THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
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The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
Unintended Consequences As a parent and teacher I read with great interest the articles "The Case Against Computers in K-13 Math Education" by Neal Koblitz and "Some Kinds of Computers for Some Kinds of Learning" by Dubinsky and Noss in the Winter 1996 Intelligencer. They reminded me of a story I tell my students. When my son was in the seventh grade he took a computer course. One of the software programs they used was intended to develop their typing skills. The program was designed as a game: faster, correct typing earned a higher score. My son discovered that if you typed the letter "a" followed by the spacebar over and over as fast as possible you could outdo those who were typing words. He didn't develop any typing skills, but he was very proud of the fact that he had the highest all-time score. S. P. Peterson Bell Laboratories 480 Red Hill Road Middletown, NJ 07748 USA e-maih
[email protected] 6
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
R e m a r k s on H i l b e r t ' s 23rd P r o b l e m Shiing-shen Chern
This is the last problem of his famous Paris address in 1900.* At the beginning he said, So far, I have generally mentioned problems as definite and special as possible, in the opinion that it is just such definite and special problems that attract us the most and from this the most lasting influence is often exerted upon science. Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture--which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due---I mean the calculus of variations.
culus of variations. In m o d e r n terminology, it can be described simply and naturally. In an n-dimensional space M with the coordinates x i, 1 0 such that for any t ~ (0, to] (1) b t defines an affine curve at such that the pair (N2, Rat) is homeomorphic to the pair (T,, L);
(2)
B t defines a projective c u r v e A t such that the pair R a t) is homeomorphic to the pair (2, ~).
(~p2,
A curve obtained by this construction is called a T-
curve. All real schemes of curves of degree --
(k - 1)(k - 2) 2
and
n > -
(k - 1)(k - 2) 2
Patchworking Harnack Curves In each area of mathematics there are objects which appear much more frequently than others. Some of them (like Dynkin diagrams) appear in several domains quite distant from each other. In the topology of real plane algebraic curves, Harnack curves play this role. It was not an accident that they were constructed in the first paper devoted to this subject. Whenever one tries to construct an M-curve, the first success provides a Harnack curve. Patchwork construction is no exception to the rule. In this section we describe, using the Patchwork Theorem, the construction of some Harnack curves of an even degree m = 2k. In what follows, all the triangulations satisfy an additional assumption: they are primitive, which means that all triangles are of area 1/2 (or, equivalently, that all integer points of the triangulated area are vertices of the triangulation). A polynominal defining a T-curve contains the maximal collection of nonzero monomials if and only if the triangulation used in the construction of the T-curve is primitive. A primitive convex triangulation of T is said to be equipped with a Harnack distribution of signs if: vertex (i, j) has the sign ..... if i, j are both even, and has the sign "+" in the opposite case. A vertex (i, j) of a triangulation of T is called even if i and j are both even, and odd otherwise. PROPOSITION. Patchworking applied to an arbitrary primitive convex triangulation of T with the Harnack distri-
O0
...
0
\,~
.-,/
~
ak(k- ~) Figure 6. The real s c h e m e of the s i m p l e s t H a r n a c k curve of degree 2k.
bution of signs produces an M-curve with the real scheme shown in Fig. 6. A n e x a m p l e of the construction u n d e r consideration is s h o w n in Fig. 7. Proof of Proposition: First, note that the n u m b e r of interior (i.e., lying in the interior of the triangle T) integer points is equal to (m - 1)(m - 2)/2, the n u m b e r of even interior points is equal to (k - 1)(k - 2)/2, a n d the n u m ber of o d d interior points is equal to 3k(k - 1)/2. Take an arbitrary e v e n interior vertex of a triangulation of the triangle T. This vertex has the sign ..... . All
adjacent vertices (i.e., the vertices connected with the vertex b y edges of the triangulation) are o d d , and thus they all h a v e the sign " + . " This m e a n s that the star of an e v e n interior vertex contains an oval of the c u r v e L. The n u m b e r of such ovals is equal to (k - 1)(k - 2)/2. Take n o w an o d d interior vertex of the triangulation. It has the sign " + . " There are t w o vertices with . . . . . and one vertex with " + " a m o n g the images of the vertex u n d e r s = sx o sy and Sx a n d sy (recall that sx a n d sy are reflections w i t h respect to the coordinate axes). Consider the i m a g e with the sign " + . " It is easy to verify that all its adjacent vertices h a v e the sign " - . " A g a i n this m e a n s that the star of this vertex contains an oval of the c u r v e L. The n u m b e r of such ovals is equal to 3k(k - 1)/2. But (k - 1)(k - 2) +
3k(k
2
-
1)
(m - 1)(m - 2)
2
2
so the c u r v e can only h a v e one m o r e oval. This oval exists because, for example, the c u r v e L intersects the coordinate axes.
~r
rd
tar
rat
rd~
~
~
~
tar
r~r
rat
rat
~
~
~
~
~
~
~
~r
rat
~ ~r
~r
rd
Figure 7. P a t c h w o r k of the s i m p l e s t Harnack curve of degree 10. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
25
oo
k.
~r
.,d
o9
20 Figure 8. Schemes of counterexamples to the Ragsdale Conjecture of degree 10. To finish the proof, we need only note that the union of the segments {x - y = -m, - m -< x,y-< m} U {x 0 such that, up to isometry, Ai = (iv, w) and Bi = (iv, 0) (see Fig. 2). This is a crude approximation of a real situation (see Fig. 3), where both "horizontal" and "vertical" distances between eyelets may vary. Even the symmetry with respect to a horizontalline can be absent, especially if the shoe is old. I want to propose a proof of minimality of the standard lacing under an assumption that seems to be much closer to the real situation:
(l)
For any k, 1, the line through A k and BI separates the sets {Ai: i < k} U {B j : j < l} and {Ai: i > k} U {B j : j > l}.
In other words, if we view the"A" eyelets from any "B" eyelet (or the "B" eyelets from any "A" eyelet), they come in the natural order: A o, AI, ... , An (or Bo, Bl , ... , Bn). In fact, even this assumption is too strong. We need only the following one: IThis lacing is called American style in Ref. 1. However, I do not recall anybody, be it in America or in Europe, having his or her shoes laced in a different way.
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THE MATHEMATICAL INTELLIGENCER VOL. 18, NO.4, 1996
Figure 1. Standard lacing.
(2)
For any i < j and k < 1, IAiBkl + IAjBII < IAiBII + IAjBkl.
To see that (l) implies (2), take i < j and k < 1, and as-' sume that (l) holds. Then as in Figure 4, the eyelets Ai and BI lie on opposite sides of the line through A j and BkJ and the eyelets Aj and Bk lie on opposite sides of the line through Ai and BI. Therefore, the segment AiBI intersects the segment AjBk. If D is the point of intersection, then by the triangle inequality we get IAiBkl
+ IAjBII < IAPI + IDBkl + IAPI + IDBII = lAm
+ IAPkl,
so (2) holds. One can try to consider an even more realistic setup, where the eyelets are points of three-dimensional space. However, a lacing should closely follow the surface of the shoe. Therefore, we can assume the eyelets to be points of the surface of the shoe, with distances measured along this surface. The reader can try to make precise measurements to verify whether in this setup his or her shoes satisfy assumption (2). But usually the piece of the shoe surface involved in lacing has a very small curvature. This means that we do not make a big error if we view this surface as a "bent" piece of plane. Smoothing it out leads us to our initial model. Now a quick glance is enough to verify whether assumption (1) is satisfied.
AO
0
o
o
o
o
a
o
Figure 2. Halton's arrangement of eyelets.
Figure 5. An arbitrary lacing.
Figure 3. A real shoe.
It is time to prove the main result of this article.
THEOREM: Assume that the set of eyelets {A o, AI' ... , An' Bo, ... , Bnl satisfies (2). Then the standard lacing is shorter than any other lacing. The idea of the proof is rather simple. Referring again to Figure 4, let us define a "move" to consist of replacing the pair of segments AiBI and AjBk by AiBk and AjB/. The only problem is that what we get as the result of such a move may not be a lacing. For instance, if we start with the lacing shown in Figure 5 and make a move that replaces segments A oB2 and An-IB I by AoB I and AIl - I B2, then what we get (Fig. 6) is not a lacing. Therefore, we will introduce objects that are more general than lacings, namely systems of arrows. In this larger class, all moves will be legitimate, and after a finite number of moves we will arrive at the system corresponding to the standard lacing. These ideas lead to the following formal proof. Proof of Theorem: Let L = Co ~ CI ~ ••• ~ C2n + 1 be a lacing (see Fig. 5). There is some k such that Ck = All' We
A.~
Figure 4. (1) implies (2).
Figure 6. Result of a move.
draw arrows from Ci to Ci + l for i = 0/ 1, ... , k - 1 and from Ci + l to Cj for i = k, k + 1, .. . 2n (see Fig. 7). Note that this system of arrows satisfies the following properties. (3)
Each arrow begins at an "A" eyelet and ends at a "B" eyelet or vice versa; all eyelets except .40, Bo, and An have one arrow coming in and one going out; eyelets A o and Bo have one arrow going out and none coming in; eyelet An has two arrows coming in and none going out.
A, J
Figure 7. A system of arrows obtained from the lacing from Figure 5. THE MATHEMATICAL INTELLIGENCER VOL. 18. NO.4, 19%
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We define the length of a system of arrows as the sum of their lengths. Although not every system of arrows comes from a lacing, if it does, then its length is equal to the length of the corresponding lacing. Let us call the system of arrows obtained from the standard lacing the standard system. We will show that it has smaller length than any other system of arrows satisfying (3). Indeed, if this is not the case, then there exists a system of arrows :A satisfying (3) that has minimallength and is not standard. If the arrows A k ~ Bk+1 and Bk ~ Ak+1 for k = 0, ... , n - 1 are in :A, then by (3), :A consists of these arrows and the arrow Bn ~ An' This means that the system :A is standard, a contradiction. Therefore, we can take the smallest k such that at least one of the arrows A k ~ B+ 1, Bk+1~ Ak+1 is not in:A. We may assume that this is Bk ~ Ak+1' Thus, we must have in :A arrows Bk ~ Ai and B1 ~ Ak+1 for some j k + 1
'*
1.
J. H. Halton, The shoelace problem, Math. Intelligencer 17: 4 (1995), no. 4, 36-40.
Department of Mathematical Sciences
IUPUI Indianapolis, IN 46202-3216, USA e-mail:
[email protected] Evariste Galois
(Endbetrag) von Ilona Bodden
(Sum Total) by Ilona Bodden translated by Kurt Bretterbauer
Die Republik-woher sollte sie?wird auch nicht mehr Verstand besitzen als ihre VorgangerSie baut sich aus ME SCHE auf. Ich sterbe mit zwanzig Jahren als Opfer der einzig unendlichen Grosse, der menschlichen Dummheit. (Denn selbst das Universum ist endlich und somit nicht un-berechenbar.) Ich sterbe.
My calculations were wrong. The republic also will not-how should it?possess more common sense than its predecessorsit is built of PEOPLE. I die aged twenty as a victim of the only infinite quantity, human stupidity. (For even the universe is finite, and hence not uncalculable.)
I die. That is an equation with two unknowns:
Das ist eine Gleichung mit zwei Unbekannten: x Tod und y Gott = das unendliche ICHTS ...
34
References
Evariste Galois*
Meine Berechnungen waren falsch.
~Reprinted by
'*
and 1 k. Since any arrow ending at Ai with j :s k begins at Bi - 1 and any arrow beginning at B1 with 1 < k ends at A 1+ 1J we have j > k + 1 and 1> k. If we now make the move replacing the arrows Bk ~ Ai and B1 ~ Ak+1 by the arrows Bk ~ Ak+1 and B1 ~ Ai' the new system will still satisfy (3). However, by (2) (with i = k + 1), its length will be smaller than the length of :A, a contradiction. This completes the proof. •
permission.
THE MATHEMATICAL lNTELLIGENCER VOL. 18, NO.4, 1996
x Death plus
yGod = the infinite OTHING. Kurt Bretterbauer Institut fiir Theoretische Geodtisie und Geophysik Gusshaussfrasse 25-29/128 1040 Vienna Austria
Lifelong Symmetry: A Conversation with H. S. M. Coxeter Istv n Hargittai
Harold Scott MacDonald Coxeter (b. 1907, London) is Professor Emeritus in the Department of Mathematics of the University of Toronto. When Buckminster Fuller published his magnum opus Synergetics, he dedicated his work to H. S. M. Coxeter, characterizing him in the following way: "By virtue of his extraordinary life's work in mathematics, Dr. Coxeter is the geometer of our bestirring twentieth century, the spontaneously acclaimed terrestrial curator of the historical inventory of the science of pattern analysis." My wife and I visited Professor Coxeter on August 1, 1995, and the following conversation was recorded on that occasion. Istv~n Hargittai (IH): You have three first names. Which is the one you like most? H. S. M. Coxeter (DC): I prefer to be known as Donald. The original intention of my parents was to call me MacDonald Scott Coxeter but some stupid godparent said that I should be named after my father and they added Harold at the beginning. That made Harold MacDonald Scott. The initials then would look like a ship, H. M. S., Her Majesty's Ship. This is why they switched the two names, and it became Harold Scott MacDonald. What I have done lately is to use H. S. MacDonald Coxeter.
IH: Grandchildren?
DC: I have five grandchildren and five great-grandchildren. IH: You wrote somewhere that your hobby was music and
travel. When you listen to music, do you relate it in any way to geometry? DC- Not directly, but the artistic feeling that one has is very much the same in both cases. Before I took up mathematics, I was very interested in music, to the extent that I tried to compose. Between the ages of 7 and 14 I did a lot of musical composition, under the guidance of Tony Galloway, an old friend of m y family who was a very expert violinist and a sadly neglected composer. He taught me about the theory. I wrote a lot of piano pieces, and songs that my father used to sing. I was even so ambitious as to write a string quartet. However, very few of them are worth preserving. Two
IH: You have a son and a daughter. Did they follow your footsteps?
DC: Not at all. My son got interested in the church and took a degree, Master of Theology. As a minister he did not fully enjoy anything except the parish visiting, looking after unfortunate people. Eventually he gave that up and got a second degree as Master of Social Work. He did something about rehabilitation of drug addicts, then got interested in geriatric hospitals and getting supplies for them and he is still in that position now in the State of New Jersey. My daughter married an accountant. She is a Registered Nurse and lives in a small place between Toronto and Hamilton. We can visit her more easily than our son who is 800 km away. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
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T h e C o x e t e r s w i t h t h e i r p r e s e n t f r o m E s c h e r . ( 9 1996 M.C. E s c h e r / C o r d o n A r t - B a a r n - H o l l a n d . A l l f i g h t s r e s e r v e d . ) 3 6 THE MATHEMATICALINTELL1GENCERVOL.18, NO. 4, 1996
samples can be seen after the biographical sketch at the beginning of my new book, Kaleidoscopes. It was edited by F. A. Sherk, one of my former students. He collected 26 of my papers that had to do with symmetry. IH: Who turned your attention to geometry? DC: It was pretty much by myself. I was always interested in the idea of symmetry. When I was 14, I was in a boarding school in England, and happened to have some trivial illness; in the school sanatorium I was put in a bed next to a boy called John Flinders Petrie and he became a firm friend. (He was the only son of Sir Flinders Petrie, the great Egyptologist.) He and I looked at a geometry textbook with an Appendix on the five Platonic solids. We thought how interesting they were and wondered why there were only five, and we tried to extend them. He said, if you can put three squares around a corner to make a cube, what about putting four squares around a comer? Of course, they'd fall flat, giving a pattern of squares filling the plane. He, being inventive in words, called it a "tesserohedron." He called the similar
arrangement of triangles a "trigonohedron." Later on he said, what about the limitation of putting four squares around the corner and why not more than four? Maybe you can put six squares around the corner if you don't mind going up and down in a zigzag formation. Thus he discovered a skew polyhedron with "holes," a kind of infinite regular sponge. He also noticed that the squares in this formation belong to the cubic lattice. He saw that it can be reciprocated so that instead of six squares at each vertex you have four hexagons. He noticed that this could be obtained from the uniform honeycomb of truncated octahedra fitting together to fill space. The hexagons of the truncated octahedra come together, four at each vertex, and continue to form a sponge filling all space; so this was a second skew polyhedron. Then I said if you can have six squares and you can have four hexagons, why not even more: why not have six hexagons at the vertex as in the space-filling of tetrahedra and truncated tetrahedra? Then we extended the Schl/ifli symbol by which the cube is called {4, 3}, and we called these new polyhedra {4, 614} and {6, 4]4}, and {6, 6[3}, the number after the stroke indicating the nature of the holes one sees in the sponge. Before we left school, we went on to consider what would happen in four or more dimensions, and other things which later we learned had been discovered before, by L. Schl/ifli in Switzerland. IH: Did your friend also continue in geometry? DC: He did, and became quite clever at it. Unfortunately, because his father belonged to University College London, and my teacher wanted me to go to Cambridge, we went to different universities. He did quite well at University College and then the War came, W.W.II; he enlisted as an officer and was taken prisoner by the Germans. He organized a choir there. After the War ended and he was released, he went to a well-known school in southwest England, Dartington Hall, and he had a rather trivial job there. He never seemed to fulfill his early promise. He just became a tutor who looked after children who were not doing well in school. But he still corresponded with me, and it was he who noticed that when you take a regular polyhedron and look at the edges, you see that there is a zigzag of edges that go round and close up; for instance, if you take those edges of a cube that do not involve one pair of opposite vertices, they form a skew hexagon. We call this the "Petrie polygon," and it is now a well known property of a regular polyhedron to have a Petrie polygon: a skew polygon in which every two consecutive edges, but no three, belong to a face. IH: Is he retired now?
