Letters to the Editor
The Mathematical lntelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
More on Vilnius
Zygmund at Mount Holyoke
I was outraged when I read the article
In the article "Vilnius Between the
titled "Vilnius between the Wars" in the
Wars" (Mathematical InteUigencer 21, 4, Fall 2000), the authors neglect to mention the five years, 1940-45, during
Fall 2000 issue of
The lnteUigencer.
Vilnius, or Wilno as the authors point out it was known in Polish, or Vilna, as
which Antoni Zygmund served on the
it was known by the 60,000 Jews who
Mount Holyoke College faculty. Indeed,
lived there-some
300-4> of the popula
tion-was a center of the flowering of
President
Ham
of
Mount
Holyoke
had been helpful in making it possi
Jewish life and culture in Poland. Jews
ble, through diplomatic channels, for
had made up a substantial proportion
Zygmund to leave Europe.
of the population for over 500 years; in
Moreover, emigration to the United
fact Napoleon had dubbed Vilna the
States for an academic at that time de
Jerusalem of Poland. All of this is doc
pended to a large degree on the as
From that Place
surance of a teaching position. Mount
and Time, a memoir by the noted his
Holyoke gave that assurance without
umented in the book
the advantage of a personal interview,
torian Lucy S. Dawidowicz. As for the University, it was re
having the courage to assume the risk,
nowned, not only as a center of learn
even though there was surely much un
ing, but as a hotbed of anti-Semitism.
certainty about Zygmund's suitability for
Its students repeatedly incited violence
an undergraduate college in which few
against
men had served as faculty members.
the Jewish
population
and
eventually pressured the administra
Norbert Wiener arranged for Zyg
tion into "ghetto bench" seating in the
mund to spend the Spring semester of
back of the classrooms for the few re maining Jewish students. The violence reached a peak in
1931 and led to the
1940 at MIT and in March drove him to him to
South Hadley to introduce
President Ham and the mathematics
death of a Polish student. Every year
chair, Marie Litzinger. In an amusing
thereafter, students from the University
notation by Marie Litzinger on a letter
would rampage in the streets on the an
from Zygmund, Wiener is identified for
niversary of his death, beating up Jews
President Ham's benefit as the son of
and breaking shop windows.
Leo Wiener, philologist and translator
None of this is mentioned in the ar ticle, which purports to be a capsule
of Tolstoy. In
1988 Mount Holyoke awarded
history not just of mathematics in
Zygmund an honorary doctorate and
Vilnius, but of the University itself, and
held a symposium in his honor at
of its role in the history of the city. This
which several of his former students
has the sad effect of appearing to make
participated, including Felix Browder,
the authors part of the ongoing attempt
Ronald
to expunge the role that Polish anti
Guido Weiss. Marshall Stone also came
Semitism has played in the unfortunate
to greet his old friend.
history of that country.
Coifman,
Peter Jones,
Upon Zygmund's death in
and
1992, the
larger part of his mathematical library,
Jacob E. Goodman
including many volumes in Polish, was
Department of Mathematics
left to the Mathematics Department at
City College, CUNY
Mount Holyoke, where it is displayed
New York, NY 1 0031
in the Department's seminar room.
USA e-mail:
[email protected]
We also have a file of correspon dence dating from his last year in
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
3
Poland (1939). A letter to J. D. Tamarkin of Brown University from Wilno (Vilnius), written on 5 November 1939, concludes with the paragraph:
I do not, of course, do any mathe matical work now. Even ifIforget my personal troubles, I cannot forget the horrors of the war and the immense suffering of people who had to leave Warsaw, sometimes with small chil dren, and were the whole time bombed by German aeroplanes. In particular, Wilno isfull of such people and I doubt if all of them will be able to survive the coming winter. Not to mention the thoughts of the barbarous treatment by the Germans of the Polish and Jewish population in the occupied territory. It is sometimes difficult to imagine the horrors. Finally, from our perusal of the College archives in preparation for the honorary degree, it became clear that Zygmund was extremely successful as a teacher at Mount Holyoke. One might have supposed that a research oriented mathematician speaking in a language not his own might be severely disadvantaged in teaching, but the record belies this. His departure at the end of WWII was deeply felt.
(the Madelung constant; see [1], [2]). Moreover, there are quite fast and sim ple formulas for r2 and r4: r2 is 4 times the number of divisors of n, unless there is a prime p congruent to 3 (mod 4) such that pm divides n, pm+l does not, and m is odd, and in that case it is 0; r4(n) is 8 times the sum of those di visors of n that are not divisible by 4. Ewell mentions that his algorithm 3 for r3(n) is O(n 12). One ought to point out that the same complexity is ob tained by simply examining all lattice points inside or on the sphere of radius Vn and counting those that are on it. But there is another approach that is much faster, having complexity O(n): count the number of representations of each possible n - i2 as a sum of two squares. Precisely,
r3(n) = r2(n) + 2
Vn
I
[r2 (n - i2)].
i=l
This function (with some further im provements as discussed in [3]) is in Mathematica as SumOfSquaresR, and it takes only about a second to deter mine the value of r3(n) for n < 107. REFERENCES
[1 ] D. Bressoud and S. Wagon , A Course in Computational Number Theory, Key College Press (Emeryville CA) and Springer-Verlag (NY), 2000.
Lester Senechal
[2] R. E. Crandall, New representations for the
Harriet Pollatsek
Madelung constant, Experimental Mathe
Mathematics Department
matics 8 (1 999), 367-379.
Mount Holyoke College
[3] E. Grosswald, Representations of Integers
So. Hadley, MA 0 1 075
as Sums of Squares, Springer-Verlag (NY),
e-mail:
[email protected]
1 985.
[email protected] Stan Wagon Macalester College St. Paul, MN 551 05 LaHice Points on Spheres,
USA
Quickly
e-mail:
[email protected]
The article by John Ewell, "Counting lattice points on spheres," in the Fall 2000 issue raises the interesting ques tion of whether there is a good algo rithm to compute r3(n), the number of representations of an integer n as a sum of three squares. This is an im portant and intriguing problem in that there is an intimate connection be tween this function and the amount of energy holding a salt crystal together
4
THE MATHEMATICAL INTELLIGENCER
John Ewell comments: To Stan Wagon's informative letter I add one observation, pertinent when very large n are in question: my algorithms are purely additive, not requiring factoring of n or any other integers involved.
Granville Sewell's Opinion piece in our Fall 2000 issue provoked many reactions from readers-too many to handle. Here are two of them; others will follow in the next issue, with Sewell's rejoinder. -EDITOR's NOTE The Credibility of Evolution
Granville Sewell ("A Mathematician's View of Evolution," v. 22, no. 4) should stick to mathematics. Biologists who read this in The Intelligencer must feel the same way we mathematicians would were an article describing how to square a circle with straightedge and compass written by a biologist/amateur mathe matician to appear in the Journal of Population Biology. Sewell's article is riddled with errors; to refute all of them would require a response much longer than the original article. Nor are the er rors new: all the ideas have previously appeared in the creationist literature. I will limit myself to a few comments and provide pointers to sources where more detailed refutations appear. I do speak partly from the biologist's standpoint. Though a mathematician/ engineer by profession (thesis in ap proximation theory, employment at uni versities and Silicon Graphics), I have spent many months doing volunteer field work in biology, much of it with the Stanford Center for Conservation Biology. I ask you: how open-minded are you toward the fellow who sends your math department a 20-page angle-trisection technique? Oh, it could be that all pro fessional mathematicians have over looked a flaw in Galois theory for a hun dred years and that we'll have to toss out most of mathematics when the ama teur proves us all wrong. Theodosius Dobzhansky has written, "Nothing in bi ology makes sense except in the light of evolution." If Sewell is right, almost all of modern biology falls apart, and there is no theory to replace it There are ex actly zero articles published in the ref ereed biological literature about "intelli gent design." The creationists claim this is due to a giant conspiracy against them. It's pretty much the same as the conspiracy we mathematicians have against angle-trisectors. Look at Sewell's references-all are from the popular lit-
erature, and the publication dates are 1996, 1987, 1960, 1982, and 1956. Behe's "irreducible complexity" the ory, which Sewell expounds, basically amounts to statements of the form, "I can't figure out how this complex sys tem arose by natural selection, so it can not have done so." When someone goes to the trouble of providing an explana tion, that example is tossed out, and a new complex system is substituted. Responding to an infinite sequence of such challenges is pointless, especially since it's so easy to make up a challenge, and so difficult to respond. Suppose I were to claim that multiplication of in tegers is not commutative, and I'll show you by providing two 100-digit integers x andy such that xy -=!= yx. Suppose you then multiplied them both ways and showed me the result was the same, and I just gave you another pair to test. And another. And another. . . . The second law of thermodynamics does not say "natural forces do not cause ext;remely improbable things to happen"; it talks about entropy. With the sun's en tropy increasing at a fantastic rate, it's easy to spare a tiny portion of that to ac count for the decrease in biological en tropy on earth. lf I flip a coin 1000 times, and tell you the result ITHTHHT . . . , my result, whatever it may be, is ex tremely unlikely. It would only happen once in 21000 times. lf you flip coins for a billion years, it'll probably never hap pen again. The second law did nothing to prevent my unlikely result. One problem with Sewell's computer code example is that he assumes the program evolves toward a specific goal. In evolution, there is no specific target just something that works better than the competition. lf the universe were restarted, Sewell is right that it is in credibly unlikely that humans would evolve again. But something would. The National Center for Science Ed ucation (www.natcenscied.org) fights legal battles all over the country to keep creationism out of the public schools. The talk.origins archive (www.talkorigins.org) maintains a huge collection of articles that refute all the points made by Sewell and other creationists. Good popular books re futing Sewell's ideas include: Climbing
Mount Improbable by Richard Dawkins, Darwin's Dangerous Idea by Daniel C. Dennett, The Red Queen by Matt Ridley, and almost any collection of Stephen Jay Gould's essays from Natural History magazine. I deplore publication of this kind of junk, especially in a reputable journal. You can be certain that Sewell's article will be referenced for the next 20 years in the creationist literature as proof that evolution has been debunked by mathematicians. The fact that you have carefully labeled his article as an opin ion piece will have no effect. In addition, such articles make politicians think that there is merit to the creationist claims, and this leads to the exclusion of evolution, the big bang theory, and other religiously unaccept able ideas from high school science texts. There is a big enough problem with scientific illiteracy as it is, and having more people who believe that antibiotics and pesticides cannot be overused since it's impossible for bac teria and insects to evolve resistance to them will not help anyone. The Catholic church and many other major Christian churches have issued statements saying that there is no con flict between evolution and Christian faith. Only those who interpret the Bible as literally true have a problem with it. Of course the Bible also says the world is flat-see Revelation 7:1, for example. Tom Davis 24603 Olive Tree Lane Los Altos Hills, CA 94024 USA e-mail: tomrdavis@earthlink. net The Credibility of Evolution 2
Tom Davis's reply to Granville Sewell effectively responds to many of Sewell's dubious objections to evolu tion. Without attempting to be com prehensive, I would like to address sev eral additional points. 1. Any biologist can tell you how sys tems that appear to be "irreducibly complex" (i.e., all parts of the system are required for the system to function) could evolve: through scaffolding. As H. Allen Orr wrote in his perceptive re view of Behe's book:
An irreducibly complex system can be built graduaUy by adding parts that, while initiaUy just advantageous, be come-because of later changes--es sential. The logic is very simple. Some part (A) initiaUy does some job (and not very weU, perhaps). Another part (B) later gets added because it helps A. This new part isn't essential, it merely improves things. But later on, A (or something else) may change in such a way that B now becomes in dispensable. This process continues asfurther parts getfolded into the sys tem. And at the end of the day, many parts may aU be required [1]. (This and other reviews of Behe's book can be found online [2].) 2. Sewell draws an analogy between bi ological systems, coded by DNA, and computer programs. He writes "to anyone who has had minimal program ming experience this idea [that a signif icantly better program can be made by accumulating small improvements] is equally implausible." Sewell seems to be entirely ignorant of the field of genetic programming, where what he claims is implausible is routinely done [3]. The analogy between computer programs and biological systems is useful, as far as it goes. But like all analogies, it is imperfect. lf we draw conclusions from our analogy that con flict with very strong evidence-<:on clusions such as that life could not have evolved over time-then we should suspect that it is the analogy, not the conclusion, that is mistaken. In particular, here are some features of biological systems that are not shared by most computer programs written by humans in modern com puter languages: their "program" codes for the assembly of physical systems; much of their "program" is historical legacy that is currently non-coding; small changes to the "program" often do not change the product (because the genetic code is robust and redun dant); and fmally, biological systems are self-reproducing and exist in an en vironment where they must interact with millions of competing systems. Sewell also seems unaware of pro-
VOLUME 23, NUMBER 3, 2001
5
grams, such as Tom Ray's Tierra [4], which simulate evolution through com peting pieces of computer code. 3. Sewell claims that the record of a computer program's development by humans "would be similar to the fossil record." But there is one similarity that we do not find. A human designing a computer program is free to incorpo rate improvements from programs de veloped by other humans in other parts of the world. On the other hand, this is precisely what we do not see in bio logical systems. (Horizontal transfer does occur, but not separated by space and time.) The genome of humans does
not include evolutionary improve ments discovered by flowering plants, for example. If today's biological di versity is the product of "intelligent de sign," it is entirely remarkable that this designer chose to make life appear to be arranged in a branching hierarchy consistent with common descent, without incorporating improvements across different branches. If life is de signed, why does it look so much as if it evolved?
Also available at http://www-polisci.mit. edu/bostonreview/BR2 1.6/orr.html. [2] Behe's empty box, http://www.world-of dawkins.com/box/behe.htm. [3] Zbigniew Michalewicz, Genetic Algorithms Data
Springer-Verlag, New York, 1994. hip.atr.co.jp/-ray/tierra/.
Jeffrey Shallit Department of Computer Science University of Waterloo
References
[1] H. Allen Orr, "Darwin v. Intelligent Design (Again)," Boston Review 2 1 (6) ( 1996-97).
Waterloo, Ontario N2L 3G 1 Canada
[email protected]
The photograph on our vol. 23, no. 2 issue, "Hyperbolic Handi work," was by Daina Taimina. She also took the photographs of crocheting accompanying the article (as well as being the princi pal crocheter).
THE MATHEMATICAL INTELLIGENCER
+
Programs,
[4] Tom Ray's Tierra home page. http://www.
GIVING CREDIT
6
Structures= Evolution
•., ·" ·"·"'
7T
Is Wrongl
Bob Palais
The ()pinion column offers
I
know it will be called blasphemy by some, but I believe that 7T is wrong.
For centuries 7T has received unlimited
7T radians, and so on.The opportunity to impress students with a beautiful and natural simplification is turned
praise; mathematicians have waxed
into an absurd exercise in memoriza
rhapsodic about its mysteries, used it
tion and dogma.An enlightening anal
as a symbol for mathematics societies
ogy is to leave clocks the way they are
and mathematics in general, and built
but define an hour to be
it into calculators and programming
that case, 15 minutes or a quarter of a
30 minutes.In
languages. Even a movie has been
clock would indeed be called half an
named after it.* I am not questioning
hour, just as a quarter of a circle is half
its irrationality, transcendence, or nu
of 7T in mathematics! Even mathemati
merical calculation, but the choice of
cally sophisticated software packages
the number on which we bestow a sym
prefer to use
bol conveying deep geometric signifi
circle rotation. We can't really blame
cance. The proper value, which does
them for the fact that 7T is wrong.
90° to indicate a quarter
deserve all of the reverence and adu
Perhaps more convincing to mathe
lation bestowed upon the current im
maticians is the litany of important the
postor, is the number now unfortu
orems and formulas into which this
nately known as 27T.
ubiquitous factor of 2 has crept and
I do not necessarily feel that 7T can
propagated: Cauchy's integral formula
or even should be changed or replaced
and Fourier series formulas all begin
with an alternative (though I've by now
the international mathematical
with
received some good suggestions!), but
the Gaussian normal distribution both
community. Disagreement and
it is worthwhile to recognize the reper
carry it, the Gauss-Bonnet and Picard
controversy are welcome. The views
cussions of the error as a warning and
theorems
a lesson in choosing good notational
(Archimedes showed that the area of
conventions to communicate mathe
the unit sphere is the area of the cylin
mathematicians the opportunity to write about any issue of interest to
and opinions expressed here, however,
�
2
,
Stirling's approximation and
have
the
mark
of
27T.
are exclusively those of the author,
matical ideas.I compare the problem
der of the same radius and height, or
and neither the publisher nor the
to
if
twice the circumference of the unit cir
editor-in-chief endorses or accepts
Leonhard Euler had defined e to be .3678 ..(the natural decay factor equal
cle: 47T = 2(27T).) The blight of factors
), in which case there would
in Maxwell's equations (Gauss's law,
responsibility for them. An ()pinion
to
what
. !
2_71 ...
would
have
occurred
of 2 even affects physics, for example
should be submitted to the editor-in
be just as many unfortunate minus signs
Ampere's law,
chief, Chandler Davis.
running around from that choice as
and Planck's constant 277• Euler's for-
there are factors of 2 from 7T = 3.14
...
The most significant consequence
mula
Coulomb's constant) h
should be ei" = 1 (or ei"12 = -1,
in which case it involves one more fun
of the misdefinition of 7T is for early
damental constant, 2, than before).
geometry and trigonometry students
Wouldn't it be nicer if the periods
who are told by mathematicians that
of the fundamental circular functions
radian measure is more natural than
cos and sin were 7T rather than 277'?
degree measure.In a sense it is, since
Wouldn't it be nicer if half-plane inte
a quarter of a circle is more naturally measured by
1.57 ...than by 90. Un
fortunately, this beautiful idea is sabo taged by the fact that 7T isn't
6.28 ...,
grals such as the Hilbert transform were indicated by the
appearance of a
factor of 2 rather than its disappear ance?
which would make a quarter of a cir
The sum of the interior angles of a
cle or a quadrant equal to a quarter of
triangle is 7T, granted.But the sum of
7T radians; a third of a circle, a third of
the
exterior angles of any polygon,
•For a non-technical movie, the mathematics was surprisingly good, except for the throwaway question "Surely you've tried all of the 216-digit numbers?" At one number per nanosecond, checking all 30-digrt numbers would take longer than the life of the universe!
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
7
sin ( x + 11r )
T
90°
=
Cn
=
sin ( x )
11r =
w
� 11r radians- a quadrant 4
=
f ( a)
1 17rr f(x)einx
-
71r
=
0
dx
z) � r f( dz 11rz}0z-a
The nth roots of unity:
i:!!!.i e n
The author's father, thinking that the formulas with
,
j
'"=
=
0, ...
, n -1
6.28 looked more wrong than sim
pler, constructed a macro in which he combined two pi's with their adjacent "legs" tied to gether as in a three-legged race! The formulas above use that symbol.
from which the sum of the interior an gles can easily be derived, and which generalizes to the integral of the cur vature of a simple closed curve, is 21T. The natural formula for area of a cir cle, 1rr , has the familiar ring of or it would have instilled good habits for representing quadratic quan tities and foreshadowed the connec tion between the area of a circle and the integral of circumference (with re spect to radius) better than 1rr2• Another way of putting it is that radius is far more convenient than diameter-
t 2 imv2;
igt2
consider what the unit circle means. If it weren't, I would agree that the tra ditional choice of1T was right. Of course you may say that none of this really matters or affects the math ematics, because we may define things however we like; and that is correct. But the analogy with e mentioned above, or the idea of redefining the symbol i to mean shows the true folly of 1T. Neither of these changes would change the mathematics, but nor would anyone deny they are ab surd. ·
7
tpj; A Source Book, L. Berggren, J. Borwein, P. Borwein. Springer-Verlag, New York 2000, p. 292.
8
THE MATHEMATICAL INTELLIGENCER
What really worries me is that the first thing we broadcast to the cosmos to demonstrate our "intelligence," is 3.14. .. .I am a bit concerned about what the lifeforms who receive it will do after they stop laughing at creatures who must rarely question orthodoxy. Since it is transmitted in binary, we can hope that they overlook what becomes merely a bit shift! I would not be surprised and would be interested to hear if this idea has been discussed previously, but I was unable to fmd any reference either in the wonderful Pi: A Source Book by Lennart Berggren, Jonathan Borwein, and Peter Borwein, or in Petr Beck mann's A History of Pi, or on the Internet. When I have suggested to peo ple that 1T has a flaw, their reactions range from surprise, amusement, and agreement, to "Of course, I knew it all along," to dismissal, to indignation. The historyt (I was surprised, along with everyone I tell, that the symbol was not in use in ancient Greece): Oughtred used the symbol1T/8 in 1647 for the ra tio of the periphery of a circle to its di ameter. David Gregory (1697) used 1rlp for the ratio of the periphery of a circle to its radius.The first to use1T as we use it now was a Welsh mathematician, William Jones, in 1706 when he stated 3. 14159 &c. = 1T. Euler, who had until then been using the letters p and c, adopted the symbol in 1737, leading to its universal acceptance. If only he or Jones had set Gregory's p to be 1 instead of Oughtred's 8, our formulas today would be much more elegant and clear. Acknowledgments
Many thanks to James Tucker, Nelson Beebe, Bill Bynum, Wayne Burleson, Micah Goodman, and Carolyn Connell for their contributions to this paper. Robert Palais 21 485 Wyoming Street Salt Lake City, UT 841 09 USA e-mail:
[email protected]
GREG N. FREDERICKSON
Geometric Dissections Now Swing and Twist
geometric dissection is a cutting of a geometric figure into pieces that we can rearrange to form another figure. As visual demonstrations of relationships such
as
the Pythagorean theorem, dissections have had a surprisingly rich his
tory, reaching back to Persian and Arabian mathematicians a millennium ago [1; 5; 29] and Greek mathematicians more than two mil lennia ago [4].As mathematical puzzles, they ef\ioyed great popularity a century ago, in newspaper and magazine columns written by the American Sam Loyd and the Englishman Henry Ernest Dudeney. Loyd and Dudeney set as a goal the minimization of the number of pieces.Their puzzles charmed and challenged readers, especially when Dudeney introduced an intriguing variation in his 1907 book, The Canterbury Puzzles [12]. After presenting the remarkable four-piece solution for the dissection of an equi lateral triangle to a square, Dudeney wrote:
I add an iUustration showing the puzzle in a rather cu rious practical form, as it was made in polished ma-
hogany with brass hinges for use by certain audiences. It wiU be seen that the four pieces form a sort of chain, and that when they are closed up in one direction they form the triangle, and when closed in the other direction they form the square. This hinged model (Fig. 1) has captivated readers ever since. It is just too nifty not to have been described in at least a dozen other books in the intervening years [3; 6; 9; 14; 17; 18; 26; 30; 32; 31; 36; 37]. Howard Eves [15] described a set of four connected tables that swing around to form either a square or a triangular top, thus accommodating card games with either three or four players! There is some thing irresistible about the idea of swinging hinged pieces
Figure 1. Hinged dissection of a triangle to a square.
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
9
hingeable dissections,
I found myself writing a book about [16]. Consequently, I have not resolved the question
another way to form an one way to form one figure, and a physical model to en need really not do You gure. fi other joy this property. Once you have examined Figure 1, you
them
will be swinging mental images of the pieces around in your
figures.
of whether hingeability restricts the class of dissectable In comparison to finding normal (unhingeable) dissec
mind.
A dissection of one figure to another is (swing)-hinge able if we can link its pieces together with swing hinges
tions, finding hingeable dissections seems much harder, like accomplishing a difficult task with one hand tied be
so that when swung appropriately on these hinges, the
hind your back The standard dissection techniques do not
pieces form one figure, and when swung around in some
seem to be adequate and need to be re-engineered for these
other way, the pieces form the other figure. It is unhinge able otherwise. For dissections of two-dimensional figures,
more challenging problems. In the end, however, suitably
it is natural to assume that the pieces stay in the plane as
strips, slides, steps, and polygonal structure, along with a
we swing them around on their hinges. Thus we tum over
goodly measure of determination and hard work, allow us
no pieces when we use swing hinges. Let's call a set of
to find a wide variety of hingeable dissections.
pieces plus the hinges that link them together a
hinged
as
semblage. A dissection of n > 1 figures to one is hingeable if the following two conditions hold. First, each of the n figures,
modified versions of techniques such as tessellations,
The triangle-to-square dissection was not the first dis section recognized to be hingeable. Kelland
[23] identified
a three-piece dissection of two attached squares to one square as hingeable. Others who have either identified dis
[2;
must have its own hinged assemblage that forms the fig
sections as hingeable or come close to doing so include
ure. Second, the set of n hinged assemblages must together
8; 10; 22; 25; 33; 38].
form the large figure. A similar notion applies to dissecting n > 1 figures into m > 1 figures, as long as no proper sub set of the n figures is of total area equal to that of some subset of the m figures. Then the dissection is hingeable if every piece is in one of m + n 1 assemblages, which to gether form the n figures and which also form the m fig ures. If a proper subset of the n figures were of area equal to that of some subset of the m figures, then we could treat
Besides the standard hinges, which we shall call swing hinges, other hinges are possible. These variations allow us to tum pieces over. I was particularly intrigued by twist hinges, which use a point of rotation on the interior of an
-
the dissection as the union of several dissections, and use fewer than
m+n
-
1 assemblages.
edge that separates two pieces. With twist hinges, we can flip one piece over relative to the other, using rotation by
180° through the third dimension. One of the earliest such dissections is by William Esser, Ill, who was awarded a U.S. patent
[13] for what was essentially the dissection of an el
lipsoid to a heart-shaped object. Esser sliced the ellipsoid
When I wrote my first book on geometric dissections
along a diagonal through its center.
A rotation of 180° then
[17], I became fascinated with hingeable dissections. I ex plicitly illustrated hingings in 18 figures and identified the
of this dissection is in Figure
hingeability of a number of others. I discovered that many
cates the position of a twist hinge. The piece that is flipped
produces the heart shape. The two-dimensional analogue
2. A small open circle indi
hingeable dissections had been published without any in
an odd number of times gets marked with an "*" in the el
dication that they are hingeable. There are enough hinge
lipse, and with a "*" in the heart. Note that after flipping
able dissections that it is tempting to ask: Given any pair
the piece, the assemblage is rotated by
of figures that are of equal area and bounded by straight
symmetry of the heart. Ernst Lurker, a sculptor and artist,
line segments, is it possible to fmd a swing-hingeable dis
discovered a similar dissection
45° to highlight the
[27].
The movement made possible by twist hinges seems
section of them? William Wallace [35], Farkas Bolyai [7], and P. Gerwien [ 19] proved the analogous property with respect to normal
rather limited. It came as a surprise that I could find a rel atively large number of twist-hingeable dissections. There
(unhinged) dissections: For any pair of figures that are of
are enough of these dissections that it is tempting to pose
equal area and are bounded by straight line segments, it is
a question similar to the one I have already posed: Given
possible to fmd a dissection between them consisting of a
any pair of figures that are of equal area and bounded by
fmite number of pieces. On the other hand, we could limit our attention to
translational dissections, in which we
move the pieces from one figure to another using only translation with no rotation. In this case, the possibilities are considerably restricted. Hadwiger and Glur
[21] gave a
simple characterization of which pairs of figures have such dissections. To develop insight into whether hingeability is restric tive in the same way that translation with no rotation is, I began searching for hingeable dissections. I chose as a nat ural goal the minimization of the number of pieces, subject to the dissection being hingeable. As I found more and more
10
THE MATHEMATICAL INTELLIGENCER
Figure 2. Twist-hingeable dissection of an ellipse to a heart.
straight line segments, is it possible to find a twist-hinge able dissection of them? And I admit once again that I have no idea whether the answer is yes or no. The material for this article is a selection from my new book [16]. Given the scope of the book, this article is only a small sampling. Succeeding parts of this article present more definitions, techniques for swing-hinged dissections, and techniques for twist-hinged dissections.
Figure 3. Swing-hingeable regular dodecagon to a square.
Definitions We consider dissections of regular polygons and regular
> 2, let {p} be a reg ular polygon with p sides. For any whole numbers p > 2 and q, with 1 < q < p/2, let {p/q} be a star polygon with p star polygons. For any whole number p
points (vertices), where each point is connected to the qth points clockwise and counterclockwise from it. Thus
{5/2)
represents our familiar five-pointed star, the pentagram. We assume that a figure such as a polygon or a star is an open set, so that its boundary is not part of the figure
to a square (see Fig.
32),
which we will make use of later.
With only two additional pieces, I have found a hingeable dissection that is related to Lindgren's unhingeable dissec tion (Fig.
3).
As in Lindgren's dissection, there are two dif
ferent edge lengths, one corresponding to the sidelength of the dodecagon, and the other to the side length of the square. All angles are multiples of
15°.
We can hinge the pieces as in Figure 4. Other ways would
itself. When we cut the figure along a sequence of line seg
work as well. For example, we could connect the lowest
ments, we effectively remove all points on those line seg
piece by attaching its upper right vertex to the lowest ver
ments, resulting in pieces that are open sets. When we as
tex of the triangle on the right. Note that we could have
semble two pieces, we "glue" them together by adding their
connected the four large pieces by three hinges in several
common boundary, minus the endpoints of the common
different ways.
boundary. We represent a hinge by the point at the center
Lindgren discovered
his dissection by superposing tes
sellations, and I did with mine too. This requires convert
of rotation of the pieces that the hinge attaches. Three (or more) pieces are also allowed to share a swing
ing the dodecagon into a set of pieces that tile the plane.
hinge, but the clockwise order of the pieces may not change.
Such a set is shown on the right in Figure 5. These six pieces
Also, two different swing hinges are allowed to abut.
are consistent with a partition of the dodecagon into squares, rhombi, and equilateral triangles, as shown on the left. The set of six pieces hinge together as shown on the
Swing-Hingeable Dissections Given a plane figure, a
tessellation of the plane is
a cover
ing of the plane with copies of the figure without gaps and without overlap [20]. The figure that we use to tile the plane
is a tessellation element and consists of one or more pieces. The technique of
superposing tessellations
is the follow
ing [17; 26]: Take two tessellations with the same pattern of repetition and overlay them so that the combined figure preserves this common pattern of repetition. The line seg ments in one tessellation induce cuts in the figure of the other, and vice versa.
left of Figure 6. They can then swing around to form the tessellation element on the right in Figure 6. We can then form the tessellation from these tessella tion elements in Figure 7. Small dots indicate the points of rotational symmetry in the tessellation. Half of these points have two-fold rotational symmetry, and the remaining ones have four-fold symmetry. We could have formed a simpler tessellation without rotating the tessellation elements, but such a tessellation would not have had the necessary points of rotational symmetry.
One more restriction makes the dissections hingeable.
rotational symmetry if rotating it 27T radians leaves it coinciding every detail with the original. It possesses n-fold rota in tional symmetry if the angle of rotation is 27Tin. Call a point about which there is rotational symmetry a symmetry point. Consider two tessellations of hinged elements with A geometric object has
by some angle smaller than
the same pattern of repetition. Suppose that we can super pose them so that points of intersection between line seg ments are at symmetry points. If they share no line seg ments of positive length in the superposition, then the induced dissection is hingeable. Harry Lindgren, an Australian patent examiner, gave a beautiful six-piece unhingeable dissection of a dodecagon
Figure 4. Swing-hinged pieces: dodecagon to square.
VOLUME 23, NUMBER 3, 2001
11
•
Figure 5. Partitioning a regular dodecagon.
Figure 6. Creating a hinged tessellation element for a dodecagon.
Figure 7. Tessellation of dodecagons.
Figure 8. Superposing dodecagons and squares.
