No title

Letters to the Editor

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.

Unfair Dice

Dawson and Finbow have shown in [ 1] that it is impossible to load a cubic die so that it will stand on only one facet, and that though the regular octahe­ dron, dodecahedron, and icosahedron are vulnerable to such loading, it is a practical impossibility.Whilst Dawson and Finbow's results do not have real­ world utility, it is worth noting that the real issue, in any game involving re­ peated throwing of dice of any de­ scription, is a small advantage that remains unknown to opponents. In games where interest is placed on the total score (as opposed to using the ci­ phers on the facets as mere labels), there are such possibilities of accruing small advantages. The traditional design for a cubic die is that each of the pairs 1 and 6, 2 and 5, 3 and 4 goes on opposite facets. This allows two possible cases, mirror images; in manufactured dice one of them predominates. Imperfections in the material from which the die is man­ ufactured might result in, say, greater density near the intersection of the facets with 4, 5, 6, thus skewing the ex­ pected distribution towards lower val­ ues. In [2], with an emphasis on dice based on the five Platonic solids, a col­ league and I sought an answer to the general question, "What distribution of the integers over the facets will min­ imise the effect of ... imperfections or of a deliberate bias?" We looked for simple criteria by which the set of in­ tegers { 1, 2, ..., n) may be distributed as uniformly as possible over the n faces for each of the Platonic solids and the semi-regular solids with 10 facets. Given that a die that rolls one num­ ber too frequently would be easier to detect, we concentrated on more gen­ eral and hence less detectable biases. Based on work by Singmaster [3] in an analysis for the design of dartboards, a simple interpretation of the minimi-

sation of the effect of irregularity in a die is to require larger numbers to lie adjacent to smaller ones, where adja­ cency means a common edge between facets. For dice this may be gener­ alised as a requirement for the max­ imisation ofS = L(ai - a1)2, where the sum is over all edges, and ai and a1 are the values on the facets sharing the edge. With this criterion we noted that S is minimised for the standard cubic die, which thus is the one most sus­ ceptibl� to potential distortion-at least by the criterion of total score over a number of throws.S is maximised when (6,5), (4,3), (1,2) are the opposite pairs. This design is thus proposed to replace the existing standard. Several other criteria for the con­ struction of dice have been envisaged. Rouse Ball [4] suggested numbering the faces of polyhedra in such a way that the faces around a vertex add to a constant sum.For regular polyhedra this is achievable only for the octahe­ dron, and three non-isomorphic cases may be identified. A generalisation of the idea is minimisation of the variance of the sums of facet values around each vertex. A further alternative is the minimisation of the variance of the sum of face values surrounding each facet.In each of these alternatives, the algebra is akin to that found in Singmaster's work on the design of dartboards, and it is necessary to in­ troduce correlations amongst non-ad­ jacent faces. The ensuing algebra ap­ pears intractable for more than six facets, although numerical approaches as in [2] could be employed. If instead we minimise the variance of sums of opposite faces, the algebra is simple; in contrast to the criterion in [2], this leads to favouring the stan­ dard die! Note that for the cube, this criterion agrees with the variance-of­ sum-around-facet criterion just dis­ cussed.

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

3

In [2] we also identified the max­ imising and minimising distributions for the regular solids with n 8, 12, 20 facets and also for the two semi-regu­ lar solids with n = 10. Between the ex­ tremes lie other labellings whose vul­ nerability to loading is intermediate. I am grateful to the various com­ mentators on this note for their many­ faceted suggestions. =

RI!F&RENCES

(1) Dawson R.J.M., and Finbow, W.A. "What Shape is a Loaded Die?" The Mathematical

lntelligencer, 21, No.2 (1 999), 32-37.

(2] Blest, D.C. and Hallam, C.B. "The Design of Dice", Bull. IMA, 32, Nos. 1/2 (1 996), 8-13.

[3) Singmaster, D., "Arranging a Dartboard." Bull. IMA ,

16, No.4(1 980), 93-97.

[4) Rouse Ball, W.W. and Coxeter, H.S.M., Mathematical

Recreations

and

Essays,

12th edition. University of Toronto Press (1974). David C. Blest School of Mathematics and Physics University of Tasmania Launceston, Tasmania Australia 7250 e-mail: [email protected] Parsing a Magic Square

Being a magic square enthusiast, I read with great delight "Alphabetic Magic Square in a Medieval Church" (lntel­ ligencer, vol. 22 (2000), no. 1, 52-53), where A. Domenicano and I. Hargittai present and comment upon a stone in­ scription on a church near Capestrano, Italy. The stone is inscribed with the Latin text "ROTAS OPERA TENET AREPO SATOR," arranged in the form of a 5 X 5 alphamagic square. In their note, Domenicano and Hargittai give the meaning and case of the words ROTAS, TENET, and SATOR, but they are not sure about the grammati­ cal case of OPERA because the word it qualifies, AREPO, "is not Latin though it recalls the Latin word ARATRO = plough (ablative)". The authors then drop this aspect of their considerations on the square by commenting that "the meaning of the text remains obscure." If one looks at the alphamagic square from the perspective of its author, how­ ever, it seems odd that he or she would

4

THE MATHEMATICAL INTELLIGENCER

introduce a strange word like AREPO, thereby ruining the intended cleverness of the whole exercise. So if AREPO is not one single Latin word, then it must be two or maybe even three Latin words, all grouped in a single line because the situation requires it. Anned with this Ansatz, let us look at the possibilities: "A REPO" and "ARE PO" are indeed divi­ sions into two Latin words, but they do not fit the present context. This leaves the following division into three words: "A RE PO," each of which is Latin. In fact this appears to be the solu­ tion, for it gives the text a reasonable meaning. Rearranging the order of the words according to the rules of English, one gets "SATOR TENET ROTAS A RE PO OPERA." The word PO is an archaic form of the adverb POTISSIMUM. With this in­ terpretation, OPERA then is in the dative case, not the nominative or the ablative, as surmised by the authors. See Dictionnaire illustre Latin-Fra'n9ais by F.Gaffiot, Hachette, Paris, 1934. The

text means that the sower looks after the wheels because of their importance, in particular for work. Finally, let me mention an astute ob­ servation made by a physicist col­ league who is an expert in optics, Dr. Jacques Gosselin. When one looks at the photograph of the stone (top, p. 53), one gets the impression that the letters are protruding. But the stone was set in the wall upside down, so to see the picture with the correct light­ ing one should look at it with the page reversed. Now one sees at once that the letters are indented, as was to be expected. This is a well-known illu­ sion; I don't know whether to call it an optical or a neurological illusion. Napoleon Gauthier Department of Physics The Royal Military College of Canada Kingston, Ontario K7K 784 Canada e-mail: [email protected]

c.m mt.J,;

A Mathematician's View of Evolution Granville Sewell

I

n 1996, Lehigh University biochemist Michael Behe published a book enti­

Darwin's Black Box

I. The cornerstone of Darwinism is

the idea that major (complex) improve­

[Free Press],

ments can be built up through many mi­

whose central theme is that every living

nor improvements; that the new organs

tled cell

is

loaded with features and bio­

and new systems of organs which gave

chemical processes which are "irre­

rise to new orders, classes and phyla de­

is, they require

veloped gradually, through many very

ducibly complex"-that

the existence of numerous complex

minor improvements. We should first

components, each essential for func­

note that the fossil record does not sup­

tion. Thus, these features and processes

port this idea, for example, Harvard pa­

cannot be explained by gradual Dar­

leontologist George Gaylord Simpson

winian improvements, because

until all

["The History of Life," in Volume

the components are in place, these as­

Evolution after Darwin,

semblages are completely useless, and

Chicago Prt;ss, 1960] writes:

I

of

University of

thus provide no selective advantage. Behe spends over 100 pages describing some of these irreducibly complex bio­

The Opinion column offers mathematicians the opportunity to write about any issue of interest to

chemical systems in detail, then sum­ marizes the results of an exhaustive search of the biochemical literature for Darwinian explanations. He concludes

the international mathematical

that while biochemistry texts often pay

community. Disagreement and

lip-service to the idea that natural se­

controversy are welcome. The views ..

and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in­ chief, Chandler Davis.

lection of random mutations can ex­ plain everything in the cell, such claims are pure "bluster," because "there is no publication in the scientific literature that describes how molecular evolution of any real, complex, biochemical sys­ tem either did occur or even might have occurred." When Dr. Behe was at the Univer­ sity of Texas El Paso in May of 1997 to give an invited talk,

I

told him that

I

thought he would fmd more support

It is a feature of the known fossil record that rrwst taxa appear abruptly. They are not, as a rule, led up to by a sequence of almost imperceptibly changing forerunners such as Darwin believed should be usual in evolution. . . . This phenomenon becomes. more universal and more intense as the hi­ erarchy of categories is ascended. Gaps among known species are spo­ radic and often small. Gaps arrwng known orders, classes and phyla are systematic and almost always large. These peculiarities of the record pose one of the most important theoretical problems in the whole history of life: Is the sudden appearance ofhigher cat­ egories a phenomenon of evolution or ofthe record only, due to sampling bias and other inadequacies?

for his ideas in mathematics, physics, and

computer

science departments

I

An April, 1982, Life

Magazine arti­

know a good

cle (excerpted from Francis Hitching's

many mathematicians, physicists, and computer scientists who, like me, are

book, The Neck of the Giraf fe: Where Darwin Went Wrong) contains the fol­

appalled that Darwin's explanation for

lowing report:

than in his own field.

the development of life is so widely ac­ cepted in the life sciences. Few of them ever speak out or write on this issue, however-perhaps because they feel the question is simply out of their do­ main. However,

I believe there are two

central arguments against Darwinism, and both seem to be most readily ap­ preciated by those in the more mathe­ matical sciences.

When you look for links between ma­ jor groups of animals, they simply aren't there. . . . ''Instead offinding the gradual unfolding of life," writes David M. Raup, a curator of Chicago's Field Museum of Natural History, "what geologists ofDarwin's time and geologists of the present day actually find is a highly uneven or jerky

© 2000 SPRINGER-VERLAG NEW YORK. VOLUME 22, NUMBER 4. 2000

5

record; that is, species appear in the fossil sequence very suddenly, show little or no change during their exis­ tence, then abruptly disappear." These are not negligible gaps. They are pe­ riods, in .aU the major evolutionary transitions, when immense physio­ logical changes had to take place. Even among biologists, the idea that new organs, and thus higher categories, could develop gradually through tiny improvements has often been chal­ lenged. How could the "survival of the fittest" guide the development of new organs through their initial useless stages, during which they obviously present no selective advantage? (This is often referred to as the "problem of novelties.") Or guide the development of entire new systems, such as ner­ vous, circulatory, digestive, respira­ tory and reproductive systems, which would require the simultaneous devel­ opment of several new interdependent organs, none of which is useful, or pro­ vides any selective advantage, by it­ self? French biologist Jean Rostand, for example, wrote [A Biologist's View, Wm. Heinemann Ltd., 1956]: ·

It does not seem strictly impossible that mutations should have intro­ duced into the animal kingdom the differences which exist between one species and the next . . . hence it is very tempting to lay also at their door the differences between classes, fami­ lies and orders, and, in short, the whole of evolution. But it is obvious that such an extrapolation involves the gratuitous attribution to the mu­ tations of the past of a magnitude and power of innovation much greater than is shown by those of today. Behe's book is primarily a challenge to this cornerstone of Darwinism at the microscopic level. Although we may not be familiar with the complex bio­ chemical systems discussed in this book, I believe mathematicians are well qualified to appreciate the general ideas involved. And although an anal­ ogy is only an analogy, perhaps the best way to understand Behe's argu­ ment is by comparing the development of the genetic code of life with the de-

6

THE MATHEMATICAL INTELLIGENCER

velopment of a computer program. Suppose an engineer attempts to de­ sign a structural analysis computer program, writing it in a machine lan­ guage that is totally unknown to him. He simply types out random characters at his keyboard, and periodically runs tests on the program to recognize and select out chance improvements when they occur. The improvements are per­ manently incorporated into the pro­ gram while the other changes are dis­ carded. If our engineer continues this process of random changes and testing for a long enough time, could he even­ tually develop a sophisticated struc­ tural analysis program? (Of course, when intelligent humans decide what constitutes an "improvement", this is really artificial selection, so the anal­ ogy is far too generous.) If a billion engineers were to type at the rate of one random character per second, there is virtually no chance that any one of them would, given the 4.5 billion year age of the Earth to work on it, accidentally duplicate a given 20character improvement. Thus our en­ gineer cannot count on making any major improvements through chance alone. But could he not perhaps make progress through the accumulation of very small improvements? The Darwinist would presumably say yes, but to anyone who has had minimal programming experience this idea is equally implausible. Major improve­ ments to a computer program often re­ quire the addition or modification of hundreds of interdependent lines, no one of which makes any sense, or re­ sults in any improvement, when added by itself. Even the smallest improve­ ments usually require adding several new lines. It is conceivable that a pro­ grammer unable to look ahead more than 5 or 6 characters at a time might be able to make some very slight im­ provements to a computer program, but it is inconceivable that he could de­ sign anything sophisticated without the ability to plan far ahead and to guide his changes toward that plan. If archeologists of some future so­ ciety were to unearth the many ver­ sions of my PDE solver, PDE2D, which I have produced over the last 20 years, they would certainly note a steady in-

crease in complexity over time, and they would see many obvious similar­ ities between each new version and the previous one. In the beginning it was only able to solve a single linear, steady-state, 2D equation in a polygo­ nal region. Since then, PDE2D has de­ veloped many new abilities: it now solves nonlinear problems, time­ dependent and eigenvalue problems, systems of simultaneous equations, and it now handles general curved 2D regions. Over the years, many new types of graphical output capabilities have evolved, and in 1991 it developed an interactive preprocessor, and more recently PDE2D has adapted to 3D and 1D problems. An archeologist attempt­ ing to explain the evolution of this computer program in terms of many tiny improvements might be puzzled to find that each of these major advances (new classes or phyla??) appeared sud­ denly in new versions; for example, the ability to solve 3D problems first ap­ peared in version 4.0. Less major im­ provements (new families or orders??) appeared suddenly in new sub-ver­ sions; for example, the ability to solve 3D problems with periodic boundary conditions first appeared in version 5.6. In fact, the record of PDE2D's de­ velopment would be similar to the fos­ sil record, with large gaps where ma­ jor new features appeared, and smaller gaps where minor ones appeared. That is because the multitude of intermedi­ ate programs between versions or sub­ versions which the archeologist might expect to fmd never existed, be­ cause-for example-none of the changes I made for edition 4.0 made any sense, or provided PDE2D any ad­ vantage whatever in solving 3D prob­ lems (or anything else), until hundreds of lines had been added. Whether at the microscopic or macroscopic level, major, complex, evolutionary advances, involving new features (as opposed to minor, quanti­ tative changes such as an increase in the length of the giraffe's neck, or the darkening of the wings of a moth, which clearly could occur gradually), also involve the addition of many in­ terrelated and interdependent pieces. These complex advances, like those made to computer programs, are not

always "irreducibly complex"-some­ times there are useful intermediate stages. But just as major improve­ ments to a computer program cannot be made 5 or 6 characters at a time, certainly no major evolutionary ad­ vance is reducible to a chain ortiny im­ provements, each small enough to be bridged by a single random mutation. II. The other point is very simple, but also seems to be appreciated only by more mathematically-oriented people. It is that to attribute the development of life on Earth to natural selection is to assign to it-and to it alone, of all lmown natural "forces"-the ability to violate the second law of thermody­ namics and to cause order to arise from disorder. It is often argued that since the Earth is not a closed system-it re­ ceives energy from the Sun, for exam­ ple-the second law is not applicable in this case. It is true that order can in­ crease locally, if the local increase is compensated by a decrease elsewhere, i.e., an open system can be taken to a less probable state by importing order from outside. For example, we could transport a truckload of encyclopedias and computers to the moon, thereby in­ creasing the order on the moon, with­ out violating the second law. But the second law of thermodynamics-at least the underlying principle behind this law-simply says that natural forces do not cause extremely improb­ able things to happen, and it is absurd

to argue that because the Earth receives energy from the Sun, this principle was not violated here when the original re­ arrangement of atoms into encyclope­ dias and computers occurred. The biologist studies the details of natural history, and when he looks at the similarities between two species of butterflies, he is understandably re­ luctant to attribute the small differ­ ences to the supernatural. But the mathematician or physicist is likely to take the broader view. I imagine vis­ iting the Earth when it was young and returning now to fmd highways with automobiles on them, airports with jet airplanes, and tall buildings full of complicated equipment, such as tele­ visions, telephones, and computers. Then I imagine the construction of a gigantic computer model which starts with the initial conditions on Earth 4 billion years ago and tries to simulate the effects that the four lmown forces of physics (the gravitational, electro­ magnetic, and strong and weak nu­ clear forces) would have on every atom and every subatomic particle on our planet (perhaps using random number generators to model quantum uncertainties!). If we ran such a sim­ ulation out to the present day, would it predict that the basic forces of Nature would reorganize the basic particles of Nature into libraries full of encyclopedias, science texts and novels, nuclear power plants, aircraft

knows all about cancer.

He's got it. Luckily, Adam has St. Jude Children's

Research Hospital, where doctors and scientists are

making progress on his

disease. To learn how you can help, call:

1-800-877-5833.

AUTHOR

GRANVILLE SEWELL

Mathematics Department University

of

Texas at E1 Paso

El Paso, TX 79968 USA

e-mail: [email protected] Granville Sewell completed his PhD at Purdue Un iversity in 1972. He has subsequently been employed by (in chronological order) Universidad Sim6n Bolivar (Caracas), Oak Ridge National Laboratory, Purdue University, IMSL (Houston), The University of Texas

Center for High Performance Com­ puting (Austin), and the University of Texas El Paso; he spent Fall 1 999 at Universidad de Tucuman in Argentina on a Fulbright grant. He has written four books on numerical analysis.

carriers with supersonic jets parked on deck, and computers connected to laser printers, CRTs, and keyboards? If we graphically displayed the posi­ tions of the atoms at the end of the simulation, would we find that cars and trucks had formed, or that super­ computers had arisen? Certainly we would not, and I do not believe that adding sunlight to the model would help much. Clearly something ex­ tremely improbable has happened here on our planet, with the origin and development of life, and especially with the development of human con­ sciousness and creativity. Granville Sewell Mathematics Department University of Texas El Paso El Paso, TX 79968 USA e-mail: [email protected]

VOLUME 22, NUMBER 4, 2000

7

MICHAEL EASTWOOD1 AND ROGER PENROSE

Drawing with Comp ex Nu m be rs •

t is not commonly realized that the algebra of complex numbers can be used in an

~

elegant way to represent the images of ordinary 3-dimensional figures, orthograph­ ically projected to the plane. We describe these ideas here, both using simple geome­ try and setting them in a broader context.

Consider orthogonal projection in Euclidean n-space

onto an m-dimensional subspace. We may

as well choose

p : �n � �m onto the first m variables. Fix a nondegenerate simplex I in �n. Two such simplices are said to be similar if one coordinates so that this is the standard projection

can be obtained from the other by a Euclidean motion to­ gether with an overall scaling. This article answers the fol­

+ 1 points in �m, when can these as the images under P of the vertices of a simplex similar to I? When n = 3 and m = 2, then P is the standard ortho­ graphic projection (as often used in engineering drawing), lowing question. Given n

points be obtained

f3 =

and we are concerned with how to draw a given tetrahe­ dron. We shall show, for example, that four points

a, {3 ,

y,

8 in the plane are the orthographic projections of the ver­ tices of a

regular tetrahedron if and only if

(a + f3 + where

y

+

8)

2

=

4(a2 + {3 2 + y +

SZ)

(1)

suppose a cube is orthographically projected and normalised

is

mapped to the origin.

If a,

{3 ,

are the images of the three neighbouring vertices, then

y

(2) again

as

a

complex equation.

satisfied, then one

can

Conversely, if this equation

is

find a cube whose orthographic image

is given in this way. Since parallel lines are seen as parallel in the drawing, equation (2) allows one to draw the general cube: 'Supported by the Australian Research Council.

8

'Y =

The result for a cube

-23

is

known

theorem of axonometry-see

a, {3 , y, 8 are regarded as complex numbers! Similarly,

so that a particular vertex

i + 2i 14 + 7i

In this example, a = 2 - 26

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

as Gauss's fundamental is stated

[3, p. 309] where it

without proof. In engineering drawing, one usually fixes three

principal axes in Euclidean three-space, and then an

orthographic projection onto a plane transverse to these

axes is known

as an

axonometric projection

(see, for ex­

ample, [8, Chapter 17]). Gauss's theorem may be regarded

as determining the degree of foreshortening along the prin­ cipal axes for a general axonometric projection. The pro­ jection corresponding to taking

a, {3 , y to be the three cube

unity is called isometric projection because the foreshortening is the same for the three principal axes.

roots of

In an axonometric drawing, it is conventional to take the image axes at mutually obtuse angles:

this diagram, the three principal axes and a are given. By drawing a perpendicular from a to one of the principal axes and marking its intersection with the remaining principal axis, we obtain P. The point Q is obtained by drawing a semi­ circle as illustrated. The point R is on the resulting line and equidistant with a from Q. Finally, f3 is obtained by drop­ ping a perpendicular as shown. It is easy to see that this con­ struction has the desired effect-in Euclidean three-space, rotate the right-angled triangle with hypotenuse Pa about this hypotenuse until the point Q lies directly above 0, in which case R will lie directly above f3 and the third vertex will lie somewhere over the line through 0 and Q. One may verify the appropriate part of Weisbach's condition In

1131

If la l =a, = b, I'Yi = c, then equation (2) is equivalent to the sine rule for the triangle with sides a2, /32, y , namely

a2

b2

sin 2A

sin 2B

c2 sin 2C'

In this form, the fundamental theorem of axonometry is due to Weisbach, and was published in Tiibingen in 1844 in the Polyte chnische Mitte ilunge n ofVolz and Karmasch. Equivalent statements can be found in modem engineer­ ing drawing texts (e.g., [7, p. 44]). Equation (2) may be used to give a ruler-and-compass construction of the general orthographic image of a cube. If we suppose that the image of a vertex and two of its neighbours are already specified, then (2) determines (up to a two-fold ambiguity) the image of the third neighbour. The construction is straightforward, except perhaps for the construction of a complex square root, for which we ad­ vocate the following as quite efficient:

a2

b2

sin 2A

sin 2B

(3)

by the following calculation. Without loss of generality we may represent all these points by complex numbers nor­ malised so that Q = 1. Then it is straightforward to check that R=

1 + i - ia ,

·a (a +a)+ 2(1 - a - a) a -a a (a +a) + 2(1 - a - a) . f3 = �, 2 - a - -a

P=

-

-

'

-

and therefore that

a2 + /32 = 4

(a - 1)(a- 1)(a +a- 1) . (a + a- 2) 2

That a2 + f32 is real is equivalent to (3). To prove Gauss's theorem more directly, consider three vectors in �3 as the columns of a 3 X 3 matrix. This ma­ trix is orthogonal if and only if the three vectors are or­ thonormal. It is equivalent to demand that the three rows be orthonormal. However, any two orthonormal vectors in �3 may be extended to an orthonormal basis. Thus, the condition that three vectors

First, C is constructed by marking the real axis at a distance llzll from the origin. Then, a circle is constructed passing through the three points C, 1, and z. Finally, the angle be­ tween 1 and z is bisected and vZ appears where this bi­ sector meets the circle. In engineering drawing, it is more usual that the images of the three principal axes are prescribed or chosen by the designer and one needs to determine the relative degree of foreshortening along these axes. There is a ruler-and­ compass construction given by T. Schmid in 1922 (see, for example, [8, § 17 . 17 -17 .19]):

in �2 be the images under p : basis of �3, is that

�3 � �2 of an

and

orthonormal

(Yl Y2 Y3)

be orthonormal in �3. Dropping the requirement that the common norm be 1, we obtain

x12 + xi + X32 = Y12 + Y22 + Y32

and X1Y1 + X2 Y2

+X3Y3 = 0.

Writing a = x1 + iy1, f3 = x + Y , y = X3 + Y3, these two 2 2 equations are the real and imaginary parts of (2). To de­ duce the case of a regular tetrahedron as described by equation (1) from the case of a cube as described by equa­ tion (2), it suffices to note that equation (1) is translation­ invariant and that a regular tetrahedron may be inscribed in a cube. Thus, we may take B = a + f3 + y and observe that (1) and (2) are then equivalent. It is easy to see that the possible images of a particular tetrahedron 2: in �3 under an arbitrary Euclidean motion folVOLUME 22, NUMBER 4, 2000

9

lowed by the projection P form a 5-dimensional space-the group of Euclidean motions is (klimensional, but translation orthogonal to the plane leaves the image unaltered. It there­ fore has codimension 3 in the 8-dimensional space of all tetrahe
X =

w (u -w .

u + iv -w

)

for

We may identify H with !R3, and, in so doing, -det X be­ comes the square of the Euclidean length. The group G of invertible 2 X 2 complex matrices of the form

A=

(� -:)

(jaj 2 + j bj 2)2 det X ,

so G acts by similarities. It is easy to check that all simi­ larities may be obtained in this way. (This trick is essen­ tially as used in Hamilton's theory of quaternions and is well known to physicists.In modem parlance it is equiva­ lent to the isomorphism of Lie groups Spin(3) = SU(2).) Therefore, an arbitrary orthographic image of a cube may be obtained by acting with A on the standard basis

and then picking out the top right-hand entries. We obtain

(� �) A( �i �) A(� �J A

N N N

(: = (: = (: =

i(a 2 + b2) *

)��a 2

: )� 2a b =

2 b

·c

+

b 2) =

f3

y

and therefore a2 + {3 2 + y = 0, as required. Conversely, is exactly the condition that a ,{3 , y may be written in this form.(Compare the half-angle formulae: if s2 + c2 = 1, then s = 2 t/( 1 + t2) and c = (1 - t2)/(1 + t2) for some t.) That Gauss [3, p. 309] makes the same observation con­ cerning the form of a ,{3 , y suggests that perhaps he also had this reasoning in mind. In general, the following terminology concerning the stan­ dard projection P : !Rn � !Rm is useful.We shall say that vh v2, . . . , Vn E !Rm are normalised e utactic if and only if there is an orthonormal basis u1 , u2, ... , Un of !Rn with v1 = Pu1 for this

10

THE MATHEMATICAL INTELLIGENCER

tic for some p, =F 0. The proof of Gauss's theorem using or­ thogonal matrices clearly extends to yield the following result

The points z1, z2, tic ifa nd only if

Theorem

Z 1 2 + Z 22 +

.

.

•

, Z E e = !R2 a re n

+

· · ·

Z

e utac ­

2= 0

n

and not a U z1 a re ze ro. There is an alternative proof for n morphism Spin(4)

=

= 4 based on the iso­

SU(2) X SU(2),

and, indeed, this is how we came across the theorem in the first place. However, a more direct route to complex numbers, and one which applies in all dimensions, is based on the obser­ vation that Gr�(IR2), the Grassmannian of oriented two-planes in !Rn, is naturally a complex manifold. When n = 3, this Grassmannian is just the two-sphere and has a complex struc­ ture as the Riemann sphere.In general, consider the mapping

eiP'n-1 \

acts linearly on H by X� AXN. Moreover, det(AXAI) =

j = 1, 2, . . . , n. We shall say that vh v2, ... , Vn E !Rm are e u­ tac tic if and only if J.LVb p, v2, . . . , J.LVn are normalised eutac­

IR IP'n- 1 �

Gr�(!Rn)

induced by en 3 z� iz/\Z . In other words, a complex vec­ tor Z = X + iy E en is mapped to the two-dimensional ori­ ented subspace of [Rn spanned by x andy, the real and imaginary parts of z. Let ( , ) denote the standard inner product on [Rn extended to en as a complex bilinear form. Then, (z, z) = 0 imposes two real equations and

(x,

y) = 0

on the real and imaginary parts.In other words, x, y is pro­ portional to an orthonormal basis for span{x,y).Hence, if z and w satisfy (z, z) = 0 = (w, w ) and defme the same ori­ ented two-plane, then w = >.z for some A E C\{0}.The non­ singular complex quadric

K=

{[z] E e1P'n-1

s.t.

(z, z) = 0)

avoids IR IP' - 1 C e1P' -b and we have shown that n n jective.It is clearly surjective. The isomorphism 7T

11]Kis in­

: K.::::.,. Gr2 + (!Rn)

respects the natural action of SO (n) on K and Gr +c�Rn). 2 The generalised Gauss theorem follows immediately, for, rather than asking about the image of a general orthonor­ mal basis under the standard projection P : !Rn � !R2, we may, equivalently, ask about the image of the standard ba­ sis e 1, e 2, ..., e under a general orthogonal projection onto n an oriented two-plane n c !Rn.Any such n is naturally com­ plex, the action of i being given by rotation by 90° in the positive sense. If n is represented by [ZI, z2, ..., Z ] E K n as above and we use x, y E TI to identify TI with e, then e1�z1and Z12 + Z 2 + 2

· · ·

+Z

2 = (z, z) = 0,

n

as required. Conversely, a solution of this complex equa­ tion determines an appropriate plane n.

For the case of a general tetrahedron or simplex and for general m and n, it is more convenient to start with Hadwiger's theorem [4] or [2, page 251] as follows. The proof is obtained by extending our orthogonal matrix proof of Gauss's theorem. Theorem (Hadwiger) Asse mble vb v2, , Vn E [Rm a s the columns ofa n m X n ma trix V. The se ve ctors a re nor­ ma lise d e uta ctic ifand only if W1 = 1 (the m X m ide n­ .

.

•

Certainly, B = PUA is a solution of these equations; but it is the only solution, because A1(AA1)-1 A has rank nand e is not in the range of this linear transformation. D

Corollary (case m = 2) Points zb z , ... , Zn, Zn+l E C 2 a re the image s unde r orthogona l proje ction of the ve rtice s ofa simplex simila r to I ifand only if

z1Qz

tity ma trix ).

Proof If v1, v2, , Vn are normalised eutactic, then as­ sembling a corresponding orthonormal basis of !Rn as the columns of an n X n matrix, we have V = P Uand fYU= 1 (the n X n identity matrix). Therefore, UU1= 1 and •

.

.

w

=

Puu tpt = ppt = 1,

as required. Conversely, if W = 1, then the columns of V1 may be completed to an orthonormal basis of !Rn, i.e., V1 = U1P1 for UU1= 1. Now, U1U= 1 and V = PU, as re­ quired. D The case of a general simplex is obtained essentially by a change of basis as follows. Suppose a 1, a 2, . . . , a n, an+l are the vertices of a non-degenerate simplex I in !Rn whose centre of mass is at the origin. In other words, the n x (n + 1) matrix A has rank n and Ae = 0 where e is the column vector all of whose n + 1 entries are 1. Form the (n + 1) X (n + 1) symmetric matrix

Q = At(AAt)-2 A,

noting that rank AA1 is invertible.

A

=

n

implies that the

mome nt ma trix

Give n b1, b2, . . . , bm bn+1 E !Rm asse mble d a s the columns of a n m X (n + 1) ma trix B, the se ve ctors a re the image s under ort hogonal proje ction P : [Rn � [Rm of the ve rtice s ofa simplex congrue nt to I ifand only if Theorem

(4) Proof The vertices of a simplex congruent to I are the columns of a matrix UA + ae1 for some orthogonal matrix U and translation vector a E !Rn. Also, note that Qe = 0. Thus, if B = P(UA + ael ), then

BQB1 = PUAQA1[Jlpt = PUAAt(A Atr 2AAtU tpt = PUfYP1 = ppt = 1,

=

0

whe re z is the column ve ctor with compone nts z1, z2, ..., Zn, Zn+l· It is, of course, possible to compute Q explicitly for any given example. If the simplex I has some degree of sym­ metry, however, we can often circumvent such computa­ tion. Consider, for example, the case of a regula r simplex. From the corollary above, we know that the image of such a simplex in the plane is characterised by a complex ho­ mogeneous quadratic polynomial. The symmetries of the regular simplex ensure that this polynomial must be in­ variant under 9'n+1, the symmetric group on n + 1 letters. Hence, it must be expressible in terms of the elementary symmetric polynomials. Equivalently, it must be a linear combination of

(zl + Z2 + ...

+

Zn + Zn+l)2

Z12

and +Z

22 +

"'

+

Zn2 + Zn+l2 ·

Up to scale, there is only one such combination that is translation-invariant, namely

(zl + Z2

+

. . . + Zn + Zn+l)2(n + 1)(zl2 + Z22 + . . . + Zn2 + Zn+l2).

(5)

It follows that the vanishing of this polynomial is an equa­ tion that characterises the possible images of a regular sim­ plex under orthogonal projection into the plane. The spe­ cial case n = 2 characterises the equilateral triangles in the plane [1, Problem 15 on page 79]. Equation (2) characterising the orthographic images of a cube, may be deduced by similar symmetry considerations. If a particular vertex is mapped to the origin and its neigh­ bours are mapped to a, {3, y, 'then, since each of these neigh­ bouring vertices is on an equal footing, the polynomial in question must be a linear combination of (a + {3 + y)2 and a2 + {32 + y . To fmd out which linear combination, we need only consider a particular projection, for example:

as required. Conversely, Qe = 0 implies that (4) is translation in­ variant. So, without loss of generality, we may suppose that b1 + b2 + . . . + bn + bn+l = 0, that is to say,Be = 0.Writing out (4) in full gives

BAt (AAt )-l (BAt (Mtrl Y = 1 so, by Hadwiger's theorem, there is an orthogonal matrix Uso that

Thus, and

Be= 0.

In this example, (a + {3 + y)2 = 2i and a2 + {32 + y = 0. Up to scale, therefore, (2) is the correct equation. The case of a regular dodecahedron is similar. Using the fact that a cube may be inscribed in such a dodecahedron [5], we may deduce a particular projection:

VOLUME 22, NUMBER 4, 2000

11

v'5- 1

thogonally projected images in the plane of the N vertices of any non-degenerate regular polytope, real or complex, will satisfy equation (6). This includes regular polygons in the plane, where the projection is vacuous.As already remarked, for polyhedra other than simplices, a quadratic equation such as (6) is no longer sufficient to characterise the orthogonal image up to scale. In general, there will also be some linear relations. For a non-degenerate N-tope in !R n there will be N- n - such relations.The simplest example is a square in !R 2, which is characterised by the complex equations

v'5+ 1.

7=-----t

4

4

1 if+{3 2 + y = a + f3+ y )2 = 3v'5)! 2 (2 - v'5)! 2. (a+ f3 + 'Y + 8)2 = a2 + {3 2+r + (a + f3+'Y? + cV5- 1)(if + {3 2 + y ) = 0.

with

(

(7In this particular case,

and

4(

Therefore, this is the correct equation in the general case. It may be used as the basis of a ruler-and-compass con­ struction of the general orthographic projection of a regu­ lar dodecahedron. It is interesting to note that if aU the vertices of a Platonic solid are orthographically projected to z 1, z 2, ..., ZN E C, then necessarily

+ · · + z� ). (1),

Cz 1 + z 2 + · · · + ZN )2 = N(z 12 + z 22

·

(6)

Only for a tetrahedron, when (6) coincides with is this condition also sufficient.To verify (6) for the other Platonic solids, first note that it is translation-invariant.Therefore, it suffi ces to impose z 1 + z 2 · · · ZN = 0 and show that z 12 + z 22 ··· + z� = 0.The case of a cube now follows immediately, as its vertices may be grouped as two regular tetrahedra. The dodecahedral case may be dealt with by grouping its vertices into five regular tetrahedra. The reg­ ular octahedron is amenable to a similar trick, but not the icosahedron. Rather than resorting to direct computation, a uniform proof may be given as follows. As before, assemble the vertices of the given solid I as the columns of a matrix A, now of size 3 X N, and consider the moment matrix M = AAt . Observe that

+ +

+

(I

i 0) M

G)

�

+ + ··· + zJ' .

z,' z,'

The moment matrix is positive definite and symmetric.In other words, it defmes a metric on IR3, manifestly invariant under the symmetries of I. If I is regular-or, more gen­ erally, er\ioys the symmetries of a regular solid (e.g., a cuboctahedron or rhombicosidodecahedron)-then its symmetry group acts irreducibly on !R3. Thus, M must be proportional to the identity matrix and the result follows. For a general solid I, the two complex numbers

±Vz 12

+ z 22+ ··· + z�

are the foci of the ellipse

(x y) R

(=) = 1,

where R is the inverse of the quadratic form obtained by restricting M to the plane of projection. This reasoning also works in higher dimensions, where it shows (as cor\iectured to us by H.S.M.Coxeter) that the or-

12

THE MATHEMATICAL INTELLIGENCER

fJ2)

a+ 'Y = f3 + and

8.