With Magdolna Hargittai, contemplating the mathematical sculptures of George Odom.
DC: No, he died. A very sad story. He married a very lovely lady and had a daughter and all went well. Then THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
37
somehow his wife got a heart attack and died. He was so distraught, missed her so terribly that he didn't k n o w where he was going, and he walked into a m o t o r w a y in England where the cars were going at a huge speed and he just didn't k n o w what was happening and one of them killed him, just two weeks after his wife died. This was about 24 years ago. II-I: Buckminster Fuller called you "the geometer of the twentieth century." How did you get to know each other? DC." This is a terribly exaggerated statement, but he was given to that sort of writing and speaking. He was a dear old man, and I was quite fond of him, but he had overblown his stars as a mathematician. He was really a very good architect and a very good engineer. His geodesic domes are really a wonderful thing. But w h e n he got into mathematics he was a little bit amateurish.
DC" As in all branches of mathematics, there is a tremendous increase in productivity. Research goes on, and m u c h of it I have no inkling of. If you only look at the development of Mathematical Reviews, w h e n they first started about 1940, it was quite a thin volume, and each m o n t h they got more a n d more. Eventually there were h u n d r e d s of thousands of papers being written, and so the later volumes are ever so much thicker than the original ones. IH" Geometry is very important in chemistry. We have simple but very helpful models of molecular geometry, but teaching them in a freshman chemistry course in the U.S. is rather hindered by the students' lack of knowledge of basic geometry. DC: It's even worse in England, where in school they teach almost no geometry.
II-I" Did he claim that he was a mathematician?
IH" Your books are full of quotations. How do you collect them?
DC" I think so, yes. He liked to invent different names for things. For instance, the cuboctahedron he called "vector equilibrium" or something like that.
DC" Just by noticing. I m u s t have read a lot, and I just remember them.
IH" How much interaction did you have with him?
IH: Do you return to books that you'd read before or just keep moving on to other books?
DC." Very little. Once Hendrina and I visited him in his
home in Southern Illinois. I have a friend w h o is a Professor of Philosophy in Carbondale, and while we were there, we visited Bucky's polyhedral house. As people passed by, they were very curious, and he finally had to build a high fence around the house so that people shouldn't see it and he could have some peace. IH" How far back can we detect the regular polyhedra in human history? DC" Of course, Plato wrote about them, and this is w h y they are called Platonic solids. Obviously the Pythagoreans k n e w them before that. Sometimes the archeologists find dodecahedral dice. That sort of thing is what I mean w h e n I say that we don't k n o w how far back they go. IH: In some of your writings you distinguish between crystallographic solids and others such as is the icosahedron and dodecahedron. Nowadays, however, this distinction is quite blurred. DC: That's true. Just look at the writings of Professor
Marjorie Senechal. I'm just reading her lovely book about Quasicrystals which refers to some recent papers of mine. IH: So even geometry is changing and evolving. 38 THEMATHEMATICAL INTELLIGENCER VOL.18,NO.4, 1996
DC: I just move on to other books. When I was y o u n g
I was very interested in stories by H. G. Wells and w h e n I was a student I was very interested in the plays of G. Bernard Shaw. II-I: You have had some connections with M. C. Escher. DC: First, at one of the International Congresses of Mathematicians which took place in Amsterdam, there was an exhibition by M. C. Escher. M y wife, being Dutch, naturally talked to him w h e n he was exhibiting his art to the mathematicians. So she got to k n o w him and that was very helpful; we kept up correspondence. Later I wrote an article for the Royal Society of Canada: m y Presidential address for Section III, on symmetry. It included a Poincar6-style model of the tessellation of (30 ~, 45 ~, 90 ~) triangles filling the hyperbolic plane so as to form a black and white pattern. Escher saw this and t h o u g h t it was just what he wanted. In some of his work he had got tired of filling the plane with congruent figures, fitting together, and he thought h o w nice it w o u l d be if they were not congruent but just similar and changed size while keeping their shape. Escher liked these things because they fulfilled his wish to m a k e a pattern in which he had fishes, for instance, of a good size near the center but getting smaller and smaller as he w e n t towards the circumference. He m a d e Circle Limit I, and then Circle Limits II, III, and IV. Circle Limit III was particularly interesting because it had four col-
ors besides black and white. It was closely related to the hyperbolic reflection group that I'd described. IH: Did you inspire him to this work? DC: That's right. He was very pleased with this idea. After he had seen that paper of mine he did Circle Limits III and IV. He had done Circle Limits I and II before. IH: Did he construct his drawings with precision?
DC: Extraordinarily well, yes. There was a very interesting apparent exception because in Circle Limit III, if you look at the rows of fishes following one another, they have white stripes along their backs so that the circle is filled with a pattern of white arcs that cross one another. It is remarkable that the spaces between the white arcs appear to form a tessellation of hexagons and squares. Yet the white arcs cross one another, three going through each vertex; therefore they cross at angles of 60 degrees. In particular, you seem to have triangles all of whose angles are 60 degrees, and that, of course, is wrong because such a triangle would be Euclidean and not hyperbolic. Bruno Ernst, in his book about Escher, The Magic Mirror, page 109, was similarly disturbed, saying, "In addition to arcs placed at right angles to the circumference (as they ought to be), there are also some arcs that are not so placed." I was interested in this and looked at it for a long time, and at last I realized what had happened. By careful measurement, I saw that all those white arcs meet the circumference at an angle which is very close to 80 degrees instead of 90 degrees. In fact, each of the white arcs does not represent a straight line in the hyperbolic plane but one branch of an equidistant curve. When you put it that way, everything falls into place, and you see that Escher did those drawings with extraordinary accuracy: when I worked it out trigonometrically I found that the angle of 80 degrees is actually arc cos [(21/4 - 2-1/4)/2] -~ 79o58'. IH: Was he aware of this? DC: Absolutely unaware. In his own words: " . . . all these strings of fish shoot up like rockets from the infinite distance at right angles from the boundary and fall back again whence they came." IH: Was it intuition?
DC: True intuition. He came to hear me give a lecture once, and I tried to make it as simple as possible; he said he didn't understand a single word. IH: Mathematicians and crystallographers recognized Escher before anybody else. What was his main appeal? DC: It was the appeal of symmetry.
IH: You give a definition of symmetry in one of your books and that definition, very geometrical, is based on congruency. How far do you think such a rigorous definition can be relaxed? DC: With Escher we've relaxed it to considering shapes that are similar instead of congruent. Groups of similarities are more general than groups of isometries. More precisely, groups of isometries occur as normal subgroups in groups of similarities. Part of the fascination for me was to look at presentations of groups. The groups have generators which satisfy certain relations. There is actually something they call a "Coxeter group," which means you have a certain number of generators of period two and you specify the periods of their products in pairs. It's a very simple idea but apparently nobody had put it like that as defining a particular family of groups. Then it turned out that some of the Coxeter groups have a relationship with Lie groups which I don't understand at all. I am very pleased, though, to see that these ideas have an application. IH: You mentioned before your wife's role in the contact with Escher. What does she do? DC: She is very artistic and appreciates music very much. She's been a wonderful wife to me, looking after me very carefully, and bringing up our children. IH: Did you know D'Arcy Thompson ? DC: He visited us about 1940. He had a tour of Canada
and actually stayed at our house. He was a wonderful man. His book On Growth and Form was very influential, and he brought out a huge second edition when he was 70 years old. He was extraordinary in combining interest in so many different things: in geometry, biology, and classical literature, languages, everything. Very remarkable. IH: How about Kepler? DC: I've been an admirer of Kepler ever since I read that it was he who invented names for all the Archimedean solids, such as the cuboctahedron. Although the names of the Platonic solids are ancient, these less regular figures were only named later. IH: One of the Archimedean solids, the truncated icosahedron, has now become very conspicuous as buckminsterfullerene, the name of the C6o molecule. Unfortunately the chemists who discovered it were not familiar with Kepler's work. DC: They thought this shape was discovered by Buckminster Fuller. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
39
IH: The story of the discovery shows how useful geometry
is, even for chemists. DC: It also illustrates the fact that people who don't know any mathematics, if they happen to play with hexagons and pentagons, inevitably make that figure. This fact was demonstrated very well by a present that I once received from Mrs. Alice Boole Stott: a lampshade made of 12 glass pentagons and 19 glass hexagons, joined together by strips of lead, as in a stained glass window. I may as well tell you a little more about her. About 150 years ago an Englishman, George Boole, started what is known today as Boolean Algebra. He wrote a famous book on finite differences. He had five daughters and they were all distinguished in various ways. The youngest daughter, Ethel, married a Pole called Wojnicz so she is known as Ethel Lillian Voynich. She wrote novels, and one of these novels was called The Gadfly. That novel somehow appealed very much to the Russians at the time of the Soviet Union, and they made a movie of this book. The music for it was composed by Shostakovich. Sometimes one hears excerpts from this music; it's quite fascinating. Another one of Boole's daughters, the middle one, was called Alice; she married an actuary, Walter Stott. I got to know her very well, as it happened, through her nephew, Geoffrey Taylor, who was a mathematician and a Fellow of the Royal Society of London. He was in Cambridge when I was a student there and he introduced me to his aunt, Mrs. Stott, because he realized that she was interested in Archimedean solids as I was. She visited me and my mother, and I visited her very often in London. She was quite elderly and I was a student, so I called her "Aunt Alice." She got to know Dutch mathematicians because her husband happened to notice some articles by a Dutchman called Pieter Hendrik Schoute. Schoute was an expert concerning regular and semiregular polytopes in any number of dimensions, following in the footsteps of Schl/ifli. She was helpful to him and he was helpful to her. Between them they made a complete classification of uniform polytopes in four dimensions. He invited her to Holland and she was given an honorary degree by the University of Groningen. She didn't have a formal education. She was self-taught until she was taught by Schoute. Quite amazing. She had such a feeling for four-dimensional geometry. It was almost as if she could work in that world and see what was happening. She was always very excited when I had things to tell her, and she helped me in what I did. Through her I was introduced to some of the Dutch mathematicians. IH: Did you keep up your interest in Archimedean solids? DC: Yes. In 1950 I was one of the three authors of a paper on uniform polyhedra: a generalization of 40
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
Archimedean solids, the idea being that you have regular faces of two or more kinds and the same arrangement at every vertex. This is characteristic of the prisms and antiprisms as well as the Archimedean solids. If you allow the faces to cross one another, as Kepler did, then you get many more: 53 of these non-convex uniform polyhedra. I wrote a joint paper on these things with Jeffrey Miller (who died long ago) and Michael LonguetHiggins. There are two brothers Longuet-Higgins: Hugh-Christopher is a psychologist and Michael is an oceanographer. It is a unique case. The two brothers are not only Fellows of the Royal Society of London but for five years they were Royal Society Professors, both of them at the same time. So we wrote this paper in 1953, enumerating the uniform polyhedra, allowing them to be non-convex. S. P. Sopov in 1968, and J. Skilling in 1975, using electronic computers, verified that our list is, in fact, complete. IH: When did you leave England? DC: In 1936, when Hendrina and I moved to Canada. Before that I was a fellow of Trinity College. At one point the Princeton topologist Solomon Lefschetz came to visit Cambridge and talked to Professor M. H. A. Newman, who knew me. He happened to mention to Lefschetz that I showed promise in geometry. Lefschetz said that he would arrange for me to get a Rockefeller Foundation Fellowship to spend a year in Princeton. So I went there and was influenced a lot by his colleague Oswald Veblen, who had written a wonderful book on Projective Geometry. While at Princeton I thought about kaleidoscopes, groups generated by reflections, and what sort of fundamental region such a group would have. During a second fellowship in Princeton I was invited to Toronto by Gilbert Robinson, who had earlier been with me in Cambridge. He was a Canadian and had a job at Toronto. So I gave a lecture, and Samuel Beatty, Chairman of the Mathematics Department, must have liked my talk. For quite unexpectedly in 1936, back in England, I received a telegram from him, asking if I'd like to come to Toronto as an Assistant Professor. That was quite startling because usually one starts as a Lecturer and not as an Assistant Professor, so it was very flattering to be asked. I consulted Professor G. H. Hardy and my father; they both said that this was an offer one shouldn't turn down: you never know what's going to happen. In 1936, people already thought that war was possibly coming; so they said, take your newlywed wife and go to Canada, which we did. I met m y wife in 1935 in an English village called Much Hadham, where she was visiting from Holland a certain Mrs. Lewis. My mother introduced me to her neighbor, Mrs. Lewis. A beautiful young Dutch lady was there: Hendrina Brouwer. We liked each other, and
I invited her to come to Cambridge to see my rooms. Later she got a job in Cambridge, and we became engaged, and finally married in 1936. We thought that we would be going back to England in a few years, but then the War came and we remained in Canada.
poem and sent it to the magazine, Nature, where it was actually published. Somehow I got to know about this, and became fascinated by it, and generalized it in an article called "Loxodromic sequences of tangent spheres."
IH: What was your father's profession ?
DC: I've had 17 graduate students who went on to get their Ph.D. and most of them have done quite well. Thirteen of them are professors.
DC: He was a businessman, and at heart an artist. He belonged to the firm of Coxeter and Son, founded by his father and grandfather. They were manufacturers of surgical instruments and compressed gases, especially anesthetics. Nitrous oxide, N20, was their specialty. My father and his partner, Leslie Hall, invented a machine that had a controlled mixture of nitrous oxide and oxygen to give to a person undergoing an operation. The anesthetist would watch the patient and gave him more oxygen if he seemed to be failing and more nitrous oxide if he seemed to be coming awake. That has been used ever since by some hospitals. I wish it were used more; it is a wonderfully safe anesthetic. IH: Have you ever met John Bernal?
DC: I visited his laboratory in London. He worked with little balls of plastic clay; rolled them up, dusted them, put them together in large numbers and squeezed them to see what shapes they formed. I visited him because of my interest in sphere packing. He was a very fascinating person. Another man in the same direction was Frederick Soddy. He was the man who invented the name "isotope." I knew him because of his interest in the Descartes circle theorem. IH: Soddy was a chemist. DC" Yes, but he was also interested in geometry, just like you. I met Soddy around 1933. I visited him in his house on the south coast of England, and had a wonderful walk with him along the beach. He wrote an article for Nature about the problem of putting circles in contact with one another. The particular problem that started it was about four circles in an ordinary plane all having contact with one another. It's very easy to make three circles have contact; the fourth one will go in between in the middle or outside. So you have four circles in mutual contact. Soddy noticed that if you don't consider the radii themselves but their reciprocals, the curvatures of the circles, then the four curvatures satisfy a nice quadratic relationship: the sum of the squares of the curvatures is half the square of their sum. He didn't know that this was already discovered hundreds of years before by Descartes. Soddy wrote the theorem in the form of a
IH: How about your pupils?
IH: You have been retired for some time now, but stayed very active. DC: I have been retired for 23 years, but the University is kind enough to let me have this little office and so I g o on.
IH: Do you need any support for your work? DC: No, just this office. Of course, I have a pension which is an annuity. Then sometimes I get a hundred dollars for a lecture, and recently I was awarded a prize for research by the Fields Institute in Toronto and the Centre de Recherches Math6matiques in Montreal. IH: You don't use a computer. DC: No, I never used a computer. I'm too busy writing with pencil and paper. Fortunately, they have a very good secretary here who does word-processing. Then she says ! mustn't mind that I am fourth in line and she may have the paper typed by next Monday, but that's all right. IH: What's your next paper about?
DC: At the moment I'm writing a paper on the trigonometry of hyperbolic tesellations. Escher may have known the solution intuitively, by trial and error, and I suppose he might have been interested in seeing precisely how to find the centers and radii of all his circular arcs. H.S.M. Coxeter Department of Mathematics University of Toronto Toronto M5S 3G3 Canada Istvdn Hargittai Department of Chemistry University of North Carolina at Wilmington 601 South College Road Wilmington, NC 28403, USA
[email protected] THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
41
The Trigonometry of Escher's Woodcut "Circle Limit III" H. S. M. Coxeter
In M. C. Escher's circular woodcuts, replicas of a fish (or cross, or angel, or devil), diminishing in size as they recede f r o m the centre, fit together so as to fill and cover a disc. Circle Limits I, II, and I V are b a s e d on Poincar6's circular m o d e l of the hyperbolic plane, w h o s e lines appear as arcs of circles orthogonal to the circular b o u n d ary (representing the points at infinity). Suitable sets of such arcs d e c o m p o s e the disc into a theoretically infinite n u m b e r of similar "triangles," r e p r e s e n t i n g congruent triangles filling the hyperbolic plane. Escher replaced these triangles b y recognizable shapes. Circle Limit III is likewise b a s e d on circular arcs, but in this case, instead of b e i n g orthogonal to the b o u n d a r y circle, they m e e t it at equal angles of almost precisely 80 ~. (Instead of a straight line of the hyperbolic plane, each arc represents one of the two branches of an "equidistant curve.") Consequently, his construction required an e v e n m o r e i m p r e s s i v e display of his intuitive feeling for geometric perfection. The present article analyzes the structure, using the elements of t r i g o n o m e t r y and the arithmetic of the biquadratic field QX/2 + X/-3): subjects of which he steadfastly claimed to be entirely ignorant.
comings of Circle Limit I are largely eliminated. We now have none but "through traffic" series, and all the fish belonging to one series have the same colour and swim after each other head to tail along a circular route from edge to edge . . . . Four colours are needed so that each row can be in complete contrast to its surroundings. As all these strings of fish shoot up like rockets.., from the boundary and fall back again whence they came, not a single component reaches the edge. For beyond that there is "absolute nothingness." And yet this round world cannot exist without the emptiness around i t . . . because it is out there in the "nothingness" that the centre points of the arcs that go to build up the framework are fixed with such geometric exactitude. ([2], p. 109) The p u r p o s e of the present article is to d e m o n s t r a t e this " g e o m e t r i c exactitude" (see Fig. 2) b y finding the radii a n d centres of the first three sets of four c o n g r u ent circles that trace the backs of the "strings of fish." I n a t u r a l l y a s s u m e that the relevant arcs of these circles cross one another at equal angles of 60 ~, d e c o m p o s e the interior of the " b o u n d a r y " into alternate triangular a n d q u a d r a n g u l a r regions, a n d all cut the b o u n d a r y at the s a m e pair of s u p p l e m e n t a r y angles 09,
"~T-- 09.