Finally, we can superpose the tessellation derived from dodecagons with a tessellation of squares, shown with dashed edges in Figure 8. The superposed tessellations have edges that intersect only at points of rotational sym metry, shown by the dots. The dashed edges indicate how to make the remaining cuts in the dodecagon. The T-strip technique is as follows [17; 26]: Cut a figure into pieces that form a strip element. Then fit copies of this element together to form a strip, rotating every second el ement in the strip by 180°. Thus every two consecutive el ements in the strip share a point of twofold rotational sym metry, called an anchor point. Similarly create a T-strip for the other figure. Then crosspose the two T-strips, forcing an anchor point in one strip either to overlay an anchor point in the other strip or to fall on a boundary edge of the other strip. William Macaulay [28] observed that the strip technique is a type of tessellation method: The crossposing of two strips induces two tessellations and their corresponding superposition. With this insight and our lrnowledge of points of rotational symmetry we can adapt the T-strip tech nique to produce hingeable dissections. Clearly, if we over lay two anchor points, we are overlaying points of twofold rotational symmetry. Furthermore, any point at which two strip boundaries cross, or at which an anchor point falls on a strip boundary, is actually a point of twofold rotational symmetry with respect to the induced tessellations. To
adapt the T-strip technique, we just require that any non boundary edge of one strip intersects a nonboundary edge of the other strip only at a common anchor point. This technique allows us to dissect two figures that have no obvious geometric relationship other than equal area. Our next example involves a Greek cross, which we form by attaching four congruent squares to the sides of a fifth square, also congruent. Harry Lindgren [26] gave several seven-piece unhingeable dissections of a Greek cross to a regular hexagon. I have found an eight-piece hingeable dis section (Fig. 9), using the T-strip technique. This lovely dis section has twofold rotational symmetry. Also, if we cut the pieces out of wood so that the wood grain lines up when forming one figure, then the grain will line up when form ing the other figure. When we open the dissection up to il-
12
THE MATHEMATICAL /NTELUGENCER
Figure 9. Swing-hingeable dissection.
lustrate its hinging, as in Figure 10, we see that it is also cyclicly hinged. It is easy to maneuver the pieces to form either the cross or the hexagon while pushing only the two large pieces. We must cut each of the figures to produce the elements with which we tile the infinite strips. For the hexagon, we can make a cut from the midpoint of one side to the end point of the opposite side. We then hinge the two pieces at the midpoint (Fig. 11). We could have swung the bottom piece in the other direction, and in fact we use both such points in our hinged dissection. We can similarly form the strip element for the Greek cross, as shown in Figure 12. Again, a careful inspection of Figure 10 reveals that we use both possible hinge points in the cyclic hinging. There is a different hinging of the pieces that use just one of the two possible hinge points from Figure 11 and one of the two from Figure 12. Can the reader find it? Finally, we cross the hexagon and Greek cross strips in Figure 13. The dots identify anclwrpoints, which are points of twofold rotational symmetry within the strip. Consistent with the requirements of this method, we have forced each anchor point to either fall on an anchor point of the other strip or to fall on the boundary of the other strip. Some polygons tile the plane directly, and others tile af ter just a few fortuitous cuts. However, some need the as sistance of another polygon-they get by with a little help from their friends. Harry Lindgren [26] called this technique completing the tessellation and surveyed some notable ex amples. The technique produces the remarkable five-piece
Figure 14. Unhingeable dissection of an octagon to a square.
dissection of an octagon to a square (Fig. 14) that appeared in the anonymous Persian manuscript, Interlocks of Similar or Complementary Figures, from approximately 1300 C.E. Apparently, this method lay forgotten, only to be redis covered over 600 years later by the Cambridge mathemati cian Geoffrey T. Bennett, as announced by Henry Dudeney [11, c]. It bettered a seven-piece effort by Henry Dudeney [11, b], who stated in [11, a] that he believed that no such dissection had been previously published. In fact, James Blaikie had earlier posed the problem of dissecting an oc tagon to a square, for which Henry Martin Taylor [34] gave an eight-piece solution.
Figure 12. Twinned strip element for a Greek cross.
I I I lA
Figure 10. Swing-hinged pieces: hexagon to cross.
I-'
I
<
,
'
Figure 11. Twinned strip element for a hexagon.
,
,
'
'
',
'
'
'
,
,
,
'
,
>
I
,f
"I I I
Figure 13. Crossposition of strips.
VOLUME 23. NUMBER 3, 2001
13
...
I I I
...... (..
...,.. ...
I I I I
...., ... ...
I I I
Figure 15. Tessellation of octagons and little squares.
We can derive this remarkable five-piece unhingeable
Figure 16. Superposition of tessellations for an octagon to a square.
on the left of Figure 17, where the two identical pieces are
dissection by completing the tessellation. An octagon and
triangles. Moreover, one triangle is adjacent to the small
a small square of the same side-length together tile the
square inside the octagon in precisely the same way as the
plane, as we see in Figure 15.
other triangle is adjacent to the small square inside the large
A pair of unequal squares also tiles the plane. As Harry
square. Thus we can merge the small square with the ad
Lindgren observed, we can superpose these two tessella
jacent triangle in the resulting dissection on the right of
tions as in Figure 16, in which the dashed lines indicate the
Figure 17.
tessellation of squares. The large square is of area equal to
We thus must pay a penalty of only two additional pieces
that of the octagon, and the small squares are congruent to
to make the dissection hingeable. The pieces are linearly
each other. The dashed lines in the interior of the octagon
hinged, as we see in Figure 18.
indicate its dissection, and the solid lines indicate the cor
Regular polygons have an internal structure of rhom
responding fitting of the pieces into the large square. The
buses and half-rhombuses that can be exploited in various
dots indicate the overlaid points of twofold rotational
dissections [17; 26]. For example, a hexagon decomposes
symmetry, which at first seem to suggest that we will be
into three 60°-rhombuses, and a hexagram into six. Thus
able to hinge the dissection. Unfortunately, the symmetry
there is a simple hinged dissection (Fig. 19). The pieces are
points allow us to hinge only four of the five pieces in
labeled to indicate how the hinged assemblage on the left
Figure 14. In doing so, we have a hinge-point to spare, since
can be swung to achieve either the hexagon or one half of
there are four available hinge-points but we need only three
the hexagram.
of them to hinge four pieces. We can use all four hinge points if we split one of the four identical pieces so that
Twist-Hingeable Dissections
the small square can slip through. When we split that piece,
There are two general techniques for converting many of
we force two of the constituent pieces to be identical, so
the swing-hingeable dissections to be twist-hingeable. Two
that we can switch their positions. Such a splitting is shown
pieces that are connected by a hinge are hinge-snug if they
Figure 17. Congruent triangles, leading to hingeable octagon to a square.
Figure 18. Hinged pieces for an octagon to a square.
14
THE MATHEMATICAL INTELLIGENCER
2
Figure 1g, Hinged dissection of a hexagram to two hexagons.
are adjacent along different line segments in each of the
Using my design, Wayne Daniel crafted for me two won
figures formed, and each such line segment has one end
derfully precise models out of beautiful hardwoods. One
point at the hinge. Suppose that we have a swing-hingeable
model is of this twist-hinged dissection, with the three right
dissection such that each pair of pieces connected by a
triangular pieces out of Ipe wood and the remaining four
hinge is hinge-snug. We can then replace each swing hinge
pieces out of Peroba Rosa wood. The other model is of the
with a new piece and two twist hinges, so that the result
swing-hinged dissection on which the twist-hinged dissec
ing dissection is twist-hingeable. The new piece results
tion is based. I enjoy demonstrating these models in talks
from swiping an isosceles triangle from each piece that is
at conferences.
connected by the swing hinge. We then merge these two isosceles triangles together to form the new piece. �For an example, we return to the swing-hingeable dis section of an equilateral triangle to a square in Figure
1.
The equilateral triangle with its cuts appears on the left in
20, with the three pairs of isosceles triangles indi In the seven-piece twist-hingeable dissection on the right in Figure 20, these isosceles trian
Figure
cated by dashed lines.
gles have been merged, and each of the new pieces is a right triangle. It is easy to see that whenever a swing-hinge attaches two pieces at vertices whose angles sum to
. . · · · · · · · · · · · · · · · · · · · · · · · ·
180°,
the merging of the two isosceles triangles results in a right
Figure 21. Intermediate configurations for the triangle to square.
triangle. Figure
21 shows how to twist the hinges to convert the
triangle to the square. First twist both twist hinges of the
The second technique converts a swing hinge to a sin
right triangle at the midpoint of the left side of the equi
gle twist hinge with no increase in the number of pieces. It
lateral triangle, giving the configuration on the left of Figure
applies when the swing hinge connects two pieces that are
21. Then twist both twist hinges of the right triangle at the
hinge-snug, and the hinged assemblage on one of the sides
midpoint of the right side of the equilateral triangle, giving the configuration on the right of Figure
21. Two pieces
of the hinge is "hinge-reflective."
hinge-reflective if when we flip all
A hinged assemblage is pieces in this hinged as
along the base of the equilateral triangle come along for the
semblage to their other side, then there is no effective
ride. Finally, twist both twist hinges of the remaining right
change to the whole hinged assemblage. Suppose we have
triangle, giving the square.
two hinge-snug pieces, such that the hinged assemblage on
Figure 20. Swiping isosceles, for a twist-hingeable triangle to square.
VOLUME 23. NUMBER 3. 2001
15
2
Figure 22. Isosceles triangles onto swing-hinged hexagram to two hexagons.
2
Figure 23. Twist-hinged hexagram to two hexagons.
one side of the swing hinge is hinge-reflective. Then we can cut an isosceles triangle from one of the pieces, merge it onto the other piece, and place a twist hinge at the mid point of the cut edge. We illustrate this technique by converting the swing hinged dissections of a hexagram to two hexagons (Fig. 19) to be twist-hinged. We copy Figure 19 to Figure 22, adding dashed edges to indicate isosceles triangles that we will transfer. In the hinged assemblage on the left, we see that piece b hanging off piece a is hinge-reflective. Thus we will cut an isosceles triangle off piece a. Similarly, piece c is hinge-reflective, so that we can cut another isosceles tri angle off piece a. We then merge the first isosceles trian gle with piece b, and the second isosceles triangle with piece c, as shown in Figure 23. By taking advantage of the reflection symmetry, we have not increased the number of pieces in the dissection.
Figure 24. Twist-hingeable parallelogram to same-angled parallelo gram.
16
THE MATHEMATICAL INTELLIGENCER
Another general technique transforms a parallelogram to another parallelogram with the same angles (Fig. 24). We call this the parallelogram twist, or P-twist. A beauti ful feature of the P-twist is that the pieces are cyclicly hinged. The range of achievable dimensions depends on the parallelogram's dimensions a and b ::; a and its nonacute angle (}. We can more than double the length, going from a up to, but not including, a + v'a2 + b2 - 2ab cos (}. The second term in this expression represents the length of the longer diagonal in the parallelogram and is derived using the law of cosines. Of course, since rectangles are paral lelograms, the P-twist can transform one rectangle to an other. One could build an unusual coffee table, which would transform from one rectangle to another. For sta bility it might need to have six legs, one attached to each vertex of the triangles.
Figure 25. Second superposition of octagons and squares.
Next is a wonderful family of dissections. For any p > 2 there is a (2p + 1)-piece twist-hingeable dissection of a {2p} to a {p }. The dissection exhibits p-fold rotational sym metry. I discovered this family by taking the two tessella tions in Figure 16 and superimposing them as in Figure 25. Readers may note that in this superposition, points of twofold rotational symmetry coincide, although not in the same way as in Figure 16. This is less important for this dissection, because we do not use such points to position
Figure 26. Twist-hinged octagon to square. Figure 27. Twist-hinged pentagram to pentagon.
swing hinges. More importantly, the centers of the octagon and the large square coincide, as do the centers of the small squares. We can locate isosceles triangles by identifying line segments between vertices of the small square in the oc tagon tessellation and nearby vertices of the small square in the tessellation of squares. If we take the corresponding dissection, cut such appropriate isosceles triangles off cer tain pieces, and add them to other pieces, we can get the twist-hinged dissection shown in Figure 26. The reader has been left with the task of identifying these isosceles trian gl�s, but this is not so hard, since there is a twist hinge at the midpoint of the base of each one. Although we use tessellations to derive this dissection, we do not need to, and this is the key to dissecting any (2p 1 to a (pl. Just overlay the (2pl and the {p} so that their cen ters coincide and each side of the (pI intersects the mid point of a side of the (2pl. Then identify the bases of cor responding isosceles triangles, and infer the pieces. We can prove correctness with the use of fairly simple trigono metric identities. Because the method is related to the com pleting the tessellation method, but in general is not based on tessellations, I call it completing the pseudo-tesseUa
tion. Remarkably, there is another family of dissections of a similar nature. Consider any integers p > 4 and 2 ::;:; q ::;:; (p + 1)/3. Then there is a (2p + 1)-piece twist-hingeable dis-
Figure 28. Superposition of hexagrams, triangles, and hexagons.
section of the (p/q I to the {p 1. The dissection exhibits p-fold rotational symmetry. As an example, a twist-hingeable dis section of a pentagram to a pentagon is shown in Figure 27. The approach is the same as what we used for the previous family of dissections, if we treat the reflex angles of the (p/q I as vertices too. To ensure that each side of the (pi inter sects a side of the (plq I at a midpoint, p and q must satisfy the condition 4 cos(q1rlp) cos((q - 1)1r/p) ;::: cos(1r/p). For positive integers q > 1 and p ;::: 2q + 1, this condition is equivalent top ;::: 3q - 1. That the same approach works for both {plql and (2pl suggests that these can be unified. Indeed, this is the case, if we relax the constraint that q be a whole number and interpret (plql appropriately. The previous family of dissections includes a thirteen piece dissection of the hexagram to the hexagon. We can do better. In [ 1 7] , I gave a seven-piece unhingeable dissec tion of a hexagram to a hexagon, which has three pieces that we must turn over. However, the pieces that we turn over are the key to a nifty twist-hingeable dissection. I de rived my seven-piece unhingeable dissection by complet ing the tessellations, as shown in Figure 28. The solid lines in Figure 29 indicate the seven-piece un hingeable dissection. We add two isosceles triangles (indi cated by dotted lines) to each of the three small triangles, giving three triangles that we can twist-hinge. Producing the three new triangles does not yield a dissection that is completely twist-hingeable, because there is an equilateral triangle that we must transfer from the center of the hexa gram to the center of the hexagon. Following an approach
Figure 29. Derivation of a twist-hingeable hexagram to a hexagon.
VOLUME 23, NUMBER 3, 2001
17
Figure 30. Twist-hingeable dissection of a hexagram to a hexagon. Figure 31 . Intermediate configuration of a hexagram to a hexagon.
similar to that in Figure 17, we identify two irregular tri angles that can swap positions, as shown with dashed edges. To make these new pieces twist-hingeable, we in troduce more isosceles triangles (dotted edges). As luck would have it, we can glue all four of these isosceles triangles together, producing a trapezoid for our ten-piece twist-hingeable dissection (Fig. 30). An observant reader will see that I have cyclicly hinged eight of the pieces, and a skeptical reader may wonder if this actually works. I was not sure myself whether something so re markable was possible until I had constructed and tested a rough model out of a thin foamboard and toothpicks. Afterwards, I verified mathematically that it does indeed work Note that there are five pieces centered on the trape zoid that play somewhat the same role as each of the two large pieces. To convert the hexagram to hexagon, flip the two large pieces and the trapezoid-centered five, rotating them simultaneously about the axes shown with dotted lines, while partially turning the small triangles so as to ac commodate the differing levels of the twist hinges on the parallel edges. Figure 31 shows a perspective view of the configuration after rotating the pieces by 90° from their po sition in the hexagram. The three dotted lines identify an imaginary equilateral triangle that stays fixed as the pieces rotate. Each vertex of this triangle is the center of a smaller equilateral triangle (not shown) adjacent to the long edge of a small triangle. Furthermore, the axis of the twist hinge between a large piece and a small triangle pierces the cen-
Figure 32. Lindgren's unhingeable dodecagon to square.
18
THE MATHEMATICAL INTELLIGENCER
ter of that smaller equilateral triangle. As the large pieces complete their turning, the small pieces return to their orig inal side up. Again, Wayne Daniel crafted a wonderfully pre cise model of this dissection for me. Oftentimes, special methods produce twist-hinged dis sections that are not possible using the general techniques. A final treat, of a twist-hinged dodecagon to a square, il lustrates this point. Harry Lindgren [24] gave a six-piece un hingeable dissection of a dodecagon to a square (Fig. 32), on which I base a twist-hingeable dissection. Dotted lines in Figure 33 indicate isosceles triangles to switch from one piece to another. Add two isosceles triangles to the equi lateral triangle, and use two twist hinges to flip the result-
Figure 33. Add twists to a dodecagon to a square.
Figure 34. Twist-hinged dodecagon to square.
REFiiRiiNCES
A U T H O R
[1 ) Abu'I-Wafa' ai-Buzjanf. Kitab ffma yahtaju al-sani' min a' mal al handasa (On the Geometric Constructions Necessary for the Artisan). Mashhad, Iran: Imam Riza 37, copied in the late 1 0th or the early 1 1 th century. Persian manuscript. (2] Jin Akiyama and Gisaku Nakamura. Dudeney dissection of polygons. Res. Institute of Educational Development, Tokai Univ., Tokyo 1 998. [3) Jin Akiyama and Gisaku Nakamura. Transformable solids exhibi tion. 32-page color catalogue, 2000. [4) George Johnston Allman. Greek Geometry from Thales to Euclid. Hodges, Figgis & Co., Dublin, 1 889. [5] Anonymous. GREG N. FREDERICKSON
Department of Computer Science West Lafayette, IN 47907 USA
Ff tadakhul
al-ashkal
al-mutashabiha aw al
mutawafiqa (Interlocks of Similar or Complementary Figures). Paris: Bibliotheque Nationale, ancien fonds. Persan 1 69, ff. 1 80r-1 99v. [6) Donald
C.
Benson.
The Moment of Truth:
Mathematical
Epiphanies. Oxford University Press, 1 999.
e-mail:
[email protected]
[7) Farkas Bolyai. Tentamen juventutem. Typis Collegii Reformatorum
Greg Frederickson was educated at Harvard (A.B. in eco nomics) and the University of Maryland (Ph.D. in computer sci
[8) Donald L. Bruyr. Geometrical Models and Demonstrations. J.
ence). He has been on the Computer Science faculty at Purdue University since 1 982. Most of his research is on the
[9] H. Martyn Cundy and A. P. Rollett. Mathematical Models. Oxford,
design and analysis of algorithms, especially approximation al
[1 0) Erik D. Demaine, Martin L. Demaine, David Eppstein, and Erich
gorithms for NP-hard problems, graph algorithms, and data structures. Formerly a tennis enthusiast and a bassoon player,
Proceedings of the 1 1 th Canadian Conf. on Computational
per Josephum et Simeonem Kali, Maros Vasarhelyini, 1 832.
he now plays squash and drives his children to piano lessons. He also creates harmonious motion in geometry.
Weston Walch, Portland, Maine, 1 963. 1 952. Friedman. Hinged dissection of polyominoes and polyiamonds. In Geometry, Vancouver, 1 999. [1 1 ] Henry E. Dudeney. Perplexities. Monthly puzzle column in The Strand Magazine (1 926) (a): vol. 7 1 , p . 4 1 6; (b): vol. 7 1 , p. 522; (c): vol. 72, p. 3 1 6 . [1 2] Henry Ernest Dudeney. The Canterbury Puzzles and Other Curious Problems. W. Heinemann, London, 1 907. [1 3] William L. Esser, Ill. Jewelry and the like adapted to define a plural
ing triangle around. Then use the conversion of two swing
ity of objects or shapes. U.S. Patent 4,542,631 , 1 985. Filed 1 983.
hinges to twist hinges, adding for each a piece that we turn
[14] Howard Eves. A Survey of Geometry, Allyn and Bacon, Boston,
over. Finally, use a twist hinge to bring the concave piece along, and slice and twist it to fit it in properly. The resulting nine-piece twist-hingeable dissection is shown in Figure
34.
1 963, vol. 1 . [1 5] Howard W. Eves. Mathematical Circles Squared. Prindle, Weber & Schmidt, Boston, 1 972. (1 6] Greg N. Frederickson. Hinged Dissections: Swinging and Twisting. Cambridge University Press, in production. [1 7] Greg N. Frederickson. Dissections Plane & Fancy. Cambridge
Conclusion With their visual and kinetic appeal, hinged dissections and their design techniques will continue to play a role in math
University Press, New York, 1 997. (1 8] Martin Gardner. The 2nd Scientific American Book of Mathematical
ematical recreation and education. They also invite sub
Puzzles & Diversions. Simon and Schuster, New York, 1 961 .
stantive research in mathematics and computer science.
[ 1 9] P. Gerwien. Zerschneidung jeder beliebigen Anzahl von gleichen
Hinges are the simplest of linkages, permitting only rela
geradlinigen Figuren in dieselben StOcke. Journal fOr die reine und
tive rotation between connected pieces; with hingeability
angewandte Mathematik (Grelle's Journal), 1 0:228-234 and Tat.
we address issues of transformation of objects which have wider relevance. In addition to the problem of generality discussed briefly in the introduction, there is the search for algorithms: procedures for determining whether a given dissection is hingeable, and for finding effectively a plan of motion that carries the hinged pieces from one of the fig ures to the other.
Ill, 1 833. [20] Branko GrOnbaum and G. C. Shephard. THings and Patterns. W. H. Freeman and Company, New York, 1 987. [21 ] H. Hadwiger and P. Glur. Zerlegungsgleichheit ebener Polygone. Elemente der Mathematik ( 1 951), 6:97-106. [22] Anton Hanegraaf. The Delian altar dissection. Elst, the Netherlands, 1 989. (23] Philip Kelland. On superposition. Part II. Transactions of the Royal Society of Edinburgh (1 864), 33:471 --473 and plate XX.
ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under grant
CCR-9731758.
(24] H.
Lindgren. Geometric dissections. Australian Mathematc i s
Teacher ( 1 95 1 ) , 7:7-1 0.
VOLUME 23, NUMBER 3, 2001
19
[25] H. Lindgren. A quadrilateral dissection. Australian Mathematics
[32] lan Stewart. The Problems of Mathematics. Oxford University
Teacher (1 960), 1 6:64-65. [26] Harry
Lindgren.
Press, Oxford, 1 987. Nostrand
[33] H. M. Taylor. On some geometrical dissections. Messenger of
[27] Ernst Lurker. Heart pill. 7-inch-tall model in nickel-plated alu
[34] Henry Martin Taylor. Mathematical Questions and Solutions from
Geometric
Dissections.
D.
Van
Mathematics, (1 905), 35:81-1 01 .
Company, Princeton, New Jersey, 1 964. minum, limited edition of 80 produced by Bayer, in Germany,
'The Educational Times, ' (1 909), 1 6:81-82, Second series. [35] William Wallace, editor. Elements of Geometry. Bell & Bradfute,
1 984. [28] W. H. Macaulay. The dissection of rectilineal figures (continued).
Edinburgh, eighth edition, 1 831 . First six books of Euclid, with a supplement by John Playfair.
Messenger of Mathematics, (1 922), 52:53-56. [29] Aydin Sayili. Thabit ibn Qurra's generalization of the Pythagorean
[36] Eric W. Weisstein. CRC Concise Encyclopedia of Mathematics. CRC Press, Boca Raton, FL, 1 998.
theorem. Isis, (1 960), 51 :35-37. [30] I. J. Schoenberg. Mathematical Time Exposures. Mathematical
[37] David Wells. The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, London, 1 991 .
Association of America, Washington, DC, 1 982. [31] Hugo Steinhaus. Mathematical Snapshots, 3rd edition. Oxford
[38] Robert C. Yates. Geometrical Tools, a Mathematical Sketch and Model Book. Educational Publishers, St. Louis, 1 949.
University Press, New York, 1 969.
S P R I N G E R F O R M AT H E M AT I C S Gregory J. Chaltln,
IBM Research Division,
PETER HILTON, State University of New York,
GEORGE M . P H I LLIPS,
Hawthorne, NY
Binghamton, NY; D ER EK HOLTON, University of
University
EXPLORING RANDOMNESS
Otago, Dunedin, New Z ealand; JEAN PEDERSEN,
Scotland
Santa Clara University, Santa Clara, CA
This essential companion volume to Chait in's highly
MATHEMATICAL VISTAS
TWO MILLENNIA OF MA'IliEMATlCS
successful books, The Unknowable and The Limits of Mathematics. also published by Springer, presents the technical core of his theory of program-size com plcJtity, also known as algorithmic information theo ry. LISP is used to present the key algorithms and to enable computer users to interact with the author's proofs and discover for themsclvc. how they work.
From a Room with Many Windows
intrigue and i n form the curious reader.
This book is a sequel to the authors' pcpular
ISBN Hl5233-417·7
independently.
be read
2001/APPROX. 350 PP. . 158 ILLUS.
James J. Callahan, Smith College, Northampton, MA
HAR OCOVER/APPROX. $49.95
THE GEOMETRY OF SPACETIME
ISBN 0-387-95064-8
Ein tei n offered a revolutionary theo relat i vi ty-to explain some of the most
In 1905. Albert
ry-spec ial
troubling problems in current physics concerning electromagnetism and motion. Soon afterwards,
Hennann Minkowski recast spec ial relativity essen tially as a new geometric structure for spacetime.
These
ideas are the subject of the first part of the
UNDERGRADUATE TEXTS IN MATHEMATICS
ALSO AVA ILABLE FROM THE AUTHORS
MATHEMATICAL REFLECTIONS In a Room with Many Mirrors
from the history of mathe matics that cau be read by undergraduate students or used as a resource by teachers. . . Seeiug these topics from the vautage poiut of the author's experieuce as mathematician and teacher is pre cisely the value of the collectiou. The clear and inviti1tg style should fiud an appreciative audieuce and poilll a curious reader towards more mathe matics and more history. . . " -Mathematical Reviews 2000/240 PP., 9 ILLUS./HAROCOVER $49.95 ISBN 0-387-95022·2 CMS BOOKS IN MATHEMATICS, VOLUME 6
1996/351 PP .. 138 ILLUS./HAR OCOVER/$44 .95 ISBN 0-387-94770-1 UNDERGRADUATE TEXTS IN MATHEMATICS
ORDER TODAY CALL: Toi�Free 1·800-SPRINGER or FAX: 201·348WRITE: Springer-Verlag New York, Inc.
book . The second part develops the main implica
4505
tion
Dept S2561, PO Box 2485, Secaucus, NJ
of Einstein's general relativity as a theory of
•
gravity rooted in the differential geometry of sur
07096-2485
face . The author has tried to encompass both the
bookstore
general and special theory by using the geometry of spaceti me as the
unifying theme of the book.
2000/456 PP., 219 ILLUS./HAR OCOVER/$49.95 ISBN 0-387-98641-3 UNOERGRADUATE TEXTS IN MATHEMATICS
20
"This is a beautifully tvritte>r collection of the author's favorite examples
book consists of nine related mathematicnl essay
book, Mathematical Reflections, and can
An Introduction to Special
to Gauss
the interest of bright people in mathematics. The
2001/176 PP./HAROCOVER/$34.95
and General Relativity
From Archimedes
The goal of Mathematical V.stas is to stimulate
which will
of St. Andrews,
THE MATHEMATICAL INTELUGENCER
•
•
•
VISIT: your local technical
EMAIL:
[email protected]
INSTRUCTORS: Call or write for information
on textbook exam copies. YOUR 30-DAY RETURN PRIVILEGE IS A LWAY S GUARANTEED 4/01
PROMOTION #S2561
M a thematic a l l y Bent
C o l i n Ad ams , Editor
Hiring Season
by offering to sign us up for the Math
Dec.
Employment Registry.
1 : Big blow. Costa went away for
Thanksgiving and never came back.
ept.
S
The proof is in the pudding.
Halls have become a wasteland, lit
People saying hello to one another at
tered with crumpled letters of recom
the
faculty mailboxes.
Past malice
mendation
and
strategically placed
tacks. No hope of deciding this easily.
by warm summer winds. Ganser and I
Ganser and I have dug in for the long
Sept.
term. Ganser is determined, but I fear
are hopeful this year may be different. Mathematical
7: All-out war has commenced.
ment seems to be getting along.
seems to have vanished, blown away
Opening a copy of The
Dec.
Rumor has it he's now an actuary.
7: Everyone in the depart
12: Classes underway. Ganser
for his health. His hands have been
has finagled us both onto the hiring
shaking. He needs caffeine, and soon.
committee. Algebraists are upset two
Dec.
topologists are members. Bullman and
man's sister.
mathematical journal, or what?" Or
Klimkee
applied,
now despise each other. Ganser and I
you may ask, "Where am /?" Or even
Bullman because she is applied, and
take the opportunity to do a celebra
Intelligencer you
may ask yourself
uneasily, "What is this anyway-a
"Who am /?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Rel03. Breathe regularly. It 's mathematical, it's a humor column, and it may even be harmless.
are
pushing
for
10: Klimkee is divorcing Bull Bullman and Klimkee
Klimkee because he is married to
tory dance in the corridor. Quite a
Oct.
show, but no one's there to see it.
Bullman's sister.
12: The hiring committee still
Dec.
15: The administration may have
can't come to agreement on whether to
to step in. Work has come to a stand
serve cookies or cheese and crackers
still.
at the meetings. Ganser just wants cof
Ganser and I are hunkered down in his
Even the students are afraid.
fee. I prefer the little frosted pink wafer
office. Departmental communication
cookies,
reduced to e-mail contact only, and
but no one
else
concurs.
Bullman and Klimkee are arguing for
most of it too coarse to repeat. Ganser
Oct.
continually paces, paces. This is al
wine.
23: The hiring committee chair
and the recording secretary are no longer on speaking terms, meaning
Dec.
most as bad as last year.
20: I tried to stop him, but Ganser
was desperate. Risked all for java run.
there are no minutes for the meetings.
Grabbed a jar of instant out of the
This allows committee members to say
lounge. Brought back my mail, includ
things they otherwise wouldn't dare.
ing the latest AMS Notices. Job listings
Nov. 2: Ganser and I have enlisted the
are meager. Both despondent.
support of Costa, who although not a
Dec.
topologist, has interests in Riemann
Nervously, Ganser and I attend the hol
surfaces. Perhaps we can broker a
iday party. Initially, everyone
deal.
Refreshments are a disappointment.
22: Cease-fire has been declared.
is civil.
Nov. 12: Algebraists are no longer co
No little frosted pink wafer cookies.
operating. Meetings are deteriorating.
Spirits low all around. We depart just
Bullman keeps kicking me under the
as yelling commences. Not much hope
table and then pretending it was acci
for the new year.
dental. It really hurts.
Jan. 2: Returnees look prepared for
Nov. 14: Ganser says faculty in his
the long haul. Several carrying cof
neighboring offices are becoming rude.
feepots. We are checking to see
He is uncomfortable entering and leav
fire marshal may prevent it.
if the
Column editor's address: Colin Adams,
ing the building. I fear for the direction
Jan. 15: Ganser and I are holed up in
Department of Mathematics, Williams
in which we are heading.
my office. Ganser is screaming for cof
College, Williamstown, MA 01 267 USA
Nov. 18: Ganser and I threaten to look
fee now. He's licked the instant jar
e-mail:
[email protected]
for jobs elsewhere. Klimkee responds
clean. I'm feeding him chocolates and
© 2001 SPRINGER-VERLAG NEW YORK. VOLUME 23, NUMBER 3, 2001
21
cola, but he's begging for Guatemalan Mocha Supreme. Jan. 22: Algebraists have broken. Waving a white flag, they file out, headed for Starbucks most likely. Now it's down to Applied versus Topology. Ganser drops in and out of lucidity. Feb. 14: There is hope. Administration has promised funding for a fluid dy namics person, half in math, half in en gineering, making Topology the high est remaining priority. Ganser is elated. Chair calls to tell us the good
news. Doesn't seem too angry that we haven't met our classes in a month. Feb. 29: On pins and needles. Depart ment meeting slated for tomorrow. All will be in attendance. This could be it. March 11: It's official. Ganser and I have received permission to hire. We are jubilant. March 29: Best candidates are gone. We had three interview talks, and it's not clear any of them know the difference between a Mobius band and an annulus. April 17: We have hired. Although he
only speaks a Kurdish dialect, and he's actually in number theory, he does seem to be familiar with the torus, or so it appears from our communication via sign language. May 13: Due to visa problems, our can didate cannot come after all. We will have to repeat the process next year. It is disappointing, but we consider it a learning process. Ganser has installed a cappucino machine in his office. All told, it could have been worse. One can only hope for the future.
T H E M AT H B O O K O F T H E N E W M I L L E N N I U M ! B. Engquist, University of Cal i fornia, Lo Angele and Wilfried Schmid, Harvard Univer ity, Cambridge, MA (eds.)
Mathematics Unlimited 200 1 and Beyond
This is a book guaranteed to delight
Content : Anrman, S.:
the reader. It not only depicts the
!.{lin ley Oden, J.: Computational Mechanics: Where is it Going?
rate of mathematic at the end of the century, bur is also full of remarkable in ights into its furure development
•
onlinear Continuum Phy ics.
•
Babuska,
Bailey, D.H./Borwein J.M.: Experimental Mathematics: Recent
Developments and Future Outlook. tions.
•
•
Darmon, H.: p-adic L-func
Fairings, G.: Diophantine Equation
.
•
Farin, G.: SHAPE.
as we enter a new millennium. True to its ride, rhe book extends
•
jorgen en }./Lang, S . : The Heat Kernel All Over the Place.
beyond the spectrum of mathematics to include contributions from
•
Kliippelberg, C.: Development in Insurance Mathematics.
other related sciences. You will enjoy readjng the many rirnularing
•
Koblitz,
.: Cryptography.
•
Marsden, j./Ccndra H./Rariu,
contributions and gain in ight into the a rounding progress of math
T.: Geometric Mechanics, Lagrangian Reduction and
ematics and the per pecrives for it future. One of the editors, Bjorn
Systems. Serre,
Engquist, is a world-renowned researcher in computational science
•
and engineering. The second editor, Wilfried Schmid, i a distingui hed
XXI
r
onholonomic
Roy, M.-F.: Four Problem in Real Algebraic Geometry.
•
.:
y rems of Con ervation Laws: A Challenge for the
Century.
•
Spencer,
J.:
Discrete Probabiliry.
•
van der Geer,
mathematician at Harvard Univer iry. Likewise, the author are all
G . : Error Correcting Codes and Curves Over Finite Fie lds.
foremo t mathematicians and
Hlvon Storch,
ienti
ts,
and their biographie and
photographs appear at the end of the book. Unique in both form and content, this is a "must-read" for every mathematician and sci entist and, in particular, for graduates still choosing their specialry.