It is interesting to investigate further the relationship be­ tween a non-degenerate simplex I in !R n and its quadratic form Q = At (AAI ) -2 A. Recall that A is the n X (n + 1) ma­ trix whose columns are the vertices of I. There are sev­ eral other formulae for or characterisations of Q.LetS de­ note the (n X (n symmetric matrix

+ 1)

1-

+ 1) 1 � � _ n+1

(

:

1

� :.1

1

)

·

It is the matrix of orthogonal projection in !R n + 1 in the di­ rection of the vector e. We maintain that Q is characterised by the equations

QG = S

and

Qe = 0,

where G = AtA. Certainly, if these equations hold, then they are enough to determine Q, because the matrix G has rank n and e is not in its range. The second equation is evident, and the first equation with Q replaced by At (AAI )- 2 A and G by AtA reads

At (AAt ) - 1A = S. To see that this holds it suffices to observe that it is clearly true after postmultiplication by At or e. We may equally well characterise Q by means of the equations

GQ = S

and

Qe = 0

These equations relate G and Q geometrically: both ma­ trices annihilate e, whilst on the hyperplane orthogonal to e they are mutually inverse.This is to say that G and Q are gene rali sed i nverses (6] of each other.Thus we write

Q = at = (AIA)t = At Att where At is the generalised inverse of A. In this case, At = At (AAt ) - 1 . This also shows how to compute Q more di­ rectly in certain cases.The matrix G has direct geometric interpretation as the various inner products of the vectors In the case of a regular simplex, for example, we know that ll aill 2 is independent of i, that ll ai - aill 2 is independent of i * j, and that a1 + a2 + . . . + an + an + l = 0.We may deduce that, with a suitable over­ all scale, G = S. Since st = S, it follows that Q = S.This is a direct derivation of (5). It is clear geometrically that G determines I up to con­ gruency.Therefore, so does Q. In other words, the possi-

a1, a2, . . . , an, an + l ·

ble quadratic forms Q that can arise give a natural para­ metrisation of the non-degenerate simplices up to congru­ ency. As to which Q do arise, certainly they eiijoy the fol­ lowing properties: • • •

Q is a real (n + 1) X (n + ll symmetric matrix. Qe = 0, and only multiples of e are in the kernel of Q. All other eigenvalues of Q are positive.

Conversely, these properties characterise the possible Q that can arise: given such a Q, we may take A1 to have as its columns a system of mutually orthogonal eigenvectors for the non-zero eigenvalues of Q, each being scaled to have length· equal to the square-root of the reciprocal of the corresponding eigenvalue. It follows easily that At(AAI)-2 A. Q It is also possible to repeat this analysis in pseudo­ Euclidean spaces. The only difference is that the condition that the non-zero eigenvalues of Q be positive is replaced by a condition on sign precisely reflecting the original sig­ nature of the inner product. Finally we should mention some possible applications. There is much current interest in computer vision. In par­ ticular, there is the problem of recognising a wire-frame object from its orthographic image. The results we have described can be used as test on such an image, for ex­ ample to see whether a given image could be that of a cube, or to keep track of a moving shape. It is clear that such =

tests could be implemented quite efficiently. Another pos­ sibility is in the manipulation of CADD2 data. Rather than storing an image as an array of vectors in IR3, it may some­ times be more efficient to store certain tetrahedra within such an image by means of the corresponding quadratic form. For orthographic imaging this may be preferable. We would like to thank H. S.M. Coxeter for drawing our attention to Hadwiger's article, R. Michaels and J. Cofman for pointing out Gauss's and Weisbach's work, E.J. Pitman for many useful conversations, and the referee for sug­ gesting several improvements. REFERiiNCES

(1 ] S. Barnard and J . M . Child, Higher Algebra, MacMillan 1 936. [2] H.S.M. Coxeter, Regular Polytopes, Methuen 1 948. [3] C.F. Gauss, Werke, Zweiter Band, Koniglichen Gesellschaft der Wissenschaften, Gottingen 1 876. (4] H. Hadwiger, Ober ausgezeichnete Vectorsteme und regulare Polytope, Comment. Math. Helv. 13 (1 940), 90--108.

(5] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea 1 952, 1983, 1 990. _ [6] R. Penrose, A generalised inverse for matrices, Proc. Camb. Phil. Soc. 51 (1 955), 406-41 3. (7] R.N. Roth and I.A. van Haeringen, The Australia n Engineering Drawing Handbook, Part One, The Institute of Engineers, Australia 1 988.

[8] R.P. Hoelscher and C.H. Springer, Engineering Drawing and Geometry, Second Edition, Wiley 1961.

AUTHORS

MICHAEL EASTWOOD

ROGER PENROSE

Department of Mathematics

Mathematical Institute

University of Adelaide

University of Oxford

South Australia 5005

Oxford OX1 3LB England

e-mail: [email protected]

e-mail: [email protected] Michael Eastwood, BA Oxford, PhD Princeton, has been at the

University of Adelaide since 1985. He is now a Senior Research Fellow of the Australian Research Council. His mathematical in­

terests are differential geometry, extending to several complex vari­ ables, integral geometry, and twistor theory. His hobbies are rock­ climbing, volleyball, and-when the aforementioned have not done excessive damage to his fingernails-playing classical guitar.

Roger

Penrose

-

since 1 994, Sir Roger Penrose-holds ap­

pointments at the University of Oxford and at Gresham College London, and a part-time appointment at the Pennsylvania State University; earlier he was at Birl
his many honors are the 1988 Wolf Prize and the Royal Society

Royal M edal. He is known especially for contributions to the study

of non-periodic tilings, and for his theory of twisters, which aims

to unite Einstein's general relativity with quantum mechanics.

Among his books are the best-selling The Emperor's NfJW Mind (1989), and a recent novel (with Brian Aldiss), White Mars.

2Computer Aided Drafting and Design.

VOLUME 22, NUMBER 4, 2000

13

i,l,ijj:i§rr6hl¥11@i§#bii,'I,J§:id

Alexan d e r S h e n , Editor

lem? Denoting the set of vertices by V, the product V X V X V is the set of or­ dered triples of vertices. We are esti­ mating m, the number of 3-cliques; each of these corresponds to 6 = 3! ordered triples. Let A, then, be the subset of V X V X V coming from 3-cliques. It has 6m elements, and we want to use (C) to estimate this. For this, we need to know the number of elements in the projections of A. Projection Axy contains only pairs (x,y) that are connected by an edge, and each edge gives two pairs, so the cardinality of the projection does not exceed 2n, where n is the number of edges.Therefore (C) says that

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

Please send all submissions to the Mathematical Entertainments Editor, Alexander Shen, Institute for Problems of

Information Transmission, Ermolovoi 1 9, K-51 Moscow GSP-4, 1 01 447 Russia; e-mail: [email protected]

14

H

(6m) 2 :5 (2n)

ere I present two rather unex­ pected solutions of a simple prob­

lem:

Let G = (V,E) be an undirected graph having n edges; to prove that the number of 3-cliques in G does not exceed (v'2! 3)n312. As usual, a 3-clique is a set of three vertices connected by three edges. It would be just as good to formulate the problem without any graph-theoretic terminology at all: Assume given a collection of n seg­ ments in the plane; to prove that there are at most (v'2i3)n312 trian­ gles whose sides belong to the col­ lection. This theorem is an easy conse­ quence of the following inequality.Let A be a finite subset of the Cartesian product X X Y X Z. Defme the sets Axy, Axz, and Ayz as the projections of A onto X X Y, X X Z, and Y X Z, re­ spectively. Then

(#A)2 ::::; #Axy · #Axz · #Ayz,

(C)

where #8 stands for the cardinality of a fmite set 8. How do we apply (C) to our prob-

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

I

· (2n) · (2n),

in agreement with the stated conclu­ sion. Now I present two proofs of in­ equality (C). (First proof) I prefer to use the fol­ lowing geometric version of (C): if A is a (measurable) subset of IR3 having vol­ ume V, and 81, 8 , 83 are the areas of 2 its two-dimensional projections (onto the three coordinate planes), then

V2 ::::; 818�3.

(G)

(If A is composed of unit cubes with integer vertices, this is exactly the statement (C), for the volume is the number of unit cubes and the area is the number of unit squares in the pro­ jection.) To prove (G), first generalize it to say

(IfI f(x,y)g(x, z )h(y, z ) dx dy d z ) 2 :5 :5 If j2(x,y) dx dy · If g2(x, z ) dx dz II h2(y, z ) dy dz (I) ·

for any non-negative!, g, and h. Iff, g, and h are equal to 1 inside the corre­ sponding projections of A and to 0 out­ side, thenf(x,y)g(x, z )h(y, z ) = 1 for all (x,y,z ) E A (and maybe for some other points), so that (I) gives (G). Now inequality (I) is a variation of the Cauchy inequality and may be re­ duced to it:

(JII f(x,y)g(x,z )h(y,z ) dx dy dz )2 :::; :::; fi j2(x,y) dx dy · II (f g(x,z )h(y,z ) dz )2 dx dy :::; :::; II f2(x,y) dx dy · II (f g2(x,z )dz I h2 (y,z )dz ) dx dy = = II f2 (x,y) dx dy · II g2 (x,z ) dx dz II h2(y,z ) dy dz .

2 log2 #A

::5

and we get (C) by exponentiation. ***

·

(Second proof) This proof of (C) is completely differ­ ent (and rather strange). It uses the notion of Shannon en­ tropy of a random variable with finite range. If a random variable g takes n values with probabilities p 1 , . . . , Pn, then the Shannon entropy of g is defined as HW = -

L Pi log2 Pi i

It does not exceed log2 n and is equal to log2 n when all values are equiprobable. If g and 17 are both random variables with finite range, then so is the pair (g,17), and its Shannon entropy H((g, 17)) is given by the general definition. The conditional entropy H(g 1 17) of g when 17 is known can be defmed as

You can easily check that this agrees with the natural de­ finition of conditional entropy of g given 17: namely, fix any value of 17 and compute H(g) using the conditional proba­ bilities of the g values in place of the Pi; and then take the weighted average of the results, weighted by the probabil­ ities of the various values of 11· It is a standard fact that H((g, 17)) :::; HW

+ log2 #Axz + log2 #Ayz,

log2 #Axy

I have recei ved thefoUowi ng letter, completi ng the pi c­ ture sketched i n an earli er column. Your column in The Mathemati cal InteUigencer for Spring, 2000, never mentions the name of the problem discussed. It is called the "majority problem" in the the­ oretical computer science literature, and the two most in­ teresting papers on the subject (he says with a blush) are

L. Alonso, E. M. Reingold, and R. Schott, "Determining the Majority," Info. Proc. Let. 47 (1993), 253-255. L. Alonso, E. M. Reingold, and R. Schott, "The Average­ Case Complexity of Determining the Majority," SIAM J. Computi ng 26 (1997), 1-14. Both papers deal with (exact, achievable) lower bounds. Edward M. Reingold Department of Computer Science University of Illinois at Urbana-Champaign Urbana, IL 61 801 -2987 USA e-mail: [email protected]

+ H(17).

What are

Tne reader not acquainted with these matters will proba­ bly ef\ioy tackling this by straightforward analysis. So now we know that

your chances of dying on your next flight,

(L)

being called

for any g and 17· Now it is easy to prove that

for jury d uty,

2H((g, 7],7)) :::; H((g, 17)) + H((g,7)) + H((7],7)).

or winning

(E)

the lottery?

Indeed, (E) can be rewritten as

In this collection

of 21 puzzles,

H(7 I (g,7])) + H(7] I (g,7)) ::5 H((7], 7)) where the right-hand side equals H(7) + H(17 I 7). It remains ) ::5 H(7), and H(17 (g,7)) :::; H(17 7). to note that H(7 (The first of these is fact (L), the second is the "condi­ tionalized" version of it. The intuitive content of these is that any conditional entropy is smaller the more we know.) Now to prove (C) using (E). Consider the random vari­ able that is uniformly distributed in the set A c X X Y X Z. It can be considered as a triple of (dependent) variables (g, 17, 7), where g E X, 17 E Y, and � E Z. The entropy of the triple (g, 7], 7) equals log2 #A. Using (E), we get that

1...

I

Paul Nahin

applies the laws

I

2 log2 #A :::; H((g, 7])) + H((g,7)) + H((7],7)). The pair (g, 17) takes values inAxy, therefore its entropy does not exceed log2 #Axy· For the same reasons H((g, 7)) ::5 log2 #Axz and H((7],7)) :5 log2 #Ayz· Therefore

of probability as they apply to a fascinating a rray of situations.

Written in an informal way and containing a wealth of

interesting historical material, Duelling

is ideal for those

who are fascinated by mathematics and the role it plays in everyday life.

65 liM i/JunntionJ. 42 computer $imulillions.

Cloth $24.95 ISBN 0·691 -00979-1 Due October

Princeton University Press

(1 -243) 779777 U.K.

•

777-4726 U.S.

•

WWW. PUP.PRINCETON.EDU

VOLUME 22. NUMBER 4. 2000

15

JOHN HOLBROOK AND SUNG SOO KIM

Bertrand ' s Parad ox Revi s ited

Random Chords- Paradox Lost?

The classic form of Bertrand's paradox concerns "random chords" of a circle [B1907, §5]. Bertrand asks for the prob­ ability p that such a chord is longer than the side of the equilateral triangle inscribed in the circle. He then com­ putes p in three different ways, obtaining three different values. His conclusion is that the question is ma l posee. Indeed, it is not hard to believe that his computations im­ plicitly interpret "random chord" in three different ways. The paradox, though disconcerting, appears to be easily re­ solved. Let us recall some details. One can argue that p = 1/3, as follows. We may fix, or regard as known, one end P1 of the chord. By symmetry, or the "principle of indifference," this knowledge cannot affect the outcome. The angle t/J between the chord P1Pz and the tangent to the circle at P1 (see Figure 1) is uni­ formly distributed over [0,7T]. The chords that form sides of an inscribed equilateral triangle correspond to t/J = 7T/3 and t/J = 27T/3, so that favourable values of t/J lie in the middle third of the inter­ val [0, 7T]. It follows that p = 1/3. A variant interpretation of "random chord" chooses the endpoints P1 and Pz of the chord independently and with uniform distribution over the circumference of the circle. The equivalence of these two models is clear upon noting that the angle subtended by the chord at the circle's centre is 2tfr (see Figure 1). On the other hand, one can argue that p = 1/2, as fol-

16

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

lows. Again by symmetry, we may consider the direction of the chord fiXed. Let us represent a fiXed diameter of the circle by the interval [ -R,R], where R is the radius of the circle. A chord perpendicular to this diameter intersects it at some point x E [-R,R], and it is easy to see that sides of an inscribed equilateral triangle correspond to x = -R/2 or x = + R/2. The favourable longer chords are obtained for x E [ -R/2,R/2], so that p = 1/2. See Figure 2. Bertrand also proposed a third model, implying that p = 114. We'll come to that in our discussion of PET-scans. Cosmic Rays - Paradox Regained?

Suppose we present Bertrand's problem as a physical ex­ periment, for which there is presumably just one correct interpretation. To be specific, imagine that cosmic rays produce tracks in some disc-shaped detector, perhaps a classic Wilson cloud chamber. If the chamber is a thin disc of radius R, it will detect, as chords of the corresponding circle, just those cosmic rays travelling in the plane of the disc. We seek the probability p that such a cosmic ray track is longer than v'3R (the length of a side of an inscribed equilateral triangle). It has been our observation that stu­ dents presented with the problem in this form may respond with rather convincing arguments for both the p = 112 model and the p = 113 model. Now, however, only one (at most!) of these models can be correct. On the one hand, those cosmic rays falling perpendicu-

· . · · ·. ·. .

. · . ·

�)

.

e

. .

. .

.

.

�f

Figure 1. Bertrand's first model, with 1/J uniformly distributed. Figure 3. Cosmic rays enter a tangent window of width ,;.

lady on a given diameter [ -R,R] of the chamber (which Figure 2 might also depict) will clearly be uniformly dis­ tributed over the interval. They make tracks of length greater than v3R exactly when they fall on the middle half of the interval; hence p = 112. On the other hand, each cos­ mic ray must enter the chamber at some point P1 (as in Figure 1), and it cannot matter which. Moreover, the isotropy of the cosmic radiation suggests that the angle of incidence should range uniformly over the interval [0, 1r] from one tangent direction to its opposite. But this angle i� essentially if; in Figure 1, and we may follow the corre­ sponding argument to claim that p = 1/3. Many puzzling probabilistic paradoxes have been in­ vented (see Szekely's rich compendium [Sz1986], for exam­ ple), and Bertrand's paradox has been examined from sev-

eral points of view (see our final section for a brief account). Yet it appears that the more urgent form of the aradox that is posed by our physical model is not widely known. Of course, this form of the paradox can also be resolved. One way to view its resolution is by refming our physi­ cal model. We may think of the incoming cosmic rays as entering the detection chamber through any one of a finite number of tiny windows, represented by the sides of a reg­ ular n-gon (n large) approximating the circular boundary. It then becomes clear that the distribution of if; for rays en­ tering a given window depends on the cross-section pre­ sented by the window to rays at angle if; (see Figure 3). Thus the distribution of if; E [0, 7T] has density proportional to sin if;. This leads to the calculation:

p

1 2' the same value obtained from the other argument (which the experienced reader will, perhaps, have picked as the "right" choice all along).

-R

-L·

,_ :

Other Rays, and PET-scans X

0

Figure 2. Bertrand's second model, and a cosmic ray.

R.

Curiously, another well-known cloud chamber experiment require the uniform distribution of if;. If a small ra­ dioactive sample is fixed to the wall of the chamber (at some point P1 as in Figure 4), then some of the particles emitted (a and {3 "rays") will form tracks with a uniform distribution of if; over [0,7T]. Thus, by the now-familiar ar­ gument, the proportion of tracks of length exceeding v3R will be 1/3. In Figure 4 we have also suggested a model that has been studied abstractly (see [Szl986, § l . l ld]) but could be viewed in terms of positron emission tomography. During

does

VOLUME 22, NUMBER 4, 2000

17

"1_

Figure 4. a-rays from radioactive sample at P1, and PET-scan y-rays.

a PET-scan, a positron-electron union may occur at any point M in the object under study. This results in the emis­ sion of a pair of y-ray photons, travelling in opposite di­ rections away from M and defining a chord AB as in Figure 4. If we assume that M is uniformly distributed with respect to area within the disc of radius 1, then the distance r from M to the centre 0 has density 2r. If we also assume that the angle 'P is uniformly distributed, we have yet another variant of our question: what, now, is the probability p that AB exceeds the length of a side of the inscribed equilateral triangle? This is just the probability that the midpoint of AB lies within distance 1/2 of the centre 0. It is an amus­ ing exercise to verify that ,

P

=

JI 0

sin- 1(1/2r) 7T/2

2r dr = .!. + v3 3

27T

=

0.609,

where sin- 1(1/2r) is interpreted as 1r/2 when r < 1/2. If, for some reason, the y-rays always trace out the short­ est possible chord through M (i.e., if 'P is always 7T/2), we arrive back at the simple model proposed by Bertrand as his third interpretation of "random chord." In this case, each random choice of M determines the unique chord CD having M as midpoint, and the probability p is just the probability that M is within distance 1/2 from the centre, namely the relative area of the disc of radius 1/2. Thus Bertrand obtained his third answer: p = 1/4. Crofton, Borel, Poincare, et al.

. . . geometric probabi li ti es have run i nto diffi culti es cul­ mi nati ng i n the paradoxes ofBertrand whi ch threatened the fledgli ng fi eld wi th bani shment from the home of Mathemati cs . . .

in many of the classic works of probability, and it is inter­ esting to trace the various interpretations and attitudes found there. Crofton, in [C1885] (written even before the first edition of Bertrand's book), describes a model for ran­ dom lines in the plane that corresponds to our cosmic ray analogy. Crofton's model is measure-theoretic but can be changed to a probability model by the device of regarding a random line as a set of parallel lines, making angle (} with some fixed direction and separated by 2R, where R is again the radius of our circle. If r is the distance from the centre of the circle to the closest line of the family, Crofton's model says that r and (} are independent and uniformly distributed on [O,R] and [0,27T], respectively, and it leads directly to the conclusion that p = 1/2 (see [C1885], [S1976]). Edgeworth, on the other hand, writing in a later edition of the Encyclopaedi a Britanni ca [E 1911] than did Crofton, seemed willing to consider random lines as those defined by equations ax + by = 1, where the parameters a and b are independent and uniform over the real line. Moreover, von Mises [vM1964] felt that even in the famous Buffon needle problem the various possible models must be dis­ tinguished with care. Borel discussed Bertrand's paradox in [Bo1909], but spent more energy on a related paradox from [B1907,§7] concerning the distribution of latitude for a point chosen at random on the surface of the earth. If we first use sym­ metry to claim that, as each longitude is equivalent to every other, we may restrict our attention to points on a single meridian, then we come to the wrong conclusion that lat­ itude A will be uniformly distributed over [ -90,90]. Borel uses a device analogous to our infmitesimal windows (those that admit cosmic rays in an earlier section), namely the narrow sectors between nearby meridians. By this means, he explains that A will have a density proportional to cos A (see Figure 5). We note that an even more classi­ cal treatment of this problem may be based on Archimedes's theorem on the areas of spherical slices be­ tween parallel planes: these areas are proportional to the

e '\. I.A t\to '("

-Mark Kac, from his foreword to [81976]

While Kac may have been indulging in provocative over­ statement, Bertrand's paradox is rather anxiously discussed

18

THE MATHEMATICAL INTELLIGENCER

Figure 5. Meridians versus infinitesimal sectors.

AU T H O R S

SUNG SOO KIM

JOHN HOLBROOK

Mathematics &

Hanyang University

Statistics Department

Ansan. Kyunggi 425-791

University of Guelph

Guelph, Ontario N1 G 2W1

Korea

[email protected]

Canada

e-mail: [email protected] John Holbrook studied mathematics at Queen's University in

Sung Soo Kim studied mathematics at Hanyang University, and did his graduate work at Kaist (Korean Advanced Institute

Ontario and at Caltech. He has taught in the United States

of Science and Technology) under Kil Hyun Kwon. He has

and in Venezuela; for many years he has been at the University of Guelph. He has worked mostly in operator theory and ma­

been at Hanyang University since and is now an Associate Professor. He and his wife Beanmoo have no children, but en­

trix analysis, but also in electron micrograph tomography, ran­

joy playing with twin n ieces Youkyung and Soobean. ,

dom balanced sampling, and computational experiment. Here he is seen with a grandson, also John Holbrook.

thickness of the slices, i.e., the distance between the planes. This means that sin A is uniformly distributed, and that A has density cos A/2. Poincare [P1912] resolves the Bertrand paradox more formally in terms of the transformation of coordinates. His view is that the model for random lines should be invari­ ant under rotations and translations, so that if (r, O) are the polar coordinates of the point on the line that is closest to the origin, then r and 0 are independent and uniformly dis­ tributed (Crofton's model). Conclusion: p = 1/2, and if the variables 1/J (from Figure 1) and 0 are introduced with the Jacobian factor (sin 1/J) relating them to r and 0, we also get p = 112. This is tantamount to our "physical" argument, reflected in Figure 3. Jaynes applied another invariance argument to Bertrand's problem (see [J1973], [Sz1986]). He argued that the size and location of the circle are not specified by Bertrand. The distribution of lines should be invariant un­ der expansion/contraction, translation, and rotation, in or­ der that the normalized distribution of chords should be identical for every circle. This leads to Crofton's model, so that according to Jaynes's invariance argument, Bertrand's problem is well-posed and p = 112.

(2) To John Milton, whose epic poems permit (perhaps) the pun: Paradox Lost/Regained. Mter putting this note to­ gether we came upon Ian Stewart's column in Scientific American (June 2000), where paradoxes are also lost and regained. One wonders when, in the 3.5-century interval since Milton's time, this little joke first occurred. We don't know, but a cursory search turned up the story "Paradox Lost" by SF writer Fredric Brown; it was first published in 1943.

Acknowledgements

[81 976) L. Santal6, Integral Geometry and Geometric Probability,

(1) To the MPC2 students at the University of Guelph, for their spirited arguments about the physical implications of Bertrand's paradox.

REFERENCES

[81 907) J. Bertrand, Calcul des 'Probabilites, Chelsea (reprint of 1 907 edition) [Bo1 909) E. Borel, Elements of the Theory of Probability (translation by J. Freund), Prentice-Hall 1 965 [C1 885] M. W. Crofton, Probability, in Encyclopaedia Britannica, 9th ed., v. 1 9, 768-788, 1 885 [E1 9 1 1 ] F. Y. Edgeworth, Probability, in Encyclopaedia Britannica, 1 1 th ed., v. 22, 376-403, 1 91 1

[J1 973) E. T. Jaynes, The well-posed problem, Foundations of Physics 3, 477-493, 1 973 [vM1 964] R. von Mises, Mathematical Theory of Probability and Statistics (H. Geiringer, ed.), Academic Press, 1 964

[P1 9 1 2) H. Poincare, Calcul des Probabilites, Gauthier-Villars, 1 91 2 Addison-Wesley, 1 976

[Sz1 986) G. Szekely, Paradoxes in Probability Theory and Mathematical Statistics, Reidel, 1 986

VOLUME 22. NUMBER 4, 2000

19

[email protected]§116'h¥1MQ.'I,'i,tl!:hh¥J

Impoverishment, F Fem inization, and Glass Ceilings: Women in Mathematics in Russia Karin Johnsgard

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

I

Marjorie Senec h a l , Editor

rom May 16th to 25th, 1998, a del­ egation of women in mathematics visited St. Petersburg and Moscow, meeting with their female counterparts at secondary schools, universities, and research institutions of the Russian Academy of Science. The delegation consisted of eight Americans and one Norwegian-some professors, some high school teachers. (At the end of this article is information about the in­ dividual members of the delegation.) The purpose of the meetings was to ex­ change information about research and education mechanisms and issues in our countries, and specifically to dis­ cuss gender-related issues in educa­ tion and professional advancement. 1 It should be understood that the par­ ticipants in the meetings (on both sides) were mathematicians, not historians or political scientists. None of the dele­ gates even spoke Russian (we relied on translators). This article, therefore, should be viewed in that light: One woman's attempt to collate a body of largely unverifiable and sometimes self­ contradictory data, impressions, and first-hand observations from an all-too­ brief visit to a country from which more reliable data is not obtainable, from her own perspective as an intelligent but ig­ norant first-time visitor.

A Delegate's Perspective, Prior to the Journey

Why should the mission focus on women mathematicians in particular? Why did we go to Russia and not some other na­ tion? And why did anyone feel that the

best way to conduct the mission was per­ sonal contact, rather thanjust looking up information on the Internet or in a book? The answers to these are interre­ lated. With regard to the choice of na­ tion: I now know that many nations would be appropriate for an exchange on gender issues in mathematics. Russia, however, does have the dis­ tinction of not maintaining data on women in mathematics, making long­ distance study impractical. Relations between Russia and the West were un­ usually good in 1998, making a journey to that nation appropriate and timely. Readers uncertain as to why the del­ egates thought that there might be any gender-sensitive issues in mathematics should consider the following. That summer of 1998, the U.S. Census Bureau reported that the average earn­ ings for an American woman were 74% of the average for men; this percentage was a new high for our nation. With re­ gard to academia in particular, the AAUP Annual Report of 1997-98 re­ ported disparities in academic salaries (appears at the bottom of the page). In 1993, women made up 45% of the U.S. workforce, but only 16% of employed scientists and engineers. Among recent graduates in science and engineering, 35% of the men were ac­ tually employed in such occupations, while only 18% of the women were. There was also an earnings gap: among 1993 college graduates, women's me­ dian starting salary was 84.2% of men's over all subjects, and 84.7% in natural sciences and mathematics.

mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department

U.S. Academic Salaries: Women's salary as a percentage of men's [AAUP] Public Institutions

Private Institutions

Rank

'82-'83

'87-'88

'92-'93

'97-'98

'82-'83

'87-'88

'92-'93

Prof.

90.5

89.4

89.3

87.7

84.9

85.2

85.5

87.2

Assoc.

94.2

93.1

93.4

92.9

92.1

91 .6

92.4

92.0

Asst.

93.6

91 . 1

92.7

93.9

91 .7

89.2

90.3

91 .9

'97-'98

(Notice that the gap has actually widened at public institutions, and remained essentially unchanged at private institutions.)

of Mathematics, Smith College, Northampton, MA 01 063, USA;

1The delegation and itinerary were organized by People-to-People International, a non-profit organization ded­

e-mail: [email protected]

icated to increasing international understanding. See website http://www.ptpi.org for more information.

20

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

1996 at U.S. colleges and univer­ sities, women earned over half of all bachelor's and master's degrees and 40% of all doctorates. However, in sci­ ence and engineering disciplines they received only 300!6 of bachelor's de­ grees and less than a quarter of ad­ vanced degrees2 [A, NCES]. In

% of U.S. degrees eamed by women In '95-'96

Overall and in math-related fields [NCES] BA/BS

MAIMS

Ph.D.

All fields

55.1

55.9

39.9

Math

45.7

38.8

20.43

Phys. Sci.

36.0

32.2

23.1

Comp. Sci.

27.5

26.7

1 4.5

Eng.

1 6. 1

1 7.2

1 2.5

The percentage of women receiving U.S. doctorates in mathematics has been increasing, especially among American recipients. (The figures shown include statistics degrees, among which 34.3% of the '97-'98 re­ cipients were female.) Despite the fact that about one in four recent math Ph.D.'s for that last decade have been female, the propor­ tion of women among the degree re­ cipients who were hired by doctorate­ granting math departments in 1998 was ocly 18.5%. (This has been fairly con­ stant over recent years.) Among the 48 departments ranked as excellent, the percentage of women among those hired was 16.2% (at private schools in this category it was 7.1%) [DMR]. Virtually all of the hires at the well­ ranked universities are temporary in­ structorships. At the time (1991) of the Jenny Harrison sex discrimination case against UC Berkeley, there were only four tenured women (to 303 tenured men) in the ten top-ranked math de­ partments in the United States, and one untenured woman to 86 untenured men [S]. (I am unaware of any significant improvement since that time. )4 The standard justification for these inequalities, of course, is that "women are not as good at math," and the proof

% of U.S. math doctorates eamed by women, by year

Over all recipients and over U.S. recipients [NCES,DMR]

'77-'78

'82-'83

All

1 4.9

1 6.4

u.s.

14

20

of this assertion is invariably given as the Scholastic Aptitude Test, Math sec­ tion. There has been a very persistent gap in the mean scores of the two gen­ ders, with girls' mean currently about 93% of the boys' (SAT]. The conclusion about their relative abilities is then clinched by examining the very high­ est scorers and noting they are dispro­ portionately male. But in studies of to­ tal population, both the highest and the very lowest scorers are disproportion­ ately male; the real difference is that the male scores have a larger standard deviation [S]. Almost none of the low tail of the curve is reflected by the SAT; in 1997, only 14% of exam-takers re­ ported GPA's lower than a B [NECS]. Nevertheless, the 1997 U.S. Dept. of Education report [BS] uses precisely this same examination of exception­ ally high scores to report, "Men score higher than women on the SAT math­ ematics . . . and . . . Advanced Placement (AP) examinations." In fact, the SAT-M consistently re­ ports a greater gender difference in math than other standardized exams. A remarkable cross-study analysis on this topic was carried out in the late 1980s [HFL]; the following is taken from the abstract of the results.

Reviewers lwve consistently concluded tlwt males perform better on mathe­ matics tests tlwn females do. To make a refined assessment of the magnitude of gender differences in mathematics performance, we performed a meta­ analysis of 100 studies. . . . Averaged over aU effect sizes and based on sam­ ples of the general population, d was - 0. 05,5 indicating tlwt females out­ performed males by only a negligible

'92-'93

'97-'98

16.1

23.8

24.4

21

28

28

'87-'88

amount. For computation, d was -0.14 (the negative value indicating superior performance byfemales). For understanding of mathematical con­ cepts, d was - 0. 03; for complex prob­ lems solving, d was 0. 08. An exami­ nation of age trends indicated tlwt girls showed a slight superiority in computation in elementary school and middle school. There were no gender differences in problem solving in ele­ mentary or middle school; differences favoring men emerged in high school (d = 0.29) and in coUege [d = 0.32). Gender differences were smallest and actuaUyfavoredfemales in samples of the general population, flrew larger with increasingly selective samples, and were largest for highly selected samples and samples of highly preco­ cious persons. The magnitude of the gender difference has declined over the years; for studies published in 1973 or earlier d was 0.31, whereas it was 0. 14 for studies published in 1974 or later. We conclude tlwt gender differences in mathematics performance are smaU. The authors of the study found that if they included the SAT-M, d overall was 0.20, but if it was excluded, d was 0.15. For SAT-M itself that year, d = 0.40.6 "The �agnitude of the gender differ­ ence favoring males grew larger as the sample was more highly selected: d was 0.33 for moderately selected sam­ ples (such as college students), 0.54 for highly selected samples (such as stu­ dents at highly selected colleges, or graduate students), and 0.41 for sam­ ples selected for exceptional mathe­ matical precocity.'' The authors con­ cluded that the SAT-M scores are an anomaly, even among moderately se-

2The percentages are significantly lower if the field of psychology is not included among these. 3247 women out of 1 209 total. The AMS annual suNey found 250 women out of 1 1 54 total, for a representation of 21 .7%. 4See the final section, "Closing thoughts, " for more on these issues. 5This d is the mean for males mi_nus the mean for females, divided by the mean of the two within-sexes standard deviations. 6Considered "moderate to large." I computed a figure of d = 0.32 for 1 999 SAT·M data [SAT], but lack the resources to do a comparable meta-analysis on other re­ cent data.

VOLUME 22, NUMBER 4, 2000

21

lected samples. They obseiVed that more girls than boys take the SAT, and the girls are overall less advantaged in terms of family income and level of ed­ ucation, and less likely to attend pri­ vate schools. 7 They also suggested that there may be problems with the test it­ self or its administration. An international study of the same period [H] testing five subject areas of math in 20 countries found inconsistent results: girls out-performed boys in some areas in some countries but out­ comes were reversed in other countries. Overall, differences between countries greatly exceeded gender differences, which were insignificant in three sub­ jects and small in the remaining two. Note that the atypical but widely publicized gender difference on the SAT-M and AP (both of which are self­ selecting exams) occurs precisely where the cross-study says the male advantage should be greatest: at the extreme high end. Even the testing ser­ vice that administrates the SAT and AP cautions that their test scores consid­ ered alone are an inadequate indicator of potential in undergraduate studies (far less a measure of future success in profession). In particular, the SAT­ M underestimates freshman GPA for girls by 0. 10, and overpredicts for boys by 0. 1 1 [SATR] . Girls continue to get better grades than boys do in college math courses such as calculus! In math courses, among students receiving the same course grade, girls had SAT-M scores 33 pts lower than the boys did. Standardized multiple choice math ex­ ams do not appear to reflect faithfully the qualities that instructors seek and evaluate in the classroom. I am unaware of any studies linking professional mathematicians and sci­ entists with exceptional SAT-M scores. (My own were high but not extraordi­ nary, and my first GRE-M was quite low. Nevertheless, I won an award for original research in graduate school, and was both a Sloan Fellow and an NSF Postdoctoral Fellow.) Of far greater relevance than SAT-M scores is the question of whether mathematically gifted girls and women actu-

ally pursue training and careers in fields emphasizing these abilities. Males are still more likely to take math and physics courses in high school (al­ though the gap has narrowed substan­ tially), and more likely to take AP test in these areas [BS]. Female representa­ tion in math and science studies de­ creases astonishingly at every step from high school to doctorate. Many studies suggest women's flagging participating during this period reflects poor early training (often resulting from inappro­ priate academic advising), unequal classroom treatment, lessening levels of interest, or self-doubt, rather than in­ nate lack of ability. Percent of U.S. mathematically talented students earning Ph.D.'s In math, engineering, and physical sciences [A] Math

Eng.