Introduction Concerning his four Circle Limit w o o d c u t s , M. C. Escher wrote: Circle Limit I, being a first attempt, displays all sorts of shortcomings ... and leaves much to be desired . . . . There is no continuity, no "traffic flow" nor unity of colour in each row . . . . In the coloured woodcut Circle Limit III, the short42
The acute angle w a p p e a r s on the side of each arc w h e r e the regions are q u a d r a n g u l a r . An earlier article ([1], p. 24) used hyperbolic t r i g o n o m e t r y to p r o v e that
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
cos w = sinh(14 log 2) sinh 0.1732868 ~ 0.1741553.
Figure 1. Escher's Circle Limit III. 9 1996 M.C. Escher/Cordon Art-Baarn-Holland. All rights reserved.
Since cos(79~ ') ~ 0.17424, co scarcely differs f r o m the value 80 ~ which can easily be m e a s u r e d in Escher's woodcut. H e r e I obtain this expression for co b y a m o r e e l e m e n t a r y procedure.
T h e A n g l e o~ at t h e B o u n d a r y Figure 2 is a sketch of the m i d d l e p a r t of Escher's " f r a m e w o r k , " s h o w i n g the centres O~, at distances d~ = A O ,
f r o m the centre A of the b o u n d i n g circle, of radius 1, a n d s h o w i n g the radii r~ = O ~ X ~ .
F r o m the triangle XIAO1, w h o s e angle co at X1 is opposite to the side A 0 1 d 1, as in Figure 3, w e h a v e =
d 2 = 1 + r 2 - xrl,
(1)
x = 2 cos co.
(2)
where
THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996 43
X-
G
Figure 2. Escher's "framework."
Similarly, the triangle X 2 A 0 2 , whose angle ~- - o~ at )(2 is opposite to d2, yields
Because the angle between two intersecting circles equals the angle between their radii to a c o m m o n point, the triangle 0 1 A C has angles 2r rr/4, and ~-/12 opposite to sides
X 1
-
CO 1 = rI
C A =- d 2 -
that is, dl
rl
(4)
r2,
o,
v
A
~
C
v
0 2
Figure 3. Triangles w i t h angles w at X a and X3, 1r - w at X2. 44
dl _ rl _ d2 - r 2 sin(21r/3) sin(1r/4) sin(~-/12)'
(3)
d 2 = 1 + r 2 + x r 2.
AO 1 = dl,
respectively, as in Figure 4. Hence, we have
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
A
C
Figure 4. T h e s i m i l a r triangles
02 01AC and
02AB.
The similar triangle 0 2 A B , with angles 27r/3 and rr/4 opposite to sides AO 2 = d2
and
V2 : 1(0_
0-1),
~
- 1 ( 0 q- 0-1),
: 1(02 _ 5).
In this field, 0 is called an integer because it satisfies a monic equation, namely
B 0 2 = r2,
respectively, yields
04-1002+1 d2
G
=0.
r2
vT
When w e assert that "factorization is unique," w e disregard, as factors, the units, which are divisors of 1; for if st = 1, any number
Thus, d 2 = 3 2r 2 v ( v = l o r 2 )
and expressions (1) and (3) for dayyield quadratic equations for G: r 2 + 2xrl - 2 = O,
r 2-
n = nst
(5)
has the trivial factorization ns x t. In our approach to (7) w e replaced (1 -[- ~ -
2xr2 - 2 = O.
Solving these equations for the positive numbers G, w e find
~V/3)2
by
2(V3-
1 ) ( V 3 - V2).
This " f a c t o r i z a t i o n " loses its element of surprise w h e n w e face the obvious fact that 1/3 - ~ is a unit: ( V 3 + X / 2 ) ( V 3 - h / 2 ) = 1.
r I = - x + V ~ + 2,
r 2 = X + ~ X 2 + 2.
(6)
The First Two Circles From (4) we have Since V ~ x2 + 2
=
V21/2 +
2 -1/2
2-1/4V3, (6) yields
=
(Nf3 - 1)rl = 2(d2 - r 2) = (V6 - 2)r2. r I = 2-'/4(1 - V 2 + V 3 ) ~ 1.1081646, r 2 = 2-1/4(V2 - 1 + V 3 ) --~ 1.8047860,
In terms of x, this equation becomes, in turn, (X/3 - 1 ) ( - x + V~x2+ 2) = ( V 6 - 2)(x + X/-~x2 + 2), ( 3 - X / 3 - V6)x = - ( 1 + V 3 -
and, from (5), d 1 = 2 - 3 / 4 ( V 3 -- G
V6)X/~x2 + 2,
d2 = 2-3/4(G
{(3 - X / 3 - X/-6)2 - (1 + X / 3 - V6)2}x 2 = 2(1 + V 3 -
X/-6)2,
(2 - 2X/-3)(4 - 2V6)x 2 = 2{(V2 - 1)(1 + V 2 - V3)} 2,
q- 3) ~ 1.3572189,
- V 3 + 3) ~ 2.2104024.
From (4), A C = d 2 - r 2 = 21-(V3- 1)r 1 = 2
3/4(1 + V 2 -
V3)
0.4056164. (g3-
1)(G-
V2X 2 • (V2-
2)x 2 = ( V 2 -
1 ) 2 ( V ~ - 1 ) ( V 3 - Xf2), Finally, the triangles C A 0 1 and B A 0 2 are similar,
1)2,
x = 2-1/4(21/2 - 1) = 21/4 - 2 -1/4 = 2 sinh(88 log 2).
(7)
r2 V3 - 1 F1-G 2-
The Biquadratic field Q (X/2 + V3)
and
The numbers (a + b X / 2 + c V 3 + d X / 6 ) / q , where a, b, c, and d are integers and q is a positive integer, are easily seen to constitute a field ([3], p. 230). This field is called Q ( V 2 + X/3) because it can be expressed as the set of all rational functions of the special n u m b e r 0 = X/2 + X/3, in terms of which
(1 +
v5)(3-
=
v5 + G ) = V 2 ( - 1 + 2~/2 + ~ -
"V6);
hence A B = 2-3/4( - 1 + 2 V 2 + V 3 - V6) ~ 0.6605975. THEMATHEMATICALINTELLIGENCERVOL.18,NO. 4, 1996 45
22.7, 37.0, 6.92,
The Third Circle Looking again at Figure 3, we see that
27.8, 4.53, 20.46.
These distances agree perfectly with actual measurements in the w o o d c u t itself.
d 2 = 1 + r 2 - xr3
and, since the third circle passes t h r o u g h B, Acknowledgments
d 3 - r3 = AB.
I am grateful to J. Chris Fisher for the numerical computations and to Catherine Crockett for Figures 2, 3, and 4.
Thus,
d3 + r3 -
(1/AB) Y3 =
=
2r 3 _ 1 -
1 - xr 3 AB '
xr 3
AB,
AB - AB
_
2 + x/AB
References
1 - AB 2 2AB + x
1. H. S. M. Coxeter, The non-Euclidean symmetry of Escher's picture "Circle Limit III," Leonardo 12 (1979), 19-25, 32. 2. B. Ernst, The M a g i c M i r r o r of M . C. Escher, New York: Random House (1976). 3. G. H. Hardy and E. M. Wright, A n Introduction to the T h e o r y of N u m b e r s , 4th ed., Oxford: Clarendon Press, 1960. D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of Toronto Toronto M 5 S 3 G 3 Canada
2-1/4
3 /
= 2-1/4 (
3 /
= 2-1/4(
-3)
= 2-1/4 (
MOVING? We need y o u r new address so that y o u do not miss any issues of
= 2-1/4 (
THE MATHEMATICAL INTELLIGENCER.
--~ 0.3375915
Please fill out the form below and send it to: Springer-Verlag N e w York, Inc. Journal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485
and d 3 = r3 + AB
= 2 -3/4
Name Old Address (or label)
0.998189.
Address City/State/Zip
Since Escher's b o u n d i n g circle has diameter 41 cm, our results
Name r I ~ 1.10816,
dl ~ 1.3572,
r 2 ~ 1.8048,
d 2 ~ 2.2104,
r 3 -~ 0.3376,
d 3 ~ 0.9982
should be multiplied b y 20.5 to obtain the distances in centimetres: 46
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
New
Address
Address City/State/Zip Please give us six weeks notice.
Nestor of Mathematicians: Leopold Vietoris Turns 105 Gilbert Helmberg and Karl Sigmund
His latest research paper--a short note on Euclidean geometry--is currently in print. Some of his other recent publications deal with statistics. Leopold Vietoris, who was born on the fourth of June 1891, can see the statistics of age distributions from the vantage point of an outlier. He looks a bit frail today, and wears a hearing aid, but otherwise is in good shape. The family name dates back to the time when German intellectuals took to latinizing their names (cf. Mercator, Regiomontanus). The father of Leopold Vietoris was a railway engineer who expected his son to follow in his steps, but after reading a book on geometry, young Leopold switched to mathematics. The First World War broke out before he could complete his studies at the University in Vienna. He was immediately called to arms, and severely wounded in 1915 on the Russian front. After his recovery, his regiment entrained for Bosnia, but then was shunted to South Tyrol and the Italian front. This was a decisive turning point in the life of Vietoris. Stationed in a snow-bound Alpine hamlet, he started working on topology and learned to ski--this at a time when both the science and the sport were in their thrilling pioneer stage. The war, of course, soon caught up with him. As a front officer and a military mountain guide, Vietoris had his full share of it right up to the end, and was detained after the Austrian collapse in an Italian camp till August 1919. The treatment there was so decent that he was able to write his Ph.D. thesis on "continuous sets" (i.e., connected sets). After his return to Austria, Vietoris became
assistant, first in Graz to Weitzenb6ck (whose ideas are experiencing a remarkable revival lately) and then in Vienna. With his very next paper he obtained the coveted title of Dozent. The 1920s were a heady decade for topologists, and Vienna was as good a place to be as any, with Hahn, Menger, Reidemeister, and later Hurewicz and N6beling
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
47
ogy was later extended to general topological spaces. When Vietoris returned to Vienna and lectured there on topology, this led to the method of the Mayer-Vietoris sequence, which allows one to compute the homology group of X U Y in terms of those of X, Y and X n Y. (Walter Mayer, 1889-1953, was at the time the owner of a small coffee-house. Later he became Einstein's assistant and collaborator in Berlin and Princeton.) The theorem of Vietoris and Begle which relates the homology groups of a compact metric space and its image is also a classic. Furthermore, Vietoris introduced (at about the same time as Lefschetz, Alexander, and Pontrjagin) the notion of cocycles. And in 1931, he wrote jointly with Tietze (who had left Vienna in 1910) an encyclopedia article on the relation between the different branches of topology which is still well worth reading. By then, the other passion of Vietoris proved decisive: the High Alps. He preferred their eerie silence to t h e opinionated debates of the Vienna Circle and to the fervid atmosphere of the Austrian capital, which Karl Kraus had described as "laboratory for the apocalypse." In 1927, Vietoris eagerly accepted a position as associate professor in Innsbruck. In 1928 he returned to Vienna to become full professor at the Technical University (and to marry), but in 1930 he was offered a
Five years ago, Leopold Vietoris celebrated his hundreth birthday. He still l o o k s the same.
around. In the general commotion many ideas emerged independently and almost simultaneously in several places. Vietoris, who always was an extremely modest person, never engaged in priority debates (quite in contrast, for instance, to his young and fiery colleague Karl Menger). But Vietoris was the first to introduce filters (which he called "wreaths") and one of the first to define compact spaces (which he called "liickenlos"), using the condition that every filter has a cluster point. He also introduced the notion of regularity, and proved that (in modern parlance) compact spaces are normal. During a stay in Amsterdam with Brouwer (which was financed by a scholarship from the Rockefeller Foundation shared with Karl Menger), Vietoris became one of the founding fathers of algebraic topology. His contribution in this field is much better recognised: Saunders MacLane, for instance, wrote some 60 years later a paper entitled "Topology becomes algebraic with Vietoris and Noether." Indeed, Vietoris adapted Brouwer's technique of simplicial approximation and used it to define the concept of homology group for compact metric spaces. This notion of Vietoris h o m o l 48
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
When topology became algebraic, Vietoris (here in a portrait from 1930) had a hand in it.
Glory to the pioneers. In the good old days, skiers had to make do with one stick only (Vietoris has the longest). Notice the fair proportion of intrepid ladies.
chair in Innsbruck and did not hesitate one minute to accept. The m o u n t a i n - b o u n d capital of Tyrol was the ideal place for him. Quickly spotted as a committed m o u n t a i n fan in Innsbruck's faculty, Vietoris became involved with the
local school of glaciologists led b y the distinguished Finsterwalder. In the role of a Gletscherknecht, he carried the h e a v y instruments for geological measurements and set u p experiments in countless scientific alpine excursions. In due time, Vietoris started publishing himself
On the rocks. Kept fresh by a life among glaciers, Vietoris (here at age 80) feels truly at home at high altitudes.
THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996 49
on the blockstream of the Hochebenkar, a glacier-like formation of rock d6bris pasted together by ice, which he had come to k n o w like no one else. He also wrote on how to use the compass as an alpinist (rather than a sailor), on "geometry in the service of the mountaineer," and on the physics of skiing, and he held patent no. 100832 for a m e t h o d of using air photographs in cartography. At the start of the Second World War, Vietoris was enrolled again, and w o u n d e d in Poland. On reaching 50, he was allowed to resume his post at the university in Innsbruck. Vietoris rarely returned to topology, but started working on mechanical integration, on probability and statistics, and on real analysis (thus he developed a method to introduce the sine by a functional equation which has found its w a y into some textbooks). Much of this was continued during five decades of peace following the war, which were filled with scientific work, increasing
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academic recognition, and a rich family life (he has six daughters and altogether 51 descendants, at current count). His interests have returned to geometry; his first paper on the subject, which was almost finished before the First World War, is still his favorite. A n d the lure of the mountains did not abate. At the university skiing championships he routinely w o n the gold medal in his age class, ultimately by being the only one at the start. Ten years ago, doctors told him that he really ought to stop skiing. Vietoris said that he would think it over. Institut far Mathematik und Geometrie Universit/tt Innsbruck 6020 Innsbruck Austria Institut far Mathematik Universitft't Wien 1090 Vienna Austria
CONTENTS:
9 Numerical Computation (Arithmetical and Numerical Analysis) 9 Equation and Inequalities (Algebra) 9 Geometry and Trigonometry in the Plane 9 Solid Geometry 9 Functions 9 Vector Calculus 9 Coordinate Systems 9 Analytic Geometry 9 Matrices, Detemainants, and Systems of Linear Equations 9 Boolean Algebra - Application in Switching Algebra 9 Graphs and Algorithms 9 Differential Calculus
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Jeremy J. Gray* Enriques and the Popularisation of Mathematics In 1906 Federigo Enriques published a successful popular work, later issued in English in 1914 as The Problems of Science. The book, and its reception, tell us a lot about the popular understanding of science at the time. The years before 1914 were an exciting period for scientists and mathematicians. After Paris in 1900 and Heidelberg in 1904, the International Congress of Mathematicians came to Rome in 1908, and Cambridge (England) in 1912, while the peripatetic philosophers went to Paris, Geneva, Heidelberg, and in 1912 to Bologna, where Enriques presided. The Congresses offered images of the Italians in various ways: at Paris, Italians appeared as logicians. Naturally, as an eloquent spokesman for this community, Enriques benefited from the depth and range of his countrymen's work, and he was indeed regarded as drawing on that collective wisdom. W h a t D o e s the P u b l i c A s k from M a t h e m a t i c i a n s ?