J.- ., and Muller, P.:
•
von Storch,
oise in Climate Models ... And
many more. 200 1 / 1 236
PP., 253 ILLUS./HARDCOVER/S44.95/ISBN 3-540-66913-2
Order Today! Call: 1 -800-SPRINGER
•
Fax: (201 )·348-4505
Visit: http://www.springer-ny.com
' Springer .
www.springer-ny.com Promotion 152560
22
THE MATHEMATICAL INTELLIGENCER
JESUS CRESPO CUARESMA
Po i nt S p l itti n g and C o n d orcet C rite ri a "You can 't always get what you want" The Rolling Stones
group offriends have to decide on how to spend some money they won in the lottery. Three alternatives are put forward: going on holiday, buying a new car, and giving the money to charity. One of them (let us call him Alan) pro poses to vote upon the three alternatives by each one of the friends dividing fifty points among the three competing choices freely, be
when there is an alternative x which obtains a majority of
ing as accurate as they want in the division of points among
votes in pairwise contest against every other alternative (a
alternatives (one may vote, for instance,
40.99 points for
Condorcet winner), x is chosen as winner.
9.01 for the second, and 0 for the third alter
In an augmented version of Alan's points voting proce
native), and then choose the alternative that receives the
dure, the number of votes of the Condorcet winner alter
highest number of points. Charles (another member of the
native
the first one,
lucky group) replies that he has something against such a
(if there is one) is multiplied by a fixed number {3 (> 1 ), independent of the size of the voting population, that
way of deciding upon their money, because he once read
can be as high as we want. Intuitively, such a voting pro
that this "points voting scheme" may not give the same out
cedure would seem likely to satisfy the Condorcet winner
come as ordering the alternatives from most to least pre
criterion and have better features than the original points
ferred by each voter and then choosing the one that beats
voting procedure. Yet I will prove that, provided that indi
the most alternatives in pairwise comparison.
viduals have the possibility of being as precise as they want
Alan thinks about Charles's criticism and refines his
in distributing their points among alternatives (that is, the
original voting procedure: if there exists an alternative that
number of points available for distribution is infinitely di
beats all the others in pairwise comparison, its score will
visible), this class of voting procedure does not
be multiplied by a high number (say
20).
fulfill the
Condorcet winner criterion and/or the (symmetrically de
This paper proves that, surprisingly, Charles's reserva tions about the original procedure also apply to the modi
fmed) Condorcet loser criterion for any value of {3. I begin by formalizing the class of voting procedures,
and stating some criteria that are satisfied by these proce
fication. dure that Alan proposed, in which a certain number of
Pareto criterion and the monotonicity criterion. Then I prove that our class of voting procedures
points are available for distribution among the candidates,
satisfies neither the Condorcet winner nor the Condorcet
has the great advantage that voters' personal intensities of
loser criterion.
From the point of view of the voter, the "points" proce
dures, namely the
preference can be represented. Yet Charles is right: points voting has been proved not to satisfy several criteria that reasonable voting procedures might be expected to fulfill , such as the
Condorcet winner criterion.
A voting proce
dure is said to satisfy the Condorcet winner criterion
The Voting Procedure
Each of
N individuals has to distribute R votes among k
alternatives
if,
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
23
All through
this article we will suppose that these individ
and for xn
uals are sincere, that is, that they will not misrepresent their preferences in order to get some other benefits from it.(For a study of the problem of preference misrepresentation in this framework, see
[3] or [8].) That means that if we de
fine for each voter an intensity-of-preference function
U(xi) > U(xj) if and only if Xi > Xj (alternative Xi is preferred to alternative Xj) , then voters would distribute Vij = R
[
Ui(xj) k =I Ui(Xh) Ih
]
(1)
is the number of votes assigned to alternative individual
Xj by each
i.
If there is a Condorcet winner (for the possibility of non
existence of a Condorcet winner, see, e.g., boost: the score for a given alternative
Xj is
[5]), it gets a
.
gJ with f3
=
{
this voting procedure.
DEFINITION 2. A voting rule satisfies the monotonicity criterion if, when x is a winner and one or more voters change their preferences in a way favorable to x (with out changing the order in which they prefer any other al ternatives), x is stiU a winner. PROPOSITION 2. The class of voting models defined by (1)-(3) fulfills the monotonicity criterion for every value of f3 > 1.
Proof If Xr is a winner under the voting procedure defined by (1)-(3), then
Vr > V8 (2)
where
gr as defmed in (3). It is straightforward to see that
under
where
their points in the following way:
with
Vr > V8, and therefore Xs would never be the social choice
Vs E { 1 , 2, ... , k - 1, k) \ {r},
and if one or more voters change their preferences in a way favourable to
Xn without changing the order in which they
prefer other alternatives, then
1
f3
if Xj is not a Condorcet winner if Xj is a Condorcet winner
(3)
where the asterisk denotes "after the change" values. On the other hand, as the preference on other variables re
> 1, fixed.
Our question is whether this augmented version of the
mains unchanged,
cumulative voting procedure has better features than sim ple cumulative voting. Some Positive Properties of the Voting Procedure
The study of the voting scheme defined above
will be done
It is straightforward to prove that
v; > v; and that there
exists no
Therefore,
Xg E X such that Vg > Vr·
Xr is still a
winner.
by analyzing which of the criteria that seem reasonable for a "good" voting procedure are satisfied. For a deeper in sight into the definitions displayed in this section and the following one see, e.g.,
[ 1 ] , [2],[4], [6],[7].
DEFINITION 1 . A voting rule satisfies the Pareto criterion if, when every voter prefers an alternative x to an alterna tive y, the voting rule does not produce y as a winner. PROPOSITION 1. The class of voting models defined by (1)-(3) fulfills the Pareto criterion for all values of f3 > 1.
Proof If every voter prefers an alternative native
Xs then
Ui(Xr) > Ui(Xs) I�=I Ui(xh) I�=I Ui(xh)
Xr to an alter
Some Negative Properties of the Voting Procedure
PROPOSITION 3. lf voters in the class of voting proce dures defined by (1)-(3) are not limited in their possi bility of discrimination among alternatives (that is, if the number of votes assigned to each voter is infinitely divisible), then for any f3 > 1 there exists a profile and a population such that this voting scheme does not satisfy the Condorcet winner criterion.
Proof An example constitutes the proof.Suppose the case in which M + 1 voters (M is an even integer) decide upon
Xi, and the preference profile is XI >j X2 >j X3 >j . . . >j Xk - I >j Xk
k alternatives
Vi ::S N,
a)
(j
which implies that
I Vir > I Vis. i i As every single voter prefers Xr to X8, Xs cannot be Condorcet winner.The number of votes for x8 is thus
24
Tl-IE MATl-IEMATlCAL INTELLIGENCER
b)
Xk >a (Xk- I
-a
Xk- 2 -a · · ·
1, 2,
=
2,
...'�+ )
-a XI)
( �+ �+ d
a
=
1 .
(4)
)
(5)
3, . . . , M + 1 .
where a >f b means "a is preferred to b by individual!," and a -f b means "individual f is indifferent between a and b." Suppose further that all individuals belonging to the
preference profile a) give a proportion to alternative
Kr of their R votes
an where Kr fulfills
Kr > 0 Kr+ l < Kr k Kr = 1 ,
Vr E { 1, 2, 3, . . . , k - 1 }, Vr E { 1, 2, 3, . . . , k - 1 },
L
(6) (7) (8)
r�l
Vr E { 1, 2, 3, . . . , k - 1 }.
Kr - Kr+ l = K (constant)
(9)
Voters in preference profile b), on the other hand, give
R, to alternative xk and zero to the rest. The relative number of votes for alternative x1 is
the totality of their votes, therefore
(!f + )
(M + 1)R '
.
tions concerning the axiomatic properties of the voting scheme studied. The four criteria studied until now are not the only ones that are usually taken into account when an alyzing voting procedures. Consider, for example, the fol lowing criterion:
DEFINITION 3. A voting rule is said to satisfy the ma jority criterion if, when most voters have an alternative x as their first choice, the voting rule chooses x as a winner.
a case such that the augmented cumulative voting proce
1 Kk
dure does not choose x1 as a winner. Thus, the voting pro
(M + 1)R
cedure defmed by (1}-(3) does not satisfy the majority cri
(4) and (5) that x1 is the Condorcet xk are
winner, therefore the scores of x1 and
{3
(!f ) +
terion. Summary and Conclusions
First impressions are not always right, especially in this
1 K1
field mostly inhabited by impossibility theorems. The ac
M+ 1
tual properties of the voting procedure constructed violate the intuition that motivated the rule. The Rolling Stones's
(!f + ) + !f 1 Kk
vk =
vk > VI. (vi > v2, . . , > vk - l is evident.) 4 have a number of further implica
Propositions 3 and
of a majority of voters, and for every value of f3 we can fmd
[(!f + ) + !f]R
VI =
sen so that
Proof See the proof of Proposition 3: x1 is the first choice
xk,
It can be seen from
C-"f + 1 against -"f) in pairwise contest. Neverthe
PROPOSITION 5. The class of voting models defined by (1)-(3) does not fulfiU the majority criterion.
1 K1R
and for alternative
alternative
less, as we saw, for any {3 the other parameters can be cho
sentence with which this paper started is quite apposite to Public Choice, where getting "what you want" is usually
M+ 1
more difficult than it seems.
I must now choose k, K, and M so that v1 <
vk. This is easy.
Set
A U T H O R
M
2 + 1 H=M + 1' _ _
= {3K1 H, vk > 1 - H. Choose k > 4{3, then choose so v1 < H/2. Now any M will K1 < 2 < do, for M 2= 2, H ::::; vk - VI > 1 H 2= 0. so that v1
K so small that
f,
i- 2�;
-�
Notice that the opposite could also be argued, namely that given a number of alternatives and voters, we could always find a
{3 so that the voting procedure fulfills the
Condorcet winner criterion. What the example proves im possible is a voting procedure that does this for any num ber of alternatives and voters�
PROPOSITION 4. lf voters in the class of voting proce dures defined by (1)-(3) are not bounded in their possi bility of discrimination among alternatives (that is, if the number of votes assigned to each voter is infinitely divisible), then for any {3 > 1 there exists a profile and a population such that this voting scheme does not satisfy the Condorcet loser criterion.
Proof The example is the same as in Proposition 3. Notice that xk is a Condorcet loser, as it is beaten by each other
JESUS CRESPO CUARESMA Department
of Economics
University
of Vienna
1 2 1 0 Vienna, Austria e-mail:
[email protected]
Jesus Crespo Cuaresma was born
in
Seville, Spain. After un
dergraduate education there, he obtained an MSc. at the Institute for Advanced Studies in Vienna. He is now an assis tant professor at the University of Vienna. His main field is econometrics and time series analysis.
VOLUME 23, NUMBER 3, 2001
25
ACKNOWLEDGMENTS The author is indebted to Frantisek Turnovec, Don Saari, Maya Dimitz, Susi Winklehner, and an anonymous referee for helpful comments.
4. P. K. Pattanaik, Voting and Collective Choice, Cambridge University Press, Cambridge, 1 971 . 5. D. G. Saari and F. Valognes, Geometry, voting and paradoxes, Mathematics Magazine 7 1 { 1 998), 243-259. 6. A. K. Sen, Collective Choice and Social Welfare, North-Holland,
REFERENCES
1 . S. J. Brams, "Voting procedures," in Handbook of Game Theory, North-Holland, Amsterdam, 1 994. 2. R. Farquharson , Theory of Voting, Yale University Press, New Haven, 1 969. 3. S. Nitzan, J. Paroush, and I. L. Shlomo, Preference expression and
26
Amsterdam, 1 984. 7. P. D. Straffin, Topics in the Theory of Votn i g, UMAP Monograph Series, Birkhauser, Boston, 1 980. 8. F. Turnovec, "Distance games and goal programming models of vot ing behaviour," In Advances in Multiple Objective and Goal Pro gramming, Proceedings of the Second International Conference on
misrepresentation in points voting schemes, Public Choice 35
Multi-Objective Programming and Goal Programming, Springer
(1 980), 421 -36.
Verlag, Berlin, 1 997.
THE MATHEMATICAL INTELLIGENCER
A. K. DEWDNEY
The Forest and the Trees : Ro man c i ng the J- cu rve
•
~
n the forest, we find a peace that is hard to describe, like coming home without realizing that we'd ever been away. Because it is a mature forest, the floor is open. We can see jar into the gloomy recesses, almost grasping it as a whole. What adventure could there be here for a mathematician?
Near at hand we can see individual trees: the smooth grey bark of the American Beech, the curling plates of Sugar Maple, the cross-veined hide of White Ash. We see another ten or so species before the peace dissolves in nagging questions. Why are there so many Beeches and Maples, why so few Basswoods and Black Cherrys? Is there some kind of pattern here? Although the exact reasons for the specific numbers of each species of tree are myriad, there can be no doubting the overall pattern. If we counted every tree in the forest, adding for good measure the shrubs and understorey species, we would see few species of high abundance in the forest and relatively many of low abundance. If we made a histogram, plotting the number of species at each abundance, from one up to the maximum, the shape would look strongly familiar, perhaps hyperbolic. Biologists have an informal term for this kind of pattern. They call it a "J-curve," owing to its resemblance to a back ward letter J. The J-curve (if I may use the definite article) is ubiquitous in nature. It appears in over 99 percent of the field biosurvey literature. Theoretical ecologists have struggled to discover the formula that lurks behind the
J-curve. Beginning in 1943, they have proposed about eight different formulas. My first encounter with the J-curve in 1995 sparked the research that has occupied much of my time since. Although my formal training was in mathematics, I have a long-standing love affair with biology. For many years I have pursued microbiology as a hobby. In 1990 I decided to get serious, learning to identify most of the ciliated pro tists that swam through my field of view, as well as algae, flagellates, amoeboids, and so on. I then selected a small creek near my hometown of London, Ontario, and began a regimen of monthly sample taking and subsequent microscopic examination. As I worked, I became increasingly curious about the various populations I met. Some organisms were very numerous, appearing in all my samples; others, rather rare, often with out a single representative on my slide. Was there a pat tern? One evening, I made a species/abundance histogram of the data from my counts. How odd! The resulting curve was shaped like a ski-jump. How could one account for such a curve? It seemed to me that some species must have few nat-
© 2001 SPRINGER-VERlAG NEW YORK. VOLUME 23, NUMBER 3, 2001
27
in natural communities. It was, indubitably, a candidate J-curve. In 1948, an American ecologist, E. F. Preston, proposed a completely different formula that involved the normal dis tribution [8]. He was aware of the peculiar J-shape that most biological samples had, so he decided that the shape was an artifact of sampling, there being a "veil-line" that blocked all species below a certain abundance from ap pearing in the sample. For the rest, a logarithmic transfor mation would give his curve the requisite taiL Thus was born the lognormal curve:
# of species 20
10
f(k)
3 2
2 4 6 8
abundance
Figure 1. A species/abundance histogram for a forest.
ural enemies and so could build up great numbers, while others had many predators and so were grazed to a mini mum. To check what seemed like a perfectly reasonable idea at the time, I wrote a computer simulation in which 100 "species" all predated equally on each other. (There are several pairs of protozoa which can mutually predate.) The populations in this mini-ecosystem should all hover about roughly equal values. Or so I thought. After its first bug free run, the program dutifully displayed its species/abun dance histogram, and the hair on my neck stood up. It looked just like the curve I had plotted a few days earlier! The simulation taught me that I was very poor at guess ing how populations ought to behave. Meanwhile, the J curve haunted me. From a botanist friend I borrowed a sur vey of plants in a nearby old-growth forest. I constructed the species/abundance histogram, only to find another J-curve staring back at me. I showed the plot to a field ecologist. "I see you have a J-curve there." Was this known? "Oh yes, we see them all the time." Was there a formula? He didn't know. In 1996 I began to delve into the ecology literature. In 1943, the British entomologist C. B. Williams had shown abundance distributions of moths to the eminent statistician R. A Fisher [3]. Williams thought the curve might be a hyperbola. Fisher kindly explained that a hy perbola couldn't be a statistical distribution because it had an infinite area under it. The thing to do was to multiply the terms of the hyperbolic formula by a convergent series. That would force the distribution to have a finite area. Williams bowed to Fisher's superior mathematical insight and the log-series distribution was born: f(k)
=
Tl-IE MATHEMATlCAL INTELLIGENCER
c exp( -ak2). ·
In this normal pdf, c and a are constants, but k is the number of the "octave," a concept that requires explana tion. Preston divided the abundance axis of his histograms into segments that doubled their lengths consecutively. Thus the first octave, reflecting the lowest abundance, might consist of a single abundance, while the second oc tave would then consist of the next two abundances, the third octave would involve the next four abundances, and so on. When plotted on standard species/abundance axes, the lognormal distribution resembles a normal distribution that has been pulled out like taffy by the logarithmic trans formation inherent in the scheme of octaves. The veil-line, as mentioned earlier, had the salutary ef fect of preventing the lognormal distribution from being laughed out of the laboratory. If you chopped the lognor mal distribution in just the right place and shifted what re mained to the origin, you would get a J-curve. Well, close anyway. I will return to the mysterious veil-line presently. Both the log-series and lognormal distributions have en joyed a continuing popularity since their introduction in the 1940s. It would be fair to ask, "What testing had the origi nal authors of these schemes done to check their formu lae?" The answer is a little shocking to those who like their science straight up: None. Neither set of authors saw fit to perform a single goodness-of-fit test of their proposed dis tribution with actual field data. Instead, both sets of au thors merely selected a few J-curves from field data, draw ing attention to the resemblances, seemingly unaware that a mere handful of field histograms would not be adequate for such a comparison, even with statistical testing.
# of species
veil line
ac kfk
In this probability density function (pdf), {c k ) is the con vergent sequence that prevents the infinite sum of terms 1/k from blowing up, while a is a constant that produces a sum of unity. The constant c tends to be slightly less than unity, usually greater than 0.99. The log-series curve does a creditable job of describing the abundances of species
28
=
abundance
Figure 2. The lognormal distribution (with "veil").
Since the 1940s, more distributions have been proposed,
Besides the basic receptor/donor cycle, the program op
all of them equally untested: the Gamma distribution, the
erates on a longer, display cycle. Every 100 iterations of the
broken-stick distribution, the Zipf distribution, the negative
basic cycle, the program displays a histogram of the cur
binomial distribution, and one or two others. To a relative
rent abundances. When the program is running on a rea
newcomer, the field of theoretical ecology appears to be in
sonably fast computer, one sees the histograms replace
a crisis that remains largely unacknowledged. The plethora
each other in rapid succession, creating a lively sense of
of distributions only serves to confuse field biologists, who
how the system behaves.
look to the theorists for guidance. Moreover, theoretical
Readers with programming experience may write their
ecologists have lived with the situation so long, they seem
own MSL system. Store the 100 abundances (or as many as
to regard it as normal.
you like) in an array. The program does not have to keep
In 1995 I knew only that my computer program [4] was
track of individuals. To choose an "individual" at random,
producing curves remarkably like the ones I was finding in
it uses a simple technique (see the procedure in the algo
the field. What was that curve? I proceeded to analyse the
rithm of Figure
computer program, a task that ought to have been simple, be
system are arranged in a long sequence, grouped by species
3). Imagine that all the individuals in the
3 displays the under
in some fixed order. If the total number of individuals is,
lying algorithm for the multispecies logistical (MSL) system.
say, 1000, the program selects a random number k between
cause the program was simple. Figure
The user chooses two values, one for J.L, the average
1 and 1000, then proceeds to count its way through the in
abundance, and one for R, the number of species. Within a
dividuals, in effect. Each time it comes to a new species, it
loop, the algorithm continually repeats a simple cycle in
looks up that species's abundance and adds it. As soon as
which two individuals (not species) are chosen at random
the sum reaches k, the species we are at is the one selected.
by the procedure Select Species (listed below the program),
Each species is selected with probability proportional to
one as receptor, the other as donor. The receptor then re
its abundance.
produces, thanks to the donor's biomass, and the donor
My version of the MSL system allows the user to select the number of species and the average abundance, J.L.
vanishes from the simulation.
In this way, the total number of individuals is preserved,
Initially, it assigns this abundance to all species. The first
a .feature that led me to dub the system "logistic," after the
histogram consists of a large spike at the value of J.L. With
well-known logistic system popularized by the British the
each subsequent display cycle, the spike spreads out, de
oretical ecologist Robert May during the chaos fad of the
(In the logistic system, a single spe cies that enjoys a continuous abundance, x, consumes a resource which is available in the quantity 1 x. Alterna 1980s and early 1990s.
-
veloping a tail that spreads to higher abundances. At the low abundance end, the histogram piles up at the origin. Eventually, the J-curve emerges, this being the equilibrium state of the system.
tively, the resource may be regarded as a second species
To return to my analysis of the MSL system, I pondered
which is prey for the first one. In repeated cycles the cur
the matter for a year, fmally arriving at a partial argument
rent value of x is replaced by the value of the expression
based on the equilibrium state of the system: At each pair
Ax(1
-
x), .A being a parameter that strongly influences the
k and k + 1, the number + k to k 1 must equal the flow
of adjacent abundance categories, of species moving from
appearance of chaos.)
in the opposite direction. In other words, at the kth and
(k + 1)st categories, input
J.L ,
R
p(k) . f(k)
repeat
d i s p l ay
.
+ 1).
category, and p (k), which equals kiN, is the probability that a species will move to the right (or left) from the kth po sition. The equation has only one non-trivial solution, namely, f(k)
h i s t o gram
key
p(k + 1) f(k
Here, f(k) represents the number of species in the kth
Receptor � Sel ect Speci es Donor � Sel ect Speci es increment abundance of Receptor decrement abundance of Donor unt i l
=
= llk. This solution alarmed me because, as
R. A. Fisher had observed, the hyperbola could not be a
pre s s ed
distribution function. Looking back over the system as a whole, I realized there was a condition I had not used yet.
select
a
Coun t �
The total number of individuals must be a constant.
Sel ect Speci es
procedure
0,
random number
s�
k
from
[1,
This global constraint must have the effect of limiting
s�s+ 1 Coun t � Coun t + return
Lk · f(k) = N.
0
repeat
unt i l
N]
the distribution to a finite domain. Since no species could abundanc e
of
Count :::: k
s
Figure 3. The multispecies logistical (MSL) system.
s
I could look for a num a that reflected the (average) maximum abundance in
have an abundance greater than N, ber
the system. As a mathematician, it was painful for me to add an ex
tra assumption to "solve" the system. Nevertheless, I as-
VOLUME 23, NUMBER 3, 2001
29
sumed the existence of a forcing function, q(k), which ex pressed the finiteness constraint as a factor in the equilib rium equation. The fact that physicists do this sort of thing with regularity was cold comfort. Starting, then, with
k . q(k) · f(k) = (k + 1) . q(k + 1 ) · f(k
+
1) ,
the solution would have to have the form
f(k) = ll(q(k) . k). The boundary condition!(�) = 0 implied that the function q(k) had an infinite value at k = �- The simplest function with this property was
q(k) = 11(1
-
8k),
where 5 = 11�- That made a total of two assumptions, the existence of the forcing function and the simplicity of q. Nevertheless, the "solution" gave me something to work with, a pdf that looked like this:
f(k) = c(I
-
5k)lk.
Here, c is a constant that yields an area of unity. An equivalent formulation, c(1/k - 5), indicates a hyperbola translated downward by a small amount, 5. The function is defmed to be zero beyond �. I could not be sure that this was, indeed, the solution to the MSL system, but I proceeded in spite of the uncertainty. I was now practising "inductive science," not "deductive science," or mathematics. I had something to test against data But would I never solve that system? For many nights until just last year, I went to sleep trying different solutions. But the year was still 1996. By chance, Laszlo Orloci, an eminent plant scientist at the University of Western Ontario, was holding a small conference on computer mod els in ecology. Knowing of my work, he invited me to attend. I presented the MSL system and discussed its similarity to field samples. How could I ac count for this resemblance? Was something going on in the system that is also going on in nature? I referred not to the incessant predation, but to the randomness of it all. I tried to explain the term "effective randomness," referring to sys tems such as the stock market, pseudo-random number generators, chaos, and so on. After the talk, one of the biology graduate students came to the front. She looked at me unblinkingly, almost hostile. "I will never believe that populations change randomly." I tried to explain that I hadn't meant, well, rwn-deterministic. Another problem with the MSL system and the theory that was beginning to emerge lay in the interpretation. Surely not all communities were like my original vision in which every organism was capable of eating every other organism. What about plants? Field samples of plants showed the same type of species/abundance distribution, a J-curve. It took several months for me to realize that the MSL system had another, much broader interpretation. It
was not about predation per se, but the transfer of biomass or energy, as the case may be. When one animal eats an other, biomass is transferred from predator to prey. Is it possible that plants prey on other plants? Well, for these purposes, yes. If you think about energy and the fact that plants compete for light, it doesn't take long to realize that one plant that shades out another, ultimately killing it, has taken energy that was otherwise destined for the "prey." Energy has been, in effect, transferred. I have constructed many versions of the MSL system, in cluding ones with fractional trophism, neighboring com munities, and so on. The basic model is what computer sci entists call "detail-hungry," being capable of seemingly limitless elaboration without failing to produce its J-curve. The most important version of the MSL system is a com partmentalized model which has species that behave like plants, herbivores, carnivores, and saprobic organisms such as fungi or bacteria. Energy enters the system via ran dom rewards of light to plants. It leaves through the ran dom departure of individual fungi. In between, plants com pete for sunlight, herbivores eat plants, carnivores eat herbivores (and each other), while fungi eat everything, sooner or later. Sure enough, the J-shape appears-indis tinguishable from the basic MSL system output. The final interpretation of the MSL system is this: Energy flows from one organism to another, whether directly or indirectly (through a food web), in a way that is effectively random. In other words, we cannot hope to predict which way a particular population will change at a given time. Nothing more than this is needed to guarantee the emer gence of the logistic.J distribution. Applied to nature, this interpretation becomes the stochastic communities hypothesis: Natural communities consist of fluctuating popula tions that are largely unpre dictable as to direction in a statistical sense. The unpre dictability implies that for each species in a community there is an equal probability of increase as of decrease. In addition, the total biomass/en ergy of the community must remain roughly constant. These requirements can be made less stringent without af fecting the conclusion. For example, the probabilities can be made only approximately equal, or allowed to fluctuate about the point of equality, without changing the end re sult. It must be pointed out that if the two probabilities for a given species remain even slightly different (say by 10 per cent) over an extended period, the species will inevitably either grow to the logistic limit or shrink to extirpation. This is true not only of the simulation but also of the real world. It boggles the mind, however, that even when the probabilities are strictly equal, species follow stochastic or bits about the mean abundance, some of them flirting with extirpation, others growing to enormous size. This finding, about which there can be no doubt, is the exact opposite of what we would intuitively expect.
Elaborate the m odel al l
you want - out comes the
J-c u rve .
30
THE MATHEMATICAL INTELLIGENCER
In 1997 I became increasingly curious about Preston's lognormal distribution, especially his use of the veil-line. I studied his original paper [8] carefully, as well as a few more recent papers that sought to justify the use of the lognormal distribution. To my horror, I discovered that Preston assumed that any species with an abundance less than a certain ("veil") threshold would simply not show up in a sample-thus the veil-line. But this was surely false! If there were enough species of abundance below this thresh old, some would show up, possibly even many. I went back to basics, rereading W. Feller's classic book [2] on proba bility. There, I discovered the hypergeometric distribution: if a species has abundance ni in its community, then the probability of it contributing k individuals to an unbiased sample of size n is
II of species
0 species
veil curv e
(a)
abundance
11 of
species
veil curve
where N is the total number of individuals in the commu nity. Preston had perhaps assumed that species of low abun dance were rare in natural communities, whereas the J curve was pointing in exactly the opposite direction. But could Preston have been right after all? I needed a general theory of sampling that would describe the relationship between abundances in a sample and those in the corre sponding community. In vain I searched the literature for such a theory. Clearly, I would have to develop one. It took a few weeks before I worked out an integral transform that would operate on the abundance distribution of a commu nity and convert it into an expected abundance distribution for a sample. I used the fact that the Poisson distribution is a very close approximation to the hypergeometric, even for low abundances. The transformation took the form
(b)
abundance
Figure 4. Veil curves for the uniform distribution (a) and the lognor mal distribution (b).
veiled) lognormal, so would any sample of it. As far as I was concerned, I had "unveiled" a fatal error at the very foundation of the lognormal distribution. The lognormal was dead. I submitted the new general theory of sampling (with applications to the veil-line) to Theoretical Popula tion Biology [5]. After it appeared, I received ten requests for reprints. In the light of a massive "experiment" that I was about to perform, Williams [3] would look increasingly prophetic with his original view that J-curves were hyper bolae. In altering the hyperbolic distribution to a conver gent series, Fisher took a wrong tum. Perhaps he was used to dealing with distributions that, like the normal distribu tion, have infinite domains. The J-curves produced by the MSL system were provably fmite. Back in May of 1996, I had framed a special, continuous version of the logistic-J distribution. It looked like this:
The log normal d i stri bution was dead .
f(k) = I (e - rfr(rx)kfk!) g(x)dx. ·
Here, f(k) is the expected number of species with abun dance k in the sample, and g(x) is the (continuous) version of the community distribution. The remaining integrand is the Poisson distribution, and the parameter r is the sam pling ratio, essentially the fraction n/N. I proved that the transformation was shape-preserving (or formula-preserving, if you like) for a wide variety of functions, including polynomials [5]. I claimed that the transformation ought to have the same effect on analytic functions (such as the lognormal) which could be approx imated to an arbitrary degree of closeness by polynomials. The general theory carne equipped with its own "veil," not a vertical line this time, but a sloping, sigmoidal curve, as illustrated in Figure 4a for a uniform distribution. This veil-curve, when applied to the lognormal distribution, far from cutting it vertically, hugs it closely, as in Figure 4b. This meant that if a community had the shape of the (un-
f(x)
= =
c(l - &c)lx; 0;
The function is thus defmed over the interval [ e, oo), which was technically necessary because I expected to encounter distributions in which the maximum abundance exceeded A. Goodness-of-fit tests would require that the theoretical distribution (the logistic-J) must have some value at such abundance levels. The continuous version of the logistic-J distribution was useful because with it, I could also treat data given as fractional densities or percentages, both com mon in the literature.
VOLUME 23. NUMBER 3, 2001
31
With the continuous logistic-J distribution in hand, I hired two biology graduate students to begin a library search. I wanted 100 "randomly selected" biosurvey papers
computer simulation) often fail to grasp is how the chi square, like other statistical tests, operates. When one uses a chi square test to compare a theoreti
(2) the num
cal distribution with an empirical distribution, one takes
bers were not order-of-magnitude figures, (3) no species
the differences between the empirical and theoretical val
in which (1) at least 30 species were reported,
were omitted because of rarity or low abundance. As for
ues, squares them, divides by the theoretically predicted
the "random" nature of the search, I told them to pick any
values, then adds up all the resulting terms. While it is cer
year that came into their heads and to search
Biological
tainly true that smaller chi square scores mean better fits,
If our library at the University of
we do not expect all scores to be low. For example, if we
Abstracts for that year.
Western Ontario had the journal, they were to make a copy
compared a theoretical distribution D with 100 samples that
of it and pass it along to me. The papers came in as fast as
originated unquestionably from
I could deal with them: Butterflies in Sumatra, lichens in
10 degrees of freedom, we would expect the average chi
the Arctic, trees in India, abyssal fish in the Pacific Ocean,
square score would be about 10. We would also expect a
fungi in Japan, bats in Guatemala, and so on. For every pa
small percentage to have rather high scores.
per that fulfilled the three criteria, I made a histogram of
D, via a chi square test at
The chi square test is meant to be a rejector of hy
In other words, if someone tested distribu D against just one empirical data set at 10 degrees of
the main data, then fitted the appropriate version of the lo
potheses only.
gistic-J distribution to it.
tion
8) dis
freedom and found a score of 19.3, he or she would be jus
tribution. I had already worked out a set of solution equa
tified in denying that distribution D somehow governed or
The logistic-J distribution is a two-parameter (e,
tions that used two parameters from the field data to ar
underlay the empirical data. The rejection would, after all,
rive at estimates for e and 8. The first of these was the
have only a small chance of being wrong. As for samples
number F1 of species in the lowest abundance category.
that achieve low scores on a chi square test, paradoxical
The second was the mean abundance, p,. The main equa
as it may sound, we can say nothing. There are infinitely
tion contained a mixed linear/logarithmic expression and
many theoretical curves that would fit the data as well or
had no closed-form solution. It took about ten minutes,
better. How could we possibly conclude that distribution
once I became good at it, to solve a typical system numer
D is the "correct" one?
ically by hand, but it became increasingly time-consuming,
To return to my story, I had realized from the start that
so I wrote a computer program [6) that would solve the
one or a handful of tests would not be enough. On the 100
equation and generate the parameters e and
8..
randomly selected biosurveys, the logistic-J had scored, on
The number F1 was useful in deriving a value for e, as
average, about as well as could be expected. Not only that,
this parameter was simply a point somewhere between 0
it had outperformed the log-series by a statistically dis
and the lowest abundance category at which the J-curve
cernible margin in a paired difference test. The margin
could be said to "start." It was not a veil-line, but corre
could only be due to chance, it turned out, with a proba
sponded to the minimum abundance in the community be
bility of 0.005. Another surprising result of the metastudy
ing sampled. Solving the equation amounted to finding from
was that the estimates of maximum abundance
what point e, the integral of the J-curve from
by the solution equations, when expressed as a percentage
e
to 1 would
a provided
of the actual maximum abundances of the field data, pro
yield an area equal to F1• My test of choice was the chi square distribution. Besides testing the logistic-J distribution against the 100 biosurveys, I decided also to test the log-series distribution,
duced an average score of 99 percent. In ecological testing, such a result is rather extraordinary. I sent a paper to the
Biological Bulletin. The review
its only serious competitor among the extant and frequently
process used up no less than three sets of referees, many
all the tests
of whom rejected the paper for reasons that were spurious
used theoretical distributions. At first I did
by hand, but, finding a tendency to commit the odd arith
and showed little understanding of statistical methods.
metical error, I automated the process with a custom
surprised me. Each time I had to explain via long letters to
This
written program [6). By the time I had accepted and tested
the editor why the referees were wrong. Reading between
100 biosurveys, I had received about 150 papers from my
the lines, I would say that at least some of the referees were
assistants, some 50 of them having been rejected by at least
rather upset by the logistic-J distribution, with its under
one of the three criteria just mentioned. Not one of the pa
lying theory of stochastic populations. I had to point out
pers accepted (nor any of the ones rejected) showed any
that whatever they thought of the theory, the paper was re
thing like a lognormal distribution.
ally about a new distribution and how well it fit the data.