Phys.

Women

0.1

0.7

0.25

Men

0.5

3.4

3. 7

So in brief, in the U.S. women are scarce in math at all levels (but decrease even further toward the higher levels), do not get hired at the best research universities, and earn less money than their male peers. Some have mentioned other problems-lack of professional opportunities, disre­ spect among one's colleagues, sexual harassment-but this quick sketch should suffice to give an impression of

THE MATHEMATICAL INTELLIGENCER

A Brief Overview of Impressions and Discoveries

The delegation's visit (May 16th to 25th) took place shortly before the July 1998 Russian economic disaster and ensuing political developments. It was a time in fact of unprecedented Russian prosperity. This circumstance should be kept in mind throughout. My own subconscious expectations before the trip were wildly mixed. The most negative of these (derived from a high school visit to Hungary in cold­ war conditions) were of peiVasive grayness of scenery and spirit: a gen­ eral poverty of beauty or extravagance of any sort, rigorous public adherence to doctrine, and personal isolation and distrust. In opposition to these were mental images of modern western European cities, and almost a fear that I would fmd Russian cities virtually in­ distinguishable from these. Of course neither extreme proved accurate. On the whole, however, the Russian cities seemed more closely akin to Newark than to cold-war Budapest. The stores were full of goods of all kinds, even

The members of the delegation with the women of Moscow school #1 03: (1-r) Eleanor Jones, Liv Berge, Olga Medovar [Moscow teacher], Mila Bolgak [translator], Sue Geller, Diana Vincent,

Audrey Leef, Karin Johnsgard, Elena Fyodorova [principal], Pamela Ferguson, Natalie Golubinsteva [assistant principal], Maureen Gavin, Lucy Dechene.

7These still hold; 7,529 more girls than boys took the SAT in 1 996 [SATR].

22

why the delegates might be curious about conditions elsewhere. I, at least, did not know whether I should expect matters to be better in Russia, worse, or not significantly different; but I was eager to investigate.

luxury items, and the streets were filled both with bustling business peo­ ple in fashionable garb and with wan­ dering panhandlers. The woefully ar­ chaic streets were jammed to bursting with cars, and the walls were �lastered with neon restaurant signs and bill­ board advertisements for cigarettes, liquor, and cosmetics. New banks, one woman told us, "spring up overnight like mushrooms." At the same time, ac­ tual production seems scant. The laws and taxes are so labyrinthine and se­ vere that itis all but impossible to start or run a business legally. Farming as it is known in the West is purely im­ practicable; thus, it is easier to import even food than to produce it locally. Everywhere we saw fabulously col­ orful old palaces and breath-taking churches, and the churches were really being used as places of worship, not as "cultural museums." (Religion, we were told, is "fashionable" currently.) A drive outside of St. Petersburg re­ vealed both impoverished collective farms and the intimidating new brick dachas of "New Russians" (i.e., mob­ sters). People we spoke to were open about their problems, hopes, and fears; we encountered some indifference, a ltttle downright rudeness, a fair amount of perplexity over the purpose of our visit, and a great deal of curiosity about ourselves and willingness to cooperate with our questions. In light of the then current Russian economic prosperity (however un­ fairly distributed and ephemeral), the delegation's basic fmding stands even more starkly illuminated. The over­ whelming issue affecting our Russian counterparts is basic subsistence fund­ ing: 900,1, of Russian scientists and edu­ cators live in poverty, even though many hold second jobs. Library, com­ puter, and technical funding at many institutions have shrunk to the point of being virtually non-existent, and are expected to decrease further. An in­ evitable corollary has been the Russian "brain leak": 80% of Russian scientists and teachers wish to go abroad, and 70,000 to 90,000 do so each year,8 with 52% of those departing being mathe­ maticians or physicists. Those leaving

Some of the professors and graduate students of St. Petersburg State University. Photo by Sue Geller.

the country (or abandoning science for more lucrative professions) have been disproportionately male, leading to "feminization" of the sciences and ed­ ucation: women accounted for 53% of Russian scientists and educators in 1992, up from 47% in 1970 and 42% in 1960 [AWSE]. (We will see that in­ creasing female representation has not ensured equity, however.) These basic trends (except feminization) were dis­ cussed everywhere we visited. With regard to gender issues: Many Russian women we spoke to denied that discrimination existed or was possible in their country, and saw no reason for the existence of any orga­ nization specifically for women in mathematics. These same women made statements (or heard without ap­ parent reaction compatriots' state­ ments) such as, "Well, girls aren't as good in math as boys are." Statistics provided by the [Russian] Association of Women in Science and Education (AWSE), employment information from the Central Economics and Mathemat­ ics Institute (CEMI), and a study of em­ ployees of the Academy of Science all reveal virtually impenetrable glass ceil­ ings for women in their professional advancement. Thus there is a substan­ tial discrepancy in earning power be-

tween men and women. In addition, because school choices with long-term effects are made for children while they are quite young, the system tends to perpetuate parental stereotypes about gender aptitudes. Based on our classroom observa­ tions and our discussions of curricu­ lum with Russian instructors, the qual­ ity of education appears to be extremely high for those who receive it, at least in the institutions visited by the delegation. The dedication and achievement of Russian educators in the face of dwindling resources is re­ markable. This level of excellence will be unsustainable if Russia continues to lose itS best minds to other nations by withholding decent living wages; to deny its institutions of learning and re­ search basic operating expenses; and to allow further erosion of a once­ excellent system of universal pre­ school training. And the delegation's reactions to all of this? In my own case at least, my mood varied with the reactions we elicited at a given meeting. At our first meeting for example (at St. Petersburg University), I felt horror (over the ap­ palling state of the facilities and lack of resources), shame (for my past dis­ interest and ignorance of these condi-

6This is as many as for the past decade in Latin America and Asia combined.

VOLUME 22, NUMBER 4, 2000

23

tions),

awkwardness

(neither

side

seemed to be completely prepared for

my youthful appearance could possi­

cussion continued in a stultified fash­

bly hold a doctorate.

ion until one of the delegates, Pam

The

of

Ferguson, inquired about CEMI's sys­

duct it), and above all, frustration: frus­

purposes in our professional meetings

tem of employee ranks. She had hit

tration over the sense that the two

was a continual disappointment during

upon

sides' purposes were at odds, that we

our trip, as much as we cherished other

seemed to open to the discussion one

were failing to communicate, that we

experiences. I felt that only one of

were asking the wrong questions and

these exchanges represented a true

by one, in order to air their grievances

that we were equally unable to elicit in­

meeting of minds and purposes: our

terest in our topics of greatest concern,

meeting

of

companied us from the AWSE meet­

that we were unwelcome, that our con­

Women in Science and Engineering

ing, gave an impromptu but passionate

the meeting or to know how to con­

imperfect understanding

with

the

Association

the

key

on the glass

point!

ceiling

The

women

conditions

at

CEMI. Ms. Vinokurova, who had ac­

tacts were either lying about or were

(AWSE). This was the one time that

address on obstacles to women's ad­

fundamentally unable to perceive what

both sides clearly shared basic work­

vancement in the mathematical sci­

the deleg_ates saw as blatant institu­

ing assumptions about being female

ences. We inquired whether any of the

tionalized gender discrimination.

and a mathematician. Finally, we had

other

I was not the only delegate to find

found contacts who understood our

AWSE, and were simply stunned to

this first meeting difficult. In a discus­

purposes and concerns; who had stud­

discover that the rest had never heard

sion afterwards over lunch, the dele­

ied the issues themselves and were

of this organization-with a chapter

gates dissected the event and our var­

happy to share their data (unobtain­

located perhaps five minutes drive

women

were

affiliated

with

I said

able otherwise); who were eager to

away!

plainly that I felt pessimistic about the

hear about our own experiences and

struck by the gulf between these back­

trip in general, suspecting that "I had

ideas. Our relief and joy were pro­

to-back

been a fool to think I would have any­

found. It was largely this meeting that

searchers no longer seemed bored; I

thing to contribute. " Another woman

makes me personally feel that our mis­

would

rebuked me, saying that it would be ar­

sion was not a failure in its most criti­

somewhat shaken and beginning to

ious dissatisfactions with it.

Even

our translator seemed

meetings. describe

The

them

CEMI as

re­

seeming

rogant to suppose that we could "con­

cal goal: fmding common ground for

question whether our visit to CEMI

tribute" anything at all, and that we

understanding.

was so inappropriate and meaningless

should be content simply to learn. We

One other meeting was notable.

all were unhappy with the way our

Immediately after our exchange with

sleepy

Russian contacts had seemed to view

AWSE, the delegates (and one AWSE

the purpose of the meeting simply as a

member)

Delegate Sue Geller described AWM

forum for us to ask questions of them,

Economics and Mathematics Institute

It

went

on

to

the

Central

after all. They certainly did not seem and

indifferent

any

more.

(Association for Women in Mathemat­

ics) to them and explained that its ori­

while showing no interest in learning

(CEMI).

was by this time late on a

gins were as humble and self-funded

anything from us. We resolved that at

Friday afternoon; the women who had

as those of A WSE, but that it now has

the next meeting (to take place at a

been assigned to meet with us there at

government funding for its projects

secondary school) we would volunteer

first seemed bored, barely masking

and carries real clout with other na­

information whether it was requested

their impatience to leave for the week­

tional mathematics organizations. We

or not. We also resolved to word our

end. For ourselves, we were starting to

departed from this meeting feeling that

own questions more carefully, giving il­

feel somewhat burnt out (it was our

at the very least, our visit had given

lustrative examples instead of using

third meeting of the day, on the heels

them food for thought. It made a sat­

terms subject to misinterpretation. As

of an overnight train ride from St.

isfying conclusion to our meetings!

a probably consequence of this resolu­

Petersburg) and a bit let down over

In the following three sections I give

tion, we did a great deal of the talking

what appeared to be another indiffer­

more detail about conditions in Russia,

at our next meeting and elicited plenty

ent reception right after our wonder­

separating these into economic effects

of sympathy, but didn't learn much!

ful high-energy meeting with AWSE.

on

One recurring source of miscom­

For example, during the round of in­

Russian educational system, and gen­

munication was due to the different

troductions one young woman's re­

der differences. For more about my

university degree structures within the

search was described to us as being so

personal reactions, see the concluding section.

the

scientific

community,

the

U.S. and Russia (explained below).

striking and original that there had

Both sides in the meetings were prone

been some talk of awarding her a doc­

to assume that the degree systems

torate. This paragon said nothing in re­

Poverty in the Russian Scientific

were comparable. Thus, the delegates

sponse to this description, but just

and Educational Community

felt belittled when our doctorates were

smiled dreadfully in a way that I could

Our

apparently being interpreted as mere

only interpret as cynical. Her silent re­

Petersburg and Moscow. However, we were provided with some more general

delegation

visited

only

St.

Master's degrees, and were shocked at

action struck me; clearly, she felt our

how few female Russian doctorates we

officially

was

data by AWSE, recording the plum­

met. On the reverse side, at least once

meaningless, not even meriting the ef­

meting economic status of scientists

a Russian was astonished that one of

fort of honest participation. Our dis-

and educators:

24

THE MATHEMATICAL INTELLIGENCER

sanctioned

meeting

Salary of general science worker (as a Year

1 940

1 960

Science salary avg.

1 42.3

1 37.3

*(60

% of the national average salary)

Summary of the Russian

1 980

1 990

1 991

1 993

1 994

1 06.3

77

65

66

76*

in Moscow)

Academy salary (as a

Expense

Although education is mandatory in Russia, the agencies that formerly en­

% of the national average salary) 1 970

Year

Educational System9

1 979

1 989

1 994

1 996

Avg. salary (All univ. prof.)

1 68

1 50

1 08

55

45

Avg. salary (Full prof./Dept. Head)

41 0

290

240

95

89

forced participation are

no

longer

funded, and many children are now working or begging instead. Kinder­ gartens were free under the Soviets,

In a survey of (mostly female) math­

falling

(245 students at the main cam­

ematicians attending a conference, the

pus, versus a previous enrollment of

highest monthly salary reported was

400) and their best and brightest stu­

2,500 rubles (about $440). For 27% of

dents go abroad afterwards. (It should

respondents, monthly earnings were

be emphasized that the programs are

600 rubles (about $104), and for 18% less than 300 rubles; some re­ ported as little as 150 rubles. Our trans­

academically rigorous-insofar as we riculum with the faculty-and the qual­

lator in St. Petersburg told us that her

ity of the education appears to be ex­

mother, a professor, earned a salary of

cellent. Almost the first thing we were

$30 a month.

told was that the school's computer

less than

could determine by discussion of cur­

A popular Russian witticism is that

team had recently won second place in

since educators continue to teach no

an international competition in the

matter how much their salaries are

United States; considering students' lim­

slashed, the government is now trying

ited access to resources, this seems in­

to devise a way to charge them for the

credible.)

but now free kindergartens are both rare and of very dubious quality. Both private and public primary and sec­ ondary schools exist; lunches for pub­ lic school students are funded by the city government, but transportation is not. Private schools may cost as much as

$600 a month (three times the aver­

age monthly Russian salary). Never­ theless cost does not guarantee quality education, or even that a school is legally entitled to present, graduates with

the

official document of educa­

tion. University tuition is free to stu­ dents passing the admission exams, but students must pay their own living expenses.

privilege of working. At CEMI, this is

Although the conditions appeared

very nearly true: a junior researcher

worst at St. Petersburg University,

Infancy through primary grades

340 rubles a month, but has to pay 180 rubles each month for a "traf­

everyone encountered by the delega­

Ideally, children aged one month to

tion spoke of funding problems, par­

three years attend nursery school, fol­

fic card" in order to get to work

ticularly in terms ofjournals and equip­

lowed by kindergarten for ages three

earns

ment, and of an inability to provide

to five (kindergarten classes are di­

signs of poverty witnessed by the dele­

fellowships,

vided

gation were at St. Petersburg State

travel grants for young mathemati­

schools (grades one through three)

University. The faculty of necessity has

cians. Travel outside the former Soviet

have entrance exams: children should

been forced to use old textbooks, as

states, once prohibited for political

be able to read, count to a hundred,

very few new ones can be obtained, and

reasons, is now generally impossible

memorize

even used ones cannot be supplied

because of economic hardship. Dr.

demonstrate

.. The most visible and distressing

to

summer

programs,

or

into

three

and

grades).

recite

set

Primary

poetry,

recognition

and (e.g.,

every student. Library funding (particu­

Yuri Matiyasevich (of Hilbert's Tenth

is all but non­

Problem acclaim) asked us to inform

ory, children have the right to attend

existent; the faculty asked us to send

others of the existence of the Euler

their local school even if they fail the

them even year-old copies of journals.

Institute in St. Petersburg, an institu­

exams, but in practice this is not the

The library holdings are not computer­

tion that serves as a contact point for

case because students who perform

ized-students do not have access

Russian and foreign scientists. This

well are accepted in preference to

larly

for journals)

to

"Which one doesn't belong?"). In the­

stacks, but must search the card catalog

converted mansion has facilities for

those who have not. Some of the pri­

and make written requests for specific

small conferences (including beautiful

mary schools are "specialized." For ex­

volumes from the over-worked staff.

rooms and dining facilities, a library,

ample, a school may offer heavy con­

The library has no copiers. Computer

computers, and electronic equipment)

centration in some subject such as

equipment at the university is outdated

and small rooms for "tete-a-tete" pair­

math

is no student access to printers, and access is difficult

ings of Russian and foreign mathe­

schools assign homework, perhaps an

maticians

hour's worth each day. Schools that

to obtain even for faculty. The tmiver­

This Institute, too, is underfunded, and

are not specialized are not appropriate

sity building interiors are chipped, bro­

the conference facilities all too often

preparation for

ken, and peeling, and only one in three

go unused.

eventually enter institutions of higher

toilets actually functions. Enrollment is

check http://www.pdmi.ras.ru/EIMI.)

and in short supply; there

doing

research

together.

(For more information,

or

foreign

languages;

students

such

who

will

learning. (Thus, parents decide when

91nformation in this section was 'obtained primarily in briefings by the group's translators; we also obtained details directly from Russian educators in response to our questions.

VOLUME 22, NUMBER 4, 2000

25

their children are six or seven whether they will prepare for a college educa­ tion, and even broadly which subjects will be potential majors.) Students are responsible for keeping the school and grounds clean. Classes include a "home craft" lesson, which consists of cooking and sewing for the girls, car­ pentry for the boys. At the end of each spring semester are exams. The scores are 1 to 5, with 5 being highest; a stu­ dent who receives all 2's must repeat the grade, and a student who fails is expelled. (Such a student may of course 'seek enrollment elsewhere; see "Addressing specialized needs.") Students finishing primary school take graduation exams in math, Russian, lit­ erature (which is an oral exam), and physical education. Secondary school

There is no "grade four" (by that name) due to an education reform. Students entering secondary school go directly into what is called "grade five" and continue to "grade eleven." Again, sec­ ondary schools may be ordinary or specialized. Homework is heavy and is referred to as "torture"; it consists (at a specialized school) of some three to four hours of work each evening, pri­ marily math and literature. Children have seven to eight lessons daily, plus extracurricular subjects such as History of Art. Specialized schools may have arrangements with nearby re­ search institutes to provide special ad­ vanced classes. There are graduation exams in the core subjects. Higher education

Seventy percent of Russian children fmishing secondary school receive fur­ ther education. This may consist of "college," which is really vocational school or educational training, or uni­ versity study (which requires passing

entrance exams). University-level in­ struction is also available from re­ search institutions such as CEMI. The university degree structures in the U.S.10 and Russia are quite dissim­ ilar, which led to confusion for the del­ egation on both sides. The Russian "candidate of science" is considered the terminal degree, although it falls between the American Master's and Doctoral degrees in terms of rigor. The Russian "doctorate" is actually an ad­ ditional step that may be conferred on established professionals after years of work experience in their field, and is based on an overview of total accom­ plishments. This degree is necessary for those wishing to rise to certain high ranks in their professions, but is not a prerequisite for employment and there­ fore is often not sought. The very high­ est scientific honor is admittance to the Academy of Science; a member is dis­ tinguished by the title, "Academician." Addressing specialized needs

There exist (often very expensive) pri­ vate schools for children with learning or behavioral problems, or offering high security for wealthy parents who believe their children at risk for kid­ napping. One of the secondary schools the delegation visited was one of four special government-subsidized board­ ing schools for the gifted (in this in­ stance, for children gifted in the nat­ ural and mathematical sciences, literature, or history). A fairly recent innovation in Russian education is night classes: an adult who did not fin­ ish secondary school may earn gradu­ ation certification through evening in­ struction, and universities now offer night classes as well. (Written sources imply that technical night classes were common under Lenin and Stalin.) CEMI provides classroom instruction in English for foreign scientists.

Gender Disparities in Russian

Education and Promotion 1 1 Birth rate

One of our translators, who had been a teacher, said that the birth ratio of girls to boys in Russia is high and used to be higher-in her mother's day, the ratio was 3:2. A teacher we met later claimed that boys are disproportion­ ately represented in her math class be­ cause more boys than girls were born that year. 12 Enrollment and distribution by discipline

We were told of girls who had been de­ nied entry to a primary school because the school already had a dispropor­ tionate number of female students. However, the secondary school for the gifted visited by the delegation has a 700;6 male enrollment. 13 Within a given classroom, representation by gender may be quite disproportionate. In one classroom observed by the delegation, six of 23 students present were girls, and in another (consisting of lOth grade biology students), eleven of the thirteen students present were fe­ male. 14 The director of the school for the gifted told us that the percentage of female students varies by major, be­ ing high in biology, about even in chemistry, and between 20 and 30 per­ cent in math. According to "unofficial data," the percentage of girls leaving secondary school interested in mathe­ matics and wishing to continue their studies is high, but over 700;6 of women in mathematical disciplines wind up in economics or other applied specializa­ tions; the speaker felt that these young women were "pushed" away from pure mathematics [V]. At the St. Petersburg State University campus for math, me­ chanics, astronomy, and computer sci­ ence, we were told that "about a third" of the students in each discipline were

101n the U.S., there are some two-year degree programs suitable for low-end job training, usually offered by "community colleges" attracting only local students. A pro­ fessional, however, needs a minimum of a four-year program or Bachelor's degree, and often requires a Master's degree (requiring typically two more years). The lat­ ter degree may involve require a thesis, but not usually original research. For professors and researchers, the Master's is optional, but they must earn a Doctorate. This "Ph.D." averages ten years (8 for math) beyond the Bachelor's of study and research work, typically seven of them at the university. This period culminates in a sub­ stantive written dissertation of original research, which must be deemed a significant contribution to the field by a committee of established experts. 1 1This section contains sometimes-contradictory reports received from different sources. 121n the United States, more male babies are conceived and bom. However, the incidence of miscarriage, infant mortality, and indeed death at all ages is higher for males. By age 1 7 , female representation in U.S. schools is typically 50.5%. 13Admission is by competitive examination; preparatory classes are available. 1 4Biology and medicine are primarily practiced by women in Russia, and doctors are not well paid.

26

THE MATHEMATICAL INTELLIGENCER

female. At Moscow University,

1 1.6% were

the

a great deal of the difference is attrib­

ment or laboratory heads,

of

utable to women delaying seeking this

women, and Academic Boards of in­

24 men; in econom­

additional degree, or that they are not

stitutes

ics, there were ten women and 20 men.

as active in research, or both. This

women members.

mathematics

students

four women and

consisted

had

3% and 9%

between

We were also told that the proportion

could be due to lower expectations,

of women students there used to be

fewer opportunities, or less scope be­

ample

higher.

cause of family responsibilities (see

Employees are paid based on their

later section,

and atti­

rank. Even the lowest rank, "junior re­

Instructors

tudes"). In any case, however, this de­

searcher," requires a high level of edu­

Over

809-6 of Russian schoolteachers

"Awareness

CEMI provided an illustrative ex­ of

promotion

discrepancy.

lay must profoundly affect a woman's

cation. With sufficient achievement,

are women, but in the secondary and

lifetime earning capability.

one may after perhaps ten years of

fall to about

or specialized schools women teachers 700A> [LF]. 15 (We were also

Research and professional

Successfully defending a thesis

told that at specialized schools, male

advancement

math or economics) entitles one to the

work achieve the rank of "researcher." (in

instructors may be the majority.) At

Professional discrimination based on

rank of "senior researcher." Defending

the Moscow secondary school visited

gender is illegal in Russia, and several

a doctoral thesis confers the rank of

800A> of the instruc­

of those interviewed felt that state­

"leader," roughly analogous to a full

tors were female, and the principal and

ment of this fact thoroughly exhausted

professor. The highest rank is "chief re­

by the delegation,

assistant principal were women. At the

the topic. However, according to a sur­

searcher."

school for the gifted, the director was

vey by A WSE, discrepancies in highest

rank forms a glass ceiling beyond

male, and he informed us that the

degree obtained and in job title ac­

school was very atypical because "at

count for an average salary for Russian

which no woman has passed. Of the 500 �rsonnel at CEMI, 70 hold doc­

700;-6

torates, but none of these is female.

The

"senior

researcher"

most schools, most of the math teach­

women in mathematics that is just

ers are female graduates of teaching

of the average for men. The AWSE

(See

universities; here, teachers are actively

president, Dr. Galina Yu. Riznichenko,

"Completion of doctorate.") Thus, not

also

preceding , subsection,

involved in research." There are more

also indicated that it is difficult for

one of the CEMI administrators is fe­

men than women teaching at voca­

women to publish in respected jour­

male.

tional schools, and this disparity is

nals and to present papers at presti­

much greater at universities. At the St.

gious conferences. Academician Olga

Awareness and attitudes

Petersburg State University we were

Ladyzhenskaya of the Steklov Institute

According to AWSE,

there are no

told that that about a fourth of the in­

confirmed that there are very few

Russian

structors in math-related disciplines

Russian women in high positions in

women in math and science.

statistics on 16 The li­

were women, but only four of these

government and science, in particular

brary of Moscow State University con­

faculty

in the Academy, and that the last honor

tains only three relevant articles, two

women

possessed

Russian­

government

level doctorates (no data was given for

appears to have been withheld inap­

of which are written in English. AWSE

comparative purposes, making analy­

propriately

has consequently carried out its own

sis difficult).

mathematicians

from

deserving

women Dr.

survey; most of the Russian statistics

Ladyzhenskaya also told us that signif­

in this paper were provided by AWSE.

in

the

past.

Completion of doctorate

icant research results of women had

On several occasions during the del­

The members of the delegation were

been altered slightly and published by

egation's visit, a Russian woman would

shocked when we were told that the

men as their own work.

make ·some sweeping statement link­

average time for a Russian candidate

Information on the composition of

to earn a doctorate was ten to fifteen

the Academy of Science was collected

years for a man, seventeen to thirty­

by Vitalina Koval in

five years for a woman [V] . Our first re­

Academy employees without an ad­

1989 [LF]. Among 41.7% were

ing gender (or race) to aptitude, and none of the Russians hearing it seemed to

fmd

this

at

all

unreasonable.

"Political correctness" in this regard seems unknown in Russia. One sec­

action was that here was blatant evi­

vanced educational degree,

dence of discrimination. (Because of

women; of those holding as highest de­

ondary school teacher in

our American model, we envisioned

gree candidate of the sciences (roughly

claimed both that girls are not as good

this period as time spent at the uni­

equivalent to

Ph.D.),

as boys in math, and that women make

versity actively seeking a terminal de­

better teachers than men (even in

gree.) However, in light of the differ­

34.4% were women; of those holding Russian doctorates, 14.9% were

ent meaning and significance -of the

women; of those holding professor­

CEMI claimed that women are better

doctorate in Russia, it seems likely that

ships,

7. 7% were women. Of depart-

at pure than applied math, reversing

an

American

Moscow

math). Interestingly, one woman at

15For comparison, in 1 993 about 74% of U.S. teachers were women, in both public and private schools. At public schools, 47% of teachers held master degree's or higher; at private schools, 34% did. About 35% of principals were women [NCES]. In 1 995, about 40% of all U.S. college faculty (full- and part-time) were women [B]. 16The U.S. National Assessment of Education Programs dates from 1 967; its publications are free and easily accessible on the Internet. The American Association of University Professors has been collecting gender data since 1 975.

VOLUME 22, NUMBER 4, 2000

27

what is the apparently prevailing con­ ception (I noticed that another woman there made a face of disagreement). In St. Petersburg, both at the uni­

versity and the secondary school, the women said that there

is no discrimi­

nation at all, not even on the level of faculty discussions. Tl).ey did not un­ derstand the purpose of an organiza­ tion like the AWSE nor see a need for it. The only particular obstacles the university faculty perceived as women in mathematics were that most men do not wish to marry mathematicians, and that those who do marry them expect their wives to do

all of the

child-rear­

ing and cookingP However, it should be noted that the secondary school teachers allowed the (male) principal to lead the discussion and to do almost

Women of the Steklov Mathematics Institute: (1-r) Natalia Kirpichnikova, Olga Ladyzhenskaya, and a "candidate of sciences" (Ph.D. equivalent).

all of the talking, and that the univer­ sity faculty did not contradict the (fe­

have to explain and justify the gender­

hopes to publish textbooks and re­

male) Registrar when she claimed that

issue portion of our mission. AWSE's

search monographs that are not avail­

males are superior in math. (Both St.

primary purposes are dissemination of

able in print because of the general

Petersburg

information on financial support avail­

poverty of the Russian scientific com­

institutions

were

also

clearly catastrophically underfunded,

able for women in science and educa­

munity. Since their own activity is vir­

and delegation members felt that the

tion, support for women's professional

tually entirely provided by dues and by volunteer work (outside funding has

overwhelming need to focus on simple

advancement by nominations to key

survival might contribute to the lack of

administrative positions, promotion of

been found only for their conferences),

interest in gender inequity.)

scientific and educational events for

this project is still unrealized.

The Moscow secondary school was better funded and had female adminis­

between

tration. The women interviewed (prin­

within

women

Russia

in

and

science

(both

internationally).

cipal, assistant principal, and teacher)

There are two striking differences be­

seemed

and

tween the activities of the AWSE and

showed no reluctance to speak up.

those of the [American] Association for

calm

and

confident

Otherwise, they seemed similar to

Women in Mathematics (AWM). First,

their

the major activity of the AWSE is the

St.

Petersburg

counterparts,

showing no interest in gender issues.

organization of entire conferences in­

The women of the Steklov and of

tended primarily for female partici­

CEMI (both of which are research in­

pants, and the publication of the at­

stitutions of the Academy of Science)

tendant proceedings. This isapparently

showed some awareness of profes­

seen as necessary because of the rar­

sional discrimination. However, the

ity of women's papers being accepted

women at CEMI seemed as shocked as

by major conferences and journals in

the delegates by the statistics cited by

the former Soviet nations. (The AWM

one of their own workers (Natalya

by contrast encourages its members to

Vinokurova, a member of AWSE), and

participate in existing conferences and

had apparently been totally unaware of

journals. It hosts one-day workshops

the existence of AWSE.

that are held in conjunction with the

The membership of AWSE were the

annual Combined Meetings, primarily

only Russian women encountered by

as a travel-fund mechanism for young

the delegation to whom we did not

female mathematicians.) Second, AWSE

1 7An

AWSE survey showed 73% of the respondents were married, and 71 % had at least one child. In addi­

tion, 64% said that the woman did all of the housework, and 1 8% said that the housework was shared equally (no one claimed that the man did all the housework).

28

THE MATHEMATICAL INTELLIGENCER

It would

women, and improved communication

Elena Novikova and Yuri Matiyasevich, out­ side the Euler International Mathe-matics Institute. Photo by Sue Geller.

Closing Thoughts Why the differences in belief?

One question that our trip left largely unanswered was why Russians, par­ ticularly

female

mathematicians,

seemed so thoroughly convinced that women are inferior at mathematics. While the annual media fanfare for the SAT-M "gender gap" keeps that hoary old refrain alive in the American pop­ ular culture, any implication of male superiority is considered simply inad­ missible in academia and in the tech­ nological business world. More than one male colleague in mathematics has assured me that he has observed that a significant majority of the best stu­ dents he trains are female. American professors

know their female students

are as good as and often better than their mal� students; why isn't this ob­ viomsto our Russian counterparts? I am indebted to Marjorie Senechal for giving me some insight-into this puz­ zle. According to Dr. Senechal, who has

had the opportunity to study honors pro­

grams for young mathematicians in

Russia, almost the only method used in Russia for measuring mathematical ex­ cellence

is highly competitive, very

stressful exams. (Here again, we see an instance where a conclusion about in­ nate gender ability in mathematics

is

based on a self-selecting sample at the extreme high end of ability.) The Euler Institute is housed in a renovated mansion that was nationalized during the

In America we do have some such ex­

Communist revolution.

ams, but for most students we look at

be difficult to exaggerate the poverty

ment decisions was unequal, and

800;6

possible) research to get a comprehen­

of the A WSE; their first attempt to sur­

felt that chances for promotion were

sive picture of potential. Our experience

vey conference attendees was greatly

unequal. One of the questions posed

in this'country is that young women sel­

hindered by the fact they could afford

was why women are a minority in

dom take highly competitive math ex­

to make only fifty copies of the ques­

leadership positions in Russian sci­

ams

tions. (However, they received

ence

sult from not being encouraged to do so

grades, letters of reference, and (where

seven

responses;

copied the form

fifty­

some

attendees

by hand

in order to

answer the questions.)

The AWSE survey reports, "The

and

sponses,

education.

In

their

re­

27% felt that women had less

if it is not mandatory. This may re­

by their teachers, discomfort because

ambition, lower qualifications, and

the existing profile of exam takers

is

less "work capacity" than their male

overwhelmingly

of

colleagues;

their abilities, or simple disinterest in

35% felt that home and

male,

self-doubt

overwhelming majority of the women

family commitments inhibited women's

such competitions. According to letters

admit that the officially stated equal­

professional advancements; and

in the

ity

and

blamed patriarchal traditions and the

women in science does not ·exist."

creation of negative images of female

tions are sometimes belittled or ha­

Less than half believed there was dis­

leadership in the mass media. How­

rassed by male competitors.

of

opportunities

for

men

38%

A WM Newsletter, young women

who do participate in math competi­

crimination in opportunity to defend

ever, virtually all the women

(96%)

What seems to me very peculiar is the

the thesis. However,

54% felt that

said that they never consider leaving

assumption that future potential for re­

there was inequity in "opportunities

their jobs, citing love of their work

for access to information,"

and responsibility to future genera­

search mathematics should necessarily

74% felt

that participation in major manage-

tions as their primary reasons [LF].

be in any way correlated to performance on time-critical competitions!

These

VOLUME 22, NUMBER 4, 2000

29

of its recruiting based on the results of such competitions. In

1994 I attended

a symposium on Women in Mathemat­ ics hosted by the NSA, at which we were asked: Why isn't the NSA at­ tracting more women hires? We told them: Stop putting so much credence in those exams! The news from MIT

In March

1999 (a year after our trip), the

Massachusetts Institute of Tech-nology

(MIT) stunned academia by publishing

a report admitting that there had been gender discrimination towards the fac­

ulty in their School of Science. The ef­ fects, described as "marginalization" of the women faculty, were observed by each female faculty member increas­ Delegate Audrey Leef enlists the assistance of translator Mila Bolgak to describe the

ingly as she became more senior. These

Association for Women in Mathematics (AWM) to Natalya Vinokurova and Zelikina Lyudmila

senior women described themselves as

of CEMI.

being excluded from having any voice in their departments and from positions

competitions in no way simulate re­

nities. The outstanding scores go to stu­

search conditions.

dents who have received special train­

If the Russian exams

are similar to American ones, exam-tak­

ing and coaching sessions. Given the

ers have no access to references, but

Russian school systems' "specialized"

many of the exam questions are never­

nature and the prevailing stereotypes,

theless based on advanced topics not

probably few girls receive the appropri­

covered in a standard curriculum. Thus,

ate mathematical training.

such exams sharply penalize any test­

I may add that at least until recently,

taker who has not had special opportu-

the National Security Agency did much

of any real power. The MIT Dean of Science has been acting on the recommendations of the Committee, including increasing the number of female faculty, and the re­ port says that the results have already been highly beneficial. (However, they also point out that because of pipeline considerations, even at the increased rate of new hires it will be

fore

40 years be­ 400/0 of the faculty in the School of

Science are female.) "Feminization" of higher education in the United States?

One aspect of the delegation's discover­ ies that I found difficult to understand was the fact that women composed a majority of Russia scientists and educa­ tors, but were still not able to achieve equitable treatment. In the U.S., the

common wisdom is that increasing rep­

resentation will bring increased influ­ ence and therefore, equity. But is this

necessarily so? According to a recent AAUP study [B], in

1995 women com­ 40% of all U.S. faculty, up from 27% in 1975. However, salary gender dis­

prised

parities "not only remain substantial but are greater in

1998 than in 1975 for half

of the categories, including 'all-institu­ Several of the members of the [Russian] Association of Women in Science and Education (AWSE), waving good-bye to the delegation. The (foreground) woman in black is Galina

tion' average salaries for full, associate,

and assistant professors." Nor can these

Reznichenko, the AWSE president whose term was just ending, and the woman on the far

disparities be attributed solely to dis­

right is Irina Gudovich, expected to be the next to hold that office.

parity in average seniority within rank

30

THE MATHEMATICAL INTELLIGENCER

As female participation in the profes­ sion increases, women remain more likely than men to obtain appoint­ ments in lower-paying types of insti­ tutions and disciplines. Indeed, even controUing for category of institution, gender disparities continue and in some cases increased, because women are more often found in tlwse specific institutions (and disciplines) thatpay lower salaries. . . . The increasing en­ try of women into the profession has sojar exceeded the improvement in the positions that women attain that the proportion of aU female faculty wlw are tenured has actuaUy declinedfrom 24 to 20 percent. . . . The report observes that during the

runs

for scientists and educators in Russia

and

have worsened further, but a second

has also been active in the Association

tenure appeal hearings. She

delegation from the U.S. would be un­

for Women in Mathematics (AWM); at­

likely to fmd a particularly warm wel­

tendees at the Combined Meetings may

come at this time.

be familiar with Dr. Geller from the

Composition of the Delegation

small

and Some of Its Contacts

women in mathematics in our culture.