Between 1900 and 1914 the American public mostly supported the progressives, who want money spent wisely on social reform. In this spirit Woodrow Wilson, on becoming President of Princeton in 1902, said, "Science and scholarship carry the truth forward from generation to generation and give the certain touch of knowledge to the processes of life" (quoted in Kevles, p. 70). James McKeen Cattell, who was one of the early American psychologists and a former student of Wilhelm Wundt in Leipzig, edited the Popular Science Monthly, from 1900. He used his influence to extol science and its uses, but the enterprise was not a lasting financial success. In 1915 the owner sold the journal, and Cattell brought out a new one, Scientific Monthly, which retained the popular science but dropped the social reform (Kevles, p. 96); reformers, led by Dewey, were no longer looking in such numbers to science. We get a sense of what was wanted from scientists by looking at what other writers asked of them. Paul Carus built up the Open Court Publishing House in Chicago *Column editor's address: Faculty of Mathematics, The Open University,MiltonKeynes,MK7 6AA, England.
and edited The Monist, where he published essays by Ernst Mach, Poincar6, and many others. Open Court published the English translation of Poincar6's books of essays, and Hilbert's Foundations of Geometry. Carus was himself a hugely prolific author, concentrating on philosophy and religion, and in these books his aim was to make a science of philosophy. His book on mathematics (The Foundations of Mathematics, 1909) opens with a lengthy, intelligent, and up-to-date account of nonEuclidean geometry. His sources are the American G.B. Halsted (translator of Bolyai and Lobachevsky into English) and the excellent text by Engel and St/ickel [1895]. So Carus had done his homework. From the existence of hyperbolic and spherical geometry he was led to ask what this can mean for our understanding of space. He wanted to retain a primitive concept of straightness (we need not discuss his attempt here), and so he had to confront the question of the true geometry of space. The authority he turned to at this point was C.S. Peirce, the wild man of American thought, who set him right by describing astronomical tests that discriminate between the three geometries. But Carus had read, indeed published, Poincar6, and knew that Poincar6 believed that the question was purely a matter of convention. Carus agreed with him too, saying that the straightness of the Euclidean line is the most convenient, allowing one to keep with Euclidean geometry, which in any case is a good approximation to the truth. This is an example of what I mean by an issue in mathematics. There is a real question: what is the nature of space? The question is mathematical: no one is out there anxiously measuring parallaxes; any pragmatic person would accept that the true geometry is Euclidean to a sufficient degree of accuracy. But apparently mathematicians have come up with other geometries, so the audience wants to know what that means, what these other geometries are, and how we can tell which is true, even though that question is entirely abstract. There is typically another issue involved, which has overtones of psychology. The late 19th century saw the start of serious professional psychology as an academic discipline. Mindful of the error into which Kant had
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO, 4 9 1996 Springer-Verlag N e w York
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Federigo Enriques fallen, psychologists took care to make sure that their theories allowed human beings to discover nonEuclidean geometry. The pioneer in this regard was the energetic if unsystematic Wilhelm Wundt. Gradually psychologists turned from the investigation of the prerequisites for any kind of thought to the study of how people think. What are concepts, what is it to know something? What sort of an intellectual activity is the study of geometry, regarded as the elementary appreciation of space? A third issue also presented itself: logic. There was a psychological side to logic, and an abstract side. At the resolutely hard-edged, anti-psychology end stood Gottlob Frege. Dedekind, Russell, and Peano wished to reduce mathematics to logic or to derive mathematics from logic. Hilbert may have wished to reverse the process, at least on occasion, and derive logic from axiomatic set theory, but he was equally austere. At the other extreme one might place Poincar6, who roundly denigrated attempts to derive the integers from abstract sets. There was a widespread feeling, in logic as in geometry, that much had happened since the Greeks, and much of that recently. These are significant issues, worthy of attention. H o w the mind works, how we come to know the simplest but most essential features of the world, what logical thinking is--are real questions. And, as the public could easily discover, the experts found them thorny too.
Against the Priority of Deductive Thinking Enriques regarded his philosophical position as critical and positive. Part of his programme was an investi52
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
gation of logic, and an integral part of that was an investigation of psychology, which he hoped would ultimately be reducible to physiology. This reduction, which was of course very sketchy, was intended to explain how the laws of thought related to the world of phenomena. Enriques argued that we acquire a knowledge of the deductive logic of the mathematician through evolution as a species, and each of us is more or less born with it. It differs from the creative process called inductive logic, which brings with it the problem of explaining "the real meaning and the means of acquiring the more general and abstract concepts of geometry" (p. 100). But Enriques did not have in mind anything as simple as a logic of discovery and a logic of proof. Enriques pointed out that psychological conviction comes with the experience of finding that a hypothesis is not refuted. Indeed, he observed, this is exactly what the creators of non-Euclidean geometry had had to g o on; the rigorous proofs came a generation later. So Enriques rejected Kant's distinction between absolute knowledge and absolute ignorance in favour of a spectrum of degrees of knowledge. Within this spectrum, knowledge is acquired, he argued, by a mixture of nativist and empiricist methods. Association of ideas and abstraction, combined with logic, give rise to the explanation of the postulates of elementary geometry and the feeling of self-evidence that accompanies them. One part of a postulate will appear as a fact, which is a condition needful for uniting infinitely many examples, the other part as an axiom of a logical kind. Drawing on the so-called associative theory of Taine and Delboeuf as well as the work of Mach, he produced an analysis of how the different spaces of sensation are united into the general concept of space. There are tactile, visual, and muscular spaces: and of these the tactile-muscular is constructed with the notion of distance, and so has circles and spheres, whereas the line belongs to visual space, which is formalized in terms of projective geometry because it is not metrical. The anomalous position of the parallel postulate arises because it combines touch and sight (p. 229). In support of this position he offers us the historical observation that Saccheri made progress by departing from the then current definition of parallels as equidistant lines on the grounds that this was a complex definition. It is complex, according to Enriques, in that it derives genetically from several other conceptions. But matters are not so simple, even for Enriques. Earlier on, he had argued that Legendre's definition of a straight line as the shortest curve which can connect two points is flawed, because from it one cannot deduce that two distinct straight lines meet in at most one point. Yet the definition is not fatally flawed. Enriques said that it does teach something, and pupils find that it gives them a precise idea. So definitions do two things. Some, usually the more abstract, really define. Others, usually the initial ones, gesture. It is because these two kinds of definition
Albert Einstein and Enriques.
are inescapable that Enriques did not regard geometry as a purely formal, axiomatic science, and rejected the idea of a logic of discovery as something that can and must be kicked away by more formal methods before one can speak of knowledge. To do so, for him, would be to reject not only personal experience but history. In some respects, Enriques's psychologising makes his position close to Poincar6's, but Enriques rejected Poincar6's conventionalism, arguing that the cooledplate universe, for example, makes temperature a geometrical feature of the world, and for that reason this geometry is truly different from ours. None of this means, of course, that Enriques thought geometry was 3-dimensional and Euclidean. Other geometries, higher dimensions, non-Euclidean and even non-Archimedean spaces enter our thinking precisely through the method of abstraction, but they cannot transcend their origins. For Enriques, mathematics was conceptual, the knowledge it provided open to change. The initial phase of research into the principles of geometry sought merely to simplify Euclid's Elements. Searching criticism began with Riemann, "the deepest philosopher of geometry. Here, through the direct influence exerted by Herbart on his disciple, a psychological criterion also enters the field as a guiding interest." From this psychological origin stem the two main fields of geometry, the metrical and the projective, as already discussed.
Enriques and Royce The Problems of Science was well received abroad. It was translated into German, and in two parts into
French. The publication of the first part, on logic, was the occasion for H.M. Sheffer at Harvard to call for an English translation (Philosophical Review 19 (1910), 462-3). As Sheffer may well have known, the English translation was by then under way. It was made by Katherine Royce, the wife of Josiah Royce, who was the senior professor of philosophy at Harvard and had taught Sheffer. Royce had met Enriques at the International Congress of Philosophers in Heidelberg in 1908 and been impressed by him. He wrote to Cattell in 1908 that Enriques's book "has the advantage over Poincar6's of going deeper into modern logical problems," and that "as the book of a modern geometer and a notable representative of the great Italian school of logic, it would occupy a novel place in the literature." Paul Carus agreed to publish the translation, but various administrative difficulties prevented Royce from finishing his wife's translation (as he had agreed to do) before the end of 1913. By then, however, Royce hoped in his Introductory Note to the book, it might be useful in combatting the recent rise of anti-intellectualism, which Royce feared would prefer easy, dramatic answers to patient, critical thought. Royce argued that although Enriques's reputation was founded on his treatise on projective geometry and his essay on the foundations of geometry (no mention of the work with Castelnuovo on classifying algebraic surfaces), as a philosopher he said much that pragmatists could accept. This was all the more surprising because the Italian book had been published in 1906, while the vogue for pragmatism had not begun until Heidelberg in 1908. Indeed, Enriques's form of pragTHEMATHEMATICALINTELLIGENCERVOL.18,NO.4,1996 53
counts. Enriques's interests in logic have nothing to do with the severe formalism of Peano, and everything to do with psychology. But these both enter in the popular perception of mathematics, and Enriques was taken to speak for both. Likewise Royce saw no problems in harmonising Enriques's ideas with any others he (Royce) happened to support. T h e V i e w from the 1990s
Josiah and Katherine Royce. Probably in their early Cambridge years. (From Josiah Royce, by Robert V. Hine. Photo courtesy of Nancy Hacker.)
matism was largely original, many-sided, and judicious. Royce went on to comment on many aspects of Enriques's diverse yet synthesising approach, before concluding by welcoming the book above all as a treatise on methodology. Royce himself is a major figure: many-sided and brilliant, in the view of Norbert Wiener. He graduated with distinction from the fledgeling university of his native California in 1875, and went to Leipzig where he studied logic and anthropology under Wundt. Then he went to G6ttingen to study under Lotze (he also took some mathematics courses). He married in 1880 and became a professor at Harvard in 1882. There he was an intimate friend of William James. Some time after 1900 he turned from philosophy and religion to the study of logic. In 1905 he published a lengthy paper, inspired by work of Kempe, on the close connection he saw between logic and geometry, citing work in the axiomatic style by Huntingdon at Harvard and Veblen at Princeton. So we see in Royce a breadth reminiscent of Wundt, who had impressed him as a young man; a turn towards mathematics; a definition of philosophy that embraced psychology as well as logic; and a sense of pragmatism. All these naturally pre-disposed him to the writings of Enriques. Insofar as they were typical--and they increasingly were---they indicate that in American intellectual circles Enriques had a ready-made audience. When a scholar is received as a spokesman, he is given credit for a point of view that is broader than his own. This happens in two ways. He is taken to represent a school to which he may barely belong, and he is assimilated to traditions that he may not subscribe to. He may be falsely praised and also falsely criticised on these ac54
THE MATHEMATICAL [NTELL[GENCER VOL. 18, NO. 4, 1996
The mathematical community has evolved sophisticated ways of reading Enriques's work in algebraic geometry, and we see most of it as either correct or easy to put right. It is harder for us today to accommodate his writing as a philosopher or populariser. He held a subtle position, according to which knowledge is inseparable from the means of knowing, logic from psychology. This has long been unfashionable in the sciences. It may be that cognitive psychology will reopen the avenues Enriques explored; there are signs that it has reached at least the philosophy of mathematics. But that field is not what it was; even its practitioners detect a widespread feeling that philosophy of mathematics has drifted out of the mainstream and is becalmed. Moreover, the English-speaking world is losing its sense that a philosopher is the right person to ask. France is different, of course, but there the work of Derrida and the iconoclastic, deconstructionist spirit of Bruno Latour, which has made him a guru on the practise and policy of science, offer no platform for popularisation. What these intellectuals propose is a social critique; their expertise is a skill, they merely claim to be quicker at unmasking presuppositions and seeing through mystiques than the rest of us. They do not offer a theory, but claim to be the enemy of all theories. They are market leaders in a nervous, disenchanted, cynical age. Enriques offered a position on the nature of knowledge that was original and sophisticated. His readers found a rare grasp of modern science, traditional philosophy, and contemporary psychology. If only someone would make a similar contribution to the present debates about the nature, meaning, and value of mathematics and science! References
Carus, P., 1909: The Foundations of Mathematics, Open Court, Chicago. Enriques, F., 1906: Problemi delle scienze, Zanichelli, Bologna. Enriques, F., 1914: The Problems of Science, translated by K. Royce with an introduction by J. Royce, Open Court, Chicago. Kevles, D. J., 1987: The Physicists; the History of a Scientific Community in Modern America, Harvard University Press, Cambridge, Mass. Engel, F., and St/ickel, P., 1895: Die Theorie der ParalMlinien yon Euklid bis auf Gauss, Leipzig, Teubner.
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Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the
famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.
The Bone that Began the Space Odyssey D. Huylebrouck
The Ishango Bone, a Mesolithic Mathematical Artifact
Classical Greek mathematics had its roots in Egypt and Mesopotamia; going further back in time, one has to turn to non-written proto-mathematics. From bones, strings, and standing stones, early mathematical reasoning can sometimes be read off quite clearly (see [Jos]). In the Lebombo mountains between South Africa and Swaziland, a 37,000-year-old baboon fibula was found, marked with 29 notches, while a 32,000-year-old bone from ex-Czechoslovakia has 57 notches (see [Bog]). These engravings are most probably simple tally marks to ease counting; a more interesting find is the so-called Ishango bone. It has carvings according to an unknown but intriguing pattern. Ishango is a little village on the shores of Lake Edward, one of the farthest sources of the River Nile, on the border of Uganda and Za~re (see Figure 1). In the surrounding mountains, Prof. Jean de Heinzelin did archeological excavations at a Mesolithic settlement about 20,000 years old. Some archaeologists believe the early Ishango-man was a pre-sapiens species. The later inhabitants of that region, who gave it the name Ishango, have no immediate ties with the primary settlement, which was buried in a volcanic eruption. Together with many stone tools and human remains, de Heinzelin excavated a little carved bone of about the 56
Fig. 1. The earliest mathematical artifact was discovered at Ishango, at the sources of the Nile.
THE MATHEMATICAL 1NTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
Fig. 2. The Ishango Bone from de Heinzelin's original representation. size of a pencil. It even looks like a writing device: it has a firmly fixed piece of quartz at one end (see Figure 2). It is generally supposed to be about 11,000 years old, but this is unsure, because the nearby volcanoes have upset the usual ratios of isotopes used by the carbon 14 method. There are three separate columns, each consisting of sets of notches arranged in distinct patterns. The upper photograph shows two rows of notches on the bone. Above, there are four groups composed of 11,
13, 17, and 19 notches (from right to left); below it, there are 11, 21, 19, and 9 notches, in that order. The third column has the indentations arranged in eight groups: 3, 6, 4, 8, 10, 5, 5, 7. The following observations suggest the Ishango man might have carved the bone according to some kind of pattern: (a) the markings on the second row obey the rule "10 + 1, 20 + 1, 20 - 1, 10 - 1";
Fig. 3. Marshack's interpretation of the bone as a lunar calendar. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
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(b) the pairs of the third row are related b y duplication or halving, except for the final 5 and 7; (c) counting the carvings on the first r o w yields the prime n u m b e r s between 10 and 20; (d) the markings on the first two rows each add u p to 60. The interpretation of these observations is more speculative. Fact (a) could indicate the use of a n u m b e r system with base ten, while the next two observations seem to indicate some arithmetic was done. The final fact suggests the bone could have been a lunar calendar, since 60 corresponds to two lunar months; 48, the sum of the n u m b e r of carvings on the last row, accounts for about a m o n t h and a half. The arithmetical g a m e interpretation was favored by de Heinzelin. H e based his option on archaeological evidence, and c o m p a r e d the Ishango h a r p o o n heads to those found in n o r t h e r n Sudan and ancient Egypt. The professor emeritus of the Ghent University is an authority in the field of African archaeology, and suggested a link b e t w e e n the Ishango arithmetic and the c o m m e n c e m e n t of mathematics in ancient Egypt: It is possible to trace the influence of the Ishango technique on other African peoples by examining harpoon points at other sites. From central Africa the style seems to have spread northward. At Khartoum near the upper Nile there is a site that was occupied considerably later than Ishango. The harpoon points found there show a diversity of styles. Some have the notches that seem to have been invented first at Ishango. Near Khartoum, at Es Shanheinab, is a Neolithic site that contains harpoon points bearing the imprint of Ishango ancestry. From there the Ishango technique moved westward, but a secondary branch went northward from Khartoum along the Nile Valley to Nagada in Egypt.[...] The first example of a well worked out mathematical table dates from the dynastic period in Egypt. There are some clues, however, that suggest the existence of cruder systems in predynastic times. Because the Egyptian number system was a basis and a prerequisite of classical Greece, and thus for many of the developments in science that followed, it is even possible that the modern world owes one of its greatest debts to the people who lived at Ishango. Whether or not this is the case, it is remarkable that the oldest clue to the use of a number system by man dates back to central Africa of the Mesolithic period. No excavations in Europe have turned up such a hint. (After de Heinzelin [Hei], p. 109 and 116). The explanation of the patterns carved on the Ishango bone as a lunar calendar was s u p p o r t e d b y A. Marshack, w h o carried out a detailed microscopic examination of the bone (see [Mar]). H e found markings of different shapes and sizes, and m a d e a diagram w h e r e the phases of the m o o n c o r r e s p o n d e d to the Ishango notches (see Figure 3). Circumstantial evidence based on what we k n o w of the religious rituals of present-day peoples w h o still follow the hunter-gatherer life style of the Ishango, might favor his explanation. Whatever the interpretation, the patterns surely show the bone was more than 58
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
Fig. 4. At the Brussels Natural Science Institute, the mathematical tourist has to confront a dinosaur on the way to the bone.
a simple tally stick. But to credit the computational and astronomical reading simultaneously w o u l d be farfetched; as Joseph stated it, "a single bone may well col-
lapse under the heavy weight of conjectures piled upon it." From Africa to Belgium
and into Space
The bone was brought to Belgium b y its discoverer, Prof. J. de Heinzelin, and stored in the treasure r o o m of the Royal Belgian Institute for Natural Science in Brussels (Belgium). A mould, and several copies, were m a d e from the petrified bone, ensuring the survival of the cryptic information contained on the small and fragile artifact. It is not exposed a n y longer to the public, but the mathematical tourist in Brussels can, b y a simple written request, ask for the m u s e u m ' s authorization to see it (see Figures 4 and 5). Prof. William A. Hawkins,
Fig. 5. The Ishango bone (center), together with some harpoon heads, in a drawer of the Institute's treasure room.
