o i n t l y p u b l i s h e d a r e s e a r c h p a p e r o r b o o k (with p o s s i b l y m o r e co-authors). As is t h e c a s e for any simple (undirected) graph, in C w e have a notion o f distance b e t w e e n t w o vertices u a n d v: d(u,v) is the n u m b e r of edges in the s h o r t e s t p a t h b e t w e e n u a n d v, if such a p a t h exists, ~ otherwise (it is u n d e r s t o o d t h a t d(u,u) = 0). In this p a p e r w e are c o n c e r n e d with the c o l l a b o r a t i o n s u b g r a p h c e n t e r e d at Paul E r d 6 s (1913-1996). F o r a res e a r c h e r v, the n u m b e r d ( P a u l Erd6s, v) is called the ErdSs n u m b e r of v. That is, Paul Erd6~ himself has E r d 6 s numb e r 0, a n d his co-authors have E r d 6 s n u m b e r 1. P e o p l e n o t having E r d 6 s nttmber 0 o r 1 w h o have p u b l i s h e d with s o m e one with E r d 6 s n u m b e r 1 have E r d 6 s n u m b e r 2, and so on. T h o s e w h o are not linked in this w a y to Paul E r d 6 s have E r d 6 s n u m b e r ~. The collection o f all individuals with a finite E r d 6 s n u m b e r constitutes the Erd6s component o f C.
The E r d 6 s c o m p o n e n t o f C is outstanding for its amazing size and for the m a n n e r in w h i c h it clusters a r o u n d Erd6s. A l m o s t 500 p e o p l e have E r d 6 s n u m b e r 1, and o v e r 5000 have E r d 6 s n u m b e r 2. In the h i s t o r y of scholarly publishing in m a t h e m a t i c s , n o one has e v e r m a t c h e d Paul Erd6s's number of collaborators or number of papers ( a b o u t 1500, a l m o s t 70% o f w h i c h w e r e j o i n t works). With his r e c e n t d e a t h the m a n w h o i n s p i r e d so m u c h mathematical thinking h a s - - t o use his t e r m i n o l o g y - - l e f t , but his legend lives on (see for e x a m p l e t w o r e c e n t b i o g r a p h i e s [20], [28]). A n d p a r t o f this legend, inside a n d o u t s i d e mathe m a t i c a l circles, is the notion o f E r d 6 s numbers. The first explicit m e n t i o n in the literature o f a p e r s o n ' s E r d 6 s n u m b e r a p p e a r s to b e [11], w h e r e the r e a d e r is ass u r e d that Paul E r d 6 s himself was, for a long time, u n a w a r e of this entertainment. But t h e first s y s t e m a t i c a t t e m p t to s t u d y the E r d 6 s c o m p o n e n t o f C w a s c a r r i e d o u t b y t h e s e c o n d a u t h o r in [16] and [18] and c o n t i n u e s on the E r d 6 s N u m b e r P r o j e c t World Wide Web site [13]. This Web site contains a list o f all p e o p l e with E r d 6 s n u m b e r I (currently 485) a n d their o t h e r co-authors with E r d 6 s n u m b e r 2 (currently 5337). The files (available also via a n o n y m o u s ftp, s e e [14]) are u p d a t e d annually. It h a s b e e n s u r m i s e d that m o s t s c i e n t i s t s m u s t have a finite E r d 6 s number, but the evidence offered in s u p p o r t
9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 3, 1999
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has not been really abundant. In [5] the first author contributed new information, and the present paper pursues the matter much further. By skimming through several bibliographic sources we have found that many important people in academic areas--other than mathematics proper-as diverse as physics, chemistry, crystallography, economics, fmance, biology, medicine, biophysics, genetics, meterology, astronomy, geology, aeronautical engineering, electrical engineering, computer science, linguistics, psychology, and philosophy do indeed have finite Erd6s numbers. We report on some of these intriguing connections here; others can be found in an expanded version of this paper, available on-line [13]. Of course, it cannot be immediately inferred that all people in the mentioned disciplines, "or related ones, must have fmite Erd6s numbers. But the names first :~esulting from this kind of browsing are among the most prominent and productive (including more than 60 Nobel Prize winners), and most have had many collaborators over the years. Thus one is led to believe that the majority of researchers in those fields, except for those working in total isolation, probably have finite Erd6s numbers. When referring to all academic or scientific fields, the last statement should be regarded as bold--though credib l e - g u e s s . If we restrict ourselves to authors publishing mathematical research, then the conjecture ( m o s t active mathematical researchers of (e) ~the twentieth century have a finite (and 1. rather small) Erd6s number seems so plausible that it has been accepted folklore. Looking at the list of those with Erd6s number -