"Micro-inequities"

skits,

illustrating

(and large) il\iustices against

Dr. Karin Johnsgard, Richard Stock­

The delegation

Leader: Dr. Pamela Ferguson, Past President of Grinnell College (lA). Dr.

Ferguson had traveled in Russia before,

ton College of New Jersey (NJ). I have been a registered Girl Scout for over

25 years, and can truly describe my in­

and graciously consented to substitute

terest in gender issues as life-long. In

for our originally intended leader, Dr.

high school, I was a People-to-People

Ms. Liv Berge, Upper Secondary

pean nations. I was one of the women

Alice Schafer, who was unable to attend.

student ambassador to several Euro­

School (Husnes, Norway). Ms. Berge,

graduate students who benefited from

the author of several articles on gen­

the AWM's one-day workshops. On this

der and mathematics, was working on

delegation, I was the youngest and only

huge upswing in women in academia,

a

untenured participant.

male participation in the profession has

Gender, and Politics." She shared with

been almost constant, and the (raw)

us this data (from the Nordic Institute

project

entitled

"Mathematics,

Dr.

El�anor Jones, Norfolk State

Univ.-(VA). Norfolk State 'has histori­ had

number of men in tenure-track posi­

for

tions has actually dropped 28%. "Simply

Research): Women have 400Al represen­

stated, fewer men are finding their pro­

tation in Norwegian government, but

ence on the delegation helped keep us

fessional futures in academe, whereas

comprise only

7% of mathematics pro­

sensitive to the virtual absence of peo-

Women's

Studies

and

Gender

female participation continues to in­

fessors. (In Sweden, the percentage of

crease despite the declining terms and

female mathematicians is even lower.)

conditions of faculty employment. . . . universities

can

successfully

offer

predominantly

African­

Dr. Lucy Dechene, Fitchburg State College

(MA). Fitchburg State, a four­

women terms of agreement that would

year liberal arts institution, has ex­

similarly qualified men."

puter science programs in China and

Personal responses

there is also a special program for un­

rtt>t be acceptable to similar numbers of

cally

American enrollment. Dr. Jones's pres­

change programs with graduate com­ Russia and is developing one in India;

Regarding the effects of the journey on

dergraduates in Bermuda. In addition

myself: I learned a great deal (basically,

to her teaching duties, Dr. Dechene su­

I spent an entire month after our return

pervises the mathematical skills cen­

simply trying to record all that I had

ter, as well as independent study pro­

learned and seen and felt). I read news

jects and undergraduate research. She

reports in a new way, and have followed

had been a past participant in People­

unfolding events in Russia with entirely

to-People programs to other nations.

new interest. I sought data from other

Ms. Maureen Gavin, Bodine High

nations to put what I had learned in a

School for Int'l Affairs (PA). Bodine, a

larger perspective. I gave presentations

magnet school, was founded in cooper­

to students at my own college, com­

ation with the World Affairs Council of

paring the situations of women in math­

Philadelphia to have as its primary fo­

ematics in America and in Russia. And

cus global studies and geography. Ms.

I wrote (and revised, and revised) this

Gavin has traveled extensively (for ex­

article, hoping to spread the effects of

ample to Tibet) and had accompanied

our journey, believing our experiences

her students on a trip to Russia just a

were important and should be shared

month before the delegation's journey.

with a wider audience. The economic and political events

Dr. Sue Geller, Texas A&M Univ.

(TX). Dr. Geller, in addition to her

The author at Moscow school #103, in the

of the past two years have led to sub­

teaching and research duties, directs

stantially more acrimonious relations

the master of science program, mentors

International Youth games hosted by the city

between Russia and ·the nations of

students and junior faculty, is involved

this year. (I am about to be handed an

It seems likely that conditions

in conflict resolution and mediation,

Olympic torch.) Photo by Sue Geller.

NATO.

"sports museum" celebrating the Olympic

VOLUME 22, NUMBER 4, 2000

31

ple of color we saw in Russia, and to the

appreciated my editor, Dr.

occasionally explicit racism. Her main

Senechal, for her guidance, insight, and

focus on the mission was pedagogical.

patience. Dr. Mary Beth Ruskai sent me

Dr. Audrey Leef (emerita), Mont-clair State Univ. (NJ). Montclair is particularly

Marjorie

some very relevant material. I also thank

my husband (Dr. Ami Silberman), both

noted for training secondary school

for accompanying me in Russia and for

teachers. Although retired, Dr. Leef still

his feedback in editing this paper. Any

teaches as an acljunct and has supervised

A U T H OR

errors herein are solely the author's.

student teachers in their fieldwork Her world travels have included Antarctica!

REFERENCES

Dr. Diana Vincent, Medical Univ. of South Carolina (SC). MUSC is a teach­

United States and multi-national

ing hospital that trains health care pro­

[A] Joe Alper, "Science education: The pipeline

fessionals, conducts basic and clinical

is leaking all the way along, " Science, Vol.

research, and provides patient care. Dr.

260, 1 6 April 1 993, pp. 409-41 1 .

Vincent described her work (in part) as

[AAU P] "Doing better: The annual report on the

a bridge between the mathematical and

economic status of the profession, 1 997-98,"

physical scientists and the medical staff.

Academe: Bulletin of the American Associa­ tio n of University Professors, Vol. 84, No. 2,

Translators

March-April 1 998, pp. 1 3-1 06.

Ms. Irina Alexandrova, St. Petersburg

[BS] Yupin Bae and Thomas M. Smith, "The

KARIN JOHNSGARD

NAMS

DMsion

Richard Stockton Col lege Pomona, NJ 08240-0195

e-mail: [email protected] Karin Johnsgard, in her teens, collab­ orated with her ornithologist father Paul Johnsgard in writing and illustrating a

Center of International Programs.

Condition of Education, 1 997. No 1 1 : Women

boo k about dragons and unicorns.

Ms. Mila Bolgak, Prospects Business

in mathematics and science," U .S. Dept. of

publications since then have mostly

Education, National Center for Education

concerned knot groups, combinatorial

Cooperation Center, Moscow. Acknowledgments

to

Her

Statistics, 1 997. (Available at website http://

group theory, and geodesics in cell

nces.ed.gov/edstats)

complexes. She has been a Sloan Doctoral Dissertation Fellow and an

the members

[B] Ernst Benjamin, "Disparities in the salaries

of the delegation and all who assisted

and appointments of academic women and

NSF Postdoctoral Fellow. Photo by

our mission. In particular, I wish to

men: An update of a 1 988 report of com­

Ralph Beam

The author is grateful

thank Dr. Sue Geller for her encourage­ ment and help with details. I also greatly

mittee W on the status of women in the aca­ demic profession," AAUP, http://www.aaup. org/Wrepup.html (1 999).

CARE plants the most wonderful seeds on earth. Seeds of self-sufficiency mar help rarving people become healthy, productive people. And we do it village by village by village. Please help us rurn cries for help imo rhe laughter of hope.

[DMR] Paul W. Davis, James W. Maxwell, and Kindra M. Remick, " 1 998 Annual Survey of the

there innate cognitive gender differences? Some comments on the evidence in re­ sponse to a letter from M. Levin," Am. J.

Mathematical Sciences (first report)," Notices

Phys. , Vol. 59, No. 1 , Jan. 1 991 , pp. 1 1 -1 4 .

of the AMS, Vol. 46, No. 2, Feb. 1 999, pp.

[S] Paul Selvin, "Does the Harrison case reveal

224-235. (Available at website http://www.

sexism in math?" Science, Vol. 252, 28 June

ams.om/employmenVsurvey.html) [H] G. Hanna, "Mathematics achievement of

1 991 , pp. 1 78 1 -83. [SAT] "National report on college-bound seniors

girls and boys in grade eight: Results from

1 999," College Entrance Examination Board.

20 countries," Educ. Stud. Math . , Vol. 20,

(Available at website http://www.clep.com/saV

1 989, pp. 225-232. [HFL] Janet Shibley Hyde, Elizabeth Fennema,

sbsenior/yr1 999/NAT/natsdm99.html) [SATR] "Common sense about SAT score dif­

and Susan J. Lamon, "Gender differences in

ferences and test validity (RN-01 ), " Research

mathematics performance: A meta-analy­

Notes,

sis," Psychological Bulletin, Vol. 1 07, No. 2,

(Available at website http://www.college­

1 990, pp. 1 39-1 55.

board.org/research/html/m index.html)

The College Board, June 1 997.

[MIT] "A study on the status of women faculty in science at MIT," The MIT Faculty Newsletter,

Russia

Vol. 1 1 , No. 4 (Special Edition), March 1 999.

[AWSE] Information copied from slides prepared

(Available at the website http://web.mit.edu.

by the [Russian] Association of Women in

/fnl/women/women.html)

Science and Education; sources unspecified.

[NCES] "Digest of Education Statistics, 1 998 edition (NCES 1 99-032)," U.S. Dept. of Education, National Center for Education

M Data as cited by Natalia A Vinokurova of CEMI and AWSE; possible source an AWSE

survey she conducted with Nana Yanson. (A

Statistics, 1 999. (Available at website http://

preliminary report on this survey was con­

nces.ed.gov/edstats)

tained in Lady Fortune.)

[R] Mary Beth Ruskai, "Guest comment: Are

[LF] Lady Fortune, publication of AWSE.

OLEKSIY ANDRIYCHENKO AND MARC CHAMBERLAND

Ite rated String s an d Ce u l ar Autom ata

� •

n 1996, Sir Bryan Thwaites [4] posed two open problems with prize money offered for solutions. The first problem (with a £1000 reward) is the well-known 3x + 1 prob­ lem which has received attention from many quarters. This easily-stated problem has eluded mathematicians for about 50 years; for more information, see Lagarias [ 1]

and Wirsching [5] . Thwaites's other problem (with a

£100

reward) has no clear origin. He states it as follows:

Take any set ofN rational numbers. Form another set by taking the positive differences of successive mem­ bers of the first set, the last such difference being formed from the last and first members of the origi­ nal set. Iterate. Then in due course the set so formed will consist entirely ofzeros if and only ifN is a power of two. Thwaites concludes his note by saying that "Although neither I, nor others who have been equally intrigued, have yet proved [the second problem], one's instinct is that here is a provable cof\iecture; and so the prize for the first suc­ cessful proof, or disproof, is a mere hundred pounds. " The present paper offers an elementary proof of this second problem. In the process, binomial coefficients and cellular automata are encountered.

an) represent a string of length n, where ai is rational for all i. Upon iteration, its succesWe will let

(a1 , a2, :

•

•

,

sor will be

Cla1 - a2l , la2 - asl , . . . , lan- 1 - ani , lan - ai l) .

A string containing only zeros will be called the zero-string, while a string containing only ones will be called the

string.

one­

The way the problem was posed by Thwaites is

somewhat imprecise: the one-string iterates to the zero­ string regardless of the string's length. We restate the (proper) theorem to be proved formally:

Theorem 1.1 AU strings of length n will eventually iter­ ate to the zero-string if and only if n = 2k for some k E Z+. Half of the proof comes easily:

If the string's length n is not a power of two, then there exist strings which will never iterate to the zero-string.

Theorem 1.2

Proof: The problem considers 0-1 strings, strings whose terms take only the values 0 or 1. Since the set of 0-1 strings is for­ ward-invariant under our iterative process, this will suffice. First we prove the case when

n is

odd. Working back­

wards, note that the only predecessor of the zero-string is

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

33

the one-string. The only predecessor of the one-string has terms which alternate between ble since

n

0 and 1, which is impossi­ 0-1 strings of odd

is odd. Therefore the only

length iterating to the zero-string are the zero-string itself and the one-string. This completes the proof when n is odd.

n is an even number which is not a power of two, it p. Create a string of length n by concatenating nip substrings of length p, each If

must have an odd prime factor, say

of which is the string starting with a one then having all zero terms. For example, if n

=

12 , take p

= 3 and create

the string

100

100

100

100

The periodic nature of the iterative process implies that each substring iterates as if it were the whole string:

100

100

100

100 � 101 100 � 101

101

101

101

Because each of the (odd-length) substrings will never it­ erate to the zero string, neither will the whole string; which

D

completes the proof.

0-1 strings. 1.1, we argue that con­

The previous proof needed only the set of To prove the other half of Theorem

sidering only 0-1 strings is sufficient. First note that by scal­ ing a string by a constant, the dynamics do not change, so multiply each element in the string by the appropriate in­ teger (the least common multiple of the denominators) to yield an integer string. Also, one interaction on a string yields a non-negative string, so we can assume from here on that the string consists only of non-negative integers. Next, we show that it is sufficient to consider strings whose values are only

0 and possibly one other (positive) if the string contains at

value. To do this, we show that

least two distinct positive values, the maximum value (de­

noted henceforth by

m) will eventually decrease. If there

is no zero value, the maximum value will automatically de­ crease after one iteration, so we may assume there is at least one zero value. Consider any substring whose terms are only zero or

m (with at least one m), and assume this

Figure 1 . Iterating the string ( 1 1 0 0 1 1 0 0 ) .

substring is maximal, so that it takes the form

At this point, it is worth pointing out that iterating a

0-1 string mirrors the dynamics used in generating the

where ak equals zero or

m (with at least one m) for all k, < b, c < m. After one iteration, the substring has one few term. Note that such substrings (with at least one m)

Sierpinski Gasket with cellular automata. Consider the

and 0

"rules" in Figure

white) of the upper squares determines the parity of the

cannot be created, so after a finite number of iterations,

lower square. Starting with an infinite row with only one

these substrings all vanish. This process forces the maxi­

black square, one generates the Sierpinski Gasket in a

2. For each rule, the parity (black or

mum of the whole string to decrease, leaving us (dynami­

stretched form. Figure 3 shows the first few rows. The

cally) with two possibilities: either this descent continues

black cells in this figure correspond to the odd terms in

until all the terms are zero, or the string iterates until all

Pascal's triangle, where the top black cell corresponds to

0 or possibly one positive value.

the apex of the triangle. Details of the mathematics may

its terms are either

et

be found in Peitgen

strings are similar, with the important difference that the

0 or 1. 1 shows iterations of the string ( 1 1 0 0 1 1 0 0 ), where the black dots represent 1 and the white dots rep­ resent 0.

string is periodic.

are only

Figure

34

THE MATHEMATICAL INTELLIGENCER

al. [3] . The dynamics of our

0-1

Dividing each term by this positive value (which leaves the string dynamically unaltered) yields a string whose terms

1.1, one is required to 0-1 strings whose length is a power of 2 even­

To finish the proof of Theorem show that

tually iterate to the zero string. The analysis is simplified

Figure 2. Cellular automata "Rules. �> if we replace

0

(resp.

1) with 1

(resp. - 1), and instead of

using the absolute value of the difference, simply consider the product. For example, before we had the successive terms

(1

0) produce [ 1 - 0[ ( - 1)(1) - 1.

1) produce

=

= 1, whereas now we have

(- 1

dynamics are equivalent; we are simply representing the

ai,j denote the value of the f11 element of the string after i iterations. For ease of notation, it will be understood that if kn <j ;::;; (k + 1)n for + some k E z , then ai ,j ai,j-kn· Lemma 1.1 If a string has length n, then with this new system. We will let

=

.

for 1

:5

i :5 n, 1

:5

=

Lemma 1.2

(�) aO,j (f) + 1 aO,j

...

(1) +i aO,j

j :5 n - i.

ek; ) 1

One may easily verify that the

group £:2 in a different way. The rest of the proof will work

· at,J

The proof is by induction.

is odd if 0 :5 j :5 2k - 1.

The proof is by expanding the binomial coefficient. We note that this lemma has a generalization (see, for example, for any prime p, p

does not divide

ipk ) j-

1

if 0

:5

j :5

pk

-

[2]):

1.

These two lemmas lead to the last step in the proof of Theorem

1.1:

Theorem 1.3 If the

1 for aUj.

string's length is n = 2k, then an,j

=

A U T H O R S

OLEKSIY ANDRIYCHENKO

MARC CHAMBERLAND

Department of Mathematics and Computer Science

Department of Mathematics and Computer Science

Grinnell College

Grinnell College

Grinnell, lA 501 1 2-1 690

Grinnell, lA 501 1 2-1 690

USA

USA

e-mail: [email protected]

Oleksiy Andriychenko is a current student at Grinnell College, majoring in mathematics and economics. He g raduated from

Marc Chamberland obtained his degrees from the University

the Ukrainian National Mathematical-Physical Lyceum in Kiev in

of Waterloo and has been at Grinnell College since 1 997. His

1995. During high school and college years, he successfully

research interests are principally in differential equations and

Ukrainian National Olympiads, the Putnam Competition, and the

3x + 1 problem or the Jacobian Conjecture) can easily lead

participated in a number of math competitions , including the Mathematical Contest in Modeling. Of all the mathematical top­

ics he has seen so far, he considers problem solving in num­

ber theory as the most fascinating . His other interests include

i

chess , ping - pong , and consumer advert sing

.

dlynamical systems, though a beautiful, tough problem Oike the

him to other waters. Outside of mathematics, he spends time

with his wife and two young sons, fulfills his passion for mu­ sic (voice, piano, and guitar) , and seeks quiet places for med­ itation.

VOLUME 22, NUMBER 4, 2000

35

Ode to Andrew Wiles, KBE Tom M . Apostol

� note­

Fermat's famous scribble-as margin

Launched thousands of efforts-too many

to quote.

Anyone armed with a few facts mathematical

Can settle the problem when it's only quadratical. Pythagoras gets credit as first to produce

Figure 3. Generating the Sierpinskl Gasket.

The theorem on the square of the hypotenuse.

Proof: The first step is to show

Euler's attempts to take care of the cubics Might have had more success if devoted to Rubik's.

an-1,j = ao,1 ao,2 . . . ao,n for j = 1, . . .

, n. Using Lenunas

1 . 1 and 1 .2 successively,

we have

If an-1 ,j

That the Fermat problem was finally trounced.

=

= =

But the very same year a letter from Kununer

ao,3tlo,j+1 · · · ao,j+n-1 ao,1a0,2 ... ao,n

Revealed the attempt by

= 1 for allj, we are finished. If all the terms are

- 1 , one more iteration forces

an, j =

1 for all j.

With a handful of primes that were in the first case.

Lame at mid-century proudly announced

n 1 n 1 J. 1) ... a(�= O 1 an-1, J a(O,jo )a(O,j+ O,j+n.

Sophie Germain then entered the race

D

Lame was a bununer.

Regular primes and Kununer's ideals Brought new momentum to fast-spinning wheels. Huge prizes were offered, and many shed tears

Epilogue

When a thousand false proofs appeared in four years.

When we presented this work to Sir Bryan Thwaites, he in­

Then high-speed computers tried more and more sam-

formed us that the problem had been solved long since. However, he expressed admiration for our method, so even without the cash prize we felt he had given his blessing to our publishing it.

ples, But no one could find any counter examples. In June '93 Andrew Wiles laid claim To a proof that would bring him fortune and fame.

ACKNOWLEDGEMENT

But, alas, it was flawed-he seemed to be stuck­

The authors would like to thank Grinnell College for fi­

When new inspiration suddenly struck.

nancially supporting O.A. to work with M.C. during the The flaw was removed with a change in approach,

sununer of 1999.

And now his new proof is beyond all reproach.

The Queen of England has dubbed him a Knight

REFERENCES

[ 1 ) J.C.

Lagarias. The 3x + 1

Problem and its Generalizations.

American Mathematical Monthly 92:3-23, 1 985. [2) I. Niven, H.S. Zuckerman and H.L. Montgomery. An Introduction to the Theory of Numbers. Wiley, 1 991 .

Springer-Verlag, 1 992. [4) B. Thwaites. Two Conjectures or how to win £1 1 00. Mathematical Gazette 80:35-36, 1 996. +

1

Function. Lecture Notes in Mathematics, 1 681 , Springer-Verlag, 1 998.

36

THE MATHEMATICAL INTELUGENCER

1 -70 Caltech

Pasadena, CA 9 1 1 25

[3) H.-0. Peitgen, H. Jurgens and D. Saupe. Chaos and Fractals.

[5) G.J. Wirsching. The Dynamical System Generated by the 3n

For being the first to show Fermat was right.

BY JOHN BRUNING, ANDY CANTRELL, ROBERT LONGHURST, DAN SCHWALBE, AND STAN WAGON

R h apsody i n Wh ite : A Vi ctory fo r M ath e m ati cs

� •

n 1 999, the Breckenridge International Snow Sculpture Championships saw its first mathematical surface: the Costa surface, whose production in snow was reported on in [2]. That effort might have set the stage, for this year another minimal surface took several awards at the same event.

Robert Longhurst has used the ideas of negative curva­

of sculpting negative curvature from Helaman Ferguson at

ture in much of his sculpting work in wood, and his piece

the 1999 event; Andy Cantrell, a sophomore at Macalester

showing an Enneper surface (Figure 1) seemed ideal for

College; and John Bruning of the Tropel Corporation, the

realization in the hard snow that Breckenridge prepares.

nonsculpting photographer for the team. It was through

The team, again sponsored by Wolfram Research, Inc.

Bruning that the rest of the team was introduced to

(makers of Mathematica), consisted of Longhurst, a wood

Longhurst's work

and stone sculptor from Chestertown, New York; Dan

We sculpted an Enneper surface of degree 2 (see [ 1 ]), a

Schwalbe and Stan Wagon, who had learned the rudiments

minimal surface that was discovered by A Enneper in 1864.

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

37

Longhurst's wooden model of an Enneper surface, carved from bubinga wood.

Inspired by the swooping curves, we named it "Rhapsody

axis and the viewpoint is on the positive z-axis. Figures 2

in White." Unlike the Costa surface, Enneper's surface does

and 3 show two views of our sculpture.

not embed in 3-space (it has self-intersections) and is topo­

There were 17 teams at the January, 2000, event, from

logically dull (it is homeomorphic to the plane). But the

England, The Netherlands, Germany, Switzerland, Finland,

process of truncating the infinite surface just before the

Russian, Mexico, Canada, and the U.S.; for images of al­

self-intersections yields a surface that, at least from a sculp­

most all of the pieces see [3]. The audience's reaction told

tural perspective, has a very beautiful shape. One must drill

us that our work had succeeded and had the desired im­

and then expand several holes, which gives a topological

pact. The curves were swooping and graceful, the over­

flavor to the work. The openness of the result yields quite

hang exciting, and the surface smooth. But how would the

a pleasant view. And the negative curvature that occurs at

art judges react to a purely mathematical shape? Would

each point gives the piece structural strength that allows it

its entrancing form win them over, or would they find it

to be built out of snow.

unimaginative? When third place was announced to the

The parametric representation .f(r, O) r sin (}

=

(

r cos (} -

1

�

38

2

medals to geometric shapes. But then the silver medal was

)

1.4 and (} in [0, 2 7T] leads

1, though in that figure the x-axis is the vertical

THE MATHEMATICAL INTELLIGENCER

Swiss team's punctured sphere, we became concerned, for we thought that the judges would not award two

r5 cos(58),

+ 5 r5 sin(58), 3 r3 cos(38)

Plottingjwith r varying from 0 to to Figure

[1) is simple:

awarded to us. First place went to a soaring Russian struc­ ture illustrating human striving. The judging seemed fair; but the next night, we became convinced that our work had thoroughly won over all viewers, when we received both the People's Choice award (voting by the approxi-

mately 1 0000 viewers who see the sculptures on the final weekend) and the Artists' Choice award (voting by all the sculptors). Snow is a fantastic medium for sculpting mathematical shapes. Readers interested in information on entering the

2001 event can contact Wagon for information. All that is

/

required is stamina, a good set of tools, an appealing de­ sign, and an understanding of snow. Here are some com­ ments from Wagon's acceptance speech. "Julia Child has said," 'II faut mettre les mains dans Ia pate': To be a baker, one must put one's hands in the dough. Four members of our team are mathematicians and we spend a lot of time looking at images on a computer screen. But, both for us and for the viewers of our work, true un­ derstanding can be obtained only by interacting with the piece in a truly three-dimensional way. This is what snow allows us to do. In a very short period of time and with a

minimum of tools we can sculpt a complicated shape and

so learn much more about it. It's a glorious opportunity and tremendous fun. " REFERENCES

[1] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica , 2nd ed., CRC Press, Boca Raton, Fla. , 1 998.

[2] C. and H. Ferguson, T. Nemeth, D. Schwalbe, and S. Wagon, Invisible Handshake, The Mathematical lntelligencer 21 :4 (Fall 1 999), Time to start: Plastic sheeting was used to mark out the initial pro­ jection (photo by J. Bruning).

30-35. [3] D. Schwalbe's web page: www.math.macalester.edu/snow2000.

Rhapsody in White, side view, with crowd (photo by J. Bruning).

VOLUME 22, NUMBER 4, 2000

39

A U T HORS

L to R: C antre ll Longhurst, Schwalbe, Wagon, Brun i ng ,

ANDY CANTRELL

ROBERT LONGHURST

DAN SCHWALBE

Department of Mathematics

Macalester College

407 Potter Brook Road

St. Paul, MN 551 05

Chestertown, NY 1 2871

Macalester College

USA

USA

St. Paul, MN 551 05

Robert Longhurst got his degree in archi-

[email protected]

[email protected]

Andy Cantrell hails from Fort Collins, Colora-

USA

lecture, Kent State University, 1 975; since

do. His interest in both mathematics and art

1 976 he has had his own sculpture studio.

blossomed at Poudre High School and gets

His distinctive and personal visual vocabu­

Flight Manual, and (with Stan Wagon) of

plenty of scope as he continues study of

lary has captivated a large audience, and his

VisuaiDSolve. He oversees the computer

mathematics at Macalester College, and pur­

pieces may be found in hundreds of private,

labs at Macalester College. His wife Kathy

sues ceramic art on the side.

corporate, and museum collections interna­

just published her first book, Information

tionally.

Dan Schwalbe is co-author of the Maple

Technology Project Management. They have three children, one of whom, the 1 3-year-old son, came along on this snow-sculpting trip to do some skiing with his father.

STAN WAGON

JOHN BRUNING

Department of Mathematics

Trope! Corporation

Macalester College

60 O'Connor Road

St. Paul, MN 55105

Fairport, NY 1 4450

USA

USA

[email protected]

[email protected]

Stan Wagon makes a mission of disseminat­

John Bruning is President and CEO of Trope!

ing mathematical ideas to the public, and

Corporation,

hopes snow sculpture serves this. He is as­

makes specialized optics used in manufac­

sistant editor of Mathematica in Education and

turing computer chips. He has degrees in

Research, for which he writes a regular col­

high-tech

company

that

electrical engineering (BS Penn State, PhD

umn. He got some notoriety recently by con­

University of Illinois) and is a member of the

structing a square-wheeled bicycle that rolls

National Academy of Engineering . He ex­

smoothly along a specially designed track. His

plores the common area of mathematics and

main non-mathematical interest is skiing; he

art, through Mathematica and woodworking.

skied to near the top of Canada's highest peak, Mt. Logan, in May 2000 .

40

a

THE MATHEMATICAL INTELLIGENCER

M ath e m a t i c a l l y B e n t

C o l i n Adams, Ed itor

The S.S. Riemann The proof is i n the pudding.

Opening a copy of The Mathematical

Intelligencer you may ask yourself uneasily, "JfJI,at is this anyway--a mathematical journal, or what?" Or

you may ask, "JfJI,ere am /?" Or even

"JfJI,o am /?" This sense of disorienta­

tion is at its most acute when you open to Colin Adam 's column. Relax. Breathe regularly. It's

mathematical, it's a humor column, and it may even be harmless.

Column editor's address: Colin Adams,

T

he S.S. Riemarm embarked on its maiden voyage from the dock of the Department of Mathematics, Yale University on April 2, 1999. Weighing in at 934 pages, and including a separate 50,000-line computer proof of the main lemma, she was the most massive the­ orem ever produced up to that time. There wasn't another theorem afloat on the mathematical ocean that compared. She had a crew of over 31, includ­ ing Captain Alphonse Huber, a full pro­ fessor and Fields medalist, five other full professors, eleven associate pro­ fessors, eight assistant professors, and six post-docs in steerage. Various grad students tagged along for the ride. Yes, she was the crown jewel in the fleet of theorems that had come out of Yale. Designed to survive any catas­ trophe, she was built with expendable lemmas shielding her bow. There were back-up lemmas and back-up lemmas to those. The proof was constructed with a graph-like structure so that if an edge were to be destroyed, there would be another path to the same point in the proof. Mathematicians marveled at the intricacy of her design. The computer proof used interval arithmetic, making it as rigorous as if it had all been done by hand. They said she was unsinkable. This first cruise was a shake-down run, to get the kinks out; just a quick trip to Berkeley and Stanford for a go­ ing over by the experts there, and then on to the University of Michigan for a week-long seminar. The subsequent voyage would be a straight shot to the

Department of Mathematics, Williams

Annals of Mathematics.

College, Williamstown, MA '01 267 USA

As she departed from the wharf in New Haven, the graduate students

e-mail: [email protected]

cried out and waved exultantly, throw­ ing streamers. Bands played exuberant marches. Administrators made promises that they knew they couldn't keep. It was a sight to behold. Once in Berkeley, they put her through her paces. The crew cranked up the logical engines, and she forged ahead. Nothing could slow her down. She sliced through questions like a scull in the Charles River. The crew oiled a proposition here, tightened a corollary there, and she lived up to her reputation as the most po�erful theo­ rem on the mathematical sea. At Stanford, her reception was rand. Wine and imported cheese on g sesame crackers, and little spanikopita hors d'oeuvres. No expense was spared. The crew reveled in the attention. After a colloquium replete with standing ovation, they turned her and headed for Michigan. Ann Arbor was cool at that time of year, but no one was overly concerned. After all, she was the queen of the ocean. They docked amid much fan­ fare. But the hubbub died down quickly, and a week-long set of lectures in the analysis seminar began. It started out fme. Huber remained at the helm at first. But soon he began to relax. She had proved herself in the Bay Area He could ease off and let other members of the crew pilot the craft. As the week wore on, the semi­ nar shrank in size, and they were down to a handful of experts. Late in the week, many of the crew had dozed off, and others had wan­ dered out for coffee. It was a post-doc, Dimmick, who was on watch when he realized there was something off the port bow, something in a question asked by the diminutive Prof. Feisberg, an expert on holomorphic functions. At first, Dimmick wasn't sure that it would amount to anything, so he did not sound the alarm. But as the issue loomed larger in the darkening semi·

© 2000 SPRINGER·VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

41

nar room, he realized how serious it was. "Counterexample, counterexample, dead ahead," he screamed out. "Full re­ verse, fu1! reverse. All hands on deck" The professors leapt to their feet. Evecyone grabbed for chalk and erasers. But the theorem ground on toward the immovable object ahead. Nothing could stop their forward momentum in time. The counterexample loomed out of the darkness, tall, white, stark against the evening sky. Some of the graduate stu­ dents remained oblivious to the im­ pending disaster, as they played intra­ mural soccer on an adjacent field. Prof. Huber tried to convince the crew that it would be all right. "She can withstand it," he said. But crew mem­ bers were leaping out the door of the seminar room at an alarming rate. When the collision occurred, it seemed to happen in slow motion. There was a grinding crunch. Lemmas sloughed off the prow. Edges of the graph-like structure buckled under the impact. The hull seemed to crumple up like the ego of a jobless Ph.D. Almost immediately, they realized she was going to go down.

Huber turned to the communica­ tions officer. "Reynolds," he said. "Use your cell phone to call the nearest functional analyst. I think it's Alder at Wisconsin. We are going to need help once we are in the water. There aren't enough hypotheses to go around." As Reynolds dialed frantically, the Captain tried to quell the growing panic. "Evecyone, I ask you to remain calm. We have radioed for help." But Reynolds turned to the Captain with tears running down his cheeks. "Captain, Captain!" he cried. "Alder is in Germany. The nearest functional an­ alyst is in Utah, and there's no way she could get here in time. We're on our own." There was pandemonium at the doorway to the seminar room, as the mob fought to get out in time. "Please," shouted the captain, "let the grad students and post-docs have the hypotheses. Show some courage." But full professors were grabbing lem­ mas and claims as they pushed post­ docs to the floor in their frantic haste to escape. Reynolds seized a corollacy but Huber stopped him. "Reynolds, that won't float."

Dimmick turned to the captain. "Sir, we should get out before it's too late." "I'm not getting out," replied the Captain gravely. "I'm going down with her." "I'll go down with her, too, sir," said Dimmick, tcying not to look frightened. "No," said Huber. "You have your whole career in front of you. Don't throw it away on this ship, as beauti­ ful as she is. Abandon her. You will sur­ vive to crew another theorem." Dimmick shook his head no, but Huber placed a firm hand on his shoul­ der. "That's an order," he said. Dimmick saluted one last time, and then scram­ bled out of the seminar room. As the afternoon light dimmed, Huber and the crew members who hadn't man­ aged to escape slowly disappeared beneath the waves, lost forever in the immeasurable ocean known as mathe­ matics. Some day a ship will leave port again, a ship with the name S.S. Riemann. And that ship will be truly indestructible. And mathematicians around the world will rejoice. But until then, remember to book your passage carefully, and bring along plenty of hypotheses.

MOVING? We need your new address so that you do not miss any issues of

THE MATHEMATICAL INTELLIGENCER. Please send your old address (or label) and new address to: Springer-Verlag New York, Inc., Journal Fulfillment Services

P.O. Box 2485, Secaucus, NJ U.S.A.

07096-2485

Please give us six weeks notice.

42

THE MATHEMATICAL INTELLIGENCER

DIDIER NORDON

Eth nog raph i c

"You spent more than

10 years in the field. In spite of

that, scholars have given a very poor reception to your de­

to a rofpay, nor does the rofpay give orders to a tudentsay. They work together with seemingly complete equality. Still,

scription of the Urematherpays.2 How do you account for

when you look closer you realize that everybody knows

the rejection you have suffered?"

perfectly who is above, and the thought would never oc­

People have their theories and they stick to them. If you

bring back observations which don't fit their theories, they

cur to any of them to challenge the hierarchy. There is no special privilege associated with being an

don't believe you. Anything to avoid having to reconsider

andarinmay. It is reward enough just to know that one is

their preconceptions.

considered a chief by the others.

"Nobody took the trouble to go and verify your obser­

"And by the other tribes?" No. Every tribe is organized in this pyramidal structure,

vations?" You have to remember that it's very difficult to get ac­

but the andarinmay of one tribe is almost a nobody to an­

cess to the Urematherpays. Very few have made it to their

other tribe. Every Urematherpay is free to choose a tribe.

country; still fewer have returned.

Once you have made your choice you have to stick to it.

"Are they such a cruel people?" Oh, no! Not specially. The problem is not that. It's just that it takes such effort to get into their culture that no­

Relations between tribes �e intensely competitive. "Are the tribes numerous?" Very. On the other hand, some have very few members.

body, or almost nobody, has enough energy left to get away

"What are their names?"

again. That's why everyone is afraid of them. Acculturating

The oldest ones are very ancient, and their names never

to them leaves you inextricably involved with them. "What is so compelling about Athermay culture? Their

name goes by complicated, fluid rules, mostly by putting

mores?" No, the mores are pretty routine. Their system of chief­ tancy

change: Eometryjay, Gebralay, Alysisnay . . . When a new tribe is formed, which happens often, the coining of its

is a network of pyramids. The Urematherpays are

divided into tribes. Each tribe has a chief, called an Andarinmay. Around each chief are sub-chiefs, called rof­

so

together names of existing tribes. "Are there no exchanges between tribes?" As

little

as

possible.