Fig. 6. Kubrick used a metaphor to express man's odyssey from the first Mesolithic reasoning to the space technology of the future. (This sketch, based conceptually on an image from the motion picture "2001: A Space Odyssey," is printed with the permission of Turner Entertainment Co.) THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
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pyramids. From the Nilotic epoch on, a mere 4000 years w e r e e n o u g h for science to reach space. M a y b e one day an inspired astronaut will allow the Ishango bone to continue its Kubrick-like voyage b y carrying the bone (or a c o p y of it) with him on a Space Shuttle mission. What w o u l d be fitting words to pron o u n c e w h e n the bone reaches its destination, space?
References [Bog] J. Bogoshi, K. Naidoo and J. Webb, The oldest mathematical artifact, Mathematical Gazette, 71 (458), 294 (1987). [Hei] J. de Heinzelin, Ishango, Scientific American, 206, June, Director of S U M M A (Strengthening Under-represented Minority Mathematics Achievements), is currently editing a poster on African and African American Pioneers in Mathematics. It includes images of the Ishango Bone and is to be distributed t h r o u g h colleges and universities in the U.S. A. Marshack, at the time of de Heinzelin's discovery, was in charge of supplying some historical and scientific b a c k g r o u n d for the NASA lunar program. H e questioned m a n y authorities in the administration and in the scientific c o m m u n i t y about the reasons for going into space. Finally, these discussions led him to the Ishango bone: My interest in space and science had been kindled, in part, because these were human activities, culturally specialized products of the human brain, and to me therefore, not much different from politics, religion, art or war. I could not in my thoughts, while writing and searching for the meaning of man and the space program, separate Dr. Jerome Wiesner, Special Assistant to the President for Science and Technology, or Yuri Gagarin and John Glenn, the first Soviet and American astronauts, or Dr. Lloyd V. Berkner, the scientist and science administrator who had suggested the International Geophysical Year to the world's scientists, from the extremely primitive natives I had met in New Guinea and Australia, or the starving farmers I had seen in India, or from the men who, thousands of years before, had hunted mammoth, reindeer, and bison and had painted the caves of Ice Age Europe, or from the later men who first farmed or built cities around the Mediterranean (after A. Marshack [Mar], p. 9-11). Incidentally, the Ishango site w h e r e de Heinzelin found the famous bone, is a w o r t h y location for scenes of movies such as 2001: A Space Odyssey. Kubrick showed our ancestor in a prehistoric landscape throwing a bone tool into the sky. By special effects, the bone gradually transformed to a space ship. This movie sequence can be seen as a metaphor to illustrate h u m a n progress from discoveries that look v e r y simple nowadays, to the technology of the space age. The picture in Figure 6 was compiled with Stanley Kubrick's images in mind. The front layer shows the Ishango excavation site, the rectangular rock being evidence of the archaeological work. The water of Lake E d w a r d behind it extends to Egypt, a t h o u g h t that led to the addition of the 60
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
105-16 (1962). [Jos] G. G. Joseph, The Crest of the Peacock:Non-European Roots of Mathematics, Penguin Books, London (1992). [Mar] A. Marshack, Roots of Civilization, The Cognitive beginnings of Man's First Art, Symbol and Notation, McGraw-Hill, New York (1972).
Aartshertogstraat 42 8400 Ostends Belgium
The English Hammer-Beam Roof David Horowitz
In touring the English countryside, if one is willing to stray boldly from the larger motorways, then a drive down almost any country road will pass through small hamlets reminiscent of bygone eras. Though each has its own distinctive character, most share a general communal theme: main street, pub, common area or park, and church. The church often stands out amid the rest. Many of those in East Anglia are large edifices built during the fourteenth to eighteenth centuries by patrons who became wealthy as the wool trade flourished throughout the region. When the industry foundered the population relocated, leaving the church to dwarf the town and serve only a handful of remaining parishioners. A familiar architectural feature that appears in many of these large churches has become known as the English hammer-beam roof. It is an example of a triangulation network, or truss, which is characterized by a beam (the hammer-beam) protruding from each wall and supported by a wall post and a lower curved brace (Figure la). A vertical strut (the hammer-post) joined to the end of each hammer-beam enables the weight of the principal rafter to be carried down to the wall, which is often buttressed at the point from which the hammerbeam protrudes. There were several motivations for this form of Gothic roofing design. First of all, it permitted the spanning of wider halls where single beams of sufficient length were difficult to find. (In general, hammer-beams have a length of one-sixth to one-fifth of the entire spanning width.) When it became necessary to span larger halls, a "double hammer-beam" roof was sometimes used (Figure 2). Secondly, it shifted the point of application of the roof's outward thrust further down the wall (via
the wall post). Thirdly, it allowed the unencumbered vision of stained glass windows or other artwork at the end of the hall. And finally, it offered the aesthetic suggestion that the entire roof was "suspended" without the utilization of intervening columns and supports. Architectural historian John Hooper Harvey claims [2] that the hammer-beam roof is "one of the greatest of all triumphs of mind over matter." Although this form of roofing was executed as early as the fourteenth century (Pilgrims' Hall at Winchester, England, circa 1325), the use of the term "hammerbeam" does not seem to appear until the early 1800's, and its origin is in doubt. Some suggest that it is derived from the French heraldic term hamade, signifying three pieces cast into a triangular shape [1]. Others perceive the form of a hammer framed by the hammer-beam (as the head) and the wall post (as the handle). The wooden structural elements which form the hammer-beam roof are often ornately carved or decorated. One of the most masterful examples is in Saint Mary's Church at Bury St. Edmunds (Figure 3). It is described [6] in the words of the British architect and polemicist A. Welby Pugin (1812-1852): "At every pair of principals are two angels as large as the human figure, bearing the sacred vessels and ornaments used in the celebration of the holy sacrifice; these angels are vested in chasubles and dalmaticks, tunicles and copes, of ancient and beautiful forms; the candlesticks, thurible, chalice, books, cruets, &c., which they bear are most valuable authorities for the form and design of those used in our ancient churches." The Church of Saint Peter and Saint Paul in the village of Knapton in Norfolk offers a beautifully carved and painted double hammer-beam roof (Figure 4). (QUERY: Does there exist an example of a
THE MATHEMATICAL INTELL1GENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
61
Figure 1. Simple Hammer-Beam Roof. (a) illustrative sketch; (b) principal structural elements; (c) equivalent truss form
triple hammer-beam roof? If not, why not?) English hammer-beam roof construction was not limited to churches; a richly polished specimen can be found in the main hall of Trinity College, Cambridge (Figure 5). The renowned example in Westminster Hall, London is not a "pure" hammer-beam roof, as it incorporates great arched ribs to supplement its support (Figure 6). The stress analysis for the members of the hammerbeam roof is of interest in the field of statics [4]. The process can be simplified by replacing each curved brace in the roof (Figure lb) by an equivalent straight member (Figure lc) having a slightly lower stiffness ratio (Young's modulus). In Figure lc arrows pointing toward each other indicate members under tension (extension); arrows pointing away from each other indicate members under compression (contraction). For a deFigure 2. Double Hammer-Beam Roof
Figure 3. Saint Mary's Church, Bury St. Edmunds (c. 1424-1446)
62 THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996
Figure 4. Church of Saint Peter and Saint Paul, Knapton (1503-1504)
tailed analysis of the stresses within a h a m m e r - b e a m roof see [3] and [5l. Maxwell's Rule is a principle of architectural design used to insure that a truss will remain stable. The rule states that for a simply-stiff truss to be non-collapsible u n d e r loading the minimal n u m b e r of m e m b e r s n required to connect j joints is given b y the formula n = 2j - 3. (This formula is not foolproof w h e n the truss is attached to the environment; absolute verification of stability requires careful vector analysis. See [7].) Figure lc shows t h a t j = 10 and n = 17 for a simple h a m m e r - b e a m roof. In a double h a m m e r - b e a m roof j = 14 and n = 25. (EXERCISE: What are the values of j and n for an mh a m m e r - b e a m roof?) Exploring and analyzing the architectural motifs f o u n d in rural English villages is one of the pleasures of touring in Britain. Each adventure is almost certain to produce m a n y delights and surprises.
References 1. Architectural Publ. Soc., The dictionary of architecture, volume 4. London, Thomas Richards, 1863. 2. Harvey, J.H., Mediaeval architect. New York: St. Martin's, 1972. 3. Heyman, J., Westminster Hall roof. Proc. Instn. Civ. Engrs. 37 (1967), 137-162. 4. Pippard, A.J.S. and Baker, J.F., The analysis of engineering structures. London: Edward Arnold, 1968. 5. Pippard A.J.S. and Glanville, W.H., Primary stresses in timber roofs with special reference to curved bracing members. Building Research Technical Paper No. 2. London: H.M. Stationery Office, 1926. 6. Pugin, A.W.N., The true principles of pointed or Christian ar-
Figure 5. Hall at Trinity College, Cambridge (1604-1605) THEMATHEMATICALINTELLIGENCER VOL.18,NO.4, 1996 63
Figure 6. Westminster Hall, London (1394-1400) chitecture; set forth in two lectures delivered at St. Marie's Oscott. N e w York: St. M a r t i n ' s , 1973. 7. S c h o d e k , D.L., Structures. E n g l e w o o d P r e n t i c e - H a l l , 1980.
Cliffs, New
Jersey:
Department of Mathematics Golden West College Huntington Beach, CA 92647-2748 USA
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Mathematical Encounters of the Second Kind P.J. Davis, Brown University, Providence, RI
Phil Davis was led to a career in mathematics by his failure as a young boy to Philip J. I)avis '; solve a certain problem in geometry and by his persistence in tackling it. Here is the problem, here is how the problem itself developed, and here are the human associations that surround it.
The Mathematical Experience Study Edition P.J. Davis, Brown University; R. Hersh, University of New Mexico & E.A. Marchisotto, California State University, Northridge 1995 487pp.
An Indispensable Tool for Teachers - -
The Companion Guide to the Mathematical Experience Study Edition 1996
In the course of his career, Davis learned that mathematics, while serving the many needs of science, technology, economics, etc., can also serve as a social connection among people of diverse origins, abilities, and interests. Here also is an introduction to a number of his mathematical encounters with individuals that range over centuries, over oceans, and vary in virtuality. The reader will meet Napoleon, Queen Hortense of Holland, Lord Rothschild, John Dee the mathematician and crystal gazer, the actress Elizabeth Bergner, and many others. 1996 Approx. 250 pp., 15 illus. Hardcover $24.95 (tent.) ISBN 0-8176-3939-X
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THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
Hardcover $38.50 ISBN0-8176-3739-7
128 pp., 21 illus. Softcover $14.95 ISBN 0-8176-3849-0
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Penrose Tiling in Helsinki and Tokyo K.H. Kuo
After the sensational discovery of quasicrystals displaying fivefold rotational symmetry but no translational periodicity, Penrose tiling became a fashion not only in the quasicrystal community but also in architectural decoration, especially in universities and science museums. The following are two cases to add to the Penrose Tiling in Northfield, Minnesota, published recently in this journal (vol. 17, no. 2, 1995). In front of the new Science Museum of Helsinki, opened Spring 1989, the pavement leading to the main gate is decorated with an aperiodic tiling of kites (dark colour) and darts (light colour) first suggested by Roger Penrose. It is noteworthy that the matching marks to guarantee an aperiodic tiling are clearly shown. For all visitors coming to this museum aiming at learning some science, the first thing they experience is the aperiodic tiling displaying fivefold symmetry. In the Munich Museum, a Penrose tiling has also been used to decorate one of the inside walls. Penrose tiling can also be used to decorate a building. On the front face of the Student Hall of the Tokyo Metropolis University there is a Penrose tiling two stories high. Two female students are admiring this beautiful pattern showing the contour of metropolitan
Figure 2. Penrose tiling in Tokyo (photographed in spring 1991).
Tokyo. The ten-spoke cartwheel (part of it is enlarged and shown as an inset) symbolises the eternal progress of Tokyo.
Figure 1. Kites and darts in Helsinki.
K.H. Kuo P.O. Box 2724 100080 Beijing, China THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York 6 5
Penrose Tiling at Miami University David E. Kullman
After the appearance of Martin Gardner's Scientific American column on Penrose tilings, many architects saw their potential for decoration, especially of scientific buildings. 1 Are we alone in having such a decoration conceived before the Gardner column came out? Milton Cox of our Department, inspired by visiting speaker John H. Conway in September 1976, made the design. Cox unfortunately found, on returning from a trip, that about a quarter of the tiles had been laid with
ISee MathematicalIntelligencervo1.17(1995),no. 2, 54. 66
a 90-degree rotation from the original position, so that the axis of symmetry no longer matched that of the building. It was deemed preferable to rotate the tiling rather than the building to bring them into agreement. Bachelor Hall was dedicated in September 1979, it is still the home of our Department, and the tiling is still a focal point. Department of Mathematics and Statistics Miami University Oxford, OH 45056-1541 USA e-mail:
[email protected] THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York
Jet Wimp* Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome be-
ing assigned a book to review, please write us, telling us your expertise and your predilections. Address the column editor, Jet Wimp.
Calculus from Graphical, Numerical, and Symbolic Points of View by Arnold Ostebee and Paul Zorn
Calculus teachers (and textbook authors) today might do well to have some answers, or some sort of identifiable position, on these issues. You can bet that the authors of this two-volume series, just like the authors of most any calculus series, are far more motivated by the appreciation of one of "our species' deepest, richest, farthest-reaching, and most beautiful intellectual achievements" than by the utility of the calculus in building bridges and skyscrapers. However, you can also bet that most of their students are motivated by other factors, the two most presentable of which are the utility of calculus as a tool in other disciplines and the belief that educated people have always studied this subject so it must be important for them. The instructor's appreciation of the calculus is in general unscathed by modern technology, both student motivations are sorely challenged by its advent. After all, we do not prepare the managers and users of a communications network to understand how satellites function, nor do we prepare auto mechanics to understand the sophisticated electronics of the computer chips that control aspects of the modern automobile. We do not prepare doctors to understand the workings of instruments for laser surgery. Common sense and societal consensus have led us to prepare the experts to understand the client, customer, or patient, to understand what tools are useful in meeting customer needs, and to understand how to use those tools. Yet there is no consensus at all as to whether these goals can best be achieved by teaching mathematics which is directly relevant to our lives and work or by teaching a mathematics driven by its own inner logic and aesthetics. Even if our goal is a strictly utilitarian one, neither history nor current research data give us reliable guidelines. Mathematics education, to be effective and satisfying, needs to be guided by real needs, both the intellectual needs of the individual student and the professional needs of the future profession for which the student is preparing. For students who somehow manage, despite all the modern obstacles, to get in touch with their intellectual needs, the rest usually takes care of itself. Perhaps fortunately, most students are not "intellectu-
Philadelphia: Saunders College Publishing, 1995. v. I: paperback, vi + 516 pp., US $15.50, ISBN 0-03-010602-8 v. II: paperback, vi + 413 pp., US $15.50, ISBN 0-03010603-6
Reviewed by Herb Clemens As Bill Davis of Ohio State University is fond of asking, "Maple or Mathematica (especially with front ends like Joy of Mathematica) enable even math/computer-phobic undergraduate students to do any calculation we ever dreamed of teaching infinitely faster and more accurately than we ever dreamed of requiring. So what should we teach and how should we teach it?" It's the calculator controversy of elementary and secondary school revisited "in spades" in higher education. On the other hand, most students do not do proofs and they remain unconvinced that proofs are good for anything except persecution (their own). "Teacher knows best," unlike a couple of generations ago, no longer works, even for the serious student. So when I was asked to review the new calculus series by Arnold Ostebee and Paul Zorn of St. Olaf College, I picked it up, worked through some random sections, and thoroughly enjoyed its lively, informal yet very coherent narrative style. I even learned a few things about Simpson's rule and cubic splines that I didn't know before. However, I found myself getting stuck again and again on an issue that presumably is not relevant to the task at hand, namely, the fundamental issue lurking in the background: what of calculus is worth teaching and to whom? Is the traditional calculus curriculum any longer important for our students? If so, should it be studied for its profundity, for its intellectual power and beauty, or for its utility? If it is useful, what makes it so? *Column Editor's address: D e p a r t m e n t of M a t h e m a t i c s , D r e x e l Univ e r s i t y , P h i l a d e l p h i a , P A 19104 USA.