The results of the metastudy surprised even me. The av
The editor was inclined to accept the paper, and after I had
erage chi square score was 10.9, fairly close to the theo
satisfied most of the reasonable demands of the last set of
retical average of 10.0 for a perfect score on a chi square
referees, he published the paper [6). At last count, this pa
with ten degrees of freedom. Some people were confused
per has generated nearly 100 requests for reprints.
by such a finding;
if the logistic-J distribution is so good,
Through all of this, I had been beset by a nagging doubt.
if the MSL system didn't actually produce what I had
why shouldn't the average chi square score be close to
What
zero? What many biologists (as well as my own students in
chosen to call the logistic-J distribution? Why had I been
32
THE MATHEMATICAL INTELUGENCER
so foolish as to conduct extensive tests on a distribution that I hadn't confirmed? In the spring of 2000, I decided to go after the problem until my brain burned out. Every night, just before going to sleep, I would attempt yet another ap proach. I fantasized that the answer would come to me one day, like the famous illumination of Henri Poincare. After many fruitless days of intense mathematical labour on the structure of the Fuchsian functions, the solution came to Poincare, literally one step at a time, as he boarded a hol iday bus in Caen, France. By the time he had taken his seat, the problem was settled. What happened in my case, six months into the effort, was a strange coincidence. A colleague in the Zoology Department at the University of Western Ontario, Stan Caveney, sent me a copy of the New Scientist with an arti cle about the revival of serious study of Pareto's law by two French physicists [ 1 ] . "I thought you might be interested. It reminded me of your J-curve theory." Vilfredo Pareto was a French economist of Italian extraction who, in the late 19th century, formulated a distribution which he claimed to fit the world's wealth. There were many poor people and very few rich people. He devised a curve of the general form
where N(k) is the number of people with k units of wealth at. their disposal. Pareto had derived his law on the basis of a random interchange of money, not unlike my random in terchange of energy. Haunted by this resemblance, I was further alarmed when the article asserted that the expo nent e lay somewhere between 2 and 3, not the logistic-J "law," at all! Could the MSL system be producing an inverse square distribution? I had checked the actual shape of av erage curves that resulted from all-day runs, involving hun dreds of billions of interactions. The shape was definitely not inverse square (or worse). But the appearance of the square in the denominator of Pareto's fraction reminded me of a result I had obtained some time ago and nearly forgotten about. It concerned the expected waiting time of an MSL species in the kth abun dance category. The expression, E(k)
=
(1 - vk)lk2,
had been derived by considering the probability p(k) that a species of abundance k would not change its abundance in the next "tick" of the simulation clock This probability is 1 - 2k!N, because there is a probability kiN of the species reproduc ing, as well as dying. Writing 2/N as v, I could simplify this expression somewhat, then apply the standard expectation formula, E(k)
=
Li
0
pj(k).
This formula says that if a phenomenon happens with prob ability p(k) after 1 time unit, then it takes, on average, E(k) time units for the phenomenon to happen. The sum could be put in closed form by using that old chestnut,
'Lxj
=
11(1 - x),
then differentiating both sides and throwing in a factor of (an exercise for the student of calculus). It was only then, after forgetting this result for so long, that I realized the formula must be intimately connected with the expected number of species of abundance k. It was like the quantum-mechanical interpretation of Schrodinger's equation: The shape of an electron's orbital reflects the prob ability of finding it in a given area at a given moment. In a day, I had the rest of the argument. It goes like this: Suppose the MSL system is running R species, and thatf(k) is the number of species occupying the kth position, on av erage, when the system is at equilibrium. Thenf(k)IR is the proportion of time spent by the average species in the kth abundance category. But this proportion can be calculated in another way. The time spent by the typical or average species at abundance k is the expected waiting time mul tiplied by the probability of the species arriving at the kth abundance category in the first place. At equilibrium, the arrival probability equals the departure probability, namely vk. The product of vk with E(k) yields the right-hand side of the following proportionality, the left-hand side being the other expression for average residence time: x
�)
oc
v(l - vk)/k.
This is the logistic-J curve. If this had been the time of Pythagoras, I would have bought an ox and sacrificed it. (The equivalent act to-day would be to buy a Mercedes-Benz and set it on fire.) Admittedly, the logistic-J formula is not Pythagoras's the orem, but I was pleased, nevertheless. The original edu cated guess had been essentially correct, even though the parameter 8 was a bit more general than the parameter v. The view of nature implied by the stochastic commu nities hypothesis resembles an exercise in one-dimen sional particle physics. Species are like particles that vi brate stochastically along the abundance axis, sometimes increasing, sometimes decreasing. Particles close to the origin vibrate very slowly, while those further out vibrate rapidly. One could call abundant species "hot" and rare ones "cold." The stochastic communities theory leads to a mathe matical question of potentially great importance: If a great many deterministic processes contribute to the value of a variable over time, under what conditions may the behav iour of the variable be accepted as "effectively random"? A single deterministic equation, for example, is responsible for the "random" numbers produced by most simulation programs including, ironically enough, the MSL system it self. The numbers thus produced pass several tests of ran domness and are perfectly adequate for simulation pro grams (a billion-dollar industry). Nature does not have such a function embedded in it, but if a single determinis tic process can produce seemingly random numbers, what about the combined effect of temperature, rainfall, and
VOLUME 23, NUMBER 3, 2001
33
A U T H O R
N or species
(b)
A. K. DEWDNEY Department of Computer Science University of Western Ontario London, ON
N6A 587
Canada
abundance
e-mail: akd@csd .uwo.edu
Figure 5. The great J-curve: samples (a) and communities (b).
Alexander Keewatin Dewdney of London, Ontario was origi nally trained as a mathematician, the sort that is called a the other meteorological factors, not to mention other envi
oretical computer scientist. He was a columnist for Scientific
ronmental variables?
American for seven years. Now that his biological interests
Those viewing the output of the MSL system might well
have blossomed into a major research project, he has two ad
feel some concern at the appearance of so many species
ditional affiliations in addition to that listed above: in Zoology
at low abundances, as though an entire ecosystem were
at the University of Western Ontario, and in Computer Science
poised on the brink of destruction. Such a shape tends to
at the University of Waterloo. But he and his wife Patricia still
be typical of samples only. In real communities the abun
live in London.
dance distribution, if we could see it, would look far less threatening. The ultimate version of the logistic-J distribution works
extinct in my little community. Why not? Natural commu
as well with samples, as does the special logistic-J described
nities lose species all the time-only to be resupplied from
earlier. The
a neighbouring community.
general logistic-J distribution,
f(x')
=
written
If beauty is one of the criteria of a good scientific the
c'(l - 8'x')lx',
ory, I would claim a share for the theory of stochastic com
looks quite similar to the special formula introduced ear
munities and the resulting logistic-J distribution; while
lier. But here the function is defined over the interval (0,
planets follow elliptical orbits, species appear to move in
) the variable x' ll + e.
hyperbolic ones. And now, I believe, I can see the forest
oo ,
equals x
The selection of
e
and
+ e,
and the constant
ll'
equals
8 amounts to placement of coor
dinate axes over a plot of the standard hyperbolic function,
y
=
1/x,
as shown in Figure
smaller and
8 larger,
for the trees.
5.
In samples,
e
tends to be
as in the distribution labelled (a) in
the figure. In actual communities, on the other hand, the reverse is true, as in the distribution labelled (b). The abundances of species increase almost (but not quite) by geometric progression. The "not quite" is crucial to the theory. It corresponds to the subtractive
8 in the for
mula, and amounts to the imposition of a fmite limit on how far the progression may continue. Viewed in this light, the logistic-J distribution is not nearly so alarming, in fact it's aesthetically pleasing. I have, of course, operated the MSL system with the ex
REFERENCES
[1] Buchanan, M. That's the way the money goes. The New Scientist, Aug 1 9 (2000), 22-26. [2] Feller, W. An Introduction to Probability Theory and its Applications. John Wiley & Sons, New York, 1 968, vol. 1 . [3] Fisher, R.A. , Corbett, S.A. , Williams, C.B. The relation between the number of species and the number of individuals in a random sam ple of an animal population. J. Anim. Eco/. 1 2 (1 943), 42-58. [4] Dewdney, A. K. A dynamical model of abundances in natural com munities. COENOSES 1 2(2-3) (1 997), 67-76. [5] Dewdney, A. K. A general theory of the sampling process with ap plications to the "veil line." Theor. Popul. Bioi. 54(3) (1 998), 294-302. [6] Dewdney, A. K. A dynamical model of communities and a new
tinction switch "on" and with the high average abundances
species-abundance distribution. Bioi. Bull. 1 98(1 ) (2000), 1 52-163.
that characterize most natural communities. The individual
[7] Magurran, A. E. Ecological Diversity and its Measurement. Princeton
species dance back and forth, often trading places, but with a sequence of average positions that strongly resemble those of Figure
34
5.
And, yes, sometimes a species becomes
THE MATHEMATICAL INTELLIGENCER
University Press. Princeton, NJ, 1 988. [8] Preston, E. F. The commonness, and rarity, of species. Ecology 29 (1 948), 254-283.
Ug's Last Theorem
It is almost certain that Ug was mistaken, but of course we cannot be certain . Proposition *4.43
Fermat's Last Theorem
1-:. a,/3 E 1, :J: a n f3 = A. = .a U f3 E 2 Dem 1-.*54 26. :J I- :.a = t' x.f3 = t' y :J : a U /3 E 2. = .x i= y. = . t' x n t' y = A [*51.231] [*13. 12] = a n f3 = A (1) 1-.(1).*11 . 11 . 35. :J l-:.(3.x,y).a = t' x.f3 = t' y. :J : a U f3 E 2. = . a n f3 = A (2) 1-.(2).*11 54. * 52. 1 . :J 1-.Prop. ·
·
He [Fermat] wrote a note in the margin of his copy of the works of Diophantus saying that he had a proof [that xn + yn = zn has no nontrivial integer solutions when n 2: 3], but that the margin was too small to contain it. It is almost certain that he was mistaken, b ut of course we cannot be certain. Underwood Dudley,
Elementary Number Theory, 1978.
2nd Edition. W. H. Freeman & Co., New York,
Sasho Kalajdzievski Department of Mathematics
From this proposition it will follow, when the arithmetic has been defmed, that Russell.
1 + 1
=
2. A. N. Whitehead and B.
Principia Mathematica, 1962.
University Press, New York,
Volume 1. Cambridge
University of Manitoba Winnipeg, R3T 2N2 Canada e-mail:
[email protected].
© 2001 SPRINGER·VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
35
R. CADDEO, S. MONTALDO, AND P. PIU
Th e MObi us Stri p and
Vivian i ' s Wi ndows Dedicated to the memory of Alfred Gray
he space curves called "Viviani 's windows " are curves that solved a celebrated geometric puzzle:
"Aenigma Geometricum de miro opificio Testudinis Quadrabilis Hemisphaericae"
(Geometric enigma on the remarkable realization of a squarable hemispherical vault). This is a (pseudo-)architectural problem proposed by Vincenzo Viviani, a disciple of Galileo, in 1692 (see [1], page 201, and [2, 3] for a complete and detailed treatment), formulated as follows: build on a hemispherical cupola four equal windows of such a size that the remaining surface can be exactly squared. Among several known solutions, the following was found by Viviani and by other eminent mathematicians of that time: the four windows are the intersections of a hemisphere of radius a with two cylinders of radius a/2 that have in com mon only a ruling containing a diameter of the hemisphere (Fig. 1). When we cut away four (or more) equal half-calottes from a hemisphere by means of four (or more) planes or thogonal to its boundary (an equator), we obtain a spe cial dome vault. In Italian this vault is called Volta a vela ("vela" meaning "sail"), because it resembles a sail filled by the wind. For this reason and for the fact that the formula for its area does not involve 7T (see [ 1 ]), Viviani gave to this surface the name Vela Quadrabile Fioren tina. Figure 2 shows a cardboard model of the support ing structure made by the student Gregorio Franzoni (University of Cagliari), and the computer-generated surface. An example of such a cupola can be admired in the in terior of the basilica of San Fedele in Milan (Fig. 3). Thus Viviani's windows give rise to special spherical curves. In general, by "Viviani's window" or "curve" is meant a curve obtained by intersecting a sphere with a
36
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
round cylinder tangent to the sphere and to one of its di ameters, a sort of spherical figure eight (see Fig. 4). The Osaka Maritime Museum (designed by Paul Andreu), under construction in 1997, is a spherical shell embellished by a metal grid generated by two families of curves (see [4]). We are not sure that these are true Viviani's curves, but certainly they could be. A Mobius Strip in Place of the Cylinder
It is interesting to note that a Viviani curve can be obtained by intersecting a sphere with the non-orientable analogue of the cylinder, that is, with a Mobius strip. Let X : (0, 27T) X ( 3, 3) � IR13 be a parametrization of the Mobius strip gen erated by rotating (along a circle of radius 1 in the xy-plane) -
Figure 1. The intersection of a hemisphere and two cylinders giving Viviani's windows.
Figure 2. Cardboard model (left) of the supporting structure and computer realization (right) of a Vela Quadrabile Fiorentina.
Figure 3. A drawing of San Fedele in
Figure 5. The Osaka Maritime Museum.
Milan.
Figure 4. A Viviani window.
Figure 6. Drawing of the Osaka Maritime Museum.
VOLUME 23, NUMBER 3, 2001
37
Figure 7. A wider Mobius strip.
Figure 8. The Mobius strip.
Figure 9. The equator the curve a(u).
Figure 10. Four views of the intersection between the sphere, the cylinder, and the Mobius strip that gives the Viviani curve.
38
THE MATHEMATICAL INTELLIGENCER
v =
0 and
a vertical line segment L (of length
6)
around the z-axis, in
such a way that when the midpoint of L makes a rotation
u in the plane xy, u/2, that is,
of angle angle
(
the line segment has rotated an
=x
(u,
- 2 cos
�)
= ( - cos2
u,
- sin
u cos u,
-sin
u).
One can easily check that a(u) is also on the cylinder 2 + y2 = and therefore it is a Viviani window.
(x +
b
{,
Figure 10 (realized with Geomview) shows four views
X(u, v) =
a(u)
cos
u
(
1
+
v cos
�}
sin
(
u 1+
v cos
�}
v sin
�}
Note that for v
E ( - 3, 3) this parametrization of the Mobius strip gives rise to self-intersections, as shown in Figure 7. To obtain the usual Mobius strip we have to reduce the interval of the v's. For example, when get the configuration shown in Figure
vE 8.
( - 0.3, 0.3) we
Now we look for the intersection of our Mobius strip with the sphere 2 § =
of the intersection between the sphere, the cylinder, and the Mobius strip that gives the Viviani curve. Comment There is, of course, a relation between the radius central circle of the Mobius strip and the radius
r of the R of the
sphere in order to obtain Viviani's curves. It is easy to check that they must be equal. REFERENCES [ 1 ] G. Loria, Curve sghembe speciafi, Ed. Zanichelli, Bologna, 1 925.
{ (x, y,
z)
E
IR3 : x2 + y2 +
[2] C. S. Roero, L 'interet international d'un problerne propose par
z2 = 1 }.
Viviani, Actes de I ' Univ. d ' Ete Hist. des Math . , I.R.E.M. Toulouse,
An easy computation gives
(
v v
+2
1 986. cos
�)
[3] C. S . Roero, The Italian challange to Leibnitzian calculus in 1 692.
= 0,
Leibnitz and Viviani: a comparison of two epistemologies, V Int. Congress Leibnitz, Hannover, 1 988.
and therefore for v = 0 we have the equator of the sphere (Fig.
[4] P. Andreu, Osaka Maritime M useum (Sotto e sopra if mare), L'ARCA,
9), while when v + 2 cos u/2 = 0 we obtain the curve
1 33, L'ARCA edizioni, Milano, January 1 999.
A U T H O R S
AENZO CADDEO
STEFANO MONTALDO
PAOLA PIU
Dipartimento di Matematica Universita degli Studi di Cagliari Via Ospedale 72, 09124 Cagliari, Italy e-mail:
[email protected] [email protected] [email protected] The authors are all natives of Sardinia, and are all on the permanent faculty of the Universita degli Studi di Cagliari , where they collabo rate on geometry of surfaces and jointly run a g raduate program in Computer Graphics Renzo Caddeo studied in Romania and France before returning to Cagliari. In 1 994 he was invited by Alfred Gray to collaborate on an expanded version, translated into Italian, of Gray's Modern Differential Geometry of Curves and Sur1aces. Their work was completed but the publication has languished since Gray's death in 1 998. In his spare time, Caddeo is an enthusiast for cultivation of olive trees and production of olive o i l , also for alpine hiking. Stefano Montaldo obtained a PhD in Leeds, England, with a thesis on "Stability of harmonic maps and morphisms." He is also doing research on minimal surfaces. He is an enthusiast for scuba-d iving and mountain-biking. Paola Piu received her doctorate in 1 988 in Mulhouse, France, on the topic "Sur certains types de distributions non-intE§grables totalement geodesiques. " She also studies contact and Riemannian geometry and nilpotent Ue groups.
VOLUME 23. NUMBER 3. 2001
39
S
P
R
I
N
G
E
R
F 0
R
S T A T
Introduction , to Grap hical� 1"\ode\ling �-.
· ··�:--:
I
S T
I
C S
Sta tlsUc•l Scoience In the Courtroom
��;1;;?�
.
GEERT
VERBEKE, Katholieke Universiteit
DAVID mWARDS, Novo NordiSk A/S, Bagsvaerd, Denmark
leuven.
Belgium. and GEERT MOLENBERGHS, limburgs Un iversitair Centrum. Belgium
This book provides a comprehensive treatment of linear mixed models for continuous longitudinal data.
ext
to model fonnulation. this edition puts major emphasis on exploratory data analysis for all aspects of the model. uch as the marginal model. subject-specific profiles. and residual covariance structure. Funher. model diag nostics and missing dala receive ex ten ive treatment cnsitivity analysis for incomplete data is given a prominent place. Most analyses were done with the I XED procedure of the SAS software package. but the data analyses arc presented in a software-independent fashion. Introduction
Longitudinal Data
•
•
E.amplcs • A Model for
Exploratory Data Analysis
Estimation of the Marginal Model Marginal Model
•
•
I n ference for the
•
I n ference for the Random Effects
Filling Linear Mixed Models with SAS • General Guidelines for Model Building • Exploring Serial orrclation • Local lnnucncc for the Linear Mixed •
Model • The Heterogeneity Model Linear Mixed Models
•
•
Conditional
Ex plori ng Incomplete Data
•
Joint Modeling of Measurements and Missingnc·s
•
Simple Missing Data Methods • Selection Model Pa11em-Mixturc Models • cnsitivity Analysis for
•
Selection Models •
•
Sensitivity Analysis for Model
How Ignorable is Missing at Random?
Expectation-Maximization Algorithm onsidcrations
•
PUBLIC POLICY AND STATISTICS
Second Edition
FOR LONGITUDINAL DATA
Co11teals:
California. los Angeles (Eds.)
GRAPHICAL MODELLING
LINEAR MIXED MODELS
•
•
This textbook provides an introduction to graphical modeling with emphasis on applications and practicali ties rather than on a fonnal development. The second edition adds two chapters, One describes the usc of directed graphs, chain graphs. and other graphs. The other summari1..e s recent work on causal inference. relevant when graphical models arc given a causal interpretation. Following an introductory chapter which sets the scene and describe some of the basic ideas of graphical mod cling. subsequent chapters describe panicular families of models. including log-linear models. Gaussian models. and model for mixed discrete and continuous variables. unher chapter. cover hypothesis testing and model selection. Chapters 7 and 8 are new to the second edi tion.
haptcr 7 dc.�cribes the usc of directed graphs.
chain graphs. and other graphs. Chapter 8 summarizes some recent work on causal inference. relevant when graphical models arc given a causal interpretation.
Co11te11ts:
Preliminaries
•
Discrete Models
•
Continuous
Models • Mixed Models • Hypothesis Tcstng • Model election and Criticism • Other Types of Graphs and Models
•
SAllY MORTON, RAND Statistics Grou p, Santa Monica, CA . and JOHN ROLPH, University of Southern
INTRODUCTION TO
Causal Interference
2000/352 PP./HAR DCOVER/$69.95/ISSN 0.387·95054-0 SPRINGER TEXTS IN STATISTICS
The
Case Studies from RAND This volume describes the rich analytical techniques and substantive applications that typify how statistical think ing has been applied at the RA
being a discerning consumer of analyses. and an applied statistician who uses the available infonnation to advise the policy-maker in the face of substantial uncenainty. Each case study has a common forn1at that describes the policy questions. the statistical questions. and the suc cessful and the unsuccessful analytic strategies. This casebook is designed for statistics courses in areas rang· ing from economics to health pol icy to the law at both the advanced undergraduate and graduate levels. Empirical researchers and policy- makers should also find this casebook informative.
Co11ten1s:
School-based Dmg Prevention: Challenges
i n Designing and Analyt.ing Social Experiments
JOSEPH GASTWIRTH, The George Washington University.
2000/600 PP./HARDCOVER/$79.95/ISSN 0.387·95027·3
Washington. DC (Ed.)
SPRINGER SERIES IN STATISTICS
STATISTICAL SCIENCE
B.S. EVERITT, Institute of Psychiatry, london. United
Suaislical Science ;, the Couruoom is a collection of arti
Kingdom
cles wrillen by statisticians and legal scholars who have
•
The
Health Insurance Experiment: Design Using the Finite Selection Model
•
Counting the Homeless: Sampling
Difficult Populations
•
Periodicity in the Global Mean
Temperature Series • Malpractice and the Impaired Physician: An Application of Matching
•
upply Delays
for F- 14 Jet Engine Repair Pans: Developing and Applying Effective Data Graphics • Hospital M nality Rates: Comparing with Adjustments for Casemix and Sample Size
Design
Case Studies
D Corporation over the
past two decades. The goal is to bridge the gap between
Uncenainty
•
•
Eye Care Su1>ply and Needs: Confronting Modeling Block Grant Formulae for
Substance Abuse Treatment 2000/304 PP./HA RDCOVER/$49.95/ISBN 0.387·98777·0 STATISTICS FOR SOCIAL SCIENCE A D PUBLIC POLICY
IN THE COURTROOM
been concerned with problems arising in the usc of statisti
CHANCE RULES
cal evidence. The twenty-two anicles include contribu
An Informal Guide to Probability, Risk, and Statistics
tions by Donald B. Rubin. eymour Gcisscr. David Pollard. COli L. Zeger. Joseph B. Kadanc. Bruce Weir.
In Clra11a Rules, aspect of chance. risk. and probabili ty. from simple games involving the toss of a coin to the usc of clinical trials in medicine and the evaluation of
Call: 1-S()().SPRINGER or Fax: ( 201} 3484505
AUGUST 2000/552 PP./HARDCOVER/$59.95/ ISSN ().387-98997-8
Write: Springer-Verlag New York , Inc., Dept. 52508.
STATISTICS FOR SOCIAL SCIENCE AND PUBUC POUCY
PO Box 2485, Secaucus. NJ 0709&2485 Visit: Your loc al technical bookstore
alternative therapies. are explored in an i n formal and cntcnaining manner.
Co111e11ts: A
Brief History of Chance
•
Tos ing Coins
and Having Babies • Rolling Dice • Gambling for
Fun: Lo11crics and Football Pools Coincidences
•
Thomas Bayes •
•
Binhdays and
Conditional Probability and the Reverend •
Puzzling Probabilities
•
Taking Risks
Statistics. Statisticians and Medicine • Alt.emativc
Thcrapies-Paneccas of Placebos? •
Serious Gambling:
•
Rouleue. Cards and Horse Racing
•
Chance i n
What is Chance? 1999/ 216 PP .. 35 ILLUS./SOFTCOVER/$26.00 ISSN 0.387·9877&2
aturc
Order Today!
and Joseph L. Gastwinh.
ANDREAS KRAUSE, Novart i s Pharma AG. Basel. Switze�and and MELVIN OlSON, Allergan, Irvine, CA
THE BASICS OF S AND S-PLUS
E-mail:
[email protected] Instructors:
Call or write for information on textbook
exam copies YOUR JO.OAY RETURN PRIVILEGE IS ALWAYS GUARANTEED.
Second Edition The second edition has been updated to incorporate the completely revised S Language and its implementation in S-PL
S.
ew chapters have been added to explain how
to work with the graphical user interface of the Windows version. how to explore relation hip in data using the powerful Trellis graphics system. and how to understand and use object-oriented programming. In addition. the programming chapter ha been extended to cover some of the more technical but imponant aspects of S-PLUS. 2000/400 PP./SOFTCOVER/$44 .95/ISSN 0.387-98961-7 STATISTICS AND COMPUTING fALL 2000
PROMOTION •52508
li,i%$Mj.i§,@ihfiii.IIIQ?ii
A Bridge Over a Ham iltonian Path Gary McGuire and Fiacre
6 Cairbre
E
D i rk H uylebro u c k ,
Ed itor
very year on the 16th of October, a group of people assemble at Dun
sink Observatory in Dublin, Ireland, and walk to a bridge over the Royal Canal. They are the staff and students of
I
as it may have been-to cut with a knife on a stone of Brougham Bridge as we passed it, the fundamental for mula with the symbols i, j, k namely i 2 = j 2 = k2 ijk = - 1. ,
=
the Department of Mathematics at the National University of Ireland, May
All traces of Hamilton's carving are
nooth, and they are commemorating
gone, but there is a plaque on the
the creation of the quatemions by the
bridge commemorating the moment.
Irish mathematician, William Rowan
The bridge is in the suburb of Cabra
Hamilton, on 16 October, 1843, at that
and is now called Broombridge. There
bridge. The walk is about three miles
is a train station there of the same
long and passes through very pleasant
name, and it is the second stop on the
countryside, on the outskirts of Dublin.
Arrow
train
route
from
Connolly
William Rowan Hamilton was born
Station, in Dublin, to Maynooth. The
in Dublin on 4 August, 1805. He spent
plaque was erected in 1958 by the
his youth in Trim, Co. Meath, where
Taoiseach
he was educated by his uncle, James
Eamon de Valera, who had a degree in
(Irish
prime
minister),
Hamilton. Hamilton lived in a large
mathematics and lectured at Maynooth
house, which is now called St. Mary's
for two years (1912-1914). De Valera
Abbey and is beautifully situated be
himself scratched the famous quater
side the Yellow Steeple, on the banks
nion
of the Boyne, across from the spectac
Kilmainham jail in Dublin while im
mathematical tourist attractions such
ular ruins of Trim castle. The house
prisoned there in 1916. One can visit
as statues, plaques, graves, the caje
also served as the local school which
the jail and take a very interesting
was run by his uncle. Hamilton was ap
guided tour, and see in the museum a
Does your hometown have any
where the famous conjecture was made, the desk where the famous initials
formula
on
his
cell
wall
in
pointed Astronomer Royal of Ireland
copy of the formula written by de
at the age of twenty-one, and thus be
Valera. In 1943 the Republic of Ireland
are scratched, birthplaces, houses, or
came a Professor while still an under
issued a commemorative stamp to cel
memorials? Have you encountered
graduate at Trinity College, Dublin. He
ebrate the centenary of the birth of the
lived at Dunsink Observatory from that
quatemions.
a mathematical sight on your travels? If so, we invite you to submit to this
point, and the mathematical tourist can
The current Taoiseach can see a
visit the observatory and see interest
statue of Hamilton near his office. The
column a picture, a description of its
ing items related to
statue is on the steps of Government
mathematical significance, and either
would frequently walk along the Royal
Buildings in Merrion Street in Dublin,
Canal to the Royal Irish Academy
and dates back to the time when the
(some possessions of Hamilton are
buildings housed the College of Science.
kept at the Academy, including some
This area may be of additional interest
notebooks and his Icosian Game). It
to the tourist, for a few hundred yards
was on one of these walks that he had
from Hamilton's statue is a monument
his famous moment of insight. He later
to
described the event to his son [ 1 ] :
Wilde. The latter statue is in Merrion
a map or directions so that others may follow in your tracks.
Hamilton.
He
another
famous
Irishman,
Oscar
Square opposite the former home of
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium e-mail:
[email protected]
An electric current seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely di rected thought and work. . . . Nor could I resist the impulse-unphilosophical
Oscar Wilde. When Hamilton first met Oscar Wilde's mother, Lady Wilde, she asked
him to be Oscar's godfather. He
declined, but later became a close friend of Lady Wilde, who was a promi nent
writer
under
the
pseudonym
© 2001 SPRINGER-YERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
41
on the 2-sphere in 3-space. It is not hard to show that the transformation x � qxq, where x is pure and q is a unit quaternion, is a rotation of 3-space. The correspondence between unit qua ternions and rotations of 3-space is two-to-one. One modern application of quater nions is that they frequently replace Euler angles in computer graphics for describing rotations of space. Thus, quaternions are now used, for exam ple, in performing attitude determina tion in spacecraft like the space shut tle,
and
in
computer
games.
The
famous character Lara Croft in the computer game "Tomb Raider" was created [2] using quaternions! Quaternions are nowadays of inter est to algebraists in the form of general Figure 1. Staff and students of NUl, Maynooth, at Dunsink Observatory on 16 October, 1999.
quaternion algebras over a field or ring
F, where the coefficients a, b, c, d come from F, and the defining relations are i 2 = a, j 2 = {3, ij k -ji, for some a and {3 in F. The Hurwitz integral quaternions, where a, b, c, d come from =
Speranza. It is curious to note that Oscar Wilde was born on
16 October
(1854). Who knows what Hamilton's re ply might have been if he had known this! A quaternion is an expression of the fonn
a + bi + cj +
dk where
a, b, c, d
With q
a + bi + cj + dk, we let a - bi - cj - dk and N(q) = qq = q a2 + b 2 + c 2 + d 2. Quaternions with a = 0 are called pure quaternions, and each one of these bi + cj + dk can =
=
=
tz, are used in number theory and lat
tice constructions.
The quaternions played an impor
be identified with a point in 3-space,
tant role in Maxwell's original for
are real numbers. Addition of quater
called
(b, c, d). Quaternions with N(q) = 1 are unit quaternions. Thus, pure
analysis,
mons is defined componentwise, and
unit
roots
physics, is an offspring of quatemions.
multiplication is defined using
of
1 and can be thought of as points
One can show that the unit quatemions
i2
=
j2
=
k2
=
ijk
=
-
-
quaternions ,
are
square
mulation of electromagnetism. Vector which is indispensable in
1.
The multiplication i s not commutative, i.e., xy is not always equal to x and
yx when y are quaternions. In Hamilton's
words: "the order of multiplication of these imaginaries is not indifferent." For example, one can show that
-ji
=
ij k. Hamilton has been called =
the Liberator of Algebra because his quaternions shattered the convention that multiplication should be commu tative in a number system. It is often said that he "freed algebra from arith metic" because of this momentous de velopment. His friend, the mathemati cian John Graves, said [ 1 ] :
I have not yet any clear view as to the extent to which we are at liberty ar bitrarily to create imaginaries, and to endow them with supernatural properties.
42
THE MATHEMATICAL JNTELLIGENCEA
Figure 2. The bridge where Hamilton created the quatemions, with the plaque visible to the left of centre.
are isomorphic to SU(2), g1vrng an
ences of the United States of America.
REFERENCES
other viewpoint on this group which is
He is buried in Mount Jerome Cemetery
[1] T. Hankins, Sir William Rowan Hamilton,
of great importance in particle phys
in
ics. Quatemions were generalised by
on Hamilton's life and works, see [ 1 ]
Graves, and independently by Cayley,
and [3].
Dublin.