One

is proud to be

an

Urematherpay, but one tries to avoid communicating with

on down to the lowest

Urematherpays of other tribes. When circumstances, like

rank, the tudentsays. The andarinmay doesn't give orders

proximity for example, result in too great a mutual com-

pays; then sub-sub-chiefs, and

1 Reprinted by permission from Gazette des Mathematiciens 67 (1 996), 43--46. This article appeared also in Deux et deux font-its quatre?, a collection of the author's essays: Pour Ia Science, Belin,, 1 999. 2-franslator's Note: The author's tribal names are concocted using verlan, so I have used the closest English equivalent, Pig Latin. Thus where the author transforms to matheux pur to purtheuma, I transform pure mathematician to pure mather to urematherpay.

mathematicien pur

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

43

prehension between tribes, they form a new tribe, which

-

AU T HO R

hastens to break all bridges with the others. "How do they go about it?" The new tribe creates its own language which the other tribes don't understand, and makes it as difficult as possi­ ble to translate. For one thing, they manage not to talk about the same objects. One tribe takes the gebra as its totem animal, talks only about the gebra and makes it the object of

all its prayers, casting of auguries, etc. Another

tribe talks about the dime and the deg, and so on. 3 These are mythical animals, of radically different natures. Those who have seen the gebra have never seen the dime or the deg, and vice versa. A statement about the gebra can not

DIDIER NORDON

be expre�sed in terms of the deg, for example; for these

47, rue du Sablonat

two creatures have n9 simultaneous existence, evolve in

33800 Bordeaux

incompatible worlds with no communication between

France

them. Every attempt at translation is bound to fail. This

e-mail: [email protected]·bordeaux.fr

makes mutual incomprehension complete. "Can it really be complete?"

Didier Norden, bom 1 946, graduated at Paris Sud, Orsay. He

People refuse to recognize the accomplishments of this

has been teaching mathematics at the University of Bordeaux

people. To be sure, all the tribes deal with the same mate­

If their different languages were designed to de­

1 since 1 970. Still there remain people in Bordeaux who un­

rial reality.

derstand nothing about mathematics. . . . A main interest of

scribe that, then translation between them would be pos­

his is the relationship between the world view of mathemati­

sible. But it is not so. The Urematherpays consider the

cians and that of the surrounding society. Among his books

material world a triviality not worth talking about. Each

are Les mathematiques pures n 'existent pas! (new edition

tribe creates a purely i maginary world and p ays no atten­

Actes Sud, 1 993) and La droite amoureuse du cercle, a col­

tion to anything else.

As

lection of fantasies, Editions Autrement, 1 997. He is a colum­

these worlds have no common

point, there is no passing from one to another, and there

nist for the monthly Pour Ia Science, the French version of

is no translating one Athermay language to another. That is why I assert that the puwose of language among the Urematherpays is to attain non-communication.

Scientific American.

"That is the conclusion that has come under attack."

lowed correctly. You're an Urematherpay, say, and you give

Naturally. It contradicts the accepted ideas about lan­

the order, "Horace! Blow up your method, and put the deg

guage. How can we imagine a language which is not in­

on a subvariety"-a typical Athermay utterance. No word

tended for communication? All right, go and live among the

in the sentence has a concrete referent. How could you

Urematherpays. I've done it for many long years, and I as­

possibly verify whether Horace has understood your com­

sure you that with them, the function of language is avoid­

mand? You can't. Now it's a well-known psychological law

ing exchange. The mythical creatures of each tribe seal it

that

off perfectly from the others.

tor understands your statement, you can be sure that in

"But you claim more than that. You say that even within each tribe mutual comprehension is poor." Yes. That is a subtler question. My hypothesis is that the

if you have no way of confmning that your interlocu­

fact he doesn't. Oh, perhaps he'll understand it once or twice. Maybe by chance. But you are tending inexorably toward incomprehension.

breaking of communication between tribes leads to break­

"And that is happening among the Uremathpays?"

ing of communication within tribes. Not that this is neces­

Very likely. They permitted me to take part in their most

is a weekly

sarily sought by the Urematherpays. It can be an unin­

important ritual, the eminarsay. The eminarsay

tended side effect.

meeting of the whole tribe. Each tribe has its own. The

"Still, they do talk to each other within a tribe?"

speaker, one of the members, gives an hour's incantation to

Oh, sure, they talk. But they don't understand.

the gods of the tribe. All my observations indicate that the

"Why not?" Well, I told you that their language deals only with imag­

communicants have little understanding of what the offi­

ciant

is saying. No response from them, little variation in

inary things. Their statements don't purport to have any

muscle tone or intellectual tone; visible langor; many doz­

agency: they are not followed by any action which would

ing. Well, what do you suppose happens at the end of the

give a check on whether such-and-such order had been fol-

incantation? The listeners add little incantations of their

3J. Alexander & A. Hirschowitz, La methode d'Horace eclatee: application a !'interpolation en degre quatre, lnventiones Mathematicae 1 07 (1 992), 585-602: "Dans

cette variants eclatee, on exploits une sous-variete de codimension quelconque: Ia dime est un enonce de rangement sur cette sous-variete, landis que Ia degue est un enonce de rangement sur Ia variete obtenue en eclatant cette sous-variete." J.-P. Serre, Gebres, L'Enseignement Mathematique 39 (1 993), 33-85: "Objet de ce texte, les enveloppes algebriques des groupes lineaires et leurs relations avec les differents types de gebres: algebres, cogebres et bigebres.'

44

THE MATHEMATICAL INTELLIGENCER

own, in the form of questions which the officiant answers.

verbiage: that, they consider a gratuitous obstacle. Every

When they paid me the compliment of letting me give the

if in spite of these struggles he remains enigmatic to the

It shows they don't need to understand to talk to each other.

Urematherpay struggles to be as clear as possible. It's only

incantation, I spoke sentences without any meaning I could

other Urematherpays that he knows he has attained the

see. The audience reaction at the end was the same as usual.

summit of richness.

I even suspect that the aim-I_repeat, the aim-of verbal ex­ change between Uremathpays

"And that is satisfying?"

is mutual incomprehension.

Yes. Look at the matter from both sides. No doubt it is

"For what purpose?"

unpleasant for an Urematherpay not to understand what is

At the foundation of the Athermay conception of the

said to him. But they're practical people: they realize that

world is the idea (maybe astonishing to you, but natural to

this unpleasantness is a small price to pay for the gratifi­

is the richer for being diffi­

cation of being recognized as incomprehensible by others.

cult to communicate. The ultimate richness, then, would

And permit me in conclusion to ask a question of you. Is

them) that an imaginary world

be found in incommunicability. But how subtle they are!

it really always all that satisfying when you do understand

Contrary to what you might suppose, they detest circuitous

what others are telling you?

EXPAN D YOUR MATHEMAT I CAL BOU N DAR IES ! ---------------------------------------------------------------------------------------

Mathematics Without Borders A

History of the International Mathematical Union

As told by Professor Olli Lehto, the history of the International Mathematical Union (IMU) is surprisingly compelling.

The twentieth century has been fraught with tremendous international conflict, but there has also been a great deal of progress made tO\IIIard cooperation among nations. Ever since Corsresses throughout the 'NOI'ld, effectively bridging cultural

love and appreciation for mathematics.

its inception in 1 897, the IMU has sponsored International

and political gaps by uniting scientists through their genuine

For anyone strictly interested in the mathematical and organizational details of the IMU Corsresses, this as an excellent resource. However, Mathematics

book will serve

Without Borders also takes time to focus on the individuals, many of

them leading mathematicians of the twentieth century, and their stories-told against the backdrop of world events. Contents: Prologue to the History of the IMU of the

•

The Old IMU (1920-1932) • Mathematical Cooperation Without the IMU (1933-1939)

New IMU (1945-1951 ) • The IMU Takes Shape (1952-1954)

(1 958-1962)

•

Consolidation of the IMU (1 963-1970)

(1 979-1986)

•

The IMU and Related Organizations

•

•

•

Expansion of the IMU (1955-1958)

North-South and

•

East-West Connections (1971-1978)

•

•

f'ax: (201)-348-4505

•

Foundation

Politics Interferes with the IMU

The IMU in a Changing World ( 1 986-1990)

1 998/368 PP., 53 FIGS./HARDCOVER/$35.00/ISBN 0-387-98358-9

Order Today! Call: 1-800-SPRING£R

•

The IMU and International Corsresses

VIsit: {http://www . sprlnger-ny.com}

f '

';. 0:·

Springer

VOLUME 22. NUMBER 4,

2000

45

The Fibonacci Chimney

F

ive years ago, residents and visi­

but nowhere to the same effect as in

tors in Turku were confronted with

Turku, where it dominates the water­

a sequence of seven-foot-high digits

front and the estuary of the Aura river.

running along the smokestack of the

It

local

intellectual

quence reflects two of the major re­

of course,

search

power

plant-an

challenge to all except,

mathematicians.

Mats Gyllenberg Karl Sigmund

In

due

time,

the

is entirely by accident that the se­ fields

of the

University

of

Turku, namely, number theory and mathematical

showpiece in the city fathers' attempt

known, Fibonacci introduced the se­

to turn Turku-the oldest university

quence at around AD 1200 to model the

town in Finland and a major calling

biology.

As

smokestack became the most salient

is

well

growth of a rabbit population.

port for midsummer cruises in the Baltic-into a capital of Conceptual

Department of Mathematics

Art. The artist responsible for the dis­

200 1 4 University of Turku

play, Mario Merz (b. 1925) from Italy,

Finland

had been obsessed by the sequence for almost 30 years. He has used it to dec­

Department of Mathematics

orate the Saint Louis chapel in Paris'

University at Vienna

Salpetriere as well as a spire in Turin,

1 090 Vienna, Austria

Does yaur hometown have any mathematical tourist attractions such as statues, plaques, graves, the cote where the famaus conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have yau encauntered a mathematical sight on yaur travels? If so, we invite yau 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 yaur tracks.

Please send all submissions to Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium

The Turun Energia Power Plant in Central Turku. Photo courtesy of Turun Energia photogra­

e-mail: [email protected]

pher Seilo Ristimiiki.

46

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

Vilnius Between the Wars Stanisfaw Domoradzki and Zofia Pawlikowska-Brozek

I

n the beginning of the 20th century, Polish mathematics rapidly expanded, in its two major mathematical centers, Lvov and Warsaw, and in Krakow, one of the oldest universities of Europe. Stanislaw Zaremba (1863-1942), Waclaw Sierpiriski (1882-1969), Hugo Steinhaus (1887-1972), Stefan Mazurkiewicz (18881945), Stefan Banach (1892-1945) and Juliusz Schauder (1899-1943) are a few of the names linked to that epoch. A center also of some importance was Vilnius. Vilnius is the name of the city in the Lithuanian language, and it is now the capital of independent Lithuania; be­ tween the wars it was in Poland and was usually designated by its Polish name, Wilno. It is a clean, beautiful, lively city, with spires of churches and newly renovated fa<;ades of historical buildings. In the present text, a pic­ turesque route is described that a mathematical tourist may follow when

arriving by train in Vilnius, going to the University. Some details are given about well-known Vilnius mathemati­ cians, reviving the glorious past of the city. Traces of these close colleagues of the so-called Polish school of math­ ematics are rare in Vilnius, because many fled the country, and only a few were buried in the city, as we shall dis­ cover. From the Railway Station, we fol­ low Gelezinkelio street (this is its Lithuanian name; Kolejowa is the Polish equivalent). Ausros Vartu street (or Ostrobramska) leads us to the Ostra Brama, a sanctuary in honor of St. Mary. Passing a Polish bookshop and St. Kazimierz's church, the visitor then reaches the Town Hall through a narrow street. Didiioji street (Wielka) leads to 11 group of buildings between Universiteto (Uniwersytecka) and Pilies (Zamkowa) streets. There, St. John's church can be

LITHUANIA

GERMANY

USSR

CZECHOSLOVAKIA

Figure 1. A map of Poland in the years 1 922-1939.

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

47

seen, attached to the University build­ ings surrounding the courtyards. They are named after the great poet Adam Mickiewicz

and

the

rectors

Piotr

isted until 1939, when World War II

Stanislaw

Krystyn

Zaremba

(1903-

started. In that period, numerous math-

1990), and Antoni Zygmund (1900-1992).

ematics and natural sciences confer­

Zygmund had come from Warsaw

ences took place in Poland, and in

University, where he got his doctoral

Skarga (1536-1612), the very first rec­

Vilnius in particular. The Congress of

degree in 1923. He was appointed as­

tor of the Vilnius Academy, and Marcin

Polish Physicians and Natural Scien-

sistant professor at the Philosophy Department of Warsaw University in

Poczobutt (1728-1810), who was in

tists, which included a mathematical

charge during reforms of the educa­

section, was held in 1929. Two years

1926 and worked there until 1929, to

tional system. Nearby is the Papal

later, the Second Congress of Polish

become

Seminary, where refreshments are of­

Mathematicians was organized by the

Department in Vilnius in 1930. Thanks

fered in a courtyard surrounded by

Polish

The

to a Rockefeller Foundation fellow­

1 7th-century cloisters and arcades.

Society had been founded in 1919, and

ship, he had gone to Oxford and

Mathematical

Society.

head

of the

Mathematics

contains

its first congress was held in 1927, in

Cambridge, where he had met G. H.

mathematicians

Hardy and J. Littlewood. There he also

who worked at the University in the pe­

Lvov. All professors working in Vilnius

at the time took part in the organization

got in touch with R. E. A. C. Paley, an

riod between the wa.rS, as well as old

of the

outstanding young English mathemati­

documents from the very beginning of

( 1866-1932), Stefan Kempisty (1892-

cian, with whom he would later pub­

the Vilnius Academy. They go back to

1940), Juliusz Rudnicki (1881-1948),

lish several works. In Vilnius, Zygmund

The

University

archives

of

Library

Polish

Congress:

Wiktor Staniewicz

1570, when the Jesuits founded a col­ lege;

later,

King

Stefan

Batory of

Poland promoted it to the rank of Academy. The Vilnius Academy be­ came the cultural center radiating to the East, South, and North, as the Krakow Academy was spreading its in­ fluence from the West. The lecturers often came from abroad, but many Polish scientists also worked at the Vilnius Academy. Towards the end of the 18th cen­ tury, when the first Polish Ministry of Education was founded, the Academy flourished

again.

The

lectures

on

higher mathematics were introduced and conducted by Franciszek Milikont Narwojsz (1742-1819), based on the works of Isaac Newton. The astronom­ ical obseiVatory was equipped with modem instruments thanks to Rector Marcin Poczobutt, a well-known as­ tronomer in Europe. Other obseiVa­ tions were made by Jan Sruadecki (1756-1830), author of Trygorwmetria kulista analitycznie wy/bZona (1817) (An Analytical Presentation of Spherical

Trigonometry), which was translated into German. Originally from Krakow University, he had come to Vilnius to become professor of astronomy and rector,

and

he

very

actively

con­

tributed to the reforms in the Polish educational

system.

In

1803

the

Academy changed into a University and flourished until the Russian occu­ pation closed it in 1832. After Poland regained independence in 1919, the University revived and be­ came the Stefan Batory University.

48

THE MATHEMATICAL INTELLIGENCER

It ex-

Figure 2. View of Vilnius streets.

Figure 3. Vilnius University today.

did not lose contact with his col­ leagues of the Warsaw school of math­ ematics. In 1938, his collaboration with Stanislaw Saks (1897-1942) led to their book Analytic Functions, which would be republished 14 years later in the Mathematical Monographs series. Zygmund's arrival in Vilnius strengthened the mathematical circle. At that time he was already the author of over forty scientific works pub­ lished abroad, as well as in the leading Polish mathematical journals, such as Fundamenta Mathematicae and Studia Mathematica. Thanks to friends such as J. Tamarkin, Zygmund would move to the USA in 1940, where his mathe­ matical career flourished. After a short time at the University of Pennsylvania in Philadelphia, he would become a professor in the Mathematics Depart­ ment of the University of Chicago, un­ til his retirement in 1980. Zygmund was awarded honorary degrees in Washing­ ton, Toruri, Paris, and Uppsala, and was a member of the National Academy of Sciences of the USA, the Polish Academy of Sciences, the Royal Academy of Sciences of Madrid, and the Academy of Sciences of Palermo. In 1986, he received the National Medal of Science of United States, 6 years be­ fore his death. One of Zygmund's students at the Stefan Batory University was J6zef Marcinkiewicz (1910-1940), who might have become a first-rank math-

ematician if destiny had been as fa­ vorable to him as it was to Zygmund. Already in 1933, he published two works: "On a theorem of trigonometric series" (Journal of the London Mathe­ matical Society) and "On a class of functions and their Fourier series"

(Reports of the Warsaw Science Society). At the age of 25, he took a

Figure 4. A drawing of Antoni Zygmund, by L. Jesmanowicz.

doctor's degree at Vilnius University. He proved that it is possible to con­ struct a continuous function whose Fourier series is uniformly convergent while the interpolation polynomials are almost everywhere divergent. J. Marcinkiewicz spent some time at the famous Lvov mathematical school and this led to several papers in the Lvov Studia Mathematica; and in 1937 he qualified as assistant professor of Stefan Banach. However, his scientific career was tragically cut short. At the beginning of the 1938-39 academic year, J. Marcinkiewicz left for London and Paris on a research grant from the National Culture Foundation. A� the end of August 1939 he interrupted the journey and came back to Poland be­ cause of the international political sit­ uation. ije took part in the defense of LvoV, and after the Soviet Army marched into the city, he was taken prisoner, to Starobielsk. Only 30 years old, he was murdered in 1940; his name would join the list of the Katyil victims. World War II took a heavy toll of Polish mathematicians. Those who survived were scattered all over the world. J. Rudnicki became the head of the Mathematics Department at Lublm University, but when he met his col­ leagues at the University of Mikolaj Kopernik in Toruri, he moved there in 1946. Similarly, Leon Jesmanowicz (1914-1989), who worked for about two years at the University of Stefan Batory, came to the Kopemik Univer­ sity. He is probably best known for his caricatures (see figure 4). Miroslaw KrzyZariski (1907-1965), who had taken a doctor's degree of mathematics at the University of Stefan Batory in 1934 and was a specialist on partial differential equations, became a professor at Krakow Technical University and the Jagiellonian University. Let us come back to the route we were following in Vilnius. At the other side of Universiteto (or Universytecka in Polish) street, there is the presiden­ tial palace, called the Prezidentura (Palac Prezydencki) . Then we con­ tinue our road crossing a flowery square, reaching the Arkikatedros aikste (Plac Katedralny) , with the mon­ umental St. Stanley's Cathedral. On the left-hand side we pass the Gedimino

VOLUME 22, NUMBER 4, 2000

49

dorobku naukowym Antoniego Zygmunda or

which surrounds a part of Vilnius and joins the Neris (Willa), the main river

On Antoni Zygmund's scientific achieve­

A trolley

ments , bibliography and portrait, Wiado­

on which the city has grown.

bus will bring us to the Rossa ceme­

moSc:i

Matematyczne

XIX.2,

p.91 -1 26,

1 976.

tery, where some Vilnius professors of mathematics are buried. Just behind

Jesmanowicz, L. Caricatures of Polish mathe­

the gate, next to the fence, lies Wiktor

maticians, 1 3th Congress of Polish Mathe­ maticians, Toruli, 1 994.

Staniewicz (1866-1932), of the Univer­ sity of Stefan Batory, while a bit fur­ ther is the grave of Tomasz Zycki, of the

Zemajtis, Z. Fiziko-Mathematiczeskije nauki w starom

Universitetie

(1579-

1 962.

From there, one can conclude the jour­ ney by paying a visit to the Lituvos

Wilniusskom

1882) , Utovskij Matematiceskij Sbornik 1 1 . 2 ,

18th-century Vilnius University.

Zygmund, A. J6zef Marcinkiewicz, Wiadomosci Matematyczne VI, p. 1 1 -41 , 1 96o-61 .

Centrinis Valstybes Archyvas, that is, the Lithuanian Central Record Office. There, documents and personal port­ folios

Zofia Pawlikowska-Broiek

of the University of Stefan

University of Mining and Metallurgy

Batory professors are gathered.

Department of Applied Mathematics

BIBLIOGRAPHY

30-059 Krakow

al. Mickiewicza 30

Figure 5. The arms of Vilnius.

Bielir'\ski, J . , Uniwersytet Wileriski (The Vilnius

bokStas (G6ra Zamkowa) or Castle Hill

Calderon, A.P., Stein, E., Antoni Zygmund

University)

(1579- 1 83 1) ,

Krakow,

Poland.

1 899Stanistaw Domoradzki

1 900.

Notices

of

the

Pedagogical University Institute of Mathematics

American

to see the architectural contrast of the

(1900-1992),

St. Ann's church and the Bernardine's

Mathematical Society, v.39, n.8, p.848-9,

ul. Rejtana 1 6a

October 1 992.

35-31 0 Rzeszow

church. We cross the Vilnia (Wilejka) River,

Fetterman, Ch. , Kahane, J . P . , Stein, E . M . , 0

Poland.

Mathema tica l Olym piad Cha l lenges Titu Andreescu, American Mathematics Competitions, University ofNebra ka, Lincoln, NE Razvan Gelca, University of Michigan, Ann Arbor. Ml

This is a comprehensive collection of problems written by two experienced and well-known mathematics educators and coaches of the U.S. International Mathematical Olympiad Team. H undreds of beautiful, challenging, and instructive problems from decades of national and international competition are presented, encouraging readers to move away from routine exer­ ci e and memorized algorithms toward creative solutions and non-standard problem-solving techniques. The work is divided into problem clu tered in self-contained sections with olution provid­ ed eparately. Along with background material, each ection include representative examples, beautiful diagrams, and li t of unconventional problems. Additionally, historical in ight and a ide are pre ented to stimulate further inquiry. The empha i

throughout is on stimulating

readers to find ingeniou and elegant olutions to problems with multiple approache . Aimed at motivated high school and beginning college students and i nstructors, this work can be used as a text for advanced problem-solving cour es, for self-study, or as a resource for teach­ ers and students training for mathematical competitions and for teacher professional develop­ ment, seminars, and workshops. From the foreword by Mark Saul: "The book weave together Olympiad problems with a com­ mon theme, so that insight become techniques, tricks become methods, and method build to mastery. . . Much is demanded ofthe reader by way ofeffort and patience, but the investment is greatly repaid."

To Order:

Call: 1 -800-777-4643

•

Fax: (20 1 ) 348-4505

•

E-mail: orders@ birkhauser.com

•

S. 0-8 1 76-41 55·6 / S29.95 0·8 1 76-4 1 90-4 / S59.95

Visit: www.birkhauser.com

orth America only and subject to change without notice.

Birkhauser Boston - International Publisher for the Mathematical Sciences

THE MATHEMATICAL INTELLIGENCER

ILL

HARDCOVER / ISB

Textbook evaluation copies available upon request, call ext. 669. Price are valid in

50

2000 / 280 PP., 85

SOFTCOVER / ISB

7100

Promotion IY1097

JOHN A. EWELL

Co u nti n g Latti ce Po i nts o n S p h e res

roblems formulated in one branch of mathematics but solved with tools of another branch have for a long time been a source of great satisfaction, especially for those among us who would guard against the tendency toward overspecialization. Noteworthy among such problems are those which lie on the boundary between geometry and arithmetic (number theory). These can be traced back to Pythagoras, and perhaps beyond (e. g., to the ancient Babylonians). In this discussion, we let n run over N : = {0, 1, 2, . . . }, and for each such n construct in IR3 a sphere of radius Vn centered about the origin (0, 0, 0), as in Fig. 1. We then set the problem of counting the lattice points (x, y, z) on the surfaces of these spheres. (A lattice point (x, y, z) is one all of whose coordinates are integers.) This problem is one of three-dimensional Euclidean geometry, where all of the numbers involved (including the radii Vn, n = 0, 1, 2, . . · ) are constructible. The tools for solving the problem come from number theory. As there are two counting functions r3 and q0, defined on N and having values in N, I shall briefly describe these: for each n E N,

where 7L := {0, ::!:: 1, ±2, . . . }; and q0(n) : = the number of par­ titions of n into distinct odd parts. By convention q0(0) : = 1. E. g., among the five unrestricted partitions of 4, viz., 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1, there is exactly one into distinct odd parts, so that q0(4) = 1. The

reader should be easily convinced that the function q0(n), n E N, is generated by the infinite product expansion 00

fl

n� l

(1 + _i!n- 1 ) =

00

L qo(n)xn,

n�o

P::l

< 1. which is valid for each complex number x such that For the sake of completeness put q0(n) : = 0 whenever n < O. The Counting Algorithm

First of all, if on the one hand n E N and r3(n) > 0, then for each of the relevant points (x, y, z) E 7L3, n = x2 + y2 + z2 if and only if Vn = Y(x - 0)2 + (y - 0)2 + (z - 0)2. On the other hand, if n E N and r3(n) = 0, then there exists no point (x, y, z) E 7L3 such that n = x2 + y2 + z2, or equiv­ alently, such that Vn = Y(x - 0)2 + (y - 0)2 + (z - 0)2. Hence, for each n E N, r3(n) is the count of all lattice points on the sphere of radius Vn centered about the ori­ gin (0, 0, 0). Next, I will state and discuss (but not prove) three arithmetical theorems.

(Legendre) If S : = {n E N : n = 4k (8m + 7), for some k, m E N }, then for each n E S, r3(n) = 0, and for each n E N - S, r3(n) > 0. Theorem 1

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

51

Figure 1. Here presented are complete descriptions of the degenerate sphere of radius

Yo and the unit sphere of radius v1, the six dots

representing all of the lattice points on the surface of the latter. To avoid clutter, descriptions of the spheres corresponding to the radii and

v3 are left incomplete. Of course, the reader must imagine the spheres having radii Vn, n

I

{C- 1r, if n = m(3m

±

1), Jor some m E N,

0, otherwise.

Theorem 3

r3(n) = +

I ( - 1)kC3k + 1)12(6k + 1)q0(n - k(3k + 1)/2)

qo(n) 3

1

1

14

3

2

0

15

4

3

16

5

4

17

5

5

18

5

6

19

6

7

···

At the present time no simple proof of Legendre's Theorem 1 is known. E.g., see [3, p. 311]. Clearly, the theorem has a striking nonintuitive interpretation relative to our problem. For a proof of Theorem 2 see [1, pp. 1-2]; and for a proof of Theorem 3 see [2].

THE MATHEMATICAL INTELLIGENCER

\12

n

13

0

( - 1)kC3k - 1Y2(6k - 1)qo(n - k(3k - 1)/2)

k ElP' = qo(n) + 5qo(n - 1) + llqo(n - 5) - 17 qo(n - 12) + + 7qo(n - 2) - 13qo(n - 7) - 19qo(n - 15) + · · ·

52

{0, 1 , 2, 3}.

TABLE 1 .

n

lf !FD : = N - {0}, then for each n E N,

qo(n) - I kElP'

-

Our recursive two-step algorithm proceeds as follows: (i) Use the recursive determination of q0 in Theorem 2 to compile a table of values of q0, as in

For each n E N, ( - 1)k(k + l)12qo(n - k(k + 1)/2) =

Theorem 2

k EN

E N

20

7

8

2

21

8 8

9

2

22

10

2

23

9

11

2

24

11

12

3

25

12

(ii) In terms of these computed values of q0, utilize Theorem 3 to compile a table of values of r3, as in

TABLE

n

2.

1

n

13

r3(n)

6

14

48

2

12

15

0

3

8

16

6

4

6

17

48

5

24

18

36

6

24

19

24

7

0

20

24

8

12

21

48

9

30

22

24

10

24

23

0

11

24

24

24

12

8

25

30

0

r3(n)

Each of these tables

A U T H O R

24

JOHN A. EWELL

Department of Mathematical Sciences Northern Illinois University DeKa/b, IL 601 1 5-2888

1 and 2 can be indefinitely extended

USA

(in the stated order) with the aid of machine computation. For a fixed but arbitrary choice of n

E N, the running time 1 here pro­

John Ewell earned his Ph.D. in 1 966 under the direction of

vides an excellent check on the accuracy of computation.

Illinois University for many years, and though recently retired

for each table is

O(n312).

Legendre's Theorem

E.G. Straus. He has been on the professorial staff at Northern he is still active there and still continues his interest in addi­ tive number theory and related fields. He is partial both to

Concluding Remarks

The problem of lattice points on spheres

opera and to baroque instrumental music.

is one of ele­

mentary geometry, easily visualized; the first few cases are easily computed. Intuition at that stage leads that the values

us to expect

r3(n) will rise steadily but irregularly with

REFERENCES

n. Legendre's Theorem steps in to show us the unexpected and striking exceptional values of n for which r3(n) 0.

1 . J.A. Ewell, "Recurrences for two restricted partition functions,"

Fibonacci Quarterly, 18 (1 980): 1 -2.

=

2. J. A. Ewell, "Recursive determination of the enumerator for sums of three squares," Int. J. of Math. and Math. Sc. (to appear).

ACKNOWLEDGMENTS

I \vould like to thank Eric Behr for producing the descrip­ tive picture of Figure

3. G. H . Hardy and E. M. Wright, An Introduction to the Theory of

1.

Numbers, Fourth edition, Clarendon Press, Oxford, 1 960.

Clear, Simple, Stimulating Undergraduate Texts from the

Trigonometry

1. M. Gelfand, Rutgers University, New Brunswick,

& M. Saul, rhe BrofiXl!i/le School, Bronxville, NY

NJ

This new text in the collec­ tion of the Gelfand School Outreach Program is written in an engaging style, and approaches the material in a unique fashion that will moti· vate students and teachers alike. All basic topics are cov­ ered with an emphasis on beautiful illustrations and '------' examples that treat elemen· tary trigonometry as an outgrowth of geometry, but stimulate the reader to think of all of mathe· matics as a unified subject. The definitions of the trigonometric functions are geometrically moti­ vated. Geometric relationships are rewritten in trigonometric form and extended. The text then makes a transition to a study of the algebraic and analytic properties of trigonometric functions, in a way that provides a solid foundation for more advanced mathematical discussions. 2000 I Approx. 280 pp., 1 8 5 illus. I Hardcover ISBN 0-8176-3914-4 I $ 1 9.95

Gelfand School Outreach Program

Algebra

I.M. Gelfand & A. Sben

"The idea behind teaching is to expect students to team why things are true, rather than have them memorize ways ofsolving a fewproblems. . . {Thisj same philosophy lies behind the current text. . . A serious yet lively look at algebra. " -The American Mathematical Monthly 1993, 3rd prinling 2000 I 160 pp. I Softcover

ISBN 0-81 76-3671-3 I

S 1 9.95

The Method of Coordinates

I.M. GeLrand, E.G. Glagoleva & A.A. Klrlllov

"High school students (or teachers) reading through these two books would learn an enor· mous amount ofgood mathematics. More impor­ tantly, they would also get a glimpse of how mathematics is done. " -The Mathematical lnteUigencer 1 990, 3rd prinling 1 996 I 84 pp. I Softcover

ISBN

O-B 1 7 6-3533·5 I

S 1 9.50

Functions and Graphs

I.M. Gelfand, E.G. Glagoleva & E.E. Sbnol

"All through both volumes, one finds a careful description of the step-by-step thinking process that leads up to the correct definition ofa concept or to an argument that dinches in the proof of a theorem We are. . . very fortunate that an account ofthis caliber has finally made it to printedpages. " -The Mathematical l ntelligencer 1 990, 51h prinling 1999 I 1 1 0 pp. / Softcover ISBH 0-81 76-3532-7 I $ 1 9.95

To Order:

Call: 1-800-777-4643 • Fax: ( 2 0 1 ) 348-4505 E-mail: [email protected] Visit: www.birkhauser.com

Textbcd evaiU01ion cOjliel avoiloble upon request, call ext 669. Prim are 1111id in N011h Arneti
SoW

i

Blrkhiiuser Boston­ International Publisher for the Mathematical Sciences

Promotioo IYI018

QfifW-J.t.i

J e remy G ray, Editor

Kepler's Critique of Algebra

A

I

lthough Johannes Kepler lived

algebra," he did not employ the new

during the initial flowering of early

symbolic mathematics, and expressed

modem algebra, he did not make much

critical reservations on the few occa­

use of it, preferring to work in the clas­

sions he mentioned it. [NA 256; KGW

sic geometric manner. Kepler consid­

9. 1 12-113,

ered Euclidean geometry to be funda­

critical of the "contrivance" of the

1 7.258]

While Viete was

mental, and thought that algebra could

Copernican alternative Kepler so pas­

not compete with geometry as a source

sionately espoused, Kepler expressed

of intelligible truths, although it was

a certain attraction but even more

a valuable, if approximate, method.

skepticism towards the use of symbols

Peter Pesic

Kepler's conception of the world also

as the central device of mathematics.

Dedicated to Curtis Wilson

rested on crucial musical assumptions.

[Swe] In a letter of May 12, 1608, writ­

As with his work in physical astron­

ten to his friend, the physician Joachim

omy, Kepler insisted on the impor­

Tanckius in Leipzig, Kepler remarked

tance of observation in judging musi­

that

cal questions.

It is from such empirical

observations of musical practice that he excludes intervals that have no geo­ metrical expression, but only algebraic ones. In this way, his musical observa­

tions support his geometrical exposi­ tion and his rejection of algebra. His critique of algebra was based on what he saw as its hidden reliance on infmite processes; his example of the heptagon shows that the solutions even of cubic equations rely on tran­ scendental functions. In a striking an­

ticipation of intuitionistic arguments,

Kepler rejected infinite processes, and conceived of geometry as finite con­ structions. Though perhaps the earliest

I too play with symbols; I have planned a little work, Geometrical Kabbala, which is about the Ideas of natural things in geometry; but I play in such a way that I do not forget that I am playing. For nothing is proved by sym­ bols, nothing hidden is discovered in natural philosophy through geometri­ cal symbols; things already known are merelyfitted [to them]; unless by sure reason it can be demonstrated that they are not merely symbolic but are descriptions of the ways in which the two things [i. e., the two terms of the analogy] are connected and of the causes of this connection. [Wal 55]

critic of symbolic algebra, Kepler al­ ready raised profound questions.

There may well be some irony here, for elsewhere Kepler remarked, "I hate

Kepler's "Geometrical Kabbala"

Kepler

phrased

his

dryly that "whoever wants to nourish

demonstrations in the classic geomet­

his mind on the mystical philosophy . . .

ric manner of Euclid. He did this for deep philosophical reasons.