THE MATHEMATICAL INTELLIGENCERVOL. 18, NO. 4 9 1996 Springer-Verlag New York 67
als," so learning calculus is not primarily about satisfying some inner drive for deep quantitative understanding and rigor. Both types of needs, intellectual and practical, are complicated. My intellectual needs as a math student may or may not be the same as yours. I may learn ritualistically, beginning with pattern recognition through repetition, and be unable, or unwilling, to "conceptualize" until a pattern or ritual is in place. You may be unable or unwilling to learn a ritual or algorithm until you understand the concept behind the ritual, and satisfy yourself that the ritual is "important." On the "practical" level, professional and personal needs are almost impossible to predict, especially during early educational years. Even for the individual college student, professional needs are difficult to predict. In these times, no job/profession is secure, and we will all be called upon to change what we do, both in our work and personal lives, every few years. The authors of this series seem to me to be firmly in the "concept-first" camp, and to operate there with considerable verbal and pedagogical sophistication. The books seem very "user-friendly," at least for the bright, articulate, verbally sophisticated undergraduates at places like St. Olaf's. The authors are trying to be the students' friends, to appeal to their sense of humor, to embellish, be whimsical, be interesting, and create a sense of shared purpose. We are in this together, they implicitly say, this exploration of one of the most profound intellectual achievements of modern humankind. Let's make a (statistically outrageous) comparison of this calculus book [OZ] with the classic Thomas and Finney (7th edition) [TF] by choosing one chapter, that on the Chain Rule. TF is four and one-half pages, OZ a page longer. TF has nine example computations, OZ eight. TF offers a "proof" of the Chain Rule halfway through (relying on the concept of "increment &t" of dubious parentage) whereas OZ "settle for plausible evidence." The OZ chapter is built around a real narrative, a kind of conversation about the Chain Rule such as one might have about a new set of running shoes. TF's few paragraphs are much tighter, serving mostly to tie "rule boxes" to examples. OZ's exercises give more emphasis to graphing and numerical solutions; TF's contain more serious applications to physics and engineering. These chapters, if typical, raise some interesting questions. By drawing the reader's attention away from computational ritual in favor of "getting the idea of what a particular calculus concept is about," could OZ possibly frustrate the "ritual-first" learner? Are the authors in effect taking the position that nothing has been lost, since the "ritual-first" learner is really a "ritual-only" learner and since computational ability per se, is no longer very useful? On the other hand, is OZ's assertion that a "fully rigorous proof of the chain rule runs quickly into delicate technical problems" as useful to those few rigorously 68 THEMATHEMATICALINTELLIGENCERVOL.18,NO.4, 1996
logical students as TF's "proof," even though it may raise further questions? The authors state in a paragraph entitled "Concept vs. rigor" in the introduction to Volume I that they "emphasize only those [proofs] which we believe contribute significantly to understanding calculus concepts." This raises the serious question of what "concept-first" means if logical rigor is only exemplified occasionally, rather than being the framework on which the course structure is built. In fairness, there is certainly something to the approach of concept without too much rigor. An awfully lot of beautiful and interesting calculus, both conceptual and computational, was done in the century or more before there was anything like rigor. (The party was over, the riches were spent, and then around 1800 the accountants moved in, some would say.) Most of us probably worry that, by eschewing logical rigor, we frustrate the relatively rare but precious student whose need for intellectual precision is his or her entryway to mathematics and science. We tell ourselves that those precious few will, by and large, fill in the logic on their own and that, in any case, we must be guided by the needs of the larger group. After a while, most of us professional mathematicians take (are reduced to?) that position in our teaching, simply because the more rigorous approach has worked out so disastrously in practice. Suppressing our lament that strict logical reasoning is just not in the cards for most of our students, we make the necessary compromises with reality. But we often do this without putting forward a very clear concept of what "the needs of the larger group" really are. We've never come up with a really good answer, and the technological challenge mentioned above only makes matters worse. These books, it seems to me, try to avoid this unanswerable, or at least unanswered, question by focusing on the student as a person and on the individual's relationship with the material. As for the focus on the student, the accessible, spiraling narrative style of presentation is attractive, as is the regular interjection of whimsical, throw-away humor. It is as if the authors are saying, "Trust us, we're going on an intellectual journey together, you're our friend and we know what we're doing . . . . And by the way, look how this is related to t h a t ! . . . And by the way, isn't this subject impressive and interesting?" Such a style projects energy, enthusiasm, and caring, so it can hardly go wrong. I'm probably not alone in believing that, if the mathematics is correct and the style is intelligent, energetic, and caring, any style of presenting mathematics, from the most traditional rote learning to the most modern forms of constructivism, will produce good results much of the time. The authors can certainly be forgiven for their failure to answer the metaquestions. Others don't either. But this approach of "exploring together" and worrying about the rigor, appropriateness, or utility of the out-
comes later, only postpones the day of reckoning. A few years from now, will I have anything left from what I have learned, and will what I have be useful, or pleasing, or profound? All of which leaves me still stuck and unable to make a reasonable evaluation of this delightfully written new calculus series!
Department of Mathematics University of Utah Salt Lake City, UT 84112-1107 USA
[email protected] Number Words and Language O r i g i n s J. Lambekt When mathematicians meet in the lounge, they don't usually chat about mathematics. Instead, they discuss hockey games, personal computers and, not infrequently, the origin of words. Mathematicians with an amateur interest in philology included Newton and Hamilton, while others pursued a more professional interest: to mind come Eratosthenes and Wallis and, most notably, Grassmann, who has been regarded as the founder of Indo-European philology. More modestly, the present author too has carried out some linguistic investigations in syntax and morphology, although his love of etymology has been confined to reading the contributions of others. Number words, like kinship terms, tend to be rather conservative and bear witness to the genetic relationship between languages. If we look at English five and French cinq, we observe that these two words have not a single sound in common, even the written letter i is pronounced quite differently in the two cases. Yet five is related to German fanf and cinq is derived from Latin quinque, which even the layman will recognize as being related. But don't infer that German f corresponds to Latin q in both places; the story is a bit more complicated. Both Latin, from which all Romance languages (Portuguese, Spanish, Catalan, French, Italian, Romanian, and a few minor ones) are descended, and English, a member of the Germanic language group (which also includes Scandinavian, Dutch, and German, plus many dialects), are members of the vast Indo-European language family, which, even before the spread of European civilization in the last few centuries, stretched all the w a y from Iceland to Ceylon and includes many other language groups, such as Celtic, Baltic, Slavic, Albanian, Greek, and most of the languages spoken in Iran, Afghanistan, Pakistan, and India. For example, the six English words describing members of the nuclear tThe author wishes to thank Merritt Ruhlen, Chandler Davis, and David Sussman for helpful comments.
family have recognizable cognates in Sanskrit and go back more than 4,000 years (see, e.g., Bhargava and Lambek [1992]). A hypothetical Indo-European proto-language has been carefully constructed by philologists; in it the original word for 5 is supposed to be *penkwe. (Reconstructed forms are marked with an asterisk; I have often simplified the spelling of such forms to make them more accessible to non-specialists.) According to the rules of Germanic sound change, discovered by nineteenth-century philologists, German f is indeed descended from proto-Indo-European p. This explains the initial consonant of fanf, but not the final one, nor is there such an explanation for the initial consonant of quinque; presumably both arose by alliteration. However, we may conjecture that behind German fiinf there lies a pre-proto-Germanic root like *ring, which also gave rise to English finger (see Klein [1971], who ascribes this hypothesis to A. Meillet). Anthropologists are alleged to know of people who have no words for numbers other than 1, 2, and 3, alias 'many." This possibly apocryphal story would not imply that these people don't know how to count, only that they do so by gestures and not verbally. Now, IndoEuropean (and many other) languages originally possessed three forms of the noun, called 'numbers,' to wit: singular, dual, and plural. The dual form has mostly disappeared, but is represented by the last vowel of Latin duo and is said to survive in the middle letter of the English word both. A dual also occurs in Hebrew sh'nayim, the masculine word for 2, Hebrew being a member of a different language family that used to be called 'Hamito-Semitic.' This family includes ancient Babylonian, modern Arabic, ancient Egyptian, the Berber languages of North Africa, and many distinct languages spoken in Ethiopia. Most people, once they learn to count beyond 3, use their fingers to do so, hence the ubiquitous scale of 10, based on the biological fact that we have ten fingers. But do we? The question is whether the thumb counts as a finger. If not, a hand contains four fingers, not five! Indeed, etymologists have surmised that, at one time, the Indo-Europeans did count by fours. While some linguists are skeptical about this theory, there being no evidence for a scale of 4 in extant languages (see, e.g., Greenberg [1990] or Zaslavsky [1973]), we shall briefly consider the evidence in its favour. English eight is related to Latin octo, which still retains the dual ending o, and in almost all Indo-European languages the word for 9 is closely related to the word for 'new,' e.g., German neun/neu; in French the word neuf means both 9 and new. This suggests that 9 was conceived as a new number after the eight fingers had been used up. According to this theory, which you find in etymological dictionaries, Indo-European *okto originally meant something like 'a pair of fours.' Admittedly, this argument would have been more convincing, had *okto THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
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retained a recognizable trace of Indo-European *qwetwor, meaning 4, or of some word meaning 'hand.' I don't know when the Indo-Europeans finally acknowledged the thumb as a finger and adopted our decimal terminology, but it must have been over 4000 years ago, before their dispersion from their original homeland. English ten, German zehn, French dix and Latin decem all go back to a reconstructed *dekm(t). Menninger [1969] speculates that this might mean something like 'two hands,' seeing two in *de and hand in *km(t). Plausible as this explanation may appear, mainstream philologists derive *dekm(t) from a word meaning 'finger,' which survives in our Latin loan-word digit and native English toe; in fact, German zehn closely resembles German Zehen, meaning 'toes.' The root of this word is quite ubiquitous and goes back to proto-IndoEuropean *deik, to show, which survives in German zeigen with the same meaning, in English teach, and in Latin dicere, to say. As we shall see later, this root may in fact be much much older. There is nothing of interest I can say about our word one. Concerning our word two, we note that it embodies the former dual ending o. Curiously, the word for 2 resembles that for 'tooth,' not only in English, but in most Indo-European languages. This is not to say that there is any obvious semantic connection between two and tooth, only that the initial consonants underwent the same changes from proto-Indo-European *duwo and *dvun respectively. By what seems a strange coincidence, Hebrew sh'nayim, the masculine word for 2, may also be read as meaning 'pair of teeth,' though it must be admitted that Hebrew sh here corresponds to two different sounds in Arabic. We shall return to the resemblance between 2 and tooth later. What about three? Menninger [1969] speculates that this originally meant something like 'many,' and he relates it to French tr~s, meaning 'very.' At least one scholar, Brunner [1969], suggests that three is related to the Semitic word for 3, as in Arabic talat, although this seems somewhat doubtful. Let us now look at six and seven, Latin sex and septem, Polish szedd and siedem, etc., going back to proto-IndoEuropean *seks and *septm. Many people have been intrigued by the analogy with Hebrew shesh and sheva, Arabic sittun and sab'un and ancient Egyptian sas and sefex. Does this mean that the Indo-European and Hamito-Semitic words for 6 and 7 go back to an earlier language from which both proto-Indo-European and proto-Hamito-Semitic are descended? According to Russian linguists, both Indo-European and Hamito-Semitic belong to a superfamily they call 'Nostratic.' However, the American linguist Greenberg [1969] disagrees with this view and incorporates IndoEuropean into a so-called Eurasiatic superfamily, which also includes Finnish, Hungarian, Turkish, Mongolian, Korean, Japanese, and even Eskimo! Although Eurasiatic has considerable overlap with Nostratic, in 70
THEMATHEMATICAL INTELLIGENCERVOL.18,NO.4, 1996
his opinion, Hamito-Semitic belongs to a quite distinct Afro-Asiatic superfamily, which also includes Hausa, a language spoken in Nigeria. He believes that any resemblance between Eurasiatic and Afro-Asiatic must go back to an earlier stage, though much later than the hypothetical origin of all languages, at least 50,000 years ago. Indeed, Ruhlen [1994a] presents evidence for a group consisting of Afro-Asiatic, Eurasiatic, and Amerind, between 20,000 and 30,000 years ago. Is the resemblance between the Indo-European and Hamito-Semitic words for 6 and 7 then a coincidence? It could also be that one group borrowed these words from the other. Menninger [1969] considers this idea, but concludes that, if so, it must have been the Semites who did the borrowing. On the other hand, according to a plausible hypothesis by Dolgopolsky [1988], it was the Indo-Europeans who borrowed a number of words from the Semites, among them the words surviving in English as six, seven, goat, star, and wine. His scenario places the homeland of the Indo-Europeans in Anatolia (modern Turkey) about five or six thousand years ago, where they were in close cultural and commercial contact with the Semites to the south. Curiously, the Bible makes a similar but wider claim in Genesis 10: that modern humans emerged from Anatolia about 5000 years ago. In particular, it asserts that the Jupiter-worshipping Indo-Europeans descended from Japhet, son of Noah, and the HamitoSemitic people from his brothers Ham and Shem. The biblical account may have been written about 1000 BC and is confined to a survey of all the people known to the Jews at the time of Solomon. It does not recognize any other language families, and it classifies Elamite as Semitic, although this extinct language is now believed to be more closely related to the Dravidian family (see Ruhlen [1987]). Dravidian is now represented by Tamil, Telugu, and other South Indian languages, but it also survives in a small pocket of Brahui speakers in Pakistan. Incidentally, the Bible in Genesis 2:2 also hints at the origin of the word for 7 when it says that God rested on the seventh day. The Hebrew words for 'rest' and 'seven' resemble one another, and the former survives in our word Sabbath. I don't know whether this was intended as a playful pun or as a serious etymology. Actually, the seven-day week goes back to the ancient Babylonians and presumably was meant to measure the intervals between the four phases of the moon, notwithstanding the theory proposed by Philo of Alexandria and later adopted by St. Augustine that God created the world in six days because 6 is a perfect number. Let us take another look at the intriguing resemblance between two and tooth, which pervades the IndoEuropean as well as the Semitic languages. We may reject as unlikely the hypothesis that the antecedent of two was borrowed like those of six and seven. Though different languages may independently represent the num-
ber 5 by a word for 'hand,' it is not plausible that they would separately come up with a word for 'tooth' representing the number 2. If not accidental, the resemblance between the words for 2 and 'tooth' must then be inherited. Greenberg and his followers believe that all languages have a common origin. In fact, Bengtson and Ruhlen produce a list of 28 words belonging to this hypothetical proto-language (see Ruhlen [1994a]). A surprising number of these words have recognizable descendants in m o d e m English; to mention just a few: hand, hound, who, queen, milk, man, mind (a Greek cognate of which is contained in the first part of mathematics), and toe, which we have met before and to which we shall return later. Looking at this reconstructed proto-language, we find the word *pal for 2. The list does not contain a word for 'tooth,' but the rival list of proto-Nostratic words contains *phal meaning 'tooth', and such a word is still preserved in Telugu palu. Is this a mere coincidence? Perhaps the best attested word on the BengtsonRuhlen list is the reconstructed word *tik, whose IndoEuropean descendants *dekm(t) and *deik we have already met. Other alleged descendants appear in over half of all the language families of the world, usually with the meaning of 'finger' and often standing for the number 1 or 10. The original Indo-Europeans also had a word for 100, usually rendered *kmto. At an early stage, they split into two groups, which are traditionally classified according to whether they preserved the initial k or transformed it into s; they are the so-called Kentum and Satem groups. Both English and French belong to the Kentum group, although English, like other Germanic languages, has transformed the k into h, while paradoxically, in French, the k (from Latin c) became s (still spelled c) after all. Both Klein [1971] and Schwartzman [1994] derive *kmto from *dekm-tom, supposedly meaning 'big ten.' While this appears to be the accepted explanation, Bengtson and Ruhlen seem to suspect that *kmto may be derived from an original form *kano, meaning 'arm' or 'hand' and surviving as English hand. As we saw, Menninger goes even further and wonders whether *dekm(t) should be explained as meaning 'two hands,' which is logical enough but would contradict the widely accepted relation to digit and the Bengtson-Ruhlen derivation from *tik. Our word thousand has cognates only in the Germanic and Balto-Slavic subfamilies of Indo-European; the latter may have borrowed their word from the former. It has been analyzed as meaning something like 'fat hundred' and its first component is said to be related to our word thumb, meaning the 'fat finger.' Bengtson [1987] derives the Indo-European words for 10, 100, and 1000 all from ancient forms meaning 'finger(s)' and 'hand(s).' In this survey, I have ignored compounds such as eleven and twenty and confined attention to traditional
number words, neglecting recent innovations with transparent etymologies such as million and billion. These were introduced during the French revolution; but there is no international agreement on the size of the billion. The cardinality of the empty set was first recognized as a number in India, though our word zero is derived from Arabic sift, meaning 'empty.' Cardinalities of infinite sets were called aleph by Cantor, after the first letter of the Phoenecian-Hebrew alphabet. This letter was so named after a Phoenecian word for 'ox,' and it resembled the head of an ox; but alef is also the Hebrew word for 1000. N e w number words are introduced whenever they are needed and may have strange etymologies. One of my then pre-school sons, after watching a rocket launching on TV, conjectured that the number below zero is called blastoff. The reader interested in finding out more about the origin of number words may wish to look at the fascinating account by Menninger [1969, 1982]. (She should be warned, though, that the English version of Menninger's book has not been proofread too carefully; for example, the letter p is frequently confused with a similar symbol standing for th.) For the origin of other words occurring in mathematics she is referred to Schwartzman [1994]; for the origin of languages, to the books by Ruhlen, in particular the popular account The Origin of Language [1994b]. While Greenberg's hypothesis that all languages have a common origin is still considered controversial in some professional circles, I for one am fairly convinced of its truth. In summary, we have looked at both recent and older speculations about the origin of number words and on how they reflect the origin of languages, as well as at the controversies still surrounding these questions. If the author dares make any suggestions for further research in this field, it is to observe that sometimes resemblances between two words of one language are more easily traced to other languages than are the individual words. This is particularly striking when the two bear no obvious semantic relationship to one another, as for the accepted pair hand and hound. More speculative is the pair tooth and two, where the resemblance may just be a coincidence.