For further
information
Johns Hopkins University Press, Baltimore, 1 980. [2] N.
to the octonions which are currently
We conclude the article with an in
Bobick,
Rotating
Objects
Using
1 998 (available online at
Quaternions,
under investigation for possible con
vitation and an open problem. We in
http://www .gamasutra.com/features/pro
nections to particle physics. Hamilton
vite mathematical tourists to join us on
gramming/1 9980703/quatemions_01 .htm).
made many other significant contribu
the walk on 16 October every year.
[3] R. Dimitric and B. Goldsmith, Sir William
tions to mathematics and physics, in
Contact either author for details, or see
Rowan Hamilton, Mathematica/ lntel/igencer,
cluding
http://www .maths.may.ie.
1 1 2 (1 989), 29-30.
his
famous
"Hamiltonian"
The
open
'
which was essential for the develop
problem concerns Hamilton's eldest
ment of Quantum Mechanics and is
son, William Edwin, who emigrated to
Department of Mathematics
ubiquitous in physics.
Canada. The last record of this son
National University of Ireland
Hamilton died at Dunsink on 2 September, 1865. In the same year, he had just become
seems to be in Chatham, where he worked for the
Ontario,
Maynooth
Planet news
Co. Kildare
the first Foreign
paper in 1891, and it is not known what
Ireland
Associate to be elected to the newly es
became of him, and indeed if he mar
e-mail:
[email protected]
tablished National Academy of Sci-
ried and had any children.
[email protected]
Mathematica l Olym piad Cha l lenges Titu Andreescu, American Mathematics Competitions, University of Nebraska, Lincoln, N£ Rlizvan Gelca, University of Michigan, Ann Arbor, Ml Thi
i
a comprehen ive collection of problem
written by two experienced and well-known
mathematics educators and coaches of the U. S . International Mathematical Olympiad Team. Hundreds of beautiful, chal lenging, and in tructive problem
from decade
of national and
international competitions are presented, encouraging readers to move away from routine exer cises and memorized a lgorithm toward creative olution and non- tandard problem- olving t clmiques. The work i divided into problems clustered in self-contained sections with solutions provid ed separately. Along with background material each ection include repre entative example beautiful diagram
,
,
and l ists of unconventional problems. Additionally, historical insights and
asides are presented to stimulate further i nqu i ry. The empha i
throughout i
on
ti mulating
reader to find ingeniou and elegant solutions to problem with mu ltiple approaches. Aimed at mot ivated high school and beginnjng college students and i nstructors, this work can be used as a text for advanced problem - olving course , for elf- tudy, or a a re ource for teach er and tudent tra in ing for mathematical competitions and for teacher professional develop ment, seminars, and workshops. From the foreword by Mark Saul : " The book weave together Olympiad p1vblems with a com mon theme, so that insights become techniques, tricks become methods, and methods build to 2000 / 280 PP., 85 ILL US.
mastery. . . Much is demanded of the reader by way ofeffort and patience, but the investment i
OfT OVER I I B
greatly repaid. "
To Order:
C al l :
HARDCOV E R / ISB
1 -800-777-4643
•
Fax :
(201) 348-4505
•
E-mail:
[email protected]
Textbook evaluation copies available upon request, call ext. 669. Prices arc valid in
•
0-8 I 76-4 1 55-6 / $29.95 0-8 1 76-4 1 90-4 / $59.95
Visit: www.birkhauser.com
orth America only and subject to change without notice.
Birkhauser Boston - International Publisher for the Mathematical Sciences
711)()
Promotion IYI097
VOLUME 23, NUMBER 3, 2001
43
Emmy Noether in Erlangen Alice Silverberg
E
mmy Noether was born and raised in the town of Erlangen, in south eastern Germany. She is remembered there as a brilliant mathematician and as one of the city's important daugh ters. Erlangen lies in Franconia, the northern region of the state of Bavaria. When Huguenot settlers arrived in the late seventeenth century they built new streets in a geometrical grid formation. A grid pattern in a block E shape even tually became the symbol of Erlangen. Today, part college town and part high tech center, Erlangen's southern half is dominated by Siemens, while the uni versity administration occupies the SchloB (palace) on the market square. Noether's birthplace is an apart ment building in the middle of the town, at HauptstraBe 23, between the Marktplatz and the Hugenottenplatz. A bronze plaque was erected there in her honor on July 29, 1997. The inscription translates:
Birth house of the [female]* mathe matician Emmy Noether. Born March 23, 1882. Emigration 1 933. Died April 14, 1935 in Bryn Mawr USA.
The University Women's Center ini tiated the project, raised the funds for the plaque, obtained the necessary per missions, and commissioned an artist. The German Mathematical Society (DMV) gave generous financial support. The circle on the plaque was chosen be cause it is a simple and easily recogniz able mathematical object. Karin Dahler, the sculptor, was a local artist whose usual medium is ceramics. An article ap peared in the local newspaper on the day of the plaque's dedication with the head line "Albert Einstein as supporter" and subheadlines: "Memorial plaque to com memorate the birthplace of the Jewish mathematician Emmy Noether" and "Posthumous appreciation as 'Genius' Discriminated against as academic be cause of ethnicity and gender."t The building dates from the begin ning of the eighteenth century. The oriel above the ornate sandstone por tal was added in 1866, and gives the building a distinctive feature. As I wrote this in August 2000, the interior of the building was being completely gutted and remodeled, after the de partment store HEKA, which had oc cupied part of the building, went out of
'In the German language, names of occupations are gender-specific. tAlbert Einstein a/s Fursprecher; Gedenktafel so// an Geburtsstatte derjudischen Mathematikerin Emmy Noether erinnem; Posthume WOrdigung als "Genie"- Wegen Herkunft und Geschlecht a/s Akademikerin diskriminiert.
Plaque at Emmy Noether's birthplace.
44
THE MATHEMATICAL INTELLJGENCER © 2001 SPRINGER-VERLAG NEW YORK
business. The plaque was temporarily removed, and the Fotoautomat in the doorway was gone, but the rococo doorlrame remained.* Max Noether joined the Erlangen mathematics faculty in 1875. He and Ida Amalie Kaufmann married in 1880, and Amalie Emmy Noether was the eldest of their four children. When she was 10 years old the family moved to a larger apartment at Ntirnberger Stra.Be 32. That building was a modem one, outside the town walls on the southern continuation of the HauptstraBe. During the city's rapid expansion after WW II, the build ing and gardens were destroyed to make room for the shopping complex Neuer Markt. The department store Horten stands there today. From age 7 until 15 Noether attended the Stadtische Hohere Tochterschule [City High School for Daughters] at Friedrichstra.Be 35, on the comer of Fahrstra.Be. The building was erected in the early eighteenth century as an aris tocrat's mansion and today houses the Sing- und Musikschule. The school was founded in 1887 as the successor to a private school that had been taken over by the city. In 1909 the school outgrew its build ing and moved to a new one at Schiller stra.Be 12. The school changed its name in 1914 to Marie-Therese-Schule, after the Royal Family paid a visit to the city. It began accepting boys in 1946, and changed its name to Marie-Therese Gymnasium in 1965. The building, which has an impressive facade, is around the comer from where the mathematics department is today. Noether passed the Bavarian exam inations for female teachers of French and Englishi in 1900. She audited courses at the university from 1900 until 1902. In Noether's time, classes were held in various buildings in the SchloBgarten, especially the Kollegien haus at Universitatsstra.Be 15. The Kollegienhaus is still used for univer sity classes. On July 14, 1903, Noether took high school graduation exams at the Konigliches Realgymnasium in the
Emmy Noether's birthplace at Hauptstrasse 23 in Erlangen.
nearby city of Ntimberg. She studied in Gottingen in the winter semester 1903104, and formally matriculated in Erlangen in the fall of 1904, when women began to overcome the barri ers to their entrance. In December 1907 she became the second woman to be granted a doctorate from the University of Erlangen. �
From the Hauptstra.Be it is a short walk around the block to a plaque on Goethestra.Be 4 dedicated to Paul Gordan, Noether's doctoral advisor. A translation of the plaque reads: "Here lived I 1890-1912 I Privy Councillor Dr. Paul Gordan I since 1875 I Professor of Mathematics I Member of 1 1 acad emies I + Breslau April 27, 1837
•The plaque has now been reinstalled, and the Fotoautomat has been replaced by a glass door leading to a staircase.
taayerische Prufungen tor Lehrerinnen der franz6sischen und der englischen Sprache.
+The first was Dixie Lee Bryant from Louisville, Kentucky, USA, who earned a doctorate in geology in 1 904.
VOLUME 23, NUMBER 3, 2001
45
Noetherstrasse street sign in Erlangen-Bruck.
t
Erlangen December 2 1 , 1912." The
university
46
THE MATHEMATICAL INTELLIGENCER
SchuhstraBe,
niversary of Gordan's birth.
Schlo:Bgarten. At the back of the dis
Universitatsstra:Be
from
the
You can ask to view the printed (in
sertation is a fold-out section of tables.
1908) version of Noether's Erlangen
Noether did not think highly of her
Ober die Bildung des Formensystems der terniiren bi quadratischen Form [On the con
called it "crap."* She is said to have
early work in Invariant Theory, and
struction of the system of foims of the
Jewish Cemetery in Erlangen.
on
across
doctoral dissertation,
Erlangen Mathematics Institute plaque.
library
plaque was erected on the l lOth an
ternary biquadratic form] , in the old
'Mist.
Emmy-Noether-Gymnasium in Erlangen-Bruck.
referred to her dissertation as a jungle of formulas and routine computations. Mter rece1vmg her doctorate, Noether worked unpaid at the univer sity until she moved to Gottingen in 1915. In 1908 she became a member of the Circolo matematico di Palermo. She joined the Deutsche Mathematiker Vereinigung (DMV) in 1909, and played an active role. Noether's first two grad uate students were Hans Falckenberg and Fritz Seidelmann. They received their doctorates in Erlangen in 191 1 and 1916, respectively. Noether's math ematical evolution toward the devel opment of abstract algebra began in Erlangen through her extensive inter action with Ernst Fischer, who took Gordan's chair in 1911.
The mathematics department of the University of Erlangen-Ntirnberg is now at BismarckstraBe 1 1/2. Halfway up the staircase leading to the large lecture hall is a stone plaque inscribed with only the three words "Max," "Emmy," and "Noether." The plaque was dedicated on February 27, 1982, at a centenary conference in her honor. The Emmy-Noether-Gymnasium (see http://emmy.nettec.de) is located on NoetherstraBe. The street was named in 1960 after both Max Noether and Emmy Noether. It lies in the suburb Erlangen-Bruck, south of the large Siemens complex and Felix-Klein-StraBe. The high school emphasizes the study of modern languages and the sciences. It opened on Liegnitzer StraBe in 1974
as the Erlanger Sudwest-Gymnasium, and moved to NoetherstraBe about 7 years later. After a competition to choose a new name, the school was re christened the Emmy-Noether-Gymna sium on March 25, 1982, the lOOth an niversary of Noether's birth. A little-known Noether fact is that the graves of two of her brothers, Alfred (1883-1918) and Robert* (1889-1928), can be found in the Erlangen Jewish Cemetery. The cemetery, opened in 1891, has been dormant since the Nazi period. It is at the north end of town, at the far side of the Burgberg, and lies within the fenced backyard of the house at RudelsweiherstraBe 85. To en ter the grounds you must ring the door bell and obtain permission from the oc-
·some sources give the name as Gustav Robert.
VOLUME 23, NUMBER 3, 2001
47
Plaque where Paul Gordan lived.
House where Paul Gordan lived.
48
THE MATHEMATICAL INTELLIGENCER
cupant. Go north on the Hauptstra.Be (which changes name along the way to Bayreutherstra.Be) until the edge of town, tum right on Bubenreuther Weg, then right again on Rudelsweiher stra13e. The Noether gravestones are at the back of the cemetery on the right edge, five rows from the back fence, and are difficult to read due to age and weathering.
Alfred studied chemistry in Erlan gen and Ttibingen, earning his doctor ate in Erlangen in 1910. Robert was mentally handicapped, and was insti tutionalized for much of his life. After Max's death, Emmy assumed responsi bility for his care. The middle son, Friedrich (Fritz), went east when his sister went west. He was accused of anti-Soviet activities and was executed
on September 10, 1941 in Orel during the advance of German General Heinz Guderian's tank divisions. Ida died in Erlangen two weeks after her daughter moved to Gottingen. An urn with her ashes was buried in Coburg. According to [4], Max and Emmy converted to Protestantism in 1920 (on November 5 and December 29, respectively). Max died a year later. He was cremated in Nfunberg and his ashes were buried there in a communal grave. Emmy's ashes were buried in the cloisters on the campus of Bryn Mawr College in a Quaker ceremony in 1982. Erlangen's Friedrich-Alexander-Uni versitiit is perhaps best known among mathematicians for faculty members Felix Klein (and his Erlangen Program), Max Noether, and Carl Georg Christian von Staudt, while Emmy Noether is most often thought of in connection with Gottingen or Bryn Mawr. How ever, the interested tourist can spend an erijoyable day in a delightful town by
tracing Emmy Noether's steps through her formative years as a girl, student, and young mathematician in Erlangen.
REFERENCES
(1] Auguste Dick, Emmy Noether 1882-1935, Elem. Math. Beiheft 13, Birkhauser Verlag, Basel, 1 970 (translated by H. I. Blocher,
ACKNOWLEDGMENTS
I thank Wulf-Dieter Geyer for informa tion about the Noether family, the buildings, and the plaques; Rainer Schmidt for information about the schools; Anette Koeppel for helpful conversations, information, and en couragement; Constance Reid and Priscilla Bremser for comments on the article; and the Frauenbeauftragten der Universitiit Erlangen-Nfunberg for ac cess to documentation. Other informa tion comes from [1], [2], [3], and [4]. I thank the Alexander-von-Humboldt Stiftung, Herbert Lange, and all the members of the Mathematisches lnsti tut der Universitiit Erlangen-Nfunberg for my very erijoyable and interesting stay in Erlangen. See my webpage for additional documents relating to Noether in Erlangen.
Birkhauser, Boston, 1 981 ). (2] Emmy Noether, Gesammelte Abhandlung en (Collected Papers), edited by Nathan Jacobson,
Springer-Verlag,
Berlin-New
York, 1 983. (3] lise Sponsel, "Spuren in Stein" - 100 Jahre lsraelitischer Friedhot in Er/angen,
30.
September 1891-30. September 1991, Stadt Erlangen, Erlanger Materialien Heft 6, Windsheimer, Erlangen, 1 991 . (4] Stadtmuseum Erlangen, Juden und Juden pogrom 1938 in Erlangen, Veroffentlichung des Stadtmuseums Erlangen, Nr. 40. Mathematics Department Ohio State University Columbus, OH 432 1 0 USA Webpage: http://www.math.ohio state.edu/-silver E-mail:
[email protected]
MOVING? We need your new address so that you do not miss any issues of
THE MATHEMATICAL INTELLIGENCER. Please send your old address (or label) and new address to: Springer-Verlag New York, Inc., J oumal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485
U.S.A. Please give us six weeks notice.
VOLUME 23, NUMBER 3, 2001
49
V(n)
V (n - V(n - 1 )) + V (n - V(n - 4)) KeUie O'Connor Gutman
Recalling a Collaboration with Greg Huber and Doug Hofstadter And now, my friends, in poetry,
A visit to each number's paid,
The lowdown on the function V,
With ne'er the welcome overstayed.
Which calls itself recursively.
Aside from four 1 's at the start-up,
My verse will mirror it, you'll see. The code pertains to how it rhymes In trios, couplets, singletons But that we'll save until the end.
Each number's tapped three times at most, And gets, as said, at least one toast. V's charm lies in its wondrous mix Of ordered chaos, as it clicks
Let's start with all those dense parens
Its way along the number line.
And minus-signs and V's and n's
No pattern's clear in its design,
That make my title hard to sing. V's formula (which yields a string Akin to Fibonacci's sequence)
Yet hidden truths are there to mine. A different way to look at V Through groups of length 1 , 2, or 3 Involves observing repetitions.
Says "Add two prior values found To get the next-thus, round and round," But V demands that one look back
If there's a value that's dunned thrice
Successively (or once, or twice), We say a "clump" is there, size 3
To distant spots along its track. To find those places that one visits,
(Or 1, or 2, respectively).
Step back by one and back by four
The list of clumps shows how V duns
But do not add these, I implore,
The integers, with 3's, 2's, 1 's. Transitions give another viewpoint;
As Fibonacci might; there's more. These merely serve as indices
They show just how the numbers climb
For two more countbacks, if you please, That yield two summands for your summing.
At most two jumps come at one time, And then, plateau. It's quite sublime!
A short example would be great,
Thus, novel views of V's quaint bumps
So here's V's startup -one through eight
We gain by listing climbs and clumps I.e., two complement'ry ways
1, 1, 1 , 1, 2, 3, 4, 5; It's V(9) we'll now derive.
With which upon V's path to gaze.
All set? Let's get those brain cells buzzing!
Replace the n's by 9; subtract To get first
Those few of us who've set our sights
8, then 5. Extract
On understanding its delights.
V's values at those spots exact. Thus V at 5 delivers 2, Whilst V at
Oh, V both baffles and unites
P.S.-For those who'd like to see A longer stretch of sequence V,
8 gives 5. Now you
Within this poem I've encrypted
Must do a wee bit more subtracting. So take these indices from 9 Get 4 and
7. These define
The list of clumps for you to find. I'll help at first, if you don't mind. The first four rhymes-jot down a "4";
The spots in V that we must add.
That digit you will see no more.
Take V(4) -this won't be had
And next we have an unrhymed For those three lines write
Our table tells us it's a 1 ,
The rest is readily decoded:
While V(7)'s 4-well done ! We're nearly finished with our
run,
When three lines rhyme (like these), write "3";
For V(9) (towards which we strive)
Write "2" for couplets; finally,
Is 4 + 1 (their sum)-thus 5.
Write
I'm sure that wasn't too demanding. A question now to contemplate: What makes this function captivate The few who've tarried in its thrall?
"1" for rhymeless lines. Let's see: 1, 2 . . . -
4, 1, 1, 1, 2, 2,
My poem's rhyme scheme.
ow go through
The clumps, translating; I'll assist. 4 ones, 1 two, 1 three . . . -the list That's on line twenty-six! This sequence
Well, first of all, from something small, A sequence starts that never ends.
In fact gives back V 's fun.ky grace,
Surprisingly, as V extends,
A lovely gem in function space.
It hits each number in succession And never ever skips a beat
75 Gardner Street
While marching up its one-way street,
West Roxbury, MA 021 32-4925
With no looks back and no retreat.
USA e-mail:
[email protected]
50
run,
"1, 1, 1".
THE MATHEMATICAL INTELUGENCER
Where Is Itt Benno Artmann
R
eaders of the Intelligencer should have no problem guessing the city (pop. 130,000) in which all these streets can be found. Can any other city do better?
Benno Artmann e-mail:
[email protected] Look for the answer in the Fall 2001 issue.
© 2001 SPRINGER·VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
51
REUBEN HERSH
Mathemati ca Menopause , or, A You ng M an ' s Game?
y mathematical career was nonstandard. I started graduate school at age 29. From
34 to 50, I produced research, much of it well
received. After 50 I could no longer create new mathematics. I popularized and philosophized from then until now, age 72. How you. You are in the happy age of productivity. When every
typical or how strange has my story been? Great authorities warn us, "Mathematics is a young man's game" (see later, under "Hardy vs. Littlewood"). My starting age for research contradicts the rule, but my con
one begins to speak well of you, you are on the downward road."
[ 14]
In a New
Yorker article,
"Mathematics and Creativity, "
Alfred Adler wrote:
cluding age seems to verify it. Albert Einstein said, "A person who has not made his great contribution to science before the age of thirty will never do so."
[6]
Andre
. . . consuming commitment can rarely be continued into middle and old age, and mathematicians, after a time,
Weil
do minor work. In addition, mathematics is continually
wrote, "Mathematical talent usually shows itself at an early
generating new concepts which seem profound to the
The French-Jewish number theoretician
age. There are examples to show that in mathematics an old
older men and must be painstakingly studied and
person can do useful work, even inspired work; but they are
learned. The young mathematicians absorb these con
rare and each case fills us with wonder and admiration."
cepts in their university studies and find them simple.
[ 13] The Bourbaki collective expels members at 50.
What is agonizingly difficult for their teachers appears
At his fiftieth birthday party the great German function
only natural to them. The students begin where the
theorist Felix Klein whispered to his English student Grace
teachers have stopped, the teachers become scholarly
Chisholm Young, said to be his favorite pupil, "Ah, I envy
observers.
52
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
[1]
On the other hand, in his biography of the Israeli American logician Abraham Robinson [5], Joseph Dauben wrote, [Robinson] was always pleased to dispel the myth that the best mathematicians were under thirty and that a mathematician did her or his best work early, at the very start of one's career. As a striking counterexample, Robinson's best mathematics was only beginning to reap the benefits of his wide experience when, suddenly at the age of fifty-five, he died. Abraham De Moivre (1667-1754) found his presumably most important result when he was 66--the "local cen tral limit theorem." . . . De Moivre had to stay competi tive as a problem-solver in order to attract rtoble coffee house frequenters as paying clients for instruction. De Moivre in old age used to sleep every day a bit longer until the sleeping phase reached 24 hours. (Ivo Schneider, e-mail communication) Weierstrass was 70 when he discovered polynomial ap proximation. The English-Jewish algebraist J. J. Sylvester pointed out that Leibniz, Newton, Euler, Lagrange, Laplace, Gauss, Plato, Archimedes, and Pythagoras all were productive un til their seventies or eighties. And of course, we would add Sylvester himself. "The mathematician lives long and lives young," he wrote. "The wings of the soul do not early drop off, nor do its pores become clogged with the earthy par ticles blown from the dusty highways of vulgar life."
"In 1896, in the eighty-second year of his age, Sylvester found a new -enthusiasm and blazed up again over the the ory of compound partitions and Goldbach's cof\iecture." (Bell, p. 405) (We would omit Plato and Pythagoras; and G. H. Hardy tells a different story about Newton.) I decided to do my own check on Hardy. Is mathemati cal aging so inexorable? I mailed out a questionnaire. Since Hardy's apology was in large part a self-evaluation, it seemed fair to base my research on self-evaluations. From the American Mathematical Society membership directory, I chose 250 names. I had known most of them somewhere, at some time. They were mostly Americans, with a sprinkling of American women. There were a few Canadian, Swedish, French, Israeli, and Japanese mathe maticians I knew who had spent time in the United States. Reflecting my training and experience, the mailing list was heayy on differential equators. Theoretical d. e.'s (both o. and p. and s. as well). Applied d.e.'s. Numerical d.e.'s. There were also stochastic processors, and a scattering of logicians, algebraists, topologists, geometers, and statisti cians. I see no reason why mathematical specialty makes much difference for these questions. I got 66 replies, which is said to be a very good response rate. They came from 23 states, plus Ontario, Alberta, British Columbia, Sweden, and Israel. California and New York led with 1 1 and 9 respondents, respectively. Ages ranged from 54 to 92 . 47 were over 60 years old, and 22 were over 70. Some of my old acquaintances were happy to get back in touch. There are many names a reader would recognize. I also include a quotation from Ivan Niven's interview in the CoUege Mathematics Journal.
t
The Questionnaire I
Here are the questions I sent:
I I
:
1. What opinions or information do you have about ag ing and mathematicians? 2. How old are you now? 3. How old were you when you started mathematical research? 4. What have been your main fields of mathematical re search? How would you compare the value and in terest of your research at the beginning of your career and that of your most recent research? How do you think the mathematical community compares them? 5. Did you find at a certain age that you had lost some zest or drive or facility for mathematical research? At what age? What happened? 6. Did you have such experiences more than once? 7. Do you attribute them to aging or other causes? 8. Did you give up your research work? Did you switch to another field of research? Which one? With what success? Did you go from pure to applied? Theoret ical to numerical? Was the new field close to the old one, or much different?
9. Did you then develop a more intense and commit ted interest in teaching mathematics? In writing trea tises and textbooks? 10. Did you collaborate more or less? Did you collabo rate withjuniors, equals, or seniors? (G.-C. Rota: "At my age the work of the collaborator is crucial.") 1 1. Did you develop a new serious commitment to non mathematical activities? Which ones? Have those ac tivities been able to replace mathematical research for you? 12. In your mature years have you tended to return to the subjects and problems of your youth? (Hille) 13. Did you feel a strong sense of loss in giving up your earlier research goals? 14. Did these experiences affect your standing in your department? How? By smaller pay raises? By less in fluence in decision-making in your department? In the math profession? 1 5. Have you suggestions for individuals or institutions to prolong the period of active research? 1 6. Other questions, comments, suggestions?
VOLUME 23, NUMBER 3, 2001
53
Hardy vs. Littlewood G. H. Hardy famously opined, "Mathematics is a young man's game." In A Mathematician 's Apology, he explains: "I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job . . . If then I find myself writing not mathematics but 'about' mathe matics, it is a confession of weakness for which I might rightly be scorned or pitied by younger and more vigor ous mathematicians . . . . I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and aban dons mathematics, the loss is not likely to be very seri ous either for mathematics or for hin1self. . . . Mathematics is not a contemplative but a creative sub ject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal any how, and it would be silly to bother about him . . . . A mathematician may still be competent enough at sixty, but it is useless to expect him to have origin.al ideas." Hardy had two great collaborators, Ramanujan and Littlewood. Ramanujan died at thirty-three. What about Littlewood? In 1941, when Hardy wrote his Apology, Littlewood was already 56. From the introduction to Littlewood's Miscellany, by Bela Bollobas: "In 1950, at the statutory age of 65, Littlewood retired and became an Emeritus Professor. The Faculty Board realized that it would be madness to lose the services of the most eminent math ematician in England, so they wrote to the General Board: 'Professor Littlewood is not only exceptionally eminent, but is still at the height of his powers. The loss of his
One recipient thought my questionnaire was biased to ward pessimism-reflecting my own depressed personal ity. Many respondents praised my project. They called it "refreshing," "provocative," "most worthy. '' One wrote, "I wish, as probably many people do, that there was some thing with a little authority written, and I am very pleased that someone of your stature is undertaking it." Responses trickled in for six months. It took another six months to absorb them and to see a way to present them. There is no claim that my choice of 250 was "typical," let alone "random." And the 66 of 250 who answered are cer tainly not typical. They are biased toward people who answer questionnaires, who like to hear from an old acquaintance, who are willing to consider some possibly painful issues, and who aren't too unlmppy or ashamed of their lot in mathe matical life. The people who don't respond to questionnaires are like the dark matter of the cosmos; we know they are out there, but we can only guess what they look like.
54
THE MATHEMATICAL INTELLIGENCER
teaching would be ineparable, and it is avoidable. Permission is requested to pay a fee of the order of 100 pounds for each tenn's course of lectures.' The response: 15 pounds per term, the fee paid to an apprentice giving his first course as a try out, to a class of 2 or 3. So Littlewood gave courses at 15 pounds for 4 years. He tried to stop once but there was a cry of distress. At the same time he turned down lucrative offers from the United States . . . Littlewood remained active in mathematics even at an advanced age: his last paper was published in 1972, when he was 87. One of his most intricate papers, conceming Van der Pol's equation and its generalizations, was written when he was over seventy: l l O pages of hard analysis, based on his joint work with Mary Cartwright. He called the paper 'The Monster' and he himself said of it: 'It is very heavy going and I should never have read it had I not written it myself.' His last hard paper, breaking new ground, was published in the first issue of the Advances in Applied Probability, when he was 84 . . . . In 1972 Littlewood had two bad falls and he fell again in January 1975. He was taken to the Evelyn ursing Home in Cambridge, but he had very little interest in life. In my desperation I suggested the problem of determining the best constant in Burkholder's weak L:! inequality (an ex tension of an inequality Littlewood had worked on). To my immense relief (and an1azement!) Littlewood became interested in the problem. He had never heard of martin gales but he was keen to learn about them. . . . All this at the age of 89 and in bad health! It seemed that mathe matics did help to revive his spirits and he could leave the nursing home a few weeks later. From then on, Littlewood kept up his interest in the weak inequality and worked hard to find suitable constructions to comple ment an improved upper bound."
Most responses didn't deal directly with Hardy's claim that if you're ever going to do anything important you must do it when you are young. A differential geometer from California pointed out that this isn't the same question as whether you are still active in old age. (Or, as people say nowadays, when you're "older.") Two of my respondents knew of earlier surveys by fa mous mathematicians. One respondent said George Mackey did a study of 50 leading mathematicians, and concluded that on average their best work was done in their late 30s. Another respondent said Gail Young did a study of people who matured very young in mathematics. He found that they generally bum out early. Young felt there was a fairly constant period during which a person could do very cre ative work Some had their period earlier, others had it later. The questionnaire invites recipients to tell as much as they like about their cunent and past situations. The an-
swers yield a glimpse at how this sample of mathematicians view their lives in mathematics. Such responses don't sub mit readily to tabulation, much less to statistical analysis. The interest is not only in the consensus but also in the many individual points of view. I wish I could quote all of them. Most of my respondents are satisfied with their life sit uation! Relevant is [12], a report by S. S. Taylor on retirees (not restricted as to field) at the universities of New Mexico and Rhode Island in the U.S. and Bath and Sussex in England. Reassuringly, perhaps surprisingly, 98% of the UNM retirees, 97% of the Rhode Island retirees, and 94% of the English retirees told Taylor they are "reasonably satis fied" or even "very satisfied" with retirement. Two-thirds of the American respondents told her they receive the same or higher income as before retirement. Most of my respondents say they continue research af ter retirement. Some think their recent research is their best ever. Some say they're doing what they're interested in, uninhibited by the judgment of the math community. I organized the answers into 7 groups. Excerpts from some answers appear in more than one group. 1. No general statement can be made about mathematical aging. 2. Mathematicians are best in youth. 3. Mathematicians may be as good or better in later years. 4. Symptoms and strategies. 5. Penalties for aging and for fol lowing one's own bent. 6. Advice for aging mathematicians. 7. Advice for mathematics departments.
a superficial view these are indistinguishable from people who lose zest at age 35, but the reason is different; when they achieve tenure their research declines, just because the pressure for it is gone." Matrix-theorist, Ontario, age 73 Group 2: Mathematicians are best in youth.
Here we hear of some sad, even tragic experiences. "One of my old, dear friends suddenly went dry in re search at age 40. It was very traumatic for him, and for me to observe." Analyst-applied mathematician, Texas, age 54 "My zest is fine, but capacity much diminished before age 55. Age and alcohol and depression." Analyst, California, age 72 "One does best between age 20 and 50. My most recent research (c. 1996) is not as good as my work in the 1950s." Differential geometer, California, age 67 "At around 55 I had lost whatever originality I once pos sessed. But not the desire to learn and try." Analyst, Maryland, age 79 "I used to work late at night, but now I'm too exhausted to do more than make calendar entries and clean up my study." Analyst, Louisiana, age almost 62 "Clearly at my age I can't keep up with the best younger people. Some old-timers have looked fool ish in their later research efforts. My hope is at least to avoid that." Applied mathematician, Rhode Island, age 71 "The vast majority of mathematicians do best before 40, and often as not, before 30. But that would be hard to sub stantiate; one would need to know the life work of a math ematician, and make reliable judgments about it. . . . The best counterexample I know is Legendre, who proved the case n = 5 of Fermat's last theorem when he was in his sev enties." Differential geometer, California, age 73 "Men age faster than we girls. It makes up for them be ing bullies earlier. How to pep them up? I try . . . People whose lives are fairly stable and satisfactory keep going a lot better. One of my colleagues gave up research at 42 when his marriage broke up. Another similar at 48." Probabilist, British Columbia, age 62 As you get older you know too much. You have all these methods, and you try all the combinations and variations you can think of. You're running down the old tracks and nothing works." (Ivan Niven, [2]) "Mathematics tends to be introverted, with unbalanced expenditure of mental energy. As one grows older there is desire for other forms of expression, which dilutes the in tensity to solve problems. 'What does it all mean?' is asked more often, which also can slow down progress." 1'Aging has two sides-your own age, and the age and aging of your subject and your contributions. This aging is brought about by the work of younger competitors." Analyst, Sweden, age 80
"As you get older
you know too m uch . "
Many responses are hard to classify. For example, "Kato, at about age 75, just published a very good paper, though he does complain bitterly about not being able to do good research any longer." Is that group 2, group 3, or both? Some painful experiences in group 5 contradict some advice in group 6. Many respondents say, follow your own bent, regardless of outside pressure; and many respondents report penalties for doing that. Some respondents don't give their age; for a few, I was unable to identify geographical location. Group 1 : No general statement can be made about mathematical aging. "All
generalizations are false, especially this one."
"Better not generalize-we are all different." Probabilist, British Columbia, age 62 "I met P6lya when he was past 70, and I thought he would go on forever, while I have met a number of promis ing mathematicians who faded before 30." Analyst, Maryland, age 79 "I don't see any general patterns. Some never ef\ioyed research, some ef\ioy it but don't want to do a lot of it. On
"
VOLUME 23, NUMBER 3, 2001
55
"The field of mathematics moves very fast. The pace has been quite extraordinary in the past 50 years. Just trying to keep up in one's specialty requires many hours of effort. One doesn't feel comfortable doing the same old thing. Some great mathematicians have been unable to handle this. When a decent problem comes along which seems ac cessible, I'm eager to jump in. The trouble is that the fron tier is moving so fast. It's not that we give up research math ematics, research mathematics gets away from us." Geometer, California Group 3: Mathematicians may be as good or better in their later years.