As

will not fmd in my book what he is

he

looking for." [KGW 6.397; Fie2, Ros]

wrote: "Geometry is unique and eter­

The "Geometrical Kabbala" he men­

nal, a reflection from the mind of God.

tions was not finished or has disap­

That mankind shares in it is because

peared, but perhaps he is referring to

man is an image of God." [Cas 380] He

some geometrical algebra of the sort

was also fascinated by the new tech­

that Descartes was later to introduce.

niques of logarithms, which he inde­

Kepler's reasoning is especially in­

pendently rederived and extended in

teresting: symbolic mathematics, while

Chilias logarithmorum (1624).

it may reveal hidden interconnections,

his

Column Editor's address:

all kabbalists." [Cas 292] Kepler noted

mathematical

He did this, however, in a Euclidean

cannot elucidate the inner nature of

Faculty of Mathematics, The Open University,

way. Although Kepler was aware of

things in themselves. This goes far be­

Milton Keynes, MK7 6AA, England

Franc;ois Viete, the "father of modem

yond his dislike of esoteric mystifica-

54

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

tions; it reflects his Platonic critique of mathematics. Plato had argued that mathematics restricts itself from a full in­ quiry into the merits of its own axioms, preferring instead to deduce further theorems and consequences of given pre­ suppositions. Accordingly, Plato assigns the highest level of knowledge not to mathematics but to philosophy, which can inquire into the good of the axioms that mathematics takes as given. [Republic 509d-51le]. However, Kepler treats mathematics, particularly geometry, as a way of gain­ ing access into nature's secrets, not merely deducing con­ sequences of unexamined axioms. Here Kepler may follow other Platonic examples (notably the Timaeus), which use geometric hypotheses as the staging ground for shaping a "likely story" to explain astronomical appearances. Beyond these Platonic recollections, though, Kepler had another important point. In his view, mathematical science works by making connections between diverse phenomena, rather than by penetrating to what lies behind the phenomena His insight anticipates Newton's avoidance of "hypotheses" in favor of faithful mathematical prediction. Kepler had explicitly responded to Petrus Ramus's challenge to find an "astronomy without hypotheses"; in The Secret of the Universe (1596), Kepler claimed that he (and Copernicus) had answered this challenge, a claim he re­ peated in the New Astronomy (1609) and Rudolphine Tables (1627). [HW xi-xii] In contrast, elsewhere in his work, Kepler did try to form fundamental hypotheses about the cause of gravity, viewing it as closely analogous to a magnetic force. Indeed, in the New Astronomy (1609) such hypotheses were helpful to Kepler as he formulated his laws, if only as heuristic devices that allowed him a dif­ ferent, fruitful perspective on a familiar problem. [NA 271-280] Kepler drew much from the magnetic analogy, and yet the quote we have been considering shows that Kepler was able to take a more skeptical, detached view of his hypothesis, at least as a mathematical construction. This may also have been Kepler's way of indicating the pri­ macy of physical considerations (the analogy with mag­ nets) over mathematical representations (the way that analogy is deployed in his astronomical work). Algebra and the Heptagon

There is only one extended passage using algebraic sym­ bolism and methods in Kepler's works; it shows both Kepler's familiarity with the "cossa" (as he called algebraic technique) and his arguments against it. The passage oc­ curs in Book I of The Harmony of the World (1619), in the context of his general discussion of the construction of reg­ ular figures. [HW 60-79] Kepler uses the case of the regu­ lar heptagon to examine the claims of algebra; in contrast to Cardano's approach to this problem, Kepler adhered to the ancient Greek notion of numbers as integers (counting numbers rather than "real" numbers, in the modern sense). [Kl, Fie3] Kepler's criterion is what he calls "knowability," correlated to the degree of irrationality of different regu­ lar polygons, following Euclid's Elements Book X. Kepler also set out the degrees of knowability, beginning with the first degree (a line equal to the diameter of a given circle,

or a surface equal to the square on that diameter). He be­ gins with a theorem asserting that it is not possible to con­ struct regular polygons with a prime number of sides greater than five; "it is on account of this result that the

Heptagon and other figures of this kind were not em­ ployed by God in ordering the structure of the World, as He did employ the knowablefigures explained in our pre­ ceding sections." Though Kepler's result was later super­ seded by Gauss's proof that polygons of sides 22n + 1 can

be constructed, his point about the heptagon remained. Here Kepler contradicts the claims of Cardano that the hep­ tagon (having as many sides as there were known planets) has cosmological significance. At this point, Kepler brought up the work of Jost Btirgi (1552-1632), a notable clock-builder and instrument­ maker, as well as a skilled mathematician, who used alge­ bra to calculate the side of the heptagon. Kepler went through Btirgi's argument in detail, using Kepler's own sim­ plified version of the Italian notation for the unknown (res or "cossa") and its powers: i, ij, iij, iiij, v, vj, . . . (which later writers would call x, x 2, x3, x4, x5, x6, ). Burgi had used a more awkward !Gillan notation; Kepler showed ingenuity in devising an apt symbolism reflecting his close study of what he calls Btirgi's "very ingenious and surpris­ ing achievements in this matter." [KGW 6.527-528] Kepler analyzed Btirgi's equation of the heptagon to show that it does not amount to a construction of the figure by com­ parison with the geometrical construction of the pentagon. He then took Burgi's result for the side of a pentagon (the root of the equation 5j - 5iij + 1v, which in modem form is 5 5x2 + x4 = 0, quadratic in x 2), and noted that •

•

.

-

Again, as for the heptagon, this does not tell us how to construct the continuous proportion for which this rela­ tionship will hold, nor does it express the lengths of the proportionals in terms of things already known, but it tells us, once the [system of continuousj proportion is set out, what relationship will follow. Kepler may have been the first (1619) to advance such a critique of Viete's symbolic mathematics, the "analytical art," a critique often ascribed mainly to Thomas Hobbes (1656), but also closely related to Newton's reservations about algebra. [Pyc, Pes3] Yet Kepler's reliance on geo­ metric construction still leaves open the possibility that al­ gebraic solutions might "express the lengths of the pro­ portionals in terms of things already known" in some other way. As J. V. Field points out, Cardano had already con­ sidered geometric and algebraic proofs to be interchange­ able in 1570. [Fie3] Kepler knew that algebraic equations can be solved ap­ proximately; he gave approximations for the regular nine­ and eleven-sided polygons and used approximation tech­ niques in his astronomical calculations. He did not con­ sider such approximations as knowledge, and (despite his keen awareness of their numerical utility) he denied them any fundamental status. Yet the equations that can be de­ rived by Btirgi's methods for the side of the pentagon (x4 -

VOLUME 22, NUMBER 4, 2000

55

5x2 + 5 = 0) and octagon (x4 - 4x2 + 2 = 0) are soluble, for they are quadratic in x2, as are the equations for the square (x4 2x2 = 0) and triangle (x4 - 3x2 = 0). Furthermore, the equations for the hexagon (x 3 - 3x + 2 = 0) and heptagon (x6 - 7x4 + 14x2 - 7 = 0) are cubic in x and x2, respectively. Kepler refers to Cardano, and it is likely that Kepler was aware of the solutions of Cardano, Tartaglia, and Ferrari for cubic and quartic equations. If so, he would have known that the heptagon equation should have a solution in closed form involving square and cube roots (the solution of the hexagon's cubic equation is sim­ ple (x = 1) because of its close relation to the triangle). Kepler could then have regarded this solution of the hep­ tagon proplem as yielding "knowledge" even according to his own definition: "T() know in geometry is to measure in terms of some known measure. In this matter of inscrib­ ing figures in circles the known quantity is the diameter of the circle." [HW 18] After all, the solution x of the hepta­ gon equation does measure its side in terms of the diame­ ter of the circle, taken as equalling 2. Field notes that "it is rather difficult to convince oneself that he is not putting arbitrary limits to God's powers by restricting Him to us­ ing only a straight edge and compasses." [Fie1 122] Yet the fact that Kepler brings this algebraic treatment forward shows that he considered that it might offer a different kind of "knowability" besides that of geometry, for he notes that "in this art the sides of all kinds of Polygon seem to be de­ terminable." [HW 66] In contrast, Galileo's complete si­ lence on algebra shows that he did not know of it or dis­ dained to compare it to synthetic geometry. [Boy1] A closer examination of the solution of this equation shows, however, that Kepler realized an important limita­ tion of algebra. Applying the Cardano-Tartaglia method to the heptagon equation yields -

Cardano himself treated quantities like � as "false" or "fictitious" (ficta) or "sophistic negatives." [Car] However, Field notes that Cardano's treatment of the heptagon is flawed by incorrect and conflicting equations. [Fie3 232-235] Rafael Bombelli (1572) had been able to extract the cube roots of certain complex numbers that arise in special cases of the cubic equation, and, by so doing, to eliminate the imaginary parts. [Boy2, Wae] However, that cannot be done in general, or even in this specific case. [Tig] If Kepler reached the above solution for x, he may rightly have been perturbed by the � In the general irreducible case, it turns out that an attempt to take this cube root by purely algebraic means will lead back to the very same cubic equation from which the sought-for cube root arose in the first place! Extracting the cube root of a complex quantity requires the use of trigonometry and the n De Moivre identity, (cos e + i sin e) = cos n e + i sin ne. Without the use of transcendental functions, the cubic equation remains generally insoluble, if one demands that the cube roots of complex quantities be expressible in .

56

THE MATHEMATICAL INTELLIGENCER

terms of their real aKd imaginary parts. After using the De Moivre identity, the solution turns out to be real and pos­ itive, x 0.8678. . . . If "knowing the solution" means de­ termining its value only through taking roots, without us­ ing trigonometric functions, the side of the heptagon is indeed not knowable. Here Kepler also pointed towards a limitation of the al­ gebraists' program. Viete proudly claimed that his analytical art solved "THE PROBLEM OF PROBLEMS: TO LEAVE NO PROBLEM UNSOLVED," including unknowns raised to ar­ bitrarily high powers. [Pes2, Pes3] Kepler raised important objections well before Descartes's La Geometrie (1635), which seems to sidestep these difficult questions of whether indeed all equations are soluble, and whether transcenden­ tal curves can really be invoked in the same way as algebraic curves. Certainly the Cardano-Tartaglia solution should be a crucial instance of algebraic solvability, but Kepler indicated that, despite their claims, the algebraists have not even solved the cubic equation in general. This objection goes beyond the formal issue of whether algebra as an analyti­ cal art can treat an unknown as if it were already given. Kepler might have allowed this analytical method, but he would not allow algebra silently to invoke infinite processes in its solutions. He objects to the algebraists' claim to have achieved trisection on just these grounds: "as rough and unshaped matter is to something which has form and as an indeterminate and indefinite quantity is to a fig­ ure, so also is the analytic method to geometrical deter­ mination . . . , " while admitting that "this Analytic of Biirgi' s tells us something general, not only about these two un­ equal chords but also about many other chords of a circle, which is useful for expressing [their lengths] in numbers." [HW 83-84] He especially objected that =

I am required

to pass over by a single act or motion some­ thing which potentiaUy involves infinite division; so that by this passage something may be attained which is con­ cealed in that potential infinity, without the light ofper­ fect knowledge, which the problems the ancients dubbed Plane do have. This kind of postulate is used frequently by Franyois Viete, a Frenchman, and Dutch Geometers of our day, in solutions of their problems, which by their very nature are not soluble except in a way that goes against the rules of the art, such as numericaUy or by Geometrical motions whose changes need to be guided by some kind of infinity. [HW 87-88] This makes explicit his objection to the hidden recourse to infinite processes implicit in modem algebra and its tran­ scendental firnctions. "We correctly maintain that the side of the Heptagon is among Non-Entities and is not suscep­ tible of knowledge. For a formal description of it is im­ possible; thus neither can it be known by the human mind, since the possibility of being constructed is prior to the possibility of being known: nor can it be known by the Omniscient Mind by a simple eternal act: because by its na­ ture it is among unknowable things." In a marginal note, Kepler emphasized that "these formal ratios of geometri-

cal entities are nothing else but the Essence of God; be­ cause whatever in God is eternal, that thing is one insepa­ rable divine essence." Kepler did not admit the possibility that God might grasp the heptagon through an infinite process, for he would not allow that divine geometry could diverge from the finite perfec_tion of a Euclidean proof.

the ancients did. They advanced to a certain point by the judgment of their ears, but soon abandoning their lead­ ership completed the rest of the journey by following er­ roneous Reason, so to speak dragging their ears astray by force, and ordering them outright to turn deaf [HW 1 64-165]

Though this precludes the powerful achievements of mod­ em algebra, it is evidence of Kepler's conviction that in

If Kepler were to acknowledge the algebraic solution of

geometry the human mind merges with the divine, as both

the heptagon as legitimate knowledge, he would have to

grasp knowledge in one "simple eternal act."

admit into music highly dissonant intervals such as 3:7 de­

The Evidence of Music

rived from the pentagon. Indeed, Kepler had had to strug­

rived from the heptagon, just as the major sixth (3:5) is de­

Kepler found confirmation of these opinions in his musi­

gle with the case of the pentagon, whose side he considered

cal experience. His musical judgments speak decisively

knowable only in the eighth degree

against algebra, and they rely on contemporary sensibili­

while the hexagon is knowable in the second degree

ties rather than the best known ancient authority. He ex­

(x =

Y5 -

v'5!

v'2), (x =

1). [HW 53] Here Kepler parted company with Ptolemy,

plicitly considered himself as both restating ancient har­

whom he often praised, for "Ptolemy still denies that the

monic knowledge and also reforming and perfecting it.

thirds and sixths, major and minor (which are covered by

[Wal] Although he had originally planned to publish a trans­

the proportions 4:5, 5:6, 3:5 and 5:8) are consonances,

count superseded it. Specifically, Kepler followed the prac­

are. On the other hand he accepts the proportions 6:7, 7:8,"

tice of his time in adopting just intonation, in which the

which would emerge fronrthe heptagon, intervhls that are

far more complex ratios assigned these intervals by

singing, even though it may be possible for strings to be

lation of Ptolemy's Harmonics, he decided that his new ac­

major third is 4:5 and the minor third 5:6, rather than the

which all musicians of today who have good ears say they

"utterly abhorrent to the ears of all men and the usages of

Boethius and Macrobius (64:81 and 2304:2144). These pre­

tuned in that way, seeing that as they are inanimate they

eminent ancient authorities (the only sources known to me­

do not interpose their own judgment but follow the hand

dieval musical theorists) accordingly classified both these

of the foolish theorist without the least resistance." [HW

intervals as dissonances (along with the major and minor

138]

sixths). Kepler knew that these intervals were treated as

Geometry as the general science of magnitude is un­

consonances (albeit imperfect ones) by contemporary

doubtedly prior to harmony, both in Boethius's account

composers and by the theorist Gioseffo Zarlino (1558), and

h� gave them precedence over Boethius.

Ptolemy had also defmed the thirds as 4:5 and 5:6, in

contrast to Boethius; though Kepler in this matter agrees

.

and Kepler's. This does not mean, though, that harmony merely follows in geometry's traces. Harmony involves sim­ ple ratios, and so excludes the complex ratios and incom­ mensurables that abound in geometry. In contrast to an­

with Ptolemy, on other points he objects to the "poetic or

cient numerological accounts, Kepler set for himself the

rhetorical rather than philosophical or mathematical" char­

problem of accounting for the harmonic ratios in terms of

acter of Ptolemy's "symbolism." [HW 499-508] Here Kepler

the properties of geometrical figures, not relying on vague

took a position in a lively contemporary debate, for

assertions of the qualities of the smallest integers. [HW 137]

Vincenzo Galilei (Galileo's father) had attempted to return

Harmony is the goal toward which his geometrical expo­

to the ancient defmition of the major third (64:81) in order

to recapture the lost powers of ancient music. Kepler knew

sition aspires. He was, after all, discoursing on the har­ mony of the world, notjustits geometry, and this harmonic

and quotes Galilei, but clearly was guided by Zarlino and

goal deeply informs his proceedings. Here, as elsewhere in

the empiricism of just intonation. At several points, Kepler

his works, Kepler approaches empirical data "theory­

emphasizes that the test of consonance is not reliance on

laden," as Curtis Wilson has put it. [Wil]

ancient authority but rather on the judgment of the ear; he

The Harmony of the World but also as early as the The Secret of the Universe (1596): expressed this fundamental principle not only in

Therefore even by reference to the sole evidence ofmy book the hearing is sufficiently for­ tified against the distraction oj the sophists, and those who dare to disparage the trustworthiness of the ears on very minute divisions, and their very subtle discrimi­ nation of consonance-especially since the reader sees that I followed the evidence of my ears at a time when, in establishing the number of the divisions, I was still struggling over their causes, and did not do the same as

The Secret of the Universe

At the beginning of the

Harmony of the World Kepler

reminded us that "I am not a geometer working on philos­ ophy, but a philosopher working on this part of geometry."

[HW 14] As Walker emphasizes, Kepler "always gave ab­

solute priority to empirical evidence; if the theoretical pat­ tern, however beautiful, did not fit the facts, it was dis­ carded." [Wall The sensual was crucial for Kepler, as in his description of musical cadence as a kind of orgasm. [HW

242]

Kepler was clearly expressing his own strong reactions to the polyphonic music of his time, which "with its thirds and sixths, excites and moves us deeply as does sexual in­ tercourse because God has modelled both on the same geo­ metric archetype," as Walker puts it. Field considers that

VOLUME 22, NUMBER 4, 2000

57

Kepler's musical preferences were "on the side of ortho­ doxy rather than standing up to be counted as a partisan of the avant garde," namely Claudio Monteverdi and the school that followed Vincenzo Galilei. [Fie3] Though that is in many ways true, by adhering to just intonation Kepler made a -pronounced departure from the best-known an­ cient sources. Kepler explicitly turned to "modem figured music" for the combined motions of the planets, even though at one point he compared the individual planets to the [monophonic] choral music of the ancients. [HW 430] No less than Vincenzo Galilei, Kepler was engaged in a di­ alogue between ancient and modem music. In the end, Kepler asserts that

etary theory. It is not coincidental that his rejection of al­ gebra agrees with his geometrical predilections; such judg­ ments are often overdetermined. As Kepler weighed the possibilities of algebraic knowledge, his musical judgments guided his mathematical choices. He must have been deeply moved that his intercourse with sensuous har­ monies converged with the inward rapture of Euclidean geometry. REFERENCES

[Boy 1 ] Carl B. Boyer, "Galilee's place in the history of mathematics" in Galileo Man of Science (Eman McMullin, ed.), New York: Basic Books

(1 967), 232-255. [Boy2] Carl B. Boyer, A History of Mathematics, 2nd ed., New York:

it is no longer surprising that Man, aping his Creator, at last found a method of singing in harmony which was unknown to the ancients, so that he might play, that is to say, the perpetuity of the whole of cosmic time in some brieffraction of an hour, by the artificial concert of several voices, and taste up to a point the satisfaction of God his Maker in His works by a most delightful sense of pleasure felt in this imitator of God, Music. [HW 447-448} has

John Wiley (1 991), 287-289. [Car] Girolamo Cardano, Ars Magna (T. Richard Witmer, tr.), New York: Dover (1 993), 1 2-22, 21 9-221 . [Cas] Max Caspar, Kepler, New York: Dover (1 993). [Cox] H. S. M. Coxeter, "Kepler and Mathematics," Vistas in Astronomy 1 8 (1 975), 661 -670. [Fie1 ] J. V. Field, Kepler's Geometrical Cosmology, Chicago: University of Chicago Press (1 988). [Fie2] Judith V. Field, "Kepler's rejection of numerology, " in [Vic 273-296].

Kepler's decision to follow the evidence of the senses is important; it led him to search for an account of thirds and sixths that shows their intelligibility in terms of the proportions of the pentagon. It is consonant with his re­ liance on precise astronomical observations and physical data elsewhere in his work, even where those data chal­ lenged his assumptions. It is also consistent with his stress on the moral and political effects of music. Kepler referred to ancient discussions of the ethical implications of music, but relied on his own judgment. He refers to the political writings of Jean Bodin on this matter, but is careful to correct what he considers Bodin's misunderstandings of harmonic ratios. [HW 255-279] Beyond the technical defi­ nitions of the different kinds of proportion, Kepler's over­ riding concern was to show how the harmonic mean is far more appropriate than the arithmetic or geometric means in matters of justice, friendship, love, punishment, repara­ tion, and the proper tempering of the state. These matters open a larger perspective on Kepler's whole project. Gerald Holton has emphasized that "so intense was Kepler's vision that the abstract and concrete merged." [Hol] In his most daring insights, Kepler joined astronomy with terrestrial physics, bridging the heavenly and the sensuous; likewise, he considered music the sexual congress of the soul with mathematical forms. Given this underlying musical agenda, it is more under­ standable why Kepler was distrustful of algebra, for it con­ flicted with the sensual evidence that Kepler took as fun­ damental. If his task was to render the harmonic intervals intelligible, Kepler had to throw out algebra, which would have validated the dissonant heptagon. Considered this way, Kepler's treatment of algebra can be viewed as his way of keeping faith with the exact data of hearing, just as he kept faith with Tycho's precise observations in his plan-

58

THE MATHEMATICAL INTELLIGENCER

[Fie3] J. V. Field, "The relation between geometry and algebra: Cardano and Kepler on the regular heptagon" in Girolamo Cardano: Philosoph, Naturforscher, Arzt (E. Kessler, ed.), Wiesbaden: Harrassowitz Verlag

(1 994), 21 9-242.

AU T H OR

PETER PESIC

St. John's College

Santa Fe, NM 87501 -4599 USA

e-mail: [email protected]

Peter Pesic received his doctorate in physics at Stanford. he has been at St. John's College in Santa Fe since 1 980, where he is Tutor and Musician-in-Residence. He writes on the his­ tory and philosophy of science. His book Labyrinth: A Search for the Hidden Meaning of Science has just been published

by MIT Press. As a concert pianist, he has performed cycles

of the complete sonatas of Beethoven and Schubert, and he

has underway a cycle of J.S. Bach's complete keyboard works.

[Hoi] Gerald Holton, "Johannes Kepler's Universe" in his Thematic Origins of Scientific Thought, 2nd ed. , Cambridge,

MA: Harvard

University Press (1 988), 69-90.

Entanglements, Cambridge: Cambridge University Press (1 997), 1 35-

1 48, 1 67-208. [Rit] Frederic Ritter, "Franr;;ois Viete, inventeur de l'algebre moderne,

[KGW] Johannes Keplers Gesammelte Werke (Max Caspar et. a/., eds.),

1 540-1 603," Revue Occidentale 1 0 (1 895), 234-274, 354-415.

Munich: C. H. Beck (1 934-), giving volume number and page.

[Ros] Edward Rosen, "Kepler's attitude towards astrology and mysti­

[HW] Johannes Kepler, The Harmony of the World (E. J. Aiton, A. M.

cism" in [Vic 253-272].

Duncan, and J. V. Field, trs.), Philadelphia: American Philosophical

[Swe] Noel M. Swerdlow, "The Planetary Theory of Franr;;ois Viete 1 .

Society (1 997).

The Fundamental Planetary Models," Journal for the History of

[NA] Johannes Kepler, New Astronomy (William H. Donahue, tr.),

Astronomy 6 (1 975), 1 85-208.

Cambridge: Cambridge University Press (1 992).

[Tig] Jean-Pierre Tignol, Galois ' theory of algebraic equations, New

[KI] Jacob Klein, Greek Mathematical Thought and the Origin of Algebra

York: Longman (1 988), 21-30.

(Eva Brann, tr.), New York: Dover (1 992).

[Vic] Occult and scientific mentalities in the Renaissance (Brian Vickers,

[Pes 1 ] Peter Pesic, "Franr;;ois Viete, Father of Modern Cryptanalysis­

ed.), Cambridge University Press (1 984), 253-296.

Two New Manuscripts," Cryptologia 2 1 (1 ) (1 997), 1 -29.

[Wall D. P. Walker, "Kepler's Celestial Music," in his Studies in Musical

[Pes2] Peter Pesic, "Secrets, Symbols, and Systems: Parallels between

Science in the Late Renaissance, Leiden: E. J. Brill (1 978), 34-62.

Cryptanalysis and Algebra, 1 580-1 700," Isis 88(4) (1 997), 674-692.

[Wae] B. L. van der Waerden, A History of Algebra: From ai-Khwarizmi

[Pes3] Peter Pesic, Labyrin th: A Search for the Hidden Meaning of

to Emmy Noether, Berlin: Springer-Verlag (1 980), 60-62.

Science, Cambridge, MA: MIT Press (2000).

[Wil] Curtis Wilson, Astronomy from Kepler to Newton, London:

[Pyc] Helena M. Pycior, Symbols, Impossible Numbers, and Geometric

Variorum (1 989).

Revisit the Birth of Mathematics . . . EU C LI D

VOLUME 22, NUMBER 4, 2000

59

Goursat, Pringsheim, Walsh, and the Cauchy Integral Theorem

The Cauchy Integral Theorem

The Cauchy Integral Theorem is the gateway to complex function theory. It is used to connect complex functions abstractly defmed with their power se­ ries expansions; it marks one of the ways in which complex function the­ ory distinguishes itself from the theory of functions of two real variables. In any modem presentation of the subject it says: Theorem: If a function f(z) is ana­ lytic in a domain D containing a re­ gion T, and C is a simple closed curve in D that is the boundary of T, then

J f(z)dz

Jeremy Gray

c

= 0.

Here, as usual, the integral along a path C in the plane of complex numbers is defmed to be

L f(z)dz C

=

f f(z(t))z(t)dt, a

where the path C is a piecewise con­ tinuously differentiable arc with equa­ tion z = z(t), a :::;; t :::;; b. The Cauchy Integral Theorem has a long history, and was given many proofs in the 19th century, starting, happily, with those by Cauchy. There are a number of obscurities attending its birth. It is not clear what sort of a path was contemplated-Cauchy him­ self changed his mind. Today, it is common to see the theorem stated for piecewise continuously differentiable curves but accompanied by remarks about its validity for any rectifiable curve homotopic to the given one. This is correct, but the integral needs to be defmed in this more general setting. As for proofs, it is possible to argue by in­ fmitesimal variation of the path (Cauchy's method), to appeal to gen­ eral theorems in the calculus of varia­ tions, to invoke Green's Theorem (Cauchy again, and, perhaps indepen­ dently, Riemann, whose method was followed by several German authors), or to argue ad hoc. Throughout the 19th century, all of these approaches were tried, and all were found problematic. In each case tacit or explicit use was made of the extra hypothesis that the derivative f(z) is itself continuous. As every modem text reveals, an essential

60

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

simplification was found by Goursat, who showed that it was unnecessary to assume that the derivative is con­ tinuous. As many a not-so-modem text reveals, the actual nature of Goursat's achievement is more complicated. Briot and Bouquet

We can start our story a mathematical generation before Goursat, with the French mathematicians Charles Auguste Briot and Jean Claude Bouquet. They were the first to write a book on the the­ ory of elliptic functions which attempted to derive that theory from Cauchy's the­ ory of complex functions. Their [1859] was a success, and they brought out a greatly revised and enlarged second edi­ tion in 1875. Only in this book did they attempt to prove the Cauchy Integral Theorem, and their argument was rather ingenious, if at times imprecise. They considered a star-shaped re­ gion, R, of the plane, and supposed without loss of generality that any point z in the region can be joined to the origin by a line segment lying en­ tirely inside R. They then argued by comparing the integral A: = with the integral A ' : =

fJCz)dz

J f(z)dz, C'

where

the contour C' is obtained from the contour C by scaling it by a factor a, and also lies inside R. They let the con­ tour C be defined by the function acf>(t) = z, 0 :::;; t :::;; l. As a increases from a to a' = a + Lla, the contour moves from Ca defmed by acf>(t) = z to C� defmed by a'(t) = z', and the differ­ ence in the corresponding integrals, A' - A =

L f(z)dz -J f(z)dz, vanishes c�

c;,

with Lla. They deduced that the integral

J f(z)dz Ca

is acontinuous function

of a, which they denoted �(a). Then they considered the difference quotient A' - A Lla =

They

I: (

a

f(z) � + f(z')) '(t)dt. Ll

f(z')

that

noted

f(z') - f(z)

z' - z

z'

-

z

• ---. A

ua

f(z') - f(z)

Now the function

e

f(z) is holomorphic, so it has a derivative f' (z),which f(z') - f(z) means that the quotient , tends to a limitas z' z -z f(z') - f(z) = f ' (z) tends to z. So e,where e tendsto zero z' - z with Moreover, J(z') = f(z) e', where e' tendsto zero

+

a

.:la. + f(z') - f(z) , With u.a, so one can wnte a Aa + f(z ') =F' (z) + e , where F(z) = zj(z) is a holomorphic function and e" = e + e'. •

d

c

•

A

Consequently

A' - A -- =

.:la

Jo

z

b

(F' (z)

+ e")f/>'(t)dt.

then a similar argument shows that

However, the integral ofF' along a closed crnve vanishes, so

A' - A

�=

Jo

z

i t.

'

e" f/>'(t)dt.

.:la tends to zero, so does the integral on the right hand side, and so the real function �:p( a) is differentiable with differential zero. Consequently the function cp(a) is constant, and its value is arbitrarily small when a is small, But as

c

so this value must be zero. The Cauchy Integral Theorem

is therefore proved.

In view of discussions later in this paper, the reader may er\ioy discovering where the above proof relies, tacitly, on the continuity of the derivative of the function!

[ It

f(z)dz =

'

ei(z - Zi)dz

[

< 4eiaiv'2

+ eiAv'2.arc(ae).

Adding these two kinds of contribution up, one fmds that

[ Jci

[

f(z)dz < 'Y/ v'2 (4 A'

+ AS),

where 'Y/ is the maximum of the ei, A ' is the area of the squares inside

C,

and

S is the total length

of

C.But this

quantity can be made arbitrarily small, and so the Cauchy Integral Theorem is proved. At least, that is what Goursat claimed. But there is a gap, which the diligent reader may have spotted, and which Goursat went on to discuss in this fashion. In truth, he said,

the proof supposes that the length A can always be taken

Goursat's proof of 1 884

Goursat, quite correctly, saw his result as illuminating

the general definition of a complex function, and he sug­ gested that it just used the mere definition of the deriv­ ative, and the fact that the theorem is true for

fc zdz.

fc dz and

He argued that a region A bounded by a simple

or multiple contour

C of finite length can be

broken up

into squares of side-length A by two families of parallel lines. Let Ci = abed be one of these squares, of area ai = A2, and zi a point inside it. If the square lies entirely in­ side C, then from the definition of the derivative it fol­ lows that

this is true, C, then

because if the derivative is continuous in A and on

lhl

given any (J" > 0 there is a 8 > 0 such that < 8 implies - f(z) f(z - f' (z) < (J" for all z inside or on C.This

l

+ h) h

[

being the case, it is enough to take A < &v'2, and the the­ orem is proved "with complete rigour." Later readers were not to be so indulgent.

Pringsheim's first critique·

f(z) - f(zi) Z - Zi

= f ' (z,) ·

Goursat's proof, and many of its predecessors, were criti­

+ e,,.

cised by Alfred Pringsheim [1895a]. He objected to the as­ sumption that Cauchy's Theorem was valid for the simple

Ic

f(z)dz = [f(zi) - zif' (zi)l

Jci

+ f' (zi)

Ic

integrals zdz, because this required a limiting dz and argument no simpler than the general one he went on to

so

Jci

small enough so that all the Bi can be made less than some arbitrarily small number given in advance. But

present. The problem was that earlier writers, and Goursat

dz

Jci

zdz

+

Jci

in particular, had assumed that the differential quotient ei(z - zi)dz.

Of these integrals, the first two gn the right-hand side van­ ish trivially, and the remaining one satisfies

+ h) h

f(z - f(z) :.._:______:_::___:_ :_ _ _ tended uniformly to the derivative!' (z) for all z in the region T bounded by the path C.But this must be proved, and it turns out to be equivalent to the conti­ nuity of the derivative. Goursat's casual assumption at the

end of his paper is equivalent to the uniform differentia­ bility of the function!

If the square contains part of the boundary, as in this fash­ ion,

Pringsheim's argument was very careful, entirely gen­ eral, and backed up by a wide-ranging historical analysis (quoted partially above). He started by defining a path in-

VOLUME 22, NUMBER 4, 2000

61

tegral in a way that did not require the path to be differ­ entiable, merely continuous. In particular, there was no re­ striction to rectifiable curves. Instead, he let P(g, YJ) be a single-valued function defmed on a curve C = { (g, YJ) : YJ = c/>(0) inside a region T. The curve has merely to be the im­ age of a -real interval under a continuous function cf>. He then defmed the path integral as follows:

He then showed how to establish the Cauchy Integral Theorem for regions bounded by curves which are step­ shaped (made up, piece-wise, of curves that are parallel to the coordinate axes). He then extended the proof to deal with continuous curves, C, that are the limits of step­ shaped curves, using im analogue of the mean value theo­ rem. He also considered when an integral of the form r (x,y) ),(xo,yo) P(g,YJ)dg + Q(g,YJ)dYJ is independent of the path in a simply or multiply connected region, and found that the

aP aYJ

aQ ag

condition - = - was necessary. In a second paper later the same year [ 1895b], he cleared up the relationship between the derivative and the differ­ ential quotient, by giving a simple proof that continuity of f'(z) implied that the differential quotient converged uni­ formly tof'(z). So, he said, Goursat's achievement was only a simpler proof of a major result, not a weakening of its conditions. Three years later Pringsheim [1899a, b] returned to the question, and sought to investigate what happens when the function j(z) fails to be analytic at a number of points, in­ deed, even on a 2-dimensional set of points. He found that there were cases when the Cauchy Integral Theorem still held. He therefore concluded that it might be the case that the mere existence of the derivative, without any assumption of continuity, was enough to establish the Cauchy Integral Theorem. As long as this possibility was not excluded by counter-examples, the question, he said, must remain open. The American debate

Pringsheim's papers provoked a flurry of comments in America. Bocher had already taken up the issue in his [1896]. Now Goursat, at Osgood's request, repeated his proof in the first issue of the Transactions of the American Mathematical Society. In fact, the proof was more so­ phisticated this time around. He now said that the closed contour C satisfies condition a with respect to a number e > 0, if there is a fixed z ' inside or on C such that

i f(z) - j(z') - (z - z') f'(z') j < lz - z' l e as z describes the contour C. He then established what came to be called Goursat's Lemma: given any e > 0, any region T bounded by a simple closed contour can be di­ vided into portions satisfying condition a with respect to

1Jordan [1 893], §§ 1 93-6.

62

THE MATHEMATICAL INTELLIGENCER

the number e. The proof was by contradiction. If the claim is false, successive subdivisions of T can be made yielding a sequence of subregions that never contain a region ap­ propriately bounded. But any sequence so obtained con­ verges to a limit point at which, however, the functionj(z) is differentiable. This implies a contradiction. The proof of the Cauchy Integral Theorem followed im­ mediately. The given region T is divided up into congruent squares so small that condition a applies to them for an ar­ bitrary but fixed e > 0. Condition a allows the integral to

be estimated by estimating values of

Jdz J d

and z z, which

were bounded by the perimeter and area of the squares. But for them the theorem is trivially true, and so, allowing e to tend to zero, the full theorem is established. In the same issue E.H. Moore gave his own proof. He defmed a continuous curve as the image in the complex plane of an interval under a continuous function, and

J

c

j(z)dz as the limit as 8k � 0 and

n

I,J(?k)(zk + l - zk),

k =O

n -"> oo

of the sum

where Zn + l = zo, ?k is any point on

the arc (zk, zk+ I), and 8k = tk + l - tk. He observed that the path integral exists if the curve is rectifiable, quoting a the­ orem in Jordan's Cours d'analyse to that effect. 1 In a foot­ note he said he had done this because Pringsheim [1895] had proved what were special cases of this result, and "seems to be unfortunately out of touch with the current no­ tion of the general rectifiable curve" as treated by Scheeffer, Ascoli, and Study. Moore now proved a theorem about any single-valued functionf which is continuous and has a single-valued de­ rivative everywhere inside and on a region R bounded by a closed, continuous, rectifiable curve C, subject to the fol­ lowing conditions: 1) the curve C meets curves parallel to the x and y axes in only fmitely many points; and, to sim­ plify the proof, 2) if a sequence of squares whose sides are parallel to the x and y axes converges to a point A on C, then the ratio of the total lengths of the arcs of C inside the squares to the perimeter of the squares is ultimately less than some constant Pc which may vary as ? traverses C (for the usual curves considered, P( = 1 for all points ?).

Theorem

J f(z)dz c

(Moore [1900}) Under the above conditions,

=

0.

The proof was the usual proof by contradiction, the various hypotheses being introduced to guarantee the existence of suitable estimates. The observation that for each z E R, f(z) = fW + (z - 0 f'W + Ll(z), where a(z) < e z wherever z is within a suitably small distance of ?, reduced the evaluation of the integral to estimating sums of inte-

j

J

j

j

�

grals of the form Ll(z)dz around suitable contours. Con­ ditions (1) and (2) control the lengths of the parts of the curve C to be considered and the behaviour of Pc· A com­ pactness argument is at work here under the surface.