References J.D. Bengtson, Notes on Indo-European '10', "100' and "1,000', Diachronica 4 (1987), 257-262. J.D. Bengtson and M. Ruhlen, Global etymologies, in: Ruhlen 1994, 277-336. M. Bhargava and J. Lambek, A production grammar for Sanskrit kinship terminology, Theoretical Linguistics 18 (1992), 45-60. L. Brunner, Die gemeinsamen Wurzeln des semitischen und indogermanischen Wortschatzes, Francke Verlag, Bern 1969. A. Dolgopolsky, The Indo-European homeland and lexical contacts of proto-lndo-European with other languages, Mediterranean Language Review 3 (1988), 7-31. THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
71
J. Greenberg, Indo-European and its closest relatives: the Eurasiatic language family, Stanford University Press 1996. , Generalizations about numeral systems, in: K. Denning and S. Kemmer (eds.), On language: Selected writings by Joseph H. Greenberg, Stanford University Press 1990. J.A. Hawkins and M. Gell-Mann, The evolution of human languages, Addison-Wesley Publishing Company, Reading, MA 1992. E. Klein, A comprehensive etymological dictionary of the English language, Elsevier, Amsterdam 1971. K. Menninger, Number words and number symbols, MIT Press, Cambridge 1969. M. Ruhlen, A guide to the world's languages L Stanford University Press 1987, 1991. , On the origin of languages, Stanford University Press 1994. , The origin of language, John Wiley and Sons, New York 1994. , Proto-Amerind numerals, Anthropological Science 103 (1995), 209-225. S. Schwartzman, The words of mathematics, The Mathematical Association of America, Washington, D.C. 1994. C. Zaslavsky, Africa counts, Prindle, Weber and Schmidt, Boston 1973. Department of Mathematics and Statistics McGill University Montreal, Quebec H3A 2K6 Canada
Eight Recent Mathematics Books Reviewed by Jet W i m p
Potential Theory in the Complex Plane by Thomas Ransford
readable for the novice, and Hille's treatment in the second volume of Analytic Function Theory (1959) is too brief and too much oriented toward function theory. The present book satisfies a palpable need. About the only real variable background required is a little knowledge of Borel measures, and what is necessary is smoothly developed in an appendix. As the author assures us, you don't need to romance Borel measures, only to have a flirting acquaintance with their properties. The author is careful to define terms that may be unfamiliar, for example, "compact exhaustion," and to reassure us when a concept is something we needn't usually worry about; for instance, open subsets of the complex plane have compact exhaustion, as (trivially) do all compact spaces. When one is trying to learn only enough about something to use it, it helps to know what to fret about and what to take for granted. The book begins with broad discussions of harmonic and subharmonic functions and then delves into potential theory. The treatment of polar sets is captivating. Next comes the Dirichlet problem, and then, in Chapter 5, capacity. This is, hands down, the most satisfying treatment of this highly intuitive yet devilishly subtle concept. The author talks about actually computing capacity and gives an amusing and useful table listing the capacity of some common sets: ellipses, intervals, triangles. The book closes with an application of potential theory to various problems in analysis. An excellent text; my compliments to the author.
Contests in Higher Mathematics: Mikl6s Schweitzer Competitions 1962-1991 edited by G~bor J. Sz6kely
Cambridge: Cambridge University Press, 1995. London Mathematical Society Student Texts 28, x + 232 pp. Hardcover, ISBN 0 5621 46120 0 $54.95; softcover, ISBN 0 5212 46654 7 $19.95.
N e w York: Springer-Verlag, 1996. vii + 569 pp. US $39.00, ISBN 0 387 94588 1
This book is a engaging addition to the estimable London Mathematical Student Text Series. These texts are generally short works, seldom over 250 pages, designed to make contemporary mathematics accessible to today's graduate students. The quality of exposition that prevails in this series is high, and this book is no exception. It is, I believe, the best treatment of potential theory available. Potential theory has proved to be of crucial importance in many areas of contemporary analysis: in special function theory, where it is used to analyze the root distribution of orthogonal polynomials and the behavior of the coefficients in their recursion formulas; in numerical analysis, for instance, estimating the error of Gaussian integration procedures; and in approximation theory, namely, the problem of uniform approximation. Unfortunately, current treatments do not recommend themselves to someone who wants to master the use of the theory as a tool. Tsuji's book, Potential Theory in Modern Function Theory (Chelsea, 1975), is all but un-
I am a sucker for two things: undefended cats and problem books in mathematics. I have a house full of the former and a library full of the latter. When I'm feeling selfcritical, I condemn my infatuation with problem books as a waste of time, something like Nintendo or surfing the web. On the other hand, I sometimes think that gratitude for such books--is this a positive addiction?--is more appropriate. Several of my published papers had their origins in intriguing mathematical problems I discovered in books I reviewed. In 1894, the Hungarian Mathematical and Physical Society initiated a high school competition for mathematical students. Among the winners of this event, at one time or another, were people with familiar names: Fej6r, Haar, K~rm~in, (Marcel) Riesz. The success of this competition led to the establishment in 1949 of a college-level contest, named after Mikl6s Schweitzer. The contest problems are suggested by prominent Hungarian mathematicians, so they tend to reflect mainstream mathematical thinking in Hungary. The problems in the current book
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THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
are a continuation of those given in a previous book, Contests in Higher Mathematics, 1949-1961, published by the Kiad6 Academy of Budapest in 1968. The present book is a member of the Springer series Problem Books in Mathematics, edited by Paul Halmos. All the books in this series are reasonably priced. The statement of the problems---usually with the original proposer identified--is predictably brief, about 50 pages. The solutions of the problems are predictably protracted, almost 500 pages. The problems are taken from all areas of mathematics and none is easy. Yet none is oppressively difficult, provided the solver knows a little about the problem area. There are many delicious morsels in this book. I'll give two of my favorites.
Let yl(x) be a continuous positive function on [0, A] and y,+](x) --
2fxoX/yn(t) dt,
n = 1, 2 , . . . .
Show limo~ yn(X) = x 2 uniformly. My first reaction, before I even attacked the problem, was predictably avid: "I can generalize that!" Alas, the problem-solver had already done it.
Let f(n) denote the maximum number of right triangles determined by n coplanar points. Show that lim fin) n2 -- ~,
n ---* ~
lim f(n) = 0. n 3
n ---> ~
The proposer of the first is not given. The proposer of the second is Paul Erd6s. Part of the success of any problem book lies in elegance of design. Are the problems well separated from each other? Is access to the solutions made easy? In short, browsing in a problem book should not be a problem. The publishers, Springer-Verlag, have not let us down with this volume: it's sleek, eumorphous, wellpaced, and scrumptiously typeset. Those sharing my disorder should either avoid this book at all costs or immediately order it. I'm not certain which piece of advice is the more responsible.
Catastrophe T h e o r y by V. I. Arnol'd Third revised and expanded edition, translated from the Russian by G. S. Wassermann, based on a translation by R. K. Thomas N e w York: Springer-Verlag, 1992. xiii + 150 pp. US $29.95, ISBN 3 540 54811 4 This unusual little paperback, translated from the Russian, has the qualities we tend to associate with Russian expository mathematical writing: a highly personal style, imaginatively presented examples, nontechnical language, sometimes puzzling asides, and de trop observations taken from the world at large.
What is catastrophe theory? Well, many of us may remember that several years ago it was the regnant mathematical fad. Almost every physical law or phenomenon-from insect mating rituals to the curdling of milk--found its interpretation as a cusp, a fold, a bifurcation. Its proponents, swept to impossible heights of hype, discovered in it an afflatus denied them by the more mundane religious passions. These paeans today make pretty uncomfortable reading. "Catastrophe theory . . . . "Ren6 Thorn enthused in 1974, "favorizes a dialectical, Heraclitian view of the universe, of a world which is the continual theater of the battle between 'logoi'.... It is a fundamentally polytheistic outlook to which it leads us: in all things one must learn to recognize the hand of God . . . . Just as the hero of the Iliad could go against the will of a God, such as Poseidon, only by invoking the power of an opposed divinity, such as Athena, so shall we be able to restrain the action of an archetype only by opposing to it an antagonistic archetype..." Thorn goes on to hint at catastrophe theory's potential for resolving such eschatological issues as failure, success, illness, and death. But I will spare the reader any more of his fervor. Well, the new religion had feet of clay, as Mark Kac opined, when he discovered a simple and common chemical reaction which refused to submit to the predictions of catastrophe theory. "Is this just nature being unkind to mathematicians?" he sneered. With all fads, the hype eventually dissipates, and we are then left to contemplate in relative detachment whatever of true value it has bequeathed to the mathematical canon. This book, despite its occasional agitation, tends to view the discipline with coolheaded and occasionally very critical hindsight. " . . . we can try to use this information to study large numbers of diverse phenomena and processes in all areas of science ... in the majority of works on catastrophe theory, however," he cautions, "a much more controversial situation is considered, where not only are the details of the mapping to be studied not known, but its very existence is highly problematical." The theory, the author points out, goes back to the work of the American mathematician Hassler Whitney, who in 1955 proved that every singularity of a smooth mapping of a surface onto a plane, after an appropriate small perturbation, can be described by folds and cusps. The topic Whitney studied came to be called singularity theory, and that theory coupled with its applications was later termed catastrophe theory by the English mathematician E. C. Zeeman. The publication of a selection of Zeeman's papers, Catastrophe Theory: Selected Papers, 1972-1977 (Addison-Wesley, 1977), started an avalanche of articles of decreasing rigor and increasing hyperbole, popularity, and delirium. As an example of one of the speculative uses of catastrophe theory, the author quotes Zeeman's analysis of the "creative personality" in terms of three parameters, T H E M A T H E M A T I C A L INTELLIGENCER VOL. 18, N O . 4, 1996
73
technical proficiency = T, enthusiasm = E, and achievement = A. The theory, applied to the resulting surface in three-dimensional (T, E, A) space, allegedly furnishes us with truths about the profound issue of human creativity. We even have catastrophe theory models of drug addiction! ~ Here and in many other instances, catastrophe theory, if it does anything, only provides a dubitable verification of conclusions we can already reach because of our knowledge of human behavior. Arnol'd will have none of this moonraking. Brutally, he tallies the shortcomings of Zeeman's construct, declaring, "The deficiencies of this model and many similar speculations in catastrophe theory are too obvious to discuss in detail. I remark that articles on catastrophe theory are distinguished by a sharp and catastrophic 2 lowering of the demands of rigor . . . . " He slyly quotes a passage from the obscure Russian novelist V. M. Doroshevich (1864-1922): "I think, dear, that all this decadence is nothing more than a way of approaching tradesmen." Arnol'd's discussion is reality-oriented. One of the first concepts he introduces is that of a catastrophe machine, a simple device consisting of a board, a cardboard disk, pins, pencil, and rubber bands, which he invites the reader to build. The apparatus exemplifies beautifully the sophisticated ideas of the bifurcation of equilibrium states in elasticity. The resulting catastrophe curve has four cusps. Next, he talks about possible bifurcation states, and then, in short but highly readable sections, he discusses loss of stability of equilibrium, caustics and wave fronts. I found the chapter "The Large Scale Distribution of Matter in the Universe" tremendously exciting. As the reader undoubtedly knows, tile inhomogeneity of the universe has long puzzled cosmologists; in its early stages of development, the universe was homogeneous. What caused the transition from homogeneity to nonhomogeneity? Zel'dovich, in 1970, proposed an explanation that is mathematically equivalent to the formation of singularities, a topic covered in this book's previous chapters. The treatment takes only four pages, and it is gripping. The remaining chapters deal with singularities in optimization problems, smooth surfaces and their projections, and applications. In a welcome appendix, Arnol'd examines the origins of catastrophe theory and shows that it did not originate with Whitney nor with any other single person. Huygens, Cayley, Jacobi, Kronecker, Poincar6, all explored similar concepts. He can't resist a swipe at the deterioration of the mathematical idiom. "The unsophisticated texts of Poincar6 are difficult for mathematicians raised on set theory," he observes, ironically. 1S. J. Guastello, Cusp and butterfly catastrophe modelling of two opponent process models: Drug addiction and work performance, Behavioral Science 29, 258-262 (1984). 2I don't know whether this pun exists in the Russian, or is merely the whimsy of the translator.
74 THE MATHEMATICALINTELLIGENCERVOL.18, NO. 4, 1996
He points out that whereas Poincar6 would have said, "The line divides the plane into two half-planes," modern mathematicians write, "The set of equivalence classes of the complement R2/R 1 of the line R 1 in the plane R 2 defined by the following equivalence relation: two points A, B ~ R2/R 1 are considered to be equivalent if the line segment AB connecting them does not intersect the line R 1, consists of two elements." He claims he is quoting from an actual textbook. I believe him.
Real Computing Made Real: Preventing Errors in Engineering and Scientific Calculations by Forman S. Acton Princeton, N.J.: Princeton University Press vii + 259 pp. US $29.95, ISBN 0 691 03663 2 "Who," I asked myself, "is Forman S. Acton?" Actually, I knew: the author of the 1957 book, Analysis of Straight Line Data (John Wiley and Sons) and the 1970 b o o k Numerical Methods That Work, recently reprinted by the Mathematical Association of America. What I meant was, "Who is Forman S. Acton to write a book on error analysis of numerical methods?" Yes, professor emeritus of computer science at Princeton, but hardly a household name. A scant three papers on numerical analysis to his credit. I briefly riffled the pages. No mention of norms, no mention of Banach spaces. No matrix inequalities, no condition numbers, no closure, no completeness. We all know you have to have such things to discuss error analysis intelligently: Isaacson, Keller, and Davis have predicated their reputations on convincing us of this. Instead, this book gives us Latin quotations, frequently intercalated photographs of Thai temple statues, a beginning chapter labeled "An Exhortation," nursery rhymes ("As I was going up the stair, I met a man who wasn't there . . . . "), text printed upside down (to discourage peeking at a solution), and headings done in a script like the titles of a Charlie Chan movie. Acton, apparently, runs a reductionist numerical atelier. "When the volume and sophistication of your problems demand [new] weapons, you will know it." He quotes approvingly Hamming's famous comment, "The purpose of computing is insight, not numbers." He provides gentle admonitions and occasional homilies. If you can't understand how to solve quadratic equationy the example of which begins his discussion--and keep the error under control, he seems to be saying, you won't understand error in normed spaces, either. It may be that in this book he is trying to make the point that the appreciation of error is as much a humanistic enterprise as it is a mathematical one. If so, he is on firm philosophical ground. I have long puzzled over the reasons w h y students find the concept of error so elusive. What could be more intuitive than the closeness of one object to another? My
students happily reel off three figures from their desk calculators w h e n they should k n o w it's not enough, seld o m try to reconcile the n u m b e r they have calculated with the realities of the problem, and blissfully give me an occasional negative n u m b e r w h e n I ask for the cosh of something. A n d m y students are engineers. I think Acton has p u t his finger on the problem, and for doing so he deserves some sort of decoration. H e says, "[numerical methods] are also the vehicle for teaching something m o r e important: how to visualize the shape of a function and ultimately, h o w to use that shape to construct algorithms that process that function accurately and efficiently." H e continues with the trenchant observation, " G e o m e t r y is as important as algebra and calculus here, with constant switching of the viewpoint between them." Yes indeed. Check it out. I'll bet you'll find that students w h o can't c o m p r e h e n d numerical error are also those with the weakest geometric intuition. This beautifully written book doesn't have a lot of fancy mathematics in it, but it can help close the gulf between blind calculation and true understanding. There are many, m a n y exercises, and sedulously devised solutions. I sense that Acton is a veteran of long standing of the classroom wars. H e knows h o w students think and h o w they fail to think, and it's to this kind of voice that those w h o value the role of numerical analysis in building a tolerable technological society should be listening.
The Queen of Mathematics: An Introduction to Number Theory by W. S. Anglin Dordrecht: Kluwer Academic Publishers, 1995. x + 389 pp.; US $155.00, ISBN 0 7923 3287 3
Each of the writers of the new books on n u m b e r theory has staked out a claim for a different part of the territory, although there is, inevitably, some overlap. Anglin's relies heavily on simple continued fractions (continued fractions whose coefficients are integers). Thus, it invites comparison with the book Continued Fractions, b y A. M. Rockett and Peter Sz/isz (World Scientific, 1992). That book has some virtues, but after beginning their discussion equably, the authors become fastidious and nit-picky b e y o n d endurance, and reading the book is like watching a hyperactive spider at work on a web. Anglin is simply a fine writer (the philosophical background coming through, perhaps?) and the material never gets away from him. Everything is u n d e r control, and each topic receives its p r o p e r modicum of attention. Nevertheless, this is a v e r y curious book; to some extent, in its curiosity lie its virtues. The book is Volume 8 in Kluwer's Graduate-Level book series. Yet it requires almost no b a c k g r o u n d - - n o t more than a d o z e n pages even use calculus. One could use it to teach a highly motivated high school class. Anglin has respect for the cultural aspects of the subject, and often gives a historical or authorial context for the result at hand. I therefore f o u n d it pretty strange that he is so lackadaisical in providing references in other places. Let m e illustrate. From page 5, In 1875 Edouard Lucas, who had been a French artillery ofricer in the Franco-Prussian war, challenged the readers of the Nouvelles Annales de Mathdmatiques to prove the following: A square pyramid of cannon balls contains a square number of cannon-balls only when it has 24 cannon-balls along its base. In other words, the only nontrivial natural number solution of 12+22+
One of the pleasures of serving as book review editor of this journal is having d u m p e d in m y lap a continual profusion of books on n u m b e r theory. N o n e of the books I have examined so far is an unqualified success, but the subject brings out the best in potential writers. The intellectual b e a u t y of the subject seems to vivify authors and stoke their mental acquisitiveness to a white heat. It is axiomatic that w h e n an author (or teacher!) is interested in the subject, w e probably will be, too. The author of the present book is a tyro---as far as I can discern, Anglin's b a c k g r o u n d is in p h i l o s o p h y and in expository mathematical writing3--but that doesn't m e a n the book should be dismissed.
interdisciplinary Ph.D. in mathematics and philosophy from McGill in 1987, and prior to his present position at McGill he was a postdoc in philosophy of religion at Notre Dame. Also the author of Mathematics: a Concise History and Philosophy (Springer-Verlag, 1994). Anyone considering the use of that book as a text should be prepared to tolerate the occasional incursion of the author's religious affections, such as his attempt to buttress up Descartes's lame ontological arguments by an appeal to authority. 3An
9
+n 2=m 2
is n = 24, m = 90. Anglin points out that the problem waited for a solution until 1918, w h e n G. N. Watson solved it by using hyperelliptic functions. Later in the book, Anglin offers an elementary proof he himself obtained b y simplifying a proof due to D. G. Ma. Anglin should have p r o v i d e d this reference. 4 The Watson reference is also missing. 5 The bibliography contains only 10 entries, far too few for a textbook with such aspirations. The absence of calculus in the book m a y have w o r k e d to the author's advantage, as there is evidence he is uncomfortable with analytic arguments. For instance, he
4Sichuan Daxue Xuebao 4, 107-116 (1985). Z h e n Fu Cao claims, in MR91e:11026 that he a n d Z. Y. Xu g a v e a n e l e m e n t a r y proof of this fact in 1985 in Kexue Tongbao. These C h i n e s e journals are only intermittently abstracted in Math Reviews. This p r o b l e m h a s a connection w i t h the f a m o u s p r o b l e m of Mordell, a b o u t w h i c h so m u c h h a s been written.