"A Young Woman's Game?", below, provides impressive testimony that women mathematicians are often at their best in their 30s, 40s, even 50s. "Mordell is supposed to have said modestly, 'I did work in my 70s many a younger man would have been proud to have done.' Among my teachers, I know that Beurling, Ahlfors, Zariski, Mackey worked intensely on research when they were quite old." Analyst, Rlwde Island, age 72 "Since I became emeritus in July 1995 my research has increased. Most of it is joint with former students and post docs. The mathematical tools are ones I've used before this is probably typical. It is a great relief to shed 9 years of department chairmanship, too many committees, and obligations to seek external grants. I no longer attend de partment meetings." Analyst-applied mathematician, Rlwde Island, age 71 "Some of my best work was done after age 4 7. Possible motivations were a bad spell of drinking and divorce from 1974 to 1977, and prostate cancer treated successfully by radiation and implants. After such trauma I tend to over accomplish." Analyst, /Uinois, age 69
"Knowledge and experience count for a lot more than CPU speed. At the minimum it improves your mathemati cal taste. My recent papers are a lot better than those just after the Ph.D." Analyst, Alabama, age 61 "Young guys may luck out but often only when some one older points the way." Applied mathematician, Colorado, age 64 "The young may find gold but cannot read the land; the older have familiarity with the landscape, which guides them to where to dig." "Recently a friend compared me with Brahms, who turned out great works throughout his life! I hope to live up to the praise. " Numerical analyst, Ohio, age 70 Group 4: Symptoms and strategies.
By this heading I mean symptoms of aging, and strate gies to cope with it. "My wife and I have been happily married for forty-four years; that's extremely important. Our garden takes a ma jor part of our time in the growing season." Applied math ematician, Rlwde Island, age 71 "My memory is not what it used to be. My work takes much longer and the need for careful notes is greater. My best work was around age 40." Analyst, Sweden, age 68 "My drive for research and my direction haven't changed dramatically. I just don't think as clearly and quickly now. But I've grown efficient in other respects, and as a result my best work has been in the last 15 years. I collaborate more, but still do a lot by myself." Applied mathematician, Utah "As I age my memory declines, making it more difficult to keep in mind all the threads of a complicated situation. Also my computational abilities decline-! take longer to get through a routine calculation, and make more mistakes.
A Young Woman's Game? Does the slogan "mathematics is a young man's game" exclude women? Or should it be "a young person's game"? Claudia Henrion [8) says no. " . . . there is a deeply entrenched belief that mathematics is a 'young man's game,' despite the fact that there is no compelling evidence to support this hypothesis; indeed, the studies that have been done suggest the contrary. But when the image and reality differ, it is often the image that can have a more powerful influence on attitudes, practices, and policies. If the focus were not so much on the young, virile mathematician, it would be easier to design pro grams with women in mind. For example, [recognizing) the fact that women are likely to have children in what is traditionally considered their prime mathematical years . . . looking at their productivity over a longer time span . . . recognizing that women may need to enter the
56
THE MATHEMATICAL INTELLIGENCER
mathematical research pipeline later in life, as Joan Birman did, or they may need to work part time for a period to balance having children with mathematical re search, as Mary Ellen Rudin did." The prominent logician Marian Pour-El told Henrion, "I've never felt that you're over the hill if you're in your late thirties. I think I've done my best work later on, by a long shot." The leading braid-theorist Joan Birman focused bet ter on math after the issues of marriage were settled, her children were older, etc. "I think doing mathematics when you're enthusiastic is important-not your age." Rudin, a famous topologist, said, "I don't think most people's best work will be done by the time they're thirty, and certainly my best work wasn't done until I was fifty-five years old."
I catch mistakes by my sense of what seems right, rather
not imagine not doing research. Then I became interested
than by repeating computations. On the other hand, I'm
in
more canny in developing effective research strategy, and
(RUME). I tried to work in both fields, but my interest and
research
in
undergraduate
mathematics
education
. . I have an intellectual
ideas for work in functional analysis disappeared. There
home with a small but active worldwide community of
are two possible factors. Interest in RUME may have
more daring in carrying it out. . scholars with similar interests."
Age 70
�
Applied mathematician,
driven out interest in functional analysis. The other was a feeling that I might do other work as good as my best, but
"I toy with retiring at the end of this year. I am nervous about it, but clearly recognize the diminishment of my abil
I could not do much better."
Math education researcher,
South Carolina, age 64
ity to do first-rate research. The main cause is inability to
"My success and pleasure at research is tied to my abil
stick with messy detailed manipulations. In the past I could
ity to travel worldwide and make connections with people
calculate for hours, but now I shy away from such grunge.
from diverse cultures. Politics has been extremely impor
I still have plenty of things to work on, but I pick them
tant for me to keep my balance, and this is much easier in
more carefully."
Canada than the U.S. I have strong human rights interests
Numerical analyst, California, age 74
"Getting old is a pain. I still do decent mathematics.
related to indigenous peoples in North America and the
However, what I do is very much related to my previous work. I do not jump into a new field, because the same intuition as earlier to 'know' it
I have not will lead to some
Some N u mbers
thing. Everything takes much longer to complete and I make more mistakes, or better, I do not know immediately when the result is wrong. So I have to check much more carefully. I have been a good thesis advisor, which I en joyed very much. Former students still speak to me, and
I
still work with them. But I have no students any more, be cause I cann ot be sure I will be around in
4 years. Also,
young people should work with young people on 'modem' problems. There can be one advantage with old age. If one is lucky and in balance with oneself, one can look at the world as an independent observer."
Numerical analyst,
Sweden/California "The principal obstacles to continuing research are: (a) Research requires energy, and this is in increasingly short supply. (b) Research requires keeping up with the litera ture, and this becomes difficult as one's mental and phys ical energy declines. (c) Good research requires breadth and flexibility, but the tendency as one ages is to concen trate on a narrow path, dominated by what one has always done, and knows well. "Collaboration is essential in maintaining research ac tivity. I have tended to collaborate with juniors, since very
Claudia Henrion ports an article by
("A Young Woman's Game?") re ancy Stem [ 1 1 ] , possibly the only
article on aging mathematicians in the research liter ature (as opposed to the anecdotal literature). Thanks to Judith Grabiner for this reference. Stem's mentor Stephen Cole [4] studied chemists,
geologists, physicists, psychologists, and sociolo gists, and found, "There are basically no differences
in the quality of work published by scientists between the ages of 30 and 50. Scientists over the age of 50
are slightly less likely to publish high quality re search ."
Stem extended Cole's work to mathematicians. She counted publications of 435 "randomly chosen"
mathematicians at Ph.D.-granting institutions, and sorted them by age. Since citations of a paper should
roughly measure mathematicians' interest in it, she also counted their citations. The numbers in paren
theses in the table that follows are the number of samn ,...
T"Y'"'�+-l-o"--'� ...; ,...: ... ...... .... '-.;
ro.
eh'-"U'O!lu.Jt:J. U1 ::>d.lll-
pled mathematicians in each age group.
many of my collaborators have been my students. The younger partner provides energy and awareness of what is
Mean number of citations
currently a 'hot topic'; the older provides perhaps greater
Mean number
of single-authored and
familiarity with the history of the topic and a larger battery
Age
of publications
first-authored work
of available methods."
< 35
5.12
2.73 (101)
"There are many useful things someone with mathe
35-39
7 .33
3.80 (96)
matical ability can do. But education and research rewards
40-44
6.24
5.79 (67)
do not encourage people to branch out and explore. They
45-49
3.49
3.44 (63)
get stuck in the frontiers of their narrow specialty. The go
50-59
5.22
5.63 (73)
ing gets rough when they no longer have the ability or will
60+
6. 1 1
5.09 (35)
5.64
4.22 (435)
Topologist, New York, age 76
ingness for the concentrated effort to do really complicated
TOTAL
technical work. I am still able to do this if I get away for a couple of weeks, but at home commitments to family and work preclude that concentration. It does get harder as you get older, from aging but also from accumulation of other responsibilities and interests."
Logician, Indiana, age 57 "My enthusiasm for research increased rapidly from 25 to 35 and stayed high for 15 years. During that time I could
Stem concludes: "The claim that younger mathe maticians (whether for physiological or sociological reasons) are more apt to create important work is un substantiated .
. . . I have found no clear relationship between age and achievement in mathematics ."
VOLUME 23, NUMBER 3, 2001
57
Third World. The extent to which these activities replaced mathematics interests, or just lived actively beside them, is extremely difficult for me to say." Numerical analyst, British Columbia, age 54 Group 5: Penalties for aging or for following one's path
There are really two topics here. Penalties of either type present an opportunity for some soul searching, by mathematics departments and organizations. "As I get closer to retirement I am consulted less about departmental matters." Numerical analyst, California, age 74 "For sure I have less influence in the department, and get (and deserve) tiny raises." Analyst, Louisiana, age almost 62 "Smaller pay raises. Less influence. No NSF grants. The mafias protect themselves. I'm treated like a half-breed in the old West." Analyst-applied mathematician, Colorado, age 64 "My recent work is more interesting and valuable. Math community isn't interested. Ecology community is." Applied mathematician, Brit ish Columbia, age 65 "I attached equal value and interest to all my research. The mathematical community at tached little of either to any of it." Analyst-historian, Washing ton State, age 73 "I have been treated well. I still have my office, 10 years after retirement." Measure theorist, California, age 80 "My department has treated me well. I still have an of fice and they pay me a small amount for looking after some graduate students. My research is worth more to me than to the department, so there is no strong reason why they should actively support it." Numerical analyst, Sweden, age 66 "By following my bliss, I gave up my opportunity to get to full professor. My most valuable professional achieve ments are not appreciated by the leadership in my depart ment." Age 62 "It was ignominious that the Fields committee turned down Wiles's achievement on account of his age. There is no formal constraint on age for the medal." "I use knowledge and experience from early areas in learning new things. Shifting fields this way causes one to lose NSF support. This can be regained after a few years, but I got sick of being funded in new 'lives' and stopped applying. The judgment of proposals is silly; many people simply propose what they have already done and try to guess what directions are politically correct. . . . I found many colleagues singularly narrow in focus and rigid in their approach to scientific discovery. It was often difficult to get anyone to look at my papers." Analyst, IUinois, age 69
"At 38 I had built enough research reputation to do some writing without serious loss of status in the research com munity. . . . At times I believe the only criterion department chairmen go by is bringing in contracts and grants. At times I think my colleagues regard me as an eccentric anomaly pursuing non-standard paths." Analyst-author, Rhode Island, age 77 "The mathematical community lost interest in my work when fashions changed and I didn't. After a period as chairman when I was 40, I lost influence in the department." Analyst, California, age 72 "My best paper was never referred to in the later litera ture. I tell myself this shows it said the last word on its sub ject." Number-theorist, Minnesota, age 62 "My chief pain has been realizing that my creativity was very limited-I could not conceivably become a significant mathematician. My seniors had higher expectations for me than were fmally justified. I have always had an intense pas sion for music, and this comforted me." Analyst, Maryland, age 79 "I did feel a loss when what I was doing was not valued by the mathematical community. It took a while for me to value it for its own sake." Logician, Indiana, age 57 "The value of my research = quite high, the interest by others = quite low. The math community doesn't pay atten tion to most mathematician's work . . . I am called on a lot to do diplomatic or administrative " jobs. I am not a very able administrator, but compared to the great majority of mathe maticians, I am an administrative genius." Applied mathe matician, Alberta, age 60 "Some of my best research has been in recent years, yet I have been getting smaller pay raises and have less influ ence in my department. The situation of some of my con temporaries is even more egregious. Mathematics depart ments and organizations don't pay attention to the older members of the profession. My department treats our re tirees shabbily: we give them a 'gold watch' when they re tire, then forget about them."
" My research is worth
more to me than to the
d epartment ,
58
THE MATHEMATICAL INTELLIGENCER
Group 6. Advice to aging mathematicians.
Recommended by many: A. Stay ill good mental and physical health. Exercise. Don't smoke. B. Do what interests you, not what others expect. C. Look for new ideas and areas that appeal to you. D. Stay away from administration. E. Don't stop-keep working. F. Collaborate with your juniors. G. Have fun. Here are some quotes: "First and foremost, you need a deep love of the sub ject." Analyst, Alberta, age 60
"It is important Sweden, age 66
to have an office."
Numerical analyst,
"Stay away from administration. It eats away your cre
ativity, and is a real plague." Applied mathematician,
Utah.
"Keep working, do not hide behind administrative du ties."
"Move administrators' pay
down.
Analyst, Sweden, age
68
Limit chairs' and
1 or 2 years, as in Japan and Europe." Applied mathematician, Colorado, age deans' terms to
64
"I have believed for a long time that a lifetime appoint ment to research in mathematics, with incidental teaching, is a mistake. A person's abilities, skills, and interests change.
Quote from I. Singer: "Keep the pencil moving . "
I've often talked about a career path involving research at
"Work hard, an d have several problems to work on."
an early age, say
Logician, age
67
"Don't stop. Once you do, it's hard to get back It's not just the field that changes but you change."
York, age
25-35, research and research supervision 35-45, writing and teach
at a research university next, say
65
Geometer, New
ing at a nonresearch university thereafter, perhaps with in volvement in preuniversity mathematics such as teacher training or high school teaching. . . . Shorten the time to get
"Maintain contact with younger colleagues and students.
a Ph.D. so that people can start research earlier, as in
. . . Whenever anyone asks you a mathematical question,
England. Shorten the undergraduate school time for talented
devote at least
people."
15 minutes to it, even if it is 'not my field.'
. . . Try to maintain high ethical standards in this competi
Applied mathematician, California, age
74
"Why not let good searchers search without specifying
tive profession. I like to think of mathematics as a collec
what they will discover? The AMS establishment spend half
tive enterprise. We contribute even by being attentive spec
their time giving awards to each other, and the selection of
tators and consumers of the constant outpouring of new
referees is capricious or worse. I have been a referee many
ideas. (In the opposite view, a career in mathematics is an
times and I know that if you expect a proposal to fly you
ego trip like downhill skiing, reserved for the youngest and
have to say at least 'exceptional' everywhere-'very good'
strongest, where only those who break records matter.)"
is not enough, and even 'exceptional' may not suffice. This
Harmonic analyst, Sweden, age 71
rating system is sublime in its "
"Publish less. Communicate
more, to better chosen read ers. Alberto Calderon told me that
important
results
are
found by graduate students be cause they consider problems afresh. His advice for middle aged
mathematicians
silliness. . . . Students should
a big issue is why
be screened carefully before
retirement is synonymous
their development is under
with severance from al l
be understood that they will
academic activities . "
lem,
taken-alternatively, it should sink or swim; it's their prob not
the
teacher's."
Analyst, Illinois, age
69
"Encourage individuals to
was,
take up a new problem area; he cited Menahem Schiffer as
take chances and follow their true interests. "
his own prescription, and he continued to fmd important results anyway." Matrix theorist, Ontario, age
mathematician, Montana, age
a model. But he did not follow
73
"Hardy advised people to do research in a prone posi tion, so that more blood flows to the head."
Illinois, age 59
Probabilist,
70f
Applied
"I don't think anyone should tell us when to give up."
Analyst, California, age
72
"Create mechanisms to keep researchers affiliated with strong communities.'' Analyst,
New York, age 54
"Do not overburden senior faculty with administrative
"My early interest in 'useless things' like algebraic geom
responsibilities. Encourage them to have research students
etry paid off handsomely later in soliton theory and strings."
and to direct Ph.D. dissertations. There is also the com
Analyst, Illinois, age
69
plementary problem-senior faculty must recognize when
"Always remember, research should be fun. If it becomes
they are no longer effective in research. But they can still
too competitive and loses its pleasure, give it up. Don't take
provide
background
and
experience,
and
familiarize
your research or yourself too seriously! I have been blessed
younger colleagues and graduate students with methods
with a good sense of humour, but how could one suggest this
which have been tried, and difficulties which have been
to others?" Numerical
met."
analyst, British Columbia, age 54
"I do what I can do, and enjoy every minute of it. The
Topologist, New York, age
76
". . . a good library, some stimulating colleagues and free
mathematical community has as little awareness of me as
dom from too many onerous chores. Stan Ulam left USC
I of them. Constantly learn new things! Do mathematics
because teaching calculus to morons was killing him."
just for the fun of it!"
Analyst, Alberta, age
Number-theorist, New York, age
77
"Stop growing older. Keep having fun. And have a beer, on me!"
Analyst, California, age
83+
Group 7: Advice for mathematics departments.
60
"Reward equally. An expert teacher and motivator who designs a great calculus lab is just as valuable as a top notch researcher.''
Combinatorialist, Colorado, age
56
"For me, a big issue is why retirement is synonymous with severance from all academic activities. The University is the
I would like to hope that the following suggestions will be
one organism that consciously believes there is nothing to
seriously pondered in departmental executive committees.
learn from the past."
Number-theorist, New York, age
77
VOLUME 23. NUMBER 3, 2001
59
My own comments:
A U T H O R
After retirement I asked for a floppy disk, and the person who was then my department administrator told me (even though I was still a part-time instructor), "You're putting me in an embarrassing position. You're not in the budget. You're emeritus now. Why don't you just
run
over to the
book store?" Shabby treatment of aging professors is not special to mathematics. After retiring from the Columbia economics department William Vickrey got a Nobel Prize for work on transportation economics. A New
York Times reporter him in a tiny office far from his department. Vickrey was grateful that after retirement Columbia allowed him any office at all. Perhaps after being written up in the Times
1 000 Camino Rancheros
he would have been granted a better office, but, sadly
Santa Fe, NM 87501
found
REUBEN HERSH
and unexpectedly, he died a few days later. (See also Littlewood, in Box
USA e-mail:
[email protected]
1.)
Sometimes emeriti are even dropped from the depart ment e-mail announcements about seminars, hiring, pro
Reuben Hersh, one of the over-age mathematicians dealt with
motions, retirements, and anything else interesting that's
in this article, is retired from the University of New Mexico. This
going on.
gives him more time to contribute to The Mathematical
Yet departments can always use extra hands. Under
lntel/igencer.
graduate advising is often understaffed. Has anybody asked, "Are there emeriti who enjoy advising?" If there's no librarian on duty in the math library, is there an emeritus who would serve?
REFERENCES
1 . A. Adler, "Mathematics and Creativity, " The New Yorker Magazine
There's always too much committee work Is there an emeritus with years of service on the undergraduate com mittee or the master's exam committee? Might he/she have
(February 1 9 , 1 972). 2. D. Albers and G. Alexanderson, "A conversation with Ivan Niven," College Mathematics Joumal 22(5)(1 991 ), 371 -402.
something to contribute there?
3. E. T. Bell, Men of Mathematics, New York: Simon and Schuster,
Summary
4. S. Cole, Age and Scientific Performance, Stony Brook (1 976).
The responses were very varied. But these five statements
5. J. Dauben, Abraham Robinson: The Creation of Nonstandard
(1 937).
would be generally accepted:
Analysis: A Personal and Mathematical Odyssey, Princeton, NJ:
1. There's tremendous variation in how mathematicians age. No one pattern describes everybody.
2. Many mathematicians have been productive in advanced age.
3. To most (not all!) mathematicians, aging brings losses in memory and computing ability. These may be com pensated by broader perspective and mature judg ment. Possibly more serious is slowness or difficulty in learning new material. Some responses were more specific.
4. Live healthy and follow your own bent, not the pressures of others.
5. Older and retired mathematicians are an under-utilized resource for the mathematics community.
Princeton University Press (1 995), 491 . 6. A. Einstein, quoted by Stern, ref. S. Brodetsky, Nature 1 50(1942), 699, as quoted in C. W. Adams "The Age at Which Scientists Do Their Best Work," Isis 36(1 946), 1 66-1 69. 7. G. H. Hardy, A Mathematician's Apology, New York: Cambridge University Press (1 940). 8. C. Henrion, Women in Mathematics: The Addition of Difference, Bloomington: Indiana University Press (1 997). 9. H. C. Lehman, Age and Achievement, Princeton, NJ: Princeton University Press {1 953). 1 0. J. Uttlewood, Littlewood's Miscellany, Preface by Bela Bollobas, New York: Cambridge University Press (1 986). 1 1 . N. Stern, "Age and Achievement in Mathematics: A Case-Study in the Sociology of Science." In: Socia/ Studies of Science, Vol. 8, London and Beverly Hills: Sage (1978), 1 27-1 40
Until we find a consensus about which advances are
1 2 . S. S. Taylor, of the University of Rhode Island's Labor Relations
"major," we can't refute Hardy's claim that no major ad
Center. In: Research Dialogues of the TIAA-CREF, Number 62
vance has been made by a mathematician over
50. But his
slogan, "Mathematics is a young man's game," is mislead ing, even harmful. So far as it may discourage people from mathematics when they're no longer young, it's unjustified and destructive.
60
THE MATHEMATICAL INTELLIGENCER
(December 1 999). 1 3. A. Wei/, 'The Future of Mathematics," American Mathematical Monthly 57 (May 1 950), 296. 1 4 . S. Wiegand, "Grace Chisholm Young," Association for Women in Mathematics Newsletter 7 (May-June 1 977), 6.
1)¥1(9-\.[.1
J e remy G ray, Editor
Richard Courant in the German Revolution Colin Mclarty
I
I
n May 1933 Richard Courant wrote
I cannot regret the turn things have
two official accounts of his time in
taken in the world. It would have
the German Revolution of 1918--1919.
been much easier and less painful
He told his Provost: "The allegation
for us, if lazy, apathetic intellectuals
that during the debacle I was a revolu
and frivolous "leaders" had not let
tionary in the Soldier's Council, is per
us stagger to the brink of the abyss.
fectly laughable." He had joined the
But I look hopefully to the future
Workers' and Soldiers' Council to keep
and think, with the end of the war
the peace. He told Hermann Weyl, di
before us, each party might say: the
rector of the Mathematics Institute,
victims were not sacrificed in vain.
that his visits to Berlin convinced him
Perhaps we still face a not quite
"the Social Democratic Party under
painless inner purification to the
men like Noske was the last remaining
highest levels. In any case, for the
bulwark against Bolshevism, complete
first time we have a truly free road
collapse of the
Wehrmacht, and the dis
solution of Germany." { 1 ]
for the things that make human life worthwhile. That this road be taken,
H e wrote these words as the Nazis
and the hope now sprouting on all
took control. The evidence shows, as
sides not dissolve, are tasks for us
common sense might suggest, that they
all who in the past only criticized.
were basically true but slanted to pro
We must stand in the first ranks if
vide him some protection. His mention
Germany is to rebuild. I am very
of Gustav Noske is the most contro
happy now, when I can take off "the
versial point. Though a Social Demo
King's coat," that I will not return to
crat, Noske in 1919 used the extreme
work in the Germany of Morsbach
rightist private armies called Freikorps
and Schroder [language professors
to crush worker demonstrations, killing
at Gottingen, both nationalists], but
thousands of people.
of Hilbert and Einstein. (Courant to
As the Great War wound down Lieu
Hilbert 12.10. 18)
tenant Richard Courant was stationed in Ilsenburg, 75 km from Gottingen,
The Kaiser and Crown Prince Max
training officers to use the earth tele
abdicated 9 November 1918 under pres
graph he invented to signal between
sure from the Entente to form a demo
trenches. By spring 1917 he felt it was
cratic government. Social Democrats
"almost too late." [4] General Ludendorff
Philipp
launched a desperate offensive in March
Ebert, already in the Kaiser's govern
and
Friedrich
1918, broken by July. The Army High
ment, declared a "German Republic."
Command sought, and got, an Armistice
They soon called a 19 January election
proposal by October, always conceal
for a National Assembly to frame a con
ing from the public both the military
stitution. Throughout November, Work
situation and the role of the military in
ers' and Soldiers' Councils arose to con
the Armistice.
trol many towns around Germany. The
Through all this Courant worked on
name imitated the Russian Soviets. Most
mathematics and planned a book se
took power peacefully, and their politics
ries with Springer. He wrote to David
ranged from Bolshevism to simply want
Hilbert on 12 October 1918 about the
ing someone in charge where no one
series and a book he would write for it
seemed to be. Social Democrats often
with Hilbert on differential and integral
led them, and Social Democrat Courant
equations. The series became Springer's
became Council President in Ilsenburg.
Methods
Everyone agreed there was a Revolu
or simply
tion. But would it end with a National
Grundlehren and the book
Column Editor's address:
Scheidemann
of Mathematical Physics,
Faculty of Mathematics, The Open University,
"Hilbert and Courant." He closed the
Assembly? Or would the Councils take
Milton Keynes, MK7 6AA, England
letter with a political vision:
it farther?
© 2001 SPRINGER-VERlAG NEW YORK, VOLUME 23. NUMBER 3, 2001
61
�irklieh e i he
treie S.�n . Deea
BoffnJmt"en , d ie .1e t.zt fi�ernll
d ieee BAhn �aehr i t te n w J rd tmd deea d ie
e}lf"�e illten , n i�h t
Auf�n ,.e von une fi l len , � ie b i eber nWl
nur
zerinnen , d �e.s iet d ie
ltri t i e iert. h.-han und nnubU
in e n te r Rtt ihe etehen m!i.uen , wenn ee do!l le111'f'•, lJtm te ehlnnde gilt .
leh f"reue :zti�h f!�l'on eut denAUf"'nbliok , r.o i ch dee • KUntgl! Rock• Ptwz iehen knnn ,
in dfi'e Deutaehl&nd von .L:orellPeb und Schrlht e r zutitckzull:ehre n ,
um n i cl'lt
eondern in dem von Hilbert tud Ein�te in �:u mUzll1' rt. i ten . line tH 1 le n •1ele ae h r r.er� licbe Gw-iieae ameh en lhre :F"r1tn
von Ihrem
Figure 1. The last page of the letter to Hilbert, 12 October 1918. (Courtesy of Niedersachsische Staats- und Universitatsbibliothek Gottingen.)
Courant wrote to Hilbert on
23
November:
things are found in
see no enthusiasm for the new or
the organ of the Spartacus Group,
The Red Flag,
der among most soldiers. They take
perhaps the most widely read paper
This evening, as I have nearly fin
it as an unalterable fact, just like the
in Berlin. For example, just to un
ished the business of dissolving my
old command relations. The only
dercut the Social Democratic Party,
squad, I can complete my project of
thing the great mass of soldiers
they demand a six-hour work day
reporting to you all I saw and heard
cares about is to get home as soon
right now, since the workers need
in Berlin. I went there last Monday
as possible and be spared any kind
more free time for politics. The
because, in the general confusion,
of war and unrest. Thus the soldiers'
Spartacists demand instant renunci
Berlin headquarters forgot we exist
natural moderating
on
ation of the war loans, immediate
here in Ilsenburg. Something had to
politics grows less and less. The
socialization of all heavy industry,
be done to get my men and me our
more radical active minority, which
and so on. No syllable is said on the
discharge orders. Sadly, I cannot
stays for strictly political business,
details of this economic transfor
say the
have
can easily gain control at least for a
mation. Yet every detailed explana
much helped or clarified the situa
time. Last Tuesday I went to a large
tion in the Social Democratic press,
tion. Even in little Ilsenburg, with
meeting called by the New Father
which obviously must reach moder
me as President of the Workers' and
land League, including Einstein. The
ate conclusions, is dismissed with
Soldiers' Council, it is hard to make
topic was: Proletarians and Intellec
derision and abuse for lacking con
anything workable from the com
tuals Unite! Eduard Bernstein was
viction. Spartacists, Independents, and Majority Socialists are enemy
Soldiers'
Councils
influence
peting forces of dilettantism, hate,
the main speaker. He gave a very
mistrust, good will, conceit, and ideal
temperate, reasoned speech. For him
camps, but the last two are really di
ism, and in general to prevent ma
the "achievements of the Revolu
vided mostly by tactics, tempera
disruption.
tion" were merely to demolish the
ment, personal conflicts, and their
Berlin has so many snags and fric
old system. He wants the rule of law
past. Anyway, the atmosphere in
tion points, it is truly miraculous
returned as soon as possible. That
Berlin is very tense. An explosion in
how everything keeps going. No one
means a National Assembly very
these conditions could be stronger
knows who is in charge of our
soon, and until then no decisive so
and
Inspection, the Soldiers' Council in
cialist measures.
9 November. Something decisive
consultation with the commanding
featured reasoned
officer, or the commanding officer
side bloody dilettantism.
in consultation with the Soldiers'
very suitable representative of the
Germany into greater and lesser
Council. No one dares take respon
"Revolutionary Students' Council"
centers, paralysis of commerce and
sibility for giving orders. The whole
proclaimed
the last remains of economic life,
demobilization is stalled by uncer
most remarkable point was election
and civil war. To conceive the situ
tainty, and by uncertain instructions
of professors by students with uni
ation in Berlin, in the administration
from
versal, direct etc.
and the Executive Council (where I
jor
62
Workers' and Soldiers' Councils]. I
disturbances
the
and
Executive
Council
THE MATHEMATICAL INTELLIGENCER
[of
his
The
discussion
insight
program,
bloodier
than
the
one
on
along
must happen soon, if we are to find
So one
any way to avoid disintegration of
whose
suffrage. Such
also spoke with people), think of an
agreeable and amiable. It seems to
GOttinger Zeitung summarized his talk
airplane high in flight, its pilot dead
me very important that some ran
[4]:
of general debility. It can fly quite a
dom student or grumbler not be the
while more, but
will crash, unless a
first to take up university reform
new pilot scrambles into the seat
and influence it decisively. I believe
and gets the controls in hand. It is
that is in no way precluded in the
Ministers'. It was imposed over his
clearer every day that the Executive
present situation. On the other hand
strong objections by Ludendorff 's
Council
this is just the moment when seri
side. It crippled the spirit of the
cannot
be
the
pilot
in
The
5 October ceasefire proposal
was not Prince Max's work or his
ous reforms could easily be put
German people and rapidly demor
through. Things you have so often
alized the Army.
declaration by the
brought up could actually get on the
due to embitterment over the sys
Entente that they will only deal with
agenda. For example, department
tem of lies which kept the people in
a truly democratic Germany. Their
committees for deciding appoint
the dark up to the last minute. If you
Germany today. I think we can expect much from a categorical
This was especially
statements on this so far are not suf
ments. But it is impossible to make
had asked people then which party
ficiently definite or official. Mean
any university reforms now without
could save them from catastrophe
while the Spartacists try to paralyze
considering social questions.
An up
an overwhelming majority would
beforehand the effects of such a
surge of elementary school students
have favored the Social Democracy.
declaration.
any
is inevitable and presumably it will
That feelings today are not all so fa
such effects as made-to-order, and
not end there. I just want to raise this
vorable to the Social Democracy, is
deliberately spread the idea that all
question now because I think it
mostly due to the short memory of
unpleasant news about the food sit
would be good
if the
the masses. Parties on the right use
They
denounce
uation, or problems with demobi
got
lization, and so on, are administra
Gottingen, and soon. As a commis
tion
sion to work out such a plan, I pro
The Social Democracy is blamed
pose Hilbert, Klein, Nelson [philoso
for three things: 1. the collapse and
pher Leonard Nelson [3]]. (Courant
the Revolution; 2. the economic ca
maneuvers to
make people
afraid of radicalism. Einstein, with whom I spoke at the close of the
a
clearer,
administration
firmer
plan
from
Germany as against the expansion
meeting, is a wonderfully noble and
him Kurt Hahn, since this
The "Independents" were the Inde
pure personality. I introduced to [educator]
pendent Social Democratic Party, an
may let Einstein contact authorities
anti-war offshoot of the
in
Social
the
administration,
and
help
mend our ties with Holland. Ein
the Social Democracy.
tastrophe; 3. the war-weariness of
to Hilbert 23. 1 1. 18)
aforementioned
this for effective agitation against
"Majority"
ist greed of her eastern and south eastern neighbors. To the contrary:
most
1: If anything is to blame for the
prominent founders were radical Karl
Revolution and collapse, it is the egoism and blindness of the ruling
Democratic
Its
stein will soon speak with [Minister
Liebknecht
for the Exterior, Wilhelm] Solf. I be
Bernstein. Liebknecht was also co
circles. Throughout the war, and es
lieve such people can accomplish
founder, with Rosa Luxemburg, of the
pecially last summer,
much more for us now than pol
Spartacus Group as a faction of the
Democratic Party warned of revolu
ished members of the diplomatic
Independents. By calling Independents
tion and, to prevent it, demanded
guild.
and Spartacists enemies, Courant takes
energetic reforms in Germany and
This next thing will interest you
and
Party. reformist
Eduard
the
Bernstein's side against Liebknecht.
conclusion
of
a
the
Social
reasonable
The Independents changed quickly
peace. It also warned the masses
matters of higher education, people
when the Armistice made an anti-war
over and over against any use of
come from every side, official and
party
the
force. The Revolution was thus not
unofficial, to suggest reforms and so
Majority. The Spartacists split to form
made by the Social Democracy, and
directly. As in all things today, so in
useless.