The Cauchy Integral Theorem followed immediately from Moore's theorem. As he observed, requiring the boundary curve to be rectifiable allowed him to avoid Goursat's Lemma. Pringsheim's second critique

In May 1901 Pringsheim presented his reply to the Amer-ican Mathematical Society at its meeting in Ithaca; it was pub­ lished in the second volume of the Transactions. He certainly did not agree that he was "out of touch." As a friend of the late Ludwig Scheeffer he could claim, he said, to be as well acquainted with the new ideas as anyone, and he referred any doubting reader to his recent articles in the Encyclopiidie der Mathematischen Wissenschaften (vol 2, p. 41).2 He now objected to Goursat's proof on the grounds that it was in­ cautiously expressed: there was not only no need to use con­ gruent squares, but if one were so restricted then only a re­ stricted class of boundary curves could be admitted. It would be necessary to allow those that were only piecewise monotonic (so their coordinate functions have only finitely many extrema). Moore's condition (1) is insufficient, as the example of y = x 2 sin(1/x) shows, to ensure that small enough squares meet the boundary curve C in at most two points. If that condition is not met, there may be curves that go back and forth through some of the squares. Pringsheim therefore proposed to subdivide only those squares for which Goursat's condition did not already hold, thus adapting the subdivision to the curve at hand, and to exclude curves for which there was no suitable assembly of squares. This gave him a proof of the Cauchy Integral Theorem for rectifiable curves based on his (new) proof of Goursat's Lemma .. Pringsheim returned to the question in 1903, when he gave the proof his obituarist (Perron) was to regard as de­ finitive3. He began by noting that Heffer4 had recently established that the integral

Jc

P(x,y)dx + Q(x,y)dy vanishes

when taken along a closed curve, provided that P(x,y)dx + Q(x,y)dy is an exact differential and satisfies the condition aP aQ - = . This result contains the Cauchy Integral Theorem ax ay as a special case. But Prinsgheim now wished to avoid his earlier use of step-shaped functions, and to give a proof immediately applicable to contours bounded by straight lines, such as triangles. To describe what he did, we need aj aj to explain his notation. He wrote !1 for and f2 for , ay ax and defmed -

f(x,ylxo,Yo) :

- f(xo,Yo) fi (xo,Yo) · (x - xo) - f2 (xo,Yo)

IY - Yol < 8 implies I Jtx,yl xo,Yo) l < eCi x - xol + IY - Yol) . He observed that uniform differentiability was a stronger condition than this. He could now state and prove the fol­ lowing result. Theorem (Pringsheim, [1903]) Let P(x,y) and Q(x,y) be differentiable in the interior and on the boundary of a triangle ..1, and suppose that P2(x,y) = Q1(x,y), then

{

(P(x,y)dx +

Proof First, an observation. Let f be a function differen­ tiable at each point of a domain T. Consider a triangle Ll lying entirely inside T, and defme the integrals (taken in

the positive direction) striction. Then:

L.

·

(y - Yo) .

He said that a function f(x,y) was (totally) differentiable at a point (xo,Yo) if and only if fr(xo,Yo) and f2(xo,Yo) have values there and Ve > 0, 3 8 > 0 such that lx - xol < 8 and

J

Jtx,y)dx and

a

f(x,y)dx

=

L.

J

a

Jtx,y)dy by re-

Jtx,ylxo, Yo)dx

+ (ftxo,Yo) - !I (xo,Yo) Xo - f2 (xo,Yo) Yo)

+ fi (Xo,Yo)

J

But since clearly that

a

L.

:glx 4- f2(xo,Yo)

dx and

J

a

L.

L. ydx

dx

xdx both vanish, it follows

J

ftx,ylxo,Yo)dx + f2(xo,Yo)

J

L. Jtx,y)dy L.

Jtx,ylxo, Yo)dy + fi (Xo,Yo)

L. xdy.

f Jtx,y)dx a

=

a

a

ydx.

Similarly, =

Now, to prove the Theorem, subdivide

Ll into four con­

gruent similar triangles. Pick one for which the integral ftx,y)dy is largest; if this is 11 1 then

It

(P(x,y)dx + Q(x,y)dy)

l

�

± IL.

(P(x,y)dx +

J

a

l

Q(x,y)dy) .

Proceed successively in this manner. One fmds

IL.

(P(x,y)dx +

Q(x,y)dy)

The nth triangle

= f(x,y)

Q(x,y)dy = 0.

l

iJ

n ::::; 4 .:l. n

(P(x,y)dx +

l

Q(x,y)dy) .

Lln has perimeter sn and the perimeters s

halve at each stage, so if Ll has perimeter s then Sn

= 2n .

The triangles converge to a point (xo,Yo) inside or on Ll. Because P and Q are differentiable, for any e > 0, there is an n such that P(x,yl xo,Yo) and Q(x,yl xo,Yo) are each less

2Pringsheim [1 899]. This article, while acute in its criticisms and citing a wide range of recent literature, is about real analysis in general; p. 41 carries a reference to Scheeffer's work but is much more to do with the types of discontinuities a function can have. 3Perron [1 952]. 4Heffter [1 902].

VOLUME 22, NUMBER

4,

2000

63

than e(l x - xol + IY - Yol) for all (x,y) servation applied to P and Q yields

J J

dn

dn

P(x,y)dx Q(FC,y)dy

But P (x,y) 2 so

l{

n

=

=

=

J J

dn

dn

E an. The above ob­

P(x,yl xo,Yo)dx + P2 (xo,Yo) Q (x,ylxo,Yo)dy + Q t(Xo,Yo)

J

Q1(x,y), and

dn

(P(x,y)dx + Q(x, y)dy)

< J E

l

dn

ydx +

J

dn

xdy

l<

dn

dn dn

ydx xdy.

d(xy)

=

0,

Cl x - xol + IY - Yol)(dx + dy) .

If now n is taken large enough so that fx Y(x - xo)2 + (y - Yo)2

J

=

J J

< 8; ,

then

IJ

xol and IY - Yol

<

(P(x,y)dx +

dn

2 n _, 2 S e · !___ 4n 2 By the inequality relating integrals around an and a, one deduces Q(x,y)dy)

n 82 fdn Cldxl + IdYl) < BSn

e

{

(P(x,y)dx + Q(x, y)dy)

<

•

=

es2.

But since e can be arbitrarily small, the sought-for result follows. The Cauchy Integral Theorem follows on letting P and Q be the real and imaginary parts of a complex function f(z); the integrability condition is one of the Cauchy­ Riemann equations. However, Pringsheim pointed out the above proof can easily be adapted directly to the complex case. By the inequality relating integrals around an and a, 4n one deduces that Cz) . Define, as above, fCzlzo)

IL f(z)dzl < ILJ dzl

=

f(z) - fCzo) - f' (zo)' (z - zo) . Then IJCzlzo) l

<

< IL f(z)dzl <

�z - zol for lz - zol

8.

The above argument, combined with a direct proof that

L dz L L -zolldzl < J Vn

e

=

Vn

0

=

lz

other words,

Vn

v

zdz,

now

� 4n , so

f(z)dz

=

L

V

shows

f(z)dz

0.

<

that

Vn

es2 for arbitrary e. In

of Act 1 The route to Goursat's proof is surprisingly intricate, and closely related to what might be called the discovery of continuity: the realisation that once a curve is not smooth but merely continuous many expected properties may lapse, or at least be hard to establish. Familiar examples from the period include the Jordan Curve Theorem and Peano's space-filling curve. The acuity with which Pringsheim pounced on what seemed like a triviality to Goursat is a good example of what has to be done. In fact, the question of how to admit general, continuous boundEnd

ary curves so that the integral story, barely begun in 1903.

64

THE MATHEMATICAL INTELLIGENCER

J

Y

f makes sense is another

Act 2, Pringsheim to Walsh Problems with the Cauchy Integral Theorem flared up again after 1929, when Pringsheim returned to rebut a charge levelled at him by Mittag-Leffler. The point at issue was a published remark of Mittag-Leffler's (in Mittag­ Leffler [ 1923]) that quoted Pringsheim out of context and seemingly in error. Pringsheim wrote to Mittag-Leffler, who agreed he had made a mistake and offered to correct his mistake at the first opportunity. That was in May 1925, but when Mittag-Leffler died in July 1927 restitution had not been made, so Pringsheim took up the issue himself. Mittag-Leffler's mistake had been to confuse Pringsheim's remarks about the proof of the Cauchy Integral Theorem with the statement of the theorem itself. As Pringsheim saw it, the so-called Riemannian proof of the theorem, by a Green's Theorem argument, was due to Cauchy before Riemann, and Riemann should be credited with introduc­ ing the theorem itself into Germany. But Mittag-Leffler had gone on to remind readers that he and others had priority over Goursat. The first was the Swedish mathematician C.J. Malmsten (in Malmsten [ 1865], which I have not seen), then Mittag-Leffler himself (Mittag-Leffler [ 1873], [1875]), and in­ dependently Briot and Bouquet. These contributions seem to have been forgotten, and so he took the occasion of yet another proof appearing (this one by Borel) to remind read­ ers of the earlier work. Indeed, as a young man Gosta Mittag-Leffler had pub­ lished a new proof of the Cauchy Integral Theorem in 1873. That article being in Swedish, he recapitulated the proof in German two years later, in the Gottinger Nachrichten. In 1895, Pringsheim had criticised it for tacitly assuming that + h) - f(z) the quantity - f' (z) converged uniformly

l f(z

I

h

to zero as h � 0. In the letter of 1925, Mittag-Leffler im­ plied that the fault lay in the German translation, and re­ ferred Pringsheim to a new, more accurate French version (of which he enclosed a copy) . Never one to be fobbed off, Pringsheim recruited a Swedish mathematician who spoke good German to make a new translation of the Swedish original. He found that the German edition amounted to the first half of the Swedish version, but where they over­ lapped they had only inessential differences. Both texts agreed in assuming 1) that the function f(x) was (in addition to being finite and continuous) such that it had a single-valued and fi­ nite derivative f'(x), and 2) in making no mention at all of the uniform convergence f(x + h) - f(x) of - f'(x) .

l

I

h

However, in the new French version, and its German trans­ lation, matters were the other way round. Now the above assumption (1) was missing, but assumption (2) now ap­ peared, in the form of an assumption that

lf(ptefhi) -p)elhi - p(elhi - efii I < fCr)(/hi)

(Pt -

fCr)(/h i) - JCpe fii)

e

held uniformly for

IP - Pll < 8, j e - el l < 8, and for all z = � p � R, 0 � 8 < 27T. This as­

peie in the annular domain Ro

-

-

1

1

(11xQ) and lim0 (11xP) exist 8->0 8 8-> 8

provided the two limits lim

sumption crucially makes no reference to the existence and

and are continuous in T. This he showed by vindicating the

equality of the two differential quotients, but only to the

exchange of the limits and integration, thus showing that

uniform vanishing of their difference (from which the

Green's formula was equivalent to the claim that

Cauchy Integral Theorem can be derived). Now, said Pringsheim, a quick look at the proof of

1873

lim

8->0

shows that the assumptions about f' (x) are used only to

� )JT(I (11xQ - 11yP)dxdy u

=

iaT Pdx + Qdy.

establish the equality of this difference in the limit, which

Lichtenstein's crucial insight was that this argument could

means that it would have been enough to assume precisely

be reversed, and Green's formula deduced without requir-

such a limiting equality.

A more precise argument then

shows that it is sufficient to establish this result that the limiting property holds uniformly.

1

and are continuous in T. Instead it was enough to show the

What to make of this muddle? Pringsheim took the shrewd view that in

1

(11xQ) and lim0 - (11xP) exist 8-> 8 8->0 8

ing that the two limits lim -

weaker requirement that

1873 the idea of uniform convergence

and the awareness of its indispensability was not yet in the shared lore of mathematicians. Even Weierstrass, who had led the way in emphasising the importance of the concept, had seen fit to explain the uniform conver­ gence of a sequence of rational functions carefully in a footnote to a paper of

1880, and in 1873 Mittag-Leffler

had yet to make his trip to Germany and hear Weierstrass lecture

for

the

first

time.

Thereafter

he

took

the

Weierstrassian approach to analysis so firmly to heart that he perhaps read into his earlier work arguments that

was a continuous function of x and

y

in T. Lichtenstein

proved the theorem by reducing it to the special case where the boundary of the region is a triangle. . Pringsheim noted that-the Cauchy Integral Theorem now followed on setting x

+

come complex functions:

iy = z and letting P and Q be­ P(x,y) = f(z), Q(x,y) = if(z).

Green's formula then says that

iaT f(z)dz

were not in fact there. So Pringsheim was inclined to credit Mittag-Leffler with being the first to have the idea that the Cauchy Integral Theorem could be proved with­

=

0

_!_ (il1xf(z) - 11yf(z)) 8->0 8

if lim

=

0.

out assuming the function to be continuously differen­

This is the Cauchy Integral Theorem without any assump­

tiable, and for being the first to have some success in that

tion about the differentiability of

dl,rection. But priority could not be claimed for the proof

Mittag-Leffler had proclaimed it.

1923, for a rigorous proof of that kind had been given by Lichtenstein in 1910.

tion. In his paper

almost exactly as

Pringsheim's paper seems to have re-opened the ques­

of

In that paper, Pringsheim explained, Lichtenstein had

f(z),

[ 1932] Kamke astutely asked what it was

that the Cauchy Integral Theorem actually said. Which of

shown how to push through a Green's Theorem approach

the following was it?

to the Cauchy Integral Theorem, first with, and then-sur­

1) If a function f(z) is regular in a simply-connected do­

prisingly-without, assumptions of uniformity. Pringsheim

main bounded by a closed continuous, rectifiable curve

argued that Lichtenstein's proof fmally showed clearly

C, then

what lay behind Goursat's proof. Lichtenstein had consid­ ered the (in Pringsheim's view inappropriately named)

aQ aP ) ( dxdy JJT ax ay

where

=

closed, rectifiable Jordan curve =

iaT Pdx + Qdy,

C, then

J f(z)dz c

=

inside and on

C, then

tial derivatives are taken to be continuous and single-val­

aT

of the region T is taken to be a

taken along it in the positive sense. He then defined

11xQ

:=

Q(x + 8,y) - Q(x,y) and

11yP

:=

P(x,y + 8) - P(x,y),

and observed that Green's formula was equivalent to the claim that

LIT 8->0 � (11xQ - 11yP)dxdy iaT Pdx + Qdy lim

u

=

and it is regular on

0;

closed, rectifiable Jordan curve

tinuous in the region T and on its boundary, and the par­

rectifiable Jordan curve, and the right-hand integral is

C,

3) If a function f(z) is regular in a domain bounded by a

P and Q are functions of real variables x and y, con­

ued. The boundary

0;

2) If a function f(z) is regular in a domain bounded by a

Green's formula:

rr

J0 f(z)dz

fc f(z)dz

C, and it is continuous =

0.

He observed that proofs of the first version could be found

[1930, p. 1 18] and Knopp [1930, 56], and of the second version also by Knopp [ 1930, p. 63]; he knew no proof of the third, although it was stated in that form in the books by Osgood [1928, p. 369] and Hurwitz-Courant [1929, p. 283]. However, Knopp's proof of (2) seemed to need some more care. Knopp had reduced (2) to (1) by the Reine-Borel in the books by Bieberbach

p.

Theorem, arguing that

C and its interior can be covered by

finitely many circles inside each of which f(z) is regular, thus giving a larger region

G

containing

C and for which

VOLUME 22, NUMBER 4, 2000

65

(2) followed.

the first result was true. Accordingly version

N

But Knopp felt this was a little glib. So he first showed that

the function f extends to a function g which is regular on

G.

To do this he covered the boundary

C by discs,

took a

�

finite subcover of the boundary, and then argued carefully

that the analytic continuation of the individual function el­

M....-----+-. A

ements yielded a single-valued function. This still left ver­

sion (3) without what Kamke presumably regarded as a sat­

isfactory proof, although he did not specify what he found

wrong with the published attempts.

His paper stimulated Del\ioy (see his [ 1933]) to prove (3) in the form: if a function f(z) is defined in a domain

C, has a fi­ nite derilmtive inside C, and is continuous inside and on C, bounded by a closed, rectifiable Jordan curve then

J

) = c f(z dz

0. To - prove this result Del\ioy took an ar-

Let

w(g) be the oscillation ofjon g and w(B) the maximum

value of

IJ

B) which have an interior E (which he called a polygonal approximation to E). He then showed that, if G is a simple, rectifiable Jordan curve of length L, any polygonal approximation to G having more than 8 sides had a perime­ ter less than 16L. (Del\ioy assumed that B is less than L.) or boundary point in common with

He argued that one could work round the boundary of

the curve tices

G and its polygonal approximation picking ver­

N common to two squares and points M which are N in the corresponding squares so that the points N and M occur in the same order. (among the) closest to

l = IJ - L l II{ w(B) · (I length(g) ) w(B) ·

c f(z)dz

<

bitrary plane set E, and considered the squares (closed, con-

taining their boundaries, of side

w(g). Then

But as

c f(z)dz

f(z) dz

=

f(z) dz

l

(16 + 16v'2 + 1)L.

<

B tends to zero so does the largest dimension of each w(B) tends to zero, and Goursat's Theorem is

g and so

proved in the form stated.

What may be the last word on the matter was then given

by the Harvard mathematician J.L. Walsh, in a one-page paper

[1933]. His proof was, as he said, "much more immediate than

that of Del\ioy, although not so elementary." Walsh began by observing that the Cauchy Integral Theorem (in the third of Kamke's forms) was true for a polynomial, because it was then

possible to replace the contour

C by a suitably chosen poly­

gon. However, the given function can be represented in the

closure of the interior of the contour as the limit of a uniformly

N

convergent sequence of polynomials, because the function

analytic inside

C and continuous

inside and on

C.

is

This se­

quence can be integrated term by term, and so the result is es­

tablished. The theorem can be extended to regions bounded

by finitely many non-intersecting rectifiable Jordan curves by

replacing the polynomial approximations with rational func­ It follows that the length of the polygonal curve defmed by

the M's is at most

L.

Del\ioy then argued by contradiction

, Ns are 8 consecutive vertices 2 of the corresponding points M are at least

that as a result, if N11 then at least

•

•

•

B apart. Consequently there are at least two points M which are less than B apart. This in turn implies the claim about the sides of the polygonal approximation.

This done, Del\ioy took a region R bounded by a Jordan

curve

C

of length

L

(in fact, Del\ioy considered fmitely

many Jordan curves, but that makes for no extra difficulty).

tions whose poles lie outside the regions considered.

More modem treatments of the Cauchy integral theo­

rem naturally couch it in the language of homology theory (which derives more from Riemann than from Cauchy).

Ahlfors's influential text (Ahlfors

1953, p. 1 18) states it in

this form: If a function j(z) is analytic in a domain D and

C is

J

a cycle in

D that is

homologous to zero in

D,

then

f(z) dz = 0. A cycle is a formal sum of continuous arcs. 0 A closed cycle C is said to be homologous to zero if the

winding number

n(C,a) = 0 for

all points

a

outside

C.

A

key stage in the proof is showing that the integral of a lo­

He let f(z) be a holomorphic function defined inside R which is continuous inside and on C. Then, if A is a point

cally exact differential is not altered if the given cycle is

main formed by the squares in the polygonal approxima­

vertical arcs. This modem formulation allows one to deal

of R and

tion to

B is sufficiently small, the boundary, u, of the do­

C that also

contain the point

proximation to the curve

C having

A is a polygonal ap­

at least 10 sides and

therefore a perimeter of length less than Goursat's Theorem applies to the domain

u.

replaced by an approximation consisting of horizontal and fmally with a vast panoply of curves that are all, somehow,

equivalent to polygons. Jordan would have been pleased.

16L. Now,

Consider the

Acknowledgments

curve g defined by the line segment NM, the arc MM' of C,

I am always grateful to Bob Burckel for his careful editing

gonal approximation after N) and the side NN' of the poly­

larly so. I am also grateful to Alan Beardon, who made

the segment M'N' (where N' is the next vertex of the poly­ gonal approximation.

66

THE MATHEMATICAL INTELLIGENCER

of this column, but in the case of this article I am particu­

many useful comments.

Continued on p.

77

OSMO PEKONEN

G e rbert of Au ri ac : M ath ematician and Pope

0

ne thousand years ago-when the world was cringing before the imminent YJK problem-an extraordinary man, Gerbert of Aurillac, was elected Pope. He is also well known in the history of mathematics, as he is credited with introducing the

Arabic number system to Europe. He is the most significant

Catalunya with him so that the lad could study mathemat­

mathematician who has ever occupied the Holy See.

ics there. Gerbert's next school was to be the monastery

Gerbert was born about 945 at or near Aurillac, in the

of Santa Maria de Ripoll, which was famous for its library.

mountainous region of Auvergne, in central France. Since

Mathematics in those days meant the quadrivium-geom­

neither his place of birth nor his parents were recorded, it

etry, astronomy, arithmetic; , and music-which he studied under Bishop Atto of Vich.

seems likely that he was of peasant origin. He must have been a young man of unusual talent,

the

Muslims then held most of Spain. Catalunya was a

Benedictines-the most successful headhunters of those

because

Christian frontier territory at the outskirts of the Muslim

days-recruited him at the age of 18 to the service of the

world, and there was considerable communication of ideas

Church. He received his first training at the monastery of

between the two civilizations. The largest Muslim city of

Saint-Gerald at Aurillac. It was a part of the "archipelago

Spain was Cordoba. With 250,000 inhabitants, it may have

of Benedictine monasteries" which dotted the map of me­

been the biggest city of the world at the tum of the mil­

dieval Europe. The Benedictine order was governed by the

lennium. It boasted, among other cultural attractions, a sci­

mighty abbots of Cluny, who themselves were subject only

entific library far better equipped than any of Christian

to the Pope.

Europe. The Muslims had fallen heir to both Greek and

Gerbert's freshman and sophomore education was the usual medieval routine: he learned his grammar, i.e., Latin,

Persian science in their initial expansion, and had trans­ lated many classics of antiquity into Arabic. At the same

and rhetoric under the tutelage of Abbot Raymond de

time, Arabic traders and travelers were in contact with

Lavaur, for whom he held a special affection for the rest

India and China, and had absorbed many of their advances.

of his life. On the other hand, the third topic of the

triv­

Muslim astronomy was the most advanced in the world, and Muslim astronomers proficient in using the astrolabe

ium-dialectic, or logic-could only be touched upon. In 967, Count Borrell of Barcelona visited the monastery, and

had done much to map the skies. The whole world still uses

the Abbot asked the Count to take Gerbert back to

the Arabic names of some major stars-Aldebaran, Altair,

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000

67

Fomalhaut, etc.-and terms of astronomy, such as al­ manac, azimuth, zenith. The Arabs were even further ad­

Otto II died in December

983, and Gerbert lost his pa­

tron and protector. He had to flee from Bobbio and hasten

vanced in arithmetic. They had adopted the concept of

back to Reims. Despite his failure at Bobbio, his reputa­

zero, which had originally emerged in India, and used a po­

tion was so great that he could reclaim his position as the

sitional numeric system much like the modem system. The

cathedrai school of Vich was able to offer Gerbert some of

this knowledge, and he took advantage of the opportunity.

master of the cathedral school of Reims and secretary to the Archbishop. He became deeply involved in the power politics of the times.

As a loyal servant of the Ottonian dy­

Popular literature about Gerbert is teeming with allu­

nasty, he defended the three-year-old Otto III against the

sions to his "Arabic" or "Muslim" teachers. It makes a beau­

pretender duke Henry of Bavaria. In France, Gerbert

tiful story for a future Pope to have been directly exposed

helped to raise Hugh Capet, the Count of Paris, to the

to Muslim scholarship, but yet, to the present author's

throne in

knowledge, there is no evidence for such a conclusion.

with a new dynasty, to be called Capetian. These were non­

In

970, Count Borrell and the Bishop of Vich made a pil­

grimage to Rome, and took young Gerbert with them. The

journey proved disastrous: the Bishop was assassinated in

987, thereby replacing the old Carolingian line

trivial matters that consumed a fair share of his time and drew him deep into the muddled waters of politics. Gerbert found little time for teaching and research any more.

Rome. Gerbert now found himself without an adviser. His

Having backed the right horses, though, he emerged as

mathematical knowledge delighted Pope John XIII, who in­

Archbishop of Reims when the turmoil was over. He turned

troduced him to the Holy Roman Emperor Otto I. The Pope

out to be a singularly self-willed Archbishop who, centuries

recommended Gerbert as a tutor for the Emperor's son,

later, was remembered as a forerunner of Gallicanism, i.e.,

the future Otto II, who was to marry a Greek princess. The

self-assertion of the church of France.

young monk attended the imperial wedding ceremony in Rome in

972. The King of France was represented by

Archdeacon Gerann, a famous logic teacher from the

Mter the death of Hugh Capet in

996, Gerbert clashed

with his successor, Robert II, whose marriage to a cousin

he judged illegal. A newly appointed bishop sided with the

cathedral school of Reims. The two learned men were in­

King and refused to be consecrated by Gerbert. Pope

troduced to each other, and Gerbert got an invitation to

Gregory V summoned Gerbert to Rome, and stripped him

pursue his studies of logic at Reims. The Emperor allowed

of his episcopal functions.

him a leave of absence.

The unfortunate former logic teacher never returned to

Gerbert soon made quite a name for himself in Reims.

Reims again, but approached the new German Emperor

De

He was invited by Archbishop Adalberon (who later or­

Otto III, then

dained him) to join the faculty. He reformed the teaching

arithmetica to the Emperor.

of logic in Reims and introduced Boethius to the curricu­

inviting him to teach the Franks mathematics, in order to

An envious colleague from Magdeburg, Otric, de­ nounced him to Emperor Otto II. In December 980 the

lum.

Emperor summoned both scholars to Ravenna and en­

16. He seems to have offered Boethius's

The Emperor responded by

awaken in them the genius of the ancient Greeks. 1 Gerbert

wrote back, praising him for appreciating the universal im­ portance of mathematics. 2

gaged them in a debate on the subject of classifying knowl­

Gerbert's intelligence charmed the Emperor who en­

edge. In modem terms, the issue was whether physics is a

gaged him into his court and chancellery in Aachen. He

branch of mathematics or an independent subject. The ve­

started in

hement argument was terminated only when the Emperor

chaplain, and court musician. He impressed the court by

997 as Otto's combined advisor, teacher, scribe,

intervened. Otto was quite impressed by the intellectual

constructing a nocturlabium. The next year he was ele­

performance of his former teacher, and he bestowed upon

vated Archbishop of Ravenna. When Pope Gregory V died

Gerbert the wealthy monastery of St. Columban of Bobbio

in

in Lombardy, Italy.

did so by appointing Gerbert pope. He was consecrated on

Bobbio was a major center of learning which possessed one of the great libraries in Western Europe.

It was close

999, Otto decided to wrest control of the papacy, and

Easter day, April

9, 999.

Gerbert was the first French Pope. He took the name

to Genoa and had benefited from the trade and commerce

Sylvester II, Sylvester I having been the advisor of the

that were beginning to enrich all of northern Italy, but it

Roman Emperor Constantine. This reflected the newly

had fallen on hard times. Incompetent abbots had depleted

elected Pope's close cooperation with Otto's ideal of a re­

its treasury, local nobles had seized its lands, and its monks had taken great liberties with their duties. Gerbert under­

Christian Roman Empire, Renovatio imperii Romanorum. There may have been some millennia! fever

took to remedy these affairs, but he turned out to be inept

about the sudden idea of re-establishing the greatness of

newed

in administration and provoked outright mutiny among

ancient Rome. At Pentecost

monks, clerics, and nobles.

grimage to the tomb of Charlemagne in Aachen: He had the

1000, Otto made a curious pil­

1 Nous voulons que, sans faire violence a notre liberte, vous chassiez de nous Ia rudesse saxonne, mais surtout que vous reveliez Ia finesse hellenique qui est en nous. . . . Aussi nous vous prions de vouloir approcher de notre modeste foyer Ia !Iamme de votre intelligence et de cultiver en nous le vivace genie des Grecs, de nous enseigner le livre de l'arithmetique, afin qu'instruits par ces enseignements, nous puissions comprendre quelque chose de Ia subtilite des Anciens. :Votre demande honnete et utile est digne de votre majeste. Si vous n'etiez pas si fermement convaincu que Ia science des nombres contient en elle ou produit les premices de toutes chases, vous ne montreriez tant d'ardeur a en prendre une connaissance entiere et parfaite.

68

THE MATHEMATICAL INTELLIGENCER

tomb opened and divested the dead man of a golden cross, some gannents, and one tooth. As a spiritual leader, Sylvester II was a morally vigor­ ous one. He took energetic measures against the abuses in the life of the clergy represented by simony and concubi­ nage, and was anxious that ouly men capable of spotless lives should receive the episcopal office. He turned out to be a shrewd diplomat, as well. His Ostpolitik was far­ reaching. He established the first independent archbish­ oprics of Poland and Hungary, and moreover granted the title of king to Stephen, ruler of Hungary, in the year 1000, and appointed him as Papal Vicar of his country. He also exchanged ambassadors with the newly converted Russia. We may wonder whether Sylvester, as a mathematician, was particularly keen on exploiting the round figure of the year 1000 to embellish his diplomatic moves. Thanks to him, one thousand years later, the Hungarians now cele­ brate the millennium of their first Christian ruler, Saint Stephen, and his crown. Many advances of science, like the construction of var­ ious astronomical instruments, were posthumously attrib­ uted to Gerbert. He was an avid collector of manuscripts, who left behind a substantial library and a legacy of learn­ ing. As for his own writings, the scholars are very much di­ vided on which of the surviving texts are attributable to Gerbert himself. The genuinely Gerbertian mathematical corpus seems to be meager compared to his writings on other topics. For instance, a text on Roman land survey­ ing, which is generally attributed to Gerbert, is rather un­ interesting in its mathematical contents. His writing on the abacus, Regulae de numerorum abaci r'ationibus, became a standard text, and included a presen­ tation of Arabic numerals. Gerbert's abacus used the posi­ tional system up to 27 decimal places, which sounds amaz­ ing. One may wonder whether octillions were really needed in the administration of the Catholic church, or whether the Pope was merely showing off with his supercomputer. And what about the end of the world? Despite a lot of later romantic history writing about the "great panic of the year 1000," there seems to have been hardly any panic at all at the tum of the first millennium, for the good reason that most of Europe's populace consisted of illiterate peas­ ants who had no access to almanacs. However, among the learned few, there may have actu­ ally been a mathematical Y1K problem in the air when the date suddenly shifted from the complicated DCCCCXCIX to the simple M. It would be amusing to conclude that a

Sipos

mathematician-pope solved the Y 1 K problem b y introduc­ ing the zero. However, there exists no contemporary doc­ ument where the date 1000 would appear written in Arabic numerals. The adoption of the zero in Europe was a much slower process. Nonetheless, it is appropriate to include a celebration of the millennium of the zero as a theme of the World Mathematical Year 2000. Rarely has a mathematician shaped political history as much as Gerbert did. During his reign, the frontiers of the Catholic church were pushed to the Danube and to the Vistula, where they have stayed ever since. Otto III died on January 23, 1002, and Sylvester II on May 12, 1003. Their departure put an end to an early dream of unified Europe­ whose fulfillment we may be witnessing today. According to an early biographer, Gerbert himself mod­ estly summed up his career saying that he passed "from R to R to R" (meaning Reims, Ravenna, Rome). Just the kind of statement to be expected from a mathematician. REFERENCES

N. Bubnov (ed.), Gerberti post� Silvestri II papae opera, mathematica

(972- 1 003),

accedunt aliorum opera ad Gerberti libel/as aestiman­

dos intelligendosque necessaria, Berlin, 1 899, repr. Hildesheim,

1 963. Gerberto: scienza, storia e mito. Atti del "Gerberti Symposium", Bobbio 25-27 /uglio 1983, Bobbio, 1 985.

P. Riche, Gerbert d'Aurillac, le pape de /'an mil, Paris, 1 987. P. Riche and J.-P. Callu (eds.), Gerbert d'Aurillac, Correspondance, 2 vols. , Paris, 1 993. 0. Guyotjeannin and E. Poulle (eds.), Autour de Gerbert d'Aurillac, le pape de /'an mil, Ecole des Chartes, Paris, 1 996.

Pictures

Although Arabic numerals do not occur in any surviving man­ uscript directly attributable to Gerbert, they do appear in an 1 1th-century manuscript called "Geometry II" (Erlangen, Universitatsbibliothek, 379, fol. 35-v), whose unknown au­ thor, called Pseudo-Boethius, must have been much influ­ enced by Gerbert. The figure below shows the Arabic nu­ merals used there, with thej.r early names, whose etymology remains mysterious. Two of the names, "Arbas" (4) and "Temenias" (8), are identifiable as deformations of the re­ spective Arabic names of numbers, and it might be the case of all of them. Our present word "zero" (as well as "cipher") is derived from the Arabic sifr, meaning void. The name "Sipos" (0) in the figure, however, might rather be related to the Greek word 1/Jfj
Arbas

Ormis

s f.. Jo q �

�

Celentis Temenias Zen is

Calctis

Quinas

Andras

I gin

I( \I/ \ I \I/ \I/ \1/ \1/ \1/ \1/ '\1/ \

(/J Figure 1 .

9

� 1 VOLUME 22, NUMBER 4, 2000

69

treatise the numerals are intended to be inscribed on the beads of an abacus. A bead carrying the numeral zero was not used in most of the early abaci; however, if this was the case in Gerbert's abacus, it would be an indication of a fully­ developed positional number system. The statue of Pope Sylvester IT at Aurillac, Gerbert's home town, was sculpted by Pierre-Jean David d'Angers and erected in 1851. Notice that the scientist-pope's right hand is not depicted in a customary blessing gesture; rather he seems to be lecturing. Indeed, the intention of the sculptor was to reconcile Religion and Enlightment. One of the reliefs of the pedestal perpetuates the ahistorical legend of Gerbert as the inventor of the mechanical clock. In reality, the first me­ chanical clocks were constructed only in the 13th century. A 20th-century ll)._ural painting at the Benedictine monastery of Pannonhalma, Hungary, illustrates a scene where Pope Sylvester II hands over the Holy Crown to emissaries of the Hungarian King Stephen for his corona­ tion at Christmas 1000. An ancient crown representing the Holy Crown (or according to the most fervent believers, the Holy Crown itself) is still venerated by Hungarians. It was recently transferred, within Budapest, from the Hungarian National Museum to the Houses of Parliament. ACKNOWLEDGMENTS

Figure 1 is reproduced from the book: Olivier Guyotjeannin & Emmanuel Poulle (eds.), Autour de Gerbert d'Aurillac, le Pape de l'An Mil, Ecole des Chartes, Paris, 1996. Figure 2 is a postcard photograph by Yves Bos. Figure 3 is an original photograph taken by Osmo Pekonen.

Figure 3.

A U T H O R

OSMO PEKONEN Department of Mathematics

Jyvaskyla Jyvaskyla Finland

University of

40351

e-mail: [email protected]

Osmo Pekonen was born in Mikkeli, Finland, in 1 960. He stud­ ied mathematics at the universities of Jyvaskyla and Paris, and wrote a PhD thesis at Jyvaskyla in 1 988. His main work has been on Riemannian geometry, Teichmuller spaces, and string theory. He is a member of the European Committee for the

World Mathematical Year 2000. Besides his mathematics, Pekonen has pursued a career in poetry. His main achieve­ ment is the first-ever verse translation of the Beowulf into

Finnish, written in collaboration with the Old English scholar Clive Tolley.

Figure 2.

70

THE MATHEMATICAL INTELLIGENCER

I d§lj l§i.Jtj

Jet W i m p , Editor

I

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David

M.

Bressoud

(3j + 1)!

(n + J) ! and therefore that

An =

CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS, 1 999, xv + 274 pp. HARDCOVER, US $74.95, ISBN 0 521 66170

6; SOFTCOVER, US $29.95 ISBN 0 521 66646 5.

Mathematical lntelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if

A

n alternating sign matrix is a square array of Os, 1s and - 1s

such that the non-zero entries of each row and of each column alternate in sign, and each row sum and column sum is 1. These matrices generalize

a book to review, please write us,

permutation matrices. Here are the

telling us your expertise and your

seven alternating sign matrices of or­ der three:

·

The validity of the latter formula is

G DG D G DG DO D G DG n

David M. Bressoud's book is the story of

thiS conjecture, c�ting in its

proof in

1995 by Doran Zeilberger.