5Messenger Math. 48, 1-22 (1918-1919). THE MATHEMATICALINTELLIGENCERVOL. 18, NO. 4, 1996 75
wants to show that CV~Tc e-L2 ~0 et2 dt = 0(1), L --~ o% c > 0. His proof requires three lemmas. Note, however, that the order of the quantity on the left is unchanged by making the lower limit of the integral, say, 1. N o w integrate by parts with U = 1/2t, dV = 2tet2dt. Also, there is a great deal of flummery about integrals of the sort
fcc+i~ eft ~ dt, c > O, o~ < O. These are,
of
course, just
Gamma functions, despite the complex contour. Chapter I is titled "Propaedeutics" (< Gk., pro = before + paideO = instruct) and contains material about primes, Bernoullii numbers, Diophantine equations, Fermat's last theorem (now out of date), and the M6bius function. Chapter 2 treats continued fractions, Chapter 3 congruences, Chapter 4 the equation x 2 - Ry 2 = C. Chapter 5 talks about classical construction problems, not a typical subject in a number theory book. Even more unusual is the inclusion of the proof of Lindemann's famous theorem about the transcendence of It. The polygonal number theorem occurs in Chapter 6, and Chapter 7 is devoted to analytic number theory. In particular, the author derives Rademacher's spectacular series for the partition function--a topic usually confined to advanced books on analytic number theory. Rademacher's result is one of the great achievements of analysis in this century, and the author's devotion to it is clearly reflected in the considerable effort it must have cost him to present it in a way that is commensurate with the undemanding level of the book. There is enough uncommon material here to lend admirable new gloss to the subject, and few books on number theory are as well written as this one. However, the astronomical price of $155 makes me wonder whether Kluwer has misjudged its market.
A Primer on Nonlinear Analysis by A. Ambrosetti and G. Prodi Cambridge: The Cambridge University Press, first paperback edition (with corrections) 1995. vii + 171 pp. US $22.95, ISBN 0 521 48573 8 I get a creepy feeling when I open a book and see the first chapter labeled "0." In the past it has usually meant that I must expect a heinous snow job. Wasn't it Halmos who, in the introduction to his Measure Theory, remarked that the reader shouldn't be discouraged if he finds he doesn't have the prerequisites to read the prerequisites? But in this case, I shouldn't have worried. There's no more economical or lucid introduction to the subject than this great little book, a paperback member #34 of the fine paperback Cambridge Studies in Advanced 76
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Mathematics. I sometimes think that this must be the golden age of expository writing, particularly with Springer-Verlag getting into the act, issuing one superb text after another. It's a great time to be a book review editor. What's noteworthy about this book is the seamless w a y the authors fuse mathematical statements with concrete and challenging physical applications--water waves, the heavy string, the restricted three-body problem. If you plan to give a rather abstract course in (primarily linear) functional analysis, you might consider following it with a segment based this material, which both demonstrates the power of functional analysis and serves as an introduction to nonlinear methods. The mathematics is robust. The zeroth chapter gives some of the background required: Banach spaces, the Fredholm alternative, function spaces, a statement of the general elliptic boundary value problem. I was amused " to find the first chapter called "Differential Calculus," the authors' little joke, perhaps. The "differential" signifies, of course, Fr6chet and Gateaux derivatives. Intoxicating stuff, indeed. Occasionally, the authors slip their leashes, threatening to menace the reader with derivatives in convex topological spaces and even pseudotopological spaces--but only as an afterthought; they soon wander back to more worldly matters, such as Nemitski operators. Chapter 2 states and proves an invaluable local inversion theorem, which guarantees that a map between Banach spaces will possess, at least locally, an inverse. This theorem places on a rigorous foundation what engineers know as the process of linearization, that is, gaining insight into a nonlinear problem locally through a study of its linear approximation. By now, the authors have developed enough tools to tackle serious issues, one of the first being a description of T-periodic solutions of the differential equation x" + g(x,x')= eh(t), where g is continuously differentiable and h is continuous and T-periodic. This would have been a good place to treat Newton's method in normed spaces, a la Kantorovich and Akilov, Functional Analysis in Normed Spaces (Pergamon, 1964). It is the cleanest application I know of the Fr6chet derivative. However, the authors seem interested only in physical applications, more specifically, physical applications modeled by differential equations. Chapter 2 closes with a discussion of the implicit function theorem, the stability of orbits, and autonomous systems. Chapter 3 treats global, as opposed to local, inversion theorems. The main theorem, sometimes called the monodromy theorem, goes back to Hadamard in the finite-dimensional case, and to Caccioppoli and Levy for general Banach spaces. In Chapter 4 we find semilinear Dirichlet problems and a wealth of their applications, to resonance problems and problems with asymmetric nonlinearities. Chapter 5 states general results on bifurcation, and Chapter 6 has many applications of the ma-
terial: to the rotating heavy string, to the B6nard problem (which involves convective motions in a heated fluid), to small oscillations for second-order dynamical systems. I would have liked to see some discussion of solitons here, via the Korteweg-deVries equation, and of nonlinear variational problems. However, incorporating all of my proposed additions would make the primer no longer a primer. I realize the authors had to draw the line somewhere. An abstract version of the Hopf bifurcation theorem starts Chapter 7, and the Lyapunov center theorem and the restricted three-body problem furnish the conclusion of this remarkable little book. Though it may be familiar to many, let me describe the restricted three-body problem. I make the point to my differential equations classes that, historically, Newtonian mechanics 6 and its formulation in terms of differential equations enabled humans for the first time to uncover truths about the physical world that were divorced from the often bogus conclusions provided by Aristotelean introspection. 7 The three-body problem provides an excellent paradigm of the power of mathematical modeling. The problem deals with three bodies, P1, P2 (called primaries), and Q with masses M1, M2 and M3, respectively, acting under Newtonian gravitation. It is assumed that M3 is insignificant compared to M1 and M2. There are several other rather minor restrictions imposed on the bodies, but the resulting motion is still of great practical interest. It can be shown that in this system there are five equilibrium points, Lj, j = 1, 2 , . . . , 5. Three are called Euler points; two are called Lagrange points. Among the several fascinating facts deduced in this chapter is that in a neighborhood of the Euler points, as well as the Lagrange points (these require several additional assumptions), the system has a family of periodic solutions. There are five pages of very challenging exercises. Overall, the English of the book is supple and unflawed--a consequence, I suspect, of diligent editing by the people at Cambridge. There are occasional indications that the editors were asleep at their desks, or unduly cowed or even hypnotized by the mathematics; witness the following passage: "Even if such a theorem is a classical result often understood in the current literature we think useful to have given here an elementary version, in the frame of Banach spaces." I have stressed to the desk editors of one of the journals of which I am an editor that they must not allow themselves to be intimidated by the mathematical expertise of the writers whose manuscripts they are scrutinizing.
6To be fair, I should also mentionthe name of Galileohere.
7From the beginning of his career, Newton was skeptical of Aristotelean science, despite his high regard for the Greek philosophers: "Amicus Plato amicus Aristoteles magis arnica veritas," he scribbled in a notebookin 1664.
Good English writing requires its own set of skills, and has its own indubitable authority.
Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists by Dennis Shasha and Cathy Lazere New York: Springer-Verlag, 1995. xi + 289 pp. US $23.00, ISBN 0 387 97992 1 Who could fail to be delighted by the title of this book? Friends who have long since tired of hearing me talk about the books I've been reading do a double take when I mention this one. Dennis Shasha is a young professor of computer science at NYU's Courant Institute, Cathy Lazere is a professional writer. The reader will conclude, correctly, that this is a book aimed at the popular market. It's issued under the colophon Copernicus, a new imprint of Springer-Verlag, and the SpringerVerlag name is tendered in such delicate lettering as to suggest the august publisher's discomfort at venturing onto the unseemly turf of bestsellerdom. The computer protogenists discussed here fall into four classes, which the authors describe as follows: 1. Linguists: How should I talk to the machine? 2. Algorithmists: What is a good method for solving a problem fast on my computer? 3. Architects: Can I build a better computer? 4. Sculptors of machine intelligence: Can I write a computer program that can find its own solutions? There are unwelcome lessons in this book for those
of us who teach. Reading a few of the biographies will force us to look at our unsuccessful students---our declared failures--in a new light. Several of these scientists were dropouts. Take the case of John Backus. Backus's father was chief chemist for Atlas Powder Company, but it seemed his scientific talent wasn't passed on to his son. Backus was born in 1924 in Philadelphia and went to the Hill School--a venerable intermediate institution--in Pottstown. "I flunked out every year," he says. "I hated studying. I was just goofing around." Next, Backus had a run at the University of Virginia, attempting to major in chemistry. He detested the labs, though. He spent most of his time partying, waiting to be drafted. He joined the Army in 1943, and because of his performance on an aptitude test, the Army consigned him to a pre-med program. Bizarrely, while attending, he was diagnosed with a tumor of the skull and had a metal plate patched into his head. Soon after, he enrolled in New York Medical College. "I hated it," he says. He quit and rented a small apartment in New York. "I really didn't know what the hell I wanted to do with my life. I decided that what I wanted was a good high-fi set . . . . " After a little experimentation with stereo systems and reading about their design, he decided that he might be THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
77
interested in math and enrolled in Columbia University's undergraduate program. He found he disliked calculus. By the spring of 1949, when he was 29 and a few months short of graduating with a B.S., he lost interest. One day he happened into the IBM computing center on Madison Avenue and was shown the SSEC, one of IBM's first electronic machines. He mentioned to his guide that he was looking for work and she urged him to talk to the director. "I said no, I couldn't. I looked sloppy and disheveled. But she insisted and so I did. I took the test and did OK." He was hired to work on the SSEC. To appreciate the ironies inherent in this drama, it helps to know that the state of computer programming in 1949 was Mesozoic. Programming was extremely costly because everything was done in assembly language, only one step removed from the machine language of binary digits. Backus, along with Harlan Herrick, created a program called Speedcoding, which supported floating-point calculations and helped to resolve some of the vexing problems caused by scaling of numerical quantities in computer programs. Working so extensively with assembly language programming convinced Backus that computers needed a high-level language, one which could translate mathematical problems into machine language. In 1953, only 4 years after being hired, he wrote a memo to that effect to his boss, Cuthbert Hurd. Interestingly, John von Neumann, who was a consultant to IBM at the time, advanced persuasive arguments against the project. He felt such a language would alienate programmers from the implications of their computations. And he didn't attach much importance to the costs of programming. Amazingly, Cuthbert sided with Backus. Von Neumann's objections were disregarded and Backus was allowed to hire an ill-assorted passle of both experienced programmers and novices straight from high school to help him. Their avowed intention was to create a language that would make programming easy. By 1957 the language that we know as FORTRAN was up and running. I find this story quite unsettling. N ow I understand w h y the medieval troubadours composed such fervent poems to Dame Fortune. The slightest breeze could have wafted Backus into a totally different profession: shoe salesman, greengrocer, copy editor. The professions in which we ultimately find ourselves, like all the events in our lives, are disturbingly contingent. Was Backus's genius translatable? Would he equally have been a genius at choosing shoes sizes for suburban matrons, judging the state of ripeness of a cantaloupe, prescribing the proper use of "that" and "which"? Who knows. And if he hadn't wandered into the office of IBM in 1949, what would be the state of computers today? There are 14 other equally mesmerizing hum a n dramas in this book. Every biography in this enthralling collection is a story, a narrative in the most profound 78
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996
sense of the word. Popularization or not, SpringerVerlag has published a book of which it should be proud.
Five Hundred Mathematical Challenges by Edward Barbeau, William Moser, and Murray Klamkin Washington, DC: The Mathematical Association of America, 1995. ix + 225 pp. US $29.50, ISBN 0 88385 519 4 I was browsing through this book on 40th Street in West Philadelphia the other day and a frenzied chorus of car honking broke out behind me. The light had changed. Well, after all, this is another problem book . . . . 8 Different, though, from the one discussed above. It's much more elementary; many of the items are accessible to high school students. The need for calculus is minimal, and the editors suggest the collection be described as "problems in pre-calculus mathematics." The problems first appeared in a series of five booklets published by the Canadian Mathematical Society. They were so popular that the present editors decided to issue an edited and revised version of all five. The editors are three of the leading problemists 9 of the day, so the problems are bound to be well chosen and the solutions lucid and well coiffed. The organization of the book makes it a superb pedagogical instrument. First come the problems, stated with no particular ordering (i.e., problems from combinatorics, arithmetic, algebra, inequalities, geometry and trigonometry) and the analyses are all jumbled up. There are 47 pages of them. There are 164 pages of solutions. Then comes a section the authors call a Toolchest, which is a 12-page list of useful facts drawn from the previous 6 branches of mathematics. This feature makes the book highly serviceable for the student. Throughout the book are interspersed fables concerning mathematicians and occasional bons roots. They are wonderful. When Leo Moser was playing in a chess tournament in Toronto in 1946, a bystander was heckling the players. "Chess is a complete waste of time," the heckler shouted. "It has no relation to any other branch of knowledge." "How about mathematics?" Moser asked him. "I have studied mathematics for many years," the man replied. "and know that chess has no relation to any of the four branches of
SThis,and countless other anecdoteswe have all traffickedin, furnish ample evidencethat the driving--and presumably other--privileges of mathematicians should be subjectto some restrictions.A great one, about A. N. Whitehead, is given in the present book. He was cautioning a student about a theory of logic. "This," he said, "should be taken with a grain of er... um... ah..." "Salt,Professor?"the student suggested. "Ah yes," Whitehead said brightly, "I knew it was some chemical." 9Maybe a new word is needed. Questilargitors?
mathematics." "What branches do you mean?" Moser asked. "You know," the heckler snapped disdainfully, "addition, subtraction, multiplication, division."
Show that for any positive integer n, [V~n + X/n + 1] = [~n-n + 2]. Find a closed-form expression for [V~] + [V2] + . . . +
The second, attributed to C. W. Trigg, editor of Mathematical Quickies (Dover, 1985), is probably as perceptive a definition as any of good mathematics. An elegant solution is generally considered to be one characterized by clarity, conciseness, logic and surprise. What are the problems like? I was d r a w n - - m a y b e m y m o o d - - t o some of the puzzlers involving the greatest integer function, [.]:
[~r
2 -- 1 ) 1 / 2 ] .
The book is a paperback, done in a large, elegantly printed format. I suggest you try it out on some of your talented undergraduate students.
Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 USA e-maih
[email protected] THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4, 1996 7 9
1996 Anniversaries The following mathematicians all celebrated anniversaries last year. 400 years: Ren6 Descartes (born 1596) has been called the father of modern philosophy. His Discours de la mdthode, a treatise on universal science, appeared in 1637; it was wrongly titled on the stamp below, but later corrected. The third appendix, entitled La g6om6trie, contains Descartes's fundamental contributions to analytical geometry, as well as material on the classification of curves and the so-called Descartes' rule
of signs. 350 years: Gottfried Wilhelm Leibniz (born 1646) aimed to build the whole of knowledge from a few basic principles. This aim led to plans for a universal language for mathematical logic and the construction of a number of calculating machines that could add, subtract, multiply, divide and find square roots. Although Newton had priority for the invention of calculus, Leibniz published his results first and his notation proved to be more reliable than Newton's.
Robin Wilson*
250 years: Gaspard Monge (born 1746) developed a method for representing three-dimensional objects in the plane, thereby initiating the subject of descriptive geometry, and wrote the first book on differential geometry. He served on the committee that established the metric system in 1791, and was director of the ]~cole Polytechnique. He was Minister for the Navy in 1792-3 and accompanied his close friend Napoleon on an expedition to Egypt in 1798. 200 years: Lambert Adolphe Jacques Quetelet (born 1796) was a Belgian statistician who proposed the notion that naturally occurring distributions tend to follow a normal curve. He established a central statistical bureau that was imitated around the world. 150 years: Friedrich Wilhelm Bessel (died 1846) was a German mathematician and astronomer whose measurements on 50,000 stars first allowed the accurate determination of interstellar distances. In 1817, while investigating a problem of Kepler, he introduced the Bessel function Jn(x) of order n; this stamp depicts the Bessel functions Jo(x) and h(x).
Descartes
Leibniz
Monge
Quetelet
Bessel
*Column editor's address: Facultyof Mathematicsand Computing,The Open University,MiltonKeynes,MK7 6AA. England. 80
THE MATHEMATICAL INTELLIGENCER VOL. 18, NO. 4 9 1996 Springer-Verlag New York