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 [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.
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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 ! ---------------------------------------------------------------------------------------
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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 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
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
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)
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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