Along the way, the author searches out and explicmes the connections between alternating sign matrices and a host of other combinatorial topics, including generating functions, partitions, deter­ minants, lattice paths, inversion num­ bers, plane partitions, symmetric func­ tions, Schur functions, Young tableaux,

0

1

0

hypergeometric series, and square ice (a

0

1

model of H20 molecules frozen in a

1

1

1

0

0

-1

0

0

1

0

0

0

1

1

0

Howard Rumsey wondered whether

An,

square lattice). The story also touches on the lives and contributions of many great mathematicians of the past, in­ cluding Leibnitz, Euler, Lagrange, Gauss, Waring, Cauchy, Jacobi, Boole, Sylves­ ter, and Ramanqjan, as well as contem­ porary researchers such as Ian Macdon­ ald, Richard Stanley, Donald Knuth, George Andrews, John Stembridge, and Greg Kuperberg (who discovered a dif­

the

ferent proof of the conjecture in 1995).

number of alternating sign matrices of

Even Lewis Carroll makes an appear­

they could find a formula for order

n.

ance (via Dodgson's algorithm). In

Notice that the definition forces

the

author's

conception,

the

each of the "borders" (first and last

process by which the conjecture was in­

columns and rows) of an alternating

vestigated and eventually proved illus­

sign matrix to consist of a single 1 and

trmes a way of looking m mathematics

An,k be the

that differs from the standard para­

all other entries 0. Letting

It is not theorem-proof-corollary; is not "scaling the peaks" (moving

number of alternating sign matrices of

digms.

n in which the curs in column k, An+ l,l = An +l,n+ l ·

it

order

Philadelphia, PA 1 91 04 USA.

1

0

In the early 1980s, mathematicians William Mills, David Robbins, and

Column Editor's address: Department of Mathematics, Drexel University,

'

Conjecture.

you would welcome being assigned

predilections.

(3j + 1)!

IJ i=O (n + J ) .

known as the Alternating Sign Matrix

REVIEWED BY MARTIN ERICKSON

Feel like writing a review for The

n- 1

first row's 1 oc­

An =

from accomplishment to greater ac­

Based on knowledge of An,k for the

ploration. By analogy with archaeology,

we have

complishment); and it is not random ex­

first twenty values of n, the three re­

Bressoud suggests a new way of de­

searchers conjectured that

scribing what mathematicians do:

© 2000 SPRINGER-VERLAG NEW YORK,

VOLUME 22, NUMBER 4, 2000

71

"I would like to consider the doing of mathematics and the fmding of proofs as analogous to the work of the archaeologist. When Mills, Robbins,

lots

of diagrams,

and much back­

timated.

Conjecture is told in Chapters

both the position (x)-momentum

story of the Alternating Sign Matrix

2

1 and 6, through 5 and 7 pro­

and Rumsey first discovered their con­

while Chapters

jecture, they were not dissimilar to the

vide related material.

archaeologist who has just unearthed

The author also supplies Mathe­ matica code so that the reader can ob­

a strange and marvelous object of un­ known provenance and purpose. What

which the angular frequency can be es­

ground information. In fact, the main

tain data and follow the discussion in

is it? What was it used for? Why is it

an active way. This is an excellent idea,

here? What does it tell us about the

and the code is simple enough to copy

people who once lived here? The real

and use in a few minutes. I enjoyed do­

work of the archaeologist is to make

ing so immensely.

It occurs in Quantum Mechanics as (p)

uncertainty relation

and the time-energy uncertainty rela­ tion

tJ.EtJ.t 2=

1 -

2

n,

h is Planck's constant and n =

connecti\ms: connections to other ob­

Proofs and Confirmations is a fas­

jects at other places_ at other times,

cinating look at mathematics in the

connections to other facts that are

making. And it is a generous guide to

Mechanics in the form of uncertainty

known about this particular site. The

additional information that is interest­

relations between extensive variables

goal of the archaeologist is to provide

ing in its own right.

) (internal energy, volume,

� � ' �' · · · )

Science

but also in other places and from other

Truman State University

times.

Kirksville, MO 63501

It provides a foundation upon

' "

jugate intensive variables ( ,

Department of Mathematics and Computer

tation of others, not just in this place,

which we construct our theories.

It appears also in Statistical

(U,V,N,

As each object comes to be

understood, it facilitates the interpre­

h/27T.

number of particles, · · · ) and their con­

a context in which we can understand this object.

where

(temperature, pressure, chemical po­ tential, · . . ), one of which is

!J.U!J.

USA

"This is the role of proof, to enrich

e-mail: [email protected]

where

the entire web of context that leads

( �)

2= k,

k is Boltzmann's constant.

The Cramer-Rao inequality exhibits

to understanding. The mathematician

the duality which exists between the

does not dig for lost artifacts of a van­

two

ished civilization but for the funda­ mental patterns that undergrid our uni­ verse, and like the archaeologist we usually find only small fragments.

As

archaeology attempts to reconstruct

Physics from Fisher Information: A Unification

fields:

Statistics.

P(yi a)

Probability Theory and Specifically,

assume that

is a probability distribution

function for the random variable

Y.

This probability distribution depends

by B.Roy Frieden

on the values of one or more parame­

a. Then, given a, it is possible to

the society in which this object was

CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS (1 999),

ters

used, so mathematics is the recon­

318 pp.

estimate the various statistics of the

struction of these patterns into terms

US $74.95, ISBN: 0521631 67X

that we can comprehend.:'

REVIEWED BY ROBERT GILMORE

random variable

I believe that Bressoud successfully illustrates his thesis in this book The details are not always easy to follow,

=

(y)

where

I

t is hard not to be seduced by a book whose very first equation is the

as there are many related conjectures

Cramer-Rao inequality. This is a pro­

and much mathematical machinery to

foundly beautiful and important result

keep track of. (Most of the conjectures

which may be written poetically as

have been proved,

jj

but the author

!J.y!J.a 2= 1.

refers to them as conjectures, rather than theorems, in order to preserve the

It i s beautiful because i t i s simple.

historical point of view.) But the au­

It is important because it is far-reach­

Y, such as its mean

!J.y, ((y - y)2). Conversely, it

and standard deviation

!J.y2 =

is possible to estimate the value of the

a from the measurements Yi· These estimates lead to a mean value a, and a standard deviation !!.a, defmed by !J.a2 = ( (a - a)2 ). The prod­

parameter

uct of these two variances, of the ran­ dom variable and the parameter esti­

mates,

are

bounded

below

by

a

nonnegative term which can often be

thor does a good job of helping us keep

ing: it manifests itself in many differ­

normalized to

the main points in mind, and his nar­

ent ways in many branches of physics.

the defmition of the random variable

ration is quite clear.

It is known to sound engineers in the

and/or the parameters. The term !J.y2 is

form of a time-frequency uncertainty

called the Fisher information

This is enjoyable history of modem mathematics of the type found in the

relation

From Error-Correcting Codes Through Sphere Packings to Simple Groups, by Thomas Thompson. As in

book

that book, we are treated to quotes from the investigators, photographs,

72

THE MATHEMATICAL INTELLIGENCER

!J.w !J.t 2=

1 -

2

,

!J.t is the time duration of a sig­ nal and !J.w = 27T!J.v is the precision to

where

!J.y2 = I(a) =

+ 1 by suitable change in

J (� dP�� yP a)

(yi a)dy.

Fisher information is the starting point of the journey on which the au­ thor of the present book embarks.

Frieden has certainly made a number

final result is some equation of statics

the probability distribution is transla­

of useful observations:

or dynamics. What makes the varia­

tion-invariant

1) Uncertainty relations play a fun­ damental role in modern physics.

tional method so attractive is that there are certain rules which severely

2) Fisher information, through the

restrict the form of the Lagrangian

Cramer-Rao inequalities, underlies all

function. This drastically reduces the guesswork required in deriving dy­

uncertainty relations.

space coordinate

P(y I a)

P( I a) X

P(x a) . He does this so that the derivative d/d a translation invariance

_

3) The dynamical laws of physics

namical laws to describe new physics.

can all be formulated as variational

Frieden encounters three problems

can be replaced by the coordinate de­

principles. These involve integrals over

in trying to formulate physics from

rivative - d/d.x. This brings the Fisher

quadratic forms in the gradients of

Fisher information. I illustrate them

suitable functions.

for a particularly simple case: the quan­

4) Fisher information has a similar

tum-mechanical description of a parti­

m

cle of mass

structural form. On the basis of these observations,

moving in one dimen­

V(x). The variational

sion in a potential

which occurs in the Fisher information

information into a form more similar to that of a physical Lagrangian. Invariances in physics are closely related to conservation laws. In the present

case,

Frieden attempts to create a vision of

problem is

suggests

physics as a stepchild of Information

I L( �. :�)d.x 8 I { � (:�Y + V(x)(�(x))2} d.x 2

which

Theory, specifically of Fisher informa­ tion. His thesis is that " . . . all physical law, from the Dirac equation to the Maxwell-Boltzmann

velocity

disper­

8

constant, is translation-invariance of

=

0.

sion law, may be unified under the um­

For simplicity, I have taken the wave

brella of classical measurement the­

function to be real. In the problem as

ory.

In

particular,

the

information

specified, there is always the trivial so­

�(x)

0. To eliminate the trivial

aspect of measurement theory-Fisher

lution

Information-is the key to unifica­

solution it is useful to impose the nor­

tion. " Frieden's efforts have created a

malization constraint

=

f(�(x))2d.x

=

1.

considerable buzz in the world of sci­

This constraint i s normally imposed

ence-a recent lead article in the jour­

through the use of Lagrange multipli­

nal

The New Scientist alluded very fa­

vorably to Frieden's book. However, speaking as a card-carrying physicist who has derived the uncertainty rela­ tions for Statistical Mechanics using

ers, so that the constrained variational problem becomes

8

(f{ � (:�t 2

the Cramer-Rao inequality, I do not be­

+

}

!(a)

laws of dynamics is a very powerful tool in the physicists' bag of tricks. The

with .V(x)· tJc:r2. Is it likely that P(xia) has the same form when a - 0 as when lal is very large?. To put the =

question in even more stark terms, consider the particle in a box, so that

V(x) = 0, 0 < x < L, but V(x) = x ::s; 0 and L ::s; x. Surely P(xla 0) =/= P(xla t L) =/= P(xla L). "

" oo ,

=

=

=

Frieden deals with the second and

)

- AI(�(x))2d.x

=

Fisher information"

J.

This functional

somehow describes the physical infor­

0.

mation which is intrinsic to the quantity measured.

Although

he

strenuously

tries to put great distance between

2 I( aP aa ) P(yla) dy

J

and constraints on physical systems

1

=

not. Think of the harmonic oscillator,

a second type of information, a "bound

V(x)(�(x))2 d.x

The Fisher Information is

The variational formulation of the

P(xia) reasonable to assume? I claim

third problems together. He introduces

lieve that Frieden has succeeded in his effort.

conservation,

V(x) = constant, or (no forces). If V(x) is not

requires

dV/d.x = 0

=

translation-invariance

momentum

which

are

invariably

treated

with

Lagrange multipliers, there is in fact no

method itself dates back to the 1 7th

At this point in the analysis we are

difference. He argues that the transfer

century, to Fermat's Principle of Least

faced with three procedural problems:

of information between

Time, and to the 18th century, in the

1) The gradients are with respect

form of Maupertuis's Principle of Least

to the spatial coordinates and the pa­

Action. The original formulations had

rameters of a probability distribution

a distinctly theological formulation.

function, respectively.

Even today, the method smells faintly sulfurous. Here I will give away one of the tricks of our trade. We put exactly "the right stuff " into a Lagrangian, that is, whatever is necessary to recover the

V(x)(�(x))2, and a con­ straint term, -A(�(x))2; the Fisher in­

tential term,

formation has neither.

3) The physical law involves a vari­ ation,

pushed and the machinery goes into

not.

gear. The machinery involves a Taylor expansion around the appropriate so­ lution, one or more integrations by parts, and then an argument about an integrand necessarily being zero. The

the

Fisher information

More specifically,

=

In summary, my very strong reser­

vations about Frieden's technical pro­

2) The physical Lagrangian has a po­

desired dynamics once the button is

I and J during 81 = 8J. 8(/ - J) 0.

a measurement is the same:

does

gram

to

reconstruct

physics

from

Fisher information are two in number: the assumption of translation-invari­ ance of the probability distribution function

P(x i a)

is incorrect, and the

constraint terms in the "bound infor­ mation"

J are put in in an ad hoc man­

Frieden attempts to solve the frrst

ner to guarantee that the appropriate

problem by assuming the probability

laws are recovered. A physicist would

P(yla) is for the distribution of the spa­ tial coordinate (x) (not the wave func­ tion �(x)). Further, he assumes that

admit this up front. It is difficult to dis­ cern that the author of the present work is guilty of this, but he is.

VOLUME 22, NUMBER 4, 2000

73

Frieden is forced to identify the pa­ rameters of the probability distribution a with space-time coordinates (x,t) in order to introduce spatial (and tempo­ ral) derivatives into the expression for the Fisher information. As a result, he is forced to focus on uncertainties in position and time, rather than on the amplitudes of the electric and mag­ netic fields, or the complex wave func­ tions of Quantum Mechanics. As a con­ sequence, his interpretations of physical phenomena differ in signifi­ cant ways from the standard interpre­ tations ol physicists. This divergence. in viewpoints is well illustrated in Fig. 1.4 in this book This shows a point source located in a screen at the left of the page, and a dif­ fraction pattern produced by this point source on a screen at the right side of the page. Every physicist has done this experiment. To us, the physics lies in the peaks and valleys (intensity max­ ima and minima), and in particular, in the ratios of heights of successive peaks, and the ratio of heights of adja­ cent peaks and valleys. We know that the pattern may be offset from its ide­ ally predicted position because of slight displacements of the point source or placement and/or disorien­ tation of the intermediate lens. The off­ set of the pattern is not important, this is an "engineering" problem. For us, the physics lies in the intensity distri­ bution. For Frieden, the physics lies in the offset. There is more to physics than dy­ namical laws of motion. Frieden finds that Newton's Second Law of Motion is a consequence of variation of the Lagrangian L(x,x) �xL V(x) (pro­ vided F = - \lV), but where is Newton's First Law, whose purpose is to define the subset of reference frames which are inertial, in which the Second Law is true? Or Newton's Third Law, the conservation of momentum? Can the principles of unitarity, equiva­ lence, covariance, or the conservation of energy, momentum, angular mo­ mentum be consequences of informa­ tion of any kind? There is in fact an important role that information theory can play in the formulation of physical theories. This can be illustrated in terms of the elec=

74

THE MATHEMATICAL INTELLIGENCER

tromagnetic field. The field can be for­ purpose of an equation is to winnow mulated in two ways, called for sim­ out the nonphysical from the physical. plicity the 19th-century formulation Wouldn't it be more elegant to build up and the 20th-century formulation. ln every allowable physical state from a the former the electric and magnetic small number of building blocks (e.g., fields, E(x,t) and B(x,t), are intro­ photon states) which obey no con­ duced. Then a system of 4 equations is straints, so that there is a 1-1 corre­ introduced (by Maxwell) which these spondence between linear superposi­ fields satisfy. This formulation has tions and physically allowable states? been called "manifestly covariant." The If there is a way that all/most/some 20th-century formulation regards the parts of physics could be formulated in electromagnetic field as composed of an information-theoretic way, it would photons with two polarization (helic­ be much more elegant to do it in this ity) states. There is essentially a 1-1 "building-up way" (Aujbauprinzip) correspondence between electromag­ than in the more classical "find-the­ netic fields and superpositions of pho­ equation-which-elirninates-the-nonphys­ ton states. The superpositions satisfy ical" approach (''tearing-down way"). no constraints. It may be possible to formulate So: what role do Maxwell's equa­ some part of physics in an information­ tions play? Maxwell's equations are ex­ theoretic setting. I do not believe the pressions of our ignorance. By intro­ formulation by Frieden is successful. ducing fields E(x,t) and B(x,t) we are introducing mathematical functions Department of Physics some of which cannot represent real Drexel University physics. The function of Maxwell's Philadelphia, PA 1 91 04 equations is to eliminate all those USA mathematical functions which de­ e-mail: [email protected] scribe nonphysical fields, and to allow only those functions which do describe physically allowable fields. The simplest way to see this is as follows. Resolve both the manifestly covariant (19th-century) description and the quantum (photon) description by Gerald E. Farin in terms of their propagation direction 4-vectors (k,k4), where k-k - c2k� 0 NATICK, MA; A K PETERS, 1 999, 267 PP in free space. For each 4-vector (k, k4) US $49.00, Hardcover, ISBN: 1 -56881 -084-9 there are 6 amplitudes Ei(k, k4), REVIEWED BY LES PIEGL Bi(k,k4) in the manifestly covariant description and just 2, one for each on-Uniform Rational B-splines, helicity, in the photon description. commonly referred to as NURBS, Choose any particular 4-vector (k, k4) have acquired a remarkable success in and compare the transformation prop­ only a decade and a half. It all started erties under the Poincare group of the in the late 1940's with Schoenberg's in­ 6 amplitudes from the first description vestigations into splines using trun­ and the 2 amplitudes from the second. cated power functions. While research This comparison identifies the 4 linear on splines remained active throughout combinations of amplitudes from the the 50's and 60's, nothing really hap­ first description which must vanish, pened in the world of computational and the 2 which describe the positive science of splines until the famous and negative photon helicities. Now Cox-de Boor algorithm was published transform this identification to any in 1972 independently by Maurice Cox other 4-vector (k', k4) by a Poincare and Carl de Boor. The Cox-de Boor al­ transformation. VioUi-Maxwell's equa­ gorithm allowed fast and reliable eval­ tions result. uation of B-splines without the trun­ The point is that every time we cated power functions and divided write down an equation in physics we differences. Bill Gordon, then at are expressing our ignorance. The only Syracuse, had Rich Riesenfeld look at

NURBS: From Projective Geometry to Practical Use

=

N

Bezier curves and surlaces to see how the new B-splines, defined by an eas­ ily computable recursive formula, can be used to defme curves and surfaces. Once Riesenfeld figured out the rela­ tionship between knots, nodes, and control points, he found a scheme that was far superior to anything used thus far. It also contained Bezier c1lTVes and surlaces as special cases. In 1973 Riesenfeld's thesis was published­ which marks the birth of B-spline curves and surlaces. Though these entities were quite nice to represent free-form shapes, common curves such as the circle were not representable by integral B­ splines. In 1975 Versprille's thesis be­ carne available, investigating B-spline curves and surlaces in homogeneous space. Upon projection of the 4-D func­ tions to 3-D, a rational form was ob­ tained, which is what we call today a rational B-spline. The work of Riesenfeld and Versprille became the basis of industrial research in the late 1970's. The CAD/CAM in­ dustry was looking for a mathematical form that was able to handle both free­ form as well as specialized curves and surfaces. Companies such as Boeing aMI SDRC (Structural Dynamics Re­ search Corporation) played a crucial role in pushing this technology for­ ward. The frrst commercial product based entirely on rational B-splines, called GEOMOD, was released by SDRC in the early 1980's. The term NURBS was coined around that time (probably by Bob Blomgren, working for Boeing at the time). Today NURBS are the de facto standards for geome­ try representation and data exchange, and are used almost exclusively in the broad field of computer-aided design and manufacturing (CAD/CAM). Farin's book on NURBS is a tremen­ dous disappointment. In the 250 pages of text, he devotes exactly 14 pages to B-splines, not NURBS! He gives very brief (a page or two) discussions on such general topics as knot insertion, the de Boor algorithm, blossoms, and derivatives. There is nothing in these pages that the designer of a NURBS system can use. The chapter does not even teach the reader how to evaluate a NURBS curve or surlace.

The rest of the book is a collection of (again very short) chapters on pro­ jective geometry, conics, Bezier formu­ lation, Pythagorean curves, rectangular patches, rational Bezier triangles, quadrics, and Gregory patches. For the mathematician who wants to learn the basis of NURBS for further research, this book is a dead end. For the serious implementer who needs algorithmic de­ tails, this book is a waste of time. Though it might provide entertaining reading for someone with a solid knowl­ edge of high school algebra and may even create the impression that the reader has learned something, it gives nothing of substance to think about. Department of Computer Science University of South Florida 4202 Fowler Avenue Tampa, FL 33620 e-mail: [email protected]

A Panorama of Harmonic Analysis by Stephen Krantz CARUS MATHEMATICAL MONOGRAPH NUMBER 27 WASHINGTON, D.C.: THE MATHEMATICAL ASSOCIATION OF AMERICA, 1 999, 368 PP. US $39.95, ISBN: 088385031 1

REVIEWED BY MARSHALL ASH

I

had some trouble with the topology section of the University of Chicago's master of science exam in May 1963. But the analysis section went very well. No small part of the credit for the lat­ ter result was due to Antoni Zygrnund who had taught me real analysis I and II, and to Alberto Calderon who had taught me complex analysis I and II. In very short order, I abandoned my plan of specializing in point-set topology and picked Fourier series as the main topic for my next hurdle, the two-topic exam. Over the next year, Bill Connant, Larry Domoff, and I read Zygrnund's Trigonometric Series in preparation for the exam. After the exam, Professor Zygrnund accepted me as his student and my career as a harmonic analyst began.

Looking through Krantz's A Pan­ orama of Harmonic Analysis feels like watching a horne movie produc­ tion entitled "The Zygrnund School of Analysis: 1965-1999." Throughout most of this period, I was lucky enough to be near the University of Chicago, where the Monday 3:45 PM Calderon­ Zygrnund seminar featured, among many other things, just about every de­ velopment in harmonic analysis men­ tioned in Krantz's book. These were exciting times in harmonic analysis, and my connection with the University of Chicago's seminar placed me near the center of the action. I have always been attracted by ques­ tions that have crisp, easily grasped statements. A good example of such was Lusin's conjecture that there could exist a real-valued square-integrable functitm defmed on the interval 1f [0,27T) whose Fourier series diverged at each point of a set of positive measure. Already in 1927 Kolmogorov had given an example where the function was in­ tegrable, but not square-integrable, and the Fourier series diverged at every point. Since giving an example seemed like it couldn't be very hard, I proposed to Zygrnund that I take the Lusin con­ jecture for my thesis problem. He im­ mediately discouraged this idea, ex­ plaining that this problem might prove to be rather difficult. Zygrnund had re­ alized that the square-integrable case was much deeper than the integrable case, even though his almost infallible intuition this time predicted the exis­ tence of an example. In the early fall of 1964 my fellow graduate student Lance Small came back from a summer visit to Berkeley carrying the news that Lennart Carleson had just proved that the Fourier series of a square-inte­ grable function actually converges al­ most everywhere. He was closely ques­ tioned by Zygrnund and Calderon, who thought that he could not have gotten the story straight. But he had and Carleson had. Several years later when I spent two months working through Richard Hunt's careful exposition of his extension of Carleson's theorem, it became clear to me that although Zygrnund's guess about the outcome of the conjecture had been wrong, his as­ sessment that an extremely high level =

VOLUME 22, NUMBER 4, 2000

75

of mathematics would be required to decide the issue was quite accurate. After finishing my degree in 1966, I was a Ritt instructor at Columbia. I was at a loss for how to begin my career as a research mathematician. My inertia was assisted by the cultural cornucopia that New York City provided, and also by the Columbia campus protest move­ ment featuring Mark Rudd, SDS, and the occupation of the math building which contained my office. I wrote to Zygmund, who suggested that I get into partial differential equations. This cer­ tainly proved prophetic. The great bulk of harmonic analysiS- being done now seems to be in connection with partial differential equations. One of the ways to see this is to note that the prepon­ derance of talks being given nowadays at the Calder6n-Zygmund seminar fits this profile. Nevertheless, most of my own interest never did move in that di­ rection. One thing I did to stay mathemati­ cally alive was to attend Stein's semi­ nar at Princeton. One of the talks I heard there was by Stein's extremely young Ph.D. student Charles Feffer­ man. At the time, I did not have a suf­ ficient overview of harmonic analysis to appreciate the depth and beauty of his mathematics, but fortunately I have heard him lecture many times since. It is also fortunate that his expository skills have improved from very good to extraordinary. For example, I consider it a high compliment when I say that Krantz's book does justice to the lec­ tures I later heard in Chicago, wherein Fefferman explained the proof of his theorem that the characteristic func­ tion of the unit ball is not a multiplier on .LP(IR2) when p of. 2. After three years at Columbia, I moved to DePaul and back to the Calder6n-Zygmund seminar. When I first arrived in Chicago, the harmonic analysts there were reading Igari's book on multiple Fourier series.[!] Grant Weiland and I immediately began doing research in this direction, and the main thrust of my mathematical career has been in this direction ever since. For this reason I have an especially strong in­ terest in chapter 3 of A Parwrama of Harmonic Analysis, which is entitled Multiple Fourier Series.

76

THE MATHEMATICAL INTELLIGENCER

Extending the work of Carleson, Richard Hunt proved that the Fourier series of an LP(T) function converges almost everywhere, provided that p > 1. What happens in dimension two? Krantz points out that if "converges" means that the partial sums include terms of the series with indices lying in the dilates of a fixed polygon, the ana­ logue of Hunt's Theorem is true, whereas if "converges" means that the partial sums are taken to include the terms with indices lying in rectangles of variable eccentricity, then there is a counterexample, due to Charles Fef­ ferman. (Larry Gluck and I later added a small "bell and whistle" to that ex­ ample.) But the most important ques­ tion of what happens when "converges" means that the partial sums include the terms with indices lying in the dilates of an origin-centered disk remains un­ solved. Fefferman's Theorem that the unit ball is not a multiplier guarantees that it is not enough for p to be greater than 1, but gives no insight as to what happens when p 2. This leaves open the question of whether the Fourier se­ ries of anL2(T 2) function has circularly convergent partial sums almost every­ where. To my way of thinking, this question is the Mount Everest of mul­ tiple Fourier series. An interesting question not dealt with in chapter 3 is the question of uniqueness. Is the trigonometric series with every coefficient equal to zero the only one that converges at every point to 0? I have spent much of my life working on this question and have been pleased to see an almost com­ plete set of answers discovered. [AW] The only thing I want to say here is that uniqueness has been shown to hold in many cases, but here the situation is opposite to that for convergence of Fourier series mentioned above. We do know that uniqueness holds for circu­ larly convergent double trigonometric series, but we don't know if it holds for square convergent double trigonomet­ ric series. Speaking of chapter 3, one thing I would like to clarify is the definition of restricted rectangular convergence. I tried to explain this very subtle defm­ ition in my 1971 paper with Weiland, and I will take another try at it here. =

Fix

a

large

{ amn lm� 1,2,

number E >> 1.

Let

. ;n� 1,2, . . . be a doubly indexed series of complex numbers and denote their rectangular partial sums by SMN ��= 1 �;i=1 amn· Then say that S = ��amn is E-restrictedly rectangu­ larly convergent to the complex num­ ber s(E) if .

.

=

lim

M,N --7 oo

SMN =

s(E).

i, < ;, < E Finally say that S is restrictedly rec­ tangularly convergent if there is a sin­ gle complex number s such that for every E, no matter how large, s(E) ex­ ists and is equal to s. An example may help to clarify this. For n 2, 3, . . . , let an2,1 = n, an2,n -n, and let amn 0 otherwise. Notice that SMN of. 0 only if there is an n > N such that n2 ::::; M so that an2 1 is included in the partial ' sum, while an2,n is not. But then N2 < n2 ::::; M, so that MIN > N. Thus if any eccentricity E is given, as soon as N ex­ ceeds E, the condition MIN < E be­ comes incompatible with SMN of. 0. In other words, s(E) is 0 for every E, so that this series is restrictedly rectan­ gularly convergent to 0. And this hap­ pens despite the fact that SN2 ,N- 1 N, so that limrnin{M,NJ SMN does not ex­ ist, which is to say that S is not unre­ strictedly rectangularly convergent. Krantz has made wonderful selec­ tion choices for all of his chapters. The chapter titles are: overview of measure theory and functional analysis, Fourier series basics, the Fourier transform, multiple Fourier series, spherical har­ monics, fractional integrals singular in­ tegrals and Hardy spaces, modern the­ ories of integral operators, wavelets, and a retrospective. In particular, I think that ending with a chapter on wavelets represents a correct analysis of which way a good part of the winds of harmonic analysis have been blow­ ing for the past few years as well as a shrewd guess as to which way they will blow in the near future. A botanist re­ cently asked me for some help in find­ ing a good mathematical representa­ tion for ferns that she has been studying. Although my work is usually not very applied, I have looked into this a little bit and it seems likely that wavelets may prove to be the right tool. =

=

=

=

_, oo

A Panorama of Harmonic Analysis is Cams Mathematical Monograph number 27. The Publisher, the Mathematical Association of America, says that books in the series "are intended for the wide circle of thoughtful people familiar with basic graduate or advanced undergraduate mathematics . . . who wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises." Krantz has done an admirable job of carrying out the publisher's intentions. The right way to read this book is quickly, with-

out too much fussing over the details. While other books, such as those by Zygmund[Z] and Stein and Weiss[SW], are probably better for a graduate stu­ dent who will need to achieve techni­ cal competence in the area, A Panorama ofHarmonic Analysis pro­ vides an excellent way of obtaining a well-balanced overview of the entire subject.

Analysis and Nonlinear Differential Equations in Honor of Victor L. Shapiro, Contemporary Math., 208(1 997), 35-71 . [I] S. lgari, Lectures on Fourier Series in Several Variables, University of Wisconsin, Madison,

1 968. [SW] E. M. Stein and G. Weiss, Introduction to Fourier

Analysis

on

Euclidean

Spaces ,

Princeton Univ. Press, Princeton, 1 971 .

[Z] A. Zygmund,

Trigonometric Series, 2nd rev.

ed., Cambridge Univ. Press, New York, 1 959.

REFERENCES

[AW] J. M. Ash and G. Wang, A survey of

Department of Mathematics

uniqueness questions in multiple trigono­

DePaul University

metric series, A Conference in Harmonic

Chicago, IL 6061 4

Continued from p. 66 BIBLIOGRAPHY

Funktion zwischen imaginaren Grenzen, Journal fur die reine und

Bieberbach, L. (1 930) Lehrbuch der Funktionentheorie, Springer Verlag,

angewandte Mathematik 152, 1 -5.

Berlin, (1 st ed. 1 92 1 ) . Bacher, M. 1 896 Cauchy's Theorem o n complex integration, Bulletin of the American Mathematical Society (2) 2, 1 46-9.

Briot, C.A.A. and Bouquet, J.C. (1 859) Theorie des fonctions double­ ment periodiques et, en particulier, des fonctions elliptiques, Paris.

Bl'iot, C.A.A. and Bouquet, J.C. (1 875) Theorie des fonctions elliptiques, Paris. Denjoy, A. (1 933), Sur les polygones d'approximation d'une courbe rectifiable, Comptes Rendus Acad. Sci. Paris 1 95, 29-32. Ahlfors, L.V. (1 953) Complex Analysis, McGraw-Hill, New York. Goursat, E. (1 884) Demonstration du theoreme de Cauchy. Acta Mathematica 4, 1 97-200.

Goursat, E. (1 900) Sur Ia definition generals des fonctions analytiques, d 'apres Cauchy, Transactions of the American Mathematical Society, 1,

1 4-1 6.

Mittag-Leffler, G. (1 873), Forsok tillett nytt bevis for en sats inom de definita integralemas teori, Svenska Vetenskaps-Akademiens Handlingar. Mittag-Leffler,

G.

(1 875)

Beweis tor den

Cauchy'schen

Satz,

Nachrichten der Koniglichen Gesellschaft der Wissenschaften zu G6ttingen, 65-73.

Moore, E. H. (1 900) A simple proof of the fundamental Cauchy-Goursat Theorem, Transactions of the American Mathematical Society 1 , 499-506. Osgood, W.F. (1 928) Lehrbuch der Funktionentheorie, Teubner, Leipzig, 5th ed (1 st ed. 1 907). Perron, 0. 1 952 Alfred Pringsheim, Jahresbericht der Deutschen Mathematiker Vereinigung 56, 1 -6.

Pringsheim,

A.

(1 895a)

Ueber den Cauchy'schen

lntegralsatz,

Sitzungsberichte der math-phys. Classe der K6nigliche Akademie der Wissenschaften zu Munchen, . 25, 39-72.

Heffter, L. (1 902) Reelle Curvenintegration, G6ttingen Nachrichten,

Pringsheim, A. (1 895b) Zum Cauchy'schen lntegralsatz, as above,

26-52. Heffter, L. (1 930) Ober den Cauchyschen lntegralsatz, Mathematische

Pringsheim, A. (1 898) Zur Theorie der Doppel-lntegrale, Sitzungs­

Zeitschrift 32, 476-480.

Hurwitz, A. and Courant, R. (1 922) Allgemeine Funktionentheorie und elliptische Funktionen. Springer Verlag, Berlin.

Jordan, C. (1 893) Cours d'analyse, 3 vols, Gauthier-Villars, Paris. Kamke, E. (1 932) Zu dem lntegralsatz von Cauchy, Mathematische Zeitschrift 33, 539-543.

Knopp, K. (1 930) Funktionentheorie, Springer Verlag, Leipzig and Berlin. Lichtenstein, L. (1 91 0) Ober einige lntegrabilitatsbedingungen zwei­ gliedriger Differentialausdrucke mit einer Anwendung auf den

Cauchyschen lntegralsatz, Sitzungsberichte der Mathematischen Gesellschaft, Berlin, 9.4, 84-1 00.

Malmsten, C.J. (1 865) Om definita integraler mellan imaginara granser, Svenska Vetenskaps-Akademiens Handlingar, 6.3.

Mittag-Leffler, G. (1 922) Der Satz von Cauchy Ober das Integral einer

295-304. berichte . . . Munchen 28, 59-74.

Pringsheim, A. (1 899) Zur Theorie der Doppel-lntegrale, Green'schen und Cauchy'schen lntegralsatzes, Sitzungsberichte . . . Munchen 29, 39-62.

Pringsheim,

A.

( 1 90 1 )

Ueber den

Goursat'schen

Beweis

des

Cauchy'schen lntegralsatzes, Transactions of the American Mathe­ matical Society 2, 4 1 3-421 .

Pringsheim, A. (1 903) Der Cauchy-Goursat'sche lntegralsatz und seine O bertragung auf reelle Kurven-lntegrale, Sitzungsberichte . Munchen 33, 673-682.

Pringsheim,

A.

(1 929)

Funktionentheorie,

Kritische-historische

Sitzungsberichte

Bemerkungen

Bayerische

Akademie

zur der

Wissenschaften zu Munchen, 281 -295.

Walsh, J.L. (1 933), The Cauchy-Goursat Theorem for rectifiable Jordan curves, Proc. Nat. Acad. Sci, of USA 1 9 , 54Q-541 .

VOLUME 22, NUMBER 4, 2000

77

k1f'I.I.M9.h.i§i

Robin Wilson

Indian Mathematics

I

A

round 250 BC King Ashoka, ruler of

maticians of the first millennium AD

most of India, became the first

were Aryabhata (b. 476) and Brahma­

Buddhist monarch. The event was cele­

gupta (b. 598). Aryabhata gave the first systematic treatment of Diophantine

brated by the construction of pillars carved with his edicts. These columns

equations (algebraic equations where

contain the earliest known appearance

we seek solutions in integers), ob­

of what would eventually become our

tained the value 3.1416 for TT, and pre­

Hindu-Arabic numerals. Unlike the com­

sented formulae for the sum of natural

plicated Roman numerals,

and the

numbers and of their squares and cubes;

Greek decimal system in which differ­

the first Indian satellite was later named

ent symbols were used for 1, 2, . . . , 9,

after him, and he is commemorated on

10, 20, . . . , 90, 100, 200, . . . , the Hindu

an Indian stamp. Brahmagupta dis­

number system uses the same ten digits

cussed the use of zero (another Indian invention) and negative numbers, and

throughout, but in a place-value system where the position of each digit indi­

described a general method for solving

cates its value. This enables calculations

quadratic equations. He also solved

to be carried out column by column.

Indian mathematics can be traced

quadratic Diophantine equations such as 92J:2 + 1 =

y2, obtaining the integer

back to around 600 Be, and a number

solution x = 120, y = 1 151.

work on arithmetic, permutations and

cians and astronomers became inter­

of Vedic manuscripts contain early

In later years Indian mathemati­

combinations, the theory of numbers,

ested in practical astronomy, and built

and the extraction of square roots.

magnificent observatories such as the

The two most outstanding mathe-

Vedic manuscript

Jantar Mantar in Jaipur.

Indian Ashoka column

Nepalese Ashoka column

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

80

THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

Aryabhata satellite

Jantar Mantar