Many
rejoined
on. I was there with some people
the German Communist Party, calling
indeed not by anyone at all. It was
from my division, about using army
for a dictatorship of the proletariat. By
not even made by the Independents
physical apparatus for instructional
late 1919 the Independents echoed that
and Spartacists, who overrate them
purposes. I saw what a flood of "sug
call. That year Emmy Noether joined
selves boasting of it. It came as a
gestions" pours in on the Ministry of
the Independents. [2, 6].
Education from the most peculiar
kind of natural event. Yes, when
By December 1918 Courant was
the Revolution arrived the Social Democratic Party took the lead.
people, and how unclear the juris
back in Gottingen as the most promi
dictions are and the conduct of
nent Social Democrat in the University.
This was less for the Party's sake
business. Herr Blankenburg, who I
On
than for the whole German people,
5 January he spoke on "Social
believe is Dean at Mtinster, is in
Democracy, Revolution, and the Na
to stave off an otherwise inevitable
charge of university matters. One
tional Assembly"
fall into chaos and civil war. If the
cannot tell whether he works under,
Biirgerpark, which held
over, or alongside Becker. He is very
in
the
restaurant
500 people and
hosted many political meetings. The
National
Assembly
convenes
so
quickly, or at all, we must in the first
VOLUME 23, NUMBER 3, 2001
63
Gestoh lene Bekleidnngsstficke derHeeresverwaltnng und unrechtmaSig erworbene
scbinden den Trlger IIDd den den tscben Ramen RetcllsYerwertungstmt, Berlin I. 8, Frledrlcbstrassa 68. eoalalbemoflrofiJd)e �erJammluag. Stt ber . !3ctfcnnntung ber !J�talbemofroti·fdien 1U�tnd €onntag nwrgen tm ,ll.\Q.rgerl'ltd.. ergrlff nocfJ ben ehdtltenben IBorlen bel �enn Jt 11 {) I e �err '!i3rofeffor e; 0 U r a n t bat IBort 6U dnem me(uat &bet' .,$03talbemo"
am
fratte.
91cb0tutton
unb
9latlonalbetfamntlUltg"'.
IBegen
$'favmangd� flnb mlr Ietbet ·gfr.5ttat. 1Jllt mit dntm fur• atn tlutaug aut btm an trtffenben. menterfungen fo fi&er.
ttld}en t:Jortrttge &U �Jt'Q�(ft. !)at ScffenfttUilanbGange&ot bom 3. Cltuitr toctr nl4t ba& mn:r bet 9Jrlft6ett 'lncq obu tlnet Fetner mntpu, fonbcru eJ h)Urbt �m tto' feind �eftlgett Stt4116en3 bD1t
�nbo�f.1 a.uf9C1lWtlQtl\. �tc 8otQe blefet !1\e�ffm� ftUlftonbfangebottt mcu: cine Sdl)muq bu Sttnt1KUnR hn beutf�!t 18o(fe u1tb �e, bie 5u dna: taf*tt l>e#totoll fotton bel tt�teHn f�tte. 3nf6efonbete tmg �lerau kl
&tten
bte t!tbltterung ilbtt baf S!ilgenf'Q(lem. tntt 'beffat �life rru:nt bctt t\oll btl �um I�tea !Ingenbtlcf i1&u bte �toe 2onc im �nft.ln gd
Figure 2. Opening of summary of a talk given by Richard Courant at the Burgerpark in Gottingen. (Courtesy of Stacltarchiv Gottingen.)
place thank the Social Democratic Party. 2: The Social Democrats are blamed for the wild strikes, sense less wage demands, and lesser will ingness to work. Against that: From the start of the Revolution the Party leaders, in speeches, newspaper ar ticles, and many pamphlets, have called the workers to reason and or der and shown that only work can save us. Social Democratic leaders are blamed for the strikes in Upper Silesia and the Ruhr. At any rate,
64
THE MATHEMATICAL INTELLIGENCER
you cannot mine coal with machine guns and cannon, as the bourgeoisie means to. 3: The Social Democrats also re gret our war-weariness. Hopefully the unity the administration now happily shows will change that. But on this question, remember how each soldier's natural desire to re turn home after four and a half years of war impedes maintenance of a great army. We point to the new regime's order to accept volunteers for a re-organized army.
To correctly judge the adminis tration's current position and perfor mance one needs not only humane understanding of the bourgeoisie and the parties on the right, but also of the radical elements among the workers. The democracies of the world, in their overwhelming num bers, have thrown us to the ground. In the name of socialism the German spirit can lead the world. GZ 7. 1 . 19 Set theorist Felix Bernstein [3] at tended as a member of the Democratic Party. In discussion he claimed the Democrats were natural allies with the Social Democrats against both right-wing parties and the left-wing Independents. But he called logician Kurt Grelling and other Social Demo crats "amateur socialists," proving his distrust of the party, according to the reporter "K.G. ", who probably was Kurt Grelling [3]. On 11 January Courant spoke on "Foundations and Goals of Socialism" for the Political Society of Freedom Minded Academics, a short-lived lo cal group for scientific, non-partisan discussion of socialism. Dr. Walter Ackermann presided (not logician Wilhelm Ackermann): Prof. Courant made the following remarks: The social interests which absorbed German students in the 90s, and weakened considerably in the prewar years, now need a pow erful revival. Unlike the prewar years, the outcome of the war, and especially the re-distribution of wealth and deepening social divi sions, brings the social problem back into the clash of interests. Precisely the academics must take fundamental, objective positions uninfluenced by party slogans and agitation in the day's political strug gle. The organizers aim to promote this specifically for the range of ideas underlying socialism. Modem socialism has two deci sive concerns: First, organic trans formation of economic life to elimi nate so-called "surplus value," that is, any gain not coming from one's own work. Second, organization of
economic life to raise production and thus the general standard of life and culture. This amounts to Karl Marx's the ory of surplus value, including that elimination of surplus value cannot fundamentally solve the social prob lem. Increased production is a second essential factor, which an entirely free market cannot accom modate under present capitalist conditions, but which requires planned organization of the whole economy. . . . His lecture closed asking acade mics to study and adopt the views of socialism with a thoroughly ex amined, serious will to truth, unin fluenced by inclinations and class interest. GZ 15. 1 . 19 The other speaker was Iris Runge, soon to be Courant's sister-in-law. She said even non-socialists like Walther Rathenau, the Jewish industrialist and wartime economic planner, saw the need to organize production and the market. She complained, "A clever businessman with enough capital can, by the right advertising, make mon strous profits on some worthless prod-
F
� � �
i
uct and reduce the meaningful part of the national workforce," taking mouth wash as her example. Then came trouble from "national minded students" who admired Gener als Hindenburg and Ludendorff. They believed Ludendorff, that Germany was never defeated on the battlefield but by socialists and "foreigners" (read Jews) at home. Unfortunately, the plain bad manners of many student representatives nearly prevented any discussion on the reserved, objective, scien tific level of the presenters. There were constant shouted interruptions. Some speakers were applauded with minutes-long ear-shattering turmoil. . . . Only after the worst disrupters were led out, after singing the song "Deutschland, Deutschland iiber Alles" and raising a cheer for the German students (?), could the meet ing resume some order. Among others, . . . student Albert Miihlestein, . . . took an objective part in the discussion. In his closing word Prof. Courant took a sharp stand against the un worthy behavior of part of the atten-
i!ii!i!iiii!ii!!i!!i!i!ii!i!i Sonnabend. den 11. Januar, nachmlttags 4 1/1 Uhr,
im grossen Saale des ,Kaiser-Kaf!ee'' :
=.
Studenten= and Akademiker=Versammluog
i
Thema:
'i '= 'i � 'i :
� Ii •
I I
�
; a
Grnndlagen n. Ziele des Sozialismns. Referenten : Prof. Courant.
Iris Runge. Aile Akademlker •lnd elngeladen. ,.. Frele Aussprache. -.
Dar JDIItlsclle Varela lrellleltllcll gesbulter Akadeder. Dr.
(Z72
A c k e rm a n n.
Zll r Deckung der Unllloeten werden trlttegeld erhoben.
•s Pfg.
Eln•
122i£2!2!2!25I5I5I5I5I353535itit515I!25li IIII555I555liII55!li to35hi
Figure 3. Announcement published in Gottinger Zeitung (1 .1 1.1919, p. 4) of meeting at which Richard Courant, Iris Runge, and Walter Ackermann were to speak.
dance, who, despite claiming to be educated, put themselves on quite the same level as the Spartacus Group. (Ibid.) The question mark on "German stu dents (?)" may query whether these were students, but certainly questions their exclusive claim to "Germanness." The GOttinger Tageblatt reported the German student (?) viewpoint: From student circles we have this report: On Saturday, academics over flowed the Kaiser Cafe and moved to the Biirgerpark to discuss social problems. Before describing the meeting I must take a moment for the flyer that invited us. A real par ody, it insulted all academics without distinction as "political sucklings," "more dangerous than Spartacus," whose "favored news papers" are collectively "dishonest." Using the familiar "Du" throughout, it spoke of phrase mongering by cer tain "enemies." We national-minded academics protested this, through Herr Gerwin, and hereby protest it again most forcefully. Others who did nothing and objected to Herr Gerwin's proceedings, may have the servile pleasure of being called po litical sucklings and their press dis honest. We only thank them for fi nally admitting it! Herr Ackermann opened the meeting and emphasized the goal of the Society of Freedom-minded Academics, to stand outside all par ties. (Then, who are the "enemies" mentioned in the flyer? I can answer that: The enemies of German acad emics are the destructive influence of foreignism and the international ideas of pompous fools!) Herr Professor Courant spoke on so cial reform and socialism. He ex pounded Marx's theory of socialism without really adding anything. Later he set himself against Prof. Oppenheimer's theory. He left plenty of gaps, but made a re spectable effort to bring the ideal world of socialism closer to his au dience's comprehension. Yet like the second speaker, Fri. Iris Runge
VOLUME 23. NUMBER 3, 2001
65
monarchy. Herr Professor Courant
on those who have had political con
less interest, he had to admit that
in
tact with you, to denounce you pub
worker's
legislation,
sadly lost in the general pushing and
licly. Since I could not reply in the
which already existed long before
shoving of people streaming in and
meeting on Friday, let me do that
the war, produced such far-reaching
out, on supposed disturbances and
here and now.
reforms that intensive efforts on
attempts to disrupt the meeting. For
You know I never reproach any
these problems no longer seem
our part, we can only attach this to
one who adopts communist ideals
pressing to many people. Fri. Runge
people affiliated with the organiz
out of pure motives. Many tender
spoke
of
ers. Those of other political views
minded
on
gave quite peaceful attention to the
irony is untranslatable] people let
whose
speech
apparently
protective
mainly
on
the
socialism,
based
Rathenau's
works.
drew
goals
primarily
closing
made
some
remarks,
"hot-tempered",
the
Unfortunately
two speakers, as well as to Herr
anger and alienation drive them to
her exposition, like the "gaily col
Miihlestein, proving their good will
radicalism, some on the right, and
ored" mouthwash ads she talked
to assure a suitable proceeding to
about, was often lost in wearisome
the affair.
cliches.
She,
GZ 14. 1.19
some on the left. I do not deny even your good faith. What I find so up setting and disgraceful about your
like Herr Courant, The Social Democrats got 38% of the
debut, is the irresponsible dema
19 January. They governed in
gogy. You misuse your gifts as a
crease wages very little. Both speak
coalition with the Democrats and the
speaker to sow pointless new dis
ers still owe us an account of the
Center Party, which got
real blessings of socialism. One was
National
admitted that dividing employers' profits among the workers would in
vote on
People's
19% each. The
Party
which
de
order, to add new fuel, and to pro voke our tortured people already
first opposing speaker, Herr Clos
100-1>, and the 8%. On 18 March 1919 Courant and Ack
war. Certainly you tried to weaken
terhalfen, quickly pushed them into
ermann spoke to a student meeting
the initial impact of your words by
the realm of actuality, where they
about their fact-finding trip to Berlin
explaining you meant war with spir
sadly faded as beautiful shadows.
for the Society of Freedom-minded
itual weapons. Now you can declare
Long bursts of applause thanked
Academics. Conservative student Georg
you never proclaimed a bloody war.
him, and disturbances broke out against him too. Speeches and
Schnath says these "known socialists"
But
disturbed the meeting by getting on the
reservation that if armed war was
left feeling these are quite ideal
spised the Republic got
ideas, indeed mere theories. The
Independents
sick at the core, to civil war. . . . On Friday you preached civil
you
immediately
made
the
stormier
agenda "explicitly against the will of the
"forced on you" the case would be
and some people took unseemly ad
majority." They reported that news of
different. . . . At the end of the meet
counter-speeches
grew
this, to charge that no
the situation in Berlin was wildly exag
ing, among many other comments,
one in the group had thought about
gerated in government lies, and warned
when you read out the latest unfor
social problems before, and again to
against using troops to fight workers.
tunate N oske edict, to provoke the
call for removing the "political suck
They said, in Schnath's words:
railway workers there to "action,"
vantage of
lings" ! Herr Miihlestein further pro
you seemed to me not quite clear on
voked the gathering by inflamma
The rule of class justice, and espe
tory outbursts. When he pointed out
cially the widely ridiculed way it is
war. You left me really expecting
the unfortunate author of the flyer,
administered, are decisive failures
you to give the signal within a few
the excitement rose so far that the
of Noske's gun-barrel politics, and
months.
great majority of academics began
greatly increase the membership of
the manner or timing of your civil
heavy protest against foreign med
Spartacus,
which, moreover, the
If you, Herr Miihlestein, really
dling and emphasized their own
speaker tried to distinguish strongly
mean well for the German people, if you are ready to commit yourself
Germanness in the strongest terms.
in
After this "uproar of the sucklings,"
Bolshevism.
every
respect
audience
entirely, then go carry the torch of
and insane pounding from Herr
showed they were not the ones to
world revolution in the countries of
Miihlestein's corner as he apparently
hear such claims, with the speakers
the Entente. Leave politics here to
took out his anger on the poor fire
taking
others.
screen, we had a rousing "Deutsch
Independents' interests. [5]
up
.
.
from
.
word
The
for
Russian
word
the
cheers for our Germanness, and the German Students, and Hindenburg, as we left the room.
A.R.
All we need add is that after wards
Herr
Professor
Willrich
GZ 3.7. 19
This is very like what anti-socialists
land, Deutschland tiber Alles" . . . and Courant
closed
his
part in the
said to
Social Democrats in
1918,
Revolution with two open letters to
and Miihlestein's reply was very like
newly Communist Hans Miihlestein.
Courant's defense in January
Courant wrote on the front page of the
did not urge civil war, but warned
Gdttinger Zeitung:
against it, and pressure from the gov
lauded the beginning of social legis
66
[or
1919: He
ernment and the military will make the
lation by the old Kaiser and Hinden
Your public debut as a Communist
workers fight. Courant's rejoinder does
burg as an act of the constitutional
agitator in Gottingen imposes a duty
not point to any difference between
THE MATHEMATICAL INTELLIGENCER
the cases nor to anything specific
teers. By July
Mtihlestein said. He treats these argu
very defenses he had
ments exactly as anti-socialists treated
for Revolution. A brief stint on the
Social Democratic arguments earlier.
Gottingen town council ended in spring
He ignores them. He and his friends will clarify what Mtihlestein said. The
1919 he was deaf to the
A U T H O R
earlier given
1920. After promoting the German vote 1921 plebiscite on whether Upper
in a
editors closed the debate with Cou
Silesia would be German or Polish, he
rant's rejoinder on 31 July, which began:
took only private interest in politics. [4]
First, I insist my representation
dropped all mention of "Marxism," as he
of Herr Mtihlestein's performance,
probably had very quickly after 1919. He
His
1933 report slanted a little. He
and especially the irresponsible pro
spoke of a "Bolshevist" menace, as the
vocative way he spoke of civil war,
right had in
corresponds entirely to the facts.
praised "the Social Democratic Party
Philosophy and Mathematics
This has been confirmed to me, both
under men like Noske." Noske was
Case Western Reserve University
before and after the letter was pub
strong in the party in
lished, by the most varied and in
criticized
1919 but he had not. And he
1919, but Courant
COLIN MCLARTY
Cleveland, OH
was unaware of what he said, and
him then. Like the reference to "Wehrmacht," a rare term before the Freikorps popularized it, and one Courant did not use in 1919, all this
Col in Mclarty is an Associate Pro
would otherwise have chosen dif
merely skewed the cast list towards Nazi
fessor of P h i losophy , and of Math
ferent words. But the content and
taste. The motives Courant claimed in
ematics, at Case Western Reserve
the effect of his speech were made
1933, he had shown in the Revolution.
disputable eye- and earwitnesses. I will happily believe Herr Mtihlestein
USA
e-mail:
[email protected]
University. After a philosophy disser
tation on J. H. Lambert and Kant, he
so clear, not only in my letter but also and in the same way by others
REFERENCES
concentrated on mathematical and
who are much closer to him, that his
[1 ] Dahms, H-J. and F. Halfmann, "Die Univer
philosophic aspects of category the
attempt to deny it now seems rather
sit�i.t Gottingen in der Revolution 1 91 8/ 1 9 , "
ory and topos theory. He is currently
strange. It would seem more re
1 9 1 8: Die Revolution in SOdhannover.
working on the history of homology
spectable to me, if Herr Mtihlestein
(H-G. Schmeling, ed.) Stildtisches Museum
from Poincare through Grothendieck's
would quietly accept the fact that he
Gottingen (1 988).
algebraic geometry. This article grew
cannot defend his performance and his speech.
GZ 31.7.19
[2] Dick, A. Emmy Noether, 1882-1935. Basel: Birkhauser ( 1 98 1 ) .
from research into Ernmy Noether's role in that. His wife, Patricia Prince
[3] Peckhaus. V . Hilbertprogramm und Krit
house, is an historian of biology. To
ische Philosophie, Das G6ttinger Modell
gether they breed and show Pyrenean
acted throughout the Revolution to
interdisziplinarer Zusammenarbeit zwischen
Sheperds and Great Pyrenees dogs.
keep order and hold off radicalism and
Mathematik und Philosophie. Gottingen:
Courant said truly in
1933 that he
civil war. By late November
1918 he
agreed with Eduard Bernstein that the Revolution should end with elections,
Vandenhoeck & Ruprecht (1 990). [4] Reid, C. Hilbert-Courant. New York: Springer Verlag (1 986). dass ein weiblicher Kopf nur ganz aus
and socialist measures be postponed
[5] Schnath, G. "Gottinger Tagebucher Okto
until then. He never publicly mentioned
ber 1 9 1 8 bis Marz 1 91 9," G6ttinger Jahr
nahmsweise in der Mathematik schop
any specific socialist measures, unless
buch (1 976), 1 71 -203.
ferisch tatig sein kann," G6ttinger Jahr
we count the call for new Army volun-
[6] Tollmien, C. "Sind wir doch der Meinung,
buch 38 (1 990) 1 53-2 1 9 .
VOLUME 23, NUMBER 3 , 2001
67
MIHALY
T. BECK
Why I s Th e re N o M ath e m ati cal N o be l Prize ?
mong the many scientific awards, the Nobel Prize is of the highest rank. The question is frequently posed: why is there no mathematical Nobel Prize? The formal answer is simple: Alfred Nobel in his wiU determined that awards should be given scientists who made the most important discoveries in the fields of physics, chemistry, physiology/medicine; further, to au thors of excellent literary works and to persons who con tributed eminently to the cause of world peace. The real problem remains: why did Nobel not decide to reward the greatest contributions to mathematics? I have not found any authentic data in the available lit erature [ 1-3], but there are snippets of gossip and conjec ture in various autobiographies. In his memoir, Theodor Karman wrote [4] that although Oscar Prandtl would have deserved it, . . . he never received the prize apparently because the Nobel Committee didn't (and still doesn't) regard the sci ence of mechanics as sublime as other branches of physics for which they have provided many prizes. Einstein for example got the prize mainly for explaining the photoelectric effect, not for the brilliant mathemat-
68
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
ics underlying his theory of relativity. I have always per sonally suspected that this curious blind spot in the Nobel Committee arose because Nobel could not forgive his mistress for running off with a mathematician. The following explanation was given to Manfred von Ardenne [5] by Prof. B. Debiesse, then chief of the center for atomic energy in France: Nobel had a girl-friend younger by thirty years, whom he found tete-a-tete with a mathematician. Supposedly, this event prompted him to leave out mathematics in de scribing the statutes of the foundation.
It is not certain that the biographies give a complete ac count of the amorous affairs of Nobel, who remained a bachelor to the end of his life. However, in connection with
these assumptions it is worth mentioning that Nobel in fact
A U T H O R
had a long, most likely not merely platonic, connection with a Viennese woman, Sophie Hess, who was indeed thirty years younger than Nobel [6]. Sophie once told
him
that
she was expecting a baby and that the father was Kapivan Kapy, a Hungarian army officer. Nevertheless, Nobel was most generous with Sophie, and even provided for her in
his will. (It is surprising that there is no speculation in the literature that Sophie's seduction by an army officer con tributed to Nobel's antimilitaristic feelings.) Other possibilities, reported by Garding and Hormander [7], included a French-American and a Swedish explana tion for Nobel's neglect of mathematics: According to the
MIHALY T. BECK
former, Mittag-Leffler had an affair with Nobel's wife, while
Department of Physical Chemistry
according to the Swedish account, Nobel realized that cre
Kossuth Lajos
ation of a mathematics prize would mean that Mittag-Leffler
University
401 0 Debrecen
would be the first recipient-something Nobel did not fa
Hungary
vor. It seems to me that the two versions together afford
e-mail:
[email protected]
an explanation, but neither works alone. Nobel was a lifelong bachelor, so version one is obvi
Mih8Jy T. Beck has dealt mostly with coordination chemistry and
ously wrong, although there could be suspicion of an affair
reaction kinetics. His other interests are ethical and method
between the wife of Mittag-Leffler and Nobel. It is much
ological problems of scientific research, and history of science.
more likely, however, if Gosta Mittag-Leffler had any role
1883 he and his wife will in which they left their villa in
in the decision of Nobel, it was that in had already drawn a
Djursholm to the Swedish Academy for the promotion of
1900, a Nobel
mathematical researches in general, but first and foremost
nowadays, or even just a few years after
in the Scandinavian countries.
would disregard the promotion of the development of math
I believe that Nobel's neglect of mathematics has a more
ematics.
prosaic foundation: his general scientific view. Nobel's pub lic schooling ended when he was
1 6, and he did not go on
to the university. He received some private instruction from Zinin, an excellent Russian organic chemist. In fact, it was Zinin who called Nobel's attention to nitroglycerol in
1855.
REFERENCES
1 . H. Schuck, R. Sohlman, A Osterling, G. Liljestrand, A Westergren, M. Siegbahn, A Schou and N.K. Stahle: Nobel, the Man and His Prizes, Elsevier, Amsterdam, 1 962.
Nobel was a typical ingenious inventor of the nineteenth
2. Erik Bergengren: Alfred Nobel, Nelson and Sons, London, 1 962.
century. His inventions needed profound knowledge of ma
3. Ragnar Sohlman: The Legacy of Alfred Nobel, the Bodley Head,
terials, resoluteness, and intuition, but not any knowledge of higher mathematics.
In
the second half of the century,
research in the field of chemistry in general did not demand higher mathematics. It is likely that Nobel's mathematical knowledge did not exceed the four arithmetic rules and the rule of three. The basic change in mathematical chemistry came about only after the death of Alfred Nobel. It is very unlikely that
London, 1 983. 4. Theodor von Karman with Lee Nelson: The Wind and Beyond, Little Brown and Co., Boston, 1 967. p. 32. 5. Manfred von Ardenne: Ein gluckliches Leben tar Technik and Forschung, Kinder Verlag, Zurich und Munchen, 1 972. p. 322. 6. Ref. 3, pp. 56-58. 7. Lars Garding and Lars H. Hormander: Mathematical lntelligercer 7 (1 985) pp. 73-74.
VOLUME 23. NUMBER 3, 2001
69
I\!/Mj,i§rr@ih$11@%§4611,J,i§,id
This column is devoted to mathematics forjun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is
Alexan d e r Shen, Editor
A T realise on
the Binomial Theorem by Prof
J.
Moriarty, M.A.
allowed to be); it may not be directed
REVIEWED BY SHALOM B. EKHAD
only at specialists; it must attract
REVISED AND ENLARGED EDITION, PRIVATELY PRINTED.
andfascinate.
CORK, 1 885
We welcome, encourage, and frequently publish contributions from readers-either new notes, or replies to past columns.
Please send all submissions to the Mathematical Entertainments Editor,
I
t has long been my hobby to find and peruse antique mathematics books and code books. The Internet makes it possible for immobile individuals such as myself to indulge this interest, by search of lists of book dealers or through electronic auction houses such as eBay. By this means I recently* acquired a copy of the rare book A Treatise on the Binomial Theorem (1885), by Prof. J. Moriarty. I had read of its existence, but did not think I would be so fortunate as ever to see a copy. The history of this work is quite cu rious. Its contents surely deserve to be better known, as they may prompt a modest rewriting of the history of mathematics. The edition was printed in a small run of only 50 copies, and the original copies of the book were ap parently distributed to a select set of British and continental mathematicians and natural philosophers, as a sort of manifesto. The copy I possess has the bookplate of L. J. Rogers. It is bound in half-calf, with some foxing of the inte rior pages, a damaged spine and rear cover, and with.all pages from p. 243 on being tom out. It could be that this is the sole surviving copy, due to singular circumstances that I shall relate. It might prove useful to trace the libraries of the other recipients above in the hope of locating a complete copy.
I
Mathematical Contents
The book is written in a difficult style, characteristic of amateur authors and mathematicians who do not wish their work to be read. The author appears to be an autodidact with an axe to grind. Prof. Moriarty's book very likely was given a careless examination by most of its recipients, who took it to be the work of a crank The treatise begins with a 120-page discussion of theories of the mind and a "Calculus of Possible Experience." The author argues that, humans being prone to error in logical reasoning, it is necessary and inevitable that all ratio cination be carried out by mechanical means. Such means, with proper exe cution, can reduce error to an unimag inably small level. The method should, moreover, be applied in all walks of life. In a prefatory "Advertisement" the author states that this book is being dis tributed to a select group of eminent natural philosophers, who are asked to join him in a secret society which shall represent the vanguard of those putting this vision of the future into practice. The author suggests that there may be considerable social disruption before they eventually prevail, and that all ob stacles should be ruthlessly removed. One perceives that this is no ordinary mathematics textbook To support his thesis, Prof. Moriarty proceeds to discuss various mathe matical and logical topics, in a some what obscure fashion. He asserts that all mathematical identities of a certain recondite form can be verified in a purely mechanical way. He asserts that his methods go far beyond what is available everi to the greatest mathe maticians of the day. He then states that he will illustrate this on two prob lems, one connected with identities for
Alexander Shen, Institute for Problems of
Information Transmission, Ermolovoi 1 9, K-51 Moscow GSP-4, 1 01 447 Russia;
*It was on April 1 , 2001 , a date of note, being an anniversary of my fabrication and the etching in of my ser
e-mail: shen@landau .ac.ru
ial number.
70
THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK
binomial coefficients and the other with certain properties of a function which can be shown to be the Riemann zeta function, which the author hints at in a rather secretive and elliptical way. The author uses his own notation for binomial coefficients and hyperge ometric series without explanation, ap parently treating it as self-evident. Once a dictionary is assembled be tween his notation and the standard one, this part of the book is surpris ingly comprehensible. On page 1 71 we discover the identities (my translation)
n2
oo
];0 (1 - q)(1 - q� . =
fl
n=O
(1
_
. . (1 - qn)
q5n + l) - l (1
_
q5n + 4) - l .
and
n2+n
oo
];O (1 =
-
fl
q)(1
n=O
(1
-
_
:2)
•
·
·
(1 - qn)
q5n + 2) - l c 1
_
q5n + 3r l.
These particular identities have since been discovered by a number of mathe maticians and are known by various names in the literature, cf. [6, pp. 90-91]. Prof. Moriarty provides short proofs of these which, although bizarre, can be checked line by line to be correct. The method, described in what can only be called crabbed prose, consists of the clever introduction of extra variables and telescoping identities, using a kind of symbolic calculus based on differen tial operators. The author supplies a long sequence of further identities and q-identities of various sorts, and for each he gives grotesque proofs whose origin is mysterious. At one point he describes these as "proof certificates, the bank notes of the future." With hindsight, it seems quite clear that Prof. Moriarty was in possession of the essence of the so-called WZ method for proving hypergeometric function identities, which is detailed in Petkovsek, Wilf, and Zeilberger [7]. A significant part of this method was cre ated by D. Zeilberger [8] in 1990, see
also [2] . The particular proof used by Prof. Moriarty for one of these identi ties is essentially identical with one published by my brother and his sili con companion [4] in 1990 using the later-to-be-named WZ-method. What methods Prof. Moriarty used to carry out his own computations are un known. In a less mathematical vein, I note that Doran Zeilberger [9] seems to be developing sociological opinions sliding in the direction of that of Prof. Moriarty. Perhaps such are the conse quences of thinking like a computer. I applaud this; but I digress. In the text that follows, beginning on page 220, Prof. Moriarty discusses the Riemann zeta function. He seems un aware of the work of Riemann, and in troduces it as a function of a real vari able, as "a curious binomial product function of the esteemed Russian sa vant Mr. L. Euler." He derives identities for the values of the function at even positive integers, and then gives various integral formulae, including one imply ing the functional equation, asserting that they were obtained by a mechani cal method for obtaining quadratures. He states that he has developed meth ods of "imaginary analysis," by which he seems to mean methods of a complex variable. Perhaps these also involve some ideas of renormalisation, because there are formulae resembling operator product expansions, but at this point the book (literally) breaks off. It seems fruitless to speculate on what the rest of the book may have contained, but I find some of the final formulae sugges tive and hope to communicate further on this subject in a year's time. The Author
My information about Prof. Moriarty's early career* is derived from Dr. J. Watson (vide [3]), who credits the fol lowing information to Mr. W. Sherlock Holmes:
[Prof Moriarty's] career has been an extraordinary one. He is a man ofgood birth and exceUent education, endowed
by Nature with a phenomonal mathe matical faculty. At the age of twenty one he wrote a treatise upon the Binomial Theorem, which has had a European vogue. On the strength of it, he won the Mathematical Chair at one of our smaller Universities, and had to aU appearance, a most briUiant ca reer before him. But the man had hereditary tendencies of a most dia bolical kind. A criminal strain ran in his blood, which, instead ofbeing mod ified, was increased and rendered infinitely more dangerous by his ex traordinary mental powers. Dark ru mours gathered round him in the University town, and eventuaUy he was compeUed to resign his Chair and to come down to London. ,
Certain features of the book under re view, however, have led me to a radi cal conclusion about its author. These features include some pencilled anno tations on page 173 and the impression of a square-toed boot of an unusual make on the inside of the back cover. My belief is that "Prof. J. Moriarty" is a pseudonym and that the true author is Mycroft Holmes. It is well known that Mr. Holmes, whose intellect was supe rior to that of his drug-addicted brother Mr. Sherlock Holmes, had presciently come to the conclusion that domination of the world by machines was inevitable. The author of the "Treatise" clearly be lieved that the creation of the new world order, however rational, would only be achieved by rivers of blood, and that a secret society to effect some guidance to the future world order was essential. Naturally enough, the book was pub lished under a pseudonym. Indeed I be lieve Mr. Holmes borrowed the title of Prof. Moriarty's earlier work to discour age uninitiated readers; the difference in tone from Prof. Moriarty's original which Prof. P. Gordan reviewed as "a capital example of mathematics, not theology"-would have been self-evident. The actual history of events, as I have reconstructed them through ex act logical deduction, closely parallels
•upon Prof. Moriarty's subsequent career, his successful investments in the stock exchange, his change of name, knighthood, and raising to the peerage, it is un necessary to dwell here.
VOLUME 23, NUMBER 3, 2001
71
this view. Mycroft and his brother, though congenial through much of their lives, had increasingly bitter ar guments during a continental tour they took in the spring of 1891. Mycroft as serted that freelance investigators of his brother's sort would soon be put out of business. He argued that "man aging your clients" was the really im portant thing for the success of any investigatory business; that logical thought was irrelevant, and was best left to machines; that actually solving cases was pernicious, though perhaps of some ephemeral intellectual amuse ment; and that drug use was interfer ing with Sherlock's ability to reason. Mycroft even claimed to have tired of Sherlock's statement "When the im possible was eliminated, what remains, however improbable, must be the truth" and that repeating it often was symptomatic of grandiosity and obses sion. Mycroft's untimely death at the Reichenbach Falls soon after brought his utopian hopes to an end. I believe that his vindictive brother tracked down and destroyed as many copies of Mycroft's book as possible. We are only fortunate that either L. J. Rogers or his manservant was present when Sherlock attempted to purloin his copy, and that in the ensuing struggle he was able to retain part of the book. Sherlock Holmes's death soon after brought an end to this whole sad story. It is true that events as I have de scribed them conflict with the narrative of Dr. Watson [3], but I am afraid his account must be dismissed as fiction. Taking the facts as he presented them, is it plausible that he would have sud denly left his young wife and a thriving medical practice that he was desirous of increasing, for a sudden trip to the Continent, of several weeks' length, at considerable expense, with an obvi ously paranoid individual? Would he not rather have diagnosed cocaine-in duced psychosis and immediately clapped his companion into a sanato-
72
THE MATHEMATICAL INTELLIGENCER
rium, where warm baths, hot milk, and cold turkey might restore him to san ity? Far more believable it is that Mr. Sherlock Holmes, having killed his brother, whether accidentally or wil fully, in a state of considerable agita tion telegraphed Dr. Watson for assis tance and money, and then disappeared for three years to Tibet. Dr. Watson's epistle, together with his fulsome obit uary to the Times of London, can only be viewed as an example of the sort of "disinformation" often produced during the Great Game. That Mycroft Holmes had a power ful intellect there is no doubt. However misguided he may have been in his pri vate life, this book proves he was a re markable mathematician. It is time to set the history of mathematics right, and pay him his due! In view of the con troversy currently raging over some of these issues [1], [5], [9], this is most timely. Sadly, my attempts to have his book reprinted, even in its mutilated form, have met with no success. REFERENCES
1 . G. E. Andrews, The death of proof? Semi rigorous mathematics? You've got to be kidding, Math. lntelligencer 16 (1 994), no. 4, 1 6-1 8. 2 . P. Cartier, Demonstration "automatique" d'i dentites et fonctions hypergeometriques (d'apres D. Zeilberger), Seminaire Bourbaki, Exp. No. 746, Asterisque No. 206 (1 992), 4 1 -91 . [3] A. C. Doyle, The Adventure of the Final Problem, The Strand Magazine, December 1 893. [4] S. B. Ekhad and S. Tre, A purely verifica tion proof of the first Rogers-Ramanujan identity, J. Comb. Theory, Series A, 54 (1 990), 309-31 1 . [5] S. B. Ekhad and D. Zeilberger, Curing the
A
=
B, With a foreword by D. E. Knuth,
A. K. Peters, Inc.: Wellesley, Mass., 1 996. [8] D. Zeilberger, A holonomic systems ap proach to special function identities. J. Comp. Appl. Math. 32 (1 990), 321-368. [9] D.
Zeilberger,
Theorems for a Price:
Tomorrow's Semi-Rigorous Mathematical Culture, Notices Am. Math. Soc. 40, No. 8 (1 993),
978-981 .
(Reprinted
Editorial Note. Shalom B. Ekhad has been incapacitated by a power surge. The review is presented just as it was re covered. I am obliged to point out that other evidence conclusively indicates that the reviewer's deduction about the identity of "Prof. J. Moriarty" is erro neous. It appears that during a visit to the University of Gottingen in 1879 Prof. Moriarty gained access to the Riemann Nachlass and may have absconded with some of its papers. The absence of any reference to Riemann's work in the "Treatise" would therefore appear to be a subterfuge. The editor's own surmise is that, enervated by onerous teaching duties and frustrated in his attempts to prove the Riemann hypothesis, Prof. Moriarty turned to a life of crime. Indeed the Riemann hypothesis is a dangerous problem to work on; let this example be a warning to the unwary. J. C . Lagarias
[6] G. H. Hardy, Ramanujan, Cambridge Univ. Press: London and New York,
1 940.
(Reprint: Chelsea). [7] M. Petkovsek, S. Wilf, and D. Zeilberger,
Math.
About the Reviewer. Shalom B. Ekhad is not to be confused with his identical twin brother, Shalosh B. Ekhad, the collaborator and sometime indentured servant of Doron Zeil berger. Being unable to travel (or even speak) without assistance, he takes great interest in the careers of other disadvantaged individuals such as the late Mycroft Holmes.
Andrews syndrome, J. Diff. Eqns. App/. 4 (1 998), 299-3 1 0 .
in:
lntel/igencer 1 6 (1 994), no. 4, 1 1 -1 4.)
A.T.&T Labs Florham Park, NJ 07932-0971 USA email:
[email protected]
I a§!ii4'4i
J et Wi m p , Editor
I
What Is Random� Chance and Order in Mathematics and Life by Edward Beltrami $22.00
ISBN:
ing large in our everyday experiences" (p.
xi). It therefore behooves us to try to
understand it.
What mathematicians
over the centuries-from the ancients, through Pascal, Fermat, Bernoulli, and
NEW YORK: SPRINGER-VERLAG, INC., US
As we are told in the preface, "ran domness is the very stuff of life, loom
0-387·98737·1
1 999,
xx +
201 pp.
de Moivre, to Kolmogorov and Chaitin have discovered, is that it's a pro foundly rich concept. The more one
REVIEWED BY JERROLD W. GROSSMAN
delves into it, the more paradoxes
M
aybe I'm just too critical to be re
unknown and vice versa. Edward Bel
viewing mathematics books writ
trami, Professor of Applied Mathemat
arise, the more the known becomes the
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if
ten for a lay audience. Here are my ex
(1) The
ics and Statistics at the State Univer
author is an expert
sity of New York at Stony Brook, has
in what he is writing about (or else a
mapped out a provocative and delight
pectations:
you would welcome being assigned
professional mathematics writer with
ful journey through this still-evolving
a book to review, please write us,
excellent contacts).
(2) The mathe
subject, taking the reader from the ba
telling us your expertise and your predilections.
matics is correct, both in substance
sic premises to speculations at the
and in detail, from the statements of
fringes of current research.
the theorems to the steps of the der
Chapter
1
contains a brief exposi
cares
tion of probability and how one can use
enough to get the typesetting right, us
the language and tools of mathematics
ivations.
(3)
The
publisher
ing appropriate fonts and layout for the
to explain uncertainty and chance
mathematical expressions. (4) The au
conveying what a random process
thor and publisher have taken the time
should be, the idea of the Law of Large
to proofread the final product so that
Numbers, and the use of statistical hy
there are very few mathematical mis
pothesis testing as a first approxima
prints
tion to detecting whether the output
(especially
confusing
ones).
(5) The exposition is understandable to
of a process is consistent with that
the target audience, both the general
process's being random. It ends with the
discussions and the technical points.
introduction of Borel's notion of normal
(6) Most of all, the book intrigues the
numbers-numbers whose decimal ex
intended reader, imparts a flavor of
pansions contain, in the limit, the ex
what we do for a living, and raises the
pected number of occurrences of every
mathematical
sequence of digits. Normality is a nec
awareness. Well, one-and-a-half out of
essary, but definitely not sufficient,
level
of the
public's
six isn't too bad-at least it's the last
What Is Random? gets bonus, this book should be
condition for randomness, as Champer
of these that
nowne's
right. As a
(Beltrami's version of it in binary is
interesting to those with the technical background to understand the finer points and to relate this material to
famous
constant illustrates
0.0 1 00 01 10 1 1 000 001 010 . . .). Chapter 2 starts with a nice explana tion of Shannon's 1948 theory of infor
their previous knowledge of mathe
mation and the definition of entropy (al
matics and statistics-mathematicians
though the presence of logarithms is
who know little of these topics, gradu
likely to challenge the less mathemati
ate students, undergraduate mathe
cally inclined readers). As usual through
Column Editor's address: Department
matics or computer science majors,
out this book, Beltrami provides good
of Mathematics, Drexel University,
maybe even a curious student in her
sound bites; here it is that "maximum
Philadelphia, PA 1 9 1 04 USA.
first calculus or statistics class.
entropy is identified with quintessen-
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
73
tial randomness" (p. 43). Some good
unique
random
list has been carefully compiled, and
examples are provided to illustrate "ap
graph [2]). The level of philosophy and
countably
infmite
reading the recommended books and
proximate entropy," a rough-and-ready
mathematics is raised once again, and
articles
tool for testing binary strings for ran
the lay reader will probably have de
busy for months.
will keep the inspired reader
domness, and there is even MATLAB
spaired by now of ever understanding
Having said all this-and let me
code for computing it for your favorite
the finer points. Nevertheless, the fla
make it clear that I think it is good that
strings.
vor of what is going on here should
What Is Random? exists-let me get to
come through.
the gripes, following the list given in
Chapter 3 contains a rather confus ing, technical, and repetitive discus
The story could end there, but the
my opening paragraph. This book is
sion of generated and disappearing
author ventures on. In his words, "The
about probability (Section 60 of the
bits, algorithms running in reverse,
last chapter is more speculative. In
AMS 2000 Mathematics Subject Classi
pseudorandom
generators,
'The Edge of Randomness' [Chapter 5]
fication), a little statistics (Section 62),
gedanken experiments in physics
I review recent work by a number of
information theory (Section 94), Kol
(involving the Szilard engine and the
thinkers that suggests that naturally
mogorov complexity (Section 68), and
and
number
Second Law of Thermodynamics). The
occurring processes seem to be bal
computability (Section 03). According
author sums it up as follows: "Knowl
anced
to Mathematical Reviews, the only one
between
tight
organization,
edge of the past and uncertainty about
where redundancy is paramount, and
of Edward Beltrami's 24 publications
the future seem to be two faces of the
volatility, in which little order is possi
with any of these section numbers
same Janus-faced coin, but more care
ble" (pp.
xv-xvi). In other words, we
listed as either a primary or secondary
ful scrutiny establishes that what ap
have come full circle: Since total ran
classification is the book under review.
pears as predictable is actually ran-
domness is now as easy to understand
The book jacket states that his area of
as total order, meaningfulness lies be
expertise is applications of dynamical
tween these two extremes. As might
systems to things like blood clotting.
n encodes all
be expected in this free-wheeling
All other things being equal, I would
conclusion, we are asked to look at,
have preferred that Gregory Chaitin
information about
for example, biological processes, the
had written a book like this. Actually
music of Bach and Mozart, fractals in
he has (it's called
all com putations.
mathematics and nature, and "small
but I'll leave it for others to review that
world" networks [4]. The fmal sec
book [3], using their own criteria. On
tion, addressing the question "What
the other hand, Dr. Beltrami has also
good is randomness?", leaves the
written many erudite articles (and even
domness in disguise. I establish that
reader with an inspiring sermon, that
a book [ 1 ] ) on wine, and a sampling of
these two faces represent a tradeoff
this journey has been worth it after all.
one such article, and its recommended
between ignorance now and disorder
The book succeeds as a cross be
Chianti, left me impressed with his
later" (p. xiv).
tween a scholarly tome, a mathemati
The Unknowable),
breadth of knowledge. As the author claims, little in the
Things get back on track in Chapter
cal text, and a murder mystery. Many
4, where the big guns are rolled out
quotations from giants of the past and
way of formal mathematical
Kolmogorov's definition of complexity
present (philosophers, scientists, and
ground
(a given string is random if no shorter
popular writers, as well as mathemati
needed," p.
string encodes a computer program
cians) are sprinkled throughout. When
tends to skip the Technical Notes.
is
required
("no
back
calculus
xvii), especially if one in
that will produce the given string); Tur
the mathematical details get burden
What disturbs me as a mathematically
ing machines to capture the notion of
some, the reader is referred to (and can
sophisticated reader are the mistakes
computability, and their inherent in
choose to skip) Technical Notes and
in the mathematics: Big things, like
ability, demonstrated by Turing him
Appendices, which occupy about a
misstating
self, to analyze the computing process;
fifth of the volume. The lively non
theorem (we read that there is no proof
Chaitin's incompleteness theorem (the
technical discussions should for the
of the statement "for any positive inte
impossibility of verifying computation
most part keep the reader's interest
ger n there is a string whose complex
ally that a string is random); and the
through this short journey. An eight
ity exceeds
Law of Large Numbers ("with proba
Chaitin's
incompleteness
n, " p. 107) and the Strong
n (roughly, the prob
page postscript, "Sources and Further
ability that a randomly chosen com
Readings," points out some excellent
bility one . . . the sample averages
puter program will halt). As a conse
books and articles for filling in prere
Sr/n of a Bernoulli p-process will dif
quence of Turing's result, we cannot
quisites, reinforcing the message of the
fer from p by an arbitrarily small
n, and yet this one
present book, or going beyond it-clas
amount for all sufficiently large
sic
Scientific American articles, prob
30)-although in all fairness it must be
amazing number
know much about
number somehow encodes all infor
n," p.
mation about all computations. (I like
ability and statistics textbooks, and im
pointed out that the author para
to think of it as one of the mathemati
portant works by the leading writers in
phrases these results several times in
cian's answers to understanding the
the subjects being explored here and
different ways, and the other versions
meaning of life, the other being the
their offshoots. The 67-item reference
are correct. Medium-sized things, like
74
THE MATHEMATICAL INTELLIGENCER
sions (not to mention a handful of
usual misunderstandings. He will have
testing
omitted or misspelled words in the ex
been exposed to some of the profound
and criminal legal proceedings ("the
position), the reader has a right to be
mathematical questions that lie at the
prosecutor sets up a null hypothesis of
annoyed with the author and publisher.
heart of biology and physics. She may
getting tween
backwards statistical
the
analogy be
hypothesis
guilt, and the defending attorneys try
To cite just a few: there are three ty
even believe that she can begin to an
to reject that assumption on the basis
pos in the last four lines on page 189;
swer the question posed in the book's
of the available evidence," p. 27, which
m and m - 1 are confused on page
184;
title. Finally, the reader may well be
is also incorrect in the source Beltrami
the recurrence on page 1 70 defines Xn
left with the same pleasure upon fin
in terms of itself; on page 33 we are
ishing this book that one sometimes
requiring or
asked to believe that the three-digit bi
gets when experiencing the arts or pon
forbidding infinite (binary) decimal
nary numeral following 010 is 100; and
dering religious questions: I may not
expansions that end in all zeros (pp.
on page 26 we learn that
0.6.
understand all the details, but I can ap
185-187). Little things, like saying that
I'm hoping that if enough reviewers air
preciate that there is something very
says he took it from), and getting con fused as to whether he
is
t + -f;
=
probability one corresponds to mea
their frustrations in these matters, then
deep and beautiful and meaningful out
sure zero (p. 163) and that the word
publishers will feel the pressure and
there, and maybe I am now a little
outcome
institute better quality control.
closer to it.
in the context of a sample
space is interchangeable with the word
experiment (p.
10). The lay reader will
either be left with the wrong impres sion, or, more likely, be confused. The publisher is identified as "an im-
The perceptive reader wi l l real ize that com p uters can 't solve al l mathemat ical p roble m s .
Even when there aren't outright mis takes, the wording occasionally left me
REFERENCES
huh? (e.g., the integer n = Um -12m- l + Um -22m-2 + . . . + a12 + ao, where each ai is 0 or 1, is "never less
1 . Edward Beltrami and Philip F. Palmedo, The
asking
Wines of Long Island, Amereon Ltd. (2000). 2. Peter J. Cameron, The random graph, in The
than zero (simply choose all the coef
Mathematics of Paul Erd6s, Vol. /1, pp.
ficients to be zero)," p. 184). Some
333-351, Algorithms and Combinatorics 14,
times, especially in the last half of the
Springer, 1 997; MR 97h:05163.
book, the discussion becomes rather
3. Gregory J. Chaitin, The Unknowable, Springer
ethereal and mystical, and I fear that
Verlag, 1 999; MR 2000h:68071 ; see also
many readers
will
start to lose the
http://www.cs.auckland.ac.nz/CDMTCS/ch
drift. But for the most part, the writ ing
is fairly clear and compelling, with
aitin/unknowable/. 4. Duncan J. Watts, Small Worlds: The Dy
pithy statements that will appeal to the
namics of Networks between Order and
lay reader and the mathematician
Randomness, Princeton University Press,
alike, such as "strings that possess log
1 999; MR 2001 a:91 064; see also this writer's
ical depth must reside somewhere be
review in The American Mathematical Monthly
tween [the extremes of] order and dis
107 (2000), 664-668.
2
order" (p. 1 0), to pick one example print of Springer-Verlag" on the title
illustrating the author's main point in
Department of Mathematics and Statistics
page. Springer is certainly a respected
Chapter 5.
Oakland University
and important publisher of mathemat
And so we come to my sixth crite
ics (including, of course, this maga
rion, the bottom line for expository
USA e-mail: grossman@oakland .edu
zine). One would think, therefore, that
mathematics for the nonmathemati
Springer's expertise would be reflected
cian: Has this little book captivated the
in the typesetting of the book under re
reader, made her think more deeply
view. Alas, fonts are inconsistent, mi
about some of the ideas that drive our
nus signs often show up as hyphens,
professional lives, raised the level of
superscripts occasionally aren't raised,
his mathematical sophistication, and
spacing
is
sometimes inappropriate
conveyed at least the spirit of some of
in general, the layout makes the math
the
ematics ugly to look at and hard to
throughs? (For a good answer, we
exciting
new
research
break
read, for both mathematicians and lay
should probably tum to a review by a
readers. Is it too much to ask that
lay critic, but I was unable to find one
mathematics copy editors and typeset
anywhere, even though the book was
ters prepare the galleys?
published in late 1999.) I think that it
Everyone would agree that there is
has. The perceptive reader will realize
no way for a mathematics book to be
that computers can't solve all mathe
virtually error-free (well, maybe with
matical
problems.
She
will
Rochester, Ml 48309-4485
Squaring the Circle/ The War between Hobbes and Wallis by Douglas M. Jesseph CHICAGO AND LONDON, UNIVERSI1Y OF CHICAGO
1999, xiv + 419 pp. $80.00 (cloth), $28.00 (paper), ISBN 0-266-39899·4 (cloth), ISBN: 0-226-39900-1 (paper) PRESS,
US
REVIEWED BY GERALD L. ALEXANDERSON
have
Knuth's
brushed up against some of the trick
books). But when misprints repeatedly
ier parts of elementary probability and
T
creep into the mathematical expres-
perhaps be less likely to harbor the
author of
the
exception
of
Donald
homas Hobbes (1588--1679), the eminent English philosopher and
Leviathan,
is remembered
VOLUME 23, NUMBER 3, 2001
75
today for his influence in politics, law, and moral philosophy. His work in mathematics is much less well known and less studied-and, as we see in the work at hand and elsewhere, perhaps justifiably so. Douglas Jesseph is a philosopher and clearly is interested in exploring the implications of the Hobbes-Wallis dispute for philosophy in Hobbes's time and subsequently. At the same time we must recognize that Hobbes's concern for method and his at tempts to apply mathematical tech niques outside mathematics itself may be among his finest achievements. Hardy Grant in [G] said that, for Hobbes, "only the mathematicians' method, only strict deduction from sure premises, would serve. But this approach, so suc cessful in geometry and physics, had never (Hobbes urged) been applied outside those fields. Geometry is 'the onely Science that it hath pleased God hitherto to bestow on mankind.' Thus he did not blush to claim his own application of its method as his toric." And indeed it was. Unfortunately, as is often the case, a person's ability to appraise his or her own work is not always objective. Hobbes wrote of his mathematical work quite glowingly, citing, as we learn in the early pages of this book, his success in squaring the circle, di viding an angle in a given ratio, de scribing a regular polygon with any number of sides, and solving some other problems of well-known diffi culty. It is not entirely clear what he meant by these claims. Did he under standjust what constraints were put on the solution? Was he fully aware of what is involved in a euclidean con struction? Grant again: "Most notori ously, he claimed with complete confi dence the duplication of the cube, 'hitherto sought in vain,' and the squar ing of the circle; we may judge his grasp of this latter problem by his declaration that an 'ordinary' man might accom plish it better than any geometer, by simply 'winding a small thread about a given cylinder.' " So much for Hobbes's understanding of the problem! As every student of mathematics knows, there are three classical eu clidean construction problems dating from antiquity: (1) the squaring of a cir-
76
THE MATHEMATICAL INTELLIGENCER
cle (i.e., constructing the side of a square that has the same area as that of a given circle, or, to put it another way, constructing a root of x2 7T); (2) doubling the size of a cube, and (3) trisecting an arbitrary angle. An other such problem is: (4) constructing a regular polygon with n sides. In the early nineteenth century (2) and (3) were proved impossible for straightedge and compass, using properties of roots of cubic equations. Problem (4) was set tled by Gauss. But problem (1) re quired more: knowledge that 7T is a transcendental number. This was not proved until 1882 by F. Lindemann. Earlier work of J. Lambert (1761) had only shown that 7T is irrational, which says nothing of the constructability of =
"Thomas H obbes . came late in l ife to [mathematics] , and understood it poorly, but loved it m uch . " V";. Of course, none of this was avail able to Hobbes. The problem of the quadrature of the circle had plagued mathematicians since Anaxagoras (400-428 B.C.E.), who, we are told, worked on the problem in prison. Many mathematicians and lay men tried to "square the circle"-even Abraham Lincoln is reported to have worked on the problem-and people continue to try even to this day. Per haps the most risible attempt was pub lished in 1934 as a book with the half title, Belwld! The Grand Problem No Longer Unsolved: The Circle Squared beyond Refutation!, written by Carl Theodore Heisel, a 33rd degree Mason, in Cleveland, Ohio. The author seems to have substituted exclamation points for rigorous mathematical arguments. It is no wonder then that in the sev enteenth century this infamous prob lem should have attracted Hobbes,
who came to mathematics relatively late in life, though he had demon strated precosity in other disciplines. At the age of 14, for example, he had translated Euripides' Medea from Greek into Latin iambic. But his dis covery of Euclid was described thus by John Aubrey in [A]: He was . . . 40 yeares old before he looked upon geometry; which happened accidentally. Being in a gentleman's library in- Euclid's El ements lay open, and 'twas the 4 7 El. libri I. He read the proposition. 'By G-,' sayd he, 'this is impossi ble!' So he read the demonstration of it, which referred him back to such a proposition; which proposi tion he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was con vinced of that trueth. This made him in love with geometry. Hobbes's conversion to mathemat ics in his 40th year did not result im mediately in a confrontation with the mathematical establishment. By 1646 his reputation in mathematics was such that he was named tutor in mathematics to the Prince of Wales. It wasn't until 1655, however, that he published his De Corpore, which he hoped would establish an honored place for him in mathematics. It con tained a collection of his mathematical achievements, including three admit tedly failed attempts at the quadrature of the circle. (Wallis eventually refuted over a dozen of Hobbes's attempts to solve this problem.) Hobbes worked in a period of great mathematical achievement: Barrow, Descartes, Mersenne, Cavalieri were all active at the time. These were ex citing times. Hobbes even met Galileo on a trip to Florence! Still, the De Cor pore was published eleven years before Newton's annus mirabilis, so greater things were yet to come. One can only conclude from this book that Hobbes was not an easy man to get along with. The focus of the book, his longtime dispute with Puri tan mathematician, John Wallis, was not the only contentious debate he en tered into. An ardent monarchist,
Hobbes "directed some of his most vit
lis did not always disagree on issues,
count of one of the great intellectual dis
riolic
no matter what the question, they
putes of history and of the social, cul
polemics
at the
universities,
largely because he saw them as en
sooner or later seemed to end up on
tural, religious, and intellectual life of
dangering civil peace by challenging
opposite sides. The dispute abated in
the period. Hobbes's mathematics may
the authority of the sovereign." Still,
1674 when Wallis may have grown
be largely discredited, but, still, Grant
the intensity of Wallis's animus toward
weary of Hobbes's verbal assaults, but
writes in [G], "Mathematicians should honor Thomas Hobbes, who came late
Hobbes is hard to understand without
Hobbes, when he died in
a larger context. Wallis went so far as
hind an unpublished manuscript writ
1679, left be
in life to their subject, and understood it
to obtain advance sheets of De Corpore
ten in his last year and still laying claim
poorly, but loved it much, and staked on
from the printer so that he could pre
to his having squared the circle. By this
the supposed sureness of its methods
pare his attack on it even before pub
time, though, Hobbes's reputation in
his hopes for the peace and good gov
lication. Wallis's intent was not only to
mathematics had been essentially de
ernment of mankind."
discredit Hobbes's mathematics but by
stroyed by Wallis and others.
extension
to
attack
Hobbes's
Jesseph
con
tention that "his doctrines were suffi
conscientiously
covers
Hobbes's treatment of magnitude, ratio,
ciently well grounded that they would
and general quadrature, relating it to the
enable a solution of all problems." And
work of Mersenne, Robival, and others,
their feud, probably complicated by
and he devotes space to Hobbes's con
views on university reform, and per
temptuous views of analytic geometry.
haps by Hobbes's
But these writings of Hobbes are not
alleged atheism,
REFERENCES
[A] Aubrey, John, Brief Uves, ed. Oliver Law son Dick. New York, Penguin Books, 1 982. [G] Grant, Hardy, Geometry and Politics: Math ematics in the Thought of Thomas Hobbes, Mathematics Magazine 63 (1 990), 1 47-154.
moved well beyond circle squaring to
likely to be of much interest to modem
"fme points of Latin grammar (such as
day mathematicians, and, as the author
the proper use of the ablative case),
admits, much of the mathematical work
problems of Greek etymology, and
is largely viewed even by philosophers
Santa Clara University
as something of an embarrasm s ent.
Santa Clara, CA 95053-0290
questions
of
ecclesiastical
govern
It is
Department of Mathematics & Computer Science
ment"! So plenty of fuel was there for
not for the mathematics that one reads
USA
a good fight. Though Hobbes and Wal-
this book; instead one reads it for its ac-
e-mail:
[email protected]
MOVING? We
need your new address so that you do not miss any issues of
THE MATHEMATICAL INTELLIGENCE&. Please send your old address (or label) and new address to: Springer-Verlag New York, Inc., Journal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485
U.S.A. Please give us six weeks notice.
VOLUME 23, NUMBER 3, 2001
77
N EW B O O KS F R O M R O B I N W I L S O N ! ROBIN WILSON, The Open University, Milton Keynes, UK
Stamping Through Mathematics
An Illustrated History of Mathematics Through Stamps
Postage stamps are an attractive vehicle for pre enting mathematics and its development. For many years the author has pre ented illustrated lectures entitled Stamping through Mathematics to school and college groups and to mathematical clubs and societies, and has written a regular "Stamps Corner" for The Mathematical Intelligencer. The book contains almost four hundred postage stamps relating to mathematics, ranging from the earliest forms of counting to the modern computer age. The stamps appear enlarged and in ful l color with full historical commentary, a n d are listed a t the e n d of the book.
200 1 / 1 28 PP./HARDCOVER/$29.95/ISBN 0-387-98949-8
ROB I N WILSON and JEREMY GRAY, both, The Open Univer ity, Milton Keynes, UK (Eds.)
Mathematical Conversations
Selections from the Mathematical lntelligencer
"This fine book is a compilation of selected articles from The Mathematical lntelligencer, Springer's mathematical magazine about mathematics, about
mathematicians, and about the history and culture of mathematics. . . If you read mathematical books like I do, you will enjoy "the first reading, " i.e., browsing, of this book, because it covers so many interesting topics, has a lot of illustrations (including photographs), and displays formulas in a clear and readable format. ''The second reading, " i.e., reading the sections and articles that currently interest you, is even more enjoyable, because each gem in this collection was created by an expert in the respective field to be appreciated by -MAA Online
the general mathematical audience. "
Since its first issue, The Mathematical lntelligencer has been the main forum for exposition and debate between some of the world's most renowned mathematicians, covering not only hi tory and hi tory-making mathematics, but al o including many controver ies that urround all facets of the subject. This volume contains forty article that were published in the journal during its first eighteen years.
2000/488 PP./HARDCOVER/$59.95/ISBN 0-387-98686-3
ORDER TODAY! CALL: 1 -8 00-SPRINGER
•
FAX: (20 1 ) 348-4505
WRITE: Springer-Verlag new York, Inc., Dept. S255 8, PO Box 2485, Secaucus, NJ 07096-2485
VISIT: Your local technical bookstore E-MAIL:
[email protected] INSTRUCTORS: call or write for info on textbook exam copies.
Springer www.springer-ny.com
YOUR JO-DAY RETURN PRIVILEGE IS ALWAYS GUARANTEED! PROMOTION NUMBER 52558
�1flrri.CQ.h.i§M
R o b i n Wi lson
Mathematics and Nature
I
M
athematics
occurs
throughout
nature-from the arithmetic of
sunflowers
and
ammonites
to
the
from one end of a golden rectangle leaves another one; this process
is il
lustrated in the Swiss stamp, which
geometry of crystals and snowflakes. If
features the related logarithmic spiral
we count the leaves around a plant
found on snail shells and ammonites.
stem until we fmd one directly above
The delicate structure of a snow
the initial leaf, the number of interven
flake has sixfold rotational symmetry,
ing leaves is often a Fibonacci number
and no two snowflakes are the same.
(for example,
Their hexagonal form was recognised
5 for oak trees, 8 for poplar trees, 13 for willow trees). The
by the Chinese in the second century
arrangements of scales on a pine cone
BCE and was later investigated by
similarly
Johannes Kepler and Rene Descartes,
13
exhibit
left-hand
8 right-hand and
spirals,
while
larger
among others. Hexagons also appear in
(34, 55, . . . ) appear
honeycombs; the Pitcairn Islands re
in the spiral arrangements of seeds
cently issued a set of hexagonal stamps
in a sunflower head. The ratios of
featuring bees.
Fibonacci numbers
Fibonacci
As liquids crystallise they assume
sequence tend to the "golden ratio"
the form of polyhedra of various types:
1.618. . . . This ratio, the ratio of a di
fluorite crystals appear as regular oc
agonal to a side of a regular pentagon,
tahedra, while lead and zinc sulphide
successive terms of the
arises throughout mathematics and na
crystals appear as cuboctahedra and
ture. In particular, removing a square
truncated tetrahedra.
Bulgaria: snowflake
Switzerland: logarithmic
Pitcairn Islands: honeycomb
. . . . . .. .. 9 W
. • • • • • • •
Great Britain: sunflower
Switzerland: fluorite crystals
spiral
.. .
W" W W . W • • � I
..
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]
Israel: pine cone
.. . . ... �
... ......... . ......
Hungary: ammonite
Germany: sulphide crystals
© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 3, 2001
79
The YOK Bug Chandler Davis
B
y now everyone, even journalists, has come to terms with the problem of millennium change: the end of 2000 marked the lapsing of the second millennium, yet it was the beginning of 2000 that was marked by a change of digit in the thousands register. Journalists, especially non-mathematician journalists, sometimes sound re sentful of this, as if a perfectly healthy year had fallen victim to a crime. They may even confound it with that other outrage, the absence of a zero-th storey in many office buildings. We who know the relation between Z and R can dis abuse them of this. The North American custom of numbering storeys 1 , 2 , 3 , . . . ascending, and lB, 2B, 3B, . . . (or th e like) de scending, with no 0 in between, is admittedly absurd. Storeys in a building are discrete and in order, thus nat urally modelled by Z; it is natural to let the ground floor be G or the rez-de-chaussee* be RC, or simply 0, to let 1, 2, 3, . . . number the storeys above it, and to let IS, 2S, . . . (or the like) number those below. Time is naturally modelled by R. Humans choose a zero point and a unit of measurement of time, whereupon every instant acquires a labelling by a real number. Though the unit of measurement is determined by the observed ma neuvers of our planet, the choice of zero point is arbitrary. The most prevalent choice arose from the history of the Christian religion (as another much-used choice arose from Islamic religion), but it has no clear theological ra tionale even for Christians, and nowadays the labelling of time is usually divested of religious reference. I will call the present year simply 2001. +2001, that is. What then is a year? The year is the interval be tween the instants - 1 and so that the year 2001 is the interval between the instant 2000 and the instant 2001 . It is only slightly surprising that the custom is dif ferent for negative n, so that the year -34 (in which
n
'An
80
n;
n
some say Jesus was born) is the interval between the instant -34 and the instant -33. In a striking but su perfluous innovation, I write this explicitly: the year is the interval between the instant and the instant l) . And n =F 0. This convention divides R into disjoint intervals, and there is no room for a zeroth year. This for the same unmysterious reason that there is no zeroth cen tury, and no zet·oth millennium. Months, hours, and seconds are handled differently. They are counted forward, even in the territory of neg atives. Is that the hidden reason why this confuses peo ple? A century before 29 March 0053 is still 29 March, but we resist the impulse to call it 29 March - 0047. It is 29 March - 0046. In the same way, if my bank balance is $53. 15 and I am so careless as to write a check for $ 100, my new balance will be $53. 15 - $ 100.00 = - $46.85. But not - $46. 15: both the dollars and the cents are counted backward (like years) rather than forward (like months). Here my own reckonings part company with my cal culator's. I update my bank balance on an abacus, which would rather subtract $ 100 without touching any column smaller than the hundreds column. I ac commodate. I regard all dollar amounts as modulo $100000.00. If my balance is $53. 15 and then $100.00 goes out, my new balance on the abacus is $99953. 15. I know no way to adapt this idea to reckoning of time, where we will surely continue to count dates for ward, abacus-like, and names of years reversibly, cal culator-like. But contrast the zeroth storey of a build ing, which (like the 13th storey in some buildings) really was the victim of an idiotic tradition. The Mystery of the Missing Year Zero is not a case need ing investigation. It lacks a corpus delecti.
n(lnl - �nl
n
unmysterious word meaning street level-except for the mystery why rez instead of ras, as that word is spelled in every other context.
THE MATHEMATICAL INTELLIGENCER